Modeling of Adhesively Bonded Joints
Lucas Filipe Martins da Silva ·
Andreas Öchsner (Eds.)
Modeling of Adhesively
Bonded Joints
123
Editors
Lucas Filipe Martins da Silva
Departamento de Engenharia Mecânica
e Gestão Industrial
Faculdade de Engenharia
Universidade do Porto
Rua Dr. Roberto Frias
4200-465 Porto
Portugal
lucas@fe.up.pt
ISBN: 978-3-540-79055-6
Prof. Dr. Andreas Öchsner
Technical University of Malaysia
Faculty of Mechanical Engineering
Department of Applied Mechanics
91310 UTM Skudai, Johor
Malaysia
andreas.oechsner@gmail.com
e-ISBN: 978-3-540-79056-3
Library of Congress Control Number: 2008927234
c 2008 Springer-Verlag Berlin Heidelberg
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liable to prosecution under the German Copyright Law.
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Preface
Adhesives have been used for thousands of years, but until 100 years ago, the vast
majority were from natural products such as bones, skins, fish, milk, and plants.
Since about 1900, adhesives based on synthetic polymers have been introduced, but
these were at first of limited use as they were expensive and had poor mechanical
properties. Since 1940, there has been a rapid expansion of the chemical knowledge of polymers from which structural adhesives can be made, with a consequent
improvement in their properties and reduction of their cost. Today, there are many
industrial uses of structural adhesives, particularly in aerospace, but increasingly in
automotive applications where the need is to join sheets of dissimilar adhesives to
produce lightweight car bodies.
In the old days, adhesive use was based on trial and error, together with experience of what was known to work, without any real means of optimisation. With
modern technological needs and assisted by modern computers and experimental
techniques, it is now possible to asses the performance of adhesively bonded joints
before committing a design to manufacture. At least, that is the intention. Reality
is such that we need continually to improve and develop these techniques as definitive and certain answers are still not available. Even now, we rely to a significant
extent on trial and error and to test prototypes or coupons to validate (or to check)
the theoretical predictions.
The objective of this book is to bring together some of the latest thinking on
available predictive technology for structural bonded joints, using internationally
renowned authors who are authorities in their fields.
There are two basic ways of analysing the performance of a joint. In the old days,
before we had advanced computers, we relied on algebraic methods, using a range
of simple or complex formulae. It was difficult or impossible to solve most of these
algebraic formulations in a closed form and so we relied to some extent on numerical solutions. Even those solutions we could obtain were often so complex that it
took several minutes to calculate a single point by hand. However, modern computers can now be programmed to solve these complex formulae on a point by point
basis since they can calculate the values in microseconds. These “old” algebraic
formulae have therefore gained a new lease of life and can, for relatively simple
v
vi
Preface
joint geometries, give a good indication of the stresses and strains in a joint. Since
1970, the numerical technique called finite element analysis has been developed
from a crude and essentially a research tool into a sophisticated commercially available system. The facilitator has been the parallel development of digital computers.
These computers have become faster and able to tackle large numerical calculations
on even a lap top computer. Indeed, a modern lap top can give results in seconds that
in 1980 would have had a turn round time of a day or more using a large main frame
computer such as might be found in a university or a large industrial company. For
example, a modern motor car contains more computing power than was used for the
first space landings in 1970.
For anyone wanting to understand how adhesive joints will behave under significant loads and how you might go about getting a design load, this book provides an
excellent review of the most up to date thinking and practice.
Department of Mechanical Engineering,
University of Bristol, Bristol BS8 1TR, UK
Robert D Adams
Contents
Part I Analytical Modeling
1
Simple Lap Joint Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Andrew D. Crocombe and Ian A. Ashcroft
2
Analysis of Cracked Lap Shear (CLS) Joints . . . . . . . . . . . . . . . . . . . . 25
Liyong Tong and Quantian Luo
3
Analytical Models with Stress Functions . . . . . . . . . . . . . . . . . . . . . . . . 53
Toshiyuki Sawa
3
Part II Numerical Modeling
4
Complex Constitutive Adhesive Models . . . . . . . . . . . . . . . . . . . . . . . . . 95
Erol Sancaktar
5
Complex Joint Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Andreas Öchsner, Lucas F.M. da Silva and Robert D. Adams
6
Progressive Damage Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Marcelo F.S.F de Moura
7
Modelling Fatigue in Adhesively Bonded Joints . . . . . . . . . . . . . . . . . . 183
Ian A. Ashcroft and Andrew D. Crocombe
8
Environmental Degradation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Andrew D. Crocombe, Ian A. Ashcroft and Magd M. Abdel Wahab
9
Non-Linear Thermal Stresses in Adhesive Joints . . . . . . . . . . . . . . . . . 243
Mustafa Kemal K. Apalak
vii
viii
Contents
10
Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Chiaki Sato
11
Stress Analysis of Bonded Joints by Boundary Element Method . . . . 305
Madhukar Vable
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
List of Contributors
Andrew D. Crocombe
Division of Mechanical, Medical and Aerospace Engineering, School of
Engineering, University of Surrey, Guildford, GU2 7XH, UK,
e-mail: A.Crocombe@surrey.ac.uk
Andreas Öchsner
Department of Applied Mechanics, Faculty of Mechanical Engineering, Technical
University of Malaysia, 81310 UTM Skudai, Johor, Malaysia; University Centre
for Mass and Thermal Transport in Engineering Materials, School of Engineering,
The University of Newcastle, Callaghan, NSW 2308, Australia,
e-mail: andreas.oechsner@gmail.com
Chiaki Sato
Precision and Intelligence Laboratory, Tokyo Institute of Technology, 4259
Nagatsuta, Midori-ku, Yokohama 226-8503, Japan, e-mail: csato@pi.titech.ac.jp
Erol Sancaktar
Professor, Polymer Engineering, Adjunct Professor, Mechanical Engineering, The
University of Akron, Akron, OH 44325-0301, e-mail: erol@uakron.edu
Ian A. Ashcroft
Wolfson School of Mechanical and Manufacturing Engineering, Loughborough
University, Loughborough, Leicestershire, LE11 3TU, UK,
e-mail: i.a.ashcroft@lboro.ac.uk
Liyong Tong
School of Aerospace, Mechanical and Mechatronic Engineering, J11- Aeronautical
Engineering Building, The University of Sydney, NSW 2006, Australia,
e-mail: ltong@aeromech.usyd.edu.au
Lucas F.M. da Silva
Departamento de Engenharia Mecânica e Gestão Industrial, Faculdade de
Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto,
Portugal, e-mail: lucas@fe.up.pt
ix
x
List of Contributors
Madhukar Vable
Mechanical Engineering - Engineering Mechanics, Michigan Technological
University, Houghton, MI 49931, USA, e-mail: mavable@mtu.edu
Magd M. Abdel Wahab
Division of Mechanical, Medical and Aerospace Engineering, University of Surrey,
Guildford, GU2 7XH, UK, e-mail: M.Wahab@surrey.ac.uk
Marcelo F.S.F de Moura
Departamento de Engenharia Mecânica e Gestão Industrial, Faculdade de
Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto,
Portugal, e-mail: mfmoura@fe.up.pt
Mustafa Kemal K. Apalak
Department of Mechanical Engineering, Erciyes University, Kayseri 38039,
Turkey, e-mail: apalakmk@erciyes.edu.tr
Quantian Luo
School of Aerospace, Mechanical and Mechatronic Engineering, J11- Aeronautical
Engineering Building, The University of Sydney, NSW 2006, Australia,
e-mail: qtluo@aeromech.usyd.edu.au
Robert D. Adams
Department of Mechanical Engineering, University of Bristol, Bristol BS8 1TR,
UK, e-mail: R.D.Adams@bristol.ac.uk
Toshiyuki Sawa
Hiroshima University, 1-4-1, Kagamiyama, Higashihiroshima, Hiroshima, Japan,
e-mail: sawa@mec.hiroshima-u.ac.jp
Part I
Analytical Modeling
Chapter 1
Simple Lap Joint Geometry
Andrew D. Crocombe and Ian A. Ashcroft
Abstract The chapter outlines the basic mechanics of adhesively bonded simple lap
joints, focusing on the analytical solutions that have been developed. This can be
viewed as foundational material for the chapters in this book that deal with numerical modelling of adhesive joints. It begins with a discussion of the shear-lag concept
which governs the development of adhesive shear stresses arising from load transfer
between substrates. From this starting point a number of design analysis approaches
are outlined, all of which seek to provide easy access to adhesive stresses within
joints under a range of service loading conditions. The chapter concludes by considering approaches that have been made to enhance the accuracy of the closed
form analytical adhesive joint models. Although improving the accuracy, these developments also result in a considerable increase in complexity with the result that
analytical solutions are more difficult to utilise.
1.1 Introduction
A large number of adhesively bonded joint configurations use an overlap in their
construction. These are generically referred to as simple lap joint geometries. The
most common of these simple lap joint geometries is the axially loaded single overlap joint (sometimes referred to simply as a single overlap joint). This chapter will
consider the analytical stress analyses of the generic range of simple lap joints. Later
chapters deal with the numerical modelling of such configurations.
Andrew D. Crocombe
Division of Mechanical, Medical and Aerospace Engineering, School of Engineering, University
of Surrey, Guildford, GU2 7XH, UK, e-mail: A.Crocombe@surrey.ac.uk
Ian A. Ashcroft
Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University,
Leicestershire, LE11 3TU, UK, e-mail: i.a.ashcroft@lboro.ac.uk
L.F.M. da Silva, A. Öchsner (eds.), Modeling of Adhesively Bonded Joints,
c Springer-Verlag Berlin Heidelberg 2008
3
4
Andrew D. Crocombe and Ian A. Ashcroft
1.1.1 Limiting Joint Strengths
Adhesives are not generally as strong as the materials they join and hence use is
often made of an overlap to increase the load carrying capacity of a joint. Generally,
when designing a bonded joint, the ideal approach is to ensure that the bonded joint
is stronger than the parts being jointed. However, it can be shown that the load
carrying capacity of an overlap joint is not proportional to the length of the overlap.
Consider the simple single overlap joint illustrated in Fig. 1.1.
The axial load is transferred from one substrate to the other through shear in the
adhesive. This shear will only occur over a finite zone of the adhesive layer and any
increase in overlap length beyond this maximum transfer zone length will not result
in any increase in joint strength. Where load transfer beyond this limiting value is
required (as in the case of very thick substrates) a different type of joint design will
be required, such as the scarf or stepped lap joint, as shown in Fig. 1.2. In both these
configurations the load is distributed more uniformly across the entire adhesive layer
than in the single lap joint.
1.1.2 Load Transfer in Lap Joints
In some lap joints complete load transfer occurs from one set of substrates to another as shown in the joints in Figs. 1.1 and 1.2. In other configurations, such as
that shown in Fig. 1.3, only partial load transfer occurs across the adhesive. However, even this partial load transfer may be sufficient to cause the bond line to fail
and thus stresses and strains in the adhesive layer are required to assess the joint
strength.
τ
Fig. 1.1 Single lap joint
configuration and shear stress
distribution in the adhesive
layer
Scarf Joint
Fig. 1.2 Scarf and stepped lap joints
Step Joint
1 Simple Lap Joint Geometry
5
Fig. 1.3 Partial load transfer
within a joint
1.1.3 Overview
Methods of determining the adhesive stress and strain in a range of lap joints will
be considered in this chapter. The next section outlines the simplest form of axial
loaded single lap joint analysis and presents results in an accessible way in order
to understand basic joint mechanics. This is followed by a section that discusses
the various tools that are available to analyse a range of practical lap joint configurations. The analyses outlined in these two sections are subject to a number of
simplifying assumptions. These tend to include:
• substrate deformation due to tension and bending only
• adhesive stresses restricted to peel and shear and assumed to be constant across
the adhesive layer
The penultimate section addresses more advanced analyses, generally of the axially
loaded single overlap joint, that make less simplifying assumptions at the cost of
analytical complexity and hence general usefulness.
1.2 Single Lap Joint Mechanics
1.2.1 Qualitative Understanding of Adhesive Stresses
The simplest approach is to assume that the substrates are effectively rigid. This
means that as the load passes from substrate to substrate a uniform shear stress
distribution is generated, as shown in Fig. 1.4.
In reality the substrates are not rigid and they stretch more nearest their loaded
end, as shown in Fig. 1.5. Thus the shear stress that is generated is not uniform but
peaks at the overlap ends where the differential substrate extension is greatest, as
illustrated in Fig. 1.1.
In practice, as well as shearing the adhesive, the offset loading causes bending in
the loaded substrate, which tries to separate from the adjacent unloaded substrate. In
τ
Fig. 1.4 Uniform adhesive
shear stress
x
6
Andrew D. Crocombe and Ian A. Ashcroft
Fig. 1.5 Non-uniform adhesive shearing
σ
x
Fig. 1.6 Peel stresses in a single lap joint
addition to the non-uniform shear stress, this bending action gives rise to transverse
direct stresses, sometimes known as peel or cleavage stresses, which are maximum
at the joint ends, as shown schematically in Fig. 1.6.
Thus what appears to be a simple joint in reality gives rise to a fairly complex
state of adhesive stresses.
1.2.2 Volkersen’s Shear Lag Analysis
This approach was developed by Volkersen (1938) and accounts for the variable
shear stress distribution shown in Fig. 1.1.
Referring to Fig. 1.7 the adhesive shear stress (τ ) at distance x from the centre
of the overlap (length L and adhesive thickness η ) is defined in terms of the shear
strain γ as
δ
(δ0 + u2 − u1 )
τ = Gγ = G = G
(1.1)
η
η
u1 and u2 represent the extension of the substrates and can be derived in terms of
substrate strains (ε1 and ε2 ) as
x
ε1 dx;
u1 =
−L/2
x
u2 =
ε2 dx
(1.2)
−L/2
These strains at distance x can be obtained from the upper and lower substrate
stresses, which can be expressed by the substrate force P, thickness ti and tensile
modulus Ei , as shown in Fig. 1.8 and Eq. (1.3).
L/2 + x + u1
P
δ
δ0
A
B
L/2 + x + u2
Fig. 1.7 Deformation in the shear lag analysis
P
1 Simple Lap Joint Geometry
7
Fig. 1.8 Substrate forces
P – Fτ
P
Fτ
Fτ
ε1 = [P − Fτ ]/E1t1 = [P −
x
τ dx]/E1t1
−L/2
ε2 = Fτ /E2t2 =
x
τ dx/E2t2
(1.3)
−L/2
If the substrates are made from the same material then E1 = E2 = Es . By substituting
Eq. (1.1) into (1.3), then (1.3) into (1.2) and then differentiating twice the following
second order differential equation governing the adhesive shear stress is obtained
d 2 τ ω 2
=
τ
dx2
L
(1.4)
where ω 2 = (1 + t1 /t2 )k and k = GL2 /(Est1 η ).
This can be solved and non-dimensionalised fairly readily to give a solution of
t1 − t2 ω sinh ω X
τ
ω cosh ω X
+
=
τa
2 sinh ω /2
t1 + t2 2 cosh ω /2
where τa is the average shear stress (P/L) and X = x/L.
The above can be considerably simplified, by considering only the peak stress at
the joint ends (X = +/ − 0.5) and assuming that the joint is sufficiently long such
that sinh ω /2 = cosh ω /2, giving
t1 − t2
τ
ω
= ±
τa
2
t1 + t2
This clearly shows that for a joint with different substrates the adhesive stress is
maximum (and thus failure most likely) at the overlap end where the loaded substrate is thinnest.
Further, the lowest adhesive stresses are obtained when the substrates are identical (t1 = t2 = t).
τ
ω
= =
τa
2
GL2
2Est η
(1.5)
This is an extremely useful formula which shows a number of very important
features about the size of the peak adhesive stress in a single overlap joint, namely
– for long joints it is independent of the joint length,
– it increases with increasing adhesive shear modulus,
– it increases with decreasing substrate modulus and thickness and adhesive thickness.
8
Andrew D. Crocombe and Ian A. Ashcroft
1.3 Simple Design Tools
1.3.1 General Adhesive Sandwich Analysis
The analysis developed by Goland and Reissner (1944) for the single lap joint has
been generalised by Bigwood and Crocombe (1989) so that it can be applied to
an arbitrary end loaded single overlap configuration. The resulting analyses can be
applied to single overlap joints but also to many other configurations, as shown in
Fig. 1.9.
The substrate loads and adhesive stresses acting on small elements of the upper
and lower substrates are shown in Fig. 1.10.
Force and moment equilibrium of the upper and lower substrates gives
dTi
= τ;
dx
ti
dMi
−Vi + τ = 0
dx
2
dVi
= σ;
dx
(1.6)
where i = 1 or 2, representing the upper and lower substrate respectively and all
other terms have been defined in Fig. 1.10.
The axial and transverse displacements of the upper and lower substrates (ui and
vi ) can be expressed in terms of the substrate surface strain and moments respectively as
(1 − νi2 )
dui
Miti2
(1 − νi2 )Mi
d 2 vi
=
=−
(1.7)
Pi +
;
2
dx
Eiti
2Ii
dx
Ei Ii
V12
V11
T11
M11
T21
M12
x
M21 V21
T12
T22
V22
M22
Fig. 1.9 Arbitrarily end loaded single overlap unit
T1 + dT1
M1
T2
Fig. 1.10 Elemental substrate
stresses in the lap joint
V1 + dV1
V1
T1
V2
τ
M1 + dM1
σ
σ
dx
τ
V2 + dV2
T2 + dT2
M2
M2 + dM2
1 Simple Lap Joint Geometry
9
where νi and Ii are respectively the Poisson’s ratio and second moment of area of
substrate i. Finally expressions for the adhesive shear and transverse direct stresses
in terms of the adhesive strains and hence substrate displacements are
τ = Gγ = G
(u1 − u2 )
;
η
σ = Eε = E
(v1 − v2 )
η
(1.8)
Equations (1.8) can be differentiated an appropriate number of times and substitutions made from (1.7). The resulting expressions can be differentiated further and
substitutions made from (1.6) to give the governing differential equations as
d5τ
d3τ
dτ
d7τ
=0
− K1 5 + K3 3 − K5
7
dx
dx
dx
dx
d6σ
d4σ
d2σ
− K1 4 + K3 2 − K5 σ = 0
6
dx
dx
dx
(1.9)
Now these can be solved with appropriate boundary conditions. This cannot be done
analytically for the general adhesive sandwich shown in Fig. 1.9 but they can be easily solved numerically. However for the case of the axially loaded single lap joint
discussed in Sect. 1.2 the 12 unknown end loads shown in Fig. 1.9 are reduced
to 6 (left end of the upper substrate and right end of the lower substrate). An analytical solution for this simplified configuration was first derived by Goland and
Reissner (1944) and this is outlined in the following section.
1.3.2 Axially Loaded Single Overlap Joints
The Goland and Reissner (1944) solution of this configuration consists of two
parts. The first is a solution for the load boundary conditions and the second is
the derivation of the equations for the adhesive shear (τ ) and peel stresses (σ ). The
generalised adhesive sandwich analysis outlined in the previous section is an extension of the second part of the Goland and Reissner analysis and hence this will
not be repeated here. Instead this section outlines the solution for the load boundary conditions or what is now known as Goland and Reissner’s bending moment
factor.
Figure 1.11 shows a single lap joint in its undeformed position. If the joint is
sufficiently long then approximate expressions for the end loads are
T11 = T22 = P
M11 = M22 = −P(t/2)
The shear forces V11 ,V22 can be determined to maintain moment equilibrium.
However, in practice, the joint will rotate on loading as illustrated in Fig. 1.6
and the line of action of the load will move closer to the substrate centre line thus
10
Andrew D. Crocombe and Ian A. Ashcroft
P
t
P
V11
T11
M11
T22
V22
M22
Fig. 1.11 Simplified single lap joint end loads
reducing the moment. This is accommodated by introducing a factor (k) to reduce
the bending moment i.e.
M11 = M22 = −kP(t/2)
In an axially loaded single overlap joint the substrate moment makes a significant
contribution to the adhesive stresses and thus it is important that the correct value is
used.
To account for the joint rotation Goland and Reissner split the joint into two
regions, the free substrate region and the overlap region as shown in Fig. 1.12 and
the transverse deflection is determined for each region.
All parameters are known except the deflection of the neutral axes of the two
regions, wa and wb . Equations for the bending moment and thus the curvature can
be written for both the regions
wa = −Ma /(EI)a = −P(α xa − wa )/(EI)a
(1.10)
wb = −Mb /(EI)b = −P(α (La + xb ) − wb − (t + η )/2)/(EI)b
(1.11)
The adhesive thickness (η ) is disregarded in the classical Goland and Reissner approach. These are 2nd order differential equations in wa and wb respectively and can
be easily solved. The boundary conditions are that the deflection is zero at the left
end of region a and the right end of region b while the deflections and slopes at the
wa
P
t
xa
La
wb
η
xb
c
Fig. 1.12 Deflection of the single overlap joint
α
1 Simple Lap Joint Geometry
11
connection of regions a and b are continuous. These can be simplified and solved
for the reasonable assumption that the free substrate length is large to give sandwich
moment (M11 ) of
(1.12)
M11 = −k(Pt/2)
where
cosh u2 c
√
cosh u2 c + 2 2sinh u2 c
u2 = 3(1 − ν 2 )P/2η Es
k=
The parameter k is a factor of the undeformed moment and varies from a value of
1 at low loads to a minimum of around 0.26 for high loads. Again the shear forces
V11 ,V22 can be determined to maintain moment equilibrium.
When these boundary conditions are used in conjunction with the analysis for
adhesive stresses the following closed form equations are derived for adhesive shear
and peel stresses in an axially loaded single overlap joint
⎤
⎡
βc x
cosh
⎥
τ
1 ⎢βc
t c
(1.13)
(1 + 3k)
+ 3(1 − k)⎥
= ⎢
⎦
⎣
βc
τav 4 t
sinh
t
where β 2 =
8tG
Es η
t 2 1 k
λx
λx
σ
=
cos
R2 λ 2 − λ k cos λ cosh λ cosh
P/t
c R1
2
c
c
k
λx
λx
sin
+ R3 λ 2 − λ k sin λ sinh λ sinh
2
c
c
where
c
λ=
t
6Et
Es η
1
4
R1 = (sinh 2λ + sin 2λ )/2
R2 = sinh λ cos λ − cosh λ sin λ
R3 = sinh λ cos λ + cosh λ sin λ
Goland and Reissner’s derivation of the bending moment factor for symmetric single
lap joints assumed that the overlap region deformed as a single block of composite
stiffness ((EI)b in Eq. (1.11)). This has been developed by a number of subsequent
researchers. Hart-Smith (1973a) argued that it was better to model the deformation
of the upper and lower substrates in the overlap region separately. His approach
12
Andrew D. Crocombe and Ian A. Ashcroft
involved the solution of the adhesive stresses and the bending moments at the same
time and provides a bending moment factor k in Eq. (1.12) of:
η
k = 1+
t
1
1+ξc+
ξ 2 c2
6
√
where ξ = u2 c.2 2 (see Eq. (1.12)).
The expression derived for the moment factor varies from unity at low loads to
zero with increasing loads. Thus at higher loads the moment and hence the adhesive stress predicted by Hart-Smith are a little lower than those predicted by Goland
and Reissner. Tsai and Morton (1994) review both these approaches and a third
by Oplinger (1994) that includes large deformation of the overlap as well as the
free substrate. All three are only applicable to joints with balanced substrates. They
also investigate the assumption that the overlap is much smaller than the free substrate length. Overall, based on comparison with numerical studies, they conclude
that Hart-Smith and Oplinger give better end moments for short and long overlaps
respectively. However with regard adhesive stresses, the Goland and Reissner approach is accurate enough for short and long overlaps.
1.3.3 Simplified Design Formulae
Although the equations outlined in the general adhesive sandwich analysis section
above require a numerical solution, a simplification (neglecting the adhesive shear
and peel stress in turn) can be made resulting in simple design formulae. This simplification decouples the governing equations presented in Eq. (1.9). In fact it turns
out that this simplification is completely valid when the substrates are the same and
this is common practice. By making the same assumptions as those in the shear lag
analysis it is possible to find approximate expressions for the stresses at the end of
the overlap in terms of simple design formulae. Unlike the shear lag analysis these
formulae give both adhesive stress components due to all possible substrate loading, not just axial loads. These formulae are shown in Fig. 1.13 and Bigwood and
Crocombe (1989) provide further detail.
T
τT =
− α1T
( α1 + α2 ) 0.5
σV =
− 2 β1V
( β1 + β2 ) 0.75
V
τV = 3V
σM =
− β1 M
( β1 + β2 ) 0 . 5
M
τM =
4t1
3α1 M
t1 ( α1 + α2 ) 0.5
Fig. 1.13 Peel and shear stress design formulae for tensile shear and moment loading in the
substrate
1 Simple Lap Joint Geometry
13
The parameters α1,2 and β1,2 are called the shear and peel compliance factors
and are defined as follows
α1 = G(1 − ν12 )/(E1t1 η );
β1 = 12E(1 − ν12 )/(E1t13 η )
with similar expressions for the lower substrates. Although a good guide, the simplifications made in the analysis are only completely valid for identical substrates
(t1 = t2 = t), in this case the design formulae simplify further still to give
σV = −V (β /2)0.25 ;
τT = −T
(α /2)0.5
;
2
σM = −M(β /2)0.5
τV = 3V /4t;
τM = M
3(α /2)0.5
t
(1.14)
It is interesting to compare τT in the above to that given by Volkersen. Substituting
in for α above gives
1
τT = P
2
G(1 − ν 2 )
2Es η t
By comparison with Eq. (1.5) it can be seen that in addition to the factor (1 − ν 2 )
(this arises by assuming plane strain rather than plane stress behaviour) the stresses
given by Volkersen are exactly twice that found from the more general analysis.
This difference is caused by the fact that, in Volkersen the substrate is assumed to
be in tension only. This is clearly wrong as, from simple static considerations, the
adhesive shear stress must induce a moment in the substrate. This acts in a way to
reduce the adhesive shear stress and the result is quite significant. Thus we may say
that Volkersen formulae overestimates the peak adhesive stresses by a factor of two
and use of the simple formulae produced by Crocombe and Bigwood should be used.
1.3.4 Design Formulae for Environmentally Degraded
Single Lap Joints
Fig. 1.14 Illustrating
moisture transport and
degradation in the joint
x
Stress
Moisture
When bonded joints are exposed to moisture, the moisture diffuses through, and
degrades the adhesive. This is illustrated in Fig. 1.14.
Increasing time
Increasing moisture
x
Strain
14
Andrew D. Crocombe and Ian A. Ashcroft
In many adhesive systems it can be shown that the ultimate adhesive strength
essentially degrades linearly with increasing moisture content (cm ) up to the maximum moisture content (cm∞ ). A simple yet effective limit state approach (Crocombe,
2007) has been applied to develop design tools. This limit state approach was first
suggested for use with adhesive joints by Crocombe (1989) using the term global
yielding. The failure load (Pmax ) is found by assuming that all the adhesive reaches
its maximum load carrying capacity (τmax , which is a function of cm /cm∞ , reducing
where the adhesive is wetter) before joint failure occurs i.e.
L
Pmax =
τmax (cm /cm∞ ) dx
(1.15)
0
The moisture distribution shown in Fig. 1.14 (cm ) can be found by using the Fickian
diffusion coefficient D. Combining this with Eq. (1.15) and the linear reduction in
adhesive strength τmax with moisture cm finally provides the following equation for
degraded joint strength
2 Pmax
1
8 ∞
−(2 j+1)2 Dπ2 t
L
= 1 − Fτ 1 − 2 ∑
e
(1.16)
τdry L
π 0 (2 j + 1)2
where Fτ = 1 −
τwet
and τwet and τdry are the saturated and dry adhesive strengths.
τdry
1.4 Advanced Analytical Models
The previous two sections have outlined lap joint analyses based on the following
assumptions
(a)
(b)
(c)
that the adhesive stresses do not vary across the adhesive thickness,
that the substrates response is a simple combination of tension and bending,
both adhesive and substrate behave elastically.
This section reviews work by various authors that go beyond these basic simplifying
assumptions.
1.4.1 Advances in Strain-Displacement Models
Renton and Vinson (1977) made the next significant contribution to the analysis of
single lap joints. The effect of shearing in the substrate is included and the substrate
is modelled as a generally orthotropic system, to extend the application to composite materials. As with previous analyses the adhesive shear and peel stress have
been assumed constant across the thickness of the adhesive. Equations were derived
1 Simple Lap Joint Geometry
τou
V1
T1
15
σou
V2
x
M1
σoL
T2
h
L
τoL
M2
Fig. 1.15 Model substrate block according to Renton and Vinson
for the stress resultants (Mx , Nx and Vx ) for the model substrate element shown
in Fig. 1.15, which shows the block end loads and the upper (τou , σou ) and lower
(τoL , σoL ) surface shear and normal stresses.
Mx = −D11
Nx = A
d 3 τoL
d2w
d 3 τou
d τou
d3φ
dφ
d τoL
+H
+ H∗
+F 3 +G
+J
+ J∗
2
3
3
dx
dx
dx
dx
dx
dx
dx
d 3 τoL
du
d 3 τou
d τou
d3φ
dφ
d τoL
+C
+C∗
+ Ā − N(x)T
+B
+D
+ D∗
3
3
dx
dx
dx
dx
dx3
dx
dx
h
Vx = (τou + τoL ) + Lφ
2
(1.17)
In these equations parameters A to L are problem specific constants, D11 is the main
flexural stiffness and φ is a function of x defining the transverse shear stress distribution. Importantly, these included a parabolic shear stress distribution across the
thickness of the block. This block is used to model both substrates in the overlap
region, Fig. 1.16.
Assuming that the adhesive shear (τo ) and peel (σo ) stress is constant across
the adhesive thickness, using the same adhesive strain-displacement equations as
Goland and Reissner, considering equilibrium of the substrate blocks and setting
appropriate surface tractions and stress resultants either to zero or to the adhesive
V1
T1
τo
M1
τo
σo
τo
τo
σo
V2
T2
M2
Fig. 1.16 Blocks in the overlap region
16
Andrew D. Crocombe and Ian A. Ashcroft
shear or peel stress, an 8th order differential equation is obtained for the adhesive
shear stress in single lap joints with different substrates. This simplifies considerably for symmetric single lap joints. From the solution of this, the adhesive peel
and shear stresses and the stress resultants in the substrate blocks can be found.
Goland and Reissner’s moment factor was used to provide boundary conditions.
When compared with the Goland and Reissner solution for a typical joint configuration, the maximum peel and shear stresses are reduced by around 20% and 40%
respectively and, unlike the Goland and Reissner analysis, the adhesive shear stress
now drops rapidly to zero at the very end of the overlap, as it should for a free
surface.
Further advances to single lap joint analyses have been made by Ojalvo and
Eidinoff (1978). They incorporate a fuller description for adhesive shear strain on
the upper and lower interfaces (i = 1 or 2 respectively)
γi =
u1 − u2 dwi
+
η
dx
(1.18)
Further, the displacements in the adhesive are assumed to vary linearly between the
values on the upper and lower interface as
w(x, z) =
(w1 + w2 ) z
+ (w1 − w2 )
2
η
(1.19)
with a similar expression for u(x, y). This allows for a linear variation across the
adhesive thickness of the shear stress while the peel stress does not vary. Substrate
shearing has not been included. Balanced joints are considered and the adhesive
shear and peel stresses both reach a peak at the end of the overlap. For typical
joint parameters the maximum shear and peel stresses are increased and decreased
respectively over the values given by Goland and Reissner (1944). It was shown
that shear stress could exhibit significant variation across the overlap at the joint
ends. However, unlike Renton and Vinson (1977), the formulation does not allow
the adhesive shear stress to drop to zero at the ends of the overlap.
Delale et al. (1981) include the adhesive longitudinal stress in addition to the
shear and peel stresses, however these adhesive stresses are assumed to be constant
across the adhesive thickness and as with all previous analyses, except Renton and
Vinson (1977), the formulation does not allow the adhesive shear stress to become
zero at the overlap ends. The formulation is derived for unbalanced orthotropic lap
joints and substrate shearing has been incorporated though substrate rotation (θi ).
The adhesive shear strain is now defined in terms of substrate mid-plane displacements (ūi ) and rotations.
1
h1
h2
γ=
ū1 − θ1 − ū2 − θ2
(1.20)
η
2
2
The adhesive longitudinal strain (εx ) is taken as the average of the upper and lower
interface strains which is expressed as
1 Simple Lap Joint Geometry
εx =
17
1
2
du1 h1 d θ1 du2 h2 d θ2
−
+
−
dx
2 dx
dx
2 dx
(1.21)
Adhesive shear stresses are expressed as a 7th order differential equation. The
boundary conditions used do not incorporate the bending moment factor and thus it
is not possible to make a comparison between this and other solutions.
Tsai et al. (1998) present a different way to model substrate shear, allowing it
to vary linearly through the substrate thickness. They argue that as the adhesive
shear stress is highest towards the end of the overlap then there will be high shear
stresses acting on the adjacent substrate that forms the interface with the adhesive.
Neglecting this substrate shear deformation can have important effects, particularly
with shear compliant substrates like laminated composites. This shear goes to zero
on the opposite free surface of the substrate and Tsai et al. assume the simplest linear
variation. The substrate in-plane displacement (ui ) due to tension now has an extra
term that varies across the substrate thickness (yi )
ui = ui,os +
τ 2
y
2Gi hi i
(1.22)
where ui,os is the displacement on the outer (shear free) surface of the substrate and τ
is the adhesive shear stress and i refers to the substrate. Whilst this is not as accurate
a representation for substrate shear as others such as Renton and Vinson (1977)
it has the advantage of being readily incorporated into the earlier solutions of
Volkersen (1938) and Goland and Reissner (1944). Only adhesive shear stress is
considered as the substrate shear was not expected to affect the adhesive peel
stresses. The result of including this parameter is to modify the governing differential equation. For example Volkersen’s equation (1.4) is modified by the term α to
d 2 τ αω 2
=
τ;
dx2
L
−1
t1
G
t2
where α = 1 +
+
η 3G1 3G2
2
(1.23)
As the substrate shear deformations become smaller α tends to 1 and the original
Volkersen equation is recovered. Similar modifications are derived for the Goland
and Reissner solution where the shear stress is still given by Eq. (1.13), repeated
again here for convenience, but now the parameter β 2 is defined differently.
⎤
⎡
βc x
cosh
⎥
τ
1 ⎢βc
t c
⎥
(1.24)
(1
+
3k)
+
3(1
−
k)
= ⎢
⎦
βc
τav 4 ⎣ t
sinh
t
1
8tG
where β 2 =
Es η 1 + 2Gt/(3Gs η )
In the Goland and Reissner’s analysis the term in parenthesis in the expression for
β 2 is unity. It is shown that including the substrate shear will smooth and reduce the
shear stress distribution along the overlap.
18
Andrew D. Crocombe and Ian A. Ashcroft
Fig. 1.17 Three strips with end loads and interface tractions (shown dashed) considered by Sawa
et al. (2000)
Sawa et al. (2000) modelled an unbalanced single lap joint by considering the
two substrates and the adhesive layer as three separate strips, Fig. 1.17, subjected to
common but unknown tractions along the interfaces (illustrated schematically) and
appropriate tractions on the other faces.
All three stress components are included in each strip (σx , σy and τxy ) thus a
complete 2D elasticity problem is formulated and solved. A series of 9 separate
polynomial and Fourier series Airy stress functions X = ΣXi were derived for each
strip involving 18 + 48N unknown coefficients, where N is the number of terms
in the Fourier series (between 50 and 60 were found to be an adequate number of
terms). Stresses in each strip can be defined in terms of the Airy stress function as
σx =
d2X
;
dy2
σy =
d2X
;
dx2
τx = −
d2X
dxdy
(1.25)
The first 18 of these unknown coefficients can be found from the known boundary
conditions and the remainder are found by solving 48 simultaneous equations for
each term included in the series. Solution of this is likely to be as complex as carrying out finite element (FE) analysis and so is of limited use as an analytical tool.
They found the same sort of trends in adhesive stress as were found in the simpler
closed form solutions but with the inclusion of sufficient terms were able to model
the bi-material stress singularity reasonably.
1.4.2 Energy Based Approaches
A number of authors, including, Chen and Cheng (1983) and Adams and Mallick
(1992) have adopted a different approach to obtain a global stress analysis of adhesive joints. They use the equilibrium equations
∂ σx ∂ τxy
+
= 0;
∂x
∂y
∂ σy ∂ τxy
+
=0
∂y
∂x
(1.26)
to develop general expressions for stresses (σx , σy and τxy ) in both substrates and adhesive. Thus tensile, bending and shearing in the substrate are all included. Boundary and continuity conditions are then imposed to partially solve these equations. As
1 Simple Lap Joint Geometry
19
the substrate and adhesive sections are treated separately the condition for zero adhesive shear stress at the overlap ends can be imposed, as Renton and Vinson (1977).
The remaining unknown parameters are found by applying the variational principle
(minimising)
of
complementary
energy.
Chen and Cheng (1983) use the Goland and Reissner (1944) moment factor and
determined the shear force using equilibrium. The substrate longitudinal stresses
are assumed to vary linearly across the substrate thickness and the shear stresses
parabolically. They assume that the adhesive shear stress does not vary across the
thickness and show that from the equilibrium equations this implies that adhesive
longitudinal stress is zero. The peel stress however varies linearly across the adhesive thickness. It is shown that all stress components can be expressed in terms of
two unknown independent functions of x (Φ and Φ ∗ ). Expressions for adhesive peel
and shear stress are given below:
dΦ h dΦ ∗
+
dx
2 dx
y
σ = Φ + (Φ − Φ ∗ )
η
τ=h
(1.27)
The complementary energy is expressed in terms of these unknowns. Minimising
this produces two coupled 4th order differential equations for these unknowns.
These can be solved and used to find adhesive and substrate stresses. Although
closed form in nature the resulting expressions are highly complex. A good match
with the solution of Goland and Reissner is found except at the overlap ends
where Chen and Cheng’s analysis correctly show the adhesive shear stress to drop
rapidly to zero and the adhesive peel stress to vary significantly across the adhesive
thickness.
Adams and Mallick (1992) extended the work of Chen and Cheng by formulating
the problem to include different substrates, thermal stresses and both single and double lap joints solutions. In addition, adhesive longitudinal stresses were included and
all stresses were allowed to vary across the adhesive thickness. Due to the increased
complexity in adhesive stresses and dissimilar substrates the entire stress field was
expressed in terms of four independent functions, rather than the two in the work of
Chen and Cheng. Owing to this added complexity the authors have not determined a
closed form solution but instead have discretised the problems by breaking the joint
into a number of intervals with the functions and their first derivatives forming the
unknown parameters at each node.
{σ } = [N]{Φ } + {C(x)} + {k}
(1.28)
Where {σ } is a vector containing the 3 stress components of each of the substrates
and the adhesive, {Φ } is the vector of the 4 stress functions (Φ11 , Φ12 , Φ21 , Φ22 ) and
their first and second derivatives with respect to x. The arrays [N], {C(x)} and {k}
are known from the configuration. The entire problem is then solved numerically as
a matrix problem similar to the FE method.
20
Andrew D. Crocombe and Ian A. Ashcroft
1.4.3 Non-linear Material Models
Many modern adhesive systems exhibit significant non-linearity in their constitutive
response and hence analyses based on linear material behaviour may be of limited
use for strength assessment. Hart-Smith (1973b) extended the double lap joint analysis modelling the adhesive as an elasto-plastic material. He showed that the actual
form of the adhesive stress strain curve was less important than the area under it
(energy dissipated) and thus used an elastic-perfectly plastic response (no strain
hardening) for the adhesive. The analyses were simplified by uncoupling the shear
and peel stress analyses. Elasto-plasticity was only included in the shear stress formulation, which was based on the shear lag approach outlined by Volkersen (1938)
and this was achieved by deriving equations for an elastic inner region and plastic
outer region and then applying boundary conditions at the overlap ends and continuity at the inner-outer region boundaries. The peel stress distribution is similar
to that of the later Volkersen (1965) analysis, which was elastic. By assuming that
joint failure occurs at a critical level of plastic strain, Hart-Smith develops a chart
that gives predicted joint strength as a function of a joint geometry parameter and
the ratio of failure to initial yield shear strains.
The elastic analysis of Bigwood and Crocombe (1989) for an arbitrarily end
loaded single overlap was extended to incorporate a full non-linear representation
of the adhesive layer where both the peel and shear stresses contribute to adhesive
yield, Bigwood and Crocombe (1990). Equations (1.6), (1.7) from the linear analysis have been rewritten expressing the adhesive shear and peel stresses as non-linear
functions of strain as
Es γ
dV1
Es ε
dM1
(h1 + η )Es γ
dT1
=
;
=
;
= V1 −
2
dx
2(1 + ν p )
dx
(1 + ν p )
dx
4(1 + ν p )
dκ
dγ
dε
= κ;
= f1 (M1 , T1 , x);
= f2 (M1 , T1 , x)
dx
dx
dx
(1.29)
Unlike the elastic analyses, here Es is the secant adhesive modulus, determined from
a hyperbolic tangent model representing the non-linear adhesive stress strain curve.
Eε
σ = A tanh
(1.30)
A
In the hyperbolic tangent model the parameter E and A will give the initial modulus
and the ultimate stress respectively. These provide a set of 6 first order differential
equations in 6 unknowns (T1 (x),V1 (x), M1 (x), ε (x), κ (x) and γ (x)). These governing
equations were solved numerically in an incremental manner. Good correlation was
found between their analyses and non-linear FE solutions. Figure 1.18 shows the
adhesive shear stress distributions arising from a linear (stress lin) and non-linear
(stress non-lin) analysis of a typical single lap joint, carrying the same load. The
stress-wet curve will be considered later. The high stress regions at the overlap ends
in the elastic analysis cannot be sustained as the adhesive yields and thus are significantly reduced. Note however that the adhesive stresses in the middle of the overlap
1 Simple Lap Joint Geometry
80
70
Shear stress, MPa
Fig. 1.18 Linear and
non-linear adhesive shear
stress distribution for a single
lap joint
21
60
stress lin
stress non-lin
stress wet
50
40
30
20
10
0
0.00
6.25
12.50
Dist along overlap, mm
are higher for the non-linear analysis as the total area under the shear stress-overlap
curves must be the same as the same load is carried by both joints.
Crocombe (2007) extended the non-linear analysis of arbitrary end loaded single
overlaps (Bigwood and Crocombe, 1990) to consider the effect of environmental
degradation. The coupling between the moisture concentration within the joint and
the mechanical properties of the adhesive is treated in a similar way to that outlined
in the section Design formulae for environmentally degraded single lap joints in
Sect. 1.3 on design tools above. However the approach outlined here is not restricted
to a limit state and can assess the adhesive joint at any given level of loading. This
is a far more useful foundation for the assessment of service life of different joint
configurations.
Referring again to Fig. 1.14 the two step process is identified. Initially the moisture diffusion within the joint is determined using Fickian moisture uptake laws.
Then the effect of the moisture on the constitutive response of the adhesive is modelled as shown schematically in Fig. 1.14. In the analysis this moisture dependent
material response is implemented by making the parameters in the hyperbolic tangent model functions of moisture concentration (cm ) i.e.
E(cm )ε
σ = A(cm ) tanh
(1.31)
A(cm )
This essentially introduces an infinitely variable non-linear response for the adhesive. Experimental data has shown that it is very reasonable to assume a linear variation of modulus, E(cm ), and ultimate stress, A(cm ) between fully dry and fully
saturated values and this has been implemented. However any variation could easily
be incorporated. Figure 1.18 also shows the stress distribution in the single lap joint
after exposure at 85% RH for 230 days (stress-wet). The moisture concentration at
the ends of the overlap was high and has thus degraded the ultimate strength of the
adhesive in this region. This is the reason that the stresses in this region are lower
than the non-linear dry joint analysis. However, after only 230 days the centre of
the joint is still relatively dry, and thus higher stresses, required to transfer the total
joint load, can be sustained in this region.
22
Andrew D. Crocombe and Ian A. Ashcroft
1.5 Conclusion
A range of analyses relevant to the adhesive lap joint have been presented. The basic
mechanics of these joints have been outlined (Sects. 1.2 and 1.3) with a reasonably
detailed discussion of some of the classic analyses. In these early analyses the substrates have been considered to carry only tension and bending and the adhesive has
been considered as having only 2 stress components (peel and shear) which do not
vary across the adhesive thickness.
Section 1.3 details some analyses that have been developed to be readily useful
in a design context to allow rapid assessment of the adhesive stress distribution.
Results are presented in the form of design formulae and more general solutions
can be easily implemented on spreadsheets.
Much of the later closed form analyses have focused on a better representation
of either the substrate and/or the adhesive stress distribution. The effect of these
extensions is to increase the complexity of the analyses thus making it more difficult
to obtain an analytical rather than numerical solution.
Non-linear material behaviour for the adhesive has been discussed separately.
A number of different approaches for incorporating this non-linear behaviour have
been discussed and the most advanced also incorporates moisture dependency.
Very little has been done by way of incorporating failure criteria into these closed
form analyses. This is currently being done in FE modelling and it would be reasonable to expect that future developments of closed form analyses might include
this capability. The advantage of this is that the simplicity of a closed form approach
makes it very suitable for preliminary joint design assessment.
References
Adams RD and Mallick V (1992) A method for the stress analysis of lap joints. J Adhes 38:
199–217
Bigwood DA and Crocombe AD (1989) Elastic analysis and engineering design formulae for
bonded joints. Intl J Adhes Adhes 9: 229–242
Bigwood DA and Crocombe AD (1990) Non linear adhesive bonded joint design analysis. Int J
Adhes Adhes 10-1: 31–41
Chen D and Cheng S (1983) An analysis of adhesively bonded single lap joints. J Appl Mech, 50:
109–115
Crocombe AD (1989) Gobal yielding as a failure criteria for bonded joints. Int J Adhes Adhes
9(3): 145–153
Crocombe AD (2008) Incorporating environmental degradation in closed form adhesive joint stress
analysis. J Adhes, in press
Delale F, Erdogan F and Aydinoglu MN (1981) Stresses in adhesively bonded joints: a closed form
solution. J Compos Mater 15: 249–271
Goland M and Reissner E (1944) The stresses in cemented joints. J Appl Mech 66: 17–27
Hart-Smith LJ (1973a) Adhesive bonded single lap joints, NASA report CR112235, Langley
Research Centre
Hart-Smith LJ (1973b) Adhesive bonded double lap joints. NASA report CR112236, Langley
Research Centre
1 Simple Lap Joint Geometry
23
Ojalvo U and Eidinoff HL (1978) Bond thickness effects upon stresses in single-lap joints. AIAA
16(3): 204–211
Oplinger DW (1994) Effects of adherend deflections in single lap joints. Int J Solids Struct 31(18):
2565–2587
Renton WJ and Vinson JR (1977) Analysis of adhesively bonded joints between panels of composite materials. J Appl Mech 44: 101–106
Sawa T, Liu J, Nakano K and Tanaka J (2000) A two-dimensional stress analysis of single lap
adhesive joints of dissimilar adherends subjected to tensile loads. J Adhes Sci Tech 14(1):
43–66
Tsai MY and Morton J (1994) An evaluation of analytical and numerical solutions to the single lap
joint. Int J Solids Struct 31(18): 2537–2563
Tsai MY, Oplinger DW and Morton J (1998) Improved theoretical solutions for adhesive lap joints.
Int J Solids Struct 35(12): 1163–1185
Volkersen O (1938) Die nietkraftverteilung in zugbeanspruchten nietverbindungen mit konstanten
laschenquerschnitten. Luftfahrtforschung 15: 41–47
Volkersen O (1965) Recherches sur la theorie des assemblages colles. Construction Metallique
4: 3–13
Chapter 2
Analysis of Cracked Lap Shear (CLS) Joints
Liyong Tong and Quantian Luo
Abstract This chapter presents analytical models for cracked lap shear joints. Two
analytical frameworks are introduced: (1) the overlap is treated as an entire beam
to find displacements and force components of the cracked lap shear joints firstly,
and then the adherends in the overlap region are treated as individual beams to find
adhesive stresses and energy release rates; (2) individual beams are considered only;
displacements, force components and adhesive stresses are determined simultaneously by solving the coupled differential equations. Geometrically-nonlinear features of cracked lap shear joints are investigated. Strength of materials and fracture
mechanics based failure criteria are discussed.
2.1 Introduction
Adhesive bonding technology has been used in aircraft structures over 70 years
(Higgins 2000). In modern aeronautical and aerospace industries, this technology
is widely employed to join similar and dissimilar materials to form load-bearing
structural joints or integrated structures (Adams et al. 1997, Tong and Steven 1999).
Adhesive bonding is especially effective to join thin metallic and/or laminated composite sheets. However, structural performance can be dramatically reduced by
debonding or interface crack. It is known that debonding is mainly caused by high
interface adhesive stresses, which typically exist at a free edge or terminating ends
of adhesive layers and also exhibit singularity features.
Liyong Tong
School of Aerospace, Mechanical and Mechatronic Engineering, J11- Aeronautical Engineering
Building, The University of Sydney, NSW 2006, Australia, e-mail: ltong@aeromech.usyd.edu.au
Quantian Luo
School of Aerospace, Mechanical and Mechatronic Engineering, J11- Aeronautical Engineering
Building, The University of Sydney, NSW 2006, Australia, e-mail: qtluo@aeromech.usyd.edu.au
L.F.M. da Silva, A. Öchsner (eds.), Modeling of Adhesively Bonded Joints,
c Springer-Verlag Berlin Heidelberg 2008
25
26
L. Tong and Q. Luo
Fig. 2.1 A cracked lap shear
specimen (ASTM round
robin)
305 mm
254 mm
c
Strap adherend
P
P
A
a
Lap adherend
B
l
Due to the presence of both shear and peel stresses in an adhesive layer, fracture
in adhesive or near an adhesive-adherend interface in an adhesive joint is generally
a mixed-mode problem. To characterize such adhesive fracture behavior, a number of test specimens have been proposed (Mangalgiri and Johnson 1986, Rao and
Acharya 1995, Alif et al. 1997). A cracked lap shear (CLS) specimen in mixedmode condition is one of the specimens that has been widely used for characterizing fracture toughness and studying fracture behavior of adhesively bonded joints.
It can also be used to simulate delamination of composite laminate and skin-flange
debonding used in practical engineering structures.
An ASTM round robin (Johnson 1986) was conducted for calculating energy
release rates of the CLS specimen as schematically shown in Fig. 2.1. The CLS
specimen is assumed to have a width of 25 mm and a varying debond length a.
In general, a CLS specimen subjected to tensile loading can have four possible
combinations of support conditions at both ends (Johnson 1986, Lai et al. 1996),
namely, (1) clamped-clamped, (2) roller-clamped, (3) roller-roller (similar to single
lap joint), and (4) free-fixed.
One distinctive feature of the CLS is the eccentric loading path that leads to
geometrical nonlinearity (Johnson 1986), and thus large deflections have to be considered in analytical and numerical analyses.
2.2 Background
The cracked lap shear specimen was firstly proposed by Brussat et al. (1977) to
investigate the mixed mode fracture behavior of shear-loaded adhesive joints. The
CLS have been used to study adhesive joint debonding (Brussat et al. 1977, Lin and
Liechti 1987, Schmueser and Johnson 1990, Cheuk and Tong 2002) and composite
delamination (Mangalgiri and Johnson 1986, Rhee 1994, Rhee and Chi 2001).
Johnson (1986) reported the ASTM round robin results conducted by nine groups
or laboratories. This work was sponsored by ASTM and all participants were asked
to calculate the energy release rate of the CLS specimens with four different debond
lengths and two different geometries. It was indicated that, although no generally applicable closed-form solutions exist for the CLS specimen, the nonlinear finite element analysis gives the lab-to-lab consistent predictions with the observed
experimental behaviors.
2 Analysis of Cracked Lap Shear (CLS) Joints
27
Analytical analysis for the CLS has attracted much attention in past several
decades to sight into its essence. The analytical solutions may be obtained by modeling adherends as beams and the adhesive as a continuous spring with shear and
peel stiffness. A spring model for the adhesive is widely used due to the facts that
its Young’s modulus is much smaller than that of the isotropic adherends, and it is
normally very thin as compared to the adherends. Brussat et al. (1977), Fernlund
and Spelt (1991a), Fernlund et al. (1994), and Lai et al. (1996) presented analytical
analyses for the CLS specimen using the method developed by Goland and Reissner (1944) for a single lap joint. In this method, the overlap is treated as an entire
beam to determine the deflections of the adhesively bonded structure and the force
components at the end of the overlap as shown in Fig. 2.2.
In the light of the found CLS deflections and the overlap force components, the
adhesive stresses and/or energy release rates of the interface crack can then be calculated to predict the CLS failure loads.
This method divides analysis into two steps and treats the overlap adherends as
an entire or one single beam in the first step and then as two separated sub-beams in
the second step (Goland and Reissner 1944). An alternative way is to combine the
two steps into one and only the two separated sub-beam model for the overlap adherends is used. This coupled formulation was proposed by Hart-Smith (1973) and
has been also widely used for analytical analysis of the CLS specimen. However, the
overlap large deflection is not included in this coupled method. Recently, Luo and
Tong (2007) presented fully coupled formulations for a single lap joint, in which
adherends of the overlap are treated as two individual beams each undergoing large
deflections. The adhesive stresses/energy release rates and the force components at
the end of the overlap were determined simultaneously.
Experimental testing is an essential tool for validating various analyses for the
CLS specimen. Not only the analytical results are required to be verified by experiments, but also the experimental test is the unique way to determine the fracture
toughness of the adhesive joint to establish the fracture envelope for the CLS specimen. The experimental investigations of the CLS specimen have been conducted by
a number of authors, e.g., Fernlund and Spelt (1991b), Papini et al. (1994), Wang
l
MA
F
c
Outer adherend
Overlap
MB
F
RA
RB
Qk
Mk
Nk
I
MB
F
(t1+t2+ta)/2
I
RB
Fig. 2.2 Loadings, reaction forces and forces in cross-section I-I
28
L. Tong and Q. Luo
et al. (1995), Benzeggaph and Kenane (1996), Rhee and Chi (2001), Rhee et al.
(2003), Cheuk and Tong (2002). Both adhesive debonding and composite delamination have been tested in most of these experiments. A noticeable experimental test
was conducted by Fernlund and Spelt (1994) as the fracture toughness is obtained
for a full range of mode ratios.
Dattaguru et al. (1984) initially used geometrically nonlinear finite element
analysis (NFEA) to numerically investigate the CLS. In the report presented by
Johnson (1986), both linear and nonlinear FEA analyses were conducted. It was
concluded that geometrically nonlinear FEA had to be used for the CLS specimen.
The FEA modeling of the CLS specimen may be classified into two types: a full
2D (or full 3D) model and a beam (or plate) model with adhesive elements. In the
full 2D (or full 3D) model, 2D (or 3D) elements are used to model both adherends
and adhesive. A full 3D FEA is an effective tool for the CLS specimen. However, it
is well known that it can be computationally expensive and may even encounter numerical difficulty for beam-type and plate/shell-type structures. As concerned about
a beam model with adhesive elements, Carpenter (1973, 1980) developed a constant
shear and extensional spring element for adhesive idealization. Kuo (1984) developed a continuous spring element with shear stiffness. Luo and Tong (2004) derived
a continuous spring element with shear and peel stiffness. These two types of FEA
models for adhesive joints can also be combined. Ko et al. (1994) used the weighted
adhesive plate element for the CLS specimen, Wu and Crocombe (1996) used beam
elements to model the adherends and 2D elements to depict the adhesive.
With the development of commercial FEA software, full 2D (or full 3D) models have been widely used (Wu and Crocombe 1996, Cheuk and Tong 2002, Yang
et al. 2003, Luo and Tong 2007). When NFEA is used for the CLS analysis, energy
release rates (ERR) and/or stress intensity factors are required to be calculated when
the fracture mechanics based failure criteria are used. In this case, virtual crack closure technique (VCCT) (Harbert and Hogan 1992, Rhee and Ernst 1993, Panigrahi
and Pradhan 2007, Yang et al. 2007) has been widely used.
2.3 Fundamental Formulations
In this section, the fundamentals of analytical solutions for the CLS specimen will
be discussed. Two theoretical frameworks mentioned in Sect. 2.2 will be discussed
in details: (1) the CLS deflections and the force components at the overlap end are
determined firstly and then the adhesive stresses and/or energy release rates (ERRs)
are calculated, and (2) the CLS displacements, the overlap forces and the adhesive
stresses/ERRs are determined simultaneously.
In the first analytical framework, analytical analysis for the cracked lap shear
consists of two parts: (1) find deflections of the CLS specimen and the force components at the overlap end by taking into account large deflections as shown in Fig. 2.2,
and (2) stress analysis and/or ERR calculation for failure prediction.
2 Analysis of Cracked Lap Shear (CLS) Joints
29
The force components at the overlap end include the applied force F, reaction
force RB and moment MB and forces in cross-section I-I: axial force Nk , shear force
Vk (or Qk ) and bending moment Mk , where Vk is the transverse shear force perpendicular to the deformed beam axis and Qk is that perpendicular to the un-deformed
beam axis. The key force component is the edge bending moment Mk at the overlap
end as the other components of cross-section I-I may be found by the equilibrium
equations or the differential relations of internal forces based on the beam theory.
The bending moment at the adhesive edge can be defined as:
t2 + ta
Mk = kF
(2.1)
2
where k is the edge moment factor; t2 and ta are thicknesses of the lap adherend
and the adhesive. It is noted that (t2 + ta /2) is the distance of two horizontal forces
(see Fig. 2.2), and when t1 = t2 = t, definitions of the edge moment and its factor
in Eq. (2.1) are consistent with those of Goland and Reissner (1944) for a single lap
joint.
The edge moment of a cracked lap shear specimen can be found on the basis of
the method proposed by Goland and Reissner (1944), who treated the overlap as an
entire Euler beam. In the light of this model, Fernlund et al. (1994) determined the
edge moment and the CLS deflections for calculating the value of J-integral. Lai
et al. (1996) also used this method to find the overlap force components and then
calculated the ERRs for the CLS specimen.
After determining the CLS deflections and the overlap force components, the
adhesive stresses can be solved based on the formulations developed by Goland
and Reissner (1994), and the energy release rates can be calculated by using the
J-integration with the mode partition (Fernlund and Spelt 1991a, Fernlund et al.
1994) and the interface crack theory (Suo and Hutchinson 1990, Lai et al. 1996).
In the second analytical framework, fully-coupled equations are treated. The edge
moment, displacements and the adhesive stresses were solved at the same time. This
framework has been used to analyze the CLS with the roller-roller boundary condition (SLJ) by Hart-Smith (1973), Oplinger (1994), and Luo and Tong (2007). In
the Hart-Smith model (1973), large deflections of the overlap were not considered.
In the Oplinger’s model (1994), large deflections of the overlap coupled with the
adhesive shear stress but not peel stress were considered. Luo and Tong (2007) considered large deflections with coupled both shear and peel stresses. The formulation
of the second framework can also be extended to the CLS with the other boundary
conditions using the same procedure.
2.3.1 Basic Equations for the Outer Adherend and Reaction Forces
To find the CLS displacements and the overlap force components, we need to consider the equilibrium, constitutive and continuity conditions of the CLS specimen.
30
L. Tong and Q. Luo
The free body diagrams of the CLS specimen and the overlap are shown in
Fig. 2.2. The applied force F is the horizontal component of load P. The equilibrium
equations of the CLS specimen are:
RB + RA = 0
MB − MA + F
t2 + ta
2
(2.2)
− RA (l + c) = 0
(2.3)
In the first framework, deflections of the CLS specimen are schematically illustrated in Fig. 2.3. The bending moment of the outer adherend and the overlap is
respectively:
M3 (x3 ) = −Fw3 + RA x3 + MA
Mo (xo ) = −Fwo − RB xo + MB
0 ≤ x3 ≤ l
−c ≤ xo ≤ 0
(2.4)
(2.5)
The constitutive relations of the outer adherend and the overlap based on the
Euler beam are:
d 2 w3
d 2 wo
(2.6)
M3 = −D11 2 ; Mo = −Do 2
dxo
dx3
where D11 and Do are bending stiffness of the outer adherend and the overlap. Substituting Eq. (2.6) into Eqs. (2.4) and (2.5) yields:
RA
MA
x3 +
F
F
RB
MB
wo = B1 sinh βo xo + B2 cosh βo xo − xo +
F
F
w3 = A1 sinh βk x3 + A2 cosh βk x3 +
(2.7)
(2.8)
In Eqs. (2.7) and (2.8), A1 , A2 , B1 and B2 are the unknown integration constants, which are to be determined by using the relevant boundary conditions; and
the eigenvalues are:
F
F
βk =
; βo =
(2.9)
D11
Do
It is noted that, in Eqs. (2.7) and (2.8), there are 4 integration constants and 4
reaction force components to be determined. To find the 8 unknowns, we need to
have 8 independent equations. By the support conditions at A and B, 4 boundary
RA
F
x3
M3
MA
N3
Qo
RB
F xo
z3
Q3
No
MB
Mo
zo
Fig. 2.3 Deflections and coordinate systems of the CLS specimen
2 Analysis of Cracked Lap Shear (CLS) Joints
31
equations can be derived; 2 equilibrium equations are given in Eqs. (2.2) and (2.3);
the other 2 equations are obtained by the continuity conditions at cross-section I-I.
The continuity conditions at cross section I-I can be written as:
u3 (l) = uo (−c);
w3 (l) = wo (−c);
dw3 (l) dwo (−c)
=
dx3
dxo
(2.10)
Equation (2.10) is the general requirements of deformation continuity; the axial continuity condition was not used for the formulation based on the method of Goland
and Reissner (1944).
The 4 integration constants and the 4 unknowns RA , MA , RB and MB can be found
by the 4 boundary condition equations that will be discussed in Subsection 2.3.3 and
the 4 equations given in Eqs. (2.2), (2.3) and (2.10). Therefore, the CLS deflections
are solved and then the edge moment Mk the edge shear force Vk are found by:
Mk = −D11
d 2 w3 (l)
;
dx32
Vk =
dM3 (l)
dx3
(2.11)
When the CLS deflections and force components of the overlap are solved, adhesive stresses and the ERRs can then be found.
2.3.2 Basic Equations for the Overlap and Adhesive Stresses
When upper and lower adherends are identical, or symmetrical adherends, solutions
of the shear and peel stresses for the prescribed force boundary conditions of the
overlap were presented by Goland and Reissner (1944). For the asymmetrical adherends, solutions of the adhesive stresses can be found in Luo and Tong (2002). In
this chapter, we only present adhesive stress analysis for the symmetrical adherends
and introduce the following variables:
⎧
⎪
⎨2us = u2 + u1 ; 2ws = w2 − w1 ; 2ua = u2 − u1 ; 2wa = w2 + w1
2Ns = N2 + N1 ; 2Qs = Q2 − Q1 ; 2Ms = M2 − M1
⎪
⎩
2Na = N2 − N1 ; 2Qa = Q2 + Q1 ; 2Ma = M2 + M1
(2.12)
The variables in Eq. (2.12) have the usual meanings used in the Euler beam theory; subscripts 1 and 2 refer to identical adherends 1 and 2 in the overlap region; the
force components are shown in the free body diagrams of the infinitesimal elements,
see Fig. 2.4.
The equilibrium equations of adherends 1 and 2 in the overlap can be derived
from Fig. 2.4, as:
32
L. Tong and Q. Luo
Fig. 2.4 Free body diagram
for stress analysis of adhesive
joints
M1
M1+dM1
N1
Adherend 1
Q1
σ
z
σ
τ
N1+dN1
Q1+dQ1
τ
Adhesive
τ
τ
M2
N2
x
M2+dM2
N2+dN2
Adherend 2
Q2
Q2+dQ2
dx
⎧
dN
⎪
⎪ 1 + τ = 0;
⎨
dx
⎪
⎪ dN2
⎩
− τ = 0;
dx
dQ1
+ σ = 0;
dx
dM1 t1
+ τ − Q1 = 0
dx
2
dQ2
− σ = 0;
dx
dM2 t1
+ τ − Q2 = 0
dx
2
(2.13)
In Eq. (2.13), τ and σ are the adhesive shear and peel stresses, whose definitions are
given by Goland and Reissner (1944):
τ=
Ga
t1 dw1 dw2
+
) ;
(u2 − u1 ) + (
ta
2 dx
dx
σ=
Ea (w2 − w1 )
ta
(2.14)
In Eq. (2.14), Ea and Ga are Young’s and shear moduli of the adhesive. By making use of the variables in Eq. (2.12), Eqs. (2.13) and (2.14) become:
⎧
dNs
dQs
dMs
⎪
⎪
− σ = 0;
− Qs = 0
⎨ dx = 0;
dx
dx
⎪
⎪
⎩ dNa − τ = 0; dQa = 0; dMa + t1 τ − Qa = 0
dx
dx
dx
2
2Ga
t1 dwa
2Ea ws
τ=
ua +
; σ=
ta
2 dx
ta
(2.15)
(2.16)
The constitutive relations of the Euler beam are:
N = A11
du
;
dx
M = −D11
d2w
dx2
(2.17)
where A11 is the extensional stiffness of the adherends.
It is noted that shear and peel stresses have been decoupled in Eq. (2.15), in
which, the 1st and the 2nd row of equations can be used to solve the peel stress and
2 Analysis of Cracked Lap Shear (CLS) Joints
33
shear stress respectively. By substituting Eq. (2.17) into (2.15), and then substituting
ua , wa and ws into Eq. (2.16), the governing differential equations for the shear and
peel stresses can be obtained as follows:
d3τ
2Ga
−
dx3 A11ta
1+
A11t12
4D11
dτ
= 0;
dx
d4σ
2Ea
+
σ =0
dx4 D11ta
(2.18)
The analytical solutions of Eqs. (2.18) are:
⎧
⎪
⎨τ = A1 sinh βa x + A2 cosh βa x + A3
σ = B1 sinh βσ x sin βσ x + B2 sinh βσ x cos βσ x
⎪
⎩
+B3 cosh βσ x sin βσ x + B4 cosh βσ x cos βσ x
(2.19)
In Eq. (2.19), Ai (i = 1, 2, 3) and B j ( j = 1, 2, 3, 4) are the integration constants,
and the eigenvalues are:
⎧
⎪
2
2
⎪
⎪
⎨βa = αa βτ ;
βτ =
⎪
⎪
(1 + αk )
⎪
⎩αa =
;
4
8Ga
;
A11ta
√ 2 4 2Ea
βσ =
2
D11ta
A11t12
αk =
4D11
(2.20)
where coefficients αa and αk may reflect influence of the lay-up sequence when
the overlap adherends are composite laminates with the symmetrical lay-up. For the
isotropic adherends, αa = 1 and αk = 3.
The integration constants in Eq. (2.19) can be determined by the prescribed force
boundary conditions of the overlap, and the force components of the overlap are
dependent on loadings, support conditions and geometrical configurations of the
CLS specimen.
In this analytical method, large deflections are considered to find the force components of the overlap but not for the adhesive stresses. The entire beam model and
the sub-beam model are applied to the overlap force determination and the adhesive
stress analysis respectively.
Hart-Smith (1973) presented the coupled formulations for a single lap joint. By
substituting Eqs. (2.16) and (2.17) into Eq. (2.15), the governing differential equations for adherend displacements can be obtained:
d 4 ws 2Ea
d 2 us
= 0; D11 4 +
ws = 0
2
dx
dx
ta
⎧ 2
d ua
2Ga
t1 dwa
⎪
⎪
−
+
u
=0
⎪
a
⎨ dx2
A11ta
2 dx
⎪ d 4 wa
⎪
Gat1 dua t1 d 2 wa
⎪
⎩−
+
+
=0
dx4
D11ta dx
2 dx2
(2.21)
(2.22)
34
L. Tong and Q. Luo
Equations (2.21) and (2.22) can be analytically solved, whose integration constants are solved by boundary conditions of the overlap. The solved integration constants include unknowns: edge bending moment Mk and rigid body motions. These
unknowns should be solved by the continuity conditions at I-I and support conditions of the CLS specimen. It is noted that all the 3 continuity equations given in
Eq. (2.10) will be used in this coupled formulations. To use the continuity condition
with respect to the axial deformation, the axial displacement of the outer adherend
needs to be used. It can be solved by referring to Fig. 2.3 and the constitutive relations, which are given by:
F
x3 +C1
(2.23)
u3 =
A11
where C1 is the integration constant.
Because large deflections of the overlap are not included in adhesive stress analysis presented in Eq. (2.13) and the solutions given in Eq. (2.19), the analytical solutions of Hart-Smith (1973) do not model the overlap large deflection.
By analyzing the Goland and Reissner’s formulation (1944) and the Hart-Smith’s
one (1973), the former considered large deflections of the overlap but ignored adhesive deformation, whereas the latter considered shear and peel strains but neglect
the overlap large deflections.
Recently, Luo and Tong (2007) presented the fully-coupled nonlinear analysis
for SLJs, in which, both large deflections and adhesive deformations are modeled.
In this method, equilibrium equations of the infinitesimal adherend elements are
derived on the basis of the geometrically nonlinear analysis. As shown in Fig. 2.5,
the equilibrium equations for the free body diagrams are:
⎧
dw1
dx
⎪
⎪
dN1 + τ (ds1 )
dx = 0
= 0; dQ1 + σ (dx) + τ
⎪
⎪
ds1
dx
⎪
⎪
⎪
⎪
⎪
⎪
⎪dM1 + t1 τ (dx) − Q1 dx = −N1 dw1 dx
⎪
⎨
2
dx
⎪
dw2
dx
⎪
⎪
⎪
⎪dN2 − τ (ds2 ) ds = 0; dQ2 − σ (dx) − τ dx dx = 0
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎩dM2 + t1 τ (dx) − Q2 dx = −N2 dw2 dx
2
dx
(2.24)
where, (ds1 )2 = (dx)2 + (dw1 )2 and (ds2 )2 = (dx)2 + (dw2 )2 .
By using the variables in Eq. (2.12), the equilibrium equations become:
⎧ dN
s
⎪
=0
⎪
⎪
⎪
dx
⎨
dQs
dwa
⎪
−σ −τ
= 0;
⎪
⎪
dx
dx
⎪
⎩
dMs
dws
dwa
− Qs = −Ns
− Na
dx
dx
dx
(2.25)
2 Analysis of Cracked Lap Shear (CLS) Joints
Fig. 2.5 Free body diagram
for nonlinear analysis of
adhesive joints
35
M1
N1
Q1
M1 + dM1
N1 + dN1
Q1 + dQ1
Adherend 1
σ
τ
z
M2
N2
τ
τ
x
dw1/dx
Adhesive
σ
τ
M2 + dM2
N2 + dN2
Adherend 2
Q2
Q2 + dQ2
dx
⎧
dNa
⎪
⎪
⎪
⎨ dx − τ = 0
⎪
⎪
⎪
⎩ dQa − τ dws = 0;
dx
dx
(2.26)
dMa t1
dwa
dws
+ τ − Qa = −Ns
− Na
dx
2
dx
dx
The nonlinear constitutive relations of the Euler beam are:
⎧
⎪
d2w
du 1 du 2 1 dw 2
⎪
⎪
N
=
A
+
+
−
B
⎪
11
11
⎪
dx 2 dx
2 dx
dx2
⎨
⎪
⎪
d2w
du 1 du 2 1 dw 2
⎪
⎪
+
M
=
B
+
−
D
⎪
11
11
⎩
dx 2 dx
2 dx
dx2
(2.27)
where the coupling extension and bending stiffness B11 is equal to zero for the symmetrical cross section. In Eqs. (2.16) and (2.25), (2.26), (2.27), 12 equations consists
of 12 variables: 6 stress resultants and 4 adherend displacements plus shear and peel
stresses.
In Eqs. (2.25) and (2.26), differentiating the bending moment equilibrium equations, into which, substituting the shear force equations, and utilizing the axial force
equilibrium equations, we have:
d 2 ws
d 2 wa
d 2 Ms
− σ = −Ns 2 − Na 2
2
dx
dx
dx
(2.28)
d 2 Ma t1 d τ
d 2 wa
d 2 ws
=
−N
+
−
N
s
a
dx2
2 dx
dx2
dx2
(2.29)
When the following higher order nonlinear terms are ignored:
dua d 2 ws
dx dx2
dua d 2 wa
dx dx2
(2.30)
36
L. Tong and Q. Luo
The simplified equilibrium equations including effects of the overlap large deflections can be obtained:
dNs
= 0;
dx
d 2 Ms
F d 2 ws
−σ = −
2
dx
2 dx2
dNa
− τ = 0;
dx
(2.31)
F d 2 wa
d 2 Ma t1 d τ
=−
+
2
dx
2 dx
2 dx2
(2.32)
In Eqs. (2.31) and (2.32), the relation Ns = (N1 + N2 )/2 = F/2 has been used.
In the definitions of adhesive shear and peel strains of Eq. (2.14), small strains and
rotations of the adherends are assumed. By using this assumption, higher order terms
in Eq. (2.27) can be neglected. Utilizing Eq. (2.12), we have:
Ni = A11
dui
dx
Mi = −D11
d 2 wi
dx2
(i = s, a)
(2.33)
By substituting Eqs. (2.16) and (2.33) into Eqs. (2.31) and (2.32), the following
governing equations in terms of displacements of adherends in the overlap region of
the CLS specimen can be obtained:
⎧ 2
d us
⎪
⎪
⎨ dx2 = 0;
4
2
⎪
⎪
⎩D11 d ws − F d ws + 2Ea ws = 0
4
dx
2 dx2
ta
(2.34)
⎧
d 2 ua 2Ga
t1 dwa
⎪
⎪
⎪
A
−
ua +
=0
⎪
⎨ 11 dx2
ta
2 dx
⎪
⎪
d 4 wa Gat1 dua t1 d 2 wa
F d 2 wa
⎪
⎪
+
=0
+
⎩−D11 4 +
dx
ta
dx
2 dx2
2 dx2
(2.35)
From a mathematical viewpoint, Eqs. (2.34) and (3.35) are linear differential equations for the given axial load F, and can be analytically solved. The closed-form
solutions of Eqs. (3.34) and (3.35) are (Luo and Tong 2008):
⎧
⎪
⎨us = As1 x + As2
ws = (Bs1 sinh βs1 x + Bs2 cosh βs1 x) sin βs2 x
⎪
⎩
+(Bs3 sinh βs1 x + Bs4 cosh βs1 x) cos βs2 x
⎧
ua = Aa1 sinh βa1 x + Aa2 cosh βa1 x
⎪
⎪
⎪
⎨
+Aa3 sinh βa2 x + Aa4 cosh βa2 x + Aa5
⎪wa = Ba1 sinh βa1 x + Ba2 cosh βa1 x
⎪
⎪
⎩
+Ba3 sinh βa2 x + Ba4 cosh βa1 x + Ba5 x + Ba6
(2.36)
(2.37)
2 Analysis of Cracked Lap Shear (CLS) Joints
37
where, As1 , As2 , Bsi (i = 1, 2, 3, 4), Aa1 and Ba j (i = 1, 2, · · ·, 5; j = 1, 2, · · ·, 6)
are the integration constants, which are determined by the boundary conditions. The
eigenvalues are respectively (Luo and Tong 2008):
βs1 =
βσ2 +
βk2
,
8
βs2 =
βσ2 −
βk2
8
(2.38)
⎧
⎪
βk2
β4
1
1
⎪
2
2
2 β 4 + (α − )β 2 β 2 + k
⎪
+
β
=
α
β
+
α
a
a
⎪
a
τ
τ
τ
a1
k
⎪
2
2
2
4
⎪
⎨
⎪
⎪
⎪
βk2
1
1 2 2 βk4
⎪
⎪
2
2
2 4
⎪
⎩βa2 = 2 αa βτ + 2 − αa βτ + (αa − 2 )βτ βk + 4
(2.39)
For the case of isotropic adherends, Eq. (2.39) degenerates:
⎧
⎪
βk2
βτ2 βk2 βk4
1
⎪
2
2
4
⎪βa1 =
+ βτ +
+
βτ +
⎪
⎪
2
2
2
4
⎪
⎨
⎪
⎪
⎪
βk2
βτ2 βk2 βk4
1
⎪
⎪
2
2
4
⎪
⎩βa2 = 2 βτ + 2 − βτ + 2 + 4
(2.39a)
By using the overlap boundary conditions, the integration constants with the unknowns of the edge moment and the rigid body motions can be solved. The unknowns can be solved by the continuity conditions in Eq. (2.10) and the support
conditions.
It can be seen that, in the second analytical framework of the coupled nonlinear
analysis, large deflections of the outer adherend and the overlap, and the adhesive
deformations have been considered. The CLS displacements, force components and
adhesive stresses can be simultaneously solved.
2.3.3 Boundary Conditions, Loading Cases
and Analytical Solution
2.3.3.1 Boundary Conditions
To analytically solve the CLS for the 4 possible boundary conditions of the CLS,
we first derive the boundary equations in terms of the displacements (Johnson 1986,
Lai et al. 1996):
(1) Clamped-Clamped
w3 (0) = 0;
dw3 (0)
dwo (0)
= 0 and wo (0) = 0;
=0
dx3
dxo
(2.40)
38
L. Tong and Q. Luo
(2) Roller-Clamped
w3 (0) = 0;
MA = 0 and wo (0) = 0;
dwo (0)
=0
dxo
(2.41)
(3) Roller-Roller (equivalent to single lap joint)
w3 (0) = 0;
MA = 0 and wo (0) = 0; MB = 0 or
d 2 wo (0)
=0
dxo2
(2.42)
(4) Free-Fixed
MA = 0;
RA = 0 and wo (0) = 0;
dwo (0)
=0
dxo
(2.43)
2.3.3.2 Loading Cases
The CLS specimen is subjected to a pair of tensile forces (P) in the experimental
test, as shown in Fig. 2.1. The axial force F in Fig. 2.2 is the project of P in the
horizontal direction, which is given by:
F=
(l + c) P
(2.44)
(l + c)2 + [(t2 + ta )/2]2
The loading cases for the CLS testing specimen are illustrated in Fig. 2.2. In this
chapter, we present analytical analysis for the CLS specimen subjected to the tensile
force only. For engineering structures with lap shear joints, they may be subjected
to transverse forces and bending moments, whose solutions can be found using the
similar procedure.
2.3.3.3 The 1st Step Solutions of the 1st Analytical Framework
In the 1st analytical framework, solution procedures of the 1st step are to determine
the 8 unknowns in Eqs. (2.7) and (2.8) using the 8 equations given in Eqs. (2.2),
(2.3), (2.10) and (2.40) or (2.41) or (2.42) or (2.43). Lai et al. (1996) derived force
components of the overlap for the boundary conditions of Clamped-Clamped and
Roller-Clamped. Goland and Reissner (1944) derived the edge moment for the
Roller-Roller boundary conditions and the symmetrical substrates. The consistent
formulations for the general CLS with the 4 possible boundary conditions are presented below.
(1) Clamped-Clamped:
Substituting Eq. (2.40) into Eqs. (2.7) and (2.8) and combining Eqs. (2.2) and
(2.3), we have:
2 Analysis of Cracked Lap Shear (CLS) Joints
⎧
MA
RA
MB
RB
⎪
⎪
⎨A2 + F = 0; βk A1 + F = 0; B2 + F = 0; βo B1 − F = 0
⎪
⎪
⎩B1 = βk A1 ; B2 = A2 + A1 βk (l + c) + t2 + ta
βo
2
39
(2.45)
By utilizing Eq. (2.45), Eqs. (2.7) and (2.8) become:
w3 = A1 (sinh βk x3 − βk x3 ) + A2 (cosh βk x3 − 1)
βk
(sinh βo xo − βo xo ) + βk (l + c) (cosh βo xo − 1)
βo
t2 + ta
(cosh βo xo − 1)
+ A2 (cosh βo xo − 1) +
2
(2.46)
wo = A1
(2.47)
By substituting Eqs. (2.46) and (2.47) into Eq. (2.10), the following algebraic
equations can be obtained:
⎧
A1 [βk βo (l + c) cosh βo c − βk sinh βo c − βo sinh βk l]
⎪
⎪
⎪
⎪
⎨ +A β (cosh β c − cosh β l) = t2 +ta β (cosh β c − 1)
o
o
o
2 o
k
2
⎪
A1 [βk (cosh βo c − cosh βk l) − βk (l + c) sinh βo c]
⎪
⎪
⎪
⎩
a
−A2 (βk sinh βk l + βo sinh βo c) = t2 +t
2 βo sinh βo c
(2.48)
The integration constants A1 and A2 are readily solved from Eq. (2.48), and
then B1 , B2 , RA , RB , MA and MB are obtained by Eq. (2.45). Since deflections
and reaction force components are determined, the edge moment is found from
Eq. (2.11)
(2) Roller-Clamped:
Substituting Eq. (2.41) into Eqs. (2.2), (2.3), (2.7) and (2.8) yields:
⎧
RB
RA
MB
⎪
⎪
⎨A2 = 0; B2 + F = 0; F = − F = βo B1
⎪
⎪
⎩ MB = −βo B1 (l + c) − t2 + ta
F
2
(2.49)
Substituting Eq. (2.49) into Eqs. (2.7) and (2.8), we have:
w3 = A1 sinh βk x3 − βo B1 x3
wo = B1 [(sinh βo x0 − βo xo ) + βk (l + c) (cosh βo xo − 1)]
t2 + t a
+
(cosh βo xo − 1)
2
(2.50)
(2.51)
A set of equations used to determine the integration constants can be obtained
by substituting Eqs. (2.50) and (2.51) into Eq. (2.10):
40
L. Tong and Q. Luo
⎧
t2 + ta
⎪
⎪
⎨A1 βk cosh βk l + B1 βo [βo (l + c) sinh βo c − cosh βo c] = − 2 βo sinh βo c
⎪
⎪
⎩A1 sinh βk l + B1 [sinh βo c − βo (l + c) cosh βo c] = t2 + ta (cosh βo c − 1)
2
(2.52)
By solving Eq. (2.52), the integration constants A1 and B1 are determined and
then B2 , RA , RB and MB are obtained by Eq. (2.49). The deflections and reaction
forces for the CLS are determined.
(3) Roller-Roller:
Substituting Eq. (2.42) into Eqs. (2.2), (2.3), (2.7) and (2.8), we have:
RA = −RB = α F; A2 = 0; B2 = 0
where, α =
t2 + ta
2(l + c)
(2.53)
Equations (2.7) and (2.8) become:
w3 = A1 sinh βk x3 + α · x3
(2.54)
wo = B1 sinh βo xo + α · xo
(2.55)
By substituting Eqs. (2.54) and (2.55) into Eq. (2.10), the integration constants
A1 and B1 are obtained:
⎧
βo (t2 + ta ) cosh βo c
⎪
⎪
A =−
⎪
⎨ 1
2 (βo sinh βk l cosh βo c + βk cosh βk l sinh βo c)
(2.56)
⎪
⎪
βk (t2 + ta ) cosh βk l
⎪
⎩B1 = −
2 (βo sinh βk l cosh βo c + βk cosh βk l sinh βo c)
The edge moment can be found from Eqs. (2.11), (2.54) and (2.56), and the
edge moment factor is given by:
k=
1
1
=
βk
Do
D11
1 + tanh βo c coth βk l
1+
tanh
βk c coth βk l
βo
D11
Do
(2.57)
When t1 = t2 = t and ta << t (i.e., D0 = 8D11 ), Eq. (2.57) becomes:
k=
√
1 + 2 2 tanh
1
βk c
√ coth βk l
2 2
(2.58)
Equation (2.58) was recovered from the formulations of Goland and Reissner
(1944) by Tsai and Morton (1994). It is noted that the edge moment factor given
by Goland and Reissner (1944) is the simplified one for the long outer adherend
(βk l >> 1).
2 Analysis of Cracked Lap Shear (CLS) Joints
(4) Free-Fixed:
Combining Eq. (2.43) with Eqs. (2.2), (2.3), (2.7), (2.8), we have:
t2 + ta
t2 + ta
RB = 0; MB = −F
; B1 = 0; B2 =
2
2
41
(2.59)
Equations (2.7) and (2.8) become:
w3 = A1 sinh βk x3 + A2 cosh βk x3
wo =
t2 + ta
(cosh βo xo − 1)
2
(2.60)
(2.61)
The integration constants A1 and A2 are readily determined By Eqs. (2.10),
(2.60) and (2.61):
⎧
βo
t2 + ta
⎪
⎪
sinh βk l (cosh βo c − 1) + cosh βk l sinh βo c
⎪
⎨A1 = − 2
βk
(2.62)
⎪
+
t
β
t
⎪
a
o
2
⎪A2 =
cosh βk l (cosh βo c − 1) + sinh βk l sinh βo c
⎩
2
βk
The edge moment factor is:
D11
k = − cosh
βk c − 1
D0
(2.63)
2.3.3.4 Adhesive Stresses
When the CLS deflections and the overlap force components are solved, the adhesive stresses and the energy release rates (ERR) can be calculated. The ERR calculations will be discussed in Sect. 2.5.
The large deflections of the overlap are not included for the adhesive stress analysis in the 1st analytical framework based on the method of Goland and Reissner (1944) and the 2nd formulation framework based on the approach of HartSmith (1973). For the prescribed force boundary conditions, the same adhesive
stresses are obtained based on the formulations of Goland and Reissner (1944) and
Hart-Smith (1973). For the symmetrical CLS with the roller-roller boundary conditions, shear and peel stresses are:
⎧
βτ (Ft1 + 6Mk ) cosh βτ x 3(Ft1 − 2Mk )
⎪
⎪
+
⎪τ =
⎨
8t1 sinh βτ c
8t1 c
(2.64)
⎪
2βσ2 (Bσ 1 sinh βσ x sin βσ x + Bσ 4 cosh βσ x cos βσ x)
⎪
⎪
⎩σ =
sinh 2βσ c + sin 2βσ c
42
L. Tong and Q. Luo
where,
⎧
Vk
⎪
⎪
⎨Bσ 1 = Mk (sinh βσ c cos βσ c + cosh βσ c sin βσ c) + βσ sinh βσ c sin βσ c
⎪
⎪
⎩Bσ 4 = Mk (sinh βσ c cos βσ c − cosh βσ c sin βσ c) + Vk cosh βσ c cos βσ c
βσ
(2.65)
The maximum shear and peel stresses are:
⎧
1
⎪
⎪
⎪
⎨τmax = 8 [(3k1 + 1) βτ coth βτ c + 3 (1 − k1 )] F
⎪
βk
⎪
⎪
coth βk l βσ2 Mk
⎩σmax = 1 +
βσ
(2.66)
where
k1 = (1 +
ta
)k
t1
(2.67)
Adhesive stresses shown in Eqs. (2.64) and (2.66) were presented by Goland
and Reissner (1944), and then Hart-Smith (1973). Because they are derived on the
basis of Fig. 2.4, large deflection effects of the overlap are not included (Tsai and
Morton 1994).
In the 2nd analytical framework of the fully-coupled nonlinear analysis, the analytical solutions of the CLS specimen are found for the roller-roller boundary
conditions (SLJ) and symmetrical adherends in the existing literatures (Luo and
Tong 2007). The simplified analytical solutions for the edge moment factor and the
maximum adhesive stresses are:
βτ c f (βa2 c) − 1
8βτ c(1 + ta /t1 )
k=
1
βτ c f (βa2 c) + 3
+
1 + (βk c) coth βk l + (βk c)2
2βσ c
8βτ c
(2.68)
⎧
1
⎪
⎪
⎪
⎨τmax = 8 [(3k1 + 1) βa1 coth βa1 c + 3 (1 − k1 ) βa2 coth βa2 c] F
⎪
βk
⎪
⎪
coth βk l βσ2 Mk
⎩σmax = 1 +
βσ
(2.69)
1 + (βk c)2
where,
f (βa2 c) =
βa2 c coth βa2 c − 1
(βa2 c)2
(2.70)
The analytical solutions based on the fully-coupled nonlinear formulations can
also be applied to the symmetrical CLS specimen with other boundary conditions.
2 Analysis of Cracked Lap Shear (CLS) Joints
43
By comparing the edge moment factor and the maximum adhesive stresses for the
analytical framework of Goland and Reissner (1944) with those based on Luo and
Tong (2007), differences of the results can be found in the edge moment factor
and the shear stress. It is noted that expressions of the maximum peel stress for
the two formulations are the same when the same force boundary conditions are
prescribed.
2.3.4 Issues for CLS Joints with Composite Adherends
When the adherends of the CLS specimen are composite laminates, analytical solutions for the CLS are very complicated. Tsai et al. (1998) indicated that the factors
such as the inherent material heterogeneity, residual stresses, free-edge effects and
relatively low transverse strength and shear stiffness impose great complexity to
bonded composite structures.
For the composite laminates widely used in engineering, effects of inherent material heterogeneity, lay-up sequence, lower transverse strength and shear stiffness
may be analytically modeled. When the material heterogeneity is considered, the
governing equations become the inhomogeneous differential equations, which can
also be solved analytically. When the adherends are composite laminates with the
symmetrical lay-ups, the formulations presented in this chapter can be extended
to the composite CLS specimen; the coefficients αa and αk may reflect influences of the lay-ups. By using the Timoshenko beam or higher order theories, the
lower transverse stiffness can be modeled and the analytical analysis can also be
conducted.
2.4 Influence of Adherend’s Large Deflections
and Adhesive Deformations
In this section, numerical results of the analytical solutions are presented and compared with the geometrically nonlinear finite element analysis (NFEA) for edge moment factor, overlap deflections and adhesive stresses. The CLS specimen with the
roller-roller boundary conditions and the symmetrical adherends are considered and
the input data used in this section are: E1 = 70 GPa, ν1 = 0.34 and t1 = 1.6 mm;
Ea /E1 = 0.04, ta /t1 = 0.078, νa = 0.4; c/t1 = 32 and l/c = 1.25. The NFEA results
were conducted by Luo and Tong (2007, 2008) using the commercial FEA package
MSC/NASTRAN, In the NFEA model, a 4-node isoparametric element was used
for both the adhesive and the adherends; 3 and 18 elements were used through the
adhesive and adherend thickness in the regions of 0.8c ≤ |x| ≤ c, 1 and 6 elements
were used in the other region of the overlap. The geometrically nonlinear FEA with
plane strain was implemented.
44
L. Tong and Q. Luo
2.4.1 Edge Moment Factor
Figure 2.6 shows the edge moment factor for the CLS with the long overlap. In this
figure, NFEA is referred to the NFEA results of MSC/NASTRAN; GR, HS, OP and
LT represent the numerical results predicted by Goland and Reissner (1944), HartSmith (1973), Oplinger (1994) and Luo and Tong (2007) analytical formulations.
These symbols are used in all relevant figures in this chapter.
Figure 2.6 illustrates that the edge moment factor predicted by Hart-Smith (1973)
is significantly lower than that predicted by the NFEA, and the predictions of
Oplinger (1994), and Goland and Reissner (1944) is obviously higher than the
NFEA results. Nevertheless, the edge moment factor predicted by the NFEA and
the fully-coupled nonlinear formulation correlates well with each other. It should be
noted that Eq. (2.57) is used to calculate k of Goland and Reissner (1944) in Fig. 2.6
for the consistency. More results of comparisons of the fully-coupled nonlinear formulation conducted by Luo and Tong (2007) and the NFEA presented by Tsai and
Morton (1994) can be found in Luo and Tong (2007).
As the geometrically nonlinear FEA can properly model behaviors of the geometric nonlinearity, the numerical results of Fig. 2.6 indicate large deflections of the
outer adherend and the overlap have to be included in the analytical analysis for the
CLS specimen.
It should be pointed out that, the used data (E1 = 70 GPa and Ea /E1 = 0.04)
denote the intermediately flexible adhesive (Tsai and Morton 1994). For the flexible
adhesive, the difference between the Goland and Reissner (1994) and the NFEA
(Tsai and Morton 1994) is larger (see Luo and Tong 2007). That is, the adhesive
deformations should also be taken into account in the analytical analysis for the
CLS specimen.
0.6
GR
OP
LT
NFEA
HS
Edge moment factor k
0.5
0.4
0.3
0.2
0.1
0
0
0.8
1.6
2.4
3.2
4
βk c
4.8
5.6
6.4
7.2
8
Fig. 2.6 Bending factor predicted by the NFEA and the analytical solutions presented by Goland
and Reissner (GR), Hart-Smith (HS), Oplinger (OP), and Luo and Tong (LT) for the CLS specimen
with the roller-roller boundary condition
2 Analysis of Cracked Lap Shear (CLS) Joints
45
2.4.2 Adherend’s Deflections
Figure 2.7 shows the deflection of adherend 1 in the overlap region at βk c = 8
predicted by Goland and Reissner (1944), the NFEA and Luo and Tong (2007).
It is noted that deflections of adherends 1 and 2 are the same in the Goland and
Reissner formulations. The non-dimensional deflection (wn = w1 /t1 ) and the nondimensional overlap axis (ξ = x/c) were used in Fig. 2.7.
Figure 2.7 indicates that the overlap deflection predicted by the fully-coupled
nonlinear formulations (Luo and Tong 2007) is almost the same as that of the NFEA,
but there exist noteworthy difference between the NFEA and the formulations separated in two steps (Goland and Reissner 1944). It is further evident that the adhesive
deformations should be considered in the analytical method for the CLS.
2.4.3 Adhesive Stresses
The shear and peel stresses predicted by Goland and Reissner (1994) the NFEA
and Luo and Tong (2007) are plotted in Figs. 2.8 and 2.9 respectively. In Figs. 2.8
and 2.9, a beam/adhesive model was used in the analytical analyses. The peak shear
and peel stresses occur at the adhesive edge (i.e., crack-tip of the CLS). In the twodimensional analysis using FEA, the peak shear stress occurs very close to the adhesive edges but shear stress is zero at the edge, while peel stresses at the adhesive
edge are singular. Therefore, the stress distributions in Figs. 2.8 and 2.9 are plotted
in the range of (−1 ≤ ξ ≤ −0.801) for the present analytical solutions and in the
range of (−0.999 ≤ ξ ≤ −0.801) for the NFEA.
0.2
Deflection wn1 (= – w1/t1)
0.15
0.1
0.05
0
–0.05
NFEA
GR
LT
–0.1
–0.15
–0.2
–1
–0.8 –0.6 –0.4 –0.2 0
0.2 0.4 0.6
Non-dimensional overlap axis ξ
0.8
1
Fig. 2.7 The overlap deflection predicted by the NFEA, Goland and Reissner (GR), and Luo and
Tong (LT) formulations for the CLS with the roller-roller boundary condition
46
L. Tong and Q. Luo
120
Shear stress (MPa)
100
80
GR
NFEA
LT
60
40
20
0
–1 –0.98 –0.96 –0.94 –0.92 –0.9 –0.88 –0.86 –0.84 –0.82 –0.8
Non-dimensional overlap axis ξ
Fig. 2.8 Shear stress predicted by the NFEA, Goland and Reissner (GR), and Luo and Tong (LT)
formulations for the CLS with the roller-roller boundary condition
The numerical results in Figs. 2.8 and 2.9 are those at βk c = 8, corresponding to a
tensile force of 659.6 N/mm. Stresses at the element centre were used in the figures.
Figure 2.8 illustrates the shear distribution predicted by Luo and Tong (2007)
correlates better with the NFEA results than that of Goland and Reissner (1944).
Even for the same edge moment, there still exists difference between the two formulations for the shear stress prediction, which can also be seen from Eqs. (2.66)
and (2.69).
Figure 2.9 is the peel stresses distribution predicted by Goland and Reissner
(1944), the NFEA and Luo and Tong (2007). It can also be seen that, as compared
to the NFEA, the fully-coupled nonlinear formulations for the peel stress prediction
140
120
Peel stress (MPa)
100
80
GR
NFEA
LT
60
40
20
0
–20
–40
–1 –0.98 –0.96 –0.94 –0.92 –0.9 –0.88 –0.86 –0.84 –0.82 –0.8
Non-dimensional overlap axis ξ
Fig. 2.9 Peel stress predicted by the NFEA, Goland and Reissner (GR), and Luo and Tong (LT)
formulations for the CLS with the roller-roller boundary condition
2 Analysis of Cracked Lap Shear (CLS) Joints
47
are superior to the formulations separated in two steps based on Goland and Reissner (1944). When the same edge moment is applied, the maximum peel stress predicted by the two formulations is the same.
In the fully-coupled nonlinear formulations (Luo and Tong 2007, 2008), solutions are found for the displacements. In the formulation process, large extensions,
large rotations and deformations (material nonlinearity) are neglected. However, the
predicted numerical results of edge moment factor, adhesive stresses, adherend deflections correlate extremely well with those of the NFEA. It indicates that large
deflections of adherends are the critical feature of the CLS specimen, which is well
represented by the fully-coupled nonlinear formulations.
2.5 Strength Prediction
The failure of a CLS specimen may occur in adherend and adhesive and at a bimaterial interface. The failure load may be predicted by the strength-of-material
based approach and/or the fracture mechanics based approach (Adams et al. 1997,
Tong and Steven 1999), depending on the material properties, adhesive joint configurations and the pre-crack features. A cohesive damage zone model has also attracted attention to predict failure loads of adhesive joints (Sheppard et al. 1998,
Yang and Thouless 2001, Blackman et al. 2003, Liljedahl et al. 2006), as there is a
cohesive or plastic deformation zone near the crack-tip or the adhesive ends of the
CLS specimen generally.
2.5.1 Strength of Material Based Approach
The strength of material based approach involves a stress analysis of the CLS specimen, and employment of stress and/or strain based failure criteria. One of the most
widely used stress/strain based failure criteria is the von Mises criterion. It has also
been considered for the adhesive joints (Kusenko and Tammzs 1981, Czarnocki and
Piekarski 1986), in which the failure criteria on the basis of the tensile experimental
testing and biaxial experimental testing have been discussed for maximum tensile
stress criterion, maximum tensile strain criterion and the modified distortion energy
criterion. Crocombe et al. (1990) addressed structural adhesive failures based on
strength of materials and fracture mechanics.
Because of the stress concentration, complexity of stress analysis and difficulty
in experimental testing, the strength of material based criteria for predicting the
failure criteria based on strength of materials for adhesive joints have not been
well-established (Lee 1991, Yang and Thouless 2001). Also, the CLS specimen is
mainly designed to test fracture behaviors of the adhesive joints. Mangalgiri and
Johnson (1986) discussed the CLS design to ensure the CLS failure occurred in adhesive and interface. Therefore, most of works on the failure prediction for the CLS
has been focused on the fracture mechanics based approach.
48
L. Tong and Q. Luo
2.5.2 Calculation of Energy Release Rates
and Failure Prediction
The fracture mechanics based approach for the CLS failure prediction includes calculation of the stress intensity factors (SIF) and/or energy release rate (ERR) or
J-integral and the experimental testing for the fracture toughness, critical ERRs or
J-integral.
When the maximum adhesive stresses are found, energy release rates GI and GII
can be calculated by (Edde and Verreman 1992, Krenk 1992):
GI =
ta
ta
(σmax )2 ; GII =
(τmax )2
2Ea
2Ga
(2.71)
It is noted that both modes I and II are found in Eq. (2.71), which can be directly
applied to the CLS with relatively long overlap (Krenk 1992).
When the CLS deflections and force components of the overlap are solved, the
ERRs and/or the J-integral can be calculated. Lai et al. (1996) calculated the ERRs
using the method developed by Hutchinson and Suo (1992) for the bi-material crack.
Fernlund and Spelt (1994) calculated J-Integral using the following procedure.
As shown in Fig. 2.10, a closed-form expression of the energy release rate can
be calculated by using the J-integral:
∂u
J=
Wn − T
dΓ
(2.72)
∂τ
Γ
where W is the strain energy density; n and τ are outward unit normal and tangential vectors to the un-deformed boundary contour; T is the traction vector acting on
Γ. When the contour is o–a–b–c–d–e–f–o, no contributions to the J-integral in o–a,
b–c, d–e and f–o boundaries as there are no tractions on the crack faces and horizontal boundaries of the upper and lower adherends. The J-integral becomes:
J = J(1) + J(2) + J(3)
(2.73)
where subscripts (1), (2) and (3) represent the boundaries a–b, c–d and e–f respectively. The total energy release rate can be found in light of Eqs. (2.72) and (2.73),
which is given by:
M32
N32
N22
N12
M12
M22
+
+
+
+
−
J=
2A(1)11 2D(1)11
2A(2)11 2D(2)11
2A(o)11 2D(o)11
(2.74)
where subscripts (1), (2) and (o) represent the upper adherend, lower adherend and
the overlap. It is noted that Eqs. (2.72), (2.73), (2.74) can be applied to general
adhesive joints. For the CLS specimen as shown in Fig. 2.1, J(2) = 0. It is also worth
noting that the shear force effects on ERRs are not included.
2 Analysis of Cracked Lap Shear (CLS) Joints
Fig. 2.10 The schematics for
J-integral (a) the J-integral
boundary (b) force
components of the overlap
49
e
f
a
b
d
o
c
(a)
Q1
M1
N1
M2
N2
Q2
M3
N3
Q3
(b)
When the J-integral of the mixed mode is found, the mode should be separated
to predict the CLS failure, because the CLS is generally a mixed mode problem.
There exist analytical methods for partitioning modes for a cracked homogeneous
beam (e.g., Hutchinson and Suo 1992). For the CLS, the NFEA is an effective tool
to calculate mode ratios (Johnson 1986).
Papini et al. (1994), Fernlund and Spelt (1994) conducted the experimental investigation on the critical energy releases rate for the CLS and presented the critical
energy release rate for the 7075-T6/Permabond ESP 310 adhesive system, as shown
in Fig. 2.11.
When the loading phase angle is found, the critical J-integral Jc can be found
from the fracture envelope such as that shown in Fig. 2.11. By comparing the found
J-integral Jc with the critical one, the failure loads of the CLS specimens can be
predicted.
The critical J -integral Jc (J/m2)
6000
5000
4000
3000
2000
1000
0
0
10
20
30
40
50
60
70
80
90
Phase angle =Atan(JII/JI) (Degrees)
Fig. 2.11 The critical energy release rate of the 7075-T6/Permabond ESP 310 adhesive system
50
L. Tong and Q. Luo
2.6 Concluding Remarks
In this chapter, analytical procedures for stress analysis and failure prediction of
the CLS specimen are presented. Because of the complexity, particularly when lap
adherend and/or lap adherend of the CLS are composite laminates, closed-form solutions of the CLS are very limited in the existing literatures. Available results show
that large deflections of the outer adherend and the overlap must be modeled.
The analytical formulation based on Goland and Reissner (1944) are available
to calculate the force components at the end of the overlap, both for symmetrical
and asymmetrical adherends, but larger errors may appear as the adhesive deformations are not modeled in the formulations, particularly for the cases of long overlaps,
thicker and/or soft adhesive, larger applied loadings and composite laminates. The
adhesive stresses of general asymmetrical adherends for the prescribed force boundary conditions can be found in Luo and Tong (2002).
The analytical solution based on the fully-coupled nonlinear formulations (Luo
and Tong 2007) correlate well with the geometrically nonlinear finite element analysis; it can also be applied to the composite laminates. Currently, the analytical
solutions are only available for the symmetrical adherends. The analytical solutions
for CLS with the clamped-clamped and roller-clamped boundary conditions are yet
to be found, and that for the CLS with asymmetrical adherends should be further
studied.
Acknowledgments The authors are grateful to the continuous support of ARC (Australia) and
AOARD/ASOSR (USA).
References
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Hall, London
Alif N, Carlsson LA, Gillespie Jr JW (1997) ASTM STP. 1242:82–106
Benzeggaph ML, Kenane M (1996) Composite Science and Technology. 56:439–449
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119:25–46
Brussat TR., Chiu ST, Mostovoy S (1977), Fracture Mechanics for Structural Adhesive Bonds,
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Crocombe AD, Bigwood DA. Richardson G (1990) International Journal of Adhesion and Adhesives. 10:167–178
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Center
Higgins A (2000) International Journal of Adhesive and Adhesion. 29:367–376
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Chapter 3
Analytical Models with Stress Functions
Toshiyuki Sawa
Abstract The interface stress distributions in adhesive butt joints subjected to
tensile and cleavage loadings are described using two-dimensional theory of elasticity. Interface stress distributions of adhesive band butt joints are also discussed. In
addition, the effects of adhesive Young’s modulus and the adhesive thickness on the
interface stress distributions are shown. For adhesive tubular butt joints, the effects
on the interface stress distributions are described using axi-symmetrical theory of
elasticity. It is shown that singular stresses occur at the edges of the interfaces. It is
also observed that the singular stresses decrease as the adhesive Young’s modulus
increases and the adhesive thickness decreases. Finally, a method of stress analysis
for bonded shrink fitted joints is described and it is demonstrated that the strengths
of bonded shrink fitted joints are larger than those of shrink fitted joints.
3.1 Introduction
In this chapter, the characteristics of adhesive butt joints of thin plates under tensile [15], and cleavage loadings [20] are described using two-dimensional theory
of elasticity. The characteristics of adhesive butt joints of solid cylinders/bars under tensile loadings [14] are also described using axi-symmetrical theory of elasticity. A stress analysis of the tubular (hollow cylinder) butt adhesive joints is
also done under tensile [12] and torsional loadings [13]. In designing these adhesive butt joints, it is important to know how to determine the material properties of the adhesive and the adhesive thickness. From the analyses, it is shown
how to determine the adhesive properties and the adhesive thickness in the design
of adhesive butt joints. In practice, bonded shrink fitted joints are applied using
anaerobic adhesive. The effects of some factors are described on the interface stress
Toshiyuki Sawa
Hiroshima University, 1-4-1, Kagamiyama, Higashihiroshima, Hiroshima, Japan,
e-mail: sawa@mec.hiroshima-u.ac.jp
L.F.M. da Silva, A. Öchsner (eds.), Modeling of Adhesively Bonded Joints,
c Springer-Verlag Berlin Heidelberg 2008
53
54
T. Sawa
distributions in the bonded shrink fitted joints [17]. In addition, it is demonstrated
that the strengths of bonded shrink fitted joints are larger than those of shrink fitted
joints.
3.2 Butt Joints
Figure 3.1 shows several types of butt joints. Figure 3.1a shows an adhesive butt
joint of thin plates [10, 15, 18, 23, 24]. Figure 3.1b shows a band adhesive joint in
which the interfaces are partially bonded [16, 21]. Figure 3.1c shows a butt joint
of solid bars/cylinders [4, 7, 8, 11, 14, 25]. Figure 3.1d shows a tubular (hollow
cylinder) joint [2, 3, 5, 6, 9, 12, 13, 22, 26]. Figure 3.1e shows butt joints subjected
to cleavage loadings [1, 19, 20] as a special case of Fig. 3.1a. Important issues for
designers are interface stress distributions and the joint strength. In addition, an
important issue is how to determine adhesive material properties, and the adhesive
thickness in designing the adhesive joints.
3.2.1 Adhesive Butt Joints of Dissimilar Adherends Subjected
to Tensile Loadings
Figure 3.2 shows a model for two-dimensional analysis of butt joints under tensile
loadings [15]. Two adherends are bonded by an adhesive. Young’s modulus, shear
modulus and Poisson’s ratio of the adherends are denoted by E1 , G1 , ν1 , E3 , G3 , ν3 ,
respectively, where Cartesian coordinates (x, y) are used. The adhesive thickness is
denoted by 2h2 . The tensile loading is applied to both ends of the adherends as the
stress distribution F(x). The stress distribution F(x) is developed by Fourier series
and the two-dimensional theory of elasticity is applied for analyzing the joint under
tensile loading F(x).
The boundary conditions are described by Eqs. (3.1), (3.2), (3.3), (3.4).
(i) on finite strip (I) (adherend)
x = ±l;
y1 = h1 ;
I =0
σxI = τxy
∞
σyI = F(x) = a0 + ∑ as cos
I =0
τxy
s=1
sπ x
l
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
(3.1)
(ii) on finite strip (II) (adhesive)
x = ±l;
II
σxII = τxy
=0
(3.2)
3 Analytical Models with Stress Functions
55
(a) The adhesive butt joint of thin plates.
(b) Band adhesive joints.
(c) Butt joint of solid bars/cylinders.
(d) Tubular (hollow cylinder) joint.
(e) Butt joints subjected to cleavage loadings
.
Fig. 3.1 Several types of butt joints
(iii) on finite strip (III) (adherend)
x = ±l;
y3 = −h3 ;
⎫
⎪
sπ ⎪
⎬
III
x
σy = G(x) = b0 + ∑ bs cos
l
⎪
s=1
⎪
⎭
III = 0
τxy
III = 0
σxIII = τxy
∞
(3.3)
56
T. Sawa
Fig. 3.2 An adhesive butt
joint of dissimilar adherends
subjected to an external tensile loading [15]
(iv) at the interface between finite strips (I) and (II)
I
σy y =−h
= σyII y =h
1
1
2
2
I
II
τxy y =−h
= τxy y =h
1
1
2
2
I
II
u y =−h
= u y =h
1
∂ vI
∂x
1
2
=
y1 =−h1
∂ v II
∂x
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
(3.4)
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
2
y2 =h2
where,
=
a0
as =
1
l
1
2l
e1
−e1
F(x)dx,
sπ x dx,
l
−e1
(s = 1, 2, 3, · · ·)
e1
F(x) cos
1
2l
1
bs =
l
b0 =
e2
−e2
e2
−e2
G(x)dx,
G(x) cos
sπ x dx,
l
The numerical calculations were done and the results are shown in Figs. 3.3, 3.4
and 3.5.
In the analysis, Airy’s stress functions are used, as in Eq. (3.1). Figure 3.3 shows
the effect of Young’s modulus on the stress distributions at the interfaces y2 = h2
and y2 = −h2 , where the ratio E1 /E3 was held constant at 3 and the ratio E1 /E2
was varied as 5, 10, and 100. The abscissa represents the ratio of the distance x to
the half length l of the adhesive, and the ordinates indicate the ratio of each stress
to the mean normal stress σym at the interface. In Fig. 3.2, it was assumed that the
tensile stresses F(x) and G(x) acted uniformly on the upper (y1 = h1 ) and the lower
(y3 = −h3 ) surfaces in the region |x| ≤ l[F(x) = G(x)]. From the results, it is seen
3 Analytical Models with Stress Functions
57
Fig. 3.3 Effect of the ratio of Young’s modulus of the adherend to that of the adhesive on the stress
distribution. l/h2 = 5.0, h1 /h2 = h3 /h2 = 2.0, E1 /E3 = 3, ν1 /ν2 = ν3 /ν2 = 0.86. F(x) and G(x)
are uniform within the region |x| ≤ l [15]
that singular stresses occur at the edges of the interfaces and each stress is larger at
y2 = h2 than at y2 = −h2 . The absolute values of the stress σy and the shear stress
τxy become larger near the edge x/l = 1.0 of the interface with an increase of the
ratio E1 /E2 .
Figure 3.4 shows the effect of the thickness 2h2 of the adhesive layer, where
h1 = h3 ; the value h1 /h2 was 2, 5, and 20; E1 /E3 was 1.5; and E1 /E2 = 3.0. From
the results, it is seen that the absolute values of τxy /σym near the edges x/l = 1.0
of the interfaces increase as the ratio h1 /h2 increases. As shown in Figs. 3.3, 3.4
and 3.5, it is found that the singular stresses occur at the edge of the interfaces. It is
necessary to examine the value of the singular stresses.
58
T. Sawa
Fig. 3.4 Effect of the thickness of the adhesive layer on the stress distribution at the interface.
l/h1 = l/h3 = 2.5, E1 /E2 = 3, E1 /E3 = 1.5, ν1 /ν2 = ν3 /ν2 = 1.0. F(x) and G(x) are uniform
within the region |x| ≤ l [15]
Figure 3.5 shows the relationship between the ratio of the normal stress σy
to the mean stress σym and the distance r from the edge in logarithmic scales
in order to examine the stress singularity at the edges. This case corresponds to
Fig. 3.3. In general, the singular stress is expressed approximately by Eq. (3.5)
below, where K is the intensity of the stress singularity and λ is the order of
the singularity. The distance from the edge is denoted by r and is expressed by
r = (l − x)/l.
σy /σym = K/(rλ )
(3.5)
The singular stresses are expressed in this form in Fig. 3.5. The distance r is
varied between 0.005 and 0.02 mm, because the singular stresses are approximately linear in logarithmic scale in the region of r mentioned above. The parameter λ varies with the ratio of Young’s moduli of the adherend to that of the
adhesive.
The results show that the difference in Young’s moduli between the two adherends must be small and the rupture occurs at the interface of the adherend with
higher Young’s modulus. It is better to increase the value of the adhesive Young’s
modulus and decrease the adhesive thickness. When the adherend material is the
same, the same trend is obtained.
3 Analytical Models with Stress Functions
59
Fig. 3.5 Singular stress at the
edge of the interface.
l/h2 = 5.0,
h1 /h2 = h3 /h2 = 2.0,
E1 /E3 = 3.0,
ν1 /ν2 = ν3 /ν2 = 0.86. F(x)
and G(x) are uniform within
the region |x| ≤ l [15]
An issue is how to increase the adhesive Young’s modulus. When the adherend
material is metal, the adhesive Young’s modulus is smaller in comparison with that
of metal. So, it is important to increase the value of adhesive Young’s modulus by
adding fillers with higher Young’s modulus than that of the adhesive.
60
T. Sawa
3.2.2 Adhesive Butt Joints of Similar Adherends Subjected
to Cleavage Loadings
Figure 3.6 shows a two-dimensional model for analysis of adhesive butt joints subjected to cleavage loadings [20]. The notations are the same as mentioned in Fig. 3.1.
The cleavage loadings are important serious conditions for the adhesive butt joints.
For analyzing the stress state of the joints under cleavage loadings, the stress distribution of cleavage loading F(x) is developed into Fourier series. In the analysis,
Airy’s stress functions are used. They are much more complicated than those in
Eqs. (3.1), (3.2). The results of the numerical calculations are shown in Figs. 3.7, 3.8
and 3.9.
Figure 3.7 shows the effect of the ratio E1 /E2 of Young’s moduli of adherends
to adhesive on the stress distributions at the interface (y2 = h2 ), where E1 /E2 was
chosen as 3, 10 and 100. It is assumed that a cleavage load acts uniformly within the
region 0.8 < x/l < 1.0. In this figure, σym represents the mean normal stress. The
abscissa represents the ratio x/l of the distance x form the centre of the joints to the
half length l of adherends and adhesive. It is seen that the distributions of σy , σx ,
and τxy tend to increase near the edge, x/l = 1.0, and the stress singularity increases
with an increase in E1 /E2 . The larger E1 /E2 is, the more uniform the distribution of
stress σy becomes.
Figure 3.8 shows the effect of adhesive thickness on the stress distribution at
the interface. The ratio h1 /h2 of the thickness of the adherend, 2h1 , to that of the
adhesive, 2h2 , is varied as 2, 20 and 100. The stress distributions of σy , σx and τxy
increase near x/l = 1.0 with an increase in the ratio h1 /h2 .
Figure 3.9 shows the effect of the cleavage load distribution F(x) on the stress
distribution at the interface. In this case the thickness of the adhesive is held constant and the value of h1 /h2 is set at 5. Numerical calculations were carried out
Fig. 3.6 Adhesive butt joint
subjected to a cleavage
loading [20]
3 Analytical Models with Stress Functions
61
Fig. 3.7 Effect of ratio of Young’s modulus of the adherend to that of the adhesive on the stress
distribution at adhesive-adherend interface [20] (y2 = h2 , h1 /l = 0.2, h1 /h2 = 2, ν1 /ν2 = 1 and
F(x) is constant over the region 0.8 < x/l < 1.0)
62
T. Sawa
Fig. 3.8 Effect of adhesive thickness on the stress distribution at adhesive-adherend interface [20]
(y2 = h2 , h1 /l = 0.2, E1 /E2 = 3, ν1 /ν2 = 1 and F(x) is constant over the region 0.8 < x/l < 1.0)
3 Analytical Models with Stress Functions
63
Fig. 3.9 Effect of cleavage stress F(x) distribution on the stress distribution at adhesive-adherend
interface [20] (y2 = h2 , h1 /h2 = 5, h2 /l = 0.1, e2 /l = 1, E1 /E2 = 65.6 and ν1 /ν2 = 0.81)
64
T. Sawa
in the cases where the uniform load F(x) acts over the regions 0 < x/l < 1.0 and
0.8 < x/l < 1.0. The effect of the load distribution can be seen clearly on the stress
distribution at the interface. It is found that the stress singularity near the points
x/l = −1.0 and x/l = 1.0 increases in the case where the load distribution F(x) acts
over the region 0.8 < x/l < 1.0. This is due to the fact that E1 /E2 and l are large in
comparison with h1 and e1 (defined in Fig. 3.9).
From the results, it can be concluded that the singular stresses decrease as the
Young’s modulus of the adhesive is increased and the adhesive thickness is decreased. In addition, it is found that the singular stress at the edge of the interfaces increases depending on the cleavage distribution. It is also observed that the
interface stress σy is compressive at the left region of the interfaces, as shown in
Figs. 3.7, 3.8 and 3.9.
3.2.3 Band Adhesive Butt Joints of Dissimilar Adherends
Subjected to External Bending Moments
Figure 3.10 shows a model for analysis of band adhesive butt joints with dissimilar
adherends subjected to external bending moments [21]. The analytical method is the
same as the cases of Figs. 3.1 and 3.2 using the two-dimensional theory of elasticity.
In the numerical calculations, a stress singularity occurs at the edge (|x1 | = c + l2 ,
|x1 | = c−l2 ) of the interface (y2 = +h2 , x2 = +l2 ), hence 50 terms in the series were
Fig. 3.10 Band adhesive butt joint of dissimilar finite strips subjected to external bending
moments [21]
3 Analytical Models with Stress Functions
65
Fig. 3.11 Effect of the ratio of Young’s moduli between the adherends and the adhesive on each
stress distribution (σx , σy and τxy ) and on the maximum principal stress distribution (σ1 ) at the
interface [21]
used to guarantee the convergence of the stresses. Figure 3.11 shows the effect of
the ratio of Young’s moduli between the adherends and the adhesive on each stress
(σx , σy and τxy ) distribution and on the maximum principal stress σ1 (plane stress)
distribution at the interfaces (y2 = +h2 , x2 = +l2 ), where the ratio E1 /E2 was 5, 10
and 60. It is assumed the bending moments act linearly on the upper (y1 = h1 ) and
the lower (y3 = −h3 ) surfaces in the region |x1 | ≤ l1 (F(x1 ) = G(x3 )). The abscissa
is the ratio of the distance x1 to the half-length l1 of the adherend. In the following
analytical results, the stress distributions are analyzed until the point 0.4% inside
the length 2l2 from the edge of the adhesive. From the results, it is seen that singular
stresses occur at the edge of the unbonded area (x1 /l1 = 0.6) with a decrease of the
ratio E1 /E2 and the stress distribution tends to approach the stress distribution F(x1 )
and G(x3 ) with an increase of the ratio E1 /E2 .
66
T. Sawa
Fig. 3.12 Effect of the ratio
of Young’s moduli between
adherends on the maximum
principal stress distribution at
the interface [21]
Figure 3.12 shows the effect of the ratios E1 /E3 of Young’s moduli between adherends on the maximum principal stress σ1 /P distribution. The maximum principal
stress at the position x1 /l1 = 1.0 of the interface (y2 = h2 ) increases with an increase
of E1 /E3 and increases near the point x1 /l1 = 1.0 of the interface (y2 = −h2 ) with a
decrease of the ratio E1 /E3 . From the results, it is predicted that joint strength in the
case of dissimilar adherends is smaller than that in the case of similar adherends.
Figure 3.13 shows the effect of the thickness 2h2 of the adhesive layer, where the
ratio h1 /h2 was 50, 100 and 200. The stress singularity at the edge of the interface
(x/l = 0.6, 1.0) increases with an increase of the ratio h1 /h2 .
Figure 3.14 shows the effect of the bonding positions on the maximum stress
distribution at the interface (y2 = h2 , y2 = −h2 ) in the case of h1 /l1 = 0.2. It is
assumed that the stress distributions F(x1 ) = G(x3 ) are linear in the region |x1 | =
|x3 | ≤ 0.5l1 and the regions of bonding are 0.2 ≤ |x1 /l1 | ≤ 0.6, 0.4 ≤ |x1 /l1 | ≤ 0.8
and 0.6 ≤ |x1 /l1 | ≤ 1.0. Figure 3.15 shows the maximum principal stress at positions
A, B, C and D (positions 0.4% inside from |x1 | = c + l2 , |x1 | = c − l2 ) near the
A and
edge of adhesive [II] at the interfaces (y2 = h2 , y2 = −h2 ). When positions C are about 0.56l1 , the values of the maximum principal stress at positions A, B,
C and D are equal. Figure 3.15b shows the intensity K of the stress singularity at
3 Analytical Models with Stress Functions
67
Fig. 3.13 Effect of the ratio
of the adhesive thickness on
the maximum principal stress
distribution at the interface
[21]
positions A, B, C and D by using the stress singularity parameters. Generally, the
maximum principal stress σ1 /P near the position where stress singularity occurs is
expressed approximately by
σ1 (r)/P = K/rλ
(3.6)
where,
σ1 (r) : maximum principal stress,
r : distance from singularity,
K : intensity of stress singularity,
λ : order of stress singularity.
From the result, when positions A and C are about 0.57l1 , the value K of the intensity of stress singularity at position A becomes equal to that at position C . From
68
T. Sawa
Fig. 3.14 Effect of the
bonding positions on the
maximum principal stress
distribution at the interface
[21]
the above results, it is concluded that the joint strength is maximal when the value
c/l1 is between 0.56 and 0.57.
Figure 3.16 shows the effect of the bonding area on the maximum principal stress
σ1 /P distribution at the interfaces (y2 = h2 , y2 = −h2 ), where F(x1 ) = G(x3 ) =
Px1 /l1 (|x1 | ≤ l1 , |x3 | ≤ l1 ) and the ratio 2l2 /l1 is set as 1.0 (bonded completely at
the interfaces), 0.6 and 0.4. From the results, it is expected that an increment of
singular stress due to an unbonded area is small at both edges (x1 /l1 = +1.0) even
if the unbonded area is at the center of the joint. Thus, it is concluded that a band
adhesive joint in which the interface is partially bonded efficiently resists external
loads if a suitable bonding area is selected.
Table 3.1 shows a comparison between the analytical and the experimental results
concerning the joint strength. The experiments were performed 30 times. The joint
was assumed to fail when the maximum principal stress at position x1 /l1 = 0.996
3 Analytical Models with Stress Functions
Fig. 3.15 Effect of bonding
positions on the maximum
principal stress distribution at
the interface [21]
Fig. 3.16 Effect of the
bonding area on the
maximum principal stress
distribution at the interface
[21]
69
70
T. Sawa
Table 3.1 Comparison concerning joint strength of butt joints [21]
Adherends
St-St (Nm)
Al-Al (Nm)
St-Al (Nm)
Complete Adhesion (2l2 /l1 = 1.0)
EXP
NUM
123
118 (329)
110
107 (297)
50
48 (134)
Band Adhesion (2l2 /l1 = 0.6)
EXP
NUM
120
116 (299)
108
104 (269)
47
44 (119)
equals to the fracture stress of the adhesive. The joint strength prediction was also
done by the von Mises criterion and the values are indicated in brackets in Table 3.1.
Table 3.1 shows that the predictions based on the maximum principal stress criterion
are consistent with the experimental results. The values predicted by the von Mises
criterion are larger than the experimental results. It is shown that the joint strength
in which the interface is partially bonded is the same as that of a butt joint in which
the interface is bonded completely. In addition, it is seen that the joint strength in
the case where dissimilar adherends (St-Al) are used is about half that in the case
where similar adherends (St-St and Al-Al) are used.
In addition, it can be assumed that the residual stress in the band adhesive joint
is reduced in bonding process while the strength is the same as a joint with interfaces completely bonded. Thus, it is better for a mechanical engineer to design band
adhesive butt joints taking into account the load distribution and the interface length.
3.2.4 Adhesive Butt Joints of Solid Cylinders Subjected to External
Tensile Loadings
The previous stress analyses are two-dimensional. In this section, a method for
axi-symmetrical stress analysis of adhesive butt joints with solid cylinder [14] is
described (see Fig. 3.17). Michell’s stress functions are used, where cylindrical coordinates (r, z) are used. The stress singularity at the edge of the interfaces is not
taken into consideration. Hence, computations were done varying the number N of
terms as 100 and 120 in order to examine the effect of the number N on the stress
distributions at the interfaces. It was seen that the difference between both results
was less than 3%. Hereafter, computations were done setting the number N of terms
as 100. Figure 3.18 shows the effect of the ratio E1 /E2 of Young’s modulus of the
adherends to that of the adhesive on the stress distributions at the interface (z2 = h2 ),
where the values E1 /E2 were varied as 1, 3, 40, 60 and infinity. In the case where
the value E1 /E2 was infinity, that is the adherends were assumed rigid, the analysis
was done under the following boundary conditions.
⎫
⎪
r = a : σr = τrz = 0
⎪
⎪
⎪
⎪
⎬
∂w
z2 = ±h2 : u = 0,
=0
(3.7)
⎪
∂r
⎪
⎪
a
a
⎪
⎭
2π r(σz )z=h2 dr =
2π rF(r)dr ⎪
0
0
3 Analytical Models with Stress Functions
71
Fig. 3.17 Adhesive butt joint
of solid cylinders subjected
to an external tensile loading
[14]
Fig. 3.18 Effect of the ratio
of Young’s modulus of the
adherend to that of the
adhesive on the stress
distributions σz , σr and τrz at
the interface [14] (z2 = h2 ,
h1 /h2 = 10, h1 /a = 0.2,
ν1 /ν2 = 1.0. F(r) is constant
within the region r ≤ a)
In numerical calculations, a tensile load F(r) was assumed to act uniformly within
the region r ≤ a on the upper surface of adherend (z1 = h1 ). In Fig. 3.18, σzm represents the mean normal stress. The ordinate represents the ratios σz /σzm , σr /σzm
of the normal stresses to the mean normal stress and τzr /σzm the shear stress to the
mean normal stress. The abscissa represents the ratio r/a of the distance r from the
72
T. Sawa
center to the radius a. It is seen that the distribution of σz tends to be averaged when
the value E1 /E2 approaches 1 and the singularity increases with an increase of the
value E1 /E2 . It is also seen that the shear stress τzr near the edge r/a = 1.0 increases
with an increase of the value E1 /E2 . A difference is not found among the distribution of σz in the cases where E1 /E2 is 40, 60 and infinity, nor in the distribution of
τzr . In this model, adherends are assumed to be rigid when the value of E1 /E2 is
more than 40.
Figure 3.19 shows the effect of the adhesive thickness on the stress distributions
σz , σr and τzr at the interface (z2 = h2 ). The thickness 2h1 of the adherends is held
constant and the value of h1 /h2 is varied as 5, 10, 20 and 100. There are no differences in the distributions σz and τzr between the cases where h1 /h2 is 20 and 100.
The distribution of σz tends to a constant value and the shear stress τzr decreases
with an increase of h1 /h2 .
Figure 3.20 shows the effect of the load distribution F(r) on the stress distributions σz , σr and τzr at the interface. With the thickness of the adhesive held constant
and the value of h1 /h2 set at 5, computations were done in the cases where the
uniform load F(r) acts on the region (c/a)2 = 0.1. There is a visible effect of the
stress distribution σz but very little effect on the distribution τzr . Figure 3.21 shows
a comparison of the analytical results obtained by this study with the results obtained by finite element method (FEM) with respect to the stress distributions in the
adhesive. The results obtained by FEM show the stresses at the centroid of the triangle elements. Therefore, the stresses obtained by the analysis were compared with
Fig. 3.19 Effect of the
adhesive thickness on the
stress distributions σz , σr and
τrz at the interface [14]
(z2 = h2 , h1 /a = 0.2,
E1 /E2 = 40, ν1 /ν2 = 1.0.
F(r) is constant within the
region r ≤ a)
3 Analytical Models with Stress Functions
73
Fig. 3.20 Effect of external
load distribution on the stress
distribution σz , σr and τrz at
the interface [14] (z2 = h2 ,
h1 /h2 = 5, h2 /a = 0.1,
E1 /E2 = 65.6, ν1 /ν2 = 0.81)
Fig. 3.21 Comparison of
analytical results with results
obtained by FEM [14]
(h1 /h2 = 10, h1 /a = 1.0,
E1 /E2 = 91.0, ν1 /ν2 = 0.91.
F(r) is constant within the
region r ≤ a)
those obtained by FEM at z1 /h2 = 0.92 and 0.17. Both results are in fairly good
agreement.
From the results, it is found that the singular stresses occur at the edge of the
interfaces and they decrease as the adhesive Young’s modulus increases and the
adhesive thickness decreases. These points are important in the actual design of
joints under static loadings.
74
T. Sawa
3.3 Tubular Joints
3.3.1 Tubular Butt Adhesive Joints Subjected to a Tensile Loading
Figure 3.22 shows a tubular (hollow cylinder) butt adhesive joint subjected to a
tensile loading [12]. Cylindrical coordinates (r, z) and Michell’s stress functions
are used in the analysis. Two adherends are replaced by the hollow cylinders with
the same dimensions and material. An external tensile loading is applied by the
stress distribution F(r). Young’s modulus and Poisson’s ratio of the adherends and
the adhesive are denoted by E1 , ν1 , E2 and ν2 , respectively. The inside diameter
of the joints is denoted by 2a, the outside diameter by 2b. The analyses are done
using the axi-symmetrical theory of elasticity. The interface stress distribution is
important for analyzing the joints. Figure 3.23 shows the effect of Young’s modulus ratio E1 /E2 on the distribution of the stress components σz , σr , τzr at the
interfaces. It is seen that each stress component is singular at both edges r = a
and r = b.
Figure 3.24 shows the effect of the adhesive thickness 2h2 on the interface
stress distributions. It is found that the shear stress decreases as the adhesive thickness decreases. Figure 3.25 shows the effect of the tensile loading F(r) distribution on the interface stress distributions. It can be seen that each stress (σz , σr
and τzr ) is singular at both the edges (outside and inside radius) of the interfaces. Figure 3.26 shows the effect of the diameter ratio a/b on the stress distributions. It is seen that the shear stress τzr increases near both the edges as
the diameter ratio a/b increases while the changes in the stress components σz
and σr are smaller. For the tubular (hollow cylinders) adhesive butt joints, the
singular stresses occur at the both edge (r = a and r = b) and they decrease
as the adhesive Young’s modulus E2 is increased and the adhesive thickness is
decreased.
Fig. 3.22 Tubular butt
adhesive joint subjected to a
tensile loading [12]
3 Analytical Models with Stress Functions
75
Fig. 3.23 Effect of the ratio
of Young’s modulus of the
adherend to that of the
adhesive on the stress
distributions σz , σr and τrz at
the interface [12] (z2 = h2 ,
a/b = 0.9, (b–a)/2h2 = 10,
h1 /h2 = 10, ν1 /ν2 = 1.0.
F(r) is uniform within the
region a ≤ r ≤ b)
3.3.2 Tubular Butt Adhesive Joints Under Torsional Loading
Figure 3.27 shows tubular (hollow cylinder) butt adhesive joints under torsional
loadings [13]. The analysis is carried out using the axi-symmetrical theory of elasticity. Cylindrical coordinates (r, θ , z) are used. The shear moduli and Poisson’s
ratios for adhesive (I), adherend (II) and (III) are designated by G1 , ν1 , G2 , ν2 , G3
and ν3 , respectively.
Numerical computations are carried out in the case where the two adherends (II)
and (III) are the same material (G2 = G3 ) and of the same dimensions (c = e, d = f ,
c/d = 0.5). In order to investigate the effect of the position of the adhesive band
on the stress distributions and the displacements, two types of adhesive joints are
examined numerically, that is, one is bonded at the outer interface, called type (A)
and the other is bonded at the inner interface, called type (B), where the bonded area
of each type is the same. Moreover, the effect of the ratio of shear modulus of the
adhesive to that of the adherend and the effect of the adhesive thickness are examined in the case of the two types mentioned above. In computations, the number of
terms, N, of the series is taken as 200, which is checked to obtain the stresses and
the displacements in satisfactory accuracy.
Figures 3.28 and 3.29 show the effect of the ratio G1 /G2 of the shear modulus
of the adhesive to that of the adherend on the shear stresses and the displacement
76
T. Sawa
Fig. 3.24 Effect of the
adhesive thickness on the
stress distributions σz , σr and
τrz at the interface [12]
(z2 = h2 , a/b = 0.9, ν2 = 0.3,
E1 /E2 = ∞)
at the interface between the adhesive and the adherend, respectively, in the case of
type (A). Also, these effects in the case of type (B) are shown in Figs. 3.30 and 3.31
From these figures, the maximum stresses become large with an increase of the ratio
G1 /G2 at the inner circumference of the interface, i.e. r = a, in the case of type (A)
and at the outer circumference of the interface, i.e. r = b, in the case of type (B). On
the other hand, the maximum displacements become small with an increase of the
ratio G1 /G2 in both cases. Moreover, comparing the results of type (A) with those
of type (B), the maximum stresses and the displacements are smaller in the case of
type (A), i.e. the case where tubular shafts are bonded at the outer interface of the
adherends.
Figure 3.32 shows the effect of the thickness of the adhesive on the stress distributions in the case of type (A). From this figure, it is seen that the singularity of the
stress becomes larger at the inner circumference, i.e. r = a, with a decrease of the
thickness 2h.
Figures 3.33 and 3.34 show the effect of the ratio G1 /G2 of shear modulus on
the shear stress distributions and on the displacements for two solid shafts bonded
at the outer interface (type A). From these figures, it is seen that both the stress and
the displacement distributions are similar to the results shown in Figs. 3.28 and 3.29
of the adhesive joint with the tubular shafts.
3 Analytical Models with Stress Functions
77
Fig. 3.25 Effect of the tensile
stress F(r) distribution on the
stress distributions σz , σr and
τrz at the interface [12]
(z2 = h2 , a/b = 0.9,
(b–a)/2h1 = 0.2,
(b–a)/2h2 = 10,
E1 /E2 = 65.6, ν1 /ν2 = 0.81)
3.4 Bonded Shrink Fitted Joints
Shrink fitting has been used widely for joining cylindrical components of many mechanical structures. At present, shrink fitting in comparison with anaerobic adhesive
is used for mechanical joints such as automobile differential gears in order to improve the joint strength and to reduce the assembly weight. Thus, it is important for
mechanical design engineers to understand the interface stress distribution and the
joint strength of the bonded shrink fitted joints. Furthermore, it is critical to determine the stress distribution and joint strength when an external load is applied to
the joints. Figure 3.35 shows a bonded shrink fitted joint subjected to torsion. Prior
to assembly, the ring is heated up and is inserted into the shaft on which an anaerobic adhesive is applied. Then the bonded shrink fitted joints are cooled to ambient
temperature and left at room temperature. Based on the axi-symmetrical theory of
elasticity, the interface stress distribution is analyzed [17].
Figure 3.36 shows the effect of Young’s modulus ratio E3 /E2 on the stress ratio
τrθ /σr at the inner surface of the adhesive layer. The stress ratio τrθ /σr at the upper
78
T. Sawa
Fig. 3.26 Effect of the ratio
of the diameter a/b on the
stress distributions σz , σr and
τrz at the interface [12]
(z2 = h2 , b/2h2 = 100,
ν2 = 0.3, E1 /E2 = ∞)
end (z/h3 = 1.0) of the adhesive layer decreases as the ratio E3 /E2 increases. It
is indicated that the joint strength increases as the rigidity of the ring increases in
comparison with that of the adhesive. Figure 3.37 shows the effect of the outer
diameter of the rings, 2h3 . It is found that the stress ratio τrθ /σr decreases as the
ratio b3 /a2 increases. Thus, it can be assumed that the joint strength increases as the
outer diameter 2b3 increases in comparison with the outer diameter of the shafts,
2b1 . Figure 3.38 shows the effect of the engagement length 2h3 . It is found that the
effect of the value of h3 /a2 on the stress ratio τrθ /σr near the upper end of the rings
(z/h3 = 1.0) is small and that the ratio τrθ /σr inside the edge (z/h3 = 1.0) decreases
as the value of h3 /a2 increases. It can be concluded that the joint strength increases
as the engagement length 2h3 increases.
Figure 3.39 shows an example of the comparison of joint strength between shrink
fitted joints and bonded shrink fitted joints. It is shown that the strengths of bonded
shrink fitted joints are greater than those of shrink fitted joints. In addition, it is
easy to reduce the interference (shrink allowance) for bonded shrink fitted joints
in comparison with shrink fitted joints. This is a big benefit for mechanical design
engineers.
3 Analytical Models with Stress Functions
Fig. 3.27 Tubular adhesive
joints subjected to torsional
loadings [13]
Fig. 3.28 Effects of the ratio
of shear modulus of the
adhesive to that of the
adherend 2 on the shear stress
distribution τθ z2 /τn , type (A)
(2h/d = 0.1, z = h,
a = ((c2 + d 2 )/2)1/2 ,
c = d/2, b = d) [13]
79
80
T. Sawa
Fig. 3.29 Effect of the ratio of
shear modulus of the adhesive
to that of the adherend 2
on the displacement Vθ 2 /Vn ,
type (A) (2h/d = 0.1, z = h,
a = ((c2 + d 2 )/2)1/2 , c = d/2,
b = d) [13]
Fig. 3.30 Effect of the ratio of
shear modulus of the adhesive
to that of the adherend 2 on
the shear stress distribution
τθ z2 /τn , type (B) (2h/d = 0.1,
z = h, a = ((c2 + d 2 )/2)1/2 ,
c = d/2, b = d) [13]
3.5 Analysis with Stress Functions
In this section, the method for analyses are described for adhesive butt joints of
thin plates (Fig. 3.2; in the case where the two adherends (II) and (III) are the same
material) using two-dimensional theory of elasticity and for adhesive butt joints of
solid cylinder (Fig. 3.17) using axi-symmetrical theory of elasticity.
3.5.1 Analysis for Two-Dimensional Problems
For analyzing the stress state of a two-dimensional elastic body, Airy’s stress functions are applied. When Airy’s stress functions χ are used, each stress component is
described by (3.8) and each displacement is described by (3.9).
3 Analytical Models with Stress Functions
81
Fig. 3.31 Effect of the ratio of
shear modulus of the adhesive
to that of the adherend 2
on the displacement Vθ 2 /Vn ,
type (B) (2h/d = 0.1, z = h,
a = ((c2 + d 2 )/2)1/2 , c = d/2,
b = d) [13]
σx =
∂ 2χ
,
∂ y2
σy =
∂ 2χ
,
∂ x2
τxy = −
1
∂χ
∂ϕ
+
·
∂x 1+v ∂y
1
∂χ
∂ϕ
+
·
2Gvx = −
∂y 1+v ∂x
2Gux = −
∂ 2χ
∂ x∂ y
(3.8)
⎫
⎪
⎬
⎪
⎭
(3.9)
where, G: shear modulus, v: Poisson’s ratio, and χ , ϕ must satisfy the following
equations,
Fig. 3.32 Effect of the
adhesive thickness on the
shear stress distribution
τθ z2 /τn , type (A)
(2h/d = 0.1, z = h,
a = ((c2 + d 2 )/2)1/2 ,
c = d/2, b = d) [13]
∇2 ∇2 χ = 0
(3.10)
∇2 ϕ = 0
(3.11)
82
T. Sawa
Fig. 3.33 Effects of the ratio
of shear modulus of the
adhesive to that of the
adherend 2 on the shear stress
distribution τθ z2 /τn , type (A)
(2h/d = 0.1, z = h,
a = ((c2 + d 2 )/2)1/2 ,
c = d/2, b = d) [13]
where, ϕ is obtained from the following equation.
∂ 2ϕ
= ∇2 χ ,
∂ x∂ y
∇2 =
∂2
∂2
+ 2
2
∂x
∂y
(3.12)
In the analysis of a two-dimensional body, for example, rectangular plates, Airy’s
stress functions χ must be determined from the bi-harmonic equation described by
Eq. (3.12).
In the analysis for Fig. 3.2 (in the case where the two adherends (II) and (III)
are the same material), the stress functions χ I and χ II are chosen as follows, where
I
I
I
I
I
I I I II II II II
AI0 , An , Bs , An , Bs , A
n , Bs , An , Bs , An , Bs , An , Bs , (n, s = 1, 2, 3 . . .) are unknown
coefficients determined from the boundary conditions.
I
I
I
I
χ1I = χ1 (AI0 , An , Bs , l, h1 , αnI , βs , Δn , Ωs , x, y1 )
I
∞
AI
A I
= 0 x2 + ∑ I n
αn lch(αnI l) + sh(αnI l) ch(αnI x) − αnI xsh(αnI l)sh(αnI x) cos(αnI y1 )
2
n=1 Δn αnI2
I
∞
B + ∑ I s {βs h1 ch(βs h1 ) + sh(βs h1 )} ch(βs y1 ) − βs y1 sh(βs h1 )sh(βs y1 ) cos(βs x)
n=1 Ωs βs2
(3.13)
Fig. 3.34 Effect of the ratio of
shear modulus of the adhesive
to that of the adherend 2
on the displacement Vθ 2 /Vn ,
type (A): (2h/d = 0.1, z = h,
a = d/(2)1/2 , b = d) [13]
3 Analytical Models with Stress Functions
83
Fig. 3.35 A bonded shrink
fitted joint subjected to torsion
[17]
I
I
I
I
χ2I = χ2 (An , Bs , l, h1 , αnI , βs , Δn , Ωs , x, y1 )
I
∞
=
An
αnI lch(αnI l)+sh(αnI l) ch(αnI x)−αnI xsh(αnI l)sh(αnI x) sin(αnI y1 )
I
n=1 Δ α I 2
n n
I
∞
Bs {βs h1 sh(βs h1 )+ch(βs h1 )} sh(βs y1 )−βs y1 ch(βs h1 )ch(βs y1 ) cos(βs x)
+
I
n=1 Ω β 2
s s
∑
∑
(3.14)
Fig. 3.36 Effect of Young’s
modulus of the rings on the
interface stress τrθ /σr
distributions [17]
(b0 /b1 = 0.71, b1 = a2 ,
E1 /E3 = 1.0, h1 /b1 = 4.57,
h3 /b1 = 0.57, δ = 0.02 mm)
84
Fig. 3.37 Effect of the outer
diameter of the rings on the
stress τrθ /σr distributions
[17] (b0 /b1 = 0.71, b1 = a2 ,
E1 /E3 = 1.0, h1 /b1 = 4.57,
h3 /b1 = 0.57, δ = 0.02 mm)
Fig. 3.38 Effect of the
engagement length on the
interface stress τrθ /σr
distributions [17]
(b0 /b1 = 0.71, b1 = a2 ,
E1 /E3 = 1.0, h1 /b1 = 4.57,
h3 /b1 = 0.57, δ = 0.02 mm)
T. Sawa
3 Analytical Models with Stress Functions
85
Fig. 3.39 Comparison of the
joint strength between the
shrink fitted joint and the
bonded shrink fitted joint [17]
(2b1 = 35 mm, 2b3 = 80 mm,
2h3 = 20 mm)
In , Ω
Is , x, y1 )
In , BIs , l, h1 , αnI , βs , Δ
χ3I = χ3 (A
∞
In A
I
I
I
I
I
I
α
lsh(
α
l)ch(
α
x)
−
α
xch(
α
l)sh(
α
x)
cos(αnI y1 )
=−∑
n
n
n
n
n
n
I
I
2
n=1 Δn αn
∞
BIs −∑
β
h
sh(
β
h
)ch(
β
y
)
−
β
y
ch(
β
h
)sh(
β
y
)
cos(βs x)
1
1
1
1
1
1
s
s
s
s
s
s
I 2
n=1 Ωs βs
(3.15)
I
I
I I
I
χ4I = χ4 (A
n , Bs , l, h1 , αn , βs , Δn , Ωs , x, y1 )
I
A
= − ∑ I n αnI lsh(αnI l)ch(αnI x) − αnI xch(αnI l)sh(αnI x) sin(αnI y1 )
n=1 Δ
n αnI2
∞
I
Bs βs h1 ch(βs h1 )sh(βs y1 ) − βs y1 sh(βs h1 )ch(βs y1 ) cos(βs x)
−∑ I
n=1 Ω
s βs 2
(3.16)
∞
II
II
II
II
χ1II = χ1 (AII0 , An , Bs , l, h2 , αnII , βs , Δn , Ωs , x, y2 )
(3.17)
II , Ω
II , x, y2 )
IIn , BIIs , l, h2 , αnII , βs , Δ
χ3II = χ3 (A
n
s
(3.18)
where,
αnI = αn (h1 ) =
βs =
sπ
,
l
nπ
,
h1
βs =
αnI = αn (h1 ) =
(2s − 1)π
2l
(2n − 1)π
,
2h1
αnII = αn (h2 ),
αnII = αn (h2 ),
86
T. Sawa
I
I
= ΔI ,
Δ
n
n
I = Δn (α I l),
Δn = Δ
n
n
I
Δn = Δn (αnI l) = sh(αnI l)ch(αnI l) + αnI l,
II = Δn (α II l)
Δ
n
n
II
Δn = ΔIn (αnII l),
I
Ωs = Ωs (βs h1 ) = sh(βs h1 )ch(βs h1 ) + βs h1 ,
II
Ωs = Ωs (βs h1 ) = sh(βs h1 )ch(βs h1 ) − βs h1 ,
I = Ωs (β h1 ),
Ω
s
s
II
Ωs = Ωs (βs h2 ),
I
= Ω (β h1 ),
Ω
s
s s
II
Ωs = Ωs (βs h2 ),
sh: sinh, ch: cosh
(i) For finite strip (I) (adherends)
x = ±l;
I =0
σxI = τxy
y1 = h1 ;
σyI = F(x) = a0 + ∑ as cos
∞
sπ
l
s=1
I =0
τxy
⎫
⎪
⎪
⎪
⎬
⎪
x
⎪
⎪
⎪
⎪
⎭
(3.19)
(ii) For finite strip (II) (adhesive)
II
σxII = τxy
=0
x = ±l;
(iii) at the interface between finite strips (I) and (II)
I
σy y =−h
= σyII y =h
1
I
τxy
I
u
1
II
= τxy
y1 =−h1
= uII
y1 =−h1
∂ vI
∂x
=
y1 =−h1
2
2
y2 =h2
y2 =h2
∂ vII
∂x
(3.20)
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
(3.21)
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
y2 =h2
where,
a0 =
1
2l
e1
−e1
F(x)dx,
as =
1
l
e1
−e1
F(x) cos
sπ x dx
l
(s = 1, 2, 3, · · ·)
Substituting Airy’s stress functions χ I and χ II into Eqs. (3.8) and (3.9), the stresses
and the displacements for the finite strips [I] and [II] are obtained. By equating
them to the boundary conditions, the relations among undetermined coefficients are
obtained as follows:
3 Analytical Models with Stress Functions
87
⎫
∞
∞ I
∞
∞ I
I
I
I I
I
I
I
I
⎪
In + ∑ BIs Sns
⎪
B
An + ∑ Bs Pns
= 0, An − ∑ Bs Pns
= 0, A
= 0, A
+
S
=
0
∑
⎪
n
s ns
⎪
⎪
s=1
s=1
s=1
s=1
⎪
⎪
⎪
∞
∞
∞
∞
⎪
I
I
II
I
II II
I
II
II
II
I
I
⎪
n + ∑ Bs Sns = 0, ∑ An Qns +Bs − ∑ An Qns + Bs = −as ⎪
⎪
An + ∑ Bs Pns = 0, A
⎪
⎪
⎪
s=1
s=1
n=1
n=1
⎪
⎪
⎪
∞
∞
⎪
I I
⎪
I
⎪
I I
I
⎪
A
R
+
B
+
R
+
B
=
0
A
⎪
∑ n ns s ∑ n ns s
⎪
⎪
n=1
n=1
⎪
⎪
⎪
∞
∞
∞
⎬
I
I
I
I I
II
I
I
II
∑ An Qns +Bs + ∑ An Qns − Bs − ∑ An Qns −Bs = 0
⎪
n=1
n=1
n=1
⎪
⎪
⎪
⎪
∞
∞
∞
I I
⎪
⎪
I I
I
I
II
II
II
⎪
A
A
R
+
B
−
R
−
B
+
R
+
B
=
0
A
⎪
∑ n ns s ∑ n ns s ∑ n ns s
⎪
⎪
⎪
n=1
n=1
n=1
⎪
⎪
⎪
∞
∞
∞
∞
I
⎪
I
I
I I
I I
⎪
I
I
I
I II
I II
⎪
⎪
∑ AnUns + BsVs + ∑ AnUns + BsVs + ∑ BmWmCms − ∑ BmWm Cms
⎪
⎪
⎪
n=1
n=1
m=1
m=1
⎪
⎪
⎪
∞
∞
⎪
II II
II II
⎪
II
II
II
II
⎪
− ∑ An Uns − Bs Vs − ∑ BmWm Cms = Ds
⎭
n=1
m=1
(3.22)
AI0 = a0 ,
AII0 = AI0
(3.23)
where,
4(−1)n+s α I2 λ sh2 (λ h )
I
II
s 1
n s
I
II
Pns
= Pns αnI , λs , h1 , Ωs =
, Pns
= Pns αnII , λs , h2 , Ωs
I
2
Ωs (αnI + λs2 )2 h1
4(−1)n+s α I2 λ ch2 (λ h )
n
s
s 1
I
I
I
II
II
IIs
,
S
=
S
α
,
λ
,
h
,
Ω
Sns = Sns αn , λs , h1 , Ωs =
ns
2
ns
s
n
I (α I2 + λ 2 )2 h1
Ω
s n
s
4(−1)n+s α I λ 2 sh2 α I l
I
II
n s
n
, QIIns = Qns αnII , λs , l, Δn
QIns = Qns αnI , λs , l, Δn =
I
2
Δn (αnI + λs2 )2 l
4(−1)n+s αnI λs2 ch2 αnI l
I =
II
, RIIns = Rns αnII , λs , l, Δ
RIns = Rns αnI , λs , l, Δ
n
n
2
I (α I + λ 2 )2 l
Δ
n n
s
I
I
I
I
I
s , QIns = Qns αnI , λs , l, Δn
= Pns αnI , λs , h1 , Ωs , Sns
= Sns αnI , λs , h1 , Ω
Pns
I
n
RIns = Rns αnI , λs , l, Δ
I
Uns
=
I
Uns (αnI , Δn , G1 , ν1 ) =
2(−1)n+s λs sh2 (αnI l)
I
2
αI
1
+ 2 n
1 + ν1 αnI + λs2
Δn αnI (αnI + λs2 )G1 l
1
ν1 − 1
I
λ
h
+
sh(
λ
h
)ch(
λ
h
)
VsI = Vs (λs , h1 , Ωs , G1 , ν1 ) =
s
s
s
1
1
1
I
1 + ν1
2Ωs λs G1
2
88
T. Sawa
I
ν1 − 1
1
I
Vs = Vs (λs , h1 , Ωs , G1 , ν1 ) =
λs h1 −
sh(λs h1 )ch(λs h1 )
I
1 + ν1
2Ωs λs G1
2
2
I
, E1 ) = 2sh (λs h1 )
I , E1 ) = 2ch (λs h1 ) ,W I = W (λ , h1 , Ω
WsI = Ws (λs , h1 , Ω
s
s
s
s
s
I
I λ E1
Ω
s s
λ E1
Ω
s s
n+s α I ch2 (α I l)
2 − α I2
(−1)
ν
λ
3
+
1
n
n
n
s
I
I I
Hns = Hns (αn , Δn , G1 , ν1 ) =
+
In αnI (αnI2 + λs2 )G1 l 1 + ν1 αnI 2 + λs2
Δ
I
II
II
I
II
= Uns (αnI , Δn , G1 , ν1 ),Uns
= Uns (αnII , Δn , G2 , ν2 ),VsII = Vs (λs , h2 , Ωs , G2 , ν2 )
Uns
I
II , E2 ), F I = λs ×Ws (λs , h1 , Ω , E1 ), F I = λs ×Ws (λs , h1 , Ω , E1 )
WsII = Ws (λs , h2 , Ω
s
s
s
s
s
I
I
II
I
, G1 , ν1 ), H II = Hns (α II , Δ
II , G2 , ν2 )
= Hns (αnI , Δ
FsII = λs ×Ws (λs , h2 , Ωs , E2 ), Hns
ns
n
n
n
I
, G1 , ν1 )
I , G1 , ν1 ), J I = λ ×V (λ , h1 , Ω
JsI = λs ×Vs (λs , h1 , Ω
s
s
s
s s
s
II
II
Js = λs ×Vs (λs , h2 , Ωs , G2 , ν2 )
1
1
1
II
Cms = Cms (λm , λs ) = −
sin λm + λs l − sin λm − λs l
l λm + λs
λm − λs
2
E1
E2
ν2 ν1
II
= Ems (λm , λs ), DIIs = AI0 (−1)s
−
, G2 =
, G1 =
Ems
λs
E2 E1
2(1 + ν1 )
2(1 + ν2 )
By solving an infinite set of simultaneous Eq. (3.22), the undetermined coeffiI
I
IIn and BIIs are determined. From Eq. (3.23), AI and AII are decients An , Bs , . . ., A
0
0
termined. Using the determined coefficients, the stresses and the displacements are
obtained.
3.5.2 Analysis for Axi-Symmetrical Elastic Bodies
For analyzing the stress states of axi-symmetrical bodies such as solid cylinders
(shafts) or hollow cylinders (tubular), Michell’s stress functions are applied. When
Michell’s stress functions Φ are used, each stress component σr, σθ , σz, τrz are described as follows.
⎫
∂
∂ 2Φ
⎪
2
⎪
σr =
ν∇ Φ − 2
⎪
⎪
∂z
∂r
⎪
⎪
⎪
⎪
⎪
⎪
∂
1 ∂Φ
⎪
2
⎪
σθ =
ν∇ Φ −
⎪
⎬
∂z
r ∂r
(3.24)
∂
∂ 2Φ ⎪
⎪
2
⎪
σz =
(2 − ν )∇ Φ − 2 ⎪
⎪
⎪
∂z
∂z
⎪
⎪
⎪
⎪
2
⎪
⎪
∂
∂
Φ
2
⎭
τrz =
(1 − ν )∇ Φ − 2 ⎪
∂r
∂z
3 Analytical Models with Stress Functions
89
⎫
⎪
⎪
⎪
⎬
1 + ν ∂ 2Φ
E ∂ r∂ z
∂ 2Φ 1 ∂ Φ ⎪
1+ν
⎪
2
⎪
⎭
wz =
(1 − 2ν )∇ Φ + 2 +
E
∂r
r ∂r
ur = −
(3.25)
where, ur is the displacement in the radial direction and wz is the displacement in
the z-direction. G is the modulus in shear (G = E / 2(1 + ν )).
Michell’s stress function Φ must satisfy the following bi-harmonic equation.
∇2 ∇2 Φ = 0
where,
∇2 =
(3.26)
1 ∂2
∂2
1 ∂
+
+
∂ r 2 r ∂ r r 2 ∂ z2
In the analysis for the case of Fig. 3.18, the stress functions ΦI and ΦII are chosen
I
I
I
I
I
I
I I II II II II II
I , CI , A
In , CsI , A
as follows, where, A0 , C0 , An , Cs , An , Cs , A
n , Cs , A0 , C0 , An , Cs , An ,
0
0
II
II
II
II
II
, C , A
n and C (n, s = 1, 2, 3, · · ·) are unknown coefficients determined from
Cs , A
0
0
0
the boundary conditions, where Jμ (r) is the first kind of Bessel function of order μ ,
Iμ (r) is the first kind of modified Bessel function of order μ and λs is the positive
root satisfying the equation J1 (λs ) = 0, and Iμ (βn a) is abbreviated as Iμ a , Iμ (βn r)
as Iμ r , Iμ (βn a) as I μ a , Iμ (βn r) as Iμ r , and sinh is abbreviated as sh and cosh as ch.
I
I
I
I
I
I
ΦI1 = Φ1 (A0 ,C0 , An ,Cs , a, h1 , βnI , γs , Δn , Ωs , ν1 , r, z1 )
3
I z1
= A0
I z1 r
+C0
2
∞
+∑
I
An
3 I
6
2
n=1 βnI Δn
× 2(1 − ν1 )I1a + βnI aI0a I0r − βnI rI1a I1r sin(βnI z1 )
(3.27)
I
Cs ν
sh(
γ
h
)
+
γ
h
ch(
γ
h
)
−
2
s
s
s
1
1
1
1
I
s=1 γs3 Ωs
× sh(γs z1 ) + γs z1 sh(γs h1 )ch(γs z1 ) J0 (γs r)
∞
+∑
I
I
I
I
ΦI2 = Φ2 (An ,Cs , a, h1 , βnI , γs , Δn , Ωs , ν1 , r, z1 )
∞
=−∑
I
An
I
n=1 β I3 Δ
n
n
2(1 − ν1 )I1a + βnI aI 0a
× I 0r − βnI rI 1a I1r cos(βnI z1 )
I
Cs −
2
ν
ch(
γ
h
)
+
γ
h
sh(
γ
h
)
s 1
s 1
s 1
1
I
s=1 γ 3 Ω
s s
× ch(γs z1 ) + γs z1 ch(γs h1 )sh(γs z1 ) J0 (γs r)
∞
+∑
(3.28)
90
T. Sawa
I
In , Ω
Is , ν1 , r, z1 )
I0 , C0I , A
In , CsI , a, h1 , βn , γs , Δ
ΦI3 = Φ3 (A
∞
3
2
I
I0 z1 + C0I z1 r + ∑ An
=A
I3 I
6
2
n=1 βn Δn
× 2(1 − ν1 )I1a + βnI aI0a I0r − βnI rI1a I1r sin(βnI z1 )
(3.29)
CsI 2(1 − ν1 )ch(γs h1 ) − γs h1 sh(γs h1 )
3 I
s=1 γs Ωs
×sh(γs z1 ) + γs z1 ch(γs h1 )ch(γs z1 ) J0 (γs r)
∞
+∑
I
I
I I
I
ΦI4 = Φ4 (A
n , Cs , a, h1 , βn , γs , Δn , Ωs , ν1 , r, z1 )
I A
n
=−∑
2(1 − ν1 )I1a + βnI aI0a I0a − βnI rI1a I1r
I
n=1 β I3 Δ
n n
∞
(3.30)
I
Cs +∑
ν
)sh(
γ
h
)
−
γ
h
ch(
γ
h
)
(1
−
2
s
s
s
1
1
1
1
I
s=1 γ 3 Ω
s s
×ch(γs z1 ) + γs z1 sh(γs h1 )sh(γs z1 ) J0 (γs r)
∞
II
II
II
II
II
II
ΦII1 = Φ1 (A0 ,C0 , An ,Cs , a, h2 , βnII , γs , Δn , Ωs , ν2 , r, z2 )
IIn , Ω
IIs , ν2 , r, z2 )
II0 , C0II , A
IIn , CsII , a, h2 , βnII , γs , Δ
ΦII3 = Φ3 (A
(3.31)
(3.32)
where,
βnI = βn (h1 ) =
nπ
,
h1
βnI = βn (h1 ) =
(2n − 1)π II
, βn = βn (h2 ), βnII = βn (h2 )
2h1
I
γs = λs /a,
Ωs = Ωs (γs h1 ) = sh(γs h1 )ch(γs h1 ) + γs h1
I
Ωs = Ωs (γs h1 ) = sh(γs h1 )ch(γs h1 ) − (γs h1 ),
I = ΩI ,
Ω
s
s
I
=Ω
Ω
s
s
I
IIs = ΩIIs
Ω
1
Δn = Δn (βnI a, ν1 ) = 2(1 − ν1 ) + (βnI a)2 I21a − (βnI a)I20a /βnI a
II
Ωs = Ωs (γs h2 ),
1
Δn = Δn (βnI a, ν1 ),
I1
Δn = Δn (βnII a, ν2 ),
1
I = Δ ,
Δ
n
n
I
= Δ1
Δ
n
n
IIn = Δn (βnII a, ν2 )
Δ
3.6 Conclusions
In this chapter, the interface stress characteristics of adhesive butt joints have been
examined. At first, several types of adhesive butt joints are shown and the effects
of some factors such as the adhesive Young’s modulus and the adhesive thickness
3 Analytical Models with Stress Functions
91
on the interface stress distributions are described. For adhesive butt joints of thin
plates, solid cylinders and hollow cylinders (tubular joints), it was shown that the
singular stresses occur at the edges of the interfaces between the adherends and
the adhesive and they decrease as the adhesive Young’s modulus increases and the
adhesive thickness decreases. It was shown that the bonded area in the band adhesive
joints is smaller than that in the adhesive butt joints while the strength of the band
adhesive joints is the same approximately as that of the adhesive butt joints.
Then, an example of method of analysis for the adhesive butt joints of thin plates
and solid cylinders using the stress functions was presented. For two-dimensional
problems, Airy’s stress functions were used while for axi-symmetrical problems,
Michel’s stress functions were used. For analyzing the singular stresses which occurred at the edges of the interfaces, an analytical method was superior to a computational method such as FEM because the singular stresses depend on the mesh
sizes in the FEM.
However, the mentioned methods of stress analyses were carried out in the elastic
deformation range. In near future, the method should be developed in the elastoplastic region for obtaining more detailed results in estimating the joint strength.
Generally, adhesive joints are used under severe conditions and bad environment.
In many applications, the joints are subjected to static loadings as well as impact
loadings. In near future, adhesive joints will be analyzed under static and impact
loading conditions in the elasto-plastic range of the adhesive.
References
1. Andersson T. and Biel A. (2006) On the Effective Constitutive Properties of a Thin Adhesive
Layer Loaded in Peel. Int J Frac 141: 227–246
2. Chen D. and Cheng S. (1992) Torsional Stress in Tubular Lap Joints. Int J Solids Struct 29(7):
845–853
3. Cheng J., Wu X., Li G., Pang S. and Taheri F. (2007) Design and Analysis of a Smart Composite Pipe Joint System Integrated with Piezoelectric Layers Under Bending. Int J Solids Struct
44: 298–319
4. Ding S. and Kumosa M. (1994) Singular Stress Behavior at an Adhesive Interface Corner. Eng
Frac Mech 47(4): 503–519
5. Hollaway L., Romhi A. and Gunn M. (1990) Optimisation of Adhesive Bonded Composite
Tubular Sections. Compos Struct 16: 125–170
6. Kim W. T. and Lee D. G. (1995) Torque Transmission Capabilities of Adhesively Bonded
Tubular Lap Joints for Composite Drive Shafts. Compos Struct 30: 229–240
7. Öchsner A. and Grácio J. (2007) An Evaluation of the Elastic Properties of an Adhesive Layer
Using the Tensile-Butt Joint Test: Procedures and Error Estimates. Int J Adhes Adhes 27:
129–135
8. Öchsner A., Stasiek M., Mishuris G. and Grácio J. (2007) A New Evaluation Procedure for the
Butt-Joint Test of Adhesive Technology: Determination of the Complete Set of Linear Elastic
Constants. Int J Adhes Adhes 27: 703–711
9. Pugno N. and Carpinteri A. (2003) Tubular Adhesive Joints Under Axial Load. Trans ASME
70: 832–839
10. Reedy Jr E. D. (1993) Asymptotic Interface Corner Solutions for Butt Tensile Joints. Int J
Solids Struct 30(6): 767
92
T. Sawa
11. Reedy Jr E. D. and Guess T. R. (1993) Comparison of Butt Tensile Strength Data with Interface
Corner Stress Intensity Factor Prediction. Int J Solids Struct 30(21): 2929–2936
12. Sawa T., Ishikawa H. and Temma K. (1987) Three-Dimensional Stress Analysis of Adhesive
Butt Joints Subjected to Tensile Loads (The Case Where Adherends are Two Hollow Cylinders) Transactions of JSME, Part A 53(492): 1685–1691
13. Sawa T., Nakano Y. and Temma K. (1987) A Stress Analysis of Butt Adhesive Joints Under
Torsional Loads. J Adhes 24: 245–258
14. Sawa T., Temma K. and Ishikawa H. (1989) Three-Dimensional Stress Analysis of Adhesive
Butt Joints of Solid Cylinders Subjected to External Tensile Loads. J Adhes 31: 33–43
15. Sawa T., Temma K., Nishigaya T. and Ishikawa H. (1995) A Two-Dimensional Stress Analysis
of Adhesive Butt Joints of Dissimilar Adherends Subjected to Tensile Loads. J Adhes Sci
Technol 9(2): 215–236
16. Sawa T. and Uchida H. (1997) Two-Dimensional Stress Analysis and Strength Evaluation of
Band Adhesive Butt Joints Subjected to Tensile Loads. J Adhes Sci Technol 11(6): 811–833
17. Sawa T., Yoneno M. and Motegi Y. (2001) J. Adhes Sci Technol 15(1): 23–42
18. Seo D. W. and Lim J. K. (2005) Tensile, Bending and Shear Strength Distributions of
Adhesive-Bonded Butt Joint Specimens. Compos Sci Technol 65: 1421–1427
19. Shahid M. and Hashim S. A. (2002) Effect of Surface Roughness on the Strength of Cleavage
Joints. Int J Adhes Adhes 22: 235–244
20. Temma K., Sawa T. and Iwata A. (1990) Two-Dimensional Stress Analysis of Adhesive Butt
Joints Subjected to Cleavage Loads. Int J Adhes Adhes 10(4): 285–293
21. Temma K., Sawa T., Nishigaya T. and Ichida H. (1994) Two-Dimensional Stress Analysis
and Strength of Band Adhesive Butt Joints of Dissimilar Adherends Subjected to External
Bending Moments. JSME Int J Series A 37(3): 246
22. Thomsen O. T. (1992) Elasto-Static and Elasto-Plastic Stress Analysis of Adhesive Bonded
Tubular Lap Joints. Compos Struct 21: 249–259
23. Wright M. D. (1978) Compos 9(4): 259–262
24. Xu L. R., Sengupta S. and Kuai H. (2004) An Experimental and Numerical Investigation of
Adhesive Bonding Strengths of Polymer Materials. Int J Adhes Adhes 24: 455–460
25. Zanni-Deffarges M. P. and Shanahan M. E. R. (1993) Evaluation of Adhesive Shear Modulus
in a Torsional Joint: Influence of Ageing. Int J Adhes Adhes 13(1): 41–45
26. Zhou H. and Rao M. D. (1993) Vicoelastic Analysis of Bonded Tubular Joints Under Torsion.
Int J Solids Struct 30(16): 2199–2211
Part II
Numerical Modeling
Chapter 4
Complex Constitutive Adhesive Models
Erol Sancaktar
Abstract A complete approach to modeling adhesives and adhesive joints needs to
include considerations for: deformation theories, viscoelasticity, singularity methods, bulk adhesive as composite material, adhesively bonded joint as composite and
the concept of the “interphase”, damage models, and the effects of cure and processing conditions on the mechanical behavior. The adherend surfaces have distinct
topographies, which result in a collection of miniature joints in micron, and even
nano scale when bonded adhesively. The methods of continuum mechanics can be
applied to this collection of miniature joints by assuming continuous, or a combination of continuous/discontinuous interphase zones.
4.1 Introduction
Adhesively bonded joints are complex composite structures with at least one of the
constituents, namely the adhesive, most often, being a composite material itself due
to the presence of secondary phases such as fillers, carriers, etc. The joint structure
possesses a complex state of stress with high stress concentrations, and often, singularities due to the terminating adhesive layer where the substrates may possess
sharp corners. Consequently, accurate analysis and modeling of adhesive materials and bonded joints require the use of the methods of composite materials and
composite mechanics. The inclusion of the “interphase” region is necessary in this
analysis as a distinct continuum. The presence of geometric discontinuities creates
stress concentrations and, possibly, singularities adding additional complexity to the
topic of adhesively bonded joints. This problem, however, can be alleviated, at least
partially, by making the proper changes in the geometry of the bonded joint. Furthermore, since most adhesive materials are polymer-based, their natural viscoelastic
Erol Sancaktar
Polymer Engineering, Adjunct Professor, Mechanical Engineering, The University of Akron,
Akron, OH 44325-0301, e-mail: erol@uakron.edu
L.F.M. da Silva, A. Öchsner (eds.), Modeling of Adhesively Bonded Joints,
c Springer-Verlag Berlin Heidelberg 2008
95
96
E. Sancaktar
behavior usually serves to reduce localized stress concentrations. In those cases
where brittle material behavior prevails or, in general, when inherent material flaws
such as cracks, voids, disbonds exist, then the use of the methods of fracture mechanics are called for. For continuum behavior, however, the use of damage models
is considered appropriate in order to be able to model the progression of localized
and non-catastrophic failures.
Obviously, a technical person involved in adhesive development and/or applications should keep the above mentioned issues in mind, along with insight into
typical joint stress distributions for adhesive joints as well. Stress distributions are
relevant from the mechanical adhesion point of view also, since they depend on surface topography, which can be considered a collection of many geometrical forms.
Therefore, mechanical adhesion depends on the stress states of different adhesive
joint geometries on the scale of the surface topography, which may include many
lap, butt and scarf joints in the interphase region.
A complete approach to modeling adhesives and adhesive joints, therefore,
needs to include considerations for: deformation theories, viscoelasticity, singularity methods, bulk adhesive as composite material, adhesively bonded joint as
composite – the concept of the “interphase”, damage models, and the effects of cure
and processing conditions on the mechanical behavior.
4.2 Deformation Theories
The deformation theory was first introduced by Hencky (1924) as reported by
Hill (1956) and Kachanov (1971) in the form:
εi j = (σkk /9 K)δi j + ψ Si j
(4.1)
where, Si j is the deviatoric stress tensor. Equation (4.1) reduces to the elastic stressstrain relations when ψ = 1/2G, where G is the elastic shear modulus. If the scalar
function ψ is defined as
ψ = (1/2 G) + Ω
(4.2)
then Eq. (4.1) can be interpreted in the form
ε i j = εi j E + εi j V + εi j P
(4.3)
εi j = (σkk /9 K)δi j + (Si j /2G) + Ω Si j
(4.4)
εi j P = Ω Si j
(4.5)
where
and,
with Ω being a scalar function of the invariants of the stress tensor, and the
superscripts E, V and P representing elastic, viscoelastic and plastic behaviors,
respectively.
4 Complex Constitutive Adhesive Models
97
Consequently, the relation
ε = (σ /E) + Λ σ
(4.6)
is obtained for uniaxial tension on the basis of Eqs. (4.4) and (4.5) with Λ = 2Ω/3.
Ramberg and Osgood (1943) used a special form of Eq. (4.6) with Λ = K σ n−1
to result in:
ε = (σ /E) + K σ n
(4.7)
where, K and r are material constants.
They reported that Eq. (4.7) could be used successfully to describe uniaxial tension and compression behavior of various metal alloys. Equation (4.7) was later
modified by McLellan (1966, 1969) to accommodate strain rate effects. McLellan
interpreted the terms E, K and n of Eq. (4.7) as material functions with the function E representing viscoelastic behavior and functions K and n representing workhardening characteristics. The terms E, K and n were all described as functions of
the strain rate (dε /dt) so that rigidity, stress and plastic flow respectively were all
affected by variations in the strain rate.
Renieri et al. (1976) used a bilinear form of rate dependent Ramberg-Osgood
equation to describe the stress-strain behavior of a thermosetting adhesive in the
bulk tensile form. The bilinear behavior was obtained when log ε p was plotted
against log σ , where ε p represents the second term on the right-hand side of
Eq. (4.7). The model adhesive they used was an elastomer modified epoxy adhesive with and without carrier cloth. They made several modifications on the form of
the equation previously used by McLellan. First, the plastic strain ε p was assumed
to be a function of the over-stress above the elastic limit stress (the development
of over-stress approach will be presented subsequently) and second, the stress levels σ ∗ defining the intersection point for the bilinear behavior were found to occur
slightly below the stress whitening stress values. The equations they developed in
this fashion are given as:
ε = σ /E,
0<σ <θ
(4.8-1)
ε = (σ /E) + K1 [σ − θ ]n1
θ < σ < σ∗
(4.8-2)
ε = (σ /E) + K2 [σ − θ ]
n2
∗
σ <σ <Y
(4.8-3)
where, θ is the elastic limit stress (yield stress) in simple tension, Y is the maximum stress, K1 , K2 , n1 , and n2 are strain rate dependent material functions. Renieri
et al. (1976) reported that the adhesive material properties were different before
and after stress-whitening due to changes in material behavior and therefore bilinear equations of the form (4.8) were needed to describe such behavior. Brinson
et al. (1975) determined that for the adhesive with the carrier cloth the parameter n,
Eq. (4.7), varied much less with strain rate in comparison to the adhesive without
the carrier cloth. They concluded that the bilinear forms they proposed, Eq. (4.8),
accurately predicted the rate dependent behavior of their model adhesive.
98
E. Sancaktar
Figures 4.1 and 4.2 show the application of the model described above to the
stress-strain and stress whitening behaviors of a bulk thermosetting epoxy film adhesive containing a non-woven nylon mat, as obtained by Sancaktar and Beachtle
(1993). In this application, two intersection points are identified by intersecting
two pairs of bilinears, each of which containing a common slope line as part
of the bilinear sets (Fig. 4.2). This procedure was called “dual bilinear fit” by
Sancaktar and Beachtle (1993). These intersection points define upper and lower
stress whitening stress limits shown in Fig. 4.1.
Ramberg-Osgood type equations were also used to describe shear behavior
of structural adhesives. Zabora et al. (1971) used a rate dependent form of the
Ramberg-Osgood equation to describe the shear behavior of structural adhesives
in the bonded form.
Simple power function type relations may be utilized in an empirical fashion
to describe nonlinear elastic behavior of structural adhesives in tensile and shear
modes when their mechanical behavior is found to be relatively rate insensitive. For
example Sancaktar and Schenck (1985-a) used a power function relation given by
τ = Kγ r
(4.9)
where τ and γ are shear stress and strain respectively, to represent the nonlinear
elastic shear behavior of a thermoplastic polyimidesulfone adhesive in the bonded
form. Sancaktar and Dembosky (1986) showed that when the stress-strain curve has
a well defined initial elastic region and a yield point, τy , then a bimodal relation
given by
Fig. 4.1 A stress-strain diagram identifying upper and lower stress whitening stress limits, σU and
σLo ∗ respectively, as well as the visually observed stress whitening stress. The modulus of elasticity, E, and the elastic limit stress, θ for the bulk adhesive tested are 3.3 × 105 and 3.5 × 103 psi,
respectively (Sancaktar and Beachtle, 1993)
4 Complex Constitutive Adhesive Models
99
Fig. 4.2 A stress-strain diagram identifying upper and lower stress whitening stress limits, σU and
σLo ∗ respectively, as well as the visually observed stress whitening stress. The modulus of elasticity, E, and the elastic limit stress, θ for the bulk adhesive tested are 3.3 × 105 and 3.5 × 103 psi,
respectively (Sancaktar and Beachtle, 1993)
γ = τ /G,
γ = Bτ m
τ < τy
τ > τy
(4.10-1)
(4.10-2)
can be utilized more appropriately.
4.3 Viscoelasticity Considerations
4.3.1 Linear Viscoelasticity
The differential forms of the fundamental constitutive relations are given as:
P{Skl } = 2Q{εkl }
P {σ } = Q {ε } − Q {AT }
(4.11)
(4.12)
where, A is the thermal expansion function, T is the temperature, Skl and εkl are
stress and strain deviators respectively and the differential operators are defined as:
u
P = ∑ ai (x,t, T )∂ i /dt i
(4.13)
Q = ∑ bi (x,t, T )∂ i /∂ t i
(4.14)
i=0
v
i=0
100
E. Sancaktar
u
P = ∑ ai (x,t, T )∂ i /∂ t i
(4.15)
i=0
v
Q = ∑ bi (x,t, T )∂ i /∂ t i
(4.16)
i=0
v
Q = ∑ bi (x,t, T )∂ i /∂ t i
(4.17)
i=0
T (x,t)
Q {AT } = ∫ Q {α (T )}dT (4.18)
T0
v
i
i
Q = ∑ b
i (x,t, T )∂ /∂ t .
(4.19)
i=0
In Eqs. (4.13) through (4.19) the terms ai , ai , bi , bi , bi and bi are material
properties and the integers u, u , v, v , v and v reflect the complexity of the material behavior, and t represents time. In Eq. (4.18) the quantity α represents the
coefficient of thermal expansion.
The same constitutive relations can also be represented in integral equation form
given by:
t
t
−∞
−∞
σ (x,t) = ∫ EV (x,t,t )[∂ ε (x,t )/∂ t ]dt − ∫ EV T (x,t,t )∂ [AT (x,t )]/∂ t dt and
(4.20)
t
Skl (x,t) = 2 ∫ E(x,t,t )∂ εkl (x,t )/∂ t dt −∞
(4.21)
where EV and EV T are the relaxation moduli as functions of time and time and
temperature, respectively, and E is the “relaxation memory function”. As shown by
Flügge (1975), the assumptions of elastic dilatational behavior or no volume change
are frequently used in three dimensional analyses of polymeric materials. As for
distortion operations, however, any mechanical model (such as Maxwell, Kelvin,
three-parameter solid, etc.) can be used.
4.3.2 Interrelationship Between the Tensile and Shear Properties
The constitutive equations proposed for characterization of solid polymer materials
in the tensile mode can be reformulated for the case of pure shear. One approach is
to replace tensile stress and strain variables by their shear counterparts.
It is usually desirable to run a simple bulk tensile test program and subsequently
predict (calculate) shear properties from their tensile counterparts. This approach requires a clearly defined relationship between shear and tensile elastic limit and yield
variables and material properties. The elastic limit and yield stress values can be related between tensile and shear conditions by using an appropriate failure criterion,
4 Complex Constitutive Adhesive Models
101
such as maximum normal stress, maximum shear stress, distortion energy criteria,
etc. A material parameter that needs to be converted in addition to the usual elastic
properties is the viscosity coefficient. This can be done by using Tobolsky’s (1960)
assumption of equivalent relaxation times in shear and tension.
Application of this assumption results in the relation
μS = μT /2(l + υ )
(4.22)
where μS and μT refer to the viscosity coefficients in shear and tension respectively,
and υ is Poisson’s ratio. An application of this procedure for a linear viscoelastic
(over-stress) model was shown by Sancaktar and Brinson (1980).
4.3.3 Time-Temperature Superposition
The time-temperature superposition principle (TTSP) is used to extend the creep and
relaxation data obtained at higher temperatures to values at lower temperatures and
longer times in the same strain and stress ranges. Provided that no structural changes
occur in the material at high temperatures, TTSP can be used for amorphous and
semi-crystalline thermoplastics and thermosets for extrapolation purposes. In order
to apply this principle, one should first construct a “master curve,” where the complete compliance-time or relaxation modulus-time behavior is plotted at a constant
temperature. For the “master curve” a reference temperature To is chosen. As shown
by Ahlonis et al. (1972), the relation for the compliance observed at any time t and
temperature To is given in terms of the experimentally observed compliance values
at different temperatures T as:
D(To ,t) = [ρ (T )T /ρ (To )To ]D(T,t/aT )
(4.23)
D(t) = ε (t)/σ
(4.24)
where
and ρ (T ) is the temperature dependent density. An illustration of this procedure is
shown in Fig. 4.3, where adjustments for any change in the material density are
ignored.
A similar relation for the relaxation modulus is given as:
E(To ,t) = [ρo (To )To /ρ (T )T ]E(T, aT t)
(4.25)
E(t) = σ (t)/ε .
(4.26)
where,
The term aT is a time shift factor which can be obtained with the application of the
famous Williams-Landel-Ferry (WLF) equation, reported by Ferry (1961).
For most polymeric materials the polymer’s glass transition temperature (Tg ) is
chosen as the reference temperature for accurate results. Even though Ferry (1961)
102
E. Sancaktar
Fig. 4.3 An illustration of time-temperature (t-T ) superposition procedure for shifts in creep compliance, D(t). Adjustments for any change in the material density are ignored
reports that the above equations have not been proven to be valid below the Tg , data
showing its application for T < Tg is available in the literature. Examples of applications to structural adhesives in the bulk form were given by Renieri et al. (1976)
and Keuner et al. (1982).
The application of the TTSP can be extended into the nonlinear viscoelastic region, as shown by Darlington and Turner (1978). Examples on the establishment
of rate-temperature superposition based on WLF equation as applied to peeling
problems have been given by Kaelble (1964, 1969), Hata et al. (1965) and Nonaker (1968). Nakao (1969) and Koizumi et al. (1970) report superposition based on
Arrhenius’ equation.
4.3.4 Nonlinear Viscoelasticity
Equations (4.11), (4.12) and (4.20), (4.21) can also be used to describe nonlinear
material behavior provided that ai , bi , etc. and E, EV are functions of stresses,
strains, their derivatives and their invariants in addition to the functional variables
shown.
On the basis of their observations on polymers, composites and adhesives, Hiel
et al. (1983), Rochefert and Brinson (1983), Tuttle and Brinson (1984) and Zhang
et al. (1985) report that the extent of nonlinearity is dependent on both the stress
level and the time scale. Constitutive relations having the form of Eqs. (4.20)
and (4.21) are in agreement with this observation. Schapery (1966, 1969a, b), reports that stress and strain dependent material properties of nonlinear constitutive
equations have a thermodynamic origin. For example the one-dimensional constitutive relation
t
ε = g0 D(0)σ + g1 ∫ ΔD(φ − φ )[d(g2 σ )/d τ ]d τ
(4.27)
o
includes the variables g0 , g1 and g2 which reflect stress dependence of the Gibbs
free energy. In Eq. (4.27) D(0) is the initial value of the linear viscoelastic creep
compliance, ΔD(φ ) is the transient component and the quantities
4 Complex Constitutive Adhesive Models
103
t
φ = φ (t) = ∫ dt /aσ
(4.28)
o
and
τ
φ = φ (τ ) = ∫ dt /aσ
(4.29)
o
are reduced time parameters and depend on the shift function aσ . “aσ ” is a function
of stress and change with entropy production and free energy changes.
An equation similar to (4.27) can be written with strain as the independent state
variable. In this equation, the strain dependent properties h0 , h1 , h2 and aε reflect
the changes in the Helmholtz free energy.
Schapery’s Eq. (4.27) was applied extensively by Cartner et al. (1978), Peretz
and Weitsman (1982) and by Tuttle and Brinson (1985) in characterization of resin
matrix composites.
In using nonlinear mechanical models, in addition to utilizing nonlinear elastic
and shear thickening or thinning dashpot elements, one can also incorporate nonlinearity by means of a perturbation technique. This is accomplished by adding small
perturbation terms which are functions of the current level of elastic strain and strain
rate to the elastic and viscous coefficients respectively. This method was originally
proposed by Davis (1964) and later applied in material characterization of bulk adhesives by Renieri et al. (1976).
Further discussions on nonlinear viscoelastic theory are given by Green and
Adkins (1960) and Hilton (1975).
4.3.5 Empirical Description of Creep Behavior
The constitutive equations obtained with the use of mechanical models can be solved
for the creep condition given by
σ (t) = σo H(t)
(4.30)
with σo and H(t) describing the level of constant stress and the unit step function respectively, to result in a creep equation. In some cases, however, empirical relations
describing time dependent strain functions are preferred over this method for practical reasons. Actual creep data can be fitted with such functions to yield accurate
analysis in conjunction with equations of the form (4.20), (4.21) or (4.27).
A variety of mathematical forms have been proposed to describe the creep behavior of adhesives and polymers in an empirical fashion. The simplest procedure
is called the log-log method based on the assumption that the logarithm of the secondary creep rate is linearly related to the logarithm of the stress level at which the
creep occurs. This assumption results in a creep relation which is a linear function
of time but nonlinear function of stress in the secondary range. An application of
this method to polymeric materials was given by Sancaktar et al. (1987-a).
104
E. Sancaktar
The creep relation which is most commonly used in the literature is the “powerlaw” compliance given by
(4.31)
D(t) = D0 + D1 t n
where, D0 is the instantaneous compliance, D1 , and n are material constants. Note
that Eq. (4.31) allows time reduction by a stress-dependent shift-factor to describe
stress enhanced creep. Equations of this form were used by researchers such as
Weitsman (1981) and Ravi-Chandar and Knauss (1984) for viscoelastic stress analysis and fracture mechanics considerations (respectively) of adhesives and polymeric
materials.
Various methods have been proposed to determine the parameters D0 , D1 and
n of Eq. (4.31). For example Findley and Peterson (1958) subtract the instantaneous strain from the total strain and plot the transient strain on log-log paper.
Other methods were described by researchers such as Boller (1957), Knauss and
Enri (1981), Dillard et al. (1982), Dillard and Brinson (1983), Becher (1984) and
Henriksen (1984). A comparison and discussion of these methods was given by
Dillard and Hiel (1985).
As shown by Sancaktar (1991), Figs. 4.4 and 4.5 illustrate the variation of adhesive shear stress at the overlap edges of double lap joints as predicted by linear
and nonlinear viscoelastic analyses. In Fig. 4.4, the linear analysis was performed
using the Maxwell model with different relaxation times, T , and the nonlinear analysis was performed using Eq. (4.31) with a stress-dependent shift factor. Figure 4.5
illustrates the ability to obtain matching results between the two models by using
the Maxwell model with variable relaxation time, T , as given by Sancaktar (1991).
Additional methods and discussions on constitutive equations for creep were
given by Odquist (1954), Findley and Lai (1967), Gittus (1975) and Findley et al.
(1976).
4.3.6 Increases in Joint Strength Due to Rate Dependent
Viscoelastic Adhesive Behavior
Due to their viscoelastic nature, most adhesives exhibit rate dependent material behavior, which can be modeled, for example, by using mechanical model characterization.
In order to describe the rate dependence of limit stress and elastic limit strain
for polymeric and adhesive materials in the bulk form, Renieri et al. (1976) and
Sancaktar and Brinson (1980) utilized a semi-empirical approach proposed by
Ludwik (1909) in the form:
dγ dγ
τult = τ + τ log
(4.32)
dt
dt
where, τult is the ultimate shear stress, d γ /dt is the initial elastic strain rate, and τ ,
τ , and dγ /dt are material constants.
4 Complex Constitutive Adhesive Models
105
Fig. 4.4 Variation of adhesive shear stress at the overlap edges of double lap joints as predicted by
linear and nonlinear viscoelastic analyses. The linear analysis was performed using the Maxwell
model with different relaxation times, T , and the nonlinear analysis was performed using Eq. (4.31)
with a stress-dependent shift factor. The joint dimensions used in analysis are also shown with δ ,
L and d representing adhesive thickness, overlap length and the main plate thickness, respectively.
Es and Ga represent the Young’s modulus for the substrate and the shear modulus for the adhesive,
respectively (Sancaktar, 1991)
Renieri et al. (1976) and Sancaktar and Brinson (1980) used the same form of
Eq. (4.32) to describe the variation of elastic limit shear stress (τel ) and strains (γel )
with initial elastic strain rates. These expressions may be written as:
dγ dγ
τel = φ + φ log
(4.33)
dt
dt
and,
dγ
γel = ζ + ζ log
dt
dγ dt
(4.34)
where, additional material constants are defined accordingly. Sancaktar et al. (1984,
1985-b) later showed applicability of Eqs. (4.32) through (4.34) for adhesives in
the bonded form and also proposed superposition of temperature effects on these
equations.
As reported by Ward and Hadley (1993), the theoretical basis of Eq. (4.32) can be
found in the Eyring Theorem, according to which the mechanical response of an adhesive is a process that has to overcome a potential barrier, (ΔE). The author thinks
106
E. Sancaktar
Fig. 4.5 Variation of maximum adhesive shear stress in double lap joints based on nonlinear viscoelastic analysis, using Eq. (4.31) with a stress-dependent shift factor, and linear viscoelastic
analysis, using the Maxwell model with variable relaxation time, T (Sancaktar, 1991)
that this barrier decreases not only with increasing stress but also with increasing
temperature. Based on the Eyring Theorem, the relationship between a limit stress,
say τel , and the strain rate can be written as:
dγ /dt = A exp[(τelV − ΔE)/RT ]
(4.35)
where, A is a pre-exponential factor, R is the gas constant, and V is the activation
volume.
Equation (4.35) can be expressed in logarithmic form as:
dγ
τel = {ΔE/V } + 2.303 RT log
A
V.
(4.36)
dt
Sharon et al. (1989) applied Eq. (4.36) to describe rate and temperature dependent
variation of four structural adhesives in the bulk form. They used the shift factor
[log(1/A)] × 10−3 on the temperature (◦ C) to describe the effect of rate. Another
example for bonded joint behavior was given by Chalkley and Chiu (1993).
If the energy barrier term ΔE/V of Eq. (4.36) is also affected by temperature then
one needs two shift factors in Eqs. (4.32) through (4.34) to describe rate-temperature
effects on limit stress-strain equations:
dγ dγ
τel = {aT 1 }θ + {aT 2 }θ log
(4.37)
dt
dt
4 Complex Constitutive Adhesive Models
and,
dγ dγ /
γel = {bT 1 }ζ + {bT 2 }ζ log
dt dt
107
(4.38)
where aTi , bTi are shift factors as functions of temperature. Figure 4.6 illustrates an
application of Eq. (4.37) by Sancaktar et al. (1984), showing the variation of elastic
limit shear stress with initial elastic strain rate and environmental temperature for an
epoxy adhesive film with polyester knit fabric carrier cloth. The adhesive was tested
in the bonded lap shear mode.
In order to analyze the effects of rate and temperature on the interfacial strength
of bonded joints, one can initially use a simple energy approach in the following
manner: A critical energy level, Wc , is used to represent interfacial failure. In the
presence of adhesive, substrate and interphase (a distinct material layer between
the substrate and the adhesive layer which transmits the rigid substrate’s energy to
the adhesive layer), the combined elastic energy can be written as:
2
2
Eadhesive } +Vip {(1/2)εinterphase
Ginterphase }
Va {(1/2)εadhesive
2
Esubstrate } = Wc
+Vs {(1/2)εsubstrate
(4.39)
where, Va , Vip and Vs represent volume fractions of the adhesive layer, interphase
and the substrate, respectively. Now, if one considers Eq. (4.39) in conjunction with
Eq. (4.38) to include rate-temperature effects, then it can be easily deduced that
higher Wc levels are obtained at high rate and/or low temperature levels since the
proportion of elastic strains increases with increasing elastic limit strains due to
high rate and/or low temperature levels.
Fig. 4.6 Variation of elastic limit shear stress with initial elastic strain rate and environmental temperature, and comparison with Ludwik’s equation for an epoxy adhesive film with polyester knit
fabric carrier cloth. The adhesive was tested in the bonded lap shear mode (Sancaktar et al. 1984)
108
E. Sancaktar
4.3.7 Reductions in Stress Concentration Due to Viscoelastic
Adhesive Behavior
If the adhesive material exhibits viscoelastic behavior, then the high stress magnitudes observed at the overlap end region of bonded joints are expected to diminish
in time due to the creep process. For example Weitsman (1981) utilized the nonlinear viscoelastic “power-law” response, which describes a stress enhanced creep
process to illustrate time dependent reductions in shear stress peaks along the adhesive layers of symmetric double lap joints. Sancaktar (1991) later illustrated the
applicability of the correspondence principle to the same problem with the use of
Maxwell chain to approximate the continuous change in the relaxation time and to
coincide with the results calculated using nonlinear viscoelastic theory, as shown in
Figs. 4.4 and 4.5.
4.4 Singularity Methods
Theoretically, geometrical singularities exist at sharp interface corners. The strength
of such singularities depends on the local geometry, including the actual corner radius, and the material properties. Barsoum (1989) considered the cases of elastic
adhesive at a rigid corner, and nonlinearly hardening adhesive at a square rigid corner. He showed that for an elastic adhesive the obtuse angle notch gives a much
weaker singularity and, hence, a greater failure load.
Assuming linear elasticity, Groth (1988) proposed an exact solution for the adhesive stresses at an interface corner in the form of a power series:
σi j =
∞
∑ Kk Hikj (φ )ζ λk + Continuous part
(4.40)
k=1
where σi j is the stress tensor, ζ = r/h is a non-dimensional radial distance from the
singularity, H k i j are functions depending on the geometry angle φ of the corner, Kk
constitutes generalized stress intensity factors, and λk defines the strength of singularities governed by the eigenvalues of the λk in the open interval Re (λk ) < 1.
Due to restrictions on finite strain energy at the singularity, values of Re (λk ) ≥ 1
are not allowed. Obviously, based on Eq. (4.40), more than one singular term may
influence the stress field close to an interface corner. Dempsey (1995) showed that
in addition to the power type singularities, i.e. 0(r−λ ) as r → 0, and λ representing the strength of singularity, power-logarithmic type singularities, i.e. 0(r−λ ln r)
as r → 0, may occur for homogeneous boundary conditions of composite wedge
problems.
Stress singularities at butt joint interface corners were discussed Reedy and
Guess (1995) and Sawa et al. (1995).
4 Complex Constitutive Adhesive Models
109
4.5 Bulk Adhesive as Composite Material
Since most adhesives contain secondary phase materials such as fillers and carrier
cloth, they should be treated as composite materials in constitutive modeling. The
interesting aspect in such treatment is the fact that such secondary phase constituents
form adhesive bonds with the adhesive matrix in small scale. Consequently, localized non-catastrophic failures within the bulk adhesive consisting of interfacial separations are expected to cause constitutive behavior changes for the adhesive bulk.
Such behavior can be properly described by using “damage mechanics models”
some of which will be described later in this chapter within the context of adhesive
materials.
It is known that particle size, shape and volume fraction all affect the mechanical
behavior of filled polymers. For example, electrically and thermally conductive adhesives which usually contain dispersed metal particles of varying aspect ratios, but
usually within the micron size range, require the methods of Eshelby (1957, 1959)
for characterization. We note that these types of adhesives sometimes require directional properties. Kerner (1956), Hashin and Shtrikman (1963), Budiansky (1965)
and Hill (1965) provided methods for calculation of elastic moduli for isotropic materials filled with spherical particles. Methods for elastic moduli of unidirectional
reinforced polymers with infinitely long fibers were given by Hill (1964) and Chow
and Hermans (1969). Bulk and shear moduli for disk and needle shaped inclusions
were considered by Wu (1966) and Walpole (1969).
As reported by Ashton et al. (1969), Halpin and Tsai introduced approximate
equations for square fiber reinforcement by reducing Hermans’ solution (1967) using numerical solutions of elasticity theory. Equations for slender rigid inclusions
at low concentrations were developed by Russel and Acrivos (1972, 1973). As for
effective thermal expansion coefficient of filled polymers, the effects of filler geometry, and constituent material properties have been studied by Kerner (1956).
Using the generalized approach of Eshelby, Chow (1977, 1978-a) derived relations for the bulk modulus, transverse and longitudinal Young’s moduli and, the two
effective shear moduli, G12 and G13 of the filled polymer containing aligned ellipsoidal particles at finite concentration. Chow (1978-b) also derived relations for the
linear and volumetric thermal expansion coefficients for filled polymers containing
aligned ellipsoidal inclusions at finite concentration. He concluded that in such filled
systems, the volumetric expansion varies only slightly with the aspect ratio of the
ellipsoid, while longitudinal and transverse linear expansions show strong dependence on the particle shape. All the modulus calculations developed by Chow are
based on the following assumptions: (i) The filler particles are well bonded to the
matrix, and consequently, the isostrain condition at the particle/matrix interface can
be used to develop the method; (ii) The shapes of particles are uniform ellipsoids
with the aspect ratio Cρ , representing the ratio of the major to minor axes; (iii) The
particles are distributed with their corresponding axes aligned.
Wei (1995) and Sancaktar and Wei (1998) applied the elastic moduli and thermal
expansion coefficient relations developed by Chow in describing the material
110
E. Sancaktar
behavior of electronically conductive (filled) adhesives in order to be able to
determine, accurately, the thermal mismatch stresses which occur between the adhesive and the substrates in the bonded form. Both finite element analysis and closed
form solutions were used for stress analysis.
We note that the results given by Chow are consistent with the well known rule
of mixtures used in long fiber reinforced composites. When the aspect ratio, Cρ , of
the particle is Cρ > 50, the moduli are not a function of Cρ . Therefore, for larger
Cρ the composite’s behavior is more like that of long fiber reinforced composite.
For intermediate aspect ratio cases, i.e. 1 < Cρ < 50, Sancaktar and Wei (1998) proposed an approximation of Chow’s elastic moduli formulas by utilizing the method
used for randomly oriented short fibers in predicting the behavior of particle filled
adhesives, so that the composite material can be treated as isotropic, as suggested
by Agarwal and Broutman (1990).
As for the thermal expansion coefficient, the isotropy assumption was accomplished simply by using one-third of the volumetric thermal expansion coefficient,
as suggested by Lipatov et al. (1991). We note, however, that in special cases of
conductive adhesive applications, directional conduction may be required, in which
case the use of non-isotropic, directional material properties would be necessary.
The constitutive models cited above for predicting filled polymer behavior all
assume perfect adhesive bond between the fillers and the polymer matrix. As such
filled polymers are loaded, however, failure of adhesive bonds is expected in proportion to the applied force and the duration of its application. This, in turn, is expected
to result in changes in the constitutive properties of the filled polymer adhesive.
Lipatov et al. (1991) reported a decrease in the modulus of elasticity of a filled
elastomer at separation of filler particles from the binder and described it with the
approximate formula
(4.41)
Ex /E f = exp−4.3 (Vx )
where E f is the initial elastic modulus for the filled polymer with unbroken bonds, Vx
is the volume fraction of unbonded filler, and Ex is the composite’s elastic modulus
when the fraction, Vx , of the filler particles are no longer bonded to the matrix.
Garton et al. (1989) discussed stiff polar molecular additives which interact
strongly with the (thermoplastic or thermoset) matrix polymer adhesive, thus decreasing the composite free volume, f , as given by the equation
f = V1 f1 +V2 f2 + k V1V2
(4.42)
where V1 and V2 are the volume fractions of the polymer matrix and additive, respectively, f1 and f2 are the corresponding fractional free volumes, and k is an interaction
parameter. The composite properties: bulk elastic modulus, bulk strength (to a lesser
degree) as well as bonded joints properties of strength and stiffness are reported by
Garton et al. (1989) to increase with reductions in composite free volume.
In another study, Sancaktar and Kumar (1999-a, 2000) obtained increases in
the lap joint strength by selectively rubber toughening the high stress concentration regions of the adhesive overlap. This novel approach to increase the lap joint
strength by using functionally gradient adhesive introduced a method different from
4 Complex Constitutive Adhesive Models
111
the traditional methods of either increasing the lap joint area or altering the joint geometry. For the different configurations of adhesive arrangements tried, the trends
obtained in finite element analysis matched the trends obtained by tensile shear testing. The use of such “functionally gradient” adhesives was later adapted by other researchers such as Pires et al. (2003, 2006), You et al. (2005), and Temiz (2006) under
the name “bi-adhesive”, and by Hu and Huang (2005), da Silva and Adams (2006)
and Lei et al. (2008) under the names “mixed-adhesive” or “mixed-resin”.
4.6 Adhesively Bonded Joint Composite, the Concept
of the Interphase
There is strong interrelationship between the properties of the adhesive and the substrates in shaping the final joint behavior. A more accurate prediction of the joint
behavior can be obtained by including a distinct “interphase” region in modeling
the overall joint behavior. An early discussion of this concept was provided by
Sharpe (1972, 1974). This concept becomes especially relevant as efficient methods for inducing mechanically distinct surface topography and chemically distinct
interlayers emerge. The use of excimer pulse lasers for surface treatment and topography inducement on metal adherends, shown by Sancaktar et al. (1995-a), cure
and post-cure of interphase regions around carbon fibers embedded in epoxy matrix
via resistive electric heating by the fiber, shown by Sancaktar and Ma (1992-a), microwave curing of carbon-epoxy composites, which produce “interfacial strengths
higher than that achievable in thermally cured system”, as shown by Agrawal and
Drzal (1989), and the production of a transcrystalline region in crystalline polymers
upon solidifying when in contact with a substrate, as shown by Hata et al. (1994),
all provide examples of current methods used to create distinct interphases. Another
main reason for the presence of an interphase region is the chemical primer (sizing)
applied to the adherend surfaces to improve adhesion. The diffusion of this sizing
into the adhesive matrix can create a concentration gradient. There are other possible mechanisms which could lead to an interphase region. Some examples are:
possible polymeric diffusion into the substrate during the adhesion process, selective adsorption of one or more of the components in the matrix before curing and
the presence of free volume differences between the bulk polymer and the polymer
near the substrate-matrix boundary, possibly created by thermal stresses. Figure 4.7
shows different “interphases” between an epoxy-based, approximately 80%wt silver
flake filled electrically conductive adhesive and different substrates.
Adequate analysis and understanding of the interphase between an adherend
and an adhesive is critical for design of efficient bonded structures and composite materials. In adhesion science and technology circles it is a well accepted
fact that the mechanical properties of the (polymeric) adhesive material is altered in regions close to the adherend due to the adhesion process. For example,
Knollman and Hartog (1985) reported approximately 10% reduction in the adhesive bulk shear modulus, in an interphase region comprising approximately 20% of
112
E. Sancaktar
Piezoelectric Device
Gold Layer
Electrically Conductive Adhesive
Stainless Steel Substrate
Fig. 4.7 Illustration of different “interphases” between an epoxy-based, approximately 80 %wt
silver flake filled electrically conductive adhesive and different substrates. The flake size is
approximately 7 μm
the bulk adhesive thickness. Consequently, a satisfactory mechanical analysis on the
adherend-adhesive interaction should include a finite thickness interphase with its
own material properties rather than assuming an interface with unknown properties
or very high strength and negligible thickness. Modeling of the adhesive-adherend
interphase becomes especially relevant in understanding and design of composite
materials with “tailored” interphases which increase the toughness of the composite
material.
Using the case of a carbon fiber embedded in epoxy matrix, Sancaktar and
Zhang (1990) and Sancaktar et al. (1992-b) reported that nonlinear viscoelastic behavior of the matrix and the interphase may have a profound effect on the mechanism of stress transfer.
Sancaktar’s analytical model involves a cylindrical interphase surrounding an
elastic fiber of finite length. The interphase region has nonlinear viscoelastic material properties. The region surrounding the interphase (which can be assumed to be
the bulk of the adhesive matrix) is also assigned nonlinear viscoelastic properties,
which are different from those of the interphase. The nonlinear viscoelastic material
behavior of the interphase zone and the matrix is represented using a stress enhanced
creep. Different interphase diameter values and different material properties for the
interphase and the matrix are used to determine their effects on the interfacial shear
stress. Results of this analysis reveal that depending on the relative magnitudes of
interphase and matrix viscoelastic material properties, it is possible to have stress
reductions or increases along the fiber and, hence, the critical fiber length obtained
will also vary accordingly, as illustrated in Fig. 4.8. This result reveals that not only
the quality of adhesion but also the material properties of the bulk matrix and the interphase, all of which can be affected by cure conditions, can affect the critical fiber
length, a concept introduced by Kelly and Tyson (1965) to gage the fiber-matrix
interfacial strength.
The analytical and experimental results presented so far reveal the possibility for
mutual reduction between rate, temperature, and interphase thickness, as shown by
Sancaktar et al. (1992-b). If we assume that application of fiber sizing results in
thicker interphases, and that the cross-head rate, v, can be related to the shear strain
rate, d γ /dt, and the interphase thickness, δ , with the relation:
4 Complex Constitutive Adhesive Models
113
Fig. 4.8 The variation of the maximum shear stress with time and matrix transient creep compliance based on a nonlinear viscoelastic stress analysis of a cylindrical interphase zone for maximum
transferred load. The nonlinear analysis was performed using Eq. (4.31) with a stress-dependent
shift factor. D0 and D1 are the interphase instantaneous and transient creep compliances, δ is the
interphase thickness, and the subscripts m and f indicate matrix and fiber, respectively (Sancaktar
and Zhang, 1990)
v= f
dγ
dt
g(δ )
(4.43)
based on classic shear lag analysis (also based on definition of shear strain), we
can then argue that for a fixed strain rate (function) the effect of increased crosshead rate can be induced by increasing the interphase thickness (i.e., application
of fiber sizing). Indeed, the experimental results reveal shorter fragment lengths for
increased cross-head rate and/or presence of fiber sizing. Based on this premise we
can write the following superposition relations for the interfacial strength:
τc (T, v, δ ) = τc (To , aT v, δ )
τc (T, v, δ ) = τc (To , v, δ /aT )
(4.44)
(4.45)
τc (T, v, δ ) = τc (T, v/aδ , δo )
(4.46)
and
In order to represent the composite behavior of the adhesive/adherend interphase,
where the polymeric adhesive is mechanically interlocked to the surface projections
114
E. Sancaktar
of a metal adherend, Sancaktar (1996) proposes a “smeared” modulus zone. Although deformation within the bulk adhesive is likely viscoelastic, simple elastic
relationships probably suffice, to describe deformation within the topographic surface region. Therefore, simple elastic relationships are used to relate strain gradient
within those regions to topography.
The equivalent elastic modulus, GIP , for the mechanically interlocked interphase
is calculated by assuming an “isostress” condition for the interphase, illustrated in
Fig. 4.9. In other words, we assume that the adhesive and the adherend surface
(projections) are subjected to the same level of shear stress, τIP , at the interphase,
while the composite interfacial shear strain, γIP , is a volumetric weighted average
of the adhesive and adherend strains, i.e.
γIP = (V f A )(γA ) + (V f S )(γS )
(4.47)
In Eq. (4.47) γA and γS are the adhesive and substrate shear strains at the interphase
and V f A and V f S are the volume fractions of the adhesive and the substrate surface
(projections) at the interphase. Assuming that the adhesive essentially fills the voids,
the adhesive volume fraction can be equated to the sum of the void volume fractions
of each lamina.
With the assumption that the deformations in both the adhesive and the adherend
are elastic, we can define the interphase shear modulus as:
GIP = τIP /γIP
Fig. 4.9 The state of isostress shear at the interphase
(4.48)
4 Complex Constitutive Adhesive Models
115
Using Eqs. (4.47) and (4.48) and Hooke’s law to represent adhesive and adherend’s
material behavior, we obtain the following expression for the interphase shear
modulus:
GS GA
GS GA
=
(4.49)
GIP =
V f A GS + V f S GA V f S (GS − GA ) + GA
where GA and GS are the elastic moduli for the adhesive and the adherend, respectively.
We now assume that failure at the interphase is more likely with higher strain gradients, dγIP /dz. Of course, in the limit, a strain discontinuity defines failure based
on mechanics principles. Based on this premise, we can derive a mathematical expression to gage the possibility of failure using the shear moduli of the adhesive
and the adherend along with the adhesive volume fraction, V f A , at the interphase.
For this purpose, we first note that differentiation of Eq. (4.48) along the interphase
thickness direction (z-direction) results in
d γIP
dGIP
−2
= − τIP γIP
(4.50)
dz
dz
An equivalent expression can be obtained by differentiating Eq. (4.49) with respect
to z:
dV f A
dGIP
GS GA (GS − GA )
=−
(4.51)
dz
dz
[V f A (GS − GA ) + GA ]2
Equating Eqs. (4.50) and (4.51) we can obtain an expression relating the shear strain
gradient, dγIP /dz at the interphase to the volume fraction, V f A , and the differential
volume fraction dV f A /dz, of the adherend voided during surface preparation. Both
parameters are experimentally determinable, as discussed by Sancaktar (1996).
Variation in mechanical properties of polymer transition layers was also considered by Silberman et al. (1991). An approach involving macroscopic characterization of the interphase regions in polymer based composite systems by integrated
moduli over the thickness is given by Cardon et al. (1993). A three-phase, axisymmetric elasticity solution to predict the sensitivity of the stress state to the interphase
material properties and temperature is presented by Sottos et al. (1994).
In order to overcome the difficulties arising from the geometrical complexity
of bonded joints, a number of studies have been carried out by Adams and Peppiatt
(1974), Cooper and Sawer (1979), Hashim et al. (1990), Groth and Nordlund (1991),
Dorn and Weiping (1993) and Papini et al. (1994), using the finite element method.
Lap and butt joints have been analyzed extensively due to their common usage in
engineering applications.
Interfaces usually constitute a weak link in the chain of load transfer in bonded
joints. Also, the discontinuity of the material properties causes abrupt changes in
stress distribution, as well as causing stress singularities at the edges of the interfaces. It is very desirable to optimize the substrate surface topography at the interfaces to maximize the load bearing capacity of bonded joints, and to improve their
deformational characteristics.
116
E. Sancaktar
During the second half of the twentieth century a considerable amount of research
work has been published related to the interface stress distributions, including Bogy
(1968, 1970, 1971-a) and Bogy and Wang (1971-b), Renton and Vinson (1977),
Mittal and McKenzie (1980), Sih (1980), Okajima and Sinclair (1986), Suhir (1988),
Sancaktar (1987-b, 1996), Groth (1988), Adams (1989), Bigwood and Crocombe
(1989), Sawa et al. (1989, 1995), Ikegami et al. (1990), Temma et al. (1990,
1991), Nakano et al. (1991), Carpenter (1991), Gao and Mura (1991), Keisler and
Lataillade (1995), Apalak et al. (1995), Maugis (1996) and Ganghoffer and Schultz
(1996). These studies focused on flat surfaces between two different materials except when the effects of surface roughness were investigated. In general, stress distributions, stress concentrations, singularities, and surface roughness effects were
studied. These studies, however, were not extended to the development of a general
methodology to include surface topography effects for varying configurations which
can be represented by different mathematical functions.
An investigation by Sancaktar and Narayan (1999-b) on adhesively bonded scarf
joints revealed that the angle between the bonded interface and the loading direction
had a significant influence on the stress distributions at the interface. Their analysis
revealed that the distributions of stresses became more uniform along the adhesive
layer when scarf angles varying from lap joint to butt joint in an increasing order
of scarf angle were considered. The transverse and shear stress distributions were
in a decreasing order for scarf angle increments of 15◦ between 30◦ and 90◦ , while
a reverse trend was observed for the normal stress distributions. The peak normal
stress was minimum for the 30◦ scarf joint; on the other hand, the magnitudes of
transverse and shear stresses were minimum when a butt joint was used. Uniformly
distributed external shear loading on the joints led to higher stresses in lap joints,
while 75◦ scarf joint had the lower values.
Sancaktar and Narayan (1999-b) stated that an attention to the type of stress
(i.e., normal, transverse or shear) provided information relevant to the type of failure (i.e., brittle or ductile type failures). The authors also introduced newly defined
design parameters: stress times substrate volume, stress divided by substrate volume, and stress (strain) gradients along and across the adhesive joint, as a general
methodology to compare adhesive joints for design optimization.
In highly deformable adhesives, most of which are elastomeric in nature, such
as those used in pressure sensitive tape and tack applications, a high amount of
deformation results in cavitation of the material under dilatation. High dilatation,
which is a 3-D concept, may lead to cavitation in localized regions of the adhesive
layer starting at a microscopic scale.
It is also desirable to know the magnitudes of shear and normal stresses individually since some adhesives are weak in shear loading (ductile adhesives) while others
are weak in normal stress loading (brittle adhesives).
High levels of stresses result in crack propagation in brittle adhesives and cavitation induced failures in deformable adhesives. Furthermore, not only the magnitudes
but also gradients of stresses and strains become an important issue in optimizing
the joint strength. Large strain gradients at adhesive-substrate interfaces may result
in bond failure.
4 Complex Constitutive Adhesive Models
117
The magnitude of strain gradients depends on the volume fractions of the adhesive and substrate materials within the effective bond region defined by the geometry
of the surface topography. The effect of bonded substrate volume, corresponding to
the volume of substrate projections beyond a common base-line on the substrate, on
the strain gradients across the adhesive layer can be found by relating the volume
fraction gradients to the strain gradients. For this purpose, the composite behavior
of the adhesive/adherend interphase, where the polymeric adhesive is mechanically
interlocked into the surface projections of an adherend can be utilized to determine
an equivalent composite modulus, as suggested by Sancaktar (1996) and discussed
above.
We note that mechanical adhesion depends on surface topography, which can be
considered a collection of many geometrical forms. Therefore, mechanical adhesion
depends on the stress states of different adhesive joint geometries on the scale of the
surface topography, which may include many lap, butt and scarf joints in the “interphase” region. To address this issue, Ma et al. (2001) compared the stress distributions in adhesive joints as functions of varying geometrical interfaces described
mathematically in polynomial or other functional forms, as well as the material
properties of the adhesive and the adherend or two different substrates joined by an
infinitesimally thin adhesive layer. For this purpose, a mathematical procedure was
developed to utilize the complementary energy method, by minimization, in order to
obtain an approximate analytical solution to the 3-D stress distributions in bonded
interfaces of dissimilar materials. In order to incorporate the effects of surface topography, the interface was expressed as a general surface in Cartesian coordinates,
i.e., F(x, y, z) = 0, and the results verified by comparison with Finite Element Analysis (FEA). Their methodology involved the following steps: 1. Choose the appropriate stress functions, Φmn , to satisfy stress boundary conditions and utilize equilibrium conditions to relate the stress functions to the stress distribution functions;
2. Use stress interface boundary conditions to reduce the unknown variables in the
stress distribution functions obtained; 3. Define a function Π as the sum of compley= f (x,z)
mentary energy I and the penalty function R
∑
i=1..3
(ui A − ui B )2 . R is chosen such
that it is the largest positive number which causes the total system complementary
energy to change by only 0.1%; 4. Use the remaining unknown variables to minimize the function Π formed in step 3, using the Newton quadratic method, and solve
all the unknown variables to obtain the complete stress distribution functions.
The continuity of displacements was enforced at the interface area, approximately, using the fundamental theorem for surface theory of differential geometry,
described by Struik (1950), instead of directly using displacements. This way, the
integrations of displacements were avoided, and the stress distributions could be
calculated more accurately.
For example, for a flat interface (on x-z plane), such as a butt joint, they utilized
the following stress functions:
ΦAxx = (z2 − 1/4)2 (y − 1/2)2 f1 (x, y, z)
(4.52)
118
E. Sancaktar
ΦAyy = (z2 − 1/4)2 (x2 − 1/4)2 f2 (x, y, z)
ΦAzz
ΦAxy
ΦAxz
ΦAyz
ΦBxx
ΦByy
ΦBzz
ΦBxy
ΦBxz
ΦBzz
(4.53)
= (x − 1/4) (y − 1/2) f3 (x, y, z)
(4.54)
= (x − 1/4)(z − 1/4) (y − 1/2) f4 (x, y, z)
(4.55)
= (x − 1/4)(z − 1/4)(y − 1/2) f5 (x, y, z)
(4.56)
= (x2 − 1/4)2 (z2 − 1/4)(y − 1/2) f6 (x, y, z)
(4.57)
= (z − 1/4) (y + 1/2) f7 (x, y, z)
(4.58)
= (z − 1/4) (x − 1/4) f8 (x, y, z)
(4.59)
= (x − 1/4) (y + 1/2) f9 (x, y, z)
(4.60)
= (x − 1/4)(z − 1/4) (y + 1/2) f10 (x, y, z)
(4.61)
= (x2 − 1/4)(z2 − 1/4)(y + 1/2)2 f11 (x, y, z)
(4.62)
= (x − 1/4) (z − 1/4)(y + 1/2) f12 (x, y, z).
(4.63)
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
Functions fn (x, y, z) are of the type: gn1 sinh(x) sinh(z) + gn2 sinh(x) cosh(z) +
gn3 cosh(x) sinh(z)+gn4 cosh(x) cosh(z) where, gni are polynomials of hni1 +hni2 x +
hni3 z, and, hni1 , hni2 and hni3 are polynomials of ani j1 + . . . ani jm ym−1 . The selection
of sinh x, cosh x, sinh z, cosh z, and y facilitated numerical calculations.
A typical distribution for the normal stress, σyy , common to both substrates A
and B at the bonded interface of a three-dimensional flat interface between these
substrates is shown in Fig. 4.10. This stress is calculated by the novel mathematical
procedure based on differential geometry and the penalty function as described by
Ma et al. (2001), and summarized above. Figure 4.11 shows the transverse stresses
Interface
A
Y
B
Z
Fig. 4.10 A three-dimensional flat interface between two substrates A and B, loaded as shown and
analyzed by FEA with the mesh pattern shown (left). A typical distribution for the normal stress,
σyy , common to both substrates A and B at the bonded interface is shown on the right, as calculated by a novel mathematical procedure based on differential geometry and the penalty function
(Ma et al., 2001)
4 Complex Constitutive Adhesive Models
119
Fig. 4.11 The transverse stresses σxx , on substrates A (left) and B (right) at the bonded interface, illustrating the discontinuity of this stress across the interface, as sometimes called the “stress jump”,
calculated by a novel mathematical procedure based on differential geometry and the penalty function (Ma et al., 2001)
σxx , on substrates A and B at the bonded interface, illustrating the discontinuity of
this stress across the interface, as sometimes called the “stress jump”. Again, these
stresses are calculated by the novel mathematical procedure based on differential
geometry and the penalty function as described by Ma et al. (2001), and summarized
above.
4.7 Damage Models
The incremental theory of plasticity assumes that the strains depend on the entire
history of loading. The total increments of deviatoric strain Δεi j is composed of a
viscoplastic component
(4.64)
Δεi j P = d λ Si j
where d λ is a variable depending on the loading history, and an elastic component
d εi j E = (dSi j /dt)/2G
(4.65)
d εi j /dt = d εi j E /dt + d εi j P /dt
(4.66)
so that
Obviously, a yield criterion is needed to determine the actual magnitudes of plastic strain increments. This theory was first proposed by Prandtl (1925) and Reuss
(1930).
Several constitutive forms were proposed to describe the viscoplastic term of
Eq. (4.66). Among these Hohenemser et al. (1932), Sokolovsky (1948), Malvern
(1951) and Perzyna (1963) proposed essentially the same idea. It involved the assumption that the viscoplastic component be a function of the over-stress above the
120
E. Sancaktar
yield point. On the basis of his experimental observations, Sokolovsky determined
that the over-stress should be defined above the elastic limit, as a function F(σ − θ ),
so that
ε = σ /E,
ε = (σ /E) + F(σ − θ )/E,
σ ≤θ
σ > θ.
(4.67-1)
(4.67-2)
The over-stress idea can be incorporated into the viscoelastic mechanical models by
means of adding sliding elements to describe yield or termination points. With such
an application, equations containing viscosity terms are obtained in a form similar to (4.67). For example Brinson et al. (1975), Renieri et al. (1976), Sancaktar
and Brinson (1979, 1980), Sancaktar (1981), Sancaktar and Padgilwar (1982),
Sancaktar et al. (1984) and Sancaktar and Schenck (1985-a) used the Modified Bingham model developed by Brinson (1974) to describe the shear and tensile material
behavior of structural adhesives in the bulk and bonded forms:
σ = Eε
d ε /dt = (d σ /dt) + (σ − θ )/μ
σ =Y
σ ≤θ
θ <σ ≤Y
σ ≥ Y.
(4.68-1)
(4.68-2)
(4.68-3)
The model describes linear elastic behavior below the elastic limit θ , linear viscoelastic behavior between θ , and the maximum stress Y and perfectly plastic behavior above Y . When the sliding element is activated at σ = θ , the model becomes
essentially a Maxwell element, as illustrated in Fig. 4.12.
A similar application was also reported first by Chase and Goldsmith (1974) and
later by Sancaktar (1981), Sancaktar and Padgilwar (1982), Sancaktar et al. (1984),
and Sancaktar and Schenck (1985-a), in modifying a three parameter solid model
with a sliding element to describe the mechanical behavior of polymeric materials
and adhesives.
An isothermal theory of separation in (thermoplastic or pressure sensitive)
polymer-solid adhering systems based on drawing of filaments is given by Good
and Gupta (1988).
Ganghoffer and Schultz (1996) stated that the influence of damage on the mechanical behavior is specified through the concept of strain equivalence. They propose that the constitutive law for the damaged material is given by that of the virgin
material if the stress tensor σ is replaced by the effective stress tensor σ /(1−d) with
the term, d, representing the extent of damage between the virgin state, d = 0 and
complete failure, d = 1. The damage parameter is further developed in Chapter 6.
Another example of empirical damage modeling in joints bonded using silver
flake filled electrically conductive adhesives was given by Gomatam and Sancaktar
(2006-a,b), who introduced a comprehensive fatigue life predictive model for single
lap joints subjected to variable loading. For the purpose of formulating a fatigue
life predictive model, a linear relationship is established first between the maximum cyclic load Pmax (or maximum principal stress, σmax , as suggested by Gomatam, 2002, and Gomatam and Sancaktar, 2004), and the number of cycles N:
4 Complex Constitutive Adhesive Models
121
Stress
σ
Modified Bingham Model
Constant Strain Rate (C.S.R.)
σ=Y
Plastic
V.E.
θ= Elastic Limit Stress
σ=θ
Maxwell behavior: when θ ≤ σ ≤ Y
Perfectly Plastic behavior: when σ ≥ Y
E
Elastic
1
Strain,
θ
E
σ
Y
µ
Stress
σ(t)
σ=Y
Maxwell
σ (t) ≤ θ
σ ( t) = E ⋅ε
Modified Bingham
σ=θ
Axis Shift for
Maxwell Model
(
⎡
σ (t) = θ + μ R⋅ 1 − e
⎢⎣
E
) T⎤
− t− t 0
1
Strain,
⎦⎥
θ<
σ (t) ≤ Y
Fig. 4.12 The Modified Bingham model and its comparison with the Maxwell model
Pmax = C + m N
(4.69)
where C and m are, respectively, the intercept and slope for this linear relation.
For the purpose of the analytical fatigue life predictive model, two separate analytical equations were proposed depending on the intensity (severe or moderate) of
the load-ratio first applied, at a maximum cyclic load level. The total fatigue life,
NTotal , under severe-to-moderate condition was predicted by the relation,
1
ΔC
1
ns
NTotal = ns +
− (Cs − P)
−
1−
(4.70)
mM
ms mM
Ns
The total fatigue life under moderate-to-severe condition, on the other hand, was
predicted by,
nM
NTotal = nM + Ns 1 −
(4.71)
NM
In Eqs. (4.70) and (4.71) we have,
ΔC = CM −Cs
where CM and Cs are the intercepts, and mM and ms are the slopes for the linear relationships between the maximum cyclic load Pmax (or maximum principal
stress, σmax ), and the number of cycles N under moderate and severe conditions,
122
E. Sancaktar
respectively. nM and ns are the number of cycles applied under moderate and severe
conditions, respectively.
As illustrated in Fig. 4.13, in going from moderate to severe conditions, we simply add to the number of cycles under moderate conditions (nM ), the predicted remaining life for the sample under the severe condition, Ns [1 − (nM /NM )], based on
experimental predictions without making any slope corrections, Eq. (4.71). In going from severe to moderate condition, however, we also make a slope correction,
[Cs − P]{(1/ms ) − (1/mM )}[1 − (ns /Ns )], to make the severe and moderate P-N
curves parallel, and shift the intercepts to add the remaining life, Δ C/mM , to that already used up under severe condition (i.e. ns ). These particular methodologies were
chosen based on experimental observations. Utilizing the above equations, the total
fatigue life of the joints subjected to varying loading conditions, namely severe-tomoderate, and moderate-to-severe were computed for a set of test cases, in addition
to experimentally determined values, for the purpose of model validation. Comparison between the experimental, and predicted values validated the accuracy, and
efficiency of the proposed model.
In order to render the model usable with different stress components (σxx , σyy ,
σzz , τxy , τyz, τxz ) instead of the maximum cyclic load values, Gomatam (2002) utilized the maximum and minimum principal and von Mises stresses obtained from
non-linear elasto-plastic finite element analyses to calculate the ratios of peak stress
values obtained at corresponding maximum load levels. These stresses were then
compared to the intercept ratios obtained from the P-N curves at given environmental conditions. This comparison procedure revealed the maximum principal stress as
the appropriate parameter to be used in the form of S-N curves, i.e.,
σmax = C + mN
mm, ms = slopes;
nm, ns = number of cycles,
under moderate, m, or
severe, s, conditions.
Cm
Δ(C) = Cm – Cs
Cs
S
(4.72)
P
Δ (C)/m m
(M)
(S)
ns
(1 – ns/N s)
Ns
N
1) Severe to Moderate: NTotal = ns + (Δ(C)/mm) – (Cs–P) ((1/mm)) (1–(ns/Ns))
2) Moderate to Severe: NTotal = nm + Ns(1–(nm/Nm))
Fig. 4.13 Illustration of the cumulative fatigue damage model (Gomatam and Sancaktar, 2006-a)
4 Complex Constitutive Adhesive Models
123
where C and m are, respectively, the intercept and slope values based on maximum
principal stress. The mathematical basis for the maximum principal stress to replace
the maximum cyclic load in Eq. (4.69) was illustrated by performing fatigue tests
under three different maximum load, Pmax , conditions, and considering the following relations:
C3
C3
=
(4.73)
C1 C1
and
C2
C2
=
C1 C1
(4.74)
where C1 , C2 , C3 relate to Eq. (4.69), and C1 , C2 , C3 relate to Eq. (4.72), as illustrated
in Fig. 4.14.
In order to account for environmental effects, such as elevated temperature and
humidity, a superposition method was applied for fatigue life prediction by shifting
the slopes and the intercepts between different environmental conditions. For this
purpose, design charts were formulated based on experimental data. These design
charts comprise of two parts, one for shifting the slope, and the other for shifting
the intercept between different environmental conditions. Once the slopes and the
intercepts are determined, Eqs. (4.69) and (4.72) would yield the fatigue life of the
joint at the new environmental condition, as illustrated in Fig. 4.15 and shown by
Gomatam and Sancaktar (2006-b).
The proposed model was also extended to bonded cases with varying stress states.
For this purpose, a non-linear elasto-plastic finite element analysis (FEA) for ambient condition, and a non-linear thermo-elasto-plastic FEA for elevated temperature
conditions were performed using two different lap joint geometries and boundary
conditions to represent two distinct states of stress. Utilizing the FEA results, relations between the stress components, slopes, and intercepts were established. The
intercepts were found to be inversely proportional to the maximum stresses, and
the slopes directly proportional to a shear stress component. Thus, by using these
relations along with a knowledge of the stress states for only two joint configurations, the slope, and intercept for any other lap joint with unknown P-N (σ − N)
Pmax = C’ + m’N
σmax = C + mN
C’3
C1
C’2
C2
PmaxC’1
σmax C
3
(MPa)
(N)
C’3/C’3 = C3/C1
N
N
Fig. 4.14 The similarity in P-N and σ − N relations (Gomatam and Sancaktar, 2006-a)
124
E. Sancaktar
Hum 90°C 50°C
Stress
Ratio
Hum
RT
90°C 50°C RT
Stress
Ratio
Slope
Intercept
Fig. 4.15 The shifting mechanism implemented for the purpose of predicting fatigue life of joints
subject to a stress ratio (R), and maximum cyclic load of Pmax , from one test environment (28◦ C,
or 50◦ C, or 90◦ C) to another at the same stress ratio, and maximum cyclic load (Gomatam and
Sancaktar, 2006-a)
behavior could be computed, from which the fatigue life of the joint could be predicted, as shown by Gomatam and Sancaktar (2006-b). Fatigue and environmental
degradation are further discussed in Chaps. 7 and 8.
4.8 The Effects of Cure and Processing Conditions
on the Mechanical Behavior
Cure and processing conditions have strong effects on the resulting mechanical
properties of structural adhesives in the bulk and bonded forms.
Variation of various mechanical properties of an epoxy adhesive with or without
a carrier cloth as functions of cure temperature, time, and cool down conditions,
and their effects on bulk tensile properties were studied by Sancaktar et al. (1983),
Jozavi and Sancaktar (1989-a, b, c) and Sancaktar (1995-b), with the work by Jozavi
and Sancaktar (1989-c) providing information on the effects of cure conditions on
the relaxation behavior of bulk thermosetting adhesives. A comparison of the degree
of cure with bulk strength was given by Jozavi and Sancaktar (1988). The effects
of the cure parameters on the stress whitening behavior were given by Jozavi and
Sancaktar (1989-a, b). The effects on the bonded joint behavior were discussed by
Shaw and Tod (1989), Sancaktar and Ma (1992-a) and Turgut and Sancaktar (1992).
In general, mechanical properties are improved with increases in cure temperature
and time and with slow cool down. Such increases, however, have an upper limit,
and further increases in cure parameter magnitudes lead to deterioration in adhesive mechanical properties. Consequently, in general, the variation of mechanical
properties with the cure parameters are bell shaped functions.
Additional information on the effects of cure conditions were provided by
Sinclair (1992).
4 Complex Constitutive Adhesive Models
125
Matsui (1990) provides some empirical relations for the variation of shear modulus and strength of joints bonded using various structural adhesives.
In the case of particle filled adhesive systems, such as electrically conductive adhesives, the viscosity, μ , during processing and cure is a function of filler volume
fraction, Φ, shear rate, d γ /dt, and resin conversion level, α . The development of
viscosity from a chemorheological point of view is important as it relates to the
wetting and diffusive characteristics of the adhesive during its cure. These characteristics ultimately determine the viability of the interphase and, therefore, the overall adhesive bond. A comprehensive model may be represented by combining the
following models: the power law (shear rate effect), described by Wildemuth and
Williams (1985), Castro-Macosko model (conversion effect), described by Castro
and Macosko (1980, 1982) and Castro et al. (1984), and Liu model (filler volume
fraction effect), described by Liu (2000). Such a combined model takes the form:
μ (φ , d γ /dt, α ) = A · (φm − φ )−2 · (d γ /dt) C0 +C1 ·α · αgel
αgel − α
D+E·α
(4.75)
where, A, C0 , C1 , D and E are model constants that can be determined by multivariable nonlinear regression analysis of isothermal data. The maximum filler volume
fraction, Φm and conversion at gel point, αgel , values are usually known for typical
thermosets.
Zhou and Sancaktar (2008) incorporated the isothermal temperature effects to
obtain a general-form comprehensive model in the form:
μ (T, φ , d γ /dt, α ) = A · exp (B/T ) · (φm − φ )−2 · (d γ /dt)C0 +C1 ·α
· αgel
αgel − α
D+E·α
(4.76)
where, B is the activation temperature and other coefficients are same as in Eq. (4.75).
In order to model realistic adhesive processing, usually with a broad temperature range, nonisothermal tests usually need to be undertaken. The comprehensive
isothermal chemoviscosity model, based on Eq. (4.76), can then be extended to
nonisothermal temperature cure cycle through nonlinear regression analysis of nonisothermal data.
In fact, nonisothermal temperature cure is a different process from isothermal
cure. The reaction kinetics, total reaction order, and even the reaction energy for
epoxy systems may not be constant and same, but process-dependent. Therefore,
modifications need to be made to reflect the effects of such a difference.
An improvement in the model predictability is expected by allowing parameters D and E in Eq. (4.76) to change with temperature during nonisothermal cure.
Furthermore, both of these parameters also show some variation with the volume
fraction (Φ) and shear rate (d γ /dt). A modified comprehensive model was thus proposed as follows (Zhou and Sancaktar 2008):
μ (T, φ , d γ /dt, α ) = A · exp B T · (φm − φ )−2 · (d γ /dt)C0 +C1 ·α
· αgel
αgel − α
D∗ +E ∗ ·α
126
E. Sancaktar
where:
D∗ = D0 + D1 · (d γ /dt) + D2 · φ + D3 · T
E ∗ = E0 + E1 · (d γ /dt) + E2 · φ + E3 · T
(4.77)
with Di and Ei values to be determined by data fitting.
4.9 Concluding Remarks
Interfaces usually constitute a weak link in the chain of load transfer in bonded
joints. Also, the discontinuity of the material properties causes abrupt changes in
stress distribution, as well as causing stress singularities at the edges of the interfaces. It is very desirable to optimize the substrate surface topography at the interfaces to maximize the load bearing capacity of bonded joints, and to improve their
deformational characteristics. In most engineering joints the adherend surfaces have
distinct topographies, which result in a collection of miniature joints in micron, and
even nano scale when bonded adhesively. If the interphase is not considered as a discrete collection of individual chemical bonds, the methods of continuum mechanics
can still be applied to this collection of miniature joints by assuming continuous,
or a combination of continuous/discontinuous interphase zones. Thus, the displacement at the interphase need not be continuous at every location. The analysis of
a miniature joint contributing to the overall adhesion in a macro joint can be performed in a fashion similar to that for the macro joint itself, which is usually studied
with the employment of the methods of elasticity, viscoelasticity, plasticity, fracture,
damage, and/or failure mechanics.
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Chapter 5
Complex Joint Geometry
Andreas Öchsner, Lucas F.M. da Silva and Robert D. Adams
Abstract The finite element method is particularly suited to analyse complex joint
geometries. Adhesively bonded joints are increasingly being used in engineering applications where the loading mode, the adherends shape and the material behaviour
are extremely difficult to simulate with a closed form approach. A detailed description of finite element studies concerning non-conventional adhesive joints is presented in this chapter. Various types of joints, local geometrical features such as the
spew fillet and adherend rounding, three dimensional analyses, hybrid joints and repair techniques are discussed. Special techniques to save computer power are also
treated. It is shown that the finite element method offers unlimited possibilities for
stress analysis but also presents some numerical problems at sharp edges.
5.1 Introduction
Stress analysis of bonded joints started 70 years ago with the well known Volkersen
shear lag model (Volkersen, 1938). Since, various analytical models have been proposed that include the geometrical non-linearity (Goland and Reissner, 1944), the
adhesive plasticity (Hart-Smith, 1973), fibre reinforced plastic adherends (Renton
Andreas Öchsner
Department of Applied Mechanics, Faculty of Mechanical Engineering, Technical University of
Malaysia, 81310 UTM Skudai, Johor, Malaysia; University Centre for Mass and Thermal Transport in Engineering Materials, School of Engineering, The University of Newcastle, Callaghan,
NSW, 2308, Australia, e-mail: andreas.oechsner@gmail.com
Lucas F.M. da Silva
Departamento de Engenharia Mecânica e Gestão Industrial, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal, e-mail: lucas@fe.up.pt
Robert D. Adams
Department of Mechanical Engineering, University of Bristol, Bristol BS8 1TR, UK,
e-mail: R.D.Adams@bristol.ac.uk
L.F.M. da Silva, A. Öchsner (eds.), Modeling of Adhesively Bonded Joints,
c Springer-Verlag Berlin Heidelberg 2008
131
132
A. Öchsner et al.
and Vinson, 1975), etc. However, most of the analytical models are for lap joints
of regular shape. Closed form analyses are very useful for an initial estimate of
the stress distribution and are generally adequate for design purposes. For adhesive joints of complex shape, numerical techniques such as the finite element (FE)
method are preferable. Simple lap joints have been modified because of the nonuniform stress distribution along the overlap. The load transfer and shear stress distribution of various single lap joints (SLJ) are schematically represented in Fig. 5.1.
It can be seen that there is a stress concentration at the ends of the overlap for the
single lap joint with a square end. Modification of the joint end geometry with a
spew fillet or a taper spreads the load transfer over a larger area and give a more
uniform shear stress distribution.
Adams and co-workers are among the first to have used the FE method for
analyzing the stresses in adhesive joints (Adams and Peppiatt, 1974; Crocombe
and Adams, 1981; Harris and Adams, 1984; Adams et al., 1986; Adams and
Davies, 2002). One of the first reasons for the use of the FE method was to assess
the influence of the spew fillet. The joint rotation and the adherends and adhesive
plasticity are other aspects that are easier to treat with a FE analysis. The study of
Harris and Adams (1984) is one of the first FE analyses taking into account these
three aspects.
The increasing use of adhesive bonding with composite materials is another typical example of the great advantage of using finite elements (Adams et al., 1986;
Adams and Davies, 2002) since the anisotropic behaviour of these materials complicates drastically any analytical approach.
But it is when irregular geometries are involved that the FE method is an indispensable tool. Any type of geometry can be modelled, i.e., variations in the
adherend shape, local adherend rounding, spew fillet, reinforcements, joints with
rivets or bolts, etc. In addition, the simulation gives not only one, two or three stress
Fig. 5.1 Load transfer and
shear stress distribution in
single lap joints
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133
components but all the stress components in all members of the joint. Three dimensional (3D) analyses for assessing width effects is another possibility offered by a
FE analysis. As the model gets more complicated in terms of geometry and material
behaviour, the computation time also increases. Also, if the user wants to change
one geometrical feature, a new model has to be created. For these reasons, parametric studies are more difficult with a FE approach than with a closed form approach.
However, special techniques are emerging for solving this problem. All these aspects are discussed in this chapter with a mention to the difficult task of dealing
with the singular points present in any type of adhesive joint.
5.2 Types of Joints
Adhesive joint studies are generally related to lap joints with flat adherends. The first
theoretical analysis of such joints was an analytical model proposed by Volkersen
(1938). Since then, many analytical models have been proposed, being the overwhelming majority about lap joints with flat adherends. This is because joints of a
more complex geometry are difficult to model using a closed form analysis. Numerical techniques, such as the FE method, can obviously be used for simple geometries
such as single or double lap joints with regular flat adherends, but it is for complex
geometries that this method of analysis is truly advantageous. Within joints with flat
adherends, one can have ‘wavy’ lap joints, ‘reverse bent’ joints, ‘tongue and groove’
joints and scarf joints (see Fig. 5.2). Other types of joints include cylindrical joints,
corner joints, T joints and peel joints (see Fig. 5.2). As for analytical models, the
majority of the FE analyses are about lap joints with flat adherends. This type of
joint is treated in detail in the following sections. However, numerous studies can
be found in the literature about other types of joints.
Ávila and Bueno (2004) analyzed a new type of single lap joints where the
adherends have a ‘wavy’ configuration along the overlap. An FE analysis and experiments show that great strength improvements can be obtained with this technique.
Fessel et al. (2007) also studied this type of joint as well as joints with bent substrates
along the overlap. An FE analysis and experiments show great joint strength improvements. Dvorak et al. (2001) show with a FE analysis and experimentally that
adhesively bonded tongue-and-groove joints between steel and composite plates
loaded are stronger than conventional strap joints.
Nakagawa and Sawa (2001) studied scarf joints using photoelasticity and a 2D
FE model. One of the conclusions is that under a static tensile load, an optimum
scarf angle exists where the stress singularity vanishes and the stress distributions
become flat near the edge of the interface, but under thermal loads, the optimum
scarf angle is not found.
Apalak and Davies (1993, 1994) studied different types of corner joints using a
linear FE analysis and proposed design guidelines based on the overall static stiffness and stress analysis. Apalak (1999, 2000) developed the previous study taking
into account the non-linear geometry. Feih and Shercliff (2005) modelled simple
corner joints with composites.
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Fig. 5.2 Types of adhesive joints
Apalak (2002) studied the geometrical non-linear response of T-joints. Apalak
et al. (2003) studied T-joints in terms of thermal and mechanical loads. da Silva
and Adams (2002) used the a FE model to predict the joint strength of T-joints
based on the plasticity of the adherend. Marcadon et al. (2006) studied T-joint
for marine applications and found that both the overlap length and also the distance between the T plywood and the base have an influence on the tearing
strength.
Kim et al. (1992) studied various configurations of tubular joints (single overlap, double overlap, adherends tapering) and found with a FE analysis and experimentally that the double lap configuration is the strongest joint. Kim and
Lee (2001, 2004) used FE models to study the effect of temperature on tubular joints. Kim et al. (1999) modelled composite tubular joints. Oh (2007) used
a FE computation to analyze tubular composite adhesive joints under torsion.
Good agreement was obtained between the predicted joint strength and the available experimental data. Serrano (2001) used a nonlinear 3D FE model to study
glued-in rods for timber structures. The model could simulate experimental
behaviour.
Peel testing is a very common test for adhesive properties. The problem is highly
non-linear in terms of geometry and material and the FE method has been widely
used, as for example in the work of Du et al. (2004).
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135
5.3 Adhesive Fillet
Various authors have shown that the inclusion of a spew fillet at the ends of the
overlap reduces the stress concentrations in the adhesive and the substrate (Adams
and Peppiatt, 1974; Crocombe and Adams, 1981; Adams et al., 1986; Adams and
Harris, 1987; Dorn and Liu, 1993; Tsai and Morton, 1995; Lang and Mallick, 1998;
Belingardi et al., 2002; Apalak and Engin, 2004; Andreassi et al., 2007; da Silva and
Adams, 2007a; Deng and Lee, 2007). The adhesive fillet permits a smoother load
transfer and decreases the peak stresses.
Adams and Peppiatt (1974) found that the inclusion of a 45◦ triangular spew fillet
decreases the magnitude of the maximum principal stress by 40% when compared to
a square end adhesive fillet. Adams and Harris (1987) tested aluminium/epoxy single lap joints with and without fillet and found an increase of 54% in joint strength
for the filleted joint. Adams et al. (1986) tested aluminium/CFRP single lap joints
and found that the joint with a fillet is nearly two times stronger than the joint without a fillet. Crocombe and Adams (1981) did similar work but included geometric
(overlap length, adhesive thickness and adherend thickness) and material (modulus
ratio) parameters. The reduction in peel and shear stresses is greatest for a low modulus ratio (low adhesive modulus), a high adhesive thickness and a low adherend
thickness.
Dorn and Liu (1993) investigated the influence of the spew fillet in plastic/metal
joints. The study includes a FE analysis and experimental tests and they conclude
that the spew fillet reduces the peak shear and peel adhesive stresses and decreases
stress and strain concentrations in the adherends in the most critical regions. They
also studied the influence of different adhesive and different metal adherends. A ductile adhesive and a more balanced joint (aluminium/plastic instead of steel/plastic)
give a better stress distribution.
Tsai and Morton (1995) studied the influence of a triangular spew fillet in laminated composite single lap joints. The FE analysis and the experimental tests (Moire
interferometry) proved that the fillet helps to carry part of the load thus reducing the
shear and peel strains.
The above analyses are limited to triangular geometry. Lang and Mallick (1998)
investigated eight different geometries: full and half triangular, full and half rounded,
full rounded with fillet, oval and arc. They showed that shaping the spew to provide
a smoother transition in joint geometry significantly reduces the stress concentrations. Full rounded with fillet and arc spew fillets give the highest percent reduction
in maximum stresses whereas half rounded fillet gives the less. Furthermore, increasing the size of the spew also reduces the peak stress concentrations.
Andreassi et al. (2007) used a two-dimensional computational fluid dynamics
model to study spew fillet formation considering the actual adhesive flow produced
during joint assembly. This is an interesting approach that allows tuning the adhesive
joint strength by changing the adhesive flow parameters.
The spew fillet is not beneficial in all situations. In effect, the spew fillet tends
to generate more thermal stresses than a square end geometry when used at low
temperatures (da Silva and Adams, 2007a).
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5.4 Adherend Rounding
Adhesive single lap joints may have several singular (infinite) stresses. The stresses
tend to infinite at crack tips and at bi-material junctions. Fracture mechanics can
be applied to treat the singular points at crack tips. As regards the bi-material
singularities, there can be two or three, depending on the joint geometry (see
Fig. 5.3). For both situations, there is an abrupt change in slope giving a sharp corner. The stress field in the vicinity of a sharp corner depends on the mesh used. A
stress limit criterion will therefore lead to arbitrary results. The first FE analyses 30
years ago used coarse meshes due to computer memory limitations and therefore
did not capture the infinite stress at singular points. The joint strength predictions
obtained with stress or strain limit based criteria compared well with the experimental results (Harris and Adams, 1984). With the computer power development, finer
meshes can be used which lead to a better stress distribution description but also to
stresses that tend to infinite at singular points. The stress or strain limit failure criteria give in this circumstance more conservative predictions and are therefore very
arguable.
Groth (1988) used a fracture mechanics approach without considering a preexisting crack. He formulated a fracture criterion based on an equivalent generalised
stress intensity factor similar to that in classical fracture mechanics. Comparing it to
a critical value, joint fracture may be predicted. However, the critical stress intensity
factor needs first to be tuned with an experimental test which makes this approach
questionable.
Another approach for dealing with the singularity points is to use a cohesive
zone model (CZM). This approach is associated to interface elements and enables
to predict crack initiation and crack growth. It is a combination of a stress limit and
fracture mechanics approach and relatively mesh insensitive. Many researchers are
using this tool with accurate results (Blackman et al., 2003; de Moura et al., 2006;
Liljedahl et al., 2007). However, the parameters associated to the CZM require previous experimental ‘tuning’ and the user needs to know beforehand where the failure
is likely to occur. This subject is discussed in detail in Chap. 6.
In practical joints the sharp corners are always slightly rounded during manufacture and the singular points will not necessarily exist (see Fig. 5.4). Adams and
Harris (1987) studied the influence of the geometry of the corners. Using a simplified model, the effect of rounding on the local stress was investigated. Rounding the
corner removes the singularity. Therefore small local changes in geometry have a
significant effect. They also performed an elastic-plastic analysis by calculating the
Fig. 5.3 Singular points in adhesive joints
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137
Fig. 5.4 Rounding of adhesive and adherend
plastic energy density. They found that the stress distribution is more uniform with a
maximum some distance from the corner. The reason they give is that as the corner
is approached, although the normal stress increases, the rigid adherend restrains the
adhesive in the transverse direction. The net hydrostatic component is increased and
yield is suppressed so that close to the corner there is a reduction in plastic energy
density. It should be born in mind that the effects are local: far away from the corner
the stress distribution is unaffected. Apart from the simplified model, Adams and
Harris (1987) tested three types of aluminium/rubber-toughened epoxy single lap
joints so that local changes at the end of the joint could be assessed. They found excellent agreement between the predicted joint strengths, with the modified models,
and the experimental values.
5.5 Adherend Shaping
Adherend shaping is a powerful way to decrease the stress concentration at the ends
of the overlap. Figure 5.5 presents typical geometries used for that purpose. Some
analytical models were proposed to have a more uniform stress distribution along
the overlap (Cherry and Harrison, 1970; Adams et al., 1973; Groth and Nordlund,
1991). However, the FE method is more appropriate for the study of adherends shaping. The concentrated load transfer at the ends of the overlap can be more uniformly
distributed if the local stiffness of the joint is reduced. This is particularly relevant
for adhesive joints with composites due to the low transverse strength of composites.
Adams et al. (1986) addressed this problem. They studied various configurations of
Fig. 5.5 Adherend shaping
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A. Öchsner et al.
double lap joints where the central adherend is CFRP and the outer adherends are
made of steel. They found with FE and experiments that the inclusion of an internal taper and an external fillet can triplicate the joint strength. The same geometry
was studied at low temperatures and it was found that the thermal stresses reduce
substantially the joint strength (da Silva and Adams, 2007a).
Hildebrand (1994) did similar work on SLJs between fibre reinforced plastic and
metal adherends. The optimisation of the SLJs was done by modifying the geometry
of the joint ends. Different shapes of adhesive fillet, reverse tapering of the adherend,
rounding edges and denting were applied in order to increase the joint strength. The
results of the numerical predictions suggest that, with a careful joint-end design, the
strength of the joints can be increased by 90–150%.
The use of internal tapers in adherends in order to minimize the maximum transverse stresses at the ends of bonded joints has also been studied by Towse (1999),
Rispler et al. (2000), Guild et al. (2001), Belingardi et al. (2002) and Kaye and
Heller (2002). An evolutionary structural optimisation method (EVOLVE) was used
by Rispler et al. (2000) to optimise the shape of adhesive fillets. EVOLVE consists
of an iterative FE analysis and a progressive removal of elements in the adhesive
which are low stressed.
Other examples of joint end modifications for joint transverse stress reduction
but using external tapers are those of Amijima and Fujii (1989), Sancaktar and
Nirantar (2003), Kaye and Heller (2005) and Vallée and Keller (2006). Kaye and
Heller (2005) used numerical optimization techniques in order to optimize the shape
of the adherends. This is especially relevant in the context of repairs using composite patches bonded to aluminium structures (see Sect. 5.9) due to the highly stressed
edges.
The FE method is a convenient technique for the determination of the optimum
adherend geometry, however the complexity of the geometry achieved is not always
possible to realise in practice.
5.6 Other Forms of Geometric Complexity
The FE method can be used to study other complex geometrical features such as
voids in the bondline, surface roughness, notches in the adherend, etc. (see Fig. 5.6).
Nakagawa et al. (1999) studied the effect of voids in butt joints subjected to thermal
stresses and found that stresses around defects in the centre of the joint are more
significant than those near the free surface of the adhesive. Lang and Mallick (1999),
Olia and Rossettos (1996), de Moura et al. (2006) and You et al. (2007) studied the
influence of gaps in the adhesive and found that a gap in the middle of the overlap
has little effect on joint strength.
Kim (2003) proposed an analytical model supported by FE modelling to study
the effect of variations of adhesive thickness along the overlap. The author showed
that the variations found in practice have little effect on the adhesive stresses along
the overlap.
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139
Fig. 5.6 Various forms of geometric complexity
Kwon and Lee (2000) studied the influence of surface roughness on the strength
tubular joints by modelling the stiffness of the interfacial layer between the adherends and the adhesive as a normal statistical distribution function of the surface
roughness of the adherends. The authors found that the optimum surface roughness
was dependent on the bond thickness and applied load.
Sancaktar and Simmons (2000) investigated the effect of adherend notching on
the strength and deformation behavior of single lap joints. The experimental results
showed a 29% increase in joint strength with the introduction of the notches, which
compared well with the FE analysis results. Yan et al. (2007) studied a similar idea,
where the notch is in the middle part of the overlap. A FE analysis showed that a
more uniform stress distribution along the overlap can be obtained.
5.7 Three Dimensional Analyses
Most of the FE analyses of adhesive joints are two dimensional. Since the width
is much larger than the joint thickness, a plane strain analysis is generally acceptable. A generalised plane strain analysis where the joint is allowed to have a uniform strain along two parallel planes in the width direction gives more realistic
results. However, for some problems, a two dimensional analysis might lead to erroneous results. Richardson et al. (1993) show that applying the average load on
two-dimensional models resulted in errors in the adhesive stresses as high as 20%.
The authors propose solutions to limit this problem. Three dimensional effects in the
width direction such as lateral straining and the anticlastic bending (see Fig. 5.7)
are especially important when analysing composites. Adams and Davies (1996)
analysed composite lap joints in three dimensions and showed that the Poisson’s
ratio effects are significant in the behavior of composite lap joints.
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A. Öchsner et al.
Fig. 5.7 Three dimensional effects in lap joints
The main problem of 3D analyses is the computational time due to the large
number of elements. One way of overcoming this problem is to use the submodelling
option, where the results from a coarse mesh of the global model are applied as
boundary conditions to a much finer mesh of a localised region of particular interest.
Bogdanovich and Kizhakkethara (1999) applied this technique to composite double
lap joints and concluded that the submodelling approach is an efficient tool although
convergence of stresses along certain paths in the joint were not satisfied. They also
used the same approach in two dimensions to study the effect of the spew fillet.
5.8 Hybrid Joints
Joints with different methods of joining are increasingly being used. The idea is to
gather the advantages of the different techniques leaving out their problems. Another
possibility is to use more than one adhesive along the overlap or varying the adhesive
and/or adherend properties. All theses cases have been grouped here under a section
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Fig. 5.8 Hybrid joints
called ‘hybrid joints’ (see Fig. 5.8). Such joints are particularly difficult to simulate
using analytical models for obvious reasons. The FE method is the preferred tool to
investigate the application of such techniques.
5.8.1 Mixed Adhesive Joints
Mixed modulus joints have been proposed in the past (Semerdjiev, 1970; Srinivas,
1975; Patrick, 1976) to improve the stress distribution and increase the joint strength
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of high modulus adhesives. The stiff, brittle adhesive should be in the middle of the
overlap, while a low modulus adhesive is applied at the edges prone to stress concentrations. Sancaktar and Kumar (2000) used rubber particles to toughen the part of the
adhesive located at the ends of the overlap and increase the joint strength. The concept was studied with the FE method and proved experimentally. Pires et al. (2003)
and Fitton and Broughton (2005) also proved with a FE analysis and experimentally
with two different adhesives that the mixed adhesive method gives an improvement
in joint performance. Temiz (2006) used a FE analysis to study the influence of two
adhesives in double lap joints under bending and found that the technique decreases
greatly the stresses at the ends of the overlap. Bouiadjra et al. (2007) used the mixed
modulus technique for the repair of an aluminium structure with a composite patch.
The use of a more flexible adhesive at the edge of the patch increases the strength
performance of the repair. The technique of using multi-modulus adhesives has been
extended to solve the problem of adhesive joints that need to withstand low and high
temperatures by da Silva and Adams (2007b, 2007c). At high temperatures, a high
temperature adhesive in the middle of the joint retains the strength and transfers the
entire load while a low temperature adhesive is the load bearing component at low
temperatures, making the high temperature adhesive relatively lightly stressed. The
authors studied various configurations with the FE method and proved experimentally that the concept works, especially with dissimilar adherends.
5.8.2 Adhesive Joints with Functionally Graded Materials
Functionally graded materials are increasingly being used in various applications
including adhesive joints. For example, Apalak and Gunes (2007) studied the effect
of a functionally gradient layer between a pure ceramic layer (Al2 O3 ) and a pure
metal layer (Ni). Gannesh and Choo (2002) studied the effect of spatial grading of
adherend elastic modulus on the peak stress and stress distribution in the singlelap bonded joint. The adherends in the overlap length was divided into ten equal
regions and material properties assign as a function of the grading. The peak shear
stresses could be reduced by 20%. The study previously referred of Sancaktar and
Kumar (2000) is effectively a functionally graded adhesive with the use of rubber
particles. This is an area that is being intensively studied and where modelling at
different scales is essential.
5.8.3 Rivet-Bonded Joints
Liu and Sawa (2001) investigated, using a three-dimensional FE analysis, rivetbonded joints and found that for thin substrates bonded, riveted joints, adhesive
joints and rivet-bonded joints gave similar strengths whilst for thicker substrates the
rivet-bonded joints were stronger. They proved this experimentally. Later, the same
5 Complex Joint Geometry
143
authors (Liu and Sawa, 2003; Liu et al., 2004) proposed another technique similar
to rivet-bonded joints: adhesive joints with adhesively bonded columns. Strength
improvements are also obtained in this case. The advantage of this technique of
that the appearance is the joint is maintained in relation to an adhesive joint. Grassi
et al. (2006) studied through-thickness pins for restricting debond failure in joints.
The pins were simulated by tractions acting on the fracture surfaces of the debond
crack.
5.8.4 Bolted-Bonded Joints
Chan and Vedhagiri (2001) studied the response of various configurations of single
lap joints, namely bonded, bolted and bonded-bolted joints by three-dimensional FE
method and the results were validated experimentally. The authors found that for the
bonded-bolted joints, the bolts help to reduce the stresses at the edge of the overlap,
especially after the initiation of failure. The same type of study was carried out by
Lin and Jen (1999).
5.8.5 Weld-Bonded Joints
Al-Samhann and Darwish (2003) demonstrated with the FE method that the stress
peaks typical of adhesive joints can be reduced by the inclusion of a weld spot
in the middle of the overlap. They studied later the effect of adhesive modulus and
thickness (Darwish and Al-Samhann, 2004). Darwish (2004) also investigated weldbonded joints between dissimilar materials.
5.9 Repair Techniques
Adhesively bonded repairs are generally associated to complex geometries and the
FE method has been extensively used for the optimization of the repair, especially
with composites. The literature review of Odi and Friend (2002) about repair techniques illustrates clearly this point. Typical methods and geometries are presented
in Fig. 5.9. Among the various techniques available, bonded scarf or stepped repairs are particularly attractive because a flush surface is maintained which permits
a good aerodynamic behavior. Odi and Friend (2004) show that an improved 2D
plane stress model is sufficient to get reliable joint strength predictions. Gunnion
and Herszberg (2006) studied scarf repairs and carried out a FE analysis to asses
the effect of various parameters. They found that the adhesive stress is not much
influenced by mismatched adherend lay-ups and that there is a huge reduction in
peak stresses with the addition of an over-laminate. Campilho et al. (2007) studied
scarf repairs of composites with a cohesive damage model and concluded that the
strength of the repair increased exponentially with the scarf angle reduction.
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Fig. 5.9 Repair techniques
Bahei-El-Din and Dvorak (2001) proposed new design concepts for the repair
of thick composite laminates. The regular butt-joint with a patch on both sides was
modified by the inclusion of pointed inserts or a ‘zigzag’ interface in order to increase the area of contact and improve the joint strength.
Soutis and Hu (1997), Okafor et al. (2005) and Sabelkin et al. (2007) studied
numerically and experimentally bonded composite patch repairs to repair cracked
aircraft aluminum panels. The authors concluded that the bonded patch repair provides a considerable increase in the residual strength.
Tong and Sun (2003) developed a pseudo-3D element to perform a simplified
analysis of bonded repairs to curved structures. The analysis is supported by a full
3D FE analysis. The authors found that external patches are preferred when the
shell is under an internal pressure while internal patches are preferred when under
an external pressure.
5.10 Special Techniques in Finite Element Simulation
One of the advantages of closed form analyses in relation to the FE method is that
parametric studies are much easier to perform. With conventional FE techniques,
every time a geometric parameter of the model (adhesive or adherend thicknesses,
5 Complex Joint Geometry
145
angle of the spew fillet, etc.) with a different value is considered, a new model is
required which is time consuming. However, recent FE libraries include connections
that can be described in a parametric form. Similar families of connections need
to be meshed only once and the users need only enter the parameters. Actis and
Szabó (2003) have used parametric models for the study of bonded and fastened
repairs. These models have associated p-type meshing, where convergence of the
solution is achieved by increasing the polynomial order of the element rather than
increasing the number of elements in the model (known as h-type meshing). More
recently, Kilic et al. (2006) present a finite element technique utilizing a global
element coupled with traditional elements. The global element includes the singular
behaviour at the junction of dissimilar materials.
Other special techniques in FE simulation for the significant reduction of the
modelling (mesh generation) and computation time are presented next, which is
critical when complex geometries are involved. The main focus is on techniques
which do not require any new code at all or are achievable by very simple routines consisting of a few lines of computer code. Advanced topics which require the
development of comprehensive new routines, for instance as in the case of new element formulation (e.g. to better approximate singularities), are not considered. The
presented techniques are realisable by actual versions of several commercial finite
elements codes.
The techniques will be applied to the example of a single lap joint problem
taken from Zhao (1991) where the influence of different degrees of rounding on the
joint strength was experimentally and numerically investigated. The general problem with geometric dimensions and boundary conditions is shown in Fig. 5.10.
The adherends were made of 3.2 mm thick aluminium sheets (Young’s modulus:
70 GPa; Poisson’s ratio: 0.33) and the brittle adhesive epoxy resin Ciba MY750
with hardener HY906 (Young’s modulus 2.8 GPa; Poisson’s ratio: 0.4) was used.
A 2D plane strain problem was modelled because of the large joint width and the
Fig. 5.10 Single lap joint: (a) joint geometry and applied boundary conditions (dimensions in mm);
(b) different degrees of rounding (the grey colour represents the adhesive) (Zhao, 1991)
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A. Öchsner et al.
ends of the joints were constrained without any rotation to account for the testing
conditions. It should be mentioned here that Zhao (1991) used a two steps approach
because of limitations in computer power: A first computation was based on a coarse
mesh for the entire specimen, followed by a refined mesh in the corner region with
the displacement result from the first analysis as the new boundary condition. The
following examples which were realised with the commercial code Marc (MSC
Software Corporation, Santa Ana, CA, USA) may be considered as an alternative
for such a two step approach.
5.10.1 Consideration of Point Symmetry
Commercial finite element codes allow to consider certain types of symmetry
conditions. Common examples are symmetry about an axis (so-called reflective
symmetry) or cyclic symmetry (structures with a geometry and a loading varying
periodically about a symmetry axis). Point symmetry (or origin symmetry, rotational symmetry by 180◦ ) with respect to the origin of the coordinate system occurs
for instance in the case of single lap joints under tensile load (cf. Fig. 5.11a). This
type of symmetry is not covered by the standard symmetry conditions which are
mentioned above. If one can consider the point symmetric deformation behaviour
of a single lap joint (cf. Fig. 5.11b), it is obvious that the mesh size and the resulting
system of equations can be reduced by 50% which results in a significant reduction
of the calculation time.
Looking at the deformation of a full single lap specimen (cf. Fig. 5.12) in the
centre plane, i.e. x = 0, one can see that a node at a distance +a from the glue
line (y = 0) moves under load for the shown arrangement to the negative x and y
direction. The movement in the negative y direction is about two orders of magnitude
smaller than the displacement in the x direction and difficult to observe in the scale
of Fig. 5.12. On the other hand, a node at distance −a from the glue line moves with
the same magnitude, however, in the positive coordinate directions. This relationship
for a pair of nodes at (0,+ a) and (0,– a) can be expressed by the following constraint
condition
Fig. 5.11 Deformation of a single lap joint under tensile load: (a) deformed S-shape of the entire
specimen; (b) consideration of the S-shape for the half specimen due to appropriate point symmetry
condition
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147
Fig. 5.12 Details of a deformed single lap specimens to illustrate node translation in the centre
plane x = 0
u
v
x=0;y=−a
=
−1.0
0
u
·
,
−1.0
v x=0;y=+a
0
(5.1)
where the vector of displacement at node y = −a is referred to as the tied (“slave”)
node while the node on the right-hand side is referred to as the retained node
(“master”). The connecting matrix is called the constrained matrix. Some commercial finite element codes allow the definition of arbitrary homogeneous constraints between nodal displacements by user subroutines (in the case of MSC Marc:
UFORMS routine). When such a routine is supplied, the user is simply replacing the
one which exists in the program using appropriate control setup. In the considered
case of point symmetry, it must be mentioned that the constraint condition must be
defined for each pair of nodes in the centre plane x = 0.
5.10.2 Connecting Dissimilar Meshes
Modern adhesives can be applied in films of several micron of thickness while the
surrounding adherends may extend to a much larger scale in the range of millimetres, centimetres etc. If accuracy in the adhesive layer is requested, several elements must be introduced over the adhesive thickness and this mesh density must
be coarsened in order to limit the total number of elements to account for limitations in computer hardware (in particular the random access memory, RAM). In
addition, regions with high stress gradients require a fine and regular mesh if these
changes should be evaluated. Such constraints may result in a quite complicated
mesh which is difficult to generate and a huge system of equations whose solution is
time-consuming. In addition, it might be necessary to introduce transition elements
which may have poor performance for specific loading conditions, e.g. bending. As
an alternative, the approach of dissimilar meshes (cf. Fig. 5.13) may be introduced.
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Fig. 5.13 Example of a finite element model composed of two dissimilar meshes (nodes are not
coincident at the interface)
The major idea is to generate different types of meshes which are not connected in
the classical way, i.e. by common nodes which belong to both touching elements.
Regions where high accuracy of the analysis is requested may be composed of a
very fine and regular mesh (Fig. 5.13, right part) while other parts can be modelled
by coarse and irregular element representations (Fig. 5.13, left part).
One method of connecting these dissimilar meshes is to use interface elements
(Schiermeier et al., 2001). Nowadays, actual versions of commercial finite element
codes allow such an application in the scope of novel contact options, i.e. to “glue”
dissimilar contacting meshes without the need for interface elements. In such a
case, by specifying that the glue motion is activated, the constraint equations are
automatically written between the two meshes and the contact region is not allowed
to separate.
In the case of the corner rounding influence, a basic single lap joint was meshed
with a quite coarse mesh (adherend length: 96.8 mm) and several refined inlays
(cf. Fig. 5.10b) were separately generated and successively “glued” to the basic
joint. Figure 5.14 shows as an example the refined inlay with the sharp corner which
is “glued” at the dissimilar interfaces to the basic joint. Similar meshes for the inlays
with different degree of rounding were obtained.
To present some numerical results, the normalised von Mises stress along the
bond line for the different configurations of corners is shown in Fig. 5.15. It
can be clearly seen that the introduction of a rounding decreases the stress peak
(as shown in Peppiatt (1974) and Chen (1985)) and thus, results in a higher
strength of the adhesive joint. The difference in the peak values are presented in
Table 5.1 and compared to experimental findings taken from Zhao (1991). It can
be seen that this numerical simulations reveals the same tendency as the experimental values. It must be noted here that the presented numerical results are based
on a pure linear elastic analysis and small deformations and an additional consideration of the non-linear material behaviour and appropriate yield conditions
of both components can improve the obtained results compared to experimental
values.
5 Complex Joint Geometry
149
Fig. 5.14 Dissimilar meshes in the case of the sharp corner: the corner region consists of a much
finer mesh
Fig. 5.15 von Mises stress k normalised by average shear stress τave along the normalised bond
line (y = 0). Stress distributions are obtained or the same external load F
Table 5.1 Comparison between joint strength prediction and experiments
Corners
FE
(Strength increase in %)
Experiment, Zhao (1991)
(Strength increase in %)
sharp
r = 0.25
r = 1.6
r = 3.2
0 (ref.)
8.54
20.55
24.28
0 (ref.)
16.50
20.00
40.15
150
A. Öchsner et al.
5.11 Conclusions
This chapter describes studies that deal with adhesive joints of complex geometry
using the FE method. The main conclusions are:
1. The FE method can give the stress distribution in the whole bonded structure and
is an efficient tool to identify the stress concentrations.
2. For this reason, the FE method is the most adequate to develop and optimize
adhesive joints.
3. FE studies of different complexity were discussed: lap joints with irregular
shapes, rounding of adherend ends, spew fillet, hybrid joints and repair techniques.
4. The stress concentrations at sharp corners, where the failure is likely to occur, are
difficult to handle using traditional stress-strain approaches because the results
are mesh dependent.
5. This problem can be solved, to some extend, rounding the edges, using a strength
singularity approach or a cohesive zone model. However, even these methods
require some kind of experimental ‘tuning’.
6. The optimization of joint strength by the use of functionally graded adhesives is
one the main challenges in adhesive joints modelling.
7. Parametric studies are more difficult to perform with the FE method due to the
re-meshing problem and the computation time. However, recent FE programs
include routines for automatic re-meshing. Submodelling is another technique
for saving computation time and change geometric parameters in an easier way.
This has been performed in the present chapter and validated with experimental
results.
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Chapter 6
Progressive Damage Modelling
Marcelo F.S.F de Moura
Abstract The application of bonded joints is increasing due to their several
advantages to alternative bonding methods. As a result, more efficient predictive
tools are necessary to increase the confidence of designers. In this context, cohesive and continuum damage models acquire special relevancy owing to their capacity to simulate damage onset and growth. Both of these methodologies combine
strength of materials with fracture mechanics, thus overcoming the limitations of
each method. A cohesive mixed-mode damage model based on interface finite elements and accounting for ductile behaviour of adhesives is presented. The cohesive
parameters of the constitutive softening law are determined using an inverse method
applied to fracture characterization tests under pure modes, I and II. In this context
a new data reduction scheme based on crack equivalent concept is developed and
applied to fracture characterization tests. Good agreement between the numerical
and experimental results was obtained for strength versus overlap length in singlelap joints. A continuum mixed-mode damage model is also presented using a triangular softening law adequate only for brittle or moderately ductile adhesives. In
these models the material properties degradation occurs inside of the solid elements,
which is advantageous relatively to cohesive methods mainly when damage propagation onset and path are not known a priori.
6.1 Introduction
Adhesive joints present several advantages over other conventional bonding methods like mechanically fastened joints. Adhesive joints present less sources of stress
concentrations, more uniform load distribution and better fatigue properties. They
Marcelo F.S.F de Moura
Departamento de Engenharia Mecânica e Gestão Industrial, Faculdade de Engenharia,
Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal,
e-mail: mfmoura@fe.up.pt
L.F.M. da Silva, A. Öchsner (eds.), Modeling of Adhesively Bonded Joints,
c Springer-Verlag Berlin Heidelberg 2008
155
156
M.F.S.F. de Moura
also provide higher flexibility to join different materials, allowing the choice of the
better material for each component of a structure. To benefit from these advantages it
is crucial to design bonded joints properly. There are two basic approaches: strength
of materials and fracture mechanics based methods. These methods are described in
more detail below.
6.1.1 Failure Prediction Classical Methods
6.1.1.1 Strength of Materials
In the strength of materials approach the maximum stress or strain criteria are the
most popular ones. They are based on the assumption that failure occurs when one
of the stress or strain tensor components attains the respective strength value. Harris
and Adams [19] used with success the maximum principal stress and the maximum
principal strain criteria in order to predict the failure of single lap joints. Ikegami
et al. [20] used the von Mises stress criterion to predict the failure of scarf joints
using metal and composite adherends and Charalambides et al. [4] applied the same
criterion to double lap joints. They did not obtain accurate results due to dependence of adhesive behaviour on the hydrostatic stress, which is not accounted for
in the von Mises criterion. Several different versions of these criteria were used by
other authors. Lee and Lee [28] used the maximum shear strain criterion on single
lap tubular joints. Crocombe and Adams [6] proposed the effective plastic strain criterion for peel tests. In these tests the triaxial strain is expressed as an effective one,
which is compared to the uniaxial failure strain of bulk specimens.
The referred criteria have a main difficulty when applied to the failure prediction of bonded joints. In fact, in bonded joints a problem of stress singularity at the
end of overlapping regions arises due to sharp corners. A stress singularity can be
defined as a point where, according to linear elastic analysis, an infinite value of
stress appears. Evidence of this problem can be detected in finite element analysis,
when stresses at the singularity point increase with the mesh refinement and convergence cannot be reached. In conclusion, it can be affirmed that the stress based
methods present mesh dependency during numerical analysis due to stress singularities. To overcome this problem the stresses obtained numerically are used in a
point stress or average stress criteria [38] in order to evaluate the occurrence of failure. In the point stress criterion the stresses are evaluated at a characteristic distance
(Fig. 6.1) whereas in the average stress criterion they are averaged over a distance.
The failure occurs when these stress values attain the respective material strength.
They can be viewed as semi-empirical criteria as the characteristic distance value is
determined from experimental data. Towse et al. [36] used the criterion of critical
strain at a given distance to predict failure of double lap joints. In this case, failure is
simulated when the strain in the neighbouring region of the singularity reaches the
respective limiting value. Although the obtained results agreed with the experimental ones, it should be emphasized that the characteristic distance was experimentally
determined, which limits its use in other types of joints. John et al. [22] argues that
6 Progressive Damage Modelling
157
Fig. 6.1 Stress or strain at a
characteristic distance from
the singularity
Position of
evaluation point
failure occurs when shear stress attains a critical value at a normalized distance from
the singularity point in a double lap joint of unidirectional composite. However, this
result implies that the critical distance varies as a function of the overlap length
which is not physically sound. The average stress criterion applied to failure prediction of bonded joints was used by Zhao [42]. The critical dimension used to evaluate
the average of the maximum principal stress was considered as being equal to the
adhesive thickness. However, Charalambides et al. [4] demonstrated that for double
lap joints the point of maximum stress occurs out of the characteristic region.
In summary it can be affirmed that the point and the average stress criteria can
provide accurate results but they depend markedly on the previous determination
of the characteristic distance, which is usually performed by experimental tests.
However, the results can not be easily applied to different types of joints that do not
use the same materials and geometry. In fact, the lack of a physical meaning of this
characteristic distance renders impossible the establishment of general predictive
tools, not depending on a parameter which is a function of type of loading, materials
involved and joint geometry.
6.1.1.2 Fracture Mechanics
In the strength of materials approach, materials are assumed to be free of defects.
Unlike to what happens in strength of materials based approaches, the fracture mechanics approach assumes the presence of an inherent defect in the material, induced
during the fabrication process or during work. In this case the objective is to know
if the defects can induce catastrophic failure or if, during the predicted structure
life, they can propagate stably maintaining their dimensions inferior to the critical
size. There are two types of fracture mechanics criteria. The criteria can be based
on stress intensity factors or on the energetic concepts.
The stress intensity factor is defined as
√
(6.1)
K = Y σR π a
where Y is a non-dimensional factor depending on the geometry and loading distribution, σR the remote applied stress and a the crack length. It is assumed that crack
propagation occurs when the stress intensity factor attains its critical value
158
M.F.S.F. de Moura
Kc = σu
√
πa
(6.2)
where σu is the strength of the material.
The energetic criterion is based on the assumption that crack growth will occur
when the energy available at the crack tip (G – strain energy release rate) and due to
the applied loading, overcomes the critical strain energy release rate (Gc ), which is
a material property. The strain energy release rate is given by
G=
dW dU
−
dA
dA
(6.3)
where W is the work realized by external forces, U the internal strain energy and dA
the variation of crack surface.
It should be noted that G and K are intrinsically related. In fact, Irwin [21],
demonstrated that in plane stress
G=
K2
E
(6.4)
and in plane strain
K 2 (1 − v2 )
(6.5)
E
where E and ν are the Young’s modulus and Poisson’s ratio, respectively. These
relationships are also valid for the respective critical values (Gc and Kc ).
Crack growth can take place under three different modes, as it can be seen in
Fig. 6.2. The mode I represents an open mode and the others (mode II and mode III)
are the shear modes. In the majority of real situations the applied loading originates
a combination of modes at the crack tip, which implies that a mixed-mode criterion
should be considered in order to better simulate the damage propagation.
Several authors applied the fracture mechanics concepts to the failure strength
prediction of bonded joints. The majority of the proposed works are based on the
concepts of strain energy release rate. It is usually assumed that damage propagation
occurs when the strain energy at the crack front is equal to the critical strain energy
release rate, which is a material property. Kinloch [24] refers that energetic criterion is advantageous relatively to stress intensity factors approach. First the strain
energy release rate has an important physical significance related to the energy absorption. Second, the value of stress intensity factors is not easily determinable,
namely when the crack grows at or near to an interface. Ripling et al. [33] also
G=
Fig. 6.2 Failure modes
6 Progressive Damage Modelling
159
proposed the use of strain energy release rates instead of stress intensity factors recognizing the non-homogeneity of the bonded joints. However, it should be referred
that the application of energetic criterion is not absent of some difficulties related
to mixed-mode crack propagation. In fact, in homogeneous and isotropic materials
cracks tend to propagate under mode I, perpendicularly to the direction of maximum
principal stress, independently of the original crack. However, in a bonded joint, the
crack growth direction is restricted by the adherends which leads, in most cases, to a
mixed-mode crack propagation (I+II). Under these circumstances it is fundamental
to use adequate energetic criteria which are generally in the form of
GI
GIc
A
+
GII
GIIc
B
=1
(6.6)
where GIc and GIIc are the critical values in mode I and mode II. The linear energetic
criterion (A = B = 1) and the quadratic one (A = B = 2) are the most used.
One of the most popular methods based on fracture mechanics concepts is the
Virtual Crack Closure Technique (VCCT), which is detailed by Krueger [25]. The
method allows obtaining the strain energy release rates and is based on the assumption that, when a crack grows the energy released in this process is equal to the work
necessary to close the crack to its initial length before propagation. Figure 6.3 represents a two-dimensional problem where the crack grows from the node l to node i.
The method can be applied by two different ways. The first one consists in two steps.
In a first run, node l is closed and the forces in the mode I and mode II directions
at this node are registered. In a second run, node l is opened originating nodes l1
and l2 and, by applying the same loading, the relative displacements between these
nodes in the respective directions are also measured. However, the method can also
be applied in a sole run. In this case, it is necessary to use a refined mesh and to
assure that self-similar crack propagation exists. The energy necessary to close the
crack is obtained by multiplying the loads at node i by the relative displacements
between nodes l1 and l2
1
Δ E = (Xi Δ ul +Yi Δ vl )
(6.7)
2
where Xi , Yi represent the loads at the closed node i and Δ ul , Δ vl the difference of
displacements between nodes l1 and l2 . The respective strain energy release rates
can then be calculated by the product of the relative displacements at the “opened
point” (nodes l1 and l2 ) and the loads at the “closed point” (node i)
1
Yi Δ vl
2bΔ a
1
Xi Δ ul
GII =
2bΔ a
GI =
(6.8)
being bΔa the area of the new surface created by an increment of crack propagation
(see Fig. 6.3).
Using the energetic criterion expressed by Eq. (6.6), propagation will occur when
the values of GI and GII obtained from Eq. (6.8) satisfy the criterion. It should be
160
Fig. 6.3 The virtual crack
closure technique (VCCT)
M.F.S.F. de Moura
y
Elements
thickness = b
ul2
ul1
vl2
l2
Yi
vl1 Xi
x
i
l1
a
Δa
Δa
noted that the method can be easily applied in three-dimensional problems, where
mode III can be obtained using the same procedure described for the other modes.
It should be referred that the use of fracture mechanics criteria depends on
the existence of some kind of defect which is usually simulated as a pre-crack.
These initial cracks are artificially introduced and intend to simulate damage or
defects originated during the fabrication process or induced in service. Thus, it can
be affirmed that fracture mechanics based criteria are more adequate for damage
growth instead of its initiation. On the other hand, there are some difficulties associated to this approach. In fact, the initial crack length size and its locus constitute
two main difficulties characteristic of the method.
6.1.2 Actual Trends
The stress and fracture mechanics based criteria present some disadvantages. The
stress based methods present mesh dependency during numerical analysis due to
stress singularities. On the other hand, the point/average stress criteria require the
definition of a critical dimension which depends on loading, materials and joint
geometry, and do not have a physical theoretical foundation. Fracture mechanics
approach relies on the definition of an initial flaw or crack length. However, in many
structural applications the damage initiation locus is not obvious. On the other hand,
the stress-based methods behave well at predicting damage onset, and fracture mechanics has already demonstrated its accuracy in the crack propagation modelling.
In order to overcome the referred drawbacks and exploit the usefulness of the described advantages, cohesive damage models and continuum damage mechanics
emerge as suitable options. These methodologies combine aspects of stress based
analysis to model damage initiation and fracture mechanics to deal with damage
propagation. Thus, it is not necessary to take into consideration an initial defect
and mesh dependency problems are overcome. Chapters 7 and 8 also deal with this
methodology.
6 Progressive Damage Modelling
161
6.2 Cohesive Damage Models
The use of cohesive damage models in fracture problems has become frequent in
the most recent years. One of the most important advantages of cohesive models
is related to their capacity to simulate onset and non-self-similar growth of damage. No initial crack is needed and damage propagation takes place without user
intervention. They are usually based on a softening relationship between stresses
and relative displacements between crack faces, thus simulating a gradual degradation of material properties. They do not depend on a predefined initial flaw unlike
conventional fracture mechanics approaches. Typically, stress based and energetic
fracture mechanics criteria are used to simulate damage initiation and growth, respectively. Usually, cohesive damage models are based on spring [7, 26] or interface
finite elements [9, 30, 31] connecting plane or three-dimensional solid elements.
Those elements are placed at the planes where damage is prone to occur which,
in several structural applications, can be difficult to identify a priori. However, an
important characteristic of bonded joints is that damage propagation is restricted to
well defined planes corresponding to the ones near or at the interfaces between adhesive and adherends or inside the adhesive, thus leading to a typical application of
cohesive methods.
In the context of adhesive joints, some works considering the cohesive approach
have been reported. Gonçalves et al. [18] considered a mixed-mode cohesive damage model to simulate the debonding process of aluminium single-lap joints loaded
in tension. A triangular traction-separation law was used. The experimental loaddisplacement curves and failure loads were accurately predicted when the plastic behaviour of the materials was included in the analysis, for the entire range of overlap
lengths considered. Blackman et al. [2] used a cohesive zone model (CZM) approach
including two parameters, Gc and σmax , to study the fracture of adhesively bonded
joints. A polynomial traction-separation law was considered. The main objective
was to investigate the physical significance of σmax . Experimental and numerical
results were compared using the Tapered Double Cantilever Beam (TDCB) and peel
tests under mode I load. It was concluded that the specimen compliance and Gc depend on the value of σmax until a relatively high value of this parameter, when the
dependence significantly diminished. A CZM was also employed by Yang et al. [39]
to study the coupling between interface fracture and plastic strain of the adherends.
For the adhesive, a trapezoidal shape traction-separation law, including plasticity,
was used. The model was validated performing T-peel tests on adhesively bonded
aluminium joints. The same authors [41] considered a similar traction-separation
law for elastic-plastic mode II crack growth modelling. End Notched Flexure (ENF)
specimens subjected to a bending load, and undergoing extensive plastic strain prior
to failure, were used to validate the model. The main fracture parameters were determined comparing numerical and experimental results for a particular geometry, and
then applied to other geometries. Yang and Thouless [40] simulated the mixed-mode
fracture of joints bonded with ductile adhesives using a mode-dependent embedded
process zone (EPZ) model. Mode I and mode II fracture laws obtained from previous
works were combined with a mixed-mode failure criterion to provide quantitative
162
M.F.S.F. de Moura
predictions of the deformation and fracture of T-peel specimens and single-lap shear
joints. A linear toughness based criterion was used to access complete failure of the
EPZ elements and subsequent damage growth. Thouless et al. [35] used a cohesivezone approach to model the mixed-mode fracture of adhesively bonded Glass Fibre
Reinforced Plastic single-lap joints. Three and two parameter damage laws were
used for mode-I and for mode-II, respectively. The three-parameter (interface cohesive strength, characteristic strength and toughness) mode-I traction–separation law
was used in order to simulate interfacial cracking followed by fibre pull-out (experimentally observed for mode-I fracture). On the other hand, mode II tests indicated
that only few fibres were pulled out during mode-II fracture. Consequently, a two parameter (interface cohesive strength and toughness) trapezoidal traction–separation
law was used to simulate the behaviour of the adhesive layer in mode-II. Experimental and numerical curves revealed excellent agreement, including both the strengths
of the joints and the failure mechanisms. Andersson and Stigh [1] used an inverse
method to determine the cohesive parameters of a ductile adhesive layer loaded
in peel using equilibrium of energetic forces acting on a Double Cantilever Beam
(DCB) specimen. This method consists on fitting the load-displacement curves, up
to a predefined agreement is achieved, thus defining the used cohesive parameters.
They verified that the stress-relative displacement curve can be divided in three
parts. Initially the stress increases proportionally to the elongation (linear elastic
behaviour of the layer), until a limit stress is achieved. A plateau region is then
observed, corresponding to the plastic behaviour of the adhesive. The curve ends
with a parabolic softening part. Leffler et al. [29] performed an identical analysis to
determine the complete stress versus deformation relation of a thin adhesive layer
loaded in shear, using the ENF specimen. The used method included the determination of the energy release rate as a function of the shear deformation at the crack tip,
followed by derivation of the traction-separation relation using an inverse method.
An approximate trapezoidal relation was obtained.
6.2.1 Cohesive Damage Model Including Plastic Behaviour
A cohesive mixed-mode (I+II) damage model was developed in order to simulate
damage initiation and growth in bonded joints. The model is implemented via interface finite elements including a constitutive softening law which relates stresses (σ)
and relative displacements (δr ) between homologous points. The constitutive law
presents a trapezoidal shape to account for the ductile behaviour typical of many
adhesives (see Fig. 6.4). It should be noted that the triangular law, which can be
used for brittle or moderately ductile adhesives, is a particular case of this trapezoidal one. Before damage starts to grow
σ = Eδr
(6.9)
where E is a diagonal matrix containing the interface stiffnesses (ei , i = I, II). These
are defined as the ratio between the Young’s (mode I) or shear modulus (mode II),
6 Progressive Damage Modelling
163
σi
Pure-mode
model
Jic
i = I, II
σu,i
σum,i
Mixedmode model
Ji
δ1m,i
δ1,i
δ2m,i
δum,i
δ2,i
δu,i
δi
Fig. 6.4 The trapezoidal constitutive law for pure-mode and mixed-mode
and adhesive thickness. The effect of adhesive thickness is then included by this
way and it could be neglected in the finite element mesh. In the pure-mode damage
model, damage initiates when the relative displacement exceeds δ1,i . From this point
up to final failure (δu,i ) a progressive softening is simulated in order to account
for the different failure processes occurring in the vicinity of the crack tip. In this
region, known as the Fracture Process Zone (FPZ), several damage processes, like
plasticity and micro-cracking, take place, which is simulated by this softening law.
Damage evolution is implemented through a damage parameter ranging between
zero (undamaged) and one (complete failure). The softening relationship can be
written as
(6.10)
σ = (I − D)Eδr
where I is the identity matrix and D is a diagonal matrix containing, in the position
corresponding to mode i (i = I, II), the damage parameter, d. In the plateau region
the damage parameter can be defined as
d = 1−
δ1,i
δi
(6.11)
where δ1,i (i = I, II) is obtained from the initial stiffness (ei ) and local cohesive
strength in each mode σu,i (i = I, II). In the stress softening part of the curve,
d = 1−
δ1,i (δu,i − δi )
δi (δu,i − δ2,i )
(6.12)
The rigorous definition of the second inflexion point (δ2,i ) in pure modes is not
necessary since experience has shown that the precise details of the softening slope
are generally not very significant. This issue will be discussed in more detail in
the following sections. The maximum relative displacement, δu,i , at which complete
failure occurs, is obtained by equating the area under the softening curve to Jic ,
which corresponds to the respective critical fracture energy
164
M.F.S.F. de Moura
Jic =
σu,i
(δ2,i − δ1,i + δu,i )
2
(6.13)
As in the majority of cases, bonded joints are subjected to mixed-mode (I+II) loading, a cohesive mixed-mode damage model, which is an extension of the pure-mode
one, is also developed (Fig. 6.4). Damage initiation is predicted using a quadratic
stress criterion
σI 2
σII 2
+
= 1 if σI > 0
(6.14)
σu,I
σu,II
σII = σu,II
if σI ≤ 0
where σi and σu,i (i = I, II) represent, respectively, the stresses and local cohesive
strengths in each mode. It is assumed that normal compressive stresses do not induce
damage. Considering Eq. (6.9), the first Eq. (6.14) can be rewritten as a function of
the relative displacements
δ1m,I
δ1,I
2
+
δ1m,II
δ1,II
2
=1
(6.15)
where δ1m,i (i = I, II) are the relative displacements in each mode corresponding to
damage initiation. Defining an equivalent mixed-mode displacement
δm = δI 2 + δII 2
(6.16)
and a mixed-mode ratio
β=
δII
δI
(6.17)
the equivalent mixed-mode relative displacement at the onset of the softening process (δ1m ) is obtained combining Eqs. (6.15), (6.16), (6.17)
1+β 2
δ1m = δ1,I δ1,II
(6.18)
δ1,II 2 + β 2 δ1,I 2
Stress softening onset (δ2m ) is predicted using a relationship between the current
relative displacements in each mode and the critical relative displacements, similar
to the one considered for the damage initiation point (Eq. 6.15),
δ2m,I
δ2,I
2
δ2m,II
+
δ2,II
2
=1
(6.19)
Using a procedure similar to the one followed for δ1m , the mixed-mode relative
displacement at the onset of the stress softening process (δ2m ) can be obtained
6 Progressive Damage Modelling
165
δ2m = δ2,I δ2,II
1+β 2
δ2,II 2 + β 2 δ2,I 2
(6.20)
Crack growth is simulated by the linear fracture energetic criterion
JI
JII
+
=1
JIc JIIc
(6.21)
When Eq. (6.21) is satisfied, damage growth occurs and stresses are completely
released, with the exception of normal compressive ones. The energy released in
each mode at complete failure can be obtained from the area of the smaller trapezoid
of Fig. 6.4
σum,i
(δ2m,i − δ1m,i + δum,i )
(6.22)
Ji =
2
Combining Eqs. (6.16), (6.17), (6.22) and (6.21) it can be written
δum =
2JIc JIIc (1 + β 2 ) − (δ2m − δ1m )δ1m (eI JIIc + β 2 eII JIc )
δ1m (eI JIIc + β 2 eII JIc )
(6.23)
which corresponds to the ultimate relative displacement in mixed-mode.
The damage parameter can now be evaluated using the critical mixed-mode displacements (Eqs. (6.18), (6.20) and (6.23)) in Eqs. (6.11) and (6.12).
6.3 Evaluation of Cohesive Parameters
The accurate evaluation of cohesive parameters of the trapezoidal law is a fundamental task in order to obtain good strength predictions. The properties measured
from tests in bulk specimens are not representative because in bonded joints the
adhesive is a thin layer and behave differently than in bulk. The solution is to obtain the cohesive properties using specimens where the thickness of the adhesive is
similar to the one used in bonded joints. This can be achieved with the Double Cantilever Beam (DCB) and End Notched Flexure (ENF) fracture characterization tests
for pure mode I and mode II loading, respectively. The use of these tests to measure
the cohesive parameters is described below.
6.3.1 Fracture Energies
In the following, the application of the DCB and ENF tests to fracture characterization of adhesives is described. A new data reduction scheme avoiding the inevitable
inaccuracies present in the measurement of the crack length during propagation is
highlighted.
166
M.F.S.F. de Moura
6.3.1.1 The DCB Test
The DCB is a standardized test (ASTM D3433-99) to measure material fracture
properties under pure mode I loading. When applied to bonded joints it consists in
two beams bonded with the required adhesive thickness with a predefined initial
crack length. The load is applied in order to promote pure mode I loading at the
crack front (Fig. 6.5).
There are two classical data reduction schemes frequently used to obtain the fracture
energy in mode I (JIc ). The Compliance Calibration Method (CCM) is based on the
Irwin-Kies equation [23]
P2 dC
(6.24)
JIc =
2b da
being C the compliance (δ /P). Using this method, the values of load, applied displacement and crack length (P, δ and a) are registered in order to calculate the fracture energy during load history. Alternatively, the Corrected Beam Theory (CBT) is
also commonly used [10]. In this case, it is assumed that JIc can be obtained from
JIc =
3 Pδ
2b(a + |Δ|)
(6.25)
where Δ is a correction for crack tip rotation and deflection. Δ is determined from a
linear regression analysis of (C)1/3 versus a data.
It should be emphasized that both methods depend on crack length measurements
during its propagation. This is not easy to perform and remarkable errors can occur,
affecting the measured JIc . In order to overcome the referred difficulties, a new data
reduction scheme based on crack equivalent concept is proposed. The method does
not require crack length measurements and accounts for the energy dissipated at the
Fracture Process Zone (FPZ). The FPZ develops ahead of the crack tip in consequence of the nucleation of multiple micro-cracks through the adhesive thickness
and plastification. In fact, a quite extensive FPZ exists when ductile adhesives are
used. This non negligible FPZ affects the measured toughness as a non negligible
amount of energy is dissipated on it. Consequently, its influence should be taken
into account, which does not occur when the real crack length is used in a classical
data reduction scheme (Fig. 6.6).
The proposed method is based on beam theory and depends decisively on the
beam compliance. Consequently, it is named Compliance Based Beam Method
P
b
y
x
h
δ
P
a
Fig. 6.5 Schematic representation of the DCB test
6 Progressive Damage Modelling
167
Fig. 6.6 Schematic representation of the FPZ and crack
equivalent (ae ) concept
(CBBM). Following strength of materials analysis, the strain energy of the specimen due to bending and including shear effects is [14]
a 2
a h/2
Mf
τ2
dx +
U =2
b dy dx
(6.26)
0 2E1 I
0 −h/2 2G13
where Mf is the bending moment, I the second moment of area of the specimen, b
the specimen width, E1 and G13 the longitudinal and shear elastic properties of a
general orthotropic material and
3 P
4 y2
τ=
(6.27)
1− 2
2 bh
h
being y the coordinate along specimen thickness (Fig. 6.5). From Castigliano theorem the displacement δ , can be written as
δ=
8 Pa3
∂U
12 Pa
=
+
3
∂P
E1 bh
5bhG13
(6.28)
This equation constitutes an approach based on beam theory and does not account
for all effects influencing the specimen behaviour. For example, it is known that
stress concentrations arise around the crack tip, which is not taken into consideration by this approach. To overcome the referred discrepancies, an equivalent flexural modulus can be obtained from Eq. (6.28) and considering two initial conditions
taken from the experimental tests: the initial crack length a0 and initial compliance
C0 . Moreover, this approach takes into account the variation of the material properties between different specimens at it is based on the experimentally measured C0 .
Thus, Ef can be obtained from Eq. (6.28)
12(a0 + |Δ|) −1 8(a0 + |Δ|)3
Ef = C0 −
5bhG13
bh3
(6.29)
where Δ is the root rotation correction for the initial crack length (Fig. 6.6). In the
beam theory it is assumed that each arm of the DCB specimen is an encastred beam
whose length is equal to the crack length a. This parameter (Δ) can be achieved by
a linear regression of C1/3 = f(a0 ). The determination of Δ can be performed by
slightly loading the specimen with three different initial crack lengths (Fig. 6.7) in
order to define the C1/3 = f(a0 ) linear regression (Fig. 6.8).
168
M.F.S.F. de Moura
Fig. 6.7 Schematic
representation of the
compliance calibration as a
function of the crack length in
the DCB test
δ1, P1
δ2, P2 δ3, P3
a03
a02
a01
Alternatively, Wang and Williams [37] proposed another form of determining the
root rotation effects by altering the crack length using the parameter ΔI
ΔI = h
√
2
Γ
Ef
Ef E3
3−2
and Γ = 1.18
11G13
1+Γ
G13
(6.30)
which can be used instead of Δ in Eq. (6.29).
Owing to the difficulties inherent to crack monitoring during propagation and to
energy dissipation in the FPZ, an equivalent crack length ae (Fig. 6.6) is defined and
used instead of the measured crack length. The equivalent crack length is obtained
from Eq. (6.28) as a function of the specimen compliance registered during the test
and considering ae = a + |Δ| + ΔaFPZ instead of a. The solution of this equation can
be found using the mathematical software Matlab and is presented in Appendix.
The fracture energy in mode I can be obtained using Eq. (6.24)
6P2 2a2e
1
+
(6.31)
JIc = 2
b h h2 Ef 5G13
The presented methodology allows obtaining the fracture energy JIc only as function
of the P–δ data. For this reason it is designated by Compliance Based Beam Method
C1/3
(mm/N)1/3
Fig. 6.8 Schematic
representation of the crack
length correction due to root
rotation in the DCB test
Δ
a01
a02
a03
a (mm)
6 Progressive Damage Modelling
169
(CBBM). Using this method it is not necessary to measure the crack length during
propagation because the calculated equivalent crack length is used instead of the real
one. Another advantage is related to the fact that ae includes the effect of the FPZ,
which is not taken into account when the real crack length is considered. Moreover,
the specimen modulus (Ef ) is not a measured property but rather a computed one
(Eq. (6.29)) as a function of the initial compliance, thus accounting for the material
variability between different specimens. The only material property required in this
approach is the shear modulus G13 . However, from Eq. (6.31) it is straightforwardly
concluded that the term containing G13 is negligible relatively to the one of Ef . This
means that a typical value of G13 can be used and that it is not necessary to measure
it for each specimen.
6.3.1.2 The ENF Test
Up to now there is no standardized test to measure the fracture energy of bonded
joints under pure mode II loading. The most used are based on interlaminar fracture
characterization tests of composite materials. In this context the End Notched Flexure (ENF), End Loaded Split (ELS) and Four-Point End Notched Flexure (4ENF)
tests should be emphasized (Fig. 6.9).
Fig. 6.9 Schematic
representation of the ENF,
ELS and 4ENF tests
170
M.F.S.F. de Moura
The ELS test presents some difficulties in the measurement of fracture energy
related to large displacements effects and to sensibility of the clamping pressure
conditions [14]. On the other hand, the 4ENF test requires a quite complex setup
and some friction effects on the measured energy were detected [34]. The ENF
test is the most simple to execute and consequently is frequently used for materials
fracture characterization under pure mode II.
There are two main data reduction methods applied to ENF test in order to measure JIIc of bonded joints (Fig. 6.10). The CCM is also based on Eq. (6.24). The
compliance calibration can be made by two different ways. One of them consists
in performing bending tests using specimens with different initial cracks. This can
be achieved using only one specimen by altering its position in the supports. Alternatively, the calibration of C can be reached by measuring the crack length during
propagation. A cubic polynomial fitting should be carried out considering
C = Co + ma3
(6.32)
3ma2 P2
2b
(6.33)
which, using Eq. (6.24), leads to
JIIc =
The Corrected Beam Theory proposed by Wang and Williams [37], can also be
applied. The equation is
9P2 (a + |ΔII |)2
JIIc =
(6.34)
16b2 E1 h3
where ΔII is a crack length correction to account for shear. The authors showed that
ΔII = 0.42 ΔI , ΔI being the correction for mode I obtained for the DCB test (see
Eq. (6.30)).
It should be emphasized that experimental difficulties in measuring the crack
length are higher in mode II fracture characterization tests. In fact, due to applied
loading the crack tends to close during propagation which hinders the clear visualization of its tip. On the other hand, the quite large effect of the FPZ size [8] is
not accounted for when the real crack length is considered [11]. To overcome these
difficulties an equivalent crack procedure, similar to the one used for DCB tests, can
be followed to obtain the fracture energy under pure mode II loading using the ENF
specimen. As in the DCB test, the method is based on beam compliance and beam
theory and, consequently it is also named as CBBM.
P
P/4
ta
b
h
a0
Fig. 6.10 Schematic
representation of the ENF
test
P/4
P/2
L
2L
6 Progressive Damage Modelling
171
Employing the beam theory and Castigliano theorem, the equation of compliance
can be obtained
3L
3a3 + 2L3
(6.35)
+
C=
3
8E1 bh
10G13 bh
The equivalent flexural modulus is obtained using the initial compliance C0 and
initial crack length a0
−1
3a30 + 2L3
3L
Ef =
C0 −
8 bh3
10G13 bh
(6.36)
This procedure takes into account the variation of the material properties between
different specimens and for several effects that are not included in the beam theory,
e.g., stress concentration at the crack tip and contact between the two arms at the precrack region. In fact, these effects influence the P–δ curve even in the elastic regime,
and their effects are included when the initial compliance is used to determine the
Ef . Using Ef (Eq. (6.36)) instead of E1 in Eq. (6.35), an equivalent crack accounting
for the FPZ effects is achieved as a function of the beam compliance (C) which
varies during propagation
Cc 3 2
ae = a + ΔaFPZ =
a +
C0c 0 3
1/3
Cc
− 1 L3
C0c
(6.37)
where Cc and C0c are given by
Cc = C −
3L
;
10G13 bh
C0c = C0 −
3L
10G13 bh
(6.38)
JIIc can now be obtained using the Irwin-Kies relation (Eq. (6.24))
JIIc =
9 P2 a2e
16b2 Ef h3
(6.39)
where ae and Ef are given by Eqs. (6.37) and (6.36), respectively.
In summary, this data reduction scheme applied to fracture characterization in
mode I and in mode II presents several advantages. Using this methodology, crack
measurements are unnecessary. Experimentally, it is only necessary to register the
values of applied load and displacement. Therefore, the method is designated as
Compliance Based Beam Method (CBBM). Using this procedure the FPZ effects,
that are pronounced in mode II tests, are included in the fracture energy measurement. Moreover, the flexural modulus is calculated from the initial compliance and
crack length, thus avoiding the influence of specimen variability on the results. The
unique material property needed in this approach is G13 . However, its effect on the
measured fracture energy was found to be negligible [14], which means that a typical value can be used rendering unnecessary to measure it.
172
M.F.S.F. de Moura
6.3.2 Cohesive Critical Displacements
P (N)
In order to define completely the cohesive damage model it is also necessary to determine the critical displacements corresponding to the inflexion points (δ1,i , δ2,i ).
The first one is intrinsically associated to adhesive local strength (σu,i ) and the second one defines the extent of the plateau region (Fig. 6.4). As already referred, these
properties do not correspond to that of the adhesive as a bulk material.
The definition of the cohesive parameters is performed using an inverse method
applied to fracture tests in mode I (DCB) and mode II (ENF). The method consists in
three main steps. Initially, the experimental R-curve is obtained from the P–δ curve,
applying the CBBM (Fig. 6.11). The plateau region of the R-curve corresponds to
the fracture energy of the adhesive.
The second step consists in a numerical simulation of the test, using the cohesive damage model to simulate damage initiation and growth. In this simulation the
measured fracture energy is an inputted parameter in the numerical approach. The
other two cohesive parameters (δ1,i , δ2,i ) can be found by a trial and error procedure.
An iterative process should be performed until a good accuracy between the numerical and experimental P–δ curves is obtained, thus defining the cohesive parameters
(δ1,i , δ2,i ). This procedure depends on the number of parameters to be fixed. In fact,
if the two referred parameters should be accurately determined, a genetic algorithm
including an optimisation strategy should be used [16]. However, in some cases only
one parameter (δ1,i ) is necessary. This happens when brittle adhesives are used. In
140
120
100
80
60
40
20
0
0
1
2
δ (mm)
3
4
J II (N/mm)
0.8
0.6
0.4
0.2
0
35
37
39
41
a eq (mm)
43
45
Fig. 6.11 A typical P–δ curve of the ENF test (above) and the respective R-curve obtained using
the CBBM (below)
6 Progressive Damage Modelling
173
these cases, the trapezoidal law converts into a triangular one. Another case occurs
when the model is not sensible to the value of δ2,i . In this situation, this parameter
can be arbitrarily defined imposing, for the third part of the softening law, a symmetrical slope relatively to the initial linear part. de Moura et al. [15], performed a
parametrical study on single lap joints and verified that even for ductile adhesives,
there is a quite large range of values of δ2,i that do not alter the predicted joint
strength. The authors concluded that δ2,i can be defined as the point that leads to
the third part of the curve being symmetrical relatively to the initial linear part. This
choice implies that δ2,i falls in the values range leading to joint maximum strength.
When only one cohesive parameter has to be determined, the iterative process can
be easily performed. In fact, two or three attempts are usually enough to obtain
an acceptable agreement between the numerical and experimental P–δ curves, thus
leading to the definition of δ1,i .
6.3.3 Fracture Characterization of Bonded Joints
The method described above was applied to fracture characterization of carbonepoxy bonded joints under pure mode I (DCB test) and pure mode II (ENF test).
The adherends were unidirectional 0◦ lay-ups with sixteen layers of carbon/epoxy
prepreg (TEXIPREG HS 160 RM) with 0.15 mm of ply thickness, whose mechanical properties are presented in Table 6.1 [3]. Curing was achieved in a press during one hour at 130◦ C and 4 bar pressure. The adhesive Araldite 2015 from
Huntsmann (Basel, Switzerland) was used (E = 1850 MPa, ν = 0.3) [3]. The bonding process included roughening the surfaces to be bonded with sandpaper and
cleaning with acetone to increase the adhesion and avoid adhesive failure, followed
by assembly and holding with contact pressure and curing at room temperature.
6.3.3.1 DCB Tests
The nominal dimensions of the DCB specimen were 2h = 5 mm, L = 120 mm,
b = 15 mm, a0 = 45 mm and adhesive thickness equal to 0.2 mm. The specimens
were tested under a tensile loading using an INSTRON testing machine at room
temperature under displacement control. The loading rate was kept constant at
2 mm/min.
During the test the P–δ curves were registered in order to obtain the respective
R-curves using the proposed CBBM. The plateau region of the R-curve defines the
Table 6.1 Carbon-epoxy elastic properties [3]
E1 = 1.09E + 05 MPa
E2 = 8819 MPa
E3 = 8819 MPa
ν12 = 0.342
ν13 = 0.342
ν23 = 0.380
G12 = 4315 MPa
G13 = 4315 MPa
G23 = 3200 MPa
174
M.F.S.F. de Moura
80
70
60
P [N]
50
40
Experimental
30
Numerical
20
10
0
0
1
2
3
4
δ [mm]
5
6
7
Fig. 6.12 Comparison between numerical and experimental P–δ curves
JIc value which is an inputted parameter of the cohesive damage model. An iterative
procedure is followed until a good agreement between numerical and experimental
P–δ curves are achieved (Fig. 6.12), thus determining the remaining cohesive properties of the trapezoidal softening law. After some attempts, it was verified that local
strength σu,I (or δ1,I ) does not have too much influence and only δ2,I was varied to
fit the P–δ curves. σu,I was set equal to 23 MPa.
Afterwards, the numerical and experimental R-curves are also compared
(Fig. 6.13) in order to check the validity of the proposed model. Generally, good
agreement was observed for all tests (Table 6.2) for the inputted and numerically
obtained JIc – plateau values corresponding to stable crack propagation. The values
of the remaining cohesive parameter are also listed in Table 6.2.
6.3.3.2 ENF Tests
JI [N/mm]
The nominal dimensions of the ENF specimen are 2h = 5 mm, L = 100 mm,
b = 15 mm, a0 = 70 mm and adhesive thickness equal to 0.2 mm. The numerical-
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
Experimental
Numerical
60
65
70
aeq [mm]
75
80
Fig. 6.13 Comparison between numerical and experimental R-curves
6 Progressive Damage Modelling
175
Table 6.2 Comparison between numerical and experimental JIc and the obtained δ2,I
1
2
3
4
5
6
Average
St. Dev.
Experimental JIc
(N/mm)
Numerical JIc
(N/mm)
δ2,I
(mm)
0.484
0.444
0.420
0.333
0.406
0.468
0.425
0.054
0.482
0.441
0.415
0.331
0.404
0.466
0.422
0.055
0.021
0.019
0.018
0.014
0.018
0.020
0.018
0.003
experimental fitting procedures of P–δ curves were more complex in the ENF than
in the DCB test. In fact, some sensibility to the two inflexion points (δ1,II , δ2,II ) was
observed, which required a larger number of iterations in order to obtain a good
agreement between the P–δ curves (Fig. 6.14).
The CBBM also provides good agreement between the inputted and measured
JIIc (plateau values of the R-curves in Fig. 6.15), which demonstrates the adequacy
of the method in order to measure the fracture energy in pure mode II of bonded
joints. The values of the cohesive parameters obtained by the inverse method are
listed in Table 6.3 for five experimental tests.
In summary, it can be affirmed that the numerical model shows that the values
of fracture energies obtained by applying the CBBM are in excellent agreement
with the ones inputted in the cohesive damage model. This proves that the referred
data reduction scheme is able to reproduce the real values of fracture energy of
the specimens. Is should be noted that generally the predicted values are a little
bit inferior to the inputted ones. This can be explained by the presence of spurious
modes at the specimens’ edges. For example, de Moura et al. [13] demonstrated
that 1.14% of the total energy available at the specimen edges comes from mode III
loading.
800
P [N]
600
Experimental
Numerical
400
200
0
0
2
4
6
8
10
12
14
δ [mm]
Fig. 6.14 Comparison between numerical and experimental P–δ curves
176
M.F.S.F. de Moura
6
JIIc [N/mm]
5
4
3
Experimental
2
Numerical
1
0
70
75
80
ae [mm]
85
90
Fig. 6.15 Comparison between numerical and experimental R-curves
Table 6.3 Comparison between numerical and experimental JIIc and the calculated cohesive
parameters (σu,II and δ2,II )
1
2
3
4
5
Average
St. Dev.
Experimental JIIc Numerical JIIc
(N/mm)
(N/mm)
σu,II
(MPa)
δ2,II
(mm)
4.88
5.11
4.72
4.36
4.35
4.68
0.33
25.3
23.5
25.6
25.0
20.0
23.9
2.31
0.151
0.169
0.146
0.056
0.153
0.135
0.045
4.85
4.98
4.72
4.24
4.35
4.63
0.32
It should also be emphasized that the average values of cohesive parameters obtained by this inverse method allow defining the constitutive cohesive laws in pure
mode I and in pure mode II. These laws can then be used in progressive damage
simulations of bonded joints with this type of adherends and adhesive in order to
predict their mechanical behaviour and strength.
6.3.4 Application to Bonded Joints
In order to verify the performance of the presented cohesive damage model on joint
strength prediction, single lap bonded joints were tested experimentally. The adherends had sixteen layers of unidirectional carbon-epoxy (Table 6.1) giving a nominal thickness of 2.5 mm. The remaining nominal dimensions of the joints were
240 mm length, 15 mm width and adhesive thickness of 0.2 mm. Eight different
overlap lengths (ranging between 10 and 80 mm) were considered. Five specimens
of each case were tested and the maximum load was registered and used to define
the joint strength. Cohesive failures were observed in all cases.
6 Progressive Damage Modelling
177
Maximum load (kN)
20
15
10
Experimental
5
Numerical
0
0
20
40
Overlap length (mm)
60
80
Fig. 6.16 Comparison between numerical and experimental single lap joint strengths as a function
of overlap length
Numerical simulations were also performed considering the average real dimensions of each group of five specimens for each tested case. The average cohesive
properties measured under pure mode I and under pure mode II in the previous sections were used as inputted data. The maximum loads obtained from these numerical
simulations are compared with the experimental ones in Fig. 6.16. Generally, good
agreement was obtained for all overlap lengths.
6.4 Continuum Damage Models
The cohesive models require the knowledge of the critical zones where damage is
prone to occur in order to place the cohesive elements in these critical regions. Continuum damage models can constitute an appealing alternative when damage propagation onset and path are not known a priori. In these models the material properties
degradation occurs inside of the solid elements which avoids the use of special interface elements. The application of continuum damage models in the context of
bonded joints can be considered interesting when adhesive thickness should be considered. In fact, it is known that adhesive thickness can influence the mechanical
properties of the joint [27]. Another example is related to crack propagation occurring near to one interface due to stress concentration [17]. This non-symmetric
propagation can induce local mixed-mode loading which is not captured by cohesive damage models where adhesive thickness is neglected. In these cases the adhesive should be simulated by solid elements with the corresponding properties. The
progressive damage is simulated by including the damage onset and propagation
criteria in the formulation of solid elements. Usually this is performed by means of
a user subroutine in standard software where the material properties are degraded
according to the chosen criteria.
An approach similar to the one used in the cohesive damage model described
in Sect. 6.3 but considering a triangular softening law (see Fig. 6.17), instead of a
trapezoidal one, is proposed by several authors [5, 12, 32]. In this case there is a
178
M.F.S.F. de Moura
σm
σum
δ1m
δum
δm
Fig. 6.17 Softening stresses/displacements relationship for the continuum damage model
softening relationship between stresses and strains instead of between stresses and
relative displacements considered in the cohesive model. Consequently, a characteristic length lc must be introduced to transform the relative displacement into an
equivalent strain
1
Jic = σu,i εu,i lc,i
(6.40)
2
This parameter is equal to the length of influence of a Gauss point in the given
direction and physically can be regarded as the dimension at which the material acts
homogeneously. It can be obtained from the coordinates of Gauss points available
during simulation. The stress–strain relation can be written considering an equation
similar to Eq. (6.10)
σ = (I − D)Cε
(6.41)
being C the stiffness matrix of the undamaged material in the orthotropic directions.
Assuming that damage occurs in mixed-mode (I+II) the model described previously
can be adopted.
For the triangular softening model (Fig. 6.17) the damage parameter is given by
d=
δum (δm − δ1m )
δm (δum − δ1m )
(6.42)
The material properties at a given Gauss point are smoothly degraded according to
the assumed criterion, thus simulating the energy releasing at the FPZ. This leads
to load redistribution for the neighbouring points, simulating a gradual propagation
process and avoiding the stress singularity effects and minimizing the mesh sensitivity effects. As in the cohesive damage model, damage onset and growth are
simulated by a quadratic stress criterion and a linear energetic one, respectively,
allowing the definition of the critical displacements to be used in Eq. (6.42). More
details of the referred model are presented in [15].
6 Progressive Damage Modelling
179
6.5 Conclusions
Adhesive bonded joints are nowadays widely used in structural applications. Consequently, the development of properly design methods is fundamental to increase
the confidence of designers in using adhesive bonded joints. Mixed-mode cohesive
damage models join the positive arguments of stress based and fracture mechanics
criteria overcoming their inherent difficulties. They are presently prominent numerical tools in order to simulate the behaviour of these joints.
In this work a cohesive mixed-mode damage model adequate to the simulation of
ductile adhesives is presented. The trapezoidal constitutive softening law simulates
the adhesive behaviour including its thickness and plastic behaviour.
An important issue of the model is related to the determination of the cohesive
properties. In fact, it is known that adhesive bulk properties are not exactly the same
as the ones of a thin adhesive layer. Therefore, they were determined considering
fracture tests (DCB and ENF) where the testing conditions are similar to the ones
existing in bonded joints. An inverse method was used to define the cohesive parameters. An iterative procedure was followed in order to fit the numerical and
experimental P–δ curves, thus defining the constitutive cohesive laws in the two
modes.
A new data reduction scheme based on crack equivalent concept was also proposed to measure the fracture energies in the two fracture characterization tests.
The method does not require crack monitoring during its propagation and account
for the energy dissipated in the fracture process zone which is not negligible, mainly
when ductile adhesives are used. The method was validated by experimental tests
and numerical simulations.
Continuum damage models can be a valuable alternative to cohesive damage
models especially when adhesive thickness plays an important role on the behaviour
of the joint. In this case the adhesive is simulated by solid elements instead of interface ones. The advantage of this approach is related to damage onset and crack
propagation path which are better simulated namely when non-symmetric paths can
influence the results. The softening laws are applied to the solid elements degrading
the material properties according to the predefined criteria. It should also be noted
that continuum damage models are more complicated and can lead to numerical
problems related to convergence difficulties. Further developments are necessary in
this context.
It should be emphasized that the two models can also be used in a bonded joint
simulation. In fact, when failure of adherends, for example in composites, is also
an issue to be considered, continuum and cohesive damage models can be used to
simulate the eventual failure of adherends and adhesive, respectively. The future
trends point to the development of a numerical tool incorporating the two kinds of
models. The two methods can coexist, although till now there are some unsolved
numerical problems. In fact, both models can be implemented via user subroutines
in commercial standard software.
180
M.F.S.F. de Moura
Acknowledgements The author thanks the Portuguese Foundation for Science and Technology
for supporting part of the work here presented, through the research project POCI/EME/56567/
2004.
The author also thanks Dr João Paulo Moreira Gonçalves, Dr José Augusto Gonçalves Chousal
and Mr Raul Duarte Salgueiral Gomes Campilho (PhD student) for their valorous collaboration,
advices and discussion about the matters included in this chapter.
Appendix
Equation (6.28) can be expressed as,
α a3e + β ae + γ = 0
(6.43)
where the coefficients α , β and γ are, respectively
α=
8
;
bh3 Ef
β=
12
;
5 bhG13
γ = −C
(6.44)
Using the Matlab software and only keeping the real solution we have,
ae =
being
1
2β
A−
6α
A
13
4β 3 + 27γ 2 α
α2
1 − 108γ + 12 3
α
(6.45)
A=
(6.46)
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Chapter 7
Modelling Fatigue in Adhesively Bonded Joints
Ian A. Ashcroft and Andrew D. Crocombe
Abstract Fatigue is widely recognised as one of the most common causes of mechanical failure and hence the issue of fatigue in bonded joints must be addressed
if adhesives are to find wider usage in structural applications. Fatigue failure is
difficult to predict accurately and fatigue in bonded joints is complicated by the
multi-component nature of bonded joints and by the complex stress distributions and
material behaviour. Various methods of modelling the fatigue behaviour of bonded
joints have been proposed, varying in complexity, applicability and degree of empiricism. The aims of the models are either to predict the number of load cycles
until a certain event occurs (such as macro-crack initiation or complete failure) or
to predict the rate of change of crack length (or some other measure of damage)
as a function of cycles. In this chapter the main approaches to modelling fatigue in
adhesive joints are outlined and examples given of the application of each approach.
It is seen that there are many practical modelling routes, the choice being dependent
on what is required and the availability of resources, data and expertise. However,
it cannot be said at present that there is a generally applicable method of modelling
fatigue in bonded joints that is robust, reliable and mechanistically accurate. However, the further development of advanced computational methods, such as finite
element analyses incorporating progressive damage, and increased understanding
of the mechanisms of fatigue in bonded joints will continue to provide drivers for
improved modelling methods.
Ian A. Ashcroft
Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University,
Loughborough, Leicestershire, LE11 3TU, UK, e-mail: i.a.ashcroft@lboro.ac.uk
Andrew D. Crocombe
Division of Mechanical, Medical and Aerospace Engineering, School of Engineering,
University of Surrey, Guildford, GU2 5XH, UK, e-mail: a.crocombe@surrey.ac.uk
L.F.M. da Silva, A. Öchsner (eds.), Modeling of Adhesively Bonded Joints,
c Springer-Verlag Berlin Heidelberg 2008
183
184
Ian A. Ashcroft, Andrew D. Crocombe
7.1 Introduction
Fatigue, in an engineering sense, is the failure of a structure under a repetitive or
cyclic loading regime in which the loads involved are considerably lower than those
involved in instantaneous, or quasi-static, failure. Fatigue is a significant subject for
study for many reasons. Firstly, fatigue loading is seen in nearly all major engineering structures, e.g. aircraft, ships, cars, buildings, bridges, as well as in many nonengineering applications, such as sports equipment and even human parts, such as
knees. Secondly, it can result in sudden catastrophic failure after years of apparently
safe service. Long periods can be spent in the initiation phase of fatigue damage,
in which there may be no outward signs of damage, failure can then quickly accelerate in the final stages. Fatigue damage can be initiated or accelerated by many
factors such as accidental impact, over-loading, corrosion, abrasion etc. Thirdly, fatigue failure is notoriously difficult to predict accurately. A problem compounded
by the fact that the in-service loading and environment are seldom known to a great
degree of accuracy. Hence, it is very difficult to design against fatigue failure without resorting to large safety factors, and hence incurring structural inefficiencies.
Monitoring fatigue damage can also be difficult, particularly if initiation is in an
inaccessible location or if the critical crack size before rapid fracture is very small.
It can be summarised, therefore, that predicting fatigue failure in a structure is both
extremely important and rather difficult.
The advantages of adhesive bonding compared to other joining techniques are
now well recognised and two of the frequently quoted benefits are that stress concentrations are more uniformly distributed than in riveted or bolted joints and that
the bonding process does not explicitly weaken the adherends (although there are
likely to be stress concentrations in the adherend in the joint area). It might be expected, therefore, that adhesively bonded joints perform well in fatigue, and indeed
this is often the case. However, a number of potential problems for adhesive joints
subjected to fatigue should also be recognised. Adhesives are, like most materials,
susceptible to fatigue and hence if fatigue loading is expected in an application then
the fatigue resistance of the adhesive should be investigated. It is well known that
adhesives and the interfacial region between adhesive and adherend are sensitive to
the environment and this will affect the fatigue resistance of the joint. Adhesives are
also susceptible to creep under certain conditions and combined with fatigue this
can lead to accelerated failure (Hart-Smith 1981; Harris and Fay 1992; Ashcroft
et al. 2001). It should also be remembered that failure in a bonded joint can occur in
the adhesive, in the adherend or in the interfacial region between the two and that the
relative fatigue resistance of the various components is dependent on many factors,
such as geometry, environment and loading, and may vary as damage progresses.
For example, in a bonded composite joint subjected to fatigue loads and moisture
we have a complex system in which we may have multiple and changing initiation and propagation sites and mechanisms as the dynamic effects of progressive
fatigue damage and moisture absorption and desorption occur (Ashcroft et al. 1997,
1999, 2000, 2001a; Abdel Wahab et al. 2001, 2001a). It is not surprising, therefore that many of the initial studies of fatigue in bonded joints were either of a
7 Modelling Fatigue in Adhesively Bonded Joints
185
highly empirical nature or involved simplified systems and it is only in recent years
that the fatigue behaviour of bonded joints under loading and environmental conditions more representative of those seen in-service have been explored (Al-Ghamdi
et al. 2003; Erpolat et al. 2004; Ashcroft et al. 2005; Nolting et al. 2008).
Before discussing the methods of modelling fatigue in bonded joints, it is useful
to discuss the reasons why one may wish to do so. Perhaps the obvious reason is to
support the design of bonded structures, to ensure that fatigue failure is not likely
to occur in service and to aid in the design of efficient fatigue resistant joints; resulting in safer, cheaper and higher performance structures. There is no doubt that
modelling can indeed help in these objectives, however, it should also be recognised that the current state of confidence in such modelling means that in most
cases the modelling must accompany a complimentary testing programme. Another
reason for modelling is to help to understand the mechanisms involved in fatigue
failure. This can be achieved through comparing experimental results from carefully designed tests with the predicted results from progressive damage models.
Although the micro-mechanisms of damage are rarely an explicit part of the models, they can be used to inform the nature of continuum damage laws incorporated
into the model and hence comparison of the experimental and modelling results
enables insight into the damage mechanisms and the development of more physically accurate models. A third reason for modelling is to support in-service monitoring and re-lifeing of structures. In this case it is necessary to correlate externally
(or internally) measurable parameters with their consequences, in terms of progressive damage of the structure. For example, in recent work (Zhang and Shang 1995;
Crocombe et al. 2002, 2005; Graner-Solana et al. 2007; Shenoy et al. 2008) it has
been shown that simple back-face strain measurement can be used to monitor the
various stages of fatigue damage in a bonded single lap joint and hence continuous
monitoring of such a signal can be used to monitor that the joint is performing as
predicted and to initiate timely intervention should the measurements indicate a potentially significant change in the structure. It should be recognised that the three
activities involving the modelling of bonded joints indicated above are intrinsically
linked.
Now we have discussed why we should model the fatigue of bonded joints the
purpose of the rest of this chapter is to discuss the ways in which this may be done.
However, the scope of the chapter should be stated first. Fatigue in bonded joints
can be modelled at many different levels and using many fundamentally differing
approaches, ranging from modelling events at a molecular level to the analysis of
complete structures. In this chapter we will mainly be looking at failure from the
view of the mechanical engineer, that is generally in the 10−4 to 100 m scale where
fracture mechanics, continuum mechanics and damage mechanics are applicable.
The main goals in the modelling of fatigue behaviour are to (i) predict the time (or
no. of cycles) for a certain event to occur (such as macro crack formation, critical
extent of damage or complete failure) or (ii) to predict the rate of change of a fatigue
related parameter such as crack length or ‘damage’. The various methods of doing
this are presented in Sect. 7.3 of this chapter and the approach used is to introduce
and describe each of the main methods that have been used to date, together with
186
Ian A. Ashcroft, Andrew D. Crocombe
one or two examples. In this way the aim is not to attempt to summarise all the work
that has been done in this field but rather to give the potential modeller an introduction to the various methods which could be used. However, before the various
methods are described it is useful to discuss some of the overarching issues pertinent
to modelling fatigue.
7.2 General Considerations
7.2.1 Fatigue Loading
Stress
It is worth considering in a little more detail what exactly is meant by a fatigue load.
The main feature is that the load varies with time and it is useful to characterise the
load spectrum in terms of peaks and troughs in the varying load, with a cycle being
defined as the time between adjacent peaks (or troughs) and the frequency being the
number of cycles in a unit time. The fatigue spectrum can be characterised in terms
of the applied load or displacement for simple samples but in more complex structures it is more useful to think in terms of the cyclic stress (or other parameter, such
as strain energy release rate) at a point of interest. It is common in laboratory experiments to represent fatigue as a constant amplitude, sinusoidal waveform, as shown
in Fig. 7.1. Key fatigue parameters are defined in Table 7.1 (noting that similar parameters can be defined in terms of load, displacement, stress intensity factor etc.
rather than stress).
Only frequency plus two of the other parameters above in Table 7.1 are needed
to characterise a constant amplitude spectrum. Important considerations for adhesively bonded joints are the mean stress and the frequency. As adhesives tend to be
viscoelastic/viscoplastic in nature, then a tensile mean load can lead to progressive
Sa
Smax
S
Smn
Smin
Time
T
Fig. 7.1 Constant amplitude, constant frequency sinusoidal waveform
7 Modelling Fatigue in Adhesively Bonded Joints
187
Table 7.1 Parameters used to describe fatigue spectra
Maximum stress
Smax
Minimum stress
Smin
Stress amplitude
Sa =
Period
Smax − Smin
2
Smax + Smin
Smn =
2
Δ S = Smax − Smin
Smin
R=
Smax
T (sec)
Frequency
f=
Mean stress
Stress range
Stress ratio
1
(Hz)
T
1.2
1
1
0.8
0.8
stress
stress
1.2
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
1
2
3
4
5
0
1
2
time
(a)
4
5
(b)
1.2
1.2
1
1
0.8
0.8
stress
stress
3
time
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
1
2
3
time
(c)
4
5
0
1
2
3
4
5
time
(d)
Fig. 7.2 Alternative waveforms: (a) square, (b) trapezium, (c) saw-tooth, (d) spike
creep of the joint over time. This is exacerbated at low frequencies, where time
under load may become as significant as number of cycles in defining failure. However, at high frequencies, hysteretic heating may also lead to premature failure. Most
bonded joints are designed for tensile loading and if the joint is subjected to accidental compressive loading then buckling may occur, from which high peel forces
arise leading to rapid fracture.
188
Ian A. Ashcroft, Andrew D. Crocombe
stress
Other idealised forms of fatigue loading that may be used in the laboratory to
investigate behaviour under specific conditions are shown in Fig. 7.2. For example
an asymmetric saw tooth waveform may be used to investigate the relative importance of the loading and unloading segments of a fatigue cycle. Crack growth tends
to occur only in the loading part of the wave, whereas creep can occur whenever
a sufficient load is applied, hence, varying loading, unloading and hold segments
in a wave can help to investigate the relative importance of creep and fatigue. In
real applications it is rare that the idealised types of loading shown in Figs. 7.1
and 7.2 will be experienced. It is more likely that frequency, amplitude, mean and
waveform will all vary with time, as illustrated in Fig. 7.3. Moreover, it is likely
that there will be different types of loading associated with different events, for example in a car there may be a variable high frequency, low amplitude fatigue load
associated with the interaction of the tyres with the road superimposed with lower
frequency, higher amplitude fatigue associated with common manoeuvres such as
cornering or braking. There are a number of ways to deal with such complex loading. One is to attach a measuring device to a part in-service to characterise typical
events and then to create a spectrum from this information to represent, for example, a typical aircraft flight, including take off, in-air manoeuvring and landing etc.
The resultant spectrum can then be run repeatedly to look at the effect of multiple
such events. The main drawback of this method is that experiments tend to be rather
lengthy and may still not be useful if ‘out of the ordinary’ events are not accounted
for. It is generally preferable, therefore, to reduce the spectrum. In metals, where
the materials are generally rate insensitive, an easy method of accelerating tests is
to compress the spectrum and test at high frequency, although any time dependent
effects, such as corrosion, will not be represented accurately in such a test. When
devising accelerated service simulation tests for adhesive joints, care must be taken
when introducing acceleration techniques that any influential time dependent or load
sequencing effects are retained in the spectrum. In recent work, modelling of both
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
d
e
c
b
a
f
g
0
10
20
time
30
40
Fig. 7.3 Spectrum illustrating various features seen in-service: (a) low rate loading ramp, (b) mean
change, (c) amplitude change, (d) frequency change, (e) hold period, (f) high rate unloading ramp,
(g) unloaded period
7 Modelling Fatigue in Adhesively Bonded Joints
189
time dependent (Al-Ghamdi et al. 2003; Ashcroft 2005) and load sequencing effects
(Erpolat et al. 2004, 2004a; Ashcroft 2004) in bonded joints have been investigated
and such modelling is an important complement to any test programme for bonded
joints using accelerative techniques.
7.2.2 Fatigue Initiation and Propagation
Fatigue is generally divided into initiation and propagation phases. In adhesively
bonded joints the differentiation between these two phases, and even if there really
are two such phases, is still a contentious issue. However, at the predictive modelling
level there is a distinction between how a propagating crack is analysed and how the
number of cycles before a macro crack has formed can be predicted and this can be
used to differentiate between the initiation and propagation phases. It is clear that fatigue initiation in adhesives is complex and mechanistically different to that in metals. It can also vary with material, geometry and environment. Adhesives themselves
tend to be multi-component systems, with filler particles, carrier mats and toughening particles typically added, and failure may involve many mechanisms, including;
matrix micro-cracking (initiation, growth and coalescence), filler particle fracture or
debonding, cavitation of rubber toughening particles and debonding of carrier mat
fibres. An intrinsic part of fatigue failure in a bonded joint may also involve failure,
or damage, in the adherend or in the interfacial region between adherend and adhesive, and hence these possible modes of failure should also be considered in any
predictive model. An additional difficulty in characterising fatigue failure in bonded
joints from a mechanistic approach is that the initial damage tends to be internal
and hence detection and experimental characterisation is difficult, especially if this
is attempted non-destructively. In terms of in-service inspection, initiation may be
linked to the detectability of flaws. In most cases a purely mechanistic definition of
the initiation and propagation phases is hence unrealistic and a more pragmatic approach must be taken. In terms of the modeller a useful differentiation is to treat the
fatigue damage as an initiation phase until a sufficient crack has formed that further
growth can be predicted using fracture mechanics.
Mechanistically there is a similar blurring between ‘damage’ and ‘cracking’ as
a region considered as damaged rather than cracked may contain micro-cracking.
In terms of finite element modelling, damage is often modelled by reducing material properties (the continuum damage mechanics approach), usually stiffness,
of the element, and cracking is modelled by detaching element at nodes, after
which fracture mechanics methods can be used to model crack propagation. In
cohesive zone modelling both damage and crack growth are represented by using specialised elements to join adjacent continuum elements (Loh et al. 2003;
Crocombe et al. 2006; Liljedahl et al. 2007, 2007). This topic is described in detail in
Chap. 6.
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Ian A. Ashcroft, Andrew D. Crocombe
7.2.3 Experimental Testing
Although this chapter is mainly concerned with how to model fatigue in bonded
joints it is impossible to do so without the use of the results from experimental
tests. Firstly, tests are required to generate the mechanical properties (and potentially physical and hygro-thermal properties) for each component of the joint. The
properties required will depend on the material constitutive model deemed appropriate for each component. A potential difficulty here is that adhesives tend to have
rate dependent mechanical properties and the strain rate experienced by an adhesive
in a joint will vary significantly both temporally (due both to the cycling and progressive damage) and spatially. Attempting to include this effect in the model may
be difficult and usually a compromise value is used. This is likely to be perfectly
adequate where the temperature is well below the glass transition point but may be
less acceptable at higher temperatures.
Experimental testing is also required to provide the parameters associated with
the particular fatigue damage model that is to be applied. For example if a crack
growth model is to be applied then a fracture mechanics test will have to be carried
out to provide the constants relating the chosen fracture mechanics parameter (typically strain energy release rate, G) and the crack propagation rate. Different tests
may have to be carried out for failure in the adhesive, in the adherend and at the
interface between the two. In addition, it may be necessary to carry out these tests
under different environmental conditions. Samples for these tests are similar to those
used for determining the fracture parameters in quasi-static loading, e.g. for mode I
loading a double cantilever beam (DCB) sample, as shown in Fig. 7.4a is typically
used. In these tests it is essential to accurately measure crack growth as a function
of cycles during the test and this may be achieved by optical means or the use of
crack gauges (e.g. Ashcroft and Shaw 2002; Erpolat et al. 2004a, Ashcroft 2004). It
may also be useful to measure displacement at the point of loading if testing is in
load control.
Another class of tests are those used to validate the fatigue modelling. These tests
may also be similar to those used in standard quasi-static testing, such as the single
lap joint (SLJ) and double lap joint (DLJ) shown in Fig. 7.4b,c respectively. In these
tests the number of cycles to failure may be simply recorded or an attempt may be
made to monitor crack (or damage) progression, e.g. though optical or back-face
strain measurements (Zhang and Shang 1995; Crocombe et al. 2002, 2005; GranerSolana et al. 2007; Shenoy et al. 2008). In addition to in-situ measurements, damage
and crack progression may be investigated in post-failure examination and it is also
instructive to examine samples that have only been partially fatigue tested, e.g. by
sectioning (Crocombe et al. 2002; Graner-Solana et al. 2007; Shenoy et al. 2008),
ultrasonics or x-radiography (Ashcroft et al. 1997, 2003; Ashcroft 2004). If crack
propagation rate is to be measured, then a longer overlap length than that used in
standard quasi-static tests may be desirable. A common test geometry used to look
at fatigue initiation and propagation in adhesively bonded joints is the lap-strap
joint (LSJ) shown in Fig. 7.4d (Ashcroft et al. 2001a, 2003, 2008; Johnson 1987;
Mangalgiri et al. 1987; Mall and Yun 1987). This tends to be more representative
7 Modelling Fatigue in Adhesively Bonded Joints
191
(a)
25mm
100mm
12.5mm
(b)
(c)
(d)
Fig. 7.4 Typical fatigue test samples: (a) DCB, (b) SLJ, (c) DLJ, (d) LSJ
192
Ian A. Ashcroft, Andrew D. Crocombe
of applications with long bondlines, such as bonded stiffeners (stringers) in aircraft
wings, than the simple lap joints. This differs from the standard fracture mechanics
test in that it is often used to look at initiation as well as propagation. Also, fracture
is mixed mode and the mode-mixity varies with crack length. Fatigue behaviour in
the lap-strap joint can be markedly different from that in simple, short overlap joints,
such as the SLJ and DLJ, because, as the strap adherend spans the loading points
there is no definitive failure point for the joint and accumulative creep is restricted.
The fatigue testing of adhesive lap joints is covered by the standards BS EN
ISO 9664:1995 and ASTM D3166-99. In the former it recommends that at least
four samples should be tested at three different stress amplitude values for a given
stress mean, such that failure occurs between 104 and 106 cycles. This standard also
advises on statistical analysis of the data. In general, fatigue data exhibits greater
scatter than quasi-static data and this needs to be taken into account when using
safety factors with fatigue data. Further advice on the application of statistics to
fatigue data is found in BS 3518-5:1966.
7.3 Modelling Methods
7.3.1 Total-Life Methods
7.3.1.1 Stress-Life Approach
In the total life approach, the number of cycles to failure (N f ) is plotted as a function
of a variable such as stress or strain amplitude. Where the loading is low enough that
the deformation is predominantly elastic a stress variable (S) is usually chosen and
the resultant plot is termed an S-N curve, or Wöhler plot, and this is known as the
stress-life approach. Under these conditions a long fatigue life would be expected
and this is termed high cycle fatigue (HCF). The S-N data is either plotted as a loglinear or a log-log plot and a characteristic equation can be obtained by empirical
curve fitting, using the forms of equation shown below.
S = C + D log N f
(7.1)
S = AN f α
(7.2)
The constants in the equations are dependent on many factors, including; material,
geometry, surface condition, environment and mean stress. Hence, caution should
be used when trying to apply S-N data beyond the samples used to generate the data.
The standard stress-life method gives no indication of the progression of damage,
although in some cases the onset of cracking is indicated on the plot in addition
to the complete failure, hence allowing the initiation and propagation phases to be
differentiated. The above factors mean that the S-N curve is of rather limited use
in predicting fatigue behaviour, however, it is still useful in fatigue modelling as a
7 Modelling Fatigue in Adhesively Bonded Joints
193
validation tool. Wöhler (1867) noted a stress below which a nominally infinite life
is seen, which is termed the fatigue or endurance limit. This concept can potentially
be used more widely in predictive modelling, as described further in Sect. 7.3.12.
An S-N curve for epoxy/CFRP (carbon fibre reinforced polymer) double lap
joints with the geometry shown in Fig. 7.4b can be seen in Fig. 7.5. In this case
N f is plotted as a function of the average shear stress amplitude. It should be noted,
however, that there is no particular relation between the average shear stress in a
joint and failure load, for quasi-static or fatigue failure. For this reason, load rather
than stress is often used in total-life plots for bonded joints, to avoid any false conclusions being drawn regarding a relationship between the average shear stress and
fatigue life. Three regions can be seen in Fig. 7.5, these can be roughly divided
into the low cycle fatigue (LCF) region below approximately 1000 cycles, the high
cycle fatigue (HCF) region between approximately 1000 and 100,000 cycles and
the endurance limit region above 100,000 cycles. Both the quasi-static strength and
fatigue resistance decreases when the temperature is raised from 22 to 90◦ C. The
ratio between fatigue limit and quasi-static strength is approximately 0.36 at 22◦ C
and 0.26 at 90◦ C, showing that the joints are also proportionally more susceptible
to fatigue at 90◦ C. Ashcroft et al. (2001) attributed the accelerated fatigue failure of
the joints at 90◦ C to the combined effects of creep and fatigue.
As mentioned above, the fatigue life of bonded joints depends not only on the
stress amplitude but also on the mean stress. This is illustrated in Fig. 7.6a for a steelepoxy SLJ. The mean level is represented by the R-ratio (min. stress/max. stress),
where an increasing value of R indicates an increasing mean for a given amplitude.
It can be seen that either increasing the mean or increasing the amplitude results
in a reduction of the fatigue life. The relationship between amplitude and mean on
the fatigue life can be illustrated in a constant-life diagram in which combinations
of mean and amplitude are plotted for a given fatigue life. A constant life diagram
plotted from the data in Fig. 7.6a is shown in Fig. 7.6b.
20
22°C
Stress amplitude (MPa)
18
90°C
16
22°C (unbroken)
14
90°C (unbroken)
12
10
8
6
4
2
LCF
HCF
threshold
0
1
10
100
1000
10000
100000 1000000
No. cycles to failure
Fig. 7.5 S-N curve for epoxy/CFRP double lap joints (data from Ashcroft et al. 2001)
194
Ian A. Ashcroft, Andrew D. Crocombe
10
R = 0.1
8
Load range, kN
Fig. 7.6 (a) Effect of R-ratio
on fatigue life, (b) Constant
life diagram (data from
Crocombe and Richardson
1999)
R = 0.5
R = 0.75
6
4
2
0
100
1000
10000
100000
Cycles to failure, Nf
1000000
(a)
6
Load range, kN
5
4
3
2
10000 cycles
1
100000 cycles
0
0
2
4
6
Mean load, kN
8
10
(b)
The stress amplitude in the constant-life diagram can be normalised with respect to
the stress amplitude for that particular life at R = −1 (Sar ), to produce a normalised
amplitude-mean diagram, which tends to consolidate the data at different lives to
a single curve. This curve can be fitted to a straight line (Goodman equation) or a
curved line, such as a parabola (Gerber equation):
Goodman equation :
Gerber equation :
Sa Sm
+
=1
Sar Su
2
Sm
Sa
+
=1
Sar
Su
(7.3)
(7.4)
These curves are often not accurate for compressive mean stresses, in which case
a conservative estimate may be made by assuming that compressive mean stresses
provide no benefit, i.e.:
7 Modelling Fatigue in Adhesively Bonded Joints
For Sm ≤ 0,
195
Sa
=1
Sar
(7.5)
As was stated earlier, total life methods are completely phenomenological, however,
in some cases efforts have been made to differentiate between the initiation and
propagation phases (Harris and Fay 1992; Zhang and Shang 1995; Crocombe
et al. 2002, 2005; Graner-Solana et al. 2007; Shenoy et al. 2008; Dessureault and
Spelt 1997, Quaresimin and Ricotta 2006). Shenoy et al. (2008) used a combination of back-face strain measurements and sectioning of partially fatigued joints to
measure damage and crack growth as a function of number of fatigue cycles. It was
seen from the sectioned joints that there could be extensive internal damage in the
joint without external signs of damage; therefore, determination from external observations alone is likely to overestimate the initiation phase. Shenoy et al. (2008)
identified three regions in the fatigue life of an aluminium/epoxy single lap joint.
An initiation period (CI) in which damage starts to accumulate, but a macro-crack
has not yet formed, a stable crack growth (SCG) region in which a macro-crack has
formed and is growing slowly and a fast crack growth region (FCG), which leads
to rapid failure of the joint. They found that the percentage of life spent in each
region varies with the fatigue load, as shown schematically in Fig. 7.7. At low loads
the fatigue life is dominated by crack initiation, whereas crack growth dominates at
high loads. The figure also shows the back-face strain signal associated with each
phase of fatigue damage and it can be seen how this can be used to monitor damage
in a lap joint. Finite element analysis (FEA) can be used to relate the strain signal
to simulations of cracking and damage in the joint. Shenoy et al. (2008) observed a
curved crack front across the specimen width, necessitating a 3D FEA to accurately
represent the evolution of damage/cracking in the joint to compare with the experimental strain, as shown in the mesh in Fig. 7.8. It was further seen, that cracking
could occur from (i) one end of the overlap, (ii) approximately equally from both
ends of the overlap or (iii) asymmetrically from the two ends. The type of growth
No. of cycles to failure
Fig. 7.7 Fatigue failure map
with superimposed back-face
strain (BFS) as a function of
number of cycles for bonded
single lap joints (Shenoy
et al. 2008)
BFS plots
FCG
SCG
CI
No. of cycles
Fatigue load
Normalised backface strain
Failure curve
196
Ian A. Ashcroft, Andrew D. Crocombe
Fig. 7.8 (a) Concave crack
front shown schematically,
(b) 2D view of the, finite
element model showing
different sized fillets, (c) 3D
view of the joint. Deformed
mesh with centre crack
(Shenoy et al. 2008)
Concave
crack front
(a)
(b)
Plane of symmetry
(centre of joint width)
(c)
will affect the back-face strain signals, as shown in Fig. 7.9, and must be represented
accurately in the FE model for meaningful comparisons to be made.
7.3.1.2 Fatigue Limit
A major limitation of the total life approach is that the crack propagation is highly
geometry dependent and hence the data from an S-N curve cannot easily be used
to predict the fatigue life for a different sample geometry. However, if a fatigue
limit is seen then it may be possible to use the data from one sample to predict
the fatigue limit for a different geometry or loading condition. The approach to
be used is similar to that for predicting failure under quasi-static multiaxial loading
and similar multi-axial failure criteria, e.g. von Mises, Tresca or maximum principal
stress, should be used. For example the von Mises effective stress for fatigue limit
prediction, σae would be:
7 Modelling Fatigue in Adhesively Bonded Joints
197
SG1
1 mm
Loaded substrate
Fixed substrate
SG2
(a)
80
Backface strain [microstrain]
70
60
50
40
30
SG1
SG2
20
10
0
0
0.2
0.4
0.6
0.8
1
1.2
Crack length [mm]
(b)
Backface strain [microstrain]
80
70
60
50
40
SG1
30
SG2
20
10
0
0
0.5
1
1.5
2
2.5
Crack length [mm]
(c)
Fig. 7.9 (a) Location of strain gauges. FEA simulated back faced strain (BFS) plots for. (b) type
(i) crack growth, (c) type (ii) crack growth, (d) type (iii) crack growth (Shenoy et al. 2008)
198
Ian A. Ashcroft, Andrew D. Crocombe
Backface strain [microstrain]
120
100
80
60
40
20
SG1 FEA
SG1 Experimental
SG2 FEA
SG2 Experimental
0
0
2000
4000
6000
8000
10000
12000
No. of cycles
(d)
Fig. 7.9 (continued)
1
σae = √
2
(σa1 − σa2 )2 + (σa2 − σa3 )2 + (σa3 − σa1 )2
(7.6)
where σa1 , σa2 and σa3 are the principal stress amplitudes. The method then is
to experimentally determine the fatigue limit using calibration samples and then
calculate the value of the selected failure parameter at the fatigue limit. The fatigue
limit can then be predicted for different geometries for failure in the same material –
in theory. However, in practice there are the same difficulties facing those wishing
to predict the quasi-static failure load of adhesive joints, i.e. which failure criterion
should be chosen and how to deal with the theoretical stress singularities. Abdel
Wahab et al. (2001a) predicted the fatigue limit (or fatigue threshold) in the lapstrap joints shown in Fig. 7.4d using a variety of stress and strain based failure
criteria, taking the value at a distance of 0.04 mm from the singularity to avoid mesh
sensitivity. They also used the plastic zone size as a failure criterion. A summary of
the results can be seen in Table 7.2.
7.3.1.3 Cumulative Damage Methods
The S-N curve is only directly applicable to constant amplitude fatigue whereas in
many cases variable amplitude fatigue spectra are experienced. A simple method to
use S-N data to predict variable amplitude fatigue was proposed by Palmgren (1924)
and then further developed by Miner (1945). Miner assumed that the total amount of
work to cause failure in a sample, W , was a constant, regardless of the amplitude of
the fatigue. Therefore if the sample was subjected to a spectrum loading consisting
of i blocks, where the work associated with each block was wi , then:
∑ wi = W
(7.7)
7 Modelling Fatigue in Adhesively Bonded Joints
199
Table 7.2 Fatigue threshold predictions for lap-strap joints (Abdel Wahab et al. 2001a)
Failure load (kN)
Experimental
Maximum principal strain
Von Mises strain
Shear stress
Peel stress
Von Mises stress
Maximum principal stress
Average principal stress
Plastic zone size (0.1%)
Plastic zone size (knee)
Testing/ageing conditions
−50◦ C/50◦ C
dry
22◦ C/50◦ C
dry
22◦ C/45◦ C,
85%RH
90◦ C/50◦ C
dry
11.0
10.89
11.16
11.1
7.83
11.14
10.35
10.26
–
–
11.0
11.8
12.2
12.01
8.29
12.03
11.02
11.1
13.5
13.18
9.0
11.8
12.3
11.97
8.82
12.0
10.99
11.04
13.5
12.0
9.0
11.2
11.3
11.4
>15
11.37
14.27
>15
14.0
13.2
He further assumed that the work absorbed in a cycle was proportional to the number
of cycles in the block, ni , and hence:
wi
ni
=
W
Nfi
(7.8)
Where N f i is the number of cycles to failure at the stress amplitude for that particular block and can be obtained from the S-N curve. Therefore from Eqs. (7.7)
and (7.8):
ni
(7.9)
∑ Nfi = 1
Equation (7.9) is termed Palmgren-Miner’s (P-M) law or the linear damage
accumulation model. It can be seen that the fatigue life of a sample in variable amplitude fatigue can be predicted from an S-N curve obtained from constant amplitude
fatigue testing of similar samples using Eq. (7.9). However, there are a number of
serious limitations to this method. It is assumed that damage accumulation is linear
and that there is no load history effect. In many cases these assumptions are not
correct. For instance it is assumed that cycles below the fatigue limit will not contribute to the damage accumulation, however, once a crack has formed by the action
of stresses above the fatigue limit then it may continue to propagate at stresses below
the fatigue limit. In metals it is often seen that an overload induces crack root plasticity which retards crack growth, leading to under-predictions of fatigue life using
the P-M rule. Another effect of load sequencing that is neglected is that failure will
actually occur when the residual strength of the sample is reduced to the maximum
stress in the fatigue spectrum. Hence an overload at the beginning of the spectrum,
when little residual strength degradation has occurred may be beneficial, whereas
when the residual strength has degraded it may cause quasi-static failure.
To account for some of these deficiencies a number of modifications to the P-M
rule have been suggested. A number of non-linear damage accumulation models
have been proposed (Marco and Starkey 1954; Henry 1955; Leve 1969; Owen and
Howe 1972; Bond 1999), as illustrated in Fig. 7.10. Load sequencing events are only
accounted for if a different damage parameter is obtained for each stress amplitude
200
Ian A. Ashcroft, Andrew D. Crocombe
1.
1.
D = f (n/Nf , S)
Damage, D
Damage, D
D = f (n/Nf )
1.0
0
S1
S2
S3
1.0
0
n/Nf
n/Nf
(a)
Linear and non-linear, stressindependent damage models
(b)
Non-linear, stress-dependent
damage models
Fig. 7.10 Various forms of damage accumulation law
seen in the spectrum, as shown in Fig. 7.11a, and to incorporate load interactions
further empirical modifications are required, as shown in Fig. 7.11b. Crack growth
below the fatigue limit can be accommodated by extending the fatigue curve below
the fatigue limit, with either the same (elementary P-M) or a reduced (extended
P-M) slope, as illustrated in Fig. 7.12. In the Relative Miner’s law the Miner’s sum,
C, on the right hand side of Eq. (7.9) is set to some value other than 1, as shown in
Eq. (7.10).
ni
(7.10)
∑ Nfi = C
C is determined experimentally and Schutz and Heuler (1989) suggested that the
experimentally determined Miner’s sum can only be applied to other spectra where
the peak stresses of the two spectra differ by no more than 20 or 30%.
S2
Damage, D
1.
Damage, D
1.
S2
S1
S1
1.0
0
1.0
0
n/Nf
n/Nf
(a)
(b)
Interaction-free, multi-stress level
damage accumulation in a nonlinear,
stress-dependent model
Multi-stress level damage
accumulation considering load
interaction effects
Fig. 7.11 Sequencing and load interaction in stress-dependent damage accumulation models
7 Modelling Fatigue in Adhesively Bonded Joints
201
S
Stress level
P-M
Extended P-M
Elementary P-M
Number of cycles to failure
log(Nf )
Fig. 7.12 Modifications to Palmgren-Miner’s rule
The various proposed modifications to the P-M rule can result in better predictions, although at the expense of increased complexity and/or more testing.
However, the basic flaw in the method, that it bears no relation to the actual progression of damage in the sample, is still not addressed. Erpolat et al. (2004) used
the P-M law and the extended P-M law, shown in Fig. 7.12, to predict failure in
an epoxy-CFRP double lap joint subjected to the variable amplitude (VA) fatigue
spectrum shown in Fig. 7.13a. The resulting Miner’s sum as a function of maximum
load can be seen in Fig. 7.13b. It can be seen that the Miner’s sum is significantly
less than 1, varying between 0.04 and 0.3, and decreases with increasing load. This
means that the load sequencing is causing damage acceleration, i.e. that the P-M
rule is non-conservative, and that the damage acceleration increases with maximum
load. It can also be seen in Fig. 7.13b that allowing the cycles below the fatigue
limit to contribute to the damage makes little difference. This implies that there is
a strong load sequencing effect in the VA fatigue testing of bonded joints and that
the P-M rule should not be used as it could lead to in-service failure well below the
predicted life.
7.3.1.4 Strain-Life Approach
Under high stress amplitudes, plastic deformation occurs and the fatigue life is considerably shortened. This is known as low cycle fatigue (LCF). Under constant stress
amplitude fatigue with strain hardening, the strain amplitude decreases after the
first cycle, and the subsequent hysteresis loop is repeated a number of times before
micro-cracking occurs. In LCF the high loads involved mean that cracks are usually
still small when failure occurs. This behaviour leads to a horizontal asymptote to
the quasi-static strength in the LCF region, as seen in Fig. 7.5. In constant strain
amplitude testing either increasing or decreasing stress amplitude can be seen, depending on whether the material is cyclic strain hardening or softening, although in
many cases this stabilises to a constant value after a number of cycles. If there is
a positive strain mean then the mean tends to decrease as the sample is fatigued, a
phenomenon known as plastic shakedown. This can be compared with the effect of
202
Ian A. Ashcroft, Andrew D. Crocombe
Fig. 7.13 (a) VA spectrum
for a peak load of 9 kN,
(b) Miner’s sum as a function
of maximum load (data from
Erpolat et al. 2004)
8
Load (kN)
6
4
2
0
2
4
6
10
8
Time (sec)
(a)
0.35
P-M
0.3
Extended P-M
Miner's Sum, C
0.25
0.2
0.15
0.1
0.05
0
8
9
10
11
Maximum Load (kN)
(b)
12
creep in constant stress amplitude testing, which leads to an increase in the mean
strain with cycling.
Coffin (1954) and Manson (1954) proposed that N f could be related to the plastic
strain amplitude, Δε p /2, in the LCF region.
β
Δε p
= B Nf
2
(7.11)
This relationship dominates at high loads whereas Eq. (7.1) or (7.2) dominates at
low cyclic loads. The total strain amplitude, Δε /2, is comprised of elastic and plastic
components:
7 Modelling Fatigue in Adhesively Bonded Joints
203
Δεe Δε p
Δε
=
+
(7.12)
2
2
2
Hence from Eqs. (7.2), (7.11) and (7.12) an expression incorporating both HCF and
LCF components of the fatigue life can be derived.
β
Δε
A α
=
Nf + B Nf
2
E
(7.13)
The strain-life approach is more difficult to implement than the stress-life method,
particularly for non-homogenous material systems such as bonded joints. Adhesively bonded joints have stress singularities and a complex stress-strain state
throughout the adhesive. Also, structural joints tend to be used in HCF applications
and hence the strain-life method has seen little application to adhesively bonded
joints.
7.3.2 Strength and Stiffness Wearout Approaches
One of the main drawbacks of the total-life methods is that they cannot be used to
monitor damage in the sample as only the final failure is characterised. An alternative phenomenological approach is to characterise fatigue damage as a function of
the reduction in the strength or stiffness of a sample during its fatigue life. The advantage of this approach is that, unlike the total-life approach, the degradation in the
sample prior to failure is characterised. From these models the residual strength or
stiffness after a period of loading can be predicted and hence the response to further
loading predicted. This makes the method more suitable for predicting the effects of
complex loading than the simple total-life models. However, more testing is required
than in the total life approach as the residual strength at various percentages of the
total life must be determined experimentally at different load or stress amplitudes.
7.3.2.1 Strength Degradation Under Constant Amplitude Loading
The residual strength of a joint after a certain number of fatigue cycles, n, is defined
as SR (n). This is initially equal to the static strength, Su , but decreases as damage
accumulates during the fatigue cycling. Failure occurs when the residual strength
equals the maximum stress of the spectrum, i.e. when SR (N f ) = Smax . The rate of
strength degradation mainly depends on Su , Smax and R (Smin /Smax ), i.e.:
SR (n) = Su − f (Su , Smax , R) nκ
(7.14)
where κ is a strength degradation parameter. Substitution of the failure criterion
(SR (N f ) = Smax ) into Eq. (7.14) gives:
f (Su , Smax , R) =
Su − Smax
N κf
(7.15)
204
Ian A. Ashcroft, Andrew D. Crocombe
and the residual strength, SR (n), can be defined as:
κ
n
SR (n) = Su − (Su − Smax )
Nf
(7.16)
7.3.2.2 Strength Degradation Under Variable Amplitude Loading
Schaff and Davidson (1997, 1997a) extended Eq. (7.16) to enable the residual
strength degradation of a sample subjected to a variable amplitude loading spectrum to be predicted.
j
n j + neff , j κ j
(7.17)
SR ∑ ni = Su − (Su − Smax, j )
Nf, j
i=1
j−1 ⎞ κ1
j
S
−
S
n
∑
u
R
i
⎜
⎟
i=1
⎟
neff , j = N j ⎜
⎝ Su − Smax, j ⎠
⎛
where
(7.18)
j represents the current stage and n j is the number of cycles elapsed in stage j. However, they noted a crack acceleration effect in the transition from one constant amplitude (CA) block to another, a phenomenon they termed the cycle mix effect, and
proposed a cycle mix factor, CM, to account for this. Hence, during the transition:
SR (n) → SR (n) −CM
CM = Cm Su
Δ Smn
SR (n)
for Δ Smn > 0
(7.19)
(Δ Smax /Δ Smn )2
(7.20)
where Δ Smn and Δ Smax are the changes in the mean and maximum load values, respectively, during the transition from one stage to another and Cm is the cycle mix
constant, which is dependent on material and geometry. Cm can be determined by
comparing variable amplitude fatigue lives under different spectra, e.g. two spectra
with and without mean stress jumps. Using the cycle mix factor, Eq. (7.18) can be
modified to:
j−1 ⎞ κ1
j
S
−
S
n
−CM
∑
u
R
i
j−1→
j
⎜
⎟
i=1
⎟
neff , j = N j ⎜
⎝
⎠
Su − Smax, j
⎛
(7.21)
Where CM j−1→ j is the cycle mix factor for the transition from stage j − 1 to stage
j if Δ Smn > 0.
Erpolat et al. (2004) assumed a linear damage accumulation for CFRP-epoxy
DLJs and formulated their damage law in terms of load/failure load rather than
7 Modelling Fatigue in Adhesively Bonded Joints
205
stress/strength for the reasons stated in Sect. 7.3.1. Residual failure load degradation
for each cycle above the fatigue limit, LD, was defined as:
LD =
Lu − LOL
N f ,OL
(7.22)
where Lu is the quasi-static failure load, LOL is the maximum load of the cycle
and N f ,OL is the fatigue life corresponding to this cycle. The residual failure load
degradation due to mean load jumps, CM, was then defined as:
Δ Lmax
CM = α (Δ Lmn )β L peak ( Δ Lmn )
(7.23)
where Δ Lmn and Δ Lmax are the changes in the mean and maximum load values during the transition and L peak is the peak load in the spectrum. The parameters α and
β are the cycle mix constants, which are dependent on material and geometry. They
can be determined by comparing VA fatigue lives under different spectra, e.g. two
spectra with and without mean load jumps. Assuming that in each loading block,
there are a number of cycles above the fatigue limit (OL1 , OL2 , . . . ) and a number
of mean load jumps (CM1 , CM2 , . . .), the failure load degradation during a single
block, Δ LRB , can be defined as:
ΔL
ΔL
β L peak
Δ LRB = α (Δ Lmn,1 )
max,1
Δ Lmn,1
β L peak
+ (Δ Lmn,2 )
max,2
Δ Lmn,2
+···
Lu − LOL1 Lu − LOL2
+
+
···
N f ,OL1
N f ,OL2
(7.24)
Equation (7.24) is illustrated schematically in Fig. 7.14. The number of blocks to
failure is given by:
Lu − L peak
NB =
(7.25)
Δ LRB
Erpolat et al. (2004) termed this the linear cycle mix (LCM) model and showed that
it was capable of more accurate predictions of fatigue life for bonded joints under
variable amplitude fatigue than Palmgren-Miner’s (P-M) law. Table 7.3 shows a
comparison of the predictions using the P-M law and the LCM model for CFRPepoxy DLJs using the VA fatigue spectra shown in Fig. 7.15. It can be seen that, the
P-M law significantly over-predicted the fatigue life when there were mean jumps
in the fatigue spectrum, whereas, the LCM model produced excellent predictions of
the fatigue life.
7.3.2.3 Stiffness Degradation
An alternative approach to the strength-based wearout models is to associate damage accumulation with stiffness degradation. Like strength degradation, the stiffness
degradation rate can be considered as a power function of the number of load cycles
(Dibenedetto and Salee 1979; Yang et al. 1990; Whitworth 1990). This relation was
defined by Yang et al. (1990) as:
206
Ian A. Ashcroft, Andrew D. Crocombe
(LR -Lpeak )/(Lu -Lpeak )
Residual Failure Load
1
LD1
CM1
LD2
CM2
LD3
0
1
N / Nf
Number of Cycles
Fig. 7.14 Residual failure load degradation during VA fatigue cycling
Table 7.3 Variable amplitude fatigue predictions for epoxy/CFRP DLJs using Palmgren-Miner
(P-M) and Linear Cycle Mix (LCM) methods (Erpolat et al. 2004)
Spectra
7 (a)
7 (b)
7 (c)
7 (d)
7 (e)
Fatigue life (cycles at Lmax = 9kN)
Damage sum, C
Experimental
P-M
LCM
P-M
LCM
200k
180k
80k
> 106
> 106
> 106
200k
390k
> 106
> 106
200k
200k
62k
> 106
> 106
0.18
0.9
0.21
–
–
1
0.9
1.28
–
–
E (n) = E (0) − E (0) (d + a2 B S) na3 +BS
d = a1 + a2 a3
(7.26)
(7.27)
where E(0) is the initial stiffness and S is the stress level, a1, a2, a3 and B are stressindependent parameters. Whitworth (1990) proposed an alternative model:
n
n
a
a
a
E
= E (0) − H [E (0) −C]
(7.28)
Nf
Nf
where C is a stress-dependent parameter but a and H are independent of the applied
stress level.
A failure criterion for a stiffness-based wearout models is not as straight forward
as that for the strength-based wearout models. One approach is to relate failure
stiffness, E(N f ), to stress, e.g.:
E(N f ) Smax
=
E (0)
Su
(7.29)
7 Modelling Fatigue in Adhesively Bonded Joints
207
Fig. 7.15 Variable amplitude fatigue spectra (Erpolat et al. 2004)
which can then be used as a failure criterion for the stiffness-based wearout
models.
7.3.3 Fracture Mechanics Based Methods
Unlike the stress and strain-life approaches, the fracture mechanics approach only
deals with the crack propagation phase. It is assumed that crack initiation occurs
during the early stages of the fatigue cycling or a pre-existing crack exists. The
208
Ian A. Ashcroft, Andrew D. Crocombe
initial crack size used in the analysis can be measured or an assumed value can
be used. The rate of fatigue crack growth, da/dN, is then correlated with an appropriate fracture mechanics parameter, such as Griffith’s (1921) strain energy release rate, G, or Irwin’s (1958) stress intensity factor, K. This relationship can
be used to predict fatigue crack growth for different sample geometries. Paris
et al. (1961) proposed that da/dN was a function of the stress intensity factor range,
Δ K(= Kmax − Kmin )
da
= C Δ Km
(7.30)
dN
C and m are empirical constants dependent on the material, fatigue frequency,
R-ratio, environment, etc. Various modifications to the Paris crack growth law have
been proposed to take R-ratio effects into account. A modified version of the relationship proposed by Forman et al. (1967) is given below.
da
C (Δ K)m (Δ K − Δ Kth )0.5
=
dN
(1 − R) Kc − Δ K
(7.31)
where Kc is the quasi-static fracture toughness and Kth is the fatigue threshold
(discussed further below). Although K is the most widely used fracture mechanics parameter for the analysis of metals, it is more complicated to apply to bonded
joints, where the constraint effects of the substrates on the adhesive layer make it
difficult to define the stress distribution around the crack tip. Therefore, G, is often
used as the governing fracture parameter for adhesives if linear elastic fracture mechanics (LEFM) is applicable (i.e. localised plasticity). If an elasto-plastic fracture
mechanics (EPFM) parameter is required, because of more widespread plasticity,
then Rice’s (1968) J-integral (J) is generally used. The maximum strain energy
release rate, Gmax , is often used for the fatigue analysis of bonded joints, in preference to the strain energy release rate range, ΔG, because facial interference of
the adhesive on the debonded surfaces may lead to the generation of surface debris, which may prevent the crack from fully closing, thus giving an artificially
high value of Gmin (Martin and Murri 1990). Gmax is directly proportional to ΔG,
such that:
(7.32)
ΔG α (1 − R2 ) Gmax
Therefore, the Paris law for adhesively bonded joints can be defined as a function of
Gmax :
da
= CP Gnmax
(7.33)
dN
A plot of experimental crack growth rate against the calculated Gmax often exhibits
three distinct regions, as shown in Fig. 7.16. Region II is described by Eq. (7.33).
Region I is defined by the threshold strain energy release rate, Gth , in which there
is little or no crack growth. In Region III there is unstable fast crack growth
as Gmax approaches the critical strain energy release rate, Gc . All three regions,
which approximate a sigmoidal curve, can be represented by the empirical equation
below:
7 Modelling Fatigue in Adhesively Bonded Joints
Fig. 7.16 Fatigue crack
growth curve
209
REGION I
REGION II
REGION II
(Linear)
(Fast)
log (da/dN ) [m/cycle]
(Threshold)
Gc
Gth
da
dN
= CP ⋅ G nmax
log (Gmax) [J/m2]
⎤n
Gth n1 3
1
−
⎢
⎥
da
G
max n2 ⎥
= CP Gnmax ⎢
⎣
⎦
dN
Gmax
1−
Gc
⎡
(7.34)
The relationship between the fatigue crack propagation rate and the relevant fracture
parameter, Γ, is termed the crack growth law and can be represented generically as:
da
= f (Γ)
dN
(7.35)
This function must be determined and this is usually achieved experimentally using a simple sample geometry, such as the DCB in Fig. 7.4a. For these joints there
is often a simple analytical solution for the determination of the fracture parameter, although numerical methods can also be used. Erpolat et al. (2004b) compare
a number of different analytical and numerical solutions for the calculation of G
and J for DCBs in fatigue. Once the empirical relationship in Eq. (7.35) has been
determined then it can be used to predict fatigue crack growth in samples and components of different geometry, as long as the materials and mechanism of failure are
the same. The number of cycles to failure is determined from
a f
Nf =
ao
da
f (Γ)
(7.36)
Where ao is the initial crack length and a f is the final crack length. Equation (7.36)
is often solved numerically, a process known as numerical crack growth integration
(NCGI). An initial crack size, ai , is assumed and the fracture parameter Γ calculated.
210
Ian A. Ashcroft, Andrew D. Crocombe
The crack propagation rate, dai /dN, is then determined from Eq. (7.35) and the
crack size after ni cycles, ai+1 , determined from:
ai+1 = ai + ni ·
dai
dN
(7.37)
This is then repeated until the crack has gone all through the sample or Γ has reached
the critical value for quasi-static fracture, in which case failure has occurred, or Γ
has reached the threshold value for negligible crack growth.
A further consideration is that the mode-mixity is likely to be different in the
sample in which fatigue crack growth is being predicted to that used to determine
Eq. (7.35) and is also likely to change as the crack grows. Hence a mixed-mode
failure criterion must be assumed. The most common mixed mode failure criterion
is probably the total strain energy release rate, GT (= GI + GII ), although GI is a
possible alternative for brittle materials and many more complex alternatives exist,
such as that in Eq. (7.38) (Quaresimin and Ricotta 2006a)
Geqv = GI +
GII
GII
GI + GII
(7.38)
However, Quaresimin and Ricotta (2006b) showed that Geqv and GT provided quite
similar predictions of fatigue crack growth in single lap joints. Abdel Wahab et al.
(2003, 2004) proposed a general method of predicting crack growth and failure in
bonded lap joints incorporating NCGI and FEA. The crack growth law was determined from tests using a DCB sample and this was used to predict the fatigue crack
growth in single and double lap joints. Figure 7.17 shows a comparison of the predicted and experimental load-life curves for a single lap joint using both GI and GT
as the mixed-mode failure criterion. It has been shown that removing the fillet can
4.5
Maximum load (kN)
4
3.5
3
2.5
2
Experimental (no fillet)
Experimental (fillet)
nGI
nGT
1.5
1
0.5
0
3
3.5
4
4.5
5
5.5
log cycles to failure
6
6.5
Fig. 7.17 Fatigue life predictions using NCGI with total strain energy release rate (nGT) and mode
I strain energy release rate (nGI) as failure criteria. Open symbols indicate unfailed samples (data
from Abdel Wahab et al. 2004)
7 Modelling Fatigue in Adhesively Bonded Joints
211
reduce or eliminate the initiation phase (Crocombe et al. 2002) and it would be expected that the fatigue life in such samples would be better predicted using a fracture
mechanics method than that in samples with the fillets intact, in which the initiation
period may be significant. This is demonstrated in Fig. 7.17 where it is seen that the
prediction using GT is a good fit to the data from samples with fillets removed but is
conservative to the samples with fillets. The good fit between the predictions using
GI is thus likely to be coincidence and demonstrates how a combination of two errors (neglecting initiation and incorrect failure criterion) can potentially cancel each
other and result in a ‘false’ good prediction.
7.3.3.1 Fatigue Threshold
Adhesives and polymer composites tend to have steeper Paris curves than metals
such as aluminium, which means that once cracks are of a detectable size, fast fracture can follow quickly. In this case it may be preferable to design for a service life
with no crack growth. Abdel Wahab et al. (2001a) proposed the prediction of loading
conditions for no crack growth using the fatigue threshold strain energy release rate,
Gth , shown in Fig. 7.16. This was validated experimentally using the lap-strap joints
shown in Fig. 7.4d. Experiments from samples with unidirectional (UD) CFRP adherends were used to define the threshold value of the fracture parameter and this
was then used to predict the fatigue threshold in samples with multidirectional (MD)
CFRP adherends. Both elastic (GT ) and elasto-plastic (JT ) fracture parameters were
investigated and the crack was placed both in the centre of the adhesive centre line
and at the lap-adhesive interface. As well as placement of the crack, prediction of
fatigue thresholds using fracture mechanics is also dependent on the initial crack
size assumed. Abdel Wahab et al. (2001a) showed that G was very sensitive to
crack size at small crack sizes but became less size independent at approximately
0.05 mm and hence this crack size was used in the predictions. The approach of
Abdel Wahab et al. (2001a) was extended to samples subjected to environmental
ageing by incorporating the method with a coupled transient hygro-mechanical finite element analysis (Ashcroft et al. 2003a). A summary of the results from (Abdel
Wahab et al. 2001a) and (Ashcroft et al. 2003a) is presented in Table 7.4.
7.3.3.2 Variable Amplitude Fatigue
The numerical integration technique described earlier can easily be adapted to the
prediction of fatigue crack growth in variable amplitude (VA) fatigue. In this case
the value of Γ in Eq. (7.36) is a function of the varying amplitude fatigue spectrum as
well as the crack length and the maximum value of ni in Eq. (7.37) must correspond
to the number of cycles that Γ can be assumed constant for a particular loading
block. Erpolat et al. (2004a) applied the NCGI technique to the prediction of crack
growth in CFRP/epoxy DCB joints subjected to periodic overloads. This method
tended to underestimate the experimentally measured crack growth, indicating crack
212
Ian A. Ashcroft, Andrew D. Crocombe
Table 7.4 Prediction of fatigue limits in CFRP/Epoxy Lap-strap joints (Abdel Wahab et al. 2001a;
Ashcroft et al. 2003a)
Ageing
conditions
Testing
conditions
Fatigue threshold load, kN
Experimental
Predicted, GT
Predicted, JT
50◦ C
−50◦ C
11.0
11.0
9.0
9.0
4.5
6.0
4.5
8.1
8.7
8.2
8.7
2.8
3.7
2.8
11.1
11.6
7.9
11.6
3.8
5.3
3.8
dry
50◦ C dry
50◦ C dry
45◦ C, 85%RH
45◦ C, 85%RH
50◦ C dry
45◦ C, 85%RH
dry
22◦ C dry
90◦ C dry
22◦ C, 95%RH
90◦ C, 97%RH
90◦ C, 97%RH
90◦ C dry
growth acceleration due to the spectrum loading, as shown in Fig. 7.18a. In addition,
an unstable, rapid crack growth regime was seen when high initial values of Gmax
were applied, as shown in Fig. 7.18b. This behaviour was attributed to the generation of increased damage in the process zone ahead of the crack tip when the
overloads were applied. Ashcroft (2004) presented evidence of these damaged regions through x-radiography and microscopy and proposed a simple extension to the
NCGI method to enable these load history affects to be predicted. This was termed
the ‘damage shift’ model and is illustrated by Fig. 7.19. In this model it is assumed
constant amplitude conditions are represented by plot CA in Fig. 7.19. If an overload is superimposed onto the CA spectrum, then the damage ahead of the crack
will increase and the resistance to crack propagation will decrease. It was proposed
that this increased damage could be represented by a lateral shift in the FCG curve,
as represented by curve OL in the figure, and hence, the effect of the overloads can
be represented by a single parameter, ψ . In the initial stages of fatigue damage it is
expected that the value of ψ will be dependent on the number and magnitude of the
overloads. However, as long as the applied strain energy release rate range, ΔGA , is
below a critical value (as discussed later), then as the crack starts to grow through
the damage zone an equilibrium position of the FCG curve will be reached that is
only dependent on ROL and the ratio of overloads to CA cycles, NR , i.e.
At equilibrium : ψE = f (NR , ROL )
(7.39)
Δ GCA
Δ GOL
ψE can easily be determined by comparing the crack growth rates under CA and
VA fatigue for a single value of Δ GA .
If Δ GA is increased, a critical point will be reached at which Gmax of the
overloads is equal to the value of G for the damage shifted FCG curve, Δ GAC in
Fig. 7.19. Unstable or quasi-static fracture then occurs. If G increases with crack
length then this will lead to catastrophic failure of the joint. However, if G decreases
with a, as when testing DCB samples in displacement control, then the crack will
eventually stop if the G arrest (Garr ) value is reached before the joint has completely
fractured. The value of Δ GA associated with the crack arrest point (Δ Garr ) will now
Where: ROL =
7 Modelling Fatigue in Adhesively Bonded Joints
78
Crack Length (mm)
Fig. 7.18 Crack growth under
VA-loading: (a) low initial G,
(b) high initial G
213
76
74
72
70
Prediction
68
Experimental
66
0
0.25
0.5
0.75
1
1.25
Number of cycles * 106
(a)
Crack Length (mm)
80
75
70
Prediction
65
Sudden crack growth
Experimental
60
0
0.1
0.2
0.3
Number of cycles * 106
Log da/dN
(b)
OL CA
ψE
(da/dN)OL
(da/dN)CA
ΔGA ΔGAC
Fig. 7.19 Damage shift model
Log ΔG
0.4
0.5
214
Ian A. Ashcroft, Andrew D. Crocombe
be much smaller and hence crack growth will be greatly reduced. This model is
entirely consistent with the observed crack growth behaviour shown in Fig. 7.18.
7.3.3.3 Creep-Fatigue
Ashcroft and Shaw (2002) used Gth from testing DCB joints to predict the 106 cycle
endurance limit in lap strap and double lap joints at different temperatures, using GT
as the failure criterion. The predictions, shown in Table 7.5, are reasonable, apart
from the double lap joint tested at 90◦ C, which fails at a far lower load than predicted. This was attributed to accumulative creep in the double lap joint, which can
be seen in plots of displacement against cycles at constant load amplitude, as shown
in Fig. 7.20. Creep in fatigue testing of bonded lap joints has also been observed by
other authors (Hart-Smith 1981; Harris and Fay 1992). It should be noted, however,
that accumulative creep is prevented in the lap-strap joint by the strap adherend.
As many adhesives can be considered as visco-elastic/visco-plastic materials
over a wide range of their operating environment, it is not surprising that combined
effects of creep and fatigue have been seen when bonded joints have been tested
in fatigue with a non-zero mean load. The effect of this superimposed creep and
fatigue produces a strong frequency and temperature dependency for crack growth
when joints are tested in fatigue and also calls into question the applicability of tradiTable 7.5 Fatigue limit predictions from DCB data (Ashcroft and Shaw 2002)
Temp.
Fatigue limit load, kN
Lap-strap joint
−50◦ C
22◦ C
90◦ C
Double lap joint
Expt.
Predicted
Error (%)
Expt.
Predicted
Error (%)
14
15
14
8
10
11
43
33
21
10.1
10.0
3.3
10.0
12.5
13.0
0
25
294
0.3
90°C
RT
–50°C
Displacement (mm)
0.25
0.2
0.15
0.1
0.05
0
0
2000
4000
6000
8000
N cycles
Fig. 7.20 Creep in double lap joints in fatigue
10000
12000
7 Modelling Fatigue in Adhesively Bonded Joints
215
tional elastic and elastic-plastic fracture parameters for characterising crack growth
under these conditions. In this case, time dependent fracture mechanics (TDFM)
may be preferable.
Landes and Begley (1976) and Nikbin et al. (1976) independently proposed a
TDFM parameter analogous to Rice’s J-integral. Landes and Begley called this new
integral C∗ . As C∗ is only applicable to extensive creep conditions, Saxena (1986)
proposed an alternative parameter, Ct , which is also applicable for small-scale and
transition creep. This is defined as:
Ct = −
1 ∂ Ut∗ (a,t, v̇c )
B
∂a
(7.40)
Where B is sample width and Ut∗ is an instantaneous stress-power parameter, which
is a function of crack length, a, time, t and the load-line deflection rate, v̇c . When
analysing creep-fatigue, an average value of Ct , Ct(ave) , can be used.
Three methods of predicting creep-fatigue crack growth in bonded joints were
proposed by Al-Ghamdi et al. (2004):
(i) The empirical crack growth law approach. In this approach a suitable fracture
mechanics parameter is selected and plotted against the FCGR or the creep
crack growth rate (CCGR = da/dt). A suitable crack growth law is fitted to the
experimental data and crack growth law constants are determined at different
temperatures and frequencies. Empirical interpolation can be used to determine
crack growth law constants at unknown temperatures and frequencies.
(ii) The dominant damage approach. This assumes that fatigue and creep are competing mechanisms and that crack growth is determined by whichever is dominant.
(iii) The crack growth partitioning approach. This assumes that crack growth can
be partitioned into cycle dependent (fatigue) and time dependent (creep) components and that the total crack growth can be determined by summing these
components.
Method (iii) can be represented by the following equation:
da
1 da
da
=
+
dN
dN fatigue f dt creep
(7.41)
Al-Ghamdi et al. (2004) used the following form of Eq. (7.41):
da
mCtq
= D(Gmax )n +
dN
f
(7.42)
The fatigue crack growth constants D and n were determined from high frequency
tests where it was assumed that creep effects were negligible and the creep crack
growth constants m and q were determined from constant load crack growth tests.
In many cases this method produced an excellent prediction of crack growth, as
illustrated in Fig. 7.21a, which illustrates fatigue creep prediction for a DCB joint
subjected to variable frequency fatigue. It can be seen that assuming creep or fatigue
216
Ian A. Ashcroft, Andrew D. Crocombe
160
Crack length mm
Experimental (Variable frequency)
140
Prediction using G and Ct
Prediction using G only
120
Prediction using Ct only
100
80
60
40
20
0
5000
10000
Number of cycles
15000
20000
(a)
160
Experimental (trapezoidal)
Prediction in term of fatigue-Creep interaction
140
Crack length mm
Prediction in term of G and Ct
Prediction in term of G
120
Prediction in term of Ct
100
80
60
40
20
0
5000
10000
number of cycles
15000
20000
(b)
Fig. 7.21 Crack growth under: (a) variable frequency fatigue at 90◦ C, (b) trapezoidal fatigue at
90◦ C
crack growth alone underestimates the crack growth whereas summation of the two
components produces a good fit to the data. In some cases, however, simply adding
the fatigue and creep components still underestimates the crack growth. In this case a
creep-fatigue interaction term can be introduced, as in the equation below (Ashcroft
et al. 2005).
da
= A(Gmax )n + mCtq +CFint.
(7.43)
dN
Where: CFint. = R pf RycC f c
7 Modelling Fatigue in Adhesively Bonded Joints
217
R f and Rc are scalar factors for the cyclic and time dependent components respectively.
Rf =
(da/dN)
(da/dN) + (da/dt)/ f
(7.44)
Rc =
(da/dt) / f
(da/dN) + (da/dt)/ f
(7.45)
p, y and C f c are empirical constants.
Figure 7.21b shows the application of the damage partition method with the interaction term for a test with a trapezoidal waveform. The time parameters used
to characterize the trapezoidal waveform were: loading time, tr = 0.1 s, hold time,
th = 30 s and unloading time, td = 0.1 s. It can be seen that the prediction including
the interaction term is closer than that without.
7.3.4 Fatigue Initiation
The fatigue crack growth approach is applicable if the fatigue life is dominated by
the fatigue propagation phase. However, if the initiation phase is significant then this
approach may considerably underestimate the fatigue life, leading to over-designed
and inefficient structures. What is desired then is to predict the number of cycles
before the macro-crack forms. This can be done empirically in a similar fashion to
the stress-life approach, with the number of cycles to fatigue initiation, Ni , being
plotted as a function of a suitable stress (or other) parameter rather than cycles to
total failure, N f . This is already demonstrated in Fig. 7.7.
Lefebvre and Dillard (1999) argued that as initiation tends to occur near interface corners, where the stress field is singular, a stress singularity parameter would
make a suitable fatigue initiation criterion. They showed that under certain defined
conditions the singular stress, σkl at point A in Fig. 7.22a could be represented by:
σkl =
Qkl
xλ
(7.46)
where Qkl is a generalised stress intensity factor, dependent on load, and λ is an
eigenvalue that can be related to the order of the singularity and is dependent on
the wedge angle, φ , in Fig. 7.22a. It was proposed that Ni could be defined in
terms of Δ Q (or Qmax ) and λ in a 3D failure map, as illustrated in Fig. 7.22b.
Lefebvre and Dillard (1999a) applied this method to bi-material wedge specimens,
with the fatigue initiation point being characterised by back-face strain measurements. Quaresimin and Ricotta (2006a) used a similar generalised stress intensity
factor to characterise the number of cycles to fatigue initiation in bonded lap joints
and proposed that this could be combined with prediction of the crack propagation
life, using the techniques discussed in Sect. 7.3.3, to predict the total fatigue life.
218
Fig. 7.22 (a) Singularities
in bonded lap joints, (b) 3D
Fatigue initiation map (after
Lefebvre and Dillard 1999)
Ian A. Ashcroft, Andrew D. Crocombe
B
A
Adherend 1
Adhesive
Adherend 2
(a)
ΔQ
λ
Log Ni
(b)
It should be noted that the approach described above is based on the characterisation of an elastic stress field and hence is not applicable in the case of widespread
plasticity or creep. Also, only the singularity at position A in Fig. 7.22a was considered in the works referenced above, whereas, failure at position B is more common
in bonded lap joints.
7.3.5 Continuum Damage Mechanics Approach
Drawbacks to combining the methods described in Sect. 7.3.3 and 7.3.4 to predict
the total fatigue life are that: (i) two separate methods need to be implemented,
(ii) progressive damage in the initiation phase is not modelled and (iii) the methods are not mechanistically accurate. The damage mechanics approach addresses
some of these problems by allowing progressive degradation and failure to be modelled, thus representing both initiation and propagation phases. Continuum damage
mechanics (CDM) requires a damage variable, D, to be defined as a measure of
the severity of the material damage (Lemaitre 1984, 1985; Kachanov 1986). It is
assumed that D is equal to 0 for undamaged material and D = 1 represents the complete rupture of the material. Between those two extremes, there is another critical
7 Modelling Fatigue in Adhesively Bonded Joints
219
value for the damage variable, Dc which characterises the macro crack initiation
and is usually between 0.2 and 0.8. The damage variable, D, is difficult to determine physically, since it is almost impossible to monitor all the micro-cracks and
voids in a material. However, a simple estimation is usually made using the stiffness
degradation:
ED
(7.47)
D = 1−
E
where E and ED are the Young’s modulus of the undamaged and damaged material,
respectively. Once the damage variable is defined, a damage equivalent effective
∗ , can be defined as:
stress, σeff
∗
σeff
=
σ∗
(1 − D)
(7.48)
where σ ∗ is the damage equivalent stress which is defined as:
σ ∗ = σeq
2
(1 + υ ) + 3 (1 − 2 υ )
3
σH
σeq
2 12
(7.49)
where σeq is the von Mises equivalent stress, σH is the hydrostatic stress and ‘u’
∗ can be used as a quasi-static failure criterion. In order to
is the Poisson’s ratio. σeff
apply the CDM approach to fatigue, Lemaitre (1984, 1985) derived the following
equation for the variation of the damage variable per cycle, δ D/δ N:
so
σH 2
2
(1 + υ ) + 3 (1 − 2 υ )
2 B0
3
σeq
δD
βo +1
βo +1
=
σ
−
σ
(7.50)
eq,max
eq,min
δN
(βo + 1) (1 − D)βo +1
where so , Bo and βo are material and temperature dependent coefficients, σeq,max
and σeq,min are maximum and minimum von Mises equivalent stresses in a CA fatigue loading, respectively. Equation (7.50) can be integrated for constant amplitude
fatigue loading. Using the boundary conditions (N = 0 → D = 0) and (N = NR
[number of cycles to rupture] → D = 1):
βo +1
βo +1 −1
(β0 + 1) σeq,max
− σeq,min
NR =
s
σH 2
2
(1 + υ ) + 3 (1 − 2 υ )
2 (βo + 2) β0
3
σeq
(7.51)
o
Abdel Wahab et al. (2001b) used CDM to predict fatigue thresholds in CFRP/epoxy
lap-strap joints and double lap joints. They found that the predictions using CDM
compared favourably with those using fracture mechanics. The method was extended to predict fatigue damage in bulk adhesive samples (Hilmy et al. 2006) and
aluminium/epoxy single lap joints (Hilmy et al. 2007).
220
Ian A. Ashcroft, Andrew D. Crocombe
7.4 Summary and Future Directions
It can be seen that a number of techniques have been used to model fatigue in
bonded joints. Although none of these have been proven to be generally applicable to the prediction of fatigue behaviour, all are useful in understanding and/or
characterising the fatigue response of bonded joints and it is expected that further
developments will continue in all the methods described. For instance, the traditional load-life approach is useful in characterising global fatigue behaviour but is
of little use in fatigue life prediction and provides no useful information on damage
progression in the joint. However, combined with monitoring techniques, such as
back-face strain or embedded sensors, and FEA, the method could potentially form
the basis of an extremely powerful in-service damage monitoring technique for industry. The fracture mechanics approach is potentially a more flexible tool than the
stress-life approach as it allows the progression of cracking to failure to be modelled
and can be transferred to different sample geometries. However, problems with the
traditional fracture mechanics approach include: selection of initial crack size and
crack path, selection of appropriate failure criteria, load history and creep effects.
Also the fracture mechanics approach does not accurately represent the accumulation and progression of damage observed experimentally in many cases. However,
recent modifications to the standard fracture mechanics method have seen many of
these limitations tackled.
In future developments it is expected that fatigue studies of bonded joints will
continue to increase our knowledge of how fatigue damage forms and progresses
in bonded joints and this will feed into the models being developed. In addition to
further developments and extension to the approaches described above it is expected
that an area for huge potential advances is in the incorporation of damage growth
laws into multi-physics finite element analysis models to develop a more mechanistically accurate representation of fatigue in bonded joint than is currently available.
It is also hoped that many of the techniques described above will reach sufficient
maturity to form the basis of useful tools for industry, both in the initial design and
in-service monitoring of bonded joints in structural applications subjected to cyclic
loading.
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Chapter 8
Environmental Degradation
Andrew D. Crocombe, Ian A. Ashcroft and Magd M. Abdel Wahab
Abstract Environmental degradation is probably the main on-going concern with
regard to the long-term integrity of adhesively bonded structures in-service. This
chapter begins by outlining why predictive modelling can make such a valuable contribution in this area and then outlines the steps involved in current state of the art environmental degradation modelling of bonded joints. Essentially, this involves three
main steps. The first step is modelling moisture transport through the joint in order to
determine the moisture concentration distribution through the joint as a function of
time. The second step involves evaluation of the transient mechanical-hygro-thermal
stress-strain state resulting from the combined effects of hygro-thermal effects and
applied loads. The final step involves incorporation of damage processes in order
to model the progressive failure of the joint and hence enable the residual strength
or lifetime of a joint to be predicted. The approach adopted here is to outline in
summary the key theoretical aspects of each of these steps followed by examples
showing how they have been implemented in typical bonded joint finite element
(FE) analyses.
Andrew D. Crocombe
Division of Mechanical, Medical and Aerospace Engineering, School of Engineering,
University of Surrey, Guildford, GU2 7XH, UK, e-mail: a.crocombe@surrey.ac.uk
Ian A. Ashcroft
Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University,
Loughborough, Leicestershire, LE11 3TU, UK, e-mail: i.a.ashcroft@lboro.ac.uk
Magd M. Abdel Wahab
Division of Mechanical, Medical and Aerospace Engineering, School of Engineering,
University of Surrey, Guildford, GU2 7XH, UK, e-mail: M.Wahab@surrey.ac.uk
L.F.M. da Silva, A. Öchsner (eds.), Modeling of Adhesively Bonded Joints,
c Springer-Verlag Berlin Heidelberg 2008
225
226
Andrew D. Crocombe et al.
8.1 Introduction
Lifetime prediction has been described as the “Holy Grail” of adhesive bonding.
This is an indication both of its importance and difficulty. A number of approaches
have been used to address this important problem, including the development of
accelerated test methods (Davies and Evrard 2007; Bowditch 1996; Ashcroft et al.
2001), research into the fundamental mechanisms of ageing (Buch and Shanahan
2000; Watts and Castle 1984) and the development of empirical and semi-empirical
engineering guidelines (Jumbo et al. 2005; Imielinska and Guillaumat 2004). However, the best route to achieving a practical and generally applicable method, that
would be useful to industry, would appear to be one based on the incorporation
of empirically quantified degradation mechanisms into a mechanical model of the
bonded structure, together with the application of a progressive failure model. There
are two main challenges to developing an accurate predictive method along these
lines. Firstly, accurate modelling of the various effects of ageing on the mechanical
model and, secondly, incorporating a failure model that accurately represents not
just the final failure load, but the complete cycle of damage initiation and propagation leading to failure. This is the aim of the work presented in this chapter and a
schematic of the approach taken is shown in Fig. 8.1.
It is generally agreed that the most common cause of environmental degradation
in bonded joints involves the absorption of moisture into the joint. This has a number
of effects that must be incorporated into the mechanical model, the most important
being:
Residual
strength or
service life
Failure
criteria
Bulk
Interface
DEGRADATION
Stress
DIFFUSION
Softening
Load
Bulk
Interface
Manufacture
Joint
Fig. 8.1 Lifetime prediction framework
Moisture
8 Environmental Degradation
227
a. plasticisation of the adhesive (and adherend in some cases), which will affect
mechanical properties and the internal stress distribution (Ashcroft et al. 2003;
Loh et al. 2005; Abdel Wahab et al. 2002),
b. hygroscopic expansion of the adhesive, which will affect residual stresses in the
joint (Yu et al. 2006; Liljedahl et al. 2007; Jumbo 2007),
c. weakening of the interface by various mechanisms, which will affect failure
(Loh et al. 2002a,b; Crocombe et al. 2006).
In order to be able to predict the effects cited above, it is first necessary to be able
to quantify moisture transport through the joint. The absorbed moisture will cause
swelling as well as degradation of the constituent parts and the interface region.
This will affect the stresses in the joint and the residual strength (Abdel Wahab
et al. 2002). Once both moisture transport and the subsequent degradation processes
have been successfully modelled, there is still the critical issue of predicting the
residual strength. Traditionally, strength of materials or fracture mechanics methods
have been used to predict the failure of bonded joints (Abdel Wahab et al. 2001b).
However, both of these approaches have limitations and neither approach can model
the evolving initiation and propagation of damage that is a characteristic of failure
in many environmentally degraded joints. Owing to these limitations, progressive
damage models are being increasingly used to model the behaviour of advanced
materials such as structural adhesives.
This chapter outlines the key elements discussed above. The next section focuses
on modelling the moisture transport. This is followed by a section that addresses the
determination of stress and strain in joints with a varying moisture content. The final
part of this chapter then outlines how progressive damage modelling techniques can
be used in conjunction with the models of the environmentally degraded joints to
determine residual strength. Each section begins with a summary of relevant theory
and then concludes with examples showing how this can be implemented.
8.2 Modelling Diffusion
8.2.1 Background
Moisture transport in structural adhesives is usually governed by Fickian diffusion
(Abdel Wahab et al. 2001a). It can be shown that the 1D moisture concentration (c)
at a certain distance from the overlap end (x) within an initially dry (c = 0) joint
of overlap length L at a time (t) is given by Eq. (8.1), where co is the saturation
moisture uptake of the adhesive and D is the diffusion coefficient. Similar but more
complex equations are available for 2D and 3D diffusion.
∞
c(x,t) = co − ∑
0
4co
(2 j + 1)π x −(2 j+1)22 Dπ 2 t
L
]e
sin[
(2 j + 1)π
L
(8.1)
228
Andrew D. Crocombe et al.
This is exactly analogous to heat transfer in a conducting solid and thus is often implemented in FE using thermal elements, where the diffusion coefficient (D) and
moisture concentration (c) are modelled using the thermal conductivity and the
temperature respectively. Occasionally transport has been shown to be controlled
by anomalous forms of Fickian diffusion and various modifications of the simple
Fickian model have been used (Loh et al. 2005; Liljedahl et al. 2005).
8.2.2 Application
Transient moisture diffusion in aluminium and carbon fibre reinforced polymer
(CFRP) single lap joints (SLJ), bonded with Cytec’s toughened epoxy adhesive
FM73, have been analysed using both 2D and 3D finite element analyses (FEA)
(Jumbo 2007). Figure 8.2 shows a comparison of predicted moisture concentration
in the adhesive layer of an aluminium SLJ using 2D and 3D FEA. The plots shown
are for the centre of the adhesive layer in the thickness and width directions. Both
analyses give similar results in this case because the overlap is smaller than the sample width and hence ignoring moisture transport in the width direction is not significant with regards to the moisture concentration in the middle of the adhesive layer.
Figure 8.3 shows the variation in predicted moisture concentration across the width
of the adhesive layer at a position halfway along the overlap. It can be seen that the
2D model cannot predict the expected variation across the width. Close to the edge
1
0.8
0.6
0.4
0.2
0
0
2
2D:1 week
3D:1 week
2D:24 weeks
3D:24 weeks
2D:48 weeks
3D:48 weeks
2D:78 weeks
3D:78 weeks
4
6
8
10
Distance along overlap length, mm
12
14
Fig. 8.2 Predicted moisture concentration along the overlap of an aluminium SLJ aged at
50◦ C/95%R.H. using 2D and 3D FEA (Jumbo 2007)
8 Environmental Degradation
229
1
0.8
0.6
0.4
2D:1 week
2D:24 weeks
2D:48 weeks
2D:78 weeks
0.2
0
0
2
3D:1 week
3D:24 weeks
3D:48 weeks
3D:78 weeks
4
6
8
10
Distance across overlap width, mm
12
14
Fig. 8.3 Predicted moisture concentration across the width of an aluminium SLJ aged at
50◦ C/95%R.H. using 2D and 3D FEA (Jumbo 2007)
of the sample it can be seen that the 3D model predicts significantly higher moisture
concentrations. The moisture concentration levels for 2D and 3D FEA in the CFRP
SLJs follow a similar pattern but saturate more quickly than the aluminium SLJ
because of the added effect of moisture transport through the adherend.
8.3 Modelling Stress and Strain
8.3.1 Background
Stresses in a bonded joint can be broadly categorised as either mechanical stresses
or residual stresses. Mechanical stresses are those arising from mechanical loads
whereas residual stresses are those remaining when the mechanical loads are removed. In a bonded joint, three main sources of residual stresses can be identified. These are; changes in temperature, leading to thermal stresses (Yu et al. 2006;
Apalak et al. 2007, Jumbo et al. 2007); (see Chapter 9) changes in moisture content, leading to hygroscopic stresses (Yu et al. 2006; Khoshbakht et al. 2006; Jumbo
2007) and changes due to chemical reactions, leading to curing stresses (Zarnik
et al. 2004; Macon 2001). The curing stresses in typical adhesively bonded joints are
small compared with the thermal and hygroscopic stresses, and in any case can be incorporated into analysis of the thermal stresses once the stress-free temperature has
been determined. In bonded joints there may be significant residual thermal stresses
230
Andrew D. Crocombe et al.
from the curing operation followed by modification of these stresses as moisture is
absorbed. The combined stresses are termed hygro-thermal stresses (Jumbo et al.
2005; Yi and Sze 1998). These residual stresses are generally modelled in a similar
way. In fact it is more accurate to refer to this effect as residual strains as this is
what is induced by the temperature and the moisture. The stresses are only generated when these strains are not allowed to develop fully. The residual strains are
modelled using expressions similar to those in Eqs. (8.2) and (8.3) for thermal and
hygroscopic induce strains respectively.
εth = αth (ΔT )
(8.2)
εhy = αhy (Δc)
(8.3)
The parameter αhy is known as the swelling coefficient and this has been measured
and reported for various adhesive systems (Loh et al. 2005; Liljedahl et al. 2005).
These strains can then be incorporated into the adhesive constitutive equations as:
[σ ] = [D][ε − εth − εhy ]
(8.4)
If only one form of residual strain is being modelled then it is possible to use the
thermal strain capability that almost all FE codes have. If both forms are to be modelled then some codes (such as ABAQUS) have user routines where coefficients of
expansion can be defined. In both cases it will be necessary to run coupled analyses,
where the output from one analysis (thermal or moisture) forms the input to another
(stress).
In addition to these forms of residual stress it may also be necessary to model
the change in adhesive material response (plasticisation) caused by the absorbed
moisture. This is most easily done by utilising the field dependent material property
facility that many FE codes have. Adhesive material properties such as modulus,
yield stress and post-yield flow can be defined as a function of a field variable,
such as temperature or moisture. Many adhesives exhibit a distinct degradation in
these material properties with increasing moisture content (Liljedahl et al. 2006a;
Crocombe et al. 2006; Hua et al. 2006).
8.3.2 Application
A two step 3D FEA method has been used to model hygro-thermal stresses in the
joints discussed in Sect. 8.2 (Jumbo 2007). The first step was a thermal residual stress analysis from the curing temperature to the environmental conditioning
temperature (50◦ C) and the second step was a non-linear, coupled stress-diffusion
analysis to simulate swelling due to moisture adsorption and the effect of moisture
dependent material properties. The effects of moisture ingress in the aluminium SLJ
on the hygro-thermal residual stresses are shown in Fig. 8.4. For the stresses at the
edge of the joint, shown in Fig. 8.4a, the swelling due to moisture ingress, together
8 Environmental Degradation
231
12
Maximum Principal Stress, MPa
10
8
6
4
Dry
1 week
12 weeks
2
24 weeks
48 weeks
0
0
2
4
6
8
10
Distance along overlap length, mm
(a)
12
14
14
Maximum Principal Stress, MPa
12
10
8
Dry
6
1 week
12 weeks
4
24 weeks
2
48 weeks
0
0
2
4
6
8
10
Distance along overlap length, mm
(b)
12
14
Fig. 8.4 Hygro-thermal stresses in aluminium SLJ aged at 50◦ C/95%R.H. (a) Sample edge and
(b) sample centre (Jumbo 2007)
232
Andrew D. Crocombe et al.
with plasticisation of the adhesive, has a significant stress reducing effect during
the first week. After this the stresses increase slightly due to further swelling of the
adhesive, but remain substantially lower than in the dry adhesive. The stress distribution in the middle of the adhesive is rather different from the edge, as shown in
Fig. 8.4b. After 1 week, the stresses at 2–3 mm from the overlap ends are higher
than for the dry case. This is because the tensile modulus is higher in this area than
at the overlap end, and thus load transfer is shifted from the fillet to the inner region.
This trend continues as moisture continues to be absorbed, with the highest stresses
in the drier areas in the centre of the joint. As the entire adhesive layer becomes saturated, the stresses become more uniformly distributed, although at a significantly
lower level than in the dry adhesive.
The effects of mechanical loading on the hygro-thermal stress state was investigated by applying a 6 kN load to the aluminium/FM73 single lap joints. In this case
the stresses shown are indicative of those just before significant damage in the joint,
which would cause a re-distribution of the stresses. Figure 8.5 shows the von Mises
stresses in the adhesive layer at 50◦ C with mechanical, mechanical + thermal and
mechanical + hygro-thermal loads at full saturation. It is clear that at the edge and
middle of the joint, combined environmental and mechanical loading increases the
stress in the adhesive.
8.4 Modelling Damage and Failure
8.4.1 Background
Two different forms of progressive damage modelling have generally been used;
cohesive zone modelling (CZM), where the failure is localised along a plane (Loh
et al. 2003; Liljedahl et al. 2006b) and continuum damage modelling (CDM), where
the failure can occur throughout the material (Hua et al. 2007). Potential sites for
damage modelling include the adhesive, the interface and the adherend. The CZM
approach is more directly applicable to interfacial failure and certain forms of adherend failure whilst the CDM is more relevant to failure in the adhesive. Both are
illustrated in Fig. 8.6.
The CZM is implemented via a zero volume element where the traction between
two nodes follows the curve shown in Fig. 8.6a. Initially, the nodes are held together
with minimal separation until the traction reaches a critical value (σU ). Following
this, the nodes separate and unload, absorbing energy as they deform. When sufficient CZM elements are used, in conjunction with elastic material properties, a
linear fracture mechanics response will be obtained. However the advantage of the
CZM approach is that crack formation and propagation will be modelled in a single
evolving analysis.
The continuum failure model is based on a full non-linear stress strain curve. The
onset and development of damage degrades the undamaged non-linear response to
a state of complete failure, as can be seen in Fig. 8.6b. It is necessary to define
8 Environmental Degradation
233
40
35
Von Mises Stress, MPa
30
25
20
15
10
6 kN Load
5
6 kN Load + Thermal Load
6 kN Load + HygroThermal Load
0
0
2
4
6
8
10
12
10
12
Distance along the overlap, mm
(a)
35
Von Mises Stress, MPa
30
25
20
15
10
6 kN Load
5
6 kN Load + Thermal Load
6 kN Load + HygroThermal Load
0
0
2
4
6
8
Distance along the overlap [mm]
(b)
Fig. 8.5 Development of hygro-thermo-mechanical stress in aluminium SLJ aged at
50◦ C/95%R.H. (a) Sample edge and (b) sample centre (Jumbo 2007)
234
σu
Andrew D. Crocombe et al.
σ (MPa)
σ
d'
c
b
o
(kJm–2)
δ (mm)
δr
(a)
d
a
δp
(b)
Fig. 8.6 (a) Cohesive zone model (CZM) and (b) continuum damage model (CDM)
damage in terms of an equivalent plastic displacement in order to achieve mesh
independence. To do this a characteristic length is required for each element and
this is used in conjunction with the element plastic strain to determine the equivalent
plastic displacement. CZM and CDM are treated in detail in Chap. 6.
8.4.2 Application
The aluminium and CFRP lap joints discussed above, bonded with either FM73
film adhesive or Hysol’s EA9321 paste adhesive have been aged and tested (Hua
et al. 2006; Liljedahl et al. 2007; Jumbo 2007). The testing indicated that the joints
bonded with FM73 tended to be controlled by interfacial failure whilst those bonded
with EA9321 tended to experience cohesive failure in the adhesive. Thus CZM
elements were used with the FM73 joints and CDM was used with the EA9321
joints. The failure parameters for both models were moisture dependent and were
determined from independent tests before being applied to either the lap joints (discussed above) or to more complex joints, more representative of those seen in typical aerospace applications. The application of each damage modelling technique is
discussed separately below.
8.4.2.1 Continuum Damage Modelling (Adhesive EA9321)
Mixed mode flexure (MMF) specimens bonded with EA9321 were aged in various
environments to produce three different levels of moisture concentration in the adhesive layer. These joints were tested to failure and the structural response recorded.
Finite element analyses incorporating moisture dependent stress-strain data and failure parameters were then undertaken and the failure parameters were chosen by
matching with the experimental data. These failure parameters were then used to
predict the residual strength of both composite and aluminium single lap joints
(SLJ) that had been exposed for various durations in two different ageing environments. Unlike the MMF joints, where the moisture concentration was constant (and
8 Environmental Degradation
235
known) over the entire overlap region, the moisture distribution in the SLJ’s was
non-uniform, being largest around the periphery of the joint and smallest in the centre, as illustrated in Figs. 8.2 and 8.3. As shown in Fig. 8.1, the failure modelling is a
two-stage process; the first stage determining the moisture distribution and the second implementing the failure analysis using moisture dependent material data and
failure parameters. As discussed above, in order to model the moisture diffusing
from all edges it is necessary to undertake three-dimensional analyses. Figure 8.7
shows one of the 3D FE models and a typical moisture distribution. When modelling the composite joint the moisture diffusion through the composite, as well as
the adhesive, was included.
The second stage (failure modelling) of the analysis was then undertaken, using
the moisture distribution results from the first stage as part of the input parameters.
In the wet joints the failure depends on both the level of strain and the moisture
content in the adhesive. The sequence of failure in a typical wet joint is illustrated
in consecutive plots in Fig. 8.8. Failure appears to initiate in the fillet region at the
corner of the joint where the adhesive has the highest moisture content and experiences large stress and strains, as shown in Fig. 8.8. Failure then progresses through
the overlap region with a faster evolution across it (3 directions) than along it (1 direction). This two-stage modelling is repeated for both joints for all exposure times,
providing predicted residual strengths. For illustration, the data for the aluminium
SLJs are shown in Fig. 8.9, along with the experimental data. The correlation
between the two is very good. A more detailed account of this and similar work
can be found in Hua et al. (2008).
Fig. 8.7 3D FE model and moisture distribution
236
Andrew D. Crocombe et al.
Fig. 8.8 Failure propagation through the adhesive (failed elements shown in white and centre of
the overlap region is the bottom left corner)
8.4.2.2 Cohesive Zone Modelling (Adhesive FM73)
The CZM failure parameters were determined by matching the measured response
of mixed mode flexure (MMF) specimens bonded with FM73 and exposed to three
levels of moisture concentration. It was necessary to include measured moisture
dependent non-linear adhesive stress-strain behaviour with the CZM to be able to
match the entire MMF loading response. An excellent match to the entire load response of the MMF specimens can be seen in Fig. 8.10.
These same moisture-dependent failure parameters were then used to model failure in the various forms of lap joint bonded with FM73. Coupled stress-diffusion
analyses incorporating progressive failure along the interface were undertaken. The
progressive failure is illustrated in Fig. 8.11, which shows a partially failed joint.
The crack initiated at the embedded substrate corner and propagated into the fillet
and then along the overlap.
The predicted and measured residual strengths for some of the FM73 lap joints
tested are shown in Fig. 8.12 and the excellent correlation between the two is evident. The aluminium joints were aged in a range of ageing environments and this
revealed two different degradation mechanisms. Joints exposed in water vapour
or by immersion in de-ionised water exhibited a thermodynamic type displacement between the adhesive and aluminium at the interface whilst joints aged in
Joint strength, kN
10
8
6
4
Predicted
Experimental
2
0
0
10
20
Exposure time, wk
30
Fig. 8.9 Predicted and experimental residual strengths for the aluminium SLJs
8 Environmental Degradation
237
1200
Dry
1000
80%RH
Load (N)
800
600
96%RH
400
200
Simulation
0
0
0.5
1
1.5
2
2.5
Displacement (mm)
3
3.5
4
Fig. 8.10 Comparison of measured and predicted environmentally degraded MMF joint response
tap water (which had a much higher conductance) exhibited a more rapid degradation, driven by cathodic delamination (Liljedahl et al. 2006a). The predictive modelling described so far has focused on the former whilst modelling of the cathodic
delamination is introduced in the context of the “L joints” described in the next
section.
As a final test of the predictive modelling a series of representative “model” element joints were tested. One of these was an aluminium L joint bonded with FM73
adhesive. The same failure parameters that were determined from the MMF joints
and subsequently applied to the lap joints were used in this modelling, thus establishing the universality of the failure parameters and the modelling approach. The
predicted and experimental residual strengths are shown in Fig. 8.13. It can be seen
Fig. 8.11 Damage propagating through the aluminium single lap joint
238
Andrew D. Crocombe et al.
13
Residual strength, kN
Fig. 8.12 Measured and
predicted residual strengths
for FM73 lap joints
CFRP exp
CFRP pred
Al 95%RH exp
Al 95%RH pred
AL water exp
Al water pred
12
11
10
9
8
7
6
0
5
10
15
20
Exposure, wk
25
30
that the modelling approach based on thermodynamic displacement underestimated
the degradation at longer exposure times. Examination of the failed specimens exposed for 32 weeks revealed evidence of the faster cathodic delamination mentioned
in the discussion of the lap joints above. In such a process a crack forms on the exposed interface and this accelerates the supply of water to the crack tip. A modelling
procedure has also been developed for this mechanism and it can be seen in Fig. 8.13
that the correlation with the experimental data is significantly better.
It should be noted that the predictive model correlated closely with not only the
residual strength but the entire loading response. This is illustrated in Fig. 8.14a
where both predictions and experimental data show a controlled progressive unloading before sudden catastrophic failure. The mismatch in the initial stiffness is
because the experimental data includes the displacement in the loading train as well
as the L joint. Figure 8.14b,c show clearly that the predicted and experimental crack
fronts are curved and more advanced at the centre of the joint. Other work using a
similar approach can be found in Loh et al. (2003).
Residual strength, kN
2.5
Expt
Pred (therm displ)
Pred (cath delam)
2
1.5
1
0
5
10
15
20
Exposure, wk
25
30
Fig. 8.13 Measured and predicted FM73 L joint strengths
35
8 Environmental Degradation
239
Prediction
Expl
2500
Load (N)
2000
1500
1000
500
0
0.5
1
1.5
Displacement (mm)
(a)
2
2.5
centre
0
toe
(b)
(c)
Fig. 8.14 (a) Load-displacement response, (b) measured and (c) predicted crack front shapes
8.5 Summary and Future Directions
This chapter has outlined the current state of the art in modelling the environmental
degradation in bonded joints. This has been achieved by outlining relevant theory
and then illustrating how this can be implemented in the context of aged adhesively
bonded joints. The process outlined involves coupling between four different types
of analysis; moisture, temperature, stress and damage. It was shown that when used
appropriately, good predictions of residual strength following long term exposure
can be obtained.
At the time of writing, such a modelling approach has only been applied to joints
that have been subjected to constant ageing conditions. The next step of the process
will be to develop modelling techniques that are applicable to the more realistic
240
Andrew D. Crocombe et al.
scenario of ageing under fluctuating conditions. This will involve considering,
determining and modelling the reversibility of the ageing processes.
A further step from here will be to combine the damage modelling from fluctuating environmental conditions with those from fluctuating loads (constant and
variable amplitude fatigue-fatigue).
Finally much of this knowledge could be encapsulated within a more accessible
design-type joint analysis, enabling rapid initial screening of joint suitability.
References
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adhesively bonded joints. J Adhes 77: 43–80
Abdel Wahab MM, Ashcroft IA, Crocombe AD, Hughes DJ and Shaw SJ (2001b) The effect of
environment on the fatigue of bonded composite joints. Part 2: Fatigue threshold prediction.
Composites Part A 32: 59–69
Abdel Wahab MM, Crocombe AD, Beevers A and Ebtehaj K (2002) Coupled stress-diffusion
analysis for durability study in adhesively bonded joints. Int J Adhes Adhes 22: 61–73
Apalak MK, Gunes R and Eroglu S (2007) Thermal residual stresses in an adhesively bonded
functionally graded tubular single lap joint. Int J Adhes Adhes 27: 26–48
Ashcroft IA, Digby RP and Shaw SJ (2001) A comparison of laboratory conditioned and naturally
weathered bonded joints. J Adhes 75: 175–202
Ashcroft IA, Abdel Wahab MM and Crocombe AD (2003) Predicting degradation in bonded composite joints using a semi-coupled finite-element method. Mech Adv Mat Struct 10: 227–248.
Bowditch MR (1996) The durability of adhesive joints in the presence of water. Int J Adhes Adhes
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Buch X and Shanahan MER (2000) Thermal and thermo-oxidative ageing of an epoxy adhesive.
Polym Degrad Stabil 68: 403–411
Crocombe AD, Hua YX, Loh WK, Wahab MA and Ashcroft IA (2006) Predicting the residual
strength for environmentally degraded adhesive lap joints. Int J Adhes Adhes 26: 325–336
Davies P and Evrard G (2007) Accelerated ageing of polyurethanes for marine applications. Polym
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Hua Y, Crocombe AD, Wahab MA and Ashcroft IA (2006) Modelling environmental degradation
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Hua Y, Crocombe AD, Wahab MA and Ashcroft IA (2007) Continuum damage modelling of
environmental degradation in joints bonded with E32 epoxy adhesive. J Adhes Sci Tech 21:
179–195
Hua Y, Crocombe AD, Wahab MA and Ashcroft IA (2008) Continuum damage modelling of environmental degradation in joints bonded with EA9321 epoxy adhesive. Int J Adhes Adhes 28:
302–313
Imielinska K and Guillaumat L (2004) The effect of water immersion ageing on low-velocity
impact behaviour of woven aramid–glass fibre/epoxy composites. Comp Sci Technol 64:
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Jumbo F (2007) Modelling residual stresses and environmental degradation in adhesively bonded
joints. PhD thesis, Loughborough University, UK.
Jumbo F, Ashcroft IA, Crocombe AD and Abdel Wahab MM (2005) Modelling of hygro-thermal
residual stresses in adhesively bonded joints. In: Proc 9th Int Conf on the Sci and Technol of
Adhesives, IOM Communications, Oxford, UK, pp 138–142
Jumbo F, Ruiz PD, Yu Y, Ashcroft IA, Swallowe G and Huntley JM (2007) Experimental and
numerical investigation of mechanical and thermal residual strains in adhesively bonded joints.
Strain 43: 1–13
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Khoshbakht M, Lin MW and Berman JB (2006) Analysis of moisture-induced stresses in an FRP
composites reinforced masonry structure. Finite Elem Anal Des 42: 414–429
Liljedahl CDM, Crocombe AD, Wahab MA and Ashcroft IA (2005) The effect of residual strains
in the progressive damage modelling of environmentally degraded adhesive joints. J Adhes Sci
Technol 19: 525–548
Liljedahl CA, Crocombe AD, Wahab MA and Ashcroft IA (2006a) Modelling the environmental degradation of the interface in adhesively bonded joints using a cohesive zone approach.
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Liljedahl CDM, Crocombe AD, Wahab MA and Ashcroft IA (2006b) Damage modelling of adhesively bonded joints. Int J Fracture 141: 147–161
Liljedahl CDM, Crocombe AD, Wahab MA and Ashcroft IA (2007) Modelling the environmental
degradation of adhesively bonded aluminium and composite joints using a CZM approach. Int
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Loh WK, Crocombe AD, Abdel Wahab MM and Ashcroft IA (2002a) Environmental degradation
of bonded joint interfacial fracture energy. Eng Fract Mech, 69: 2113–2128
Loh WK, Crocombe AD, Abdel-Wahab MM, Watts JF and Ashcroft IA (2002b) The effect of moisture degradation on the failure locus and fracture energy of an epoxy-steel interface. J Adhes
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Loh WK, Crocombe AD, Abdel Wahab MM and Ashcroft IA (2003) Modelling interfacial degradation using interfacial rupture elements. J Adhes 79: 1135–1160
Loh WK, Crocombe AD, Abdel Wahab MM and Ashcroft IA (2005) Modelling anomalous moisture uptake, swelling and thermal characteristics of a rubber toughened epoxy adhesive. Int
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Chapter 9
Non-Linear Thermal Stresses in Adhesive Joints
Mustafa Kemal K. Apalak
Abstract Adhesive joints serve under thermal loads as well as structural loads.
The different thermo-mechanical properties of the adhesive and adherend materials cause incompatible thermal strains along the adhesive and adherend interfaces.
Consequently, nonuniform thermal stress distributions are observed in the adhesive
joints. In case the adhesive joints are constrained, these thermal stresses become
more evident even for a uniform temperature distribution. In practice the adhesive
joints interact with fluids (air) with different temperature and velocity and experience conductive and convective heat transfers. The variable thermal boundary
conditions introduce a new non-linearity in addition to the geometrical and material non-linearities which are observed in adhesive joints. The transient temperature distributions can be obtained by solving the energy equation under the thermal
boundary conditions. The corresponding thermal strain and stress distributions can
be calculated using the transient temperature distributions. The large displacement
and rotations occurring in adhesive joints require the implementation of the small
strain-large displacement theory to the elastic stress analysis. This study introduces
a simple method for calculating the heat transfer coefficients between the fluid and
adhesive joints and explains the implementation of the finite element method to the
thermal analysis and the geometrical nonlinear stress analysis. The thermal stresses
in the metal-metal adhesive single lap, tubular, metal-metal and composite tee joints
were analysed for the variable thermal and specified structural boundary conditions.
The variable thermal boundary conditions cause non-uniform temperature distributions through the adhesive lap joints; consequently, non-uniform thermal strain and
stress distributions occur. The adherend and adhesive regions in the neighborhood
of the adhesive-adherend interfaces were subjected to high stress concentrations.
Mustafa Kemal Apalak
Department of Mechanical Engineering, Erciyes University, Kayseri 38039, Turkey,
e-mail: apalakmk@erciyes.edu.tr
L.F.M. da Silva, A. Öchsner (eds.), Modeling of Adhesively Bonded Joints,
c Springer-Verlag Berlin Heidelberg 2008
243
244
Mustafa K. Apalak
9.1 Introduction
Today’s adhesive agents are suitable to the high static and dynamic loadings in engineering designs. Metal and composite adherends can be joined successfully using
the adhesive bonding technique. A reasonable joint strength can be achieved with
a suitable adhesive selection and a well-designed adhesive joint for certain loading
conditions. In order to understand the adhesion mechanism, the stress and deformation states of the adhesive joints and to predict the mechanical properties of different adhesives the single- or double lap, tubular and butt joints, were widely used
as simple test configurations. The present analytical and experimental studies consider only the adhesive layer, or the adhesive and metal or composite adherends as
an elastic, elasto-plastic or visco-elastic media. The axial or torsional loadings on
the adhesive joints cause stress and strain concentrations around their adhesive free
ends due to the geometrical discontinuities and mechanical mismatches of the adhesive layer and adherends. This is known as the edge effect, which can be relieved
by modifying the adherend edges [4, 38]. The analytical studies assume the adhesive free ends to be square in order to simplify the mathematical models; however,
some amount of adhesive (spew fillet) accumulates around the adhesive free ends
due to the pressure applied to the adherends. The shape and sizes of the adhesive
spew fillets play an important role in the stress concentrations around the adhesive
free ends as well as the rounded adherend corners due to etching process of the
adherends [4, 38].
Two types of non-linearities, i.e. geometrical and material, affect stress and deformation states of adhesive joints considerably. The material non-linearity is due
to the nonlinear behavior of the adhesive and adherend materials. However, the geometrical non-linearity may arise in case of the joint geometry causing large displacements and rotations when the strains are small. The bending moment occurs
due to the load eccentricity and to the adherend deformations of a single lap joint as
the load is applied, and the dependence of the moment on the applied load makes
the problem geometrically non-linear [53]. Adams [3], and Harris and Adams [30]
analyzed the failure modes and loads of single lap joints with adherends and adhesives with different mechanical properties under a tensile load considering the geometrical and material non-linearities. Adams et al. [5] also investigated the shear
and transverse tensile stresses in carbon fiber reinforced plastic/steel double lap
joints using an elastic-plastic model for the rubber-modified epoxy adhesive. They
showed significant increases in the joint strength could be achieved by modifying
the local geometry of the critical zones at the edges of the overlap region of the
double lap joint. Adams and Harris [6] also investigated the influence of local geometry changes at the adhesive free edges in single lap joints considering material
and geometrical non-linearities, and showed that significant increases in the joint
strength could be achieved by filleting the adhesive at the free edges of the adhesive
layer. Reddy and Roy [49] showed that the large displacement gradients affected
the adhesive stresses at the ends of the overlap region of a single lap joint under
different loading and boundary conditions. Edlund and Klarbring [28] presented a
3D general analysis method for determining the adhesive and adherend stresses and
9 Non-Linear Thermal Stresses in Adhesive Joints
245
deformations of adhesively bonded joints considering the geometrical nonlinearity.
Recently, Andruet et al. [8] developed two and three dimensional finite elements
for the geometrical non-linear stress analysis of an adhesively bonded lap joint, and
showed that the geometrical non-linearity (the large displacements) affected considerably the peak adhesive stresses. Apalak and Engin [10] applied the small strainlarge displacement theory to an adhesively bonded double containment corner joint
using the incremental finite element method, and showed that the small strain-small
displacement theory was misleading in predicting the stress and deformation states
of adhesive joints for some loading and boundary conditions. Apalak [11, 12, 13, 14]
also showed that the geometrical non-linearity had a considerable effect on the stress
distributions in the adherends and adhesive layer of different types of corner and tee
joints.
In practice, adhesive joints also experience thermal loads. A uniform temperature distribution results in a uniform thermal strain distribution in a unconstrained
single material; in contrast, a non-uniform temperature distribution causes a single material to have a non-uniform thermal strain distribution. In this case, a single material also experiences thermal stresses due to this non-uniform temperature
distribution. The different mechanical and thermal properties of the adhesive and adherends result in a non-uniform temperature distribution. The thermal strains become
incompatible along the bi-material interfaces whereas the interfacial total strains are
compatible. Therefore, these non-uniform strain distributions are also main reason
of non-uniform thermal stress distributions through the adhesive joint and they can
peak near the adhesive-adherend interfaces. However, the thermal stress and strain
distributions are completely dependent on the thermal boundary conditions of adhesive joints. The present thermal stress analyses of adhesive joints concentrate on
(i) the prediction of failure modes of the adhesive layer considering residual stresses
induced by fabrication [36, 37, 42, 50], (ii) thermal stress singularities at the adherend corners for wedge angles of a bi-material system [33], (iii) steady state or
transient thermal stresses along the peripheries of adhesive defects, i.e. holes or fillers
inserted into the adhesive layer or near the adhesive-adherend interfaces [47, 48],
(iv) transient thermal stress distributions in the adhesive joints for different thermal expansion coefficients and Young’s modulus ratios of adherends and adhesive
[35], and (v) effect of the adhesive surface geometry on the thermal stress fields [1].
The residual thermal stresses induced by fabrication affect failure modes of the
adhesive layer in lap joints [42]. Kim and co-workers [36, 37] presented a failure
model for the stresses occurring in a tubular lap joint considering the non-linear
adhesive properties and thermal residual stresses due to the fabrication. Reedy and
Guess [50] showed that the effect of the residual stresses in an adhesive butt joint
generated by cooling on the joint strength could be much smaller than would be
predicted by a linear analysis. Thus, the peak adhesive stresses in the yield zone
at the interface corner could decay significantly when given sufficient time. Ioka
et al. [33] studied thermal residual stress distributions at the interface and around
the intersections of the interface and the free surfaces of bonded dissimilar materials and found that the thermal stress singularity disappears for a certain range
of wedge angles of a pair of materials. Nakano et al. [48] carried out a thermal
246
Mustafa K. Apalak
stress analysis of an adhesive butt joint containing circular holes and rigid fillers in
its adhesive layer and subjected to a non-uniform temperature field. They showed
that the size and location of the circular holes and rigid fillers had an evident
effect on the thermal stresses at the interface and at the hole and filler peripheries. Nagakawa et al. [47] investigated thermal stress distributions in an adhesive butt joint containing some hole defects under uniform temperature changes.
The thermal stresses around hole defects located near the centre of the adhesive
were larger than those around the hole defects located near the free surface of
the adhesive. Katsuo et al. [35] carried out transient thermal stress analysis of an
adhesive butt joint assuming that the upper and lower end surfaces of the joint were
in different temperatures at a certain instant. The thermal expansion coefficient and
Young’s modulus ratios of the adhesive and adherends had evident effects on the
transient thermal stress distribution. Abedian and Szyszkowski [1] investigated the
effects of surface geometry of composites on the thermal stress distributions. The
stress state in the composite at and near the free surface was very sensitive to the
geometric features of the surface: thus, an ideally flat free surface resulted in large
stress concentrations whereas these stresses decreased substantially in case the difference between the fibre and the matrix heights was filled with the matrix material.
Anifantis et al. [9] investigated steady state thermal stress and strain concentrations
in unidirectional fibre reinforced composites considering the concept of an inhomogeneous interphase between the fibre and matrix phases. In the study the stresses
had stronger discontinuities than the strains, exhibiting a high sensitivity to temperature changes and the location of peak stress was independent of the degree of
adhesion developed between fibre and matrix while strains were slightly dependent.
Inhomogeneous temperature distributions or transient thermal loads might increase
these phenomena drastically. Cho et al. [24] studied the effect of curing temperature
on the adhesion strength of polyamideimide/copper joints and showed the adhesion
strength decreased as the thermal stress increased with the increase of both curing
temperature and time. Humfeld and Dillard [32] investigated thermal cycling effects
on the residual stresses in viscoelastic polymeric materials bonded to stiff elastic
substrates. The residual stresses incrementally shifted over time when subjected to
thermal cycling, and damaging tensile axial and peel stresses developed over time
due to viscoelastic response to thermal stresses induced in the polymeric layer.
In general, these studies assume that a steady-state conductive heat transfer
occurred through the adhesive joint under a specific thermal condition; thus, a constant temperature along the outer surfaces of the adhesive joint was imposed. Therefore, a uniform temperature distribution was obtained through the adhesive joint and
the residual strain distributions were found around the adhesive free ends as a result
of thermal stress analyses based on the small strain-small displacement theory. The
effects of the geometrical and material non-linearities, and the presence of the adhesive fillets were not considered. However, in practice the adhesive joints usually
interact with a moving fluid at a variable or constant temperature and velocity. As
a result, the heat transfer by convection occurs between the fluid and the adherends
and adhesive layer whereas it occurs by conduction through the adhesive joint. The
heat transfer coefficient between the fluid and adherend or adhesive material, which
9 Non-Linear Thermal Stresses in Adhesive Joints
247
evidently affects the convective heat transfer, is completely dependent on the velocity, temperature-dependent properties, and the flow direction of the fluid with
respect to plate or adhesive surface. This is the third type of (thermal) nonlinearity due to variable thermal boundary conditions, which affect considerably thermal
strain and stress distributions as well as the temperature distributions in the adhesive
joint. Apalak and co-workers [15, 17] investigated thermal stresses in adhesive single lap and tubular lap joints taking into account both the convective and conductive
heat transfers, and the large displacement and rotation effects under variable thermal boundary conditions and for different adherend end conditions. They showed
that variable thermal boundary conditions caused non-uniform temperature distributions through the adhesive lap joints; consequently, non-uniform thermal strain
and stress distributions occurred. The adherend and adhesive regions in the neighborhood of the adhesive-adherend interfaces were subjected to high stress distributions. Finally, a full knowledge of the thermal stress-deformation states of adhesive
joints would allow us to design adhesive joints that can withstand the peak adhesive
stresses.
9.2 Small Strain-Large Displacement Theory
The displacements and their gradients are assumed to be infinitesimal in the small
strain-small displacement theory, and the current configuration of a body is compared with its initial state. The components of the unit relative displacement vector
are defined as
∂ ui dX j
dui
=
(9.1)
dS
∂ X j dS
where dX is an arbitrary infinitesimal line vector in the initial position. The small
strain tensor is
1 ∂ ui ∂ u j
εi j =
+
(9.2)
2 ∂ x j ∂ xi
and the rotation tensor is
Ωi j =
1
2
∂ ui ∂ u j
−
∂ x j ∂ xi
(9.3)
respectively. The large displacement-gradient components make the strain to be
characterized from the initial state more difficult than in the small-strain case.
Lagrangian formulation allows the finite-strain to be defined based on the material coordinates in the undeformed configuration. The deformation equations for the
movement of a particle from its initial position X to the current position x are given as
x = x(X,t) or
xi = xi (X1 , X2 , X3 ,t)
(9.4)
An arbitrary infinitesimal material vector dX at X can be associated with a vector
dx at x as follows
248
Mustafa K. Apalak
dx = F · dX = dX · FT
(9.5)
using the deformation gradient vector
←
F=x∇
(9.6)
In terms of the strain tensor the change in the squared length of the material vector
dX is given as follows
(ds)2 − (dS)2 = 2 dX · E · dX
(9.7)
(ds)2 − (dS)2 = 2 dXI EIJ dXJ
(9.8)
or
2
The new squared length (ds) of the element is written in terms of the Green deformation tensor C referred to the undeformed configuration as follows
(ds)2 = dX · C · dX
(9.9)
(ds)2 = dXI CIJ dXJ
(9.10)
or
Comparison of Eqs. (9.7) and (9.9) gives the relationship between the strain tensor
E and the Green deformation tensor C
2 E = C−1
(9.11)
or
2 EIJ = CIJ − δIJ
(9.12)
2
where δIJ is Kronecker delta. The new squared length (ds) of the element is written
using the deformation gradient tensor as follows
(ds)2 = dx · dx = dX · FT · (F · dX) = dX · FT · F · dX
(9.13)
Comparison of Eqs. (9.9) and (9.13) shows that the Green deformation tensor
C = FT · F
or
CIJ =
(9.14)
∂ xk ∂ xk
∂ XI ∂ XJ
(9.15)
From Eq. (9.11) the strain tensor becomes
E=
or
1
EIJ =
2
1 T
F ·F−1
2
∂ xk ∂ xk
− δIJ
∂ XI ∂ XJ
(9.16)
(9.17)
9 Non-Linear Thermal Stresses in Adhesive Joints
249
Using the same reference axes for both xi and Xi , and using lower-case subscripts
for both, we have
(9.18)
xi = Xi + ui
with displacement components
ui = ui (X1 , X2 , X3 ,t)
(9.19)
In terms of the displacements, the general expression for Ei j in Eq. (9.17) takes
the form
1 ∂ ui ∂ u j ∂ uk ∂ uk
Ei j =
+
+
(9.20)
2 ∂ X j ∂ Xi ∂ Xi ∂ X j
When Eqs. (9.2) and (9.20) are compared it is evident that the large displacement theory is an extension of the small displacement theory since it includes
the squares of the displacement gradients. In case the displacement gradients are
not small compared to unity, the large displacement theory should be used in order to consider their non-linear effects on the stress and deformation states of the
structures [46, 52].
9.3 Non-linear Equilibrium Equations
In order to establish the non-linear equilibrium equation for the deformed body subjected to the external loads, the virtual internal and external works done on the body
are equated. The non-linear equilibrium equation from the virtual-work equation
given in terms of the Lagrangian coordinate system is given as [26, 39, 57, 59]
δ ET σ dV =
≈
V
ρ δ uT q dV +
V
δ uT qo dA
(9.21)
A
where the external work is due to the virtual displacements δ u acting on the surface
tractions, extending over the initial undeformed surface and given by
qo = qo1 qo2 qo3
T
(9.22)
and the body forces per unit mass, acting within the undeformed volume V, given by
q = q1 q2 q3
T
(9.23)
ρ being the density of the undeformed body. The total Lagrangian virtual work
Eq. (9.21) can be approximated by the finite element idealization as follows
δ pT
BT σ dV = δ pT
≈
V
V
ρ NT q dV + δ pT
NT qo dA
A
(9.24)
250
Mustafa K. Apalak
where σ is stress tensor. Since the virtual nodal displacements δ p are arbitrary,
≈
Eq. (9.24) can be written as
BT σ dV =
≈
V
ρ NT q dV +
V
NT qo dA
(9.25)
A
where N is a shape function array whose coefficients are functions of the initial
position x within the element included in the expression of displacement within an
element
u=Np
(9.26)
where the nodal displacement vector
p = u1 u2 · · · un
T
(9.27)
and the strain matrix
B = B0 + BL
(9.28)
where B0 is the linear strain matrix being a function of the shape functions only,
while BL is the non-linear strain matrix being a function of the shape functions and
displacements. The components of Green’s strain vector can be written as
1
E = B0 + BL p
(9.29)
2
in terms of the linear and non-linear Green’s strain vectors
E0 = B0 p
(9.30)
and
1
EL = BL p
2
The non-linear equilibrium Eq. (9.25) can be written as
ψ (p) =
BT σ dV − R = w − R = 0
≈
(9.31)
(9.32)
V
where R is the right hand side of Eq. (9.25) for convenience and ψ (p) is termed
the residual. The Newton-Raphson method is used for the solution of the assembled
non-linear equations. The solution is achieved when ψ (p) is reduced to zero or a
given convergence criterion is satisfied [26, 39, 57, 59].
In case the body is subjected to initial thermal strains the stress tensor becomes
σ = D E − Eth
≈
where D is stiffness matrix and Eth is thermal strain tensor.
(9.33)
9 Non-Linear Thermal Stresses in Adhesive Joints
251
9.4 Thermal Model and Finite Element Formulation
Application of the first law of thermodynamics to a differential control volume
yields
...
∂T
T
+ v ∇T = q + ∇ (D∇T )
ρc
(9.34)
∂t
with the Fourier’s law
q = −D∇T
(9.35)
where vT = ...vx , vy , vz is the velocity vector for mass transport of heat, q is the heat
flux vector, q is the heat generation rate per unit volume, and D is the conductivity matrix. The present problems assume that the adhesive joint is subjected to the
specified convection surfaces (Newton’s law of cooling) as
q · n = −hm (T∞ − TS )
(9.36)
nT D∇T = hm (T∞ − T )
(9.37)
where n is the unit outward normal vector, hm is the film coefficient, T∞ is the bulk
temperature of the adjacent fluid and TS is the temperature at the surface of the
model. Pre-multiplying Eq. (9.34) by a virtual change in temperature δ T , integrating
over the volume of the element, and combining with Eq. (9.35) yield
∂T
+ vT LT + LT δ T (DLT ) dV
ρ cδ T
∂t
V
=
δ T hm (T∞ − T ) dS +
...
δ T q dV
(9.38)
V
S
where
LT =
∂ ∂ ∂
∂x ∂y ∂z
(9.39)
is a vector operator. The temperature is dependent on both space and time. Therefore, the temperature can be written in terms of element shape functions N and nodal
temperature vector Te as
(9.40)
T = NT Te
and the time derivatives of Eq. (9.40) may be written as
∂T
∂ Te
= NT
= NT Ṫe
∂t
∂t
(9.41)
δ T and LT also become
δ T = {δ Te }T N
and
LT = BTe
(9.42)
252
Mustafa K. Apalak
where B = LNT . The variational form of Eq. (9.38) can be written as
ρ c {δ Te }T N NT Ṫe dV +
V
+
ρ c {δ Te }T N vT BTe dV
V
T
{δ Te } B DBTe dV =
V
+
{δ Te }T N hm T∞ − NT Te dS
T
S
...
{δ Te } N q dV
T
(9.43)
V
If ρ is assumed to remain constant over the volume of the element, and {δ Te }T is
dropped, Eq. (9.43) is reduced to the final form as [59]
ρ
cN NT Ṫe dV + ρ
V
=
S
cN vT BTe dV +
V
T∞ hm N dS −
BT DBTe dV
V
T
hm N N Te dS +
...
N q dV
(9.44)
V
S
9.5 Thermal Boundary Conditions
The previous studies prescribe the final temperature distributions along the boundary of the adhesive joints and in the adhesive joints and then compute the thermal
strains using the temperature differences relative to the initial uniform joint temperature [1, 9, 24, 32, 33, 35, 36, 37, 42, 47, 48, 50]. In fact, the materials forming the
adhesive joints have different heat conduction/convection capabilities. Therefore,
the heat transfer through the adhesive joint members should be analysed in detail
for the thermal conditions allowing different heat transfer mechanisms.
The heat transfer through the adhesive joint occurs by means of the convection
(from fluid to adherends or adhesive), and the conduction (through adherends and
adhesive). The heat transfer by convection requires the computation of the heat
transfer coefficients between the air and the adherends or the adhesive. Their computations are dependent on how the air flows along the surfaces (vertically or horizontally). Since each surface has different geometry and especially flow conditions,
the heat transfer coefficients should be computed for each surface, as follows. In
case of a vertical air flow, the heat transfer coefficient is given as [34];
U∞ Deqv 0.731 λair
(9.45)
hm = 0.205
ν
Deqv
where U∞ is the air velocity (m/s), Deqv the equivalent diameter (m), ν the kinematic
the thermal conductivity of the air (W/m◦ C). In case of a
viscosity (m2 /s) and λair
horizontal air flow, the heat transfer coefficient is given as [34]:
hm =
Nu λ
L
(9.46)
9 Non-Linear Thermal Stresses in Adhesive Joints
253
where λ is the thermal conductivity of the air (kcal/mh◦ C), L is the plate length (m)
and Nu Nusselt number defined as
Nu = 0.836 · Re1/2 · Pr1/3
(9.47)
where Re and Pr are Reynolds and Prandtl numbers, respectively and are described as;
U∞ L
(9.48)
Re =
ν
cp μ
(9.49)
Pr =
λ
where c p is the specific heat (kcal/kg◦ C) and μ the dynamic viscosity (kg/ms). The
previous coefficients describing the air properties are values based on an average
temperature as follows
T∞ + T0
(9.50)
Tf =
2
where T∞ and T0 are the air and the plate temperatures, respectively. The thermal
coefficients of the air and the heat transfer coefficients are given in [15, 17].
9.6 Single Lap and Tubular Lap Joints
In general the thermal stress and strain distributions in adhesive joints were analysed assuming that the adherends and adhesive layer were subjected to a uniform
temperature distribution or the temperature distribution along the boundaries of the
adhesive joint was prescribed. The conductive heat transfer through the adhesive
joint as well as the convective heat transfer between the adhesive joint and the surrounding fluid were ignored. Therefore, the transient temperature distributions were
not considered.
Apalak and Gunes investigated the non-linear thermal stresses in an adhesively
bonded adhesive single lap joint which is subjected to fluid flows in different temperature and velocity for different adherend edge conditions [15]. They used adherends
made of medium carbon steel (1040) with a rubber modified epoxy adhesive and
considered the adhesive fillets around the adhesive free edges and the rounded adherend corners (Fig. 9.1) inside the adhesive fillets occurring as a result of etching
Fig. 9.1 Mesh details of the finite element model of a single lap joint [15]
254
Mustafa K. Apalak
T1 = 120°C
1
•
•
10
Upper adherend
T0 = 20°C
•
9 •
8
•
7
U1 = 1 m /s
2
•
3
•
•4
Lower adherend
T0 = 20°C
5
•
•
6
Y
Z
X
Adhesive, T0 = 20°C
T2 = 20°C
U2 = 1 m/s
Fig. 9.2 Thermal boundary conditions of the single lap joint [15]
the adherend surfaces in order to obtain a better bonding surface. The peak adhesive stress and strains occur around these adherend corners. The sharp adherend
corners cause stress singularities due to geometrical discontinuities, and the stress
levels around the rounded adherend corners are lower than those around the sharp
corners [1, 4, 6]. Apalak and Gunes also allowed the heat transfer to occur by conduction throughout the joint and by convection from the joint surfaces to the fluids,
or the fluids to the joint surfaces. The convective heat transfer between the fluid and
adherend surfaces depends on heat transfer coefficient (see Sect. 9.5). These thermal
boundary conditions (Fig. 9.2) make the heat transfer problem of the adhesive joint
complex. Since the thermal strain distributions in the adhesive joints are dependent
on the temperature distributions, they carried out first the thermal analysis and later
the geometrically non-linear stress analysis of the adhesive joint based on the small
strain-large displacement theory. They assumed a steady state heat transfer and obtained a non-uniform temperature distribution in the adhesive joint (Fig. 9.3). Due to
the thermal boundary conditions the temperature increased from the lower adhesive
fillet toward the right upper adhesive fillet and become maximum here.
Based on the non-linear stress analysis the adhesive joint exhibited different deformation geometries (large rotations through the overlap region) depending on
the edge conditions of the lower and upper adherends (Fig. 9.4), and the thermal
and mechanical mismatches of the adhesive and adherends caused high stress concentrations through adhesive regions close to adhesive-adherend interfaces around
the adhesive free ends (Fig. 9.5). The peak thermal stresses and strains in the
59.98
61.66
63.34
65.01
66.69
68.37
70.04
71.72
73.39
75.07
Fig. 9.3 Temperature distribution in the single lap joint (in ◦ C) [15]
9 Non-Linear Thermal Stresses in Adhesive Joints
255
(a)
(b)
(c)
Y
(d)
Z
X
Fig. 9.4 Deformed geometries (not scaled) for boundary conditions: (a) (BC-I), fixed edgehorizontally free edge, (b) (BC-II), only one corner of both plates fixed, (c) (BC-III), both edges
fixed, (d) (BC-IV), fixed edge-vertically free edge [15]
(a) BC -I
8.2
9.1
10.1
11.0
12.0
13.0
13.9
14.9
15.8
16.8
9.4
10.6
11.7
12.9
14.1
15.2
16.4
17.5
18.7
19.8
(b) BC -II
23.2
26.4
29.5
32.7
35.9
39.0
42.2
45.4
48.5
51.7
25.3
28.7
32.0
35.3
38.6
41.9
45.2
48.6
51.9
55.2
(c) BC -III
28.2
32.1
36.0
39.9
43.8
47.7
51.6
55.5
59.4
63.4
30.5
34.6
38.7
42.7
46.8
50.9
55.0
59.1
63.2
67.3
(d) BC -IV
11.3
13.5
15.7
17.9
20.1
22.3
24.5
26.7
28.9
31.1
12.6
15.1
17.6
20.1
22.6
25.1
27.6
30.1
32.6
35.1
Fig. 9.5 von Mises stress distributions in the left and right adhesive fillets (All stresses in MPa) [15]
256
Mustafa K. Apalak
1.000
Normalized von Mises stress, (σ eqv / σ max)
BC - I
0.950
BC - IV
0.900
0.850
BC - III
BC - II
0.800
0.750
0.700
0.08
σmax = 20.00 MPa
σmax = 69.54 MPa
σmax = 78.61 MPa
σmax = 37.49 MPa
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
0.26
0.28
0.30
0.32
0.34
Overlap length - joint length ratio
Fig. 9.6 Effect of the overlap length on the von Mises stress σeqv in different critical locations
inside the right adhesive fillet for the boundary conditions I–IV [15]
adhesive layer occurred at the free ends of the upper adherend-adhesive and the
lower adherend-adhesive interfaces, and these thermal strain and stress conditions
exceeded yield point for some plate end conditions. The elastic analyses showed that
increasing the overlap length is not beneficial in reducing the peak stresses in the
critical adhesive (Fig. 9.6) and adherend regions for two adherend end conditions.
This study showed that the thermal and material mismatches of the adhesive and
adherends result in serious thermal strains along the adhesive-adherend interfaces,
especially around the adhesive free ends. In case the variable thermal conditions
along the outer surfaces of the adhesive joint exist, non-uniform temperature distribution occurs in the adhesive joint, consequently, non-uniform thermal strain distribution. This makes thermal stress distribution in the adhesive joint more complex.
Therefore, actual variable thermal conditions as well as the large rotations and displacements occurring in the joint should be considered in the stress analysis and in
the design of adhesive joints.
Another adhesive joint used in load transmission of thin-walled structures is the
tubular single lap (TSL) joint. Since epoxy adhesives are usually rubber modified
in order to obtain higher toughness, they present non-linear behaviour under loads
and the exact solutions of the stress and deformation of TSL joints become difficult.
The torque transmission capabilities, elastic stress states, the effects of the adhesive
thickness and adherend roughness on the torsional static and fatigue strengths of the
adhesively bonded tubular lap joints, the failure models, the non-linear properties
and the viscoleastic behaviour of the adhesive layer in these joints were investigated in detail for structural loads [2, 7, 25, 29, 31, 41, 43, 56, 58]. In practice,
adhesive tubular joints also experience thermal loads as well as structural loads.
Since the thermal loads cause different thermal strains in both tubes and adhesive
layer, in case the degrees of freedom (translation and rotation) along the free edges
of the tubular joint are restrained the thermal strains would cause thermal stresses
9 Non-Linear Thermal Stresses in Adhesive Joints
T2 = 20°C
257
U2 = 1 m/s
T0 = 20°C
T0 = 20°C
T1 = 120°C
1•
A
•
10
B
T0 = 20°C
9•
8•
•
7
T0 = 20°C
T2 = 20°C
CL
U1 = 1 m/s
•2
•3
•4
C
T0 = 20°C
U2 = 1 m/s
•5
•6
θ
r
z
Fig. 9.7 Thermal boundary conditions of a tubular single lap joint (A: Inner tube, B: Adhesive
layer, C: Outer tube, T0 : Initial temperature) [17]
in the tubular joint. Since the adhesive and tubes have different thermal and mechanical properties they present different deformation states. The strains should be
compatible along the adhesive and tube interfaces, therefore the thermal stresses
are unavoidable. First, Kukovyakin and Skory considered the effect of the adhesive
and adherend thermal expansion coefficients in the thermal elastic stress analysis
of a bonded cylindrical joint [40]. Kim et al. investigated the effect of the thermal
residual stresses arising in adhesively bonded metal-metal tubular joints [36, 37].
However, these thermal stress analyses assume a uniform temperature distribution,
i.e. a constant temperature distribution in all joint regions, or along the geometrical
boundaries and do not consider conductive and convective heat transfers through the
tubes and adhesive layer, which occur due to variable thermal boundary conditions.
Apalak et al. carried out a geometrically non-linear thermal stress analysis of an
adhesively bonded (steel) tubular lap joint subjected to air flows in different temperature and velocity over its inner and outer surfaces (Fig. 9.7, see Sect. 9.5) [17].
Based on the thermal analysis they obtained a nonuniform temperature distribution
in the tubular joint. The temperature is minimal in the inner adhesive fillet and increases nonuniformly toward the outer adhesive fillet and then becomes maximal
here (Fig. 9.8). As a result of the non-uniform temperature distribution and different
CL
r
max
min
Left free edge
Right free edge
55.6
57.2
58.8
60.4
61.9
63.6
65.2
66.7
68.3
69.9
Fig. 9.8 Temperature distributions in the left and right free edges of overlap region in the tubular
single lap joint (all temperatures in ◦ C) [17]
258
Mustafa K. Apalak
CL
(a)
r
(b)
(c)
Fig. 9.9 Deformed geometries (not scaled) for boundary conditions: (a) (BC-I), only one corner
of two tubes fixed, (b) (BC-II), the free ends of two tubes fixed, and (c) (BC-III), the free edge of
the inner tube fixed and the right free edge of the outer tube free only in the axial direction [17]
material properties, the thermal strains at the different locations of the adhesive joint,
even in zones with similar material properties, become nonuniform. Consequently,
the deformation geometries of the tubular joint were affected considerably for different tube edge conditions (Fig. 9.9). The stress concentrations were observed inside
both inner and outer adhesive fillets around the adhesive free edges (Fig. 9.10). The
(a)
(b)
(c)
– 45.8
– 40.9
– 36.1
– 31.2
– 26.4
– 21.5
– 16.7
– 11.8
– 6.94
– 2.09
– 35.8
– 32.2
– 28.6
– 24.9
– 21.3
– 17.7
– 14.0
– 10.4
– 6.78
– 3.15
– 68.2
– 61.3
– 54.4
– 47.4
– 40.5
– 33.6
– 26.6
– 19.7
– 12.8
– 5.84
– 68.5
– 61.4
– 54.4
– 47.3
– 40.2
– 33.1
– 26.1
– 19.0
– 11.9
– 4.83
– 25.6
– 23.7
– 21.9
– 20.0
– 18.2
– 16.3
– 14.5
– 12.6
– 10.8
– 8.90
– 26.8
– 24.6
– 22.4
– 20.2
– 18.0
– 15.8
– 13.6
– 11.4
– 9.18
– 6.98
Fig. 9.10 (a) radial σrr , (b) normal σzz , and (c) shear σrθ stress variations inside the left and right
adhesive fillets for the BC-II (stresses in MPa) [17]
9 Non-Linear Thermal Stresses in Adhesive Joints
259
1.00
Normalized von Mises stress, (
eqv /
max)
BC - III
0.90
max=
63.29 MPa
74.36 MPa
max= 19.10 MPa
max=
0.80
0.70
0.60
BC - II
BC - I
0.50
0.09
0.15
0.20
0.28
0.33
Overlap length - joint length ratio
Fig. 9.11 Effect of the overlap length on the von Mises stress σeqv in different critical locations
inside the right adhesive fillet for the boundary conditions I–III [17]
most critical stress states appeared in cases the tube edges are fully constrained. The
radial and circumferential stresses were compressive. The peak adhesive and tube
stresses occurred at the free ends of the adhesive-inner and outer tube interfaces,
and the lowest stresses at the adhesive fillets neighboring the tube edges. In addition, the adhesive stress distributions around the rounded tube corners were smooth.
Increasing the overlap length had an important reducing effect in the peak adhesive
stresses (Fig. 9.11) and tube stresses in case the tube edges were fixed.
9.7 Adhesively Bonded Tee Joints
In structural adhesive joints, another category is tee joints, in which the joint members are bonded at a right angle or at some other angle. Generally, the joints are
subjected to loadings either in the plane of the plate or transverse to it. The analysis
and design of tee joints are more complicated than for lap joints or tubular joints.
Adams and Wake showed some possibilities of tee joints which may be encountered
in practice [4]. Shenoi and Violette used adhesively bonded composite tee joints in
small boats and investigated the influence of joint geometry on the ability transfer
out-of-plane loads [54]. However, the stresses in tee joints, especially in the adhesive
layer, and the effect of joint geometry on the adhesive stresses and on the joint stiffness have not been investigated. Apalak et al. presented a design of a tee joint with
a double support [22] or with a single support plus angled reinforcement [23] and
showed that the peak adhesive stresses occurred around the adhesive free ends and
around the lower corners of the vertical plate, and that the side loading was the most
critical among the loading conditions used for the tee joint. They also investigated
260
Mustafa K. Apalak
the effects of support lengths on the peak adhesive stresses and the overall joint
stiffness for various loading and boundary conditions, and gave the appropriate joint
dimensions based on the analyses. Li et al. also presented similar results related to
tee joints [44, 45]. da Silva and Adams developed simple mathematical models for
predicting the strength of different tee joints [27]. Apalak investigated the effect of
geometrical non-linearity on the stress states of adhesive tee joint with double support [16] and with single support plus angled reinforcement [13], and showed that
the stresses in the critical joint regions, especially in adhesive fillets, were overestimated with the linear elasticity theory. Apalak et al. studied the thermal behaviour
of adhesively bonded tee joint with a single support plus angled reinforcement [18].
They considered the tee joint configurations bonded to a rigid base and a flexible
plate (Fig. 9.12) and applied the fluid flows in different temperatures and velocities
along the outer surfaces of the plates, supports and adhesive layer (Fig. 9.13) and
allowed the convective heat transfer between the outer surfaces and air as well as
the conductive heat transfer through the tee joint members. The thermal analysis
showed that non-uniform temperature distributions occurred in both tee joint configurations. The temperature distributions inside the adhesive fillets were different
depending on the rigid base (Fig. 9.14) or the flexible plate configurations. The air
flow along the lower surface of the horizontal flexible plate caused a reduction of the
temperature levels and the temperature contours to be parallel to the horizontal plate.
The tee joint bonded to an elastic plate can translate and rotate more in comparison
with the tee joint bonded to a rigid base (Fig. 9.12). This large rotation and displacements can be evaluated easily based on the small strain-large displacement theory.
The peak adhesive stresses occurred inside the adhesive fillets (Fig. 9.15). Consequently, the thermal stresses in the adhesive fillets in the tee joint bonded to a rigid
base were higher (Fig. 9.16). The stress concentrations were observed around the
rounded corners of the left support and the angled reinforcement and reached peak
levels at the adhesive interface-rounded adherend corners. As a result, the variable
thermal boundary conditions cause a non-uniform temperature distributions; thus, a
third type of (thermal) non-linearity in the thermal stress analysis of adhesive joints
1
10
1
To = 20°C
U∞= 1 m / s
Ta =120 °C
U∞ = 1 m / s
Ta = 120 °C
U∞ = 1 m / s
Ta = 20 °C
4
a)
U∞ = 1 m/s
Ta = 20 °C
2
4
3
8
9
5
6
2
3
b)
5
To = 20 °C
7
10
7
To = 20 °C
11
12
13
14
15
18
6
11
12
13
14
15
8
9
19
16
17
U∞ = 0.5 m / s
Ta = 60 °C
Fig. 9.12 Thermal boundary conditions of a tee joint with single support plus angled reinforcement
bonded to (a) a rigid base, and (b) a flexible plate [18]
9 Non-Linear Thermal Stresses in Adhesive Joints
261
Y
Z
X
b)
a)
Fig. 9.13 Boundary conditions and deformed geometries of a tee joint with single support plus
angled reinforcement bonded to (a) a rigid base (BC-I), and (b) a flexible plate (BC-II) [18]
appears in addition to the geometrical and material non-linearities. The edge conditions of the adherends or bonding the adhesive joint to a rigid or flexible base affect
considerably the stress and deformation states of the adherends and adhesive layer,
especially inside the adhesive fillets. In addition Apalak et al. investigated the effect
of the support length on the peak thermal stresses inside the adhesive fillets and in
the adherends [18]. In the case of the tee joint bonded to rigid base, increasing the
support length reduced (by 50–60%) the peak stresses inside the left and right horizontal fillets whereas it causes an increase of 35% in the peak stresses inside the
vertical adhesive fillet which is most critical region.
Apalak et al. also investigated thermal stresses occurring in adhesively bonded
tee joints with double support [19]. The tee joint configurations were bonded to a
flexible plate or a rigid base (Fig. 9.17) and subjected to the fluid flows parallel or
tangential to the outer surfaces of the steel adherends and adhesive layer (Fig. 9.18).
They considered the conductive and convective heat transfers through all members
of the adhesive joint and presented a simple calculation method for the heat transfer
coefficients between the adherend surfaces and the fluids (see Sect. 9.5). First, the
steady-state heat transfer analysis of the adhesively bonded tee joint was carried out
for each of the two joint configurations. They obtained temperature distributions in
the adhesive fillets for the tee joint bonded to a rigid base (Fig. 9.19). Due to the ther-
A
B
C
D
E
F
G
H
a)
=
=
=
=
=
=
=
=
88
90
91
93
95
96
98
99
A
B
C
D
E
F
G
H
b)
=
=
=
=
=
=
=
=
45
48
50
52
54
56
58
60
A
B
C
D
E
F
G
H
=
=
=
=
=
=
=
=
68
70
73
75
77
79
82
84
c)
Fig. 9.14 Temperature distributions in critical regions of a tee joint with single support plus angled
reinforcement bonded to a rigid base (BC-I): (a) left horizontal adhesive fillet, (b) right horizontal
adhesive fillet, and (c) vertical adhesive fillet (all temperatures in ◦ C) [18]
262
A
B
C
D
E
F
G
H
I
Mustafa K. Apalak
=
=
=
=
=
=
=
=
=
25
31
38
44
50
56
63
69
75
a)
A
B
C
D
E
F
G
H
I
=
=
=
=
=
=
=
=
=
15
18
22
25
29
32
36
40
43
A
B
C
D
E
F
G
H
I
b)
=
=
=
=
=
=
=
=
=
14
20
25
30
35
40
45
50
55
c)
Fig. 9.15 von Mises stress σeqv distributions in critical regions of a tee joint with single support
plus angled reinforcement bonded to a rigid base (BC-I): (a) vertical adhesive fillet, (b) left horizontal adhesive fillet, and (c) right horizontal adhesive fillet (all stresses in MPa) [18]
mal boundary conditions, the left lower and left vertical adhesive fillets experienced
higher temperatures than the adhesive fillets on the right-hand side of the adhesive
tee joints. Thus, temperatures decrease uniformly from surfaces interacting with
the air at high temperature to ones interacting with the air at low temperature. The
temperature differences through the adhesive fillets are between 3 and 9◦ C. This
temperature difference is large enough to cause the thermal strains causing the thermal stresses in the adhesive joints. In the adhesive tee joint bonded to a flexible plate
Normalized von Mises stress, (
eqv /
max )
1.10
1.00
0.90
(a)
(b)
(c)
0.80
0.70
max =
102.0 MPa
82.9 MPa
max = 73.3 MPa
max =
0.60
0.50
0.40
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Support length - joint length ratio
0.18
0.20
0.22
Fig. 9.16 Effect of the support length on the normalized von Mises σeqv /σmax stresses in the
critical adhesive locations: (a) vertical adhesive fillet, (b) left horizontal adhesive fillet, and (c) right
horizontal adhesive fillet (Rigid base, BC-I) [18]
9 Non-Linear Thermal Stresses in Adhesive Joints
263
Y
Z
a)
X
b)
Fig. 9.17 Boundary conditions and deformed geometries of a tee joint with double support bonded
to (a) a rigid base (BC-I) and (b) a flexible plate (BC-II) [19]
the joint regions interacting with the fluid with high-temperature also experience a
higher temperature than other regions. Similar temperature distributions are also observed in the adhesive fillets. The variable thermal boundary conditions (air streams
with different temperature and velocity) along the outer surfaces of the adhesive
joint resulted in a non-uniform temperature distribution through the adhesive layer
and other joint members. As a result, a non-uniform thermal strain distribution occurred.
The geometrical non-linear thermal stress analysis showed that large rotations
occurred in the bonding region of the adhesive tee joint and the free regions of
the vertical and horizontal plates deformed considerably (Fig. 9.17). In addition,
the free edges of the adhesive layers experience stress concentrations and the von
Mises stresses peak at the rounded corners inside the adhesive fillets. The adhesive
fillets interacting the fluids with higher temperatures, such as the left lower and
vertical adhesive fillets, contain considerable stress concentrations (Fig. 9.20). Both
adherend and adhesive stresses in the tee joint bonded to a flexible plate were smaller
than those in the joint bonded to a rigid base. The flexible horizontal plate allows the
1
10
2
11
3
12
U∞ = 1 m/s
Ta = 50 °C
4
a)
5
7
8
9
6
13
U ∞ = 1 m/s
U ∞ = 1 m/s
Ta = 20 °C
Ta = 50 °C
15
4
5
7
16
17
18
21
10
2
3
11
U∞ = 1 m/s
b)
14
1
6
12
Ta = 20 °C
13
14
15
16
17
18
8
9
22
19
20
U∞ = 1 m /s , Ta = 30 °C
Fig. 9.18 Thermal boundary conditions of a tee joint with double support bonded to (a) a rigid
base, and (b) a flexible plate [19]
264
Mustafa K. Apalak
A
B
C
D
E
F
G
H
A
B
C
D
E
F
G
H
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
35.2
35.9
36.5
37.1
37.8
38.4
39.0
39.7
40.8
41.3
41.8
42.3
42.8
43.3
43.8
44.2
A
B
C
D
E
F
G
H
b)
a)
=
=
=
=
=
=
=
=
30.1
30.7
31.4
31.9
32.6
33.2
33.9
34.5
A
B
C
D
E
F
G
H
c)
=
=
=
=
=
=
=
=
25.6
26.1
26.5
27.0
27.5
27.9
28.5
28.9
d)
Fig. 9.19 Temperature distributions in the critical regions of an adhesively bonded tee joint with
double support bonded to a rigid base: (a) left horizontal adhesive fillet, (b) left vertical adhesive
fillet, (c) right vertical adhesive fillet, and (d) right horizontal adhesive fillet (all temperatures in
◦ C) [19]
tee joint to deform in the transverse direction; therefore, the internal load distribution
reduce partly. Thus, the rigid adhesive joints experience serious stress levels.
The variable thermal boundary conditions and the thermal-mechanical mismatches of the adhesive and adherends result in a non-uniform temperature
distribution in the adhesive joint; consequently a non-uniform thermal strain and
stress distributions. These stress concentrations around the adhesive free ends of
A
B
C
D
E
F
G
H
I
=
=
=
=
=
=
=
=
=
a)
7.81
9.53
11.2
13.0
14.7
16.4
18.1
19.8
21.5
A
B
C
D
E
F
G
H
I
=
=
=
=
=
=
=
=
=
A
B
C
D
E
F
G
H
I
4.78
6.32
7.87
9.41
11.0
12.5
14.0
15.6
17.1
b)
c)
=
=
=
=
=
=
=
=
=
4.05
4.95
5.84
6.74
7.64
8.53
9.43
10.3
11.2
A
B
C
D
E
F
G
H
I
=
=
=
=
=
=
=
=
=
3.70
4.87
6.04
7.21
8.38
9.54
10.7
11.9
13.1
d)
Fig. 9.20 von Mises stress σeqv distributions in the critical regions of an adhesively bonded tee
joint with double support bonded to a rigid base: (a) left horizontal adhesive fillet, (b) left vertical
adhesive fillet, (c) right vertical adhesive fillet, and (d) right horizontal adhesive fillet (All stresses
in MPa) [19]
9 Non-Linear Thermal Stresses in Adhesive Joints
265
the adhesive joints subjected to structural loads can be reduced by increasing the
bonding surface (overlap length) or tapering the adherend edges [4]. Therefore, the
effect of the support length on the peak stresses inside the adhesive fillets and at the
critical plate regions was investigated for the adhesive tee joints bonded to a rigid
base and a flexible plate. Increasing the support length results in increases in the
normalised equivalent stresses at the critical adhesive locations (Fig. 9.21).
The detailed analysis of the stress components showed that the normal stress
σxx is compressive and dominant among all stress components at the critical adhesive locations whereas the normal stress σyy is compressive and dominant in the
critical vertical plate locations (not shown). However, the compressive state of the
normal stresses may be advantageous in cases the adhesive tee joint experiences
tensile loading conditions since the tensile stress levels can be reduced. The stress
components and equivalent stress are increased as the support length is increased.
Thus, the vertical plate tends to buckle as a result of thermal strains and its one fixed
end. The bending moment occurring at the left free end of the horizontal adhesive
layer increases since the vertical plate buckles. However, the equivalent stress at
only the left horizontal adhesive fillet increases 1.27 times. The adhesively bonded
tee joint was analysed for two specific boundary conditions causing buckling in the
vertical and horizontal plates. Increasing the support length makes the joint region
relatively stiffer. However, the extensional thermal elongations are restrained. In this
case most of the load is transferred to the support via the adhesive fillets. Since the
normal stresses inside the adhesive fillets are compressive, in case the tee joint experiences an additional tensile load this would be advantageous in order to reduce
the total stresses.
max)
0.92
Normalized von Mises stress, (
0.96
eqv /
1.00
0.88
0.84
0.80
max=
29.30 MPa
20.80 MPa
max= 13.75 MPa
max= 14.73 MPa
(a)
(b)
(c)
(d)
0.76
0.72
max=
0.68
20
30
40
50
60
70
80
Support length (mm)
Fig. 9.21 Effect of the support length on the normalized von Mises σeqv /σmax stresses in the
critical adhesive locations: (a) left horizontal adhesive fillet, (b) left vertical adhesive fillet, (c) right
vertical adhesive fillet, and (d) right horizontal adhesive fillet (Rigid base, BC-I) [19]
266
Mustafa K. Apalak
9.8 Micromechanics of Composite Materials
The engineering constants of unidirectional fiber-reinforced lamina can be computed by using the micro-mechanics approach based on the engineering constants
of fiber reinforcement ( f ) and matrix (m) in the material coordinates (x1 , x2 , x3 )
as [51]
E1 = E f V f + EmVm
(9.51)
E f Em
E f Vm + EmV f
(9.52)
υ12 = υ f V f + υmVm
(9.53)
G f Gm
G f Vm + GmV f
(9.54)
E2 =
G12 =
where Vm and V f are the volume fractions of the matrix and fibre, E1 is the longitudinal modulus, E2 is the transverse modulus, υ12 is the major Poisson’s ratio, G12 is
the shear modulus, and the shear moduli of the fiber and the matrix
Ef
Gf = 2 1+υf
(9.55)
Em
2 (1 + υm )
(9.56)
Gm =
respectively. The following reciprocal relations also exist
υ21 υ12
=
,
E2
E1
υ31 υ13
=
,
E3
E1
υ32 υ23
=
E3
E2
or
υi j
υ ji
=
Ei
Ej
(9.57)
In case of a transversely isotropic material with the 2 − 3 plane as the plane of
isotropy
E2 = E3 ,
G12 = G13
and
υ12 = υ13
(9.58)
The generalized Hooke’s law for an anisotropic material under isothermal conditions is given by
σi j = Ci jkl εkl
(9.59)
or in contracted notation [51]
σi = Ci j ε j
(9.60)
If the coordinate planes are chosen parallel to the three orthogonal planes of material symmetry, Eq. (9.60) can be written in matrix form in the material coordinates
(x1 , x2 , x3 ) as
9 Non-Linear Thermal Stresses in Adhesive Joints
⎡
⎤
⎡
C11 C12
σ1
⎢ σ2 ⎥
⎢ C21 C22
⎢ ⎥
⎢
⎢ σ3 ⎥
⎢
⎢ ⎥ = ⎢ C31 C32
⎢ σ4 ⎥
⎢ 0 0
⎢ ⎥
⎢
⎣ σ5 ⎦
⎣ 0 0
σ6 m
0 0
C13
C23
C33
0
0
0
0
0
0
C44
0
0
267
0
0
0
0
C55
0
⎤⎡
⎤
0
ε1
⎢ ε2 ⎥
0 ⎥
⎥⎢ ⎥
⎢ ⎥
0 ⎥
⎥ ⎢ ε3 ⎥
⎢ ⎥
0 ⎥
⎥ ⎢ ε4 ⎥
0 ⎦ ⎣ ε5 ⎦
ε6 m
C66
(9.61)
The substitution of the engineering constants into the inverse relation of Eq. (9.61)
gives
⎡ 1
⎤
υ31
υ21
⎡ ⎤
0 0 ⎡ ⎤
E1 − E2 − E3 0
ε1
⎢ υ12 1
⎥ σ1
υ
⎢ − E1 E2 − E323 0 0 0 ⎥ ⎢ σ2 ⎥
⎢ ε2 ⎥
⎢ υ13 υ23 1
⎥⎢ ⎥
⎢ ⎥
⎢
⎥⎢ ⎥
⎢ ε3 ⎥
⎢ ⎥ = ⎢ − E1 − E2 E3 01 0 0 ⎥ ⎢ σ3 ⎥
(9.62)
⎢ ⎥
⎢ 0
⎢ ε4 ⎥
0
0 G23 0 0 ⎥
⎢
⎥ ⎢ σ4 ⎥
⎢ ⎥
⎢
⎥
⎣ ε5 ⎦
0
0
0 G113 0 ⎦ ⎣ σ5 ⎦
⎣ 0
ε6 m
σ6 m
0
0
0
0 0 G112
or
{ε }m = [S]m {σ }m
(9.63)
In order to determine the strain-stress relations of a laminate, it is necessary to
convert strain components referred to the material (lamina) coordinate system
(x1 , x2 , x3 ) to those referred to the problem (laminate) coordinate system (x, y, z) as
⎡
⎤
⎤⎡ ⎤
⎡ 2
m
n2 0 0 0 −mn
εxx
ε1
⎢ εyy ⎥
⎢ n2 m2 0 0 0 mn ⎥ ⎢ ε2 ⎥
⎢
⎥
⎥⎢ ⎥
⎢
⎢ ⎥
⎢ εzz ⎥
⎢
0 1 0 0
0 ⎥
⎢
⎥ =⎢ 0
⎥ ⎢ ε3 ⎥
(9.64)
⎢ 2εyz ⎥
⎥ ⎢ ε4 ⎥
⎢ 0
0
0
m
n
0
⎢
⎥
⎥⎢ ⎥
⎢
⎣ 2εxz ⎦
⎣ 0
0 0 −n m
0 ⎦ ⎣ ε5 ⎦
2εxy p
ε6 m
2mn −2mn 0 0 0 m2 − n2
where m = cos θ , n = sin θ , and similarly
{ε } p = [R]T {ε }m
(9.65)
{σ } p = [R]T {σ }m
(9.66)
In order to relate compliance coefficients in the two coordinate systems, Eqs. (9.63)
and (9.65) are substituted into Eq. (9.66) as
{ε } p = [R]T {ε }m = [R]T ([S]m {σ }m ) = [R]T [S]m [R] {σ } p
(9.67)
{ε } p = [S] p {σ } p
(9.68)
S = [R]T [S] [R]
(9.69)
where [S] p ≡ S and [S]m ≡ [S]
268
Mustafa K. Apalak
The thermal expansion coefficients αi j are also transformed (α12 = α13 = α23 = 0)
as
αxx = α11 m2 + α22 n2
(9.70)
αyy = α11 n2 + α22 m2
(9.71)
2αxy = 2 (α11 − α22 ) mn
(9.72)
2αxz = 0,
2αyz = 0,
αzz = 0
(9.73)
9.9 Adhesively Bonded Composite Tee Joints
Adhesive bonding technique is used successfully for joining carbon fibre reinforced
plastics to metals or composite structures. A good design with either simple or more
complex geometry requires stress and deformation states to be known for different
boundary conditions. In case the adhesive joint is subjected to thermal loads, the
thermal and mechanical mismatches of the adhesive and adherends cause thermal
stresses. The plate-end conditions may also result in the adhesive joint to undergo
large displacements and rotations whereas the adhesive and adherends deform elastically (small strain). Apalak et al. carried out thermal and geometrically non-linear
stress analyses of an adhesively bonded composite tee joint with single support plus
an angled reinforcement made of unidirectional CFRPs using the non-linear finite
element method [20]. In the stress analysis, they considered the effects of the large
displacements using the small displacement-large displacement theory and investigated the stress states in the plates and the adhesive layer of the tee joint configurations bonded to a rigid base and a composite plate (Fig. 9.22). An initial uniform
temperature distribution was attributed to the adhesive joint for a stress free state,
and then variable thermal boundary conditions, i.e. air flows with different velocity
and temperature were specified along the outer surfaces of the tee joints (Fig. 9.12).
Since the adhesive tee joints consist of adhesive layer and adherends having different
Y
Z
a)
X
b)
Fig. 9.22 Boundary conditions and deformed geometries of a composite tee joint with single
support plus angled reinforcement bonded to (a) a rigid base (BC-I), and (b) a flexible plate
(BC-II) [20]
9 Non-Linear Thermal Stresses in Adhesive Joints
A
B
C
D
E
F
G
H
a)
=
=
=
=
=
=
=
=
63.2
64.1
64.9
65.9
66.7
67.6
68.5
69.4
A
B
C
D
E
F
G
H
b)
=
=
=
=
=
=
=
=
40.6
42.4
44.2
45.9
47.7
49.5
51.3
53.1
269
A
B
C
D
E
F
G
H
=
=
=
=
=
=
=
=
53.1
54.2
55.4
56.6
57.7
58.9
60.0
61.2
c)
Fig. 9.23 Temperature distributions in critical regions of a composite tee joint with single support plus angled reinforcement bonded to a rigid base: (a) left horizontal adhesive fillet, (b) right
horizontal adhesive fillet, and (c) vertical adhesive fillet (all temperatures in ◦ C) [20]
mechanical and thermal properties, non-uniform temperature distributions occurred
in the adhesive tee joints, and consequently, non-uniform thermal strain distributions
(Fig. 9.23). In addition, the edges of the horizontal plate and vertical plate were restrained partly or completely (Fig. 9.22), and then the thermal stress distributions of
two adhesive tee joints were determined based on their non-uniform thermal strain
distributions using the small strain-large displacement theory.
In case of the rigid base, the vertical plate of the adhesive tee joint buckled and
the upper free end of the vertical adhesive layer experienced large displacements.
The tee joint with flexible base had similar deformations for the vertical plate and
the free ends of the vertical adhesive layer; moreover, its horizontal plate was deformed (buckled) considerably. Thus, the left and right free ends of the horizontal
adhesive layer and the middle region of the horizontal plate had evident deformations. High normal stresses occurred in the horizontal and vertical sections of the
joint regions of both tee joints. Furthermore, the horizontal and vertical adhesive fillets were subjected to high stress concentrations in both rigid (Fig. 9.24) and flexible
bases. In general, the stress levels reached a maximum at the free ends of the horizontal plate or vertical plate-adhesive interfaces and distributed uniformly through
adhesive fillets towards the rounded corners of the support and the angled reinforcement member. However, the most critical adhesive regions were the left horizontal
and vertical adhesive fillets. In case of the adhesive tee joint with a rigid base, the
support length has an effect of decreasing the peak stresses in the right horizontal
adhesive fillet whereas its effect on the peak stresses in the vertical adhesive fillet
and at the vertical plate (Table 9.1). The support length had an effect of decreasing
the peak stresses at the critical locations inside the left horizontal and vertical adhesive fillets, and in the horizontal plate whereas its effect on the peak stresses in
the vertical adhesive fillet and in the vertical plate is insignificant. Apalak et al. also
presented a practical outline of how to carry out the thermal analysis and stress analysis of adhesively bonded joints subjected to variable thermal boundary conditions
requiring the heat transfer to take place by conduction throughout the joint members and convection between fluid and adherend surfaces [21]. They investigated
the temperature and stress states of an adhesively bonded composite tee joint with
270
A
B
C
D
E
F
G
H
I
b)
Mustafa K. Apalak
=
=
=
=
=
=
=
=
=
8.51
10.4
12.3
14.2
16.0
17.9
19.8
21.7
23.6
A
B
C
D
E
F
G
H
I
a)
=
=
=
=
=
=
=
=
=
7.96
9.06
10.2
11.3
12.4
13.5
14.5
15.6
16.7
A
B
C
D
E
F
G
H
I
=
=
=
=
=
=
=
=
=
5.66
6.42
7.19
7.95
8.71
9.48
10.2
11.0
11.8
c)
Fig. 9.24 von Mises stress σeqv distributions in critical regions of a composite tee joint with single
support plus angled reinforcement bonded to a rigid base (BC-I): (a) left horizontal adhesive fillet,
(b) vertical adhesive fillet, and (c) right horizontal adhesive fillet (all stresses in MPa) [20]
double support by applying variable thermal boundary conditions to its outer surfaces. Air streams with different velocity and temperature are specified parallel and
perpendicular to the outer surfaces of the joint members (Fig. 9.18). The thermal
analyses of the tee joints bonded to rigid and flexible bases showed that the temperature distributions were non-uniform through the adhesive joint. The temperature
distributions were also non-uniform in the adhesive fillets at the adhesive free edges
for both rigid and flexible bases (Figs. 9.25 and 9.26). In addition, the geometrical
non-linear analysis based on the small strain-large displacement theory predicted
that non-uniform temperature distributions caused non-uniform thermal strain and
stress distributions and considerable deformations in the tee joints (Fig. 9.17).
The critical adhesive and plate regions appear as the adhesive free ends as well
as the middle section of the horizontal and vertical plates, respectively. Due to the
plate edge conditions, the vertical and flexible horizontal plates are forced to buckle;
especially, the horizontal plate is deformed considerably undergoing large displacement and rotations (Fig. 9.17). The left horizontal and vertical adhesive fillets are
the most critical adhesive regions where the von Mises stresses achieve significant
levels (Figs. 9.27 and 9.28). The normal stresses σxx and σyy are dominant in the
stress states of the tee joints rather than the shear stress σxy . The normal stresses are
usually compressive in the critical adhesive fillets; consequently, these stress states
in the adhesive fillets would contribute to joint strength in cases where the tee joint
is subjected to the loading conditions which cause tensile stresses in these regions.
The joint failure can be expected along the composite plates surfaces as well as inside the adhesive fillets in cases where toughened adhesives are used [55]. In case of
tee joints made of adherends, which may yield, the adherend yielding causes the free
ends of the adhesive layer to undergo higher deformations; therefore, the adhesive
9 Non-Linear Thermal Stresses in Adhesive Joints
271
Table 9.1 Effect of the support and angled reinforcement length (a) on the peak stress components
(in MPa) at the critical adhesive and plate locations of the adhesively bonded composite tee joint
with a rigid base (PV: Percentage variation) [20]
a
σxx
PV
σyy
PV
σxy
PV
Left horizontal adhesive fillet
20
30
40
50
60
−21.20
0.00
−20.90
−1.42
−20.70
−2.36
−22.30
5.19
−20.80
−1.89
Right horizontal adhesive fillet
−17.60
−15.70
−14.90
−13.40
−14.50
0.00
−10.80
−15.34
−23.86
−17.61
−13.60
−13.00
−12.80
−13.10
−12.60
0.00
−4.41
−5.88
−3.68
−7.35
20
30
40
50
60
−11.80
−11.20
−11.10
−11.10
−11.20
Vertical adhesive fillet
0.00 −12.70
0.00
−5.08 −11.40 −10.24
−5.93 −10.80 −14.96
−5.93
−9.46 −25.51
−5.08 −10.60 −16.54
−7.06
0.00
−6.76 −4.25
−6.60 −6.52
−6.72 −4.82
−6.56 −7.08
20
30
40
50
60
−12.30
−12.40
−12.30
−11.10
−12.10
Bottom adhesive layer
0.00 −14.50
0.81 −14.70
0.00 −14.70
−9.76 −15.50
−1.63 −14.60
0.00
1.38
1.38
6.90
0.69
−9.14
−9.20
−9.21
−9.45
−9.15
0.00
0.66
0.77
3.39
0.11
20
30
40
50
60
−15.30
−15.40
−15.50
−15.50
−15.50
Vertical plate
0.00
0.65
1.31
1.31
1.31
−15.10
−15.20
−15.20
−15.20
−15.10
0.00
0.66
0.66
0.66
0.00
−4.41
−4.24
−4.17
−4.13
−4.11
0.00
−3.85
−5.44
−6.35
−6.80
20
30
40
50
60
−4.47
−4.49
−4.49
−4.40
−4.48
0.00
0.45
0.45
−1.57
0.22
−33.80
−34.00
−34.30
−35.10
−34.90
0.00
0.59
1.48
3.85
3.25
−3.79
−3.81
−3.82
−3.76
−3.82
0.00
0.53
0.79
−0.79
0.79
failure is more probable rather than the adherend failure. For that reason, bonding
an additional composite sheet along the most deformed section of the plates can reinforce the buckled plates of the tee joints. In addition, increasing the bonding area
could relieve the peak adhesive stresses at the adhesive free ends. However, the thermal non-linear elastic stress analyses of two tee joints for different supports showed
that increasing the support length do not have the same reducing effect of normal
and shear stresses, it could reduce the von Mises stresses only by 10–15% in the
most critical adhesive and plate locations. Also, it results in decreases of up to 35%
in the von Mises stresses in the less critical adhesive and plate regions. Modifying
the edge geometry of the composite plates would be more effective in reducing the
272
Mustafa K. Apalak
A
B
C
D
E
F
G
H
A
B
C
D
E
F
G
H
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
A
B
C
D
E
F
G
H
50.1
51.3
52.6
53.8
55.0
56.3
57.5
58.7
60.9
61.9
62.9
63.9
64.9
65.9
66.9
67.9
b)
=
=
=
=
=
=
=
=
40.4
41.6
42.8
43.9
45.2
46.4
47.8
48.8
A
B
C
D
E
F
G
H
c)
a)
=
=
=
=
=
=
=
=
31.4
32.4
33.4
34.3
35.3
36.3
37.3
38.3
d)
Fig. 9.25 Temperature distributions in the critical regions of an adhesively bonded tee joint with
double support bonded to a rigid base: (a) left horizontal adhesive fillet, (b) left vertical adhesive
fillet, (c) right vertical adhesive fillet, and (d) right horizontal adhesive fillet (all temperatures
in ◦ C) [21]
A
B
C
D
E
F
G
H
A
B
C
D
E
F
G
H
a)
=
=
=
=
=
=
=
=
50.7
51.4
52.2
52.9
53.7
54.4
55.1
55.9
=
=
=
=
=
=
=
=
48.9
49.9
50.8
51.7
52.7
53.6
54.5
55.5
A
B
C
D
E
F
G
H
c)
b)
=
=
=
=
=
=
=
=
41.0
42.0
43.
43.9
44.9
45.9
46.9
47.9
A
B
C
D
E
F
G
H
=
=
=
=
=
=
=
=
39.4
39.9
40.3
40.8
41.3
41.8
42.3
42.8
d)
Fig. 9.26 Temperature distributions in the critical regions of an adhesively bonded tee joint with
double support bonded to a flexible base: (a) left horizontal adhesive fillet, (b) left vertical adhesive
fillet, (c) right vertical adhesive fillet, and (d) right horizontal adhesive fillet (all temperatures
in ◦ C) [21]
9 Non-Linear Thermal Stresses in Adhesive Joints
A
B
C
D
E
F
G
H
A
B
C
D
E
F
G
H
=
=
=
=
=
=
=
=
7.93
10.0
12.1
14.2
16.2
18.3
20.4
22.5
=
=
=
=
=
=
=
=
273
A
B
C
D
E
F
G
H
7.31
8.48
9.65
10.8
12.0
13.2
14.3
15.5
4.55
5.78
7.00
8.22
9.45
10.7
11.9
13.1
A
B
C
D
E
F
G
H
c)
b)
a)
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
2.85
3.45
4.06
4.66
5.27
5.87
6.48
7.08
d)
Fig. 9.27 von Mises stress σeqv distributions in the critical regions of an adhesively bonded tee
joint with double support bonded to a rigid base: (a) left horizontal adhesive fillet, (b) left vertical
adhesive fillet, (c) right vertical adhesive fillet, and (d) right horizontal adhesive fillet (All stresses
in MPa) [21]
peak stresses rather than increasing the support length for these thermal and structural boundary conditions.
Finally, the thermal loads may result in non-uniform temperature distributions
in the adhesive joints due to the different thermal properties of the adhesive and
composite plates, consequently non-uniform thermal strain distributions arise. In
A
B
C
D
E
F
G
H
A
B
C
D
E
F
G
H
a)
=
=
=
=
=
=
=
=
6.93
8.16
9.40
10.6
11.9
13.1
14.3
15.6
=
=
=
=
=
=
=
=
6.86
7.91
8.95
10.0
11.0
12.1
13.1
14.2
A
B
C
D
E
F
G
H
b)
c)
=
=
=
=
=
=
=
=
4.65
5.73
6.81
7.89
8.97
10.1
11.1
12.2
A
B
C
D
E
F
G
H
=
=
=
=
=
=
=
=
4.18
5.11
6.05
6.98
7.91
8.85
9.78
10.7
d)
Fig. 9.28 von Mises stress σeqv distributions in the critical regions of an adhesively bonded tee joint
with double support bonded to a flexible base: (a) left horizontal adhesive fillet, (b) left vertical
adhesive fillet, (c) right vertical adhesive fillet, and (d) right horizontal adhesive fillet (All stresses
in MPa) [21]
274
Mustafa K. Apalak
case some additional structural constraints are imposed to adhesive joints, they may
experience considerable high stress and strain distributions. In case large displacements and displacement gradients are observed, the small strain-large displacement
theory would predict reasonably accurate stress and deformation states of the adhesive joints providing that the adhesive and composite adherends have small strains.
9.10 Conclusions
In case adhesive joints are subjected to the thermal loads, the different thermal and
mechanical properties of the adhesive layer and adherends cause incompatible thermal strains along the adhesive-adherend interfaces. Consequently, the non-uniform
thermal stress distributions around these regions occur even if the temperature distribution is uniform and the adhesive joint is not constrained. The variable thermal
boundary conditions cause a third non-linearity affecting the stress and strain distributions in the adhesive joints. The interaction of the adhesive joint with a fluid
having a specific velocity and temperature results in heat transfer to take place
by convection between the fluid-adherends or adhesive layer, and by conduction
through the adhesive joint. After the transient temperature distribution in the adhesive joint was determined for the thermal boundary conditions, the corresponding
thermal strain and stress distributions can be found. The adhesive joints, especially
the bonding region, experience large displacements and rotations; thus, deform considerably. In order to consider the effects of the large displacements on the stress and
deformation states of the adhesive joints, the small strain-large displacement theory
should be implemented to the thermal stress problem since the small strain-small
displacement theory overestimates the stresses as well as the displacements. Based
on the thermal stress analyses of adhesively bonded single lap, tubular lap and tee
joints with metal or composite adherends under variable thermal boundary conditions, non-uniform temperature distributions, and consequently, non-uniform thermal strain and stress distributions were observed. The peak stresses occurred around
the adherend corners inside the adhesive fillets or along the adherend-adhesive interfaces depending on the structural boundary conditions. Increasing the overlap area
did not reduce these peak adhesive stresses for all boundary conditions, on contrary,
caused the peak stresses to increase.
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Chapter 10
Impact
Chiaki Sato
Abstract This chapter treats mechanical aspects of adhesively bonded joints
subjected to impact loads. Fundamentals of impact mechanics of solids are introduced in Sect. 10.2, where stress wave phenomena are explained. A simple formulation of lap joints subjected to an impact stress, which is based on the Volkersen
model, is given and calculated numerically with the discrete element finite difference method in Sect. 10.3. Fundamentals of the finite element method for dynamic
problems are introduced in Sect. 10.4, where two types of methods: implicit and explicit schemes are explained. In Sect. 10.5, a calculation result of a single lap joint
with the finite element method is shown as an example. Experimental methods for
evaluating the impact strength are discussed in Sect. 10.6. Stress distributions and
their variation as a function of time for joints subjected to impact loads are given by
calculations based on analytical models and the finite element method.
10.1 Introduction
Adhesively bonded joints in actual products may be sometimes subjected to impact
loading. For example, passenger cars can be subjected to impact loading in case of
crash. Recent car structures of steel have many adhesively bonded parts such as door
panels, engine bonnets, and in addition, main structures in some cases. Almost all
the inner panel and the outer panel of a door are joined together with the combination of adhesion and metal plastic working, which is called hemming technique.
The engine bonnet has an outer skin and an inner stiffener that are joined with an
adhesive and spot welding. The technique is called weld bonding, and it can also be
applied to the main structures of recent expensive cars instead of using spot welding
only. Even for the joints with plastic parts including bumper, adhesion is employed
Chiaki Sato
Precision and Intelligence Laboratory, Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku,
Yokohama 226-8503, Japan, e-mail: csato@pi.titech.ac.jp
L.F.M. da Silva, A. Öchsner (eds.), Modeling of Adhesively Bonded Joints,
c Springer-Verlag Berlin Heidelberg 2008
279
280
C. Sato
quite often. In order to prove the crashworthiness of the structures, the evaluation
and estimation of the impact strength of such joints are indispensable for appropriate design. Particularly, when the joints are used in critical parts, passenger safety
depends strongly on their design. Therefore, adhesively bonded joints have become
very important recently, and automobile engineers have aimed at establishing design
rules.
Mobile electronic devices such as mobile phones or pocket PCs are the other
typical case, and they have many joints bonded with adhesives or pressure sensitive
adhesives. They are also subjected to impact loading in case of drop, and the situation is not rare as almost all users have experienced such a case. It is, therefore,
necessary to ensure the impact durability of the joints in the design process.
Despite the facts, much research on impact phenomena of adhesively bonded
joints has not been conducted so far due to various difficulties. Impact phenomena
of materials are different from static ones from two points of view. One of them
is the change of material properties because of high strain rate. The other is the
presence of stress waves that propagate in elastic bodies as if waves would do on
the surface of water. The former needs complicated experimental setups to assess
material properties under high rate loading. The latter requires difficult analysis to
determine the stress distribution and its variation with respect to time. Both are not
present in the case of static investigation, and they make dynamic investigation more
difficult.
Many procedures to assess the impact performance of materials have been devised because of interest in a wide variety of applications and some of them have
been adopted as standard. The methods using a pendulum hammer, like the Charpy
or Izod tests, are the most usual techniques for impacts of relatively low velocity.
The ASTM Block Impact Test (ASTM D950-78) is a modification of such pendulum hammer methods to be suitable for the evaluation of adhesive joints, and it is the
most usual and popular. Energy consumed to break the specimen can be determined
by the method. Though it is useful only to compare quantitatively the performance of
adhesives, those methods cannot be applied directly to obtain mechanical properties
of materials such as Young’s modulus, Poisson’s ratio and strength of the adhesive
and adherends. In addition, the variation of the material constants with respect to
strain rate is also unknown from the methods. Thus, more sophisticated methods
are required to evaluate the material properties or the impact strength of joints, even
though they tend to be more difficult and more expensive than static tests.
Although dynamic stress analysis of adhesively bonded joints is possible with a
closed-form approach, the governing equations are difficult to solve because they
involve partial differential equations. Therefore, numerical calculation is essential
even for the closed-form approach. Finite Element Method (FEM) simulations are
more versatile to calculate stress states in bodies of complicated shape. However,
dynamic FEM is more difficult than static FEM, and know-how is necessary to
carry out it properly. As another inherent difficulty of adhesively bonded joints, the
stress singularity is to be treated even in case of dynamic analysis.
This chapter shows how to estimate stress distribution in adhesively bonded joints
subjected to impact loading or high-rate loading, and how to assess the impact
10 Impact
281
strength of the joints. At first, fundamentals of impact dynamics on elastic bodies
and adhesively bonded joints will be introduced briefly. Next, a simple formulation
of single lap joints and the calculated results will be shown. The theory of numerical
dynamic analysis using FEM will be discussed and a result of stress calculation for
a single lap joint will be shown as an example. Finally, impact strength evaluation
of joints will be shown, where experimental methods and a failure criterion will be
considered.
10.2 Fundamentals of Impact Dynamics of Elastic Bodies
Stress analysis of solid bodies subjected to impact loads is different from the static
case in terms of the time variation of the stress distribution. Stress in a solid body
subjected to an impact is not only variable in time at a point, but also the stress
translates in the body as a wave propagates in a media. The phenomenon is called
“stress wave propagation”. The stress wave occurs because of the combination of
body mass, in other words, density distribution, and the elasticity of the body. The
dominant stress waves are the dilatational and the shear waves. There are similar
kinds of stress waves such as the longitudinal wave in narrow rods or the deflection
wave in flexural beams. Another type of waves is the surface acoustic wave (SAW)
including Rayleigh waves occurring on the surface of bodies. The dilatational and
the shear waves are most influential for the stress distribution in bulk materials,
and the longitudinal and the deflection waves are influential only in narrow rods
or flexural beams. The surface waves do not affect much the stress state in a body.
Thus, the dilatational and the shear waves are discussed firstly, and the longitudinal
and the deflection waves are discussed later in the section.
The kinetic equation of solids can be written:
ρ
∂ σ ji
Dvi
=∑
+ Ki
Dt
j ∂xj
(10.1)
where ρ , v, σ and K indicate density, particle velocity, stress and body force per
volume of a point in the solid respectively, and suffixes i and j denote the directions
in Cartesian space. Since v is a vector and σ is a tensor, v has a suffix and σ has
two suffixes. The symbol D/Dt denotes the operator of Lagrangian derivative. The
particle velocity can be expressed with displacement as follows:
vi =
∂ ui
∂ ui
Dui
=
+ vj
Dt
∂t ∑
∂
xj
j
(10.2)
To obtain a linear equation, trivial terms in Eq. (10.2) are neglected. That is equal to
regarding the operator of Lagrangian derivative as partial derivative.
D ∼ ∂
=
Dt
∂t
(10.3)
282
C. Sato
Then, Eq. (10.1) can be approximated as
ρ
∂ σ ji
∂ 2 ui
=∑
+ Ki
2
∂t
j ∂xj
(10.4)
The constitutive equation of a linear elastic solid can be given by:
σi j = λ ∑ εkk δi j + 2Gεi j
(10.5)
k
where λ and G are the Lamé constant and the shear modulus of the solid respectively. In Eq. (10.5), ∑ εkk indicates the summation of εxx , εyy and εzz , where εi j is
k
the strain tensor which can be defined as the following:
1 ∂ u j ∂ ui
εi j =
+
2 ∂ xi ∂ x j
(10.6)
Now, combining Eqs. (10.4), (10.5) and (10.6), the kinetic equation of a linear elastic solid, which is called the Navier equation, is obtained.
ρ
∂ 2u j
∂ 2 ui
∂ 2 ui
= (λ + G) ∑
+G∑
+ Ki
2
∂t
j ∂ xi ∂ x j
j ∂ x j∂ x j
(10.7)
Equation (10.7) can also be expressed using symbols of vector analysis as follows:
ρ
∂ 2u
= (λ + G) ∇ (∇ · u) + G∇2 u + K
∂ t2
(10.8)
Now, we neglect the body force K to simplify the situation. Using the relation ∇ ×
(∇ × A) = ∇(∇ · A) − ∇2 A, where A is an arbitrary vector, the following equation is
obtained.
∂ 2u
ρ 2 = (λ + 2G) ∇ (∇ · u) − G∇ × (∇ × u)
(10.9)
∂t
The displacement vector field u can be described with scalar potential φ (x,t) and
vector potential ψ (x,t) as u = ∇φ + ∇ × ψ . Therefore, Eq. (10.9) can be written as:
2
∂ 2φ
∂ ψ
2
2
ψ
=0
∇ ρ 2 − (λ + 2G) ∇ φ + ∇ × ρ 2 − G∇
∂t
∂t
(10.10)
Since Eq. (10.10) is an identical equation, we obtain two important results:
ρ
∂ 2φ
= (λ + 2G) ∇2 φ
∂ t2
(10.11)
∂ 2ψ
= G∇2 ψ
∂ t2
(10.12)
ρ
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283
Since Eqs. (10.11) and (10.12) are wave equations, they can be expressed with sound
velocities c1 and c2 as:
∂ 2φ
= c1 2 ∇2 φ
(10.13)
∂ t2
∂ 2ψ
= c2 2 ∇2 ψ
(10.14)
∂ t2
where c1 is (λ + 2G)/ρ , c2 is G/ρ , φ is the field of dilatational waves, and ψ
is the field of shear waves.
A one-dimensional dilatational wave propagating in a quasi-infinite solid media
of one-dimension such as a very long bar is treated next because three dimensional
wave propagation is too difficult to solve mathematically. Such a stress wave is
called a longitudinal wave. Dilatational waves are spherical waves and longitudinal
waves are plane waves. Therefore, the velocity c1 is different from the velocity c0
of longitudinal waves. In other words, no wave propagation in transversal
directions
to the axis is assumed. The longitudinal wave velocity c0 is given by E/ρ , where
E is the Young’s modulus of the solid. Here, the influence of Poisson’s ratio can be
neglected. Thus, the governing equation of a longitudinal stress wave translating in
the x-direction can be obtained as follows:
∂ 2u
1 ∂ 2u
=
∂ x2
c0 2 ∂ t 2
(10.15)
where u is the displacement in the x-direction. The wave equation can be solved
mathematically and the so called d’Alembert’s solution is obtained, given by u(x,t) =
η (x − c0t) + ξ (x + c0t), where η ( ) and ξ ( ) are arbitrary functions indicating wave
forms. The solution implies that a displacement field consists of two components:
a “progressive wave” η (x − c0t) and a “retrograde wave” ξ (x + c0t), as shown in
Fig. 10.1, and the superposition law is applicable.
Fig. 10.1 Schematic diagram of a progressive wave and a retrograde wave
284
C. Sato
If the one-dimensional media has an end, stress waves reflect there. There are
two sorts of special ends in terms of reflection: “free ends” and “fixed ends”. Now
consider the situation in which a positive progressive wave propagates along the media to the end. If it is a free end, the stress value must be zero there, and a negative
retrograde wave must occur so that the positive progressive wave could be canceled,
as shown in Fig. 10.2. In the case of a fixed end, the displacement of the end must
be zero, and a positive retrograde wave is generated to compensate the virtual displacement that could occur if the media had no end and were continuous, as shown
in Fig. 10.2. Such “free end reflection” and “fixed end reflection” can be described
as “negative reflection” and “positive reflection” respectively. However, such perfectly free or fixed ends are extreme cases, and ordinary ends have intermediate
properties. Stress wave reflection may also occur at an interface between dissimilar
materials having different acoustic impedances, which are given by ρ c0 .
The deflection wave can propagate along a flexural beam. The kinetics of a beam
part can be written as follows:
ρ Adx
∂ 2w ∂ F
dx
=
∂ t2
∂x
(10.16)
where A and w indicate the cross section and the deflection of a beam, and F is
the shear force applied to the cross section of the part, as shown in Fig. 10.3.
The relation between bending moments M and cross sectional shear loads F is
given by:
∂M
+F = 0
(10.17)
∂x
Assuming the deflection w is small, the well known following relation is obtained:
Fig. 10.2 Free end reflection (top) and fixed end reflection (bottom)
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285
Fig. 10.3 Shear force and bending moment applied to a part of a beam
∂ 2w
M
=
∂ x2
EI
(10.18)
where E and I are the Young’s modulus and the area moment of inertia of the beam
respectively. From Eqs. (10.16), (10.17) and (10.18), the governing equation is obtained and it can be written as:
ρA
∂ 2w
∂ 4w
+
EI
=0
∂ t2
∂ x4
(10.19)
This formulation is called the Bernoulli-Euler beam theory, but the equation is not
an ordinary wave equation. The velocity of waves derived from Eq. (10.19) depends
on wavenumber. The phenomenon is called “dispersion”. However, such dispersion phenomenon does not occur in actual beams. The mistake is caused because
the shear deformation is not taken into account. Another formulation, which is so
called the Timoshenko beam theory, can explain the proper motion of any beam
with consideration of the shear deformation of the beam.
10.3 Simple Kinetic Model of Lap Joints Subjected
to Impact Loads
Correct stress analysis is necessary to estimate the strength of adhesively bonded
joints. Much research on the stress distribution in joints has been carried out since
the first theory emerged: Volkersen’s shear lag model [14]. Since the stress distribution in joints depends on dimensions, types of materials and particularly configuration, suitable designs to reduce stress concentration has been pursued. Stress
analysis using FEM has a history of over 30 years and almost all joint configurations
have been investigated already with the method. For instance, Adams and Peppiatt
[2] conducted FEM analysis of lap joints and compared them with the closed forms
of Goland-Reissner models [6]. Other researches treated the joints considering the
elasto-plastic or visco-elastic properties of adherends and adhesives. However, dynamic analyses of the joints are still rare because they are more difficult than static
ones. The difficulty is due to the fact that dynamic analysis needs repeated calculations and consumes much of computer resources.
A simple kinetic model of a single lap joint subjected to impact loading is shown
next in order to calculate dynamic response of stress distribution in the adhesive
286
C. Sato
layer [11]. The Volkersen model, which is the simplest one for lap joints, is used
here. As shown in Fig. 10.4, the condition of force equilibrium, the compatibility of
deformation and the constitutive relations of the materials of each element can be
expressed as the following:
t1 W
∂ σ1
+W τA = 0,
∂x
ε1 =
∂ u1
,
∂x
E1 ε1 = σ1 ,
ε2 =
t2 W
∂ u2
,
∂x
E2 ε2 = σ2 ,
∂ σ2
−W τA = 0
∂x
(10.20)
u2 − u1
tA
(10.21)
GA γA = τA
(10.22)
γA =
where t1 , t2 and tA are the thickness of adherend 1, adherend 2 and adhesive layer,
u1 , u2 and W indicate the particle displacements of adherend 1, adherend 2 and the
width of the joint. The other symbols, ε , σ , γ , τ , E and G denote normal strain, normal stress, shear strain, shear stress, Young’s modulus and shear modulus respectively, and the suffixes, 1, 2 and A indicate adherend 1, adherend 2 and adhesive
layer, respectively. Here, the shear strain is engineering strain.
From these equations, the Volkersen’s equation is obtained as follows:
∂ 2 γA 2GA
−
γA = 0
∂ x2
EtstA
(10.23)
Here, to simplify the equation, it is assumed that the adherends have the same thickness, Young’s modulus and density, and they are denoted as ts , E and ρ respectively.
Equation (10.20) can be modified into a dynamic model adding the inertia force
for each element as follows:
Fig. 10.4 Load balance in a single lap joint
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287
t1 W
∂ σ1
∂ 2 u1
+W τA = t1W ρ1 2
∂x
∂t
t2 W
∂ σ2
∂ 2 u2
−W τA = t2W ρ2 2
∂x
∂t
(10.24)
where ρ1 and ρ2 are the density of adherend 1 and adherend 2 respectively. The following governing equation is obtained, which can be called the dynamic Volkersen
model:
∂ 2 γA 2GA
ρ ∂ 2 γA
−
γA =
(10.25)
2
∂x
EtstA
E ∂ t2
This can be written by stress as follows:
∂ 2 τA 2GA
ρ ∂ 2 τA
−
τ
=
A
∂ x2
EtstA
E ∂ t2
(10.26)
Equations (10.25) and (10.26) are a type of partial differential equation called “telegraph equation” and they can be solved fully analytically in some cases of simple
boundary and initial conditions. Unfortunately, single lap joints have quite complicated boundary conditions, and it is difficult to solve the equations analytically.
Now, a numerical approach is applied to the equation assuming a case of single
lap joints. Equations (10.25) and (10.26) are inconvenient to define proper boundary
conditions for single lap joints. Therefore, Eqs. (10.24) are used as the governing
equation. The equations can be simplified as follows:
∂ 2 u1
ρ1 ∂ 2 u1
GA
+
(u
−
u
)
=
2
1
∂ x2
E1t1tA
E1 ∂ t 2
∂ 2 u2
ρ2 ∂ 2 u2
GA
−
(u
−
u
)
=
2
1
∂ x2
E2t2tA
E2 ∂ t 2
(10.27)
It is assumed that a stress wave having a step wave form propagates in adherend 1
from the left to the right as shown in Fig. 10.4. The boundary conditions of the joint
are quite complicated because reflection of stress waves occurs at the overlap edges.
They can be written as:
t
t
σ U(λ )
∂ u1 (x, λ ) u1 |x=− L = −2 cs1
d λ + cs1
dλ
2
E1
∂ x x=− L
0
0
2
∂ u2 =0
∂ x x=− L
2
∂ u1 =0
∂ x x= L
2
t
∂ u2 (x, λ ) dλ
(10.28)
u2 |x= L = − cs2
2
∂ x x= L
0
2
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C. Sato
where σ is the amplitude of stress wave, U(t) is the unit step function, cs1 and cs2 are
the longitudinal
stress wave
velocities of adherend 1 and 2 respectively defined as
cs1 = E1 /ρ1 and cs2 = E2 /ρ2 . Equations (10.27) can be transformed to discrete
equations using the central difference for space and the backward difference for time
shown as:
2
u1 (xn+1 ,tn )
cs1
u1 (xn−1 ,tn )
−
+
k
+
+ k1 u2 (xn ,tn )
u1 (xn ,tn ) +
1
2
2
2
(Δx)
(Δx)
(Δt)
(Δx)2
cs1
=
(−2u1 (xn ,tn−1 ) + u1 (xn ,tn−2 ))
(Δt)2
2
u2 (xn+1 ,tn )
cs2
u2 (xn−1 ,tn )
−
+ k2 +
+ k2 u1 (xn ,tn )
u2 (xn ,tn ) +
(Δx)2
(Δx)2
(Δt)2
(Δx)2
cs2
=
(−2u2 (xn ,tn−1 ) + u2 (xn ,tn−2 ))
(10.29)
(Δt)2
where k1 = GA /(E1t1tA ) and k2 = GA /(E2t2tA ). Stepwise time integration of
Eqs. (10.29) gives displacement vectors u1 and u2 at each time step. Based on the
results, distribution of shear stress τA can be calculated. An example of MATLAB
program to solve this problem is shown in Appendix.
Figure 10.5 shows a calculated result of shear stress distribution in the adhesive
layer of a single lap joint subjected to a step load of 1 MPa in amplitude. Here, the
single lap joint is 40 mm in lap length, 4 mm in thickness of the adherends, which
Fig. 10.5 Shear stress distribution and variation along bondline of a single lap joint calculated
based on the dynamic Volkersen model (see Appendix)
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289
are made of aluminum alloy having a tensile modulus of 72 GPa and the density
of 2.8 × 103 kg/m3 . The thickness and the shear modulus of the adhesive layer are
0.1 mm and 1.0 GPa respectively. It is shown in Fig. 10.5 that the stress near the
load input side (Position = −L/2) increases initially and the stress at the opposite
side (Position = L/2) increases with a delay, which is due to the duration of stress
wave propagation between the sides. At the load input side, there is stress oscillation,
but it decays soon.
In case of static loading, the maximum shear stress at the both sides of the joint,
denoted as τAmax , can be calculated from the static Volkersen model:
τAmax =
GA
kL
σ coth( )
EtA k
2
(10.30)
where k = 2GA /(EtstA ). The value of τAmax is 0.527 MPa when a static load σ of
1 MPa is applied to the joint, and this is smaller than the maximum stress value of
1 MPa in Fig. 10.5. Therefore, the dynamic stress concentration factors of the joint
are greater than those in static condition. In Fig. 10.5, the shear stress values at both
edges decrease gradually, and converge to τAmax after enough time.
10.4 Theory of Numerical Dynamic Analysis by FEM
10.4.1 Static Analysis
The finite element method for solids is based on the principle of virtual work, and it
is given by:
∑ ∑ σi j δ εi j dV =
V i
j
S
q · δ udS+
V
b · δ udV
(10.31)
where σi j and δ εi j indicate stress tensor and virtual variation of strain tensor, and
q, b and δ u are surface force, body force applied to the solid and virtual displacement occurring in the solid respectively. The subscripts V and S indicate total volume and total surface of the solid respectively. Other suffixes i and j denote the
direction of x, y and z axes. The principle of virtual work has the same meaning as
the equilibrium equation which is one of the fundamental relations in solid mechanics. The stress and strain vectors are expressed as:
⎞
⎛ ⎞
σx
εx
⎜ σy ⎟
⎜ εy ⎟
⎜ ⎟
⎜ ⎟
⎜ σz ⎟
⎜ εz ⎟
⎟
⎟
⎜
σ = ⎜ ⎟ and ε = ⎜
⎜ γxy ⎟
τ
⎜ xy ⎟
⎜ ⎟
⎝ τyz ⎠
⎝ γyz ⎠
τzx
γzx
⎛
(10.32)
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C. Sato
Note that the components of strain vector are not tensor strains but engineering
strains. Though stress and strain are tensor quantities, the vector notation is convenient to calculate virtual work. Equation (10.31) can be simplified by inner product
of vector, and it is shown as follows:
V
σ · δ ε dV =
S
q · δ udS +
V
b · δ udV
(10.33)
A displacement vector u in a finite element is associated with the nodal displacement
vector a of the element via the shape function matrix N of the element as follows:
u = Naa
(10.34)
The strain vector ε in the finite element is also associated with the nodal displacement vector a via the strain-displacement relation matrix B of the element as
follows:
ε = Ba
(10.35)
Therefore, Eq. (10.33) can be approximated by the following:
∑V σ · Bδ aΔV = ∑S q · Nδ aΔS + ∑V b · Nδ aΔV
(10.36)
Since Eq. (10.36) is an identical equation which is independent on δ a, the equation
can be modified as follows:
∑V BT σ ΔV = ∑S NT qΔS + ∑V NT bΔV
(10.37)
When the constitutive relation of the solid is linear elastic, stress vector σ is
given by:
σ = Dε = DBa
(10.38)
where D is the local stiffness matrix. In case of isotropic materials, it can be expressed as:
D=
E (1 − ν )
(1 + ν ) (1 − 2ν )
⎡ 1 ν /(1 − ν ) ν /(1 − ν )
⎢
⎢
×⎢
⎢
⎣
1
sym.
0
ν /(1 − ν )
0
1
0
(1 − 2ν )/2 (1 − ν )
0
0
0
0
(1 − 2ν )/2 (1 − ν )
⎤
0
0
⎥
⎥
0
⎥
⎥
0
⎦
0
(1 − 2ν )/2 (1 − ν )
(10.39)
where, E is Young’s modulus and ν is Poisson’s ratio. Equation (10.37) can be
modified using matrix D and is shown as follows:
∑V BT DBaΔV = ∑S NT qΔS + ∑V NT bΔV
(10.40)
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291
The left term in Eq. (10.40) includes nodal displacement vector a, and it can be
extracted from summation. Thus, we obtain the final form of the equation for the
finite element method:
(10.41)
Ka = F + B
where K = ∑V BT DBΔV , F = ∑ S NT qΔS, B = ∑V NT bΔV , and a is the total nodal
displacement vector of the whole structure, and it is the sum of local displacement
vectors a defined in each element. Solving Eq. (10.41) gives a , and then ε is calculated using Eq. (10.35). The stress σ is obtained using Eq. (10.38).
10.4.2 Dynamic Analysis with Implicit Solver
In case of dynamic conditions, few modifications of the static equation are necessary. That can be done by adding the term of inertia force to the equation following
d’Alembert’s principle. Equation (10.31) is modified to the following:
∑ ∑ σi j δ εi j dV =
V i
j
S
q · δ udS +
V
b · δ udV −
V
ρ ü · δ udV
(10.42)
where ü is acceleration vector. Therefore, Eq. (10.31) becomes:
Ka = F + B − Mä
(10.43)
where M = ∑V NT ρ NΔV , and is called the mass matrix.
To solve Eq. (10.43), the integration in time space has to be carried out. For the
purpose, many schemes for time integration have been presented. Implicit schemes
use futures status of a that is extrapolated by functions such as Taylor expansion.
The simplest extrapolation in implicit schemes is to use a difference quotient of ä
given by:
a − 2an + an−1
ä n+1 = n+1
(10.44)
(Δt)2
where an+1 is the future status because the present time step is n, and än+1 is nothing
but an approximation at this step. Substituting ä in Eq. (10.43) with Eq. (10.44), the
following is obtained:
−1 Fn+1 + Bn+1 + M(2an − an−1 )Δt 2
an+1 = M/(Δt)2 , +K
(10.45)
and an+1 , i.e., a of the next time step can be calculated. Although this scheme
is simpler than other sophisticated implicit ones, it shows common characteristics
of implicit schemes. The biggest problem of implicit schemes is that the inverse
matrix calculation has to be done in each time step. If the time step Δt needs to
be changed during calculation, not only backward substitution, but also forward
elimination has to be carried out. Therefore, implicit schemes are less efficient than
explicit methods. The other problem is the huge amount of memory consumption
292
C. Sato
because the matrix M/(Δt)2 + K has to be stored in memory. The memory should
be solid-state to reduce calculation time. Use of hard disk or virtual storage is not
recommended. Therefore, the memory consuming nature of the schemes is very
expensive. On the other hand, implicit schemes are very stable, so that longer time
steps than explicit schemes can be selected with reasonable results.
Actual implicit schemes used for practical application are more complicated than
those mentioned above. One popular scheme is the Newmark-β method, which uses
Taylor expansion for extrapolation. The method is given by:
an+1 = an + Δt ȧn +
where,
(Δt)2 än + β (Δt)2 än+1 − än
2
ȧ n+1 = ȧ n + Δt γ ä n+1 + (1 − γ ) ä n
(10.46)
(10.47)
−1
ä n+1 = M + (1 + α ) β (Δt)2 K
1
2
K ȧa n − (1 + α )
− β (Δt) K ä n
× Fn+1 + Bn+1 − Kan − (1 + α ) ΔtK
2
(10.48)
α = 0, β = 0.25 and γ can take different values, determining the stability of the
scheme. The parameters are selectable to tune up the compromise of stability and
response. For instance, in the Hilber-Hughes-Tailor method, which is more stable
with shorter time steps, has the parameters α = −0.1, β = 0.3025 and γ = 0.6 [9].
10.4.3 Dynamic Analysis with Explicit Solver
Explicit schemes are more efficient than implicit ones and their use is increasing in
many applications such as crash analysis of car bodies in the last two decades. The
equation of explicit schemes is given by:
(10.49)
ä n = M−1 Fn + Bn − Kan
To obtain an+1 , additional calculations have to be carried out using the central difference as follows:
(10.50)
ȧ n+0.5 = ȧ n−0.5 + Δtn ä n
an+1 = an + Δtn+0.5 ȧ n+0.5
(10.51)
where Δtn+0.5 = tn+1 − tn and Δtn = tn+0.5 − tn−0.5 . As seen in Eq. (10.49), if a
lumped weight matrix for M is used, inverse matrix calculation can be omitted.
In addition, there is no need to store the entire matrix K −1 which includes many
fill-ins or zeros after the forward elimination, and calculation of only Kan is a less
10 Impact
293
memory consuming process. Because of such advantages, explicit schemes have
recently become the only choice to treat huge scale applications. As a matter of
fact, crash analysis of whole car structures, which is one of the biggest calculation these days, can be completed using only a PC of high performance within few
weeks. Many packages of explicit finite element analysis are commercially available now. LS-DYNA (LSTC, Livermore, U.S.A.), which is derived from previous
DYNA3D (Lawrence Livermore National Laboratory, Livermore, U.S.A.), PAMCRASH (ESI, Paris, France), RADIOSS (Altair Engineering, Troy, U.S.A.) and
DYTRAN (MSC, Santa Ana, U.S.A.) are famous explicit programs. Though these
programs aim at car design mainly, they can be applied to other purposes. A famous
implicit program: ABAQUS (SIMULIA, Providence, U.S.A.) has also developed
an explicit solver recently. Such programs are made based on the Lagrange method.
Therefore, they are also suitable for large deformation problems.
The disadvantage of explicit scheme is the necessity of using a lumped matrix
for M. Using a lumped matrix decreases the accuracy of results, and the use of
high-order elements is also ineffective. Finer meshes are more suitable for explicit
schemes than high-order elements.
10.5 Numerical Modeling of Simple Lap Joints
The result shown in Sect. 10.3 is based on the dynamic Volkersen model which
neglects the adherend deflection caused by the bending moment due to the offset of
the adherends. The deformation is considerable if thin adherends are used. Another
model of lap joints was presented by Goland and Reissner [6]. The adherend bending was considered in the model. However, it is hard to obtain a closed form solution
of a dynamic Goland-Reissner model, and even a numerical solution by the discrete
element finite difference method is too complex. Direct simulation using the finite
element method is easier to perform.
Sato and Ikegami investigated the deformation and stress distribution of single
lap joints, taper lap joints and scarf joints subjected to impact loads using the finite element method [13]. In the analysis, the adhesive layer was treated as a viscoelastic material and the impact load was a step function. In the calculated results,
sharp and impulsive stress concentrations occur at the edge of the load input side
as shown in Fig. 10.6. After the transient process, stresses at both overlap edges
increase gradually, although the applied stress wave is a step function and the amplitude is constant. The phenomenon occurs due to the bending moment caused by
the stress wave propagation through the “cranked path” in the offset bonded joint.
Higuchi et al. conducted a dynamic analysis of butt joints of cylindrical steel rods
subjected to tensile impact loads using the three dimensional finite element method
and showed the stress variation in the adhesive layer with respect to time and the
presence of a stress singularity at the circumferential edge of the adhesive layer [8].
Figure 10.7 shows the calculated results by FEM of impact deformation of a single lap joint. This is a three-dimensional elastic analysis using the explicit solver
294
C. Sato
Fig. 10.6 Stress distribution and variation with respect to time of a single lap joint subjected to
impact step load calculated by the dynamic finite element method
Fig. 10.7 Stress distribution and deformation of a single lap joint subjected to a impact step load
calculated by PAM-CRASH
10 Impact
295
PAM-CRASH (ESI, Paris, France). The adherends of the joint are 100 mm long,
25 mm wide and 4 mm thick. The overlap length is 12.5 mm and the adhesive thickness is 0.1 mm. The material constants are the same as the joint treated with the
dynamic Volkersen model in Sect. 10.3. In addition, the Poisson’s ratios νs , νA of
the adherend and the adhesive layer are 0.3 and 0.4 respectively. A tensile stress of
1 MPa step function is applied to the left edge of the specimen in Fig. 10.7, and the
right edge is fixed perfectly. The front of a stress wave reaches the overlap at 0.02
ms, so that there is no stress in the overlap at 0.01 ms as shown in Fig. 10.7. After
0.02 ms, the stress wave propagates in the overlap and the joint begins bending. Although the applied stress is constant, the deflection of the joint continues to increase
due to dynamic effects.
10.6 Impact Strength Evaluation of Adhesively Bonded Joints
Not only are analytical methods, but also experimental methods very important to
investigate the properties of materials used for adhesively bonded joints and the
strength of the joints. In this section, trends in experimental methods are discussed.
In addition, impact strengths of several adhesives are shown based on the results
presented in previous papers.
10.6.1 Pendulum Test
As mentioned in the introduction, the block impact test (ASTM D950-78) has been
the only standard used to test adhesively bonded joints under impact loading for
long time. In the test, a small block is bonded to a larger block which is fixed to the
base of the testing machine and an impact load is applied to the small block due to
the collision of a pendulum hammer, as shown in Fig. 10.8. At impact, the joint is
Fig. 10.8 ASTM block
impact test (D950-78)
296
C. Sato
subjected to a high rate shear loading and is fractured. After the pendulum hammer
hits the specimen, its velocity decreases. The impact energy absorbed by the joint
specimen can be calculated from the difference of the pendulum height between the
initial position before the test and the maximum height that is subsequently reached
after the collision.
If the pendulum is instrumented, a load-displacement curve can be obtained. In
the technique, a load sensor and a displacement sensor are installed in the pendulum
tester. The load sensor, which is usually a piezo-electric loadcell, is inserted between the pendulum and the tooth of the hammer. The displacement sensor, which
is often rotational and angular, is attached to the axis of the pendulum. The instrumented technique gives important data, such as maximum loads applied to the
specimen, maximum displacement of the small block by which maximum shear
strain of the adhesive layer can be calculated, and a more precise absorbed energy
determined.
Adams and Harris calculated the stress distribution in the specimen of the block
impact test using the finite element method [1]. Unfortunately, the stress distribution
in the adhesive layer highly depends on the position at which the tip of the impactor
hits the specimen. The position deviates occasionally from the objective point because of misalignment of the experimental setup. Thus, it is concluded that this test
only gives a quantitative information of the ability of adhesives to withstand high
loading rate and cannot be used for design purpose. Nevertheless, the results of the
test are convenient for estimating approximately the maximum stress occurring in
the joints. In the experiments, strength assessment of adhesives was carried out using
four different types of epoxy adhesives, MY750 (Ciba-Geigy, Switzerland) which
consisted of a diglycidyl ether of bisphenol A with anhydride hardener HY906
and a tertiary amine catalyst DY062, AY103 (Ciba-Geigy, Switzerland) plasticised
with an amine hardener HY956, ESP105 (Permabond, Winchester, UK) which is
a single part toughened epoxy, and toughened MY750 modified with CTBN, i.e.
a synthetic rubber carboxyl-terminated butadiene-acrylonitrile. The brittle unmodified MY750 presented the lowest absorbed energy. The plasticised epoxy AY103
also showed low absorbed energy, but greater than that of unmodified MY750. The
rubber modified epoxy and the single part toughened epoxy ESP105 showed much
higher absorbed energy compared with unmodified MY750 and AY103. Thus, the
results show the importance of adhesives ductility in order to withstand impact
loading.
Pendulum testers are available not only for impact blocks but also for other types
of specimens including lap joints. Harris and Adams carried out impact tests on single lap joints having aluminium alloy adherends bonded with epoxy adhesives [7].
Recently, another kind of pendulum test using a wedge was adopted as a standard: Impact Wedge-Peel (WP) test which is shown in Fig. 10.9. An impact load is
applied to the specimen through the wedge. The method can be thought of a modified version of the Boeing wedge test for impact loading. Blackman et al. carried
out precise experiments of the IWP test using a fast hydraulic testing machine and
a piezoelectric loadcell [4].
10 Impact
297
Fig. 10.9 Configuration of
the impact wedge-peel (IWP)
specimen ISO 11343
10.6.2 Split Hopkinson Bar (Kolsky bar)
The pendulum tests described above are easy to use, but a high strain rate cannot be
realized. A strain rate of up to about 102 s−1 can be obtained. When a higher strain
rate is necessary, the split Hopkinson bar technique is often used because a strain
rate of 102 s−1 – 103 s−1 can be realized easily and using special experimental
setups strain rates as high as 104 s−1 can be obtained. A typical Hopkinson bar
equipment is used for impact compression tests, and it has two steel bars (input
bar and output bar) between which a specimen is inserted, as shown in Fig. 10.10.
Fig. 10.10 Basic configuration of split Hopkinson bar apparatus for compressive (top) and tensile
(bottom) impact tests of materials
298
C. Sato
A striker, which is usually accelerated with a gas gun, collides with the end of the
input bar to cause a stress wave that transmits to the specimen as an impact load. The
applied load and its variation with respect to time are measured with strain gauges
bonded on the opposite surfaces of the input bar. Deformation of the specimen can
also be calculated from the velocities of the edges of the input bar and the output
bar that can be measured form the gauge data. Since a relatively higher strain rate
can be realized easily and the deformation of specimens can be measured without
any additional displacement sensors, the Hopkinson bar method is the most used
technique for impact tests. The ingenious apparatus was first introduced by Kolsky,
and is also called the Kolsky bar [10].
For the strength evaluation of adhesively bonded joints subjected to impact loads,
an aspect of Hopkinson bar – that it is suitable only for compression tests – is
sometimes an obstacle because tensile impact tests are very often required. Therefore, compression loads should be transformed into tensile loads by any means.
Yokoyama used the split Hopkinson bar technique shown in Fig. 10.10 to apply a
tensile load to butt joints bonded with a cyano-acrylic adhesive [15]. In the experiment, the tensile stress wave was caused by the collision of a tubular impactor
against a loading block, fixed at the end of an input bar. The tensile wave was applied
to the specimen as a tensile impact load. Yokoyama and Shimizu also conducted impact shear tests of adhesive joints using a compressive setup of split Hopkinson bar
and pin-and–collar joint specimens [16]. The same configuration of specimens was
also used by Bezemer et al. [3].
10.6.3 Other Special Methods
Since the adhesive layer in practical joints is usually subjected to a combination of
shear and tensile stresses even under impact loading, tests for combined high rate
loading are important to obtain a strength criterion, but are not easy to carry out.
Sato and Ikegami measured the strength of butt joints of steel tubes, bonded with
an epoxy adhesive Scotch Weld 1838 (Sumitomo 3M, Tokyo, Japan) hardened with
a polyamide resin, when subjected to a combination of tensile and torsional impact
loads using a clamped Hopkinson bar equipment [12]. The strength of the butt joints
under combined impact loading is shown in Fig. 10.11 with approximation curves
fitted with von Mises law and 2nd polynomials. The curve of 2nd polynomials shows
better fitting than that of von Mises law. The tensile strength of the joints, which is
equal to the peel strength of the joints, is approximately 80 MPa, and it is larger than
the shear strength that is about 50 MPa. The strength of the joints measured under
impact loading was much higher and approximately twice than the static strength
as shown in Fig. 10.11. However, the shapes of failure locus were similar in both
cases. Therefore, the same type of failure criterion, in this case a quadratic criterion,
can be used for static and impact loadings
10 Impact
299
Fig. 10.11 Comparison
between impact strengths and
static strength of tubular butt
joints obtained from
combined impact tests and
approximation curves fitted
by von Mises law and a
quadratic criterion
Cayssials and Lataillade used an inertia wheel equipment to carry out impact
tests of lap shear specimens which consisted of galvanized steel sheet bonded with
two kinds of epoxy adhesive: an unmodified DGEBA hardened with dicyandiamide
and its modified version with a block copolymer and fillers [5].
10.7 Conclusion
The stress state in adhesively bonded joints is complicated even they are subjected
to static loads. When impact loads are applied to them, the stress analysis becomes
too difficult with a closed-form approach because the stress state changes from a
field distribution to a wave translation. The partial differential equations found in
dynamic models are difficult to solve analytically. In addition, the effect of wave reflection and boundary conditions needs to be evaluated. Ignorance on actual impact
loads applied to structures is another problem. It will not be solved so soon even if
the other problems can be solved.
Instead of a closed-form approach, using a finite element analysis is more adequate to impact applications. As seen in the present chapter, finite element codes
have become more powerful and more affordable recently.
However, some difficulties such as the presence of stress singularities, impact properties of materials, and also deficient experimental methods still exist. Therefore, it is
very hard to predict impact strength of adhesively bonded joints still now. Since few
papers have treated the subject previously, there is no good result yet. Some of the
problems under impact are also present in static conditions. Thus, further progress in
adhesion science is required. Not only the development of new adhesives, but also the
progress of impact mechanics of adhesively bonded joints is indispensable.
300
C. Sato
10.8 Appendix
%------------------------------------------------% Dynamic Volkersen model solution for lap joints
% This is a m-file for MATLAB(c) 5.0 or later.
% by Chiaki Sato, 2/10/2008
% CAUTION! POSITION AXI is inverted.
% 0.04 m and 0 m indicate -L/2 and L/2 respectively.
clear;
EN = 41;
TT = 30e-6;
DT = 0.05e-6;
PT = 0.3e-6;
E1 = 72e9;
E2 = 72e9;
GA = 1e9;
RHO1 = 2.8e3;
RHO2 = 2.8e3;
T1 = 4e-3;
T2 = 4e-3;
TA = 0.1e-3;
LL = 40e-3;
ASTRS = 1e6;
%Element number + 1
%Total time (s)
%Time step (s)
%Display print time step (s)
%Young’s modulus of adherend 1 (Pa)
%Young’s modulus of adherend 2 (Pa)
%Shear modulus of adhesive (Pa)
%Density of adherend 1 (Kg/mˆ3)
%Density of adherend 2 (Kg/mˆ3)
%Thickness of adherend 1 (m)
%Thickness of adherend 2 (m)
%Thickness of adhesive (m)
%Lap length (m)
%Applied stress to adherend 1 (Pa)
DX = LL/(EN - 1);
%x step (m)
Cs1 = sqrt(E1/RHO1); %Stress wave velocity in adherend 1
Cs2 = sqrt(E2/RHO2); %Stress wave velocity in adherend 2
%k1,k2,k3,k4,k5 are elements of total K matrix.
K1 = 1.0/DXˆ2;
K2 = -2.0/DXˆ2 - GA/(E1*T1*TA) - 1.0/(Cs1ˆ2*DTˆ2);
K3 = GA/(E1*T1*TA);
K4 = GA/(E2*T2*TA);
K5 = -2.0/DXˆ2 - GA/(E2*T2*TA) - 1.0/(Cs2ˆ2*DTˆ2);
%Generating total K matrix
TK(1,1) = 0.0; TK(1,2) = 0.0;
TK(1,4) = 0.0; TK(1,5) = 0.0;
TK(2,1) = 0.0; TK(2,2) = 0.0;
TK(2,4) = -1.0; TK(2,5) = 0.0;
TK(1,3)
TK(1,6)
TK(2,3)
TK(2,6)
=
=
=
=
1.0;
0.0;
0.0;
1.0;
10 Impact
301
for I=3:2:2*EN+1
TK(I,I-2) =K1;
TK(I,I+1) =K3;
TK(I+1,I-2)=0.0;
TK(I+1,I+1)=K5;
end
I = 2*EN+3;
TK(I,I-4)
TK(I,I-1)
TK(I+1,I-4)
TK(I+1,I-1)
=
=
=
=
-1.0;
0.0;
0.0;
1.0;
TK(I,I-1) =0.0;
TK(I,I+2) =K1;
TK(I+1,I-1)=K1;
TK(I+1,I+2)=0.0;
TK(I,I-3)
TK(I,I)
TK(I+1,I-3)
TK(I+1,I)
=
=
=
=
TK(I,I)
=K2;
TK(I,I+3) =0.0;
TK(I+1,I) =K4;
TK(I+1,I+3)=K1;
0.0;
0.0;
0.0;
0.0;
TK(I,I-2)
TK(I,I+1)
TK(I+1,I-2)
TK(I+1,I+1)
=
=
=
=
%Transforming the TK matrix to a sparse matrix
sparse(TK);
%Initializing valiables
%DISP1 and 2 are displacements of adherend edges.
%U, Ut and Utt are displacement vectors,
%where t indicates delay of one step in time.
%F is a nodal force vector.
%STRESS is a container of results
DISP1 = 0.0; DISP2 = 0.0;
U
= zeros(2*EN+4,1);
Ut = zeros(2*EN+4,1);
Utt = zeros(2*EN+4,1);
F
= zeros(2*EN+4,1);
STRESS = zeros(1,EN);
%Time integration loop
for I=0:DT:TT
%setting displacement of adherends edges
DISP1 = DISP1 - 2.0 * Cs1* ASTRS / E1 *DT ...
+ Cs1 *(Ut(5)-Ut(3))/DX*DT;
DISP2 = DISP2 - Cs2 *(Ut(2*EN+2)-Ut(2*EN))/DX*DT;
%setting boundary condition
F(1) = DISP1;
F(2) = 0.0;
F(2*EN+3) = 0.0;
F(2*EN+4) = DISP2;
for J=3:2:2*EN+1
F(J)
= (-2.0*Ut(J)+Utt(J))/(Cs1ˆ2*DTˆ2);
1.0;
0.0;
0.0;
0.0;
302
C. Sato
F(J+1) = (-2.0*Ut(J+1)+Utt(J+1))/(Cs2ˆ2*DTˆ2);
end
%Inverse matrix calculation
U = TKˆF;
%calculating shear stress tau
for J=1:1:EN TAUA(J)= GA*(U(2*J+2)-U(2*J+1))/TA; end
%Storing displacement vectors to previous ones
Utt=Ut; Ut=U;
%Thinning out result data
if mod(I,PT)==0 STRESS = [STRESS;TAUA]; end
end
%Graphic output as a 3D figure
STRESS = fliplr(STRESS);
[X,Y]=meshgrid(0:PT:TT+PT,0:DX:(EN-1)*DX);
surf(X,Y,STRESS’);
xlabel(’TIME (sec)’); ylabel(’POSITION (m)’);
zlabel(’SHEAR STRESS (Pa)’);
%Figure modification
[row,col]=size(STRESS);
[val,num]=max(reshape(STRESS,1,row*col));
xlim([0,TT]); ylim([0,(EN-1)*DX]); zlim([0,val*1.2]);
colormap(ones(32,3))
%-------------------------------------------------
References
1. Adams RD, Harris JA (1996) A critical assessment of the block test for measuring the impact
strength of adhesive bonds. Int. J. Adhes. Adhes. 16: 61–71
2. Adams RD, Peppiatt NA (1974) Stress analysis of adhesively-bonded lap joints. J. Strain.
Anal. 9: 185–196
3. Bezemer AA, Guyt CB, Volt A (1998) New impact specimen for adhesive: optimization of
high-speed-loaded adhesive joints. Int. J. Adhes. Adhes. 18: 255–260
4. Blackman BRK, Kinloch AJ, Taylor AC, Wang Y (2000) The impact wedge-peel performance
of structural adhesives. J. Mater. Sci. 35: 1867–1884
5. Cayssials F, Lataillade JL (1996) Effect of the secondary transition on the behaviour of epoxy
adhesive joints at high rates of loading. J. Adhes. 58: 281–298
6. Goland M, Reissner E (1947) Stresses in cemented joint. ASME J. Appl. Mech. 11: A17–27
10 Impact
303
7. Harris JA and Adams RD (1985) An assessment of the impact performance of bonded joints
for use in high energy absorbing structures. Proc. Instn. Mech. Engrs. 199: C2, 121–131
8. Higuchi I, Sawa T, Okuno H (1999) Three-dimensional finite element analysis of stress
response in adhesive butt joints subjected to impact tensile loads. J. Adhes. 69: 59–82
9. Hilber HM, Hughes TJR, Taylor RL (1977) Improved numerical dissipation for time integration algorithms in structural dynamics. Earth Eng. Struct. Dyn. 5: 283–292
10. Kolsky H (1949) An investigation of the mechanical properties of materials at very high rates
of loading. Proc. Phys. Soc. Series B 62: 676–700
11. Sato C (2005) Impact behaviour of adhesively bonded joints. In Adams RD (ed) Adhesive bonding, Science, technology and applications. Woodhead Publishing, Cambridge,
pp. 164–187
12. Sato C, Ikegami K (1999) Strength of adhesively-bonded butt joints of tubes subjected to
combined high-rate loads. J. Adhes. 70: 57–73
13. Sato C, Ikegami K (2000) Dynamic deformation of lap joints and scarf joints under impact
loads. Int. J. Adhes. Adhes. 20: 17–25
14. Volkersen O (1938) Die niet kraft vertelung in zug bean spruchten. Luftfahrt forschung 15:
41–47
15. Yokoyama T (2003) Experimental determination of impact tensile properties of adhesive butt
joints with the split Hopkinson bar. J. Strain Anal. 38: 233–245
16. Yokoyama T, Shimizu H (1998) Evaluation of impact shear strength of adhesive joints with
the split Hopkinson bar. JSME Int. J. Series A 41: 503–509
Chapter 11
Stress Analysis of Bonded Joints by Boundary
Element Method
Madhukar Vable
Abstract Boundary element method (BEM) has proven to have very good resolution
of large stress gradients such as in front of cracks and in regions of stress concentration, yet its application in analysis of bonded joints is practically non-existent even
though large stress gradients exist in the bonded region and bonded joints are one
of the critical technology in modern design. This is because application of BEM to
bonded joints is not simple or straight forward.
This chapter describes the commonality and differences between BEM and other
approximate methods, advantages of BEM application to bonded joints, the research
challenges, BEM formulation, the discretization process and the sources of errors,
the mesh refinement techniques, and some numerical results.
11.1 Introduction
Finite Element Method (FEM), Finite Difference Method (FDM), and Boundary
Element Method (BEM) are the three major numerical methods for solving partial
differential equations in science and engineering. Unlike FEM and FDM, which are
very versatile and general, BEM is a more specialized numerical tool that can yield
significant advantages for a class of problems such as bonded joints.
There are three features of BEM that makes it attractive for analysis of stresses
in bonded joints:
(i) It has proven to have very good resolution of stress gradients such as those that
appear in front of cracks or in regions of stress concentrations and thus possibly
will do the same with strong stress gradients in bonded joints;
Madhukar Vable
Mechanical Engineering – Engineering Mechanics, Michigan Technological University, Houghton,
MI 49931, USA, e-mail: mavable@mtu.edu
L.F.M. da Silva, A. Öchsner (eds.), Modeling of Adhesively Bonded Joints,
c Springer-Verlag Berlin Heidelberg 2008
305
306
M. Vable
(ii) It requires discretization only of the boundary which makes remeshing easy in
shape optimisation or parametric study such as in rounding, tapering or shaping
of the adherend or adhesive fillet;
(iii) It implicitly satisfies the conditions at infinity, thus potentially eliminate the
discretization of large segment of adherends.
Given the above potential advantages, one would expect a pervasive use of BEM
in the analysis of bonded joints. This is not so. Search of several data bases reveal
there is a paucity of papers in which BEM is used in conjunction with analysis of
bonded joints. The review paper of Adams [4] while describing several modeling
techniques for analysis of bonded joints has no mention of BEM. This is because
application of BEM to bonded joints is not simple or straight forward.
This chapter describes the commonality and differences between BEM and other
approximate methods, advantages of BEM application to bonded joints, the research
challenges, BEM formulation, the discretization process and the sources of errors,
the mesh refinement techniques, and some numerical results.
11.2 Advantages of BEM Application to Bonded Joints
We first consider the commonalities and differences between various approximate
methods to elaborate the potential advantages of BEM in application to bonded
joints.
Nearly all approximate methods convert a boundary value problem into a set of
algebraic equations. This is usually accomplished by representing the field variable
u (for example displacement or temperature) by a series
n
u=
∑ c jφ j
(11.1)
j=1
where c j are the constants to be determined, and φ j are an independent and a complete set of n approximating functions. If there are no additional conditions on φ j ,
then we will generate three types of error:
— error in the differential equation (ed ) that is distributed in the domain Ω shown
in Fig. 11.1;
— error in natural boundary conditions (en ) distributed on the boundary (Γn ) where
natural boundary conditions (conditions on traction, heat flux, etc.) are imposed;
— error in essential boundary conditions (ee ) distributed on the boundary (Γe ) where
essential boundary conditions (conditions on displacements, temperature etc.) are
imposed.
Clearly we have an analytical solution if all three errors ed , en , and ee are zero.
Thus, in approximate methods at least one of the error is not zero. This non-zero
error is minimized in some manner. The most general minimizing principle is the
weighted residue in which the error is made orthogonal to a weighting function and
can be written in the general form
11 Stress Analysis of Bonded Joints
307
Γe
Fig. 11.1 Homogenous body
Γn
Ω
(d)
ψi ed dxdy +
Γe
Ω
(d)
(e)
(n)
(e)
ψi ee ds+
(n)
ψi en ds = 0
(11.2)
Γe
where ψi , ψi , and ψi are the weighting functions, chosen by some criteria
that depends upon the approximate method. Generally speaking, these weighting
functions are related to the approximating function φ j through the operators of the
differential equation or boundary conditions. On substituting the errors ed , en , and
ee into Eq. (11.2) we obtain a linear system of algebraic equations in the unknown
constants c j , whose coefficients are the integrals such as shown in Eq. (11.2).
In the domain methods such as FEM and FDM the functions φ j are chosen such
that the error en or ee is forced to be zero. For example, ee is forced to zero in the
stiffness version of FEM and en is forced to zero in the flexibility version of FEM.
In either case we are left with the evaluation of integral over Ω requiring discretization of the domain. However in boundary methods, such as BEM, the functions
φ j are chosen such that error ed is forced to zero, leaving evaluation of integrals
only over the boundary. Hence, in BEM only the boundary needs to be discretized.
Meshing and remeshing is significantly simpler in BEM than in FEM (and other domain methods) for problems of shape optimisation or parametric study (rounding,
tapering or shaping of the adherend or adhesive fillet) in which the boundary shape
changes.
The discretization process converts the integrals in Eq. (11.2) to a sum of integrals over the elements. Thus, in FEM and domain methods when we set the domain
integral over the element to zero in Eq. (11.2), we imply that the differential equation is satisfied in an average sense over the element. But in BEM it is satisfied
exactly as ed is forced to zero. If there are stress gradients in the direction of the domain interior, then BEM can potentially yield better resolution by choosing points
of stress evaluation very close to each other, while in domain methods the resolution
is dictated by the size of the elements. For this reason BEM has proven to have very
good resolution of stress gradients for problems of stress concentration and fracture
mechanics. Thus, BEM can potentially give good resolution of stress gradients in
the thickness direction of the adherend and the adhesive. However, if the stress gradients are along the boundary, as is the case on the adherend-adhesive interface, then
BEM like FEM requires a good graded mesh. One research challenge in application
of BEM to bonded joints is the development of a mesh refinement scheme for the
interface problems.
In BEM, the functions φ j are the fundamental solutions of the differential equations, hence implicitly satisfy the zero stress conditions at infinity. If there is a
308
M. Vable
uniform stress at infinity then we simply add its value to the integral equations.
Thus in BEM we need only discretize boundary of a hole in an infinite medium and
do not have to discretize the exterior as is the case in domain methods. However
joints have boundaries that start at the adhesive end and extends to infinity. Such
boundaries will need specialized infinite boundary elements.
The above discussion shows that BEM has the potential of high accuracies in
resolution of stress gradients in bonded joints but there are also some research challenges. To further elaborate the difficulties and the on going research it is necessary
to discuss BEM formulation and discretization process. The initial discussion is for
homogenous materials which is then extended to multiple materials.
11.3 BEM Formulation for Homogenous Materials
There are several versions of BEM and variety of approaches for formulating the
integral equations of BEM. Cheng and Cheng [16] gives a description of the history
of BEM, along with various means of formulating the problem. We will confine our
discussion to the two most used versions: the Indirect and the Direct BEM.
The displacement ui and the stress σi j at any point in the body can be represented [6, 8] by the integral representation shown in Eqs. (11.3) and (11.4). This representation will be used to describe two versions of Indirect BEM and Direct BEM.
ui =
(uF)ik tki − tko ds + (uc)ik uik − uok ds + (u∞ )i
Γ
σi j =
Γ
(σF)i jk tki − tko ds +
(11.3)
Γ
(σc)i jk uik − uok ds + (σ∞ )i j
(11.4)
Γ
where,
— tk and uk represent traction and displacement in the k direction, and the superscripts i and o represent the values of these variables just inside and just outside
the boundary Γ of the body which is considered inscribed in an infinite body;
— (σ∞ )i j are the uniform stress components applied at infinity;
— (u∞ )i is the displacement field associated with the uniform stress at infinity;
— the influence functions (uF)ik and (σF)i jk relate the displacement ui and stress
σi j at any field point in the body to the force singularity Fk applied at a source
point on the boundary Γ;
— the influence functions (uc)ik and (σc)i jk relate the displacement ui and stress
σi j at any field point in the body to the displacement singularity ck applied at a
source point on the boundary Γ;
— a repeated index implies summation.
Assuming linear-elastic-isotropic-homogenous material, the following relationships
between the influence functions can be established [6, 8].
11 Stress Analysis of Bonded Joints
309
(σF)i jk = μ (uF)ik, j + (uF) jk,i + λδi j (uF)mk,m
(11.5)
(σc)i jk = μ (uc)ik, j + (uc) jk,i + λδi j (uc)mk,m
(11.6)
(uc)ik = − (σF)i jk n j
(11.7)
where,
— μ and λ are the modulus of rigidity and Lame’s constant for the material;
— δi j represents Kronecker’s delta;
— n j are the direction cosines of the unit normal at the source point on the
boundary Γ;
— a comma implies differentiation with respect to the field point.
Equations (11.3) and (11.4) have two displacements and two tractions on the inside and the outside of Γ as variables, resulting in a total of 8 variables. Two of
the variables are known and two can be determined from the boundary conditions.
Thus four variables must be eliminated from these two equations. We consider three
possibilities of eliminating the additional variables.
Indirect Force Singularity BEM: In this formulation, as one moves from the inside of the body to the outside (infinite), continuity of displacements uo k = ui k
across the boundary Γ is enforced. The traction jump across the boundary Fk =
t i k − t o k is the unknown density function that is determined from the boundary
conditions.
Indirect Displacement Singularity BEM: Also known as Displacement Discontinuity [17] formulation. In this formulation, as one moves from the inside of the body
to the outside (infinite), continuity of traction t o k = t i k across the boundary Γ is enforced. The displacement jump ck = ui k − uo k is the unknown density function that
is determined from the boundary conditions.
Direct BEM: Also known as Rizzo’s Method [32]. In this formulation, it is assumed
the outside (infinite) body is undeformed, i.e., uo k = 0 and t o k = 0. Only Eq. (11.3)
is used in determining the unknown. If displacements ui k are specified then traction
t i k are the unknown density functions and if traction t i k are specified then ui k are the
unknown density functions.
There are other formulations such as Dual BEM [27], or Direct BEM in which
tangential derivatives of the displacements are the unknowns [19]. Many more are
possible, particularly for the Indirect BEM [21]. Which particular formulation for
which particular situation would lead to the smallest error, that is, require the least
number of unknowns for a specified error? Answer to this question is not currently
available in the literature. But with regard to the above three formulations, the results
presented in [7, 8, 42] provide some answers that are discussed below.
For smooth boundaries, the Indirect BEM with force singularity has been shown
to have the fastest convergence. For boundaries with corners, the Direct BEM
has the fastest convergence. For boundaries that represent cracks in the interior
of the materials, the indirect BEM with displacement discontinuity has the fastest
convergence.
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M. Vable
11.4 Discretization and Sources of Errors
The integrals in Eqs. (11.3) and (11.4) are reduced to algebraic expressions through
a discretization process. The discretization process is an approximation of the actual
geometry and the behavior of known and unknown density functions (displacements
and tractions). Thus, the discretization process is the process by which numerical
errors are introduced into the methodology. This section briefly describes the discretization and the associated errors. For additional details see [45].
A single integral in Eq. (11.3) is considered below to briefly describe the discretization process. Point Q is the point where a quantity is being evaluated and S is
the integrating point on the boundary.
Ii (Q) =
(uc)ik (Q, S) uk (S) ds (S)
(11.8)
Γ
The boundary Γ is sub-divided into NE elements and the integral is replaced by
a summation of integrals over each element as shown below:
NE
Ii (Q) =
∑
(uc)ik (Q, S) uk (S) ds (S)
(11.9)
n=1
Γn
where Γn is the boundary of the nth element.
11.4.1 Interpolation Error
On each element the density functions (tk and uk ) are represented by interpolation
functions (Nq ). The constants of the interpolation functions are the nodal values of
the density functions (Tkq ). The result is an algebraic expression in (Tkq ) as shown
below:
⎡
⎤
Ii (Q) =
NE
Kn
n=1
q=0
∑ ⎣ ∑ Tkq
(uc)ik (Q, S) Nq (S) ds (S)⎦
(11.10)
Γn
where Kn is the order of polynomial (interpolation function) in the nth element. Two
errors are introduced at this stage: mesh error discussed in Sect. 11.4.6 and an error
due to the location of the nodes (collocation points) where the nodal values of the
density function are prescribed or to be found. If the density functions are discontinuous at both ends of the element, then the optimum location of collocation points
are the roots of the Chebyshev polynomial of the second kind [23]. Traditionally in
BEM, corners, or points at which boundary condition values jumps, or points where
boundary condition changes from essential to natural or vice versa, are points where
density functions are permitted to be discontinuous during mesh creation. In [43, 44]
an algorithm is described for determining the optimum location of the collocation
11 Stress Analysis of Bonded Joints
311
points for different types of interpolation functions. These interpolation functions
could be up to fifteenth order polynomials that could be discontinuous at the element end or have continuity up to the seventh derivative of the density function.
The optimum locations of the collocation points were presented in tables for use by
other researchers. Based on the study in [43, 44], the recommendations for selecting
interpolation functions are the following:
(i) For elements that are continuous at both ends, it is better to increase the polynomial order by increasing the continuity order rather than by adding additional
nodes inside the element.
(ii) For elements that are discontinuous at one end, it is better to increase the polynomial order by adding nodes inside the element rather that by increasing continuity at the other end of the element.
11.4.2 Integration Error
The integration error arises from the evaluation of integrals in Eq. (11.10). These
integrals can be evaluated numerically or semi-analytically as described below.
In numerical integration, the integrand is approximated by known functions, usually polynomials such as in Gauss Quadrature. The singular nature of the influence
functions require, the singularity in the element be extracted analytically if possible,
or the integrand be made smooth by processing the integrand in some manner. Reference [35] documents the many approaches to the numerical integration error. The
strength of numerical integration is in its generality – that is, the algorithms are not
specific to the functions in the integrand. The integration error is from the approximation of the integrand but the shape of the integration path (boundary) does not
affect the integration error.
In semi-analytical integration schemes [2, 3, 9, 39], the integration path is usually
approximated by a series of straight lines. Analytical expressions of the integrals
over the straight line segment can be obtained. Thus, if the original boundary is
made up of straight line segments, then no integration error is introduced into the
analysis. If the boundary is curved then it leads to integration error. The creation
of corners where two straight line segment join has a particularly deleterious effect
on the integration error when the field point approaches the corner. The study in [9]
showed that:
(i) Large integration errors are obtained when the field point is within one segment
length from the boundary.
(ii) Increasing the number of tangent points (number of line segments) decreases
the magnitude of the error.
(iii) The integration errors rapidly decrease with distance from the boundary and
are nearly negligible at two segment lengths.
(iv) The error decreases as one moves towards the tangent points and away from the
corners.
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M. Vable
Based on these observations an algorithm is described in [9] that creates an integration path of straight line segments to reduce the integration error dramatically. The
advantages of the semi-analytical scheme for bonded joints are:
(i) For straight boundaries, which are most of the boundaries in bonded joints,
there is no integration error.
(ii) The singular nature of the influence function does not pose any additional difficulties when the stresses are evaluated at the interface or close to it as in the
middle of the adhesive.
11.4.3 Continuity Error
For quantities of interest, stresses and displacements, to be bounded on the boundary, the density functions (tk and uk ) must satisfy certain continuity conditions at
the element ends which depend upon the order of singularity in the influence functions used in formulating the integral equations. If the influence function varies as
(1/rn ), where r is the radial distance from the singularity then n is the order of singularity. The influence function (uF)ik has a zero order singularity and thus imposes
no continuity requirement on the density function. The influence functions (σF)i jk
and (uc)ik have a first order singularity, hence requires the density function to be
continuous on smooth boundary points. The influence (σc)i jk has a second order
singularity, hence requires the density function and its derivative to be continues on
smooth boundary points. Quite often the continuity conditions, particularly on the
derivatives of the density functions, are relaxed at the element ends for the sake of
simplicity in approximating the unknown density functions. This can impact at two
points in the BEM analysis:
(i) All requisite continuity conditions are not met during enforcement of the
boundary conditions. This can lead to errors in the density function, resulting
in large errors in the analysis as discussed in the next section.
(ii) Assuming no continuity error during enforcement of boundary conditions, then
a large spike in error will be seen in stresses close to the point where continuity
is violated and this error propagates inwards up to one element length as shown
in [8].
11.4.4 Collocation Error
The boundary conditions can be satisfied in a collocation sense or in Galerkin sense.
In collocation schemes the boundary conditions are satisfied at finite number of discrete points. The decision on the location of these points affects the error in the
analysis and is referred to as the collocation error. Galerkin schemes minimize the
collocation error by integrating the boundary condition over the elements, but this
11 Stress Analysis of Bonded Joints
313
increases the computation cost significantly as two integrals have to be evaluated and
the complexity of these integrals usually results in numerical integration schemes
and the associated integration error. The location of the boundary conditions collocation points has three effects:
(i) it affects the distribution of the error over the body.
(ii) it affects the matrix conditioning of the algebraic equations and hence the matrix conditioning error discussed Sect. 11.4.5.
(iii) it couples with the continuity error to produce extremely large error in the
analysis.
In this work, the collocation points are chosen as the same points as those
that minimized the interpolation error. When the boundary conditions and interface conditions are satisfied in a collocation sense, the algebraic expressions of the
type in Eq. (11.10) result in algebraic equations which can be symbolically represented as
[A] {w} = {r}
(11.11)
where, {w} and {r} are the unknown and known nodal values of the density function, respectively, and [A] is a known matrix with coefficients that are summations
of integrals of the type shown in Eq. (11.10).
The solution of Eq. (11.11) gives the displacements and tractions on the boundary. In a lap joint this implies that we know both the peel stress and the shear stress
on the interfaces of the adhesive and the adherends. The equivalent algebraic form
of Eqs. (11.3) and (11.4) are used to determine the stresses and displacements at any
point Q in the interior of the materials.
11.4.5 Matrix Conditioning Error
The sensitivity of the solution of Eq. (11.11) to the errors in the right hand side
vector, depends upon the conditioning of the matrix in the algebraic equation. The
amplification of the error in the input data due to the matrix conditioning [13, 40, 41]
is referred to as the matrix conditioning error.
A measure of matrix conditioning is defined below:
Matrix Condition Number = A × A−1
(11.12)
where A and A−1 are the norm of the matrix and its inverse respectively. The
norm of the matrix is defined as
A = max ∑ Ai j
i
j=1
(11.13)
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M. Vable
The conditioning of the matrix can be improved by using the equilibrium equations [13, 40, 41].
11.4.6 Mesh Error
Mesh error refers to the error in the analysis that is generated by the choice of
number of elements, the size of elements, end location of the elements, and the
choice of the order of polynomial in the elements. This error is addressed by mesh
refinement techniques.
There are several mesh refinement schemes in BEM. In the h-method [29, 31]
the order of polynomial is kept fixed and an element is subdivided to improve accuracy. However, the total number of unknowns can become very large. In the pmethod [30, 37, 38] the element size is kept fixed, while the polynomial order of
the interpolation function is increased in order to improve the accuracy. Though the
convergence rate for the p-method is better than the h-method for smooth functions,
the method may not converge [36] near a singularity. The hp-method [11, 28, 30]
is a combination of the h-method and the p-method that overcomes some of the
convergence problems of the p-method near the singularity, but the location of the
singularity has to be specified and several parameters have to be specified or evaluated to determine the neighborhood of the singularity. Furthermore the p-method
and the hp-method are usually used in conjunction with the Galerkin method due to
the difficulty posed in selecting new collocation points as the order of polynomial
increases. In the r-method [14, 15, 22, 24] the total number of elements and the
order of polynomial are kept fixed, but the spacing of the elements is adjusted to
minimize error. If the initial mesh does not have sufficient degrees of freedom then
the desired accuracy may not be obtained with r-method [33]. The hr-method [10,
33, 34] is a combination of the h-method and the r-method in which the polynomial
order is fixed. The number of unknowns in the hr-method can become large near a
singularity. Now it is known [20] that a graded mesh in which the polynomial order
increase away from singularity has a higher convergence rate than a graded mesh
with a uniform polynomial order. Such hpr-method exist in FEM but none to date
has been published for BEM.
The above discussion highlight that mesh refinements schemes do well for
smooth density functions but have difficulties when density functions have singularities, particularly when the location of singularities are not known. Successful
application of BEM for fracture mechanics problem is primarily due to the fact
that the stress singularity is in front of the crack, a region that is not discretized
in BEM. However, problems such as lap joints contain strong gradients in the
density function along the interface and the location of the maximum value cannot be prescribed during mesh construction. Thus developing an hpr-method for
BEM that is applicable to bonded joints is one of the research challenge that has to
be met.
11 Stress Analysis of Bonded Joints
315
11.5 Multiple Materials
Bonded joints have corners, thus the Direct BEM is the appropriate formulation and
will be the only one described in this paper for multiple materials.
Figure 11.2 shows a general bi-material problem. Ω1 and Ω2 refer to material 1
and material 2, respectively. Γ1 and Γ2 are the non-interface boundaries of Ω1 and
Ω2 , respectively and Γint is the interface boundary between the two materials. The
integration on the interface in each material is in opposite direction as shown in
Fig. 11.2.
The boundary integral equations for displacements and stresses at point Q that are
written for a homogenous material can be written for the mth material [18, 46] as:
(m)
ui
=
(uF)ik tk ds +
Γm
Γint
+
(m)
Γm
+
(m)
(uc)ik uk ds + (u∞ )i
(11.14)
Γint
(σ F)i jk tk ds +
=
(uc)ik uk ds +
Γm
σi j
(uF)ik tk ds
(σ F)i jk tk ds
Γint
(σ c)i jk uk ds +
Γm
(m)
(σ c)i jk uk ds + (σ∞ )i j
(11.15)
Γint
The boundary conditions in the local normal and tangential (n, t) coordinate system can be written as:
(m)
tk
(m)
(m)
= tk
or
(m)
uk
(m)
= uk
k = n,t
on
Γm
(11.16)
(m)
where, tk and uk are the specified tractions and displacements in the local normal
and tangential directions on the non-interface boundaries of each material.
Γ2
Ω1
Fig. 11.2 Bi-material
geometry
Γ1
Ω2
Γint
Γint
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M. Vable
Theoretically speaking, there are 7 possible interface conditions [46] that describe the jump in tractions or displacements across the interface in each direction.
In this paper we consider only perfectly bonded interface conditions that requires a
continuity of displacements and tractions in the local normal and tangential coordinates at the interface as shown below.
(1)
(2)
tk = tk
(1)
(2)
uk + uk = 0 k = n,t
on
Γint
(11.17)
In Eqs. (11.14) and (11.15) the integration is on the boundary of mth material and
the density functions tk and uk that affect the stresses and displacements inside the
mth material are those that are defined on the boundary of that particular mth material. The values of tk and uk of the other material boundaries do not appear in the
expressions of stresses and displacements in Eqs. (11.14) and (11.15). Thus, once
the density functions are known, the accuracy of results inside a material is determined by the accuracy with which the density function on the boundary (interface
or non-interface) is determined.
The incorporation of the stress at infinity in the Eqs. (11.14) and (11.15) work effectively [25, 48] when an inclusion is embedded in a different infinite material subjected to the uniform stresses at infinity. In bonded joints however the boundary of
the adherend starts at the adhesive end and extends to infinity, thus requires use of infinite boundary elements. Reference [12] describes an infinite boundary element that
can be used with numerical integration schemes. Research is on going to develop an
infinite boundary element that can be used with semi-analytical integration schemes.
11.6 Numerical Results
If not the only one, [47] is one of the very few papers in which BEM is used
for analysis of bonded joints. Some of the results from [47] are presented here.
The results were generated using program BEAMUP that has been developed by
the author and his students. Details of the program can be found on the webpage
http://www.me.mtu.edu/%7Emavable/BEAMUP/index.html.
Figure 11.3 shows a lap joint with various variables used in the numerical problems. The angle θ is the spew angle. The far field roller supported boundary with
stress has to be modelled because the infinite boundary elements for analytical integration scheme used here are still being developed.
The dimensions and material properties of the lap joint used by Pickett and
Hollaway [26] is used as the basic geometry. Pickett and Hollaway did not consider
any spew, that is θ = 90◦ . The remaining geometric parameters are:
L1 = L2 = 12.7 mm
h1 = 0.15 mm
L = 76.2 mm
h2 = 1.6 mm
(11.18)
The stresses were non-dimensionalized with respect to the far field stress σ .
The material properties of adherend and the adhesive are shown in Table 11.1. In
11 Stress Analysis of Bonded Joints
317
y
0.5 L
I
L2
h1
C
A
σ
x
J θ
B
H
G
E
F
D
K
σ
h2
L1
L
Fig. 11.3 Lap joint geometry
the analysis, the material parameters were non-dimensionalized with respect to the
modulus of elasticity of the adhesive.
11.6.1 Discretization of Joint Boundaries into Sub-Boundaries
A corner is a point where tractions are usually discontinuous and the modelling of
density function must account for this discontinuity. Similarly one must provide for
the possibility that the density function may become discontinuous at a point where
the nature of boundary condition changes. To permit modelling of discontinuities
in the density function, the boundary of each region is assumed to be made up of
sub-boundaries. The end points of sub-boundary are corners and points where the
nature (type) of boundary condition changes. These points in Fig. 11.3 are identified
by the letters C through K. Continuity of density function is assumed at all points
on the sub-boundary except the end points where the density function is permitted
to be discontinuous. The last element on the sub-boundary is thus a discontinuous
element [44] discussed in Sect. 11.4.1.
The boundaries of bottom and top adherend of a lap joint are modelled as seven
sub-boundaries: CD, DE, EF, FG, GH, HI, and IC as shown in Fig. 11.3. Each side
of the adhesive boundary is modelled as a sub-boundary.
11.6.2 Graded Mesh for BEM
In [47] a study was conducted to establish the importance of graded mesh along
the interface for FEM as well as BEM. As the authors had only hr-mesh refinement
scheme [10] for homogenous material, they described a procedure of constructing
the graded mesh on the interface that is not discussed here. Stresses were evaluated
Table 11.1 Material properties
Adhesive
Adherend
Modulus of elasticity (GPa)
Poisson’s ratio
3
69
0.36
0.32
318
M. Vable
along line AB in both adherends to ensure that symmetry of results were not lost due
to graded mesh construction.
Figure 11.4 shows the graded mesh of quadratic elements on each sub-boundary
of the adherend and the adhesive. The numbers of elements on each sub-boundary
are shown in brackets in Fig. 11.4. The coordinate along each sub-boundary is
non-dimensionalized with respect to the sub-boundary length in order to show the
relative mesh distribution on all sub-boundary simultaneously. The sub-boundary
meshes show a very fine mesh in the regions of large stress gradients (near point D
along CD). In adherend, gradation of mesh is also seen near points H and E. Stress
results close to the line HE show that there is a variation of stress due to bending that
is being perturbed by the discontinuity of traction at these points. As these points
have little impact on the stresses in the adhesive, it may be possible to eliminate this
gradation by decreasing the number of elements for modeling the unknowns in the
future. Note that the presence of bending stresses cause the mesh gradation to be
different along sub-boundaries EF and GH.
The mesh gradation in the adhesive sub-boundaries KD and CJ in Fig. 11.4 is
small but influenced by the proximity of the maximum shear and peel stress. The
differences in mesh gradation for sub-boundaries KD and CJ is a consequence of
the procedure used in creating the graded mesh in [47]. Given the mesh is not very
fine on these sub-boundaries, it is unlikely these differences, which are an artifact of
the procedure of [47], has significant overall impact on the stress values, but is yet
(a)
(5) I
C
(20) H
I
(10) G
H
(5) F
G
(10) E
F
(10) D
E
(30) C
D
0.00
0.20
0.40
0.60
0.80
Non-dimensionalized boundary length
1.00
(b)
K
D (5)
J
K (30)
C
J (5)
D
C (30)
0.00
0.20
0.40
0.60
0.80
Non-dimensionalized boundary length
1.00
Fig. 11.4 Mesh of quadratic elements. (a) Adherend. (b) Adhesive boundary
11 Stress Analysis of Bonded Joints
319
another reason for the development of mesh refinement scheme for multiple materials. The total number of unknowns for modelling the lap joint are 1075. Results are
also included for some cases for a uniform BEM mesh having the same number of
elements on each sub-boundary as shown in Fig. 11.4.
A graded mesh of 240 four noded quadrilateral elements is used for FEM results
for comparison. The graded mesh was produced by starting from the free edges
of the bonded region and increasing the length of successive elements by a factor of 1.25. The results were obtained using the commercial finite element package
ABAQUS [1].
Stresses were calculated through the center of the adhesive. Plots were made of
peel stress (σyy ) and shear stress (τxy ) for each case. Note the stress results will be
for the non-dimensionalized stresses with respect to the far field stress σ . As both
FEM and BEM are highly susceptible to the gradation in the meshes, the results
should be viewed with caution in deducing conclusions about relative merits of either methodology. The purpose of comparing results of FEM and BEM is only to
establish viability of BEM as an alternative in the analysis of bonded joints.
11.6.3 Single Lap Joint with No Spew
Figure 11.5 shows the results for shear and peel stress through the middle of the
adhesive obtained by FEM and BEM with uniform and graded mesh. The results
show very similar trends. The BEM graded mesh has a better stress gradient resolution than the BEM uniform mesh. The peak values for stresses by FEM are higher
than the BEM uniform and BEM graded mesh. Neither FEM nor BEM have optimum meshes, hence too much should not be read into the relative merits of the two
methodology based on the results shown in Fig. 11.5. The only conclusive observations that can be read from the results of Fig. 11.5 are:
(i) BEM is a viable analysis technique for lap joints;
(ii) research is needed for developing a mesh refinement scheme to produce optimum meshes for multiple material problems.
11.6.4 Double Lap Joint with No Spew
From analysis perspective, the primary difference between a single lap joint and
double lap joint is the boundary condition imposed on the sub-boundary of the lower
adherend shown in Fig. 11.3. In the input of the computer program the boundary
condition was changed from a free boundary to a roller boundary with zero normal
displacement and zero tangential traction. The stress results are shown in Fig. 11.6.
Once more the FEM and BEM results show similar trends. Again it must be emphasized that neither BEM nor FEM has optimum graded meshes and the differences in
320
M. Vable
1.00
0.87
Normal Stress
0.73
FEM
0.60
0.47
BEM graded
0.33
0.20
BEM uniform
0.07
–0.07
–0.20
–7.00 –5.25 –3.50 –1.75 0.00
1.75
3.50
5.25
7.00
3.50
5.25
7.00
x-coordinate in mm
0.60
0.51
Shear Stress
0.42
FEM
BEM graded
0.33
0.24
BEM uniform
0.16
0.07
–0.02
–0.11
–0.20
–7.00 –5.25 –3.50 –1.75 0.00
1.75
x-coordinate in mm
Fig. 11.5 Results for single lap joint with no spew
results can be an outcome of the gradation rather than something intrinsic to either
methodology.
11.6.5 Single Lap Joint with Spew
The lap joint with two spew angles of θ = 30◦ and θ = 45◦ were solved. The mesh
on the adherend was not changed from that shown in Fig. 11.4a. The differences
in mesh gradation for sub-boundaries KD and CJ is once more an artifact of the
procedure used for creating graded mesh in [47]. The graded meshes are shown in
Fig. 11.7.
Figures 11.8 and 11.9 shows the stress results for various spew angles of θ = 30◦,
θ = 45◦ and θ = 90◦ . The maximum peel stress and shear stress decrease significantly from no spew (θ = 90◦ ) to a spew with an angle of 45◦ . But the change
in maximum stress is negligible when the spew angle is changed from θ = 45◦ to
θ = 30◦ .
11 Stress Analysis of Bonded Joints
321
0.30
0.25
Normal Stress
0.20
FEM
0.15
0.10
0.05
0.00
–0.05 BEM
–0.10
–0.15
–7.00 –5.25 –3.50 –1.75 0.00 1.75 3.50
x-coordinate in mm
5.25
7.00
5.25
7.00
0.30
0.26
BEM
Shear Stress
0.21
0.17
0.12
0.08
FEM
0.03
–0.01
–0.06
–0.10
–7.00 –5.25 –3.50 –1.75 0.00 1.75 3.50
x-coordinate in mm
Fig. 11.6 Results for double lap joint with no spew
(a)
K
D
(30) J
K
C
J
(30) D
C
(5)
(5)
0.00
(b)
1.00
0.20
0.40
0.60
0.80
Non-dimensionalized boundary length
K
D (5)
J
K (30)
C
J (5)
D
C (30)
0.00
0.20
0.40
0.60
0.80
1.00
Non-dimensionalized boundary length
Fig. 11.7 Adhesive meshes for spew angle of (a) θ = 30◦ (b) θ = 45◦
322
(a)
M. Vable
0.70
0.60
Normal Stress
0.50
0.40
θ = 90°
θ = 45°
0.30
0.20
0.10
θ = 30°
–0.00
–0.10
–0.20
–7.00 –5.25 –3.50 –1.75 0.00 1.75 3.50
x-coordinate in mm
5.25
7.00
(b)
0.70
0.63
θ = 90°
Normal Stress
0.57
0.50
θ = 45°
0.43
0.37
0.30
θ = 30°
0.23
0.17
0.10
–6.50 –6.44 –6.38 –6.31 –6.25 –6.19 –6.12 –6.06 –6.00
x-coordinate in mm
Fig. 11.8 Normal stress results for spew angles. (a) Over entire bond. (b) Magnified near the end
of bond
In Fig. 11.7, notice that the impact of spew angle on mesh gradation on boundaries KD and CJ is negligible. In other words, modelling of spew poses no additional
complexity in BEM, which is certainly not true for FEM where the entire domain
needs to be discretized with a fine mesh in the corners. The ease of modelling changing boundary shapes is a demonstrable reason for considering BEM in analysis of
bonded joints. With similar ease, the rounding of adherend corners can be accommodated.
11.7 Research on the Horizon at Time of Publication
The results in this chapter show that BEM is a viable technique for stress analysis of
bonded joints. It has the potential of producing good resolution of stress gradients
and is robust enough for parametric study of joint parameters. Further improvements
in the resolution of stress gradients requires development of an hpr-mesh refinement
scheme for material interfaces and development of infinite boundary elements for
use with semi-analytical integration schemes. The hpr-mesh refinement scheme for
material multiple materials [25] has been developed and its application to bonded
11 Stress Analysis of Bonded Joints
(a)
0.50
0.40
Shear stress
323
0.30
θ = 90°
θ = 45°
θ = 30°
0.20
0.10
–0.00
–0.10
–7.00 –5.25 –3.50 –1.75 0.00 1.75 3.50
x-coordinate in mm
5.25
7.00
(b)
0.50
θ = 45°
Shear stress
0.40
0.30
0.20
θ = 30°
θ = 90°
0.10
–0.00
–0.10
–6.50 –6.44 –6.38 –6.31 –6.25 –6.19 –6.12 -6.06 –6.00
x-coordinate in mm
Fig. 11.9 Shear stress results for spew angles. (a) Over entire bond. (b) Magnified near the end of
bond
joints is expected in the near future. Research is ongoing in the development of
infinite boundary elements.
The design of mechanically fastened joints is facilitated by stress concentration
factor curves which are drawn as a function of various geometric parameters. If
a similar design approach is to be developed with the concepts of stress intensity
factors [5, 49] usage in bonded joints, then BEM with its high accuracies and ease
of modelling geometric features can well become a methodology of choice for stress
analysis of bonded joints.
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Index
A
Acceleration vector, 291
Accelerative techniques (fatigue), 189
Accelerated test methods, 226
Activation temperature, 125
Adherend-adhesive interaction, 112
Adherend rounding, 131, 136–137, 148, 150,
244, 253, 254, 256–260, 263, 269,
306–307, 322
Adherend shaping, 135, 137–138, 306–307
Adherend thickness, 135, 144
Adherend yielding, 270
Adhesion, 111–112, 126
Adhesive thickness, 14, 19, 53, 58, 135, 138,
163, 177, 228
Adhesive thickness (effect), 57–58, 60, 62,
65–66, 72–76, 80, 90
Ageing, 226
Airy’s stress functions, 16, 56, 60, 80, 82,
86, 91
Aluminium adherend, 135, 137, 142, 145, 161,
211, 219, 228–237, 289, 296
Anaerobic adhesive, 77
Anticlastic bending, 139
Arrhenius’ equation, 102
Average shear stress, 149, 193
Average stress criterion, 156–157, 160
Axi-symmetrical stress analysis, 70, 74–75, 91
B
Back-face strain measurement, 185, 190,
195–196, 217, 220
Backward substitution, 291
Band adhesive (butt joint), 64–70, 91
Beam model, 33
Beam theory, 29, 166–167, 170
Bending moment (butt joint), 64–70
Bending moment factor, 9–12, 29, 35–38, 44
Bending moment, 9, 29–30, 38, 167, 244, 265,
284–285, 293
Bent substrates, 133
Bessel function, 89
Bilinear behavior, 97–98
Bi-material interface, 245, 315
Bi-material wedge, 217, 245
Block impact test, 280, 295
Boeing wedge test, 296
Bolted-bonded joints, 128, 143
Bolted joints, 184
Bonded shrink fitted joints, 77–78
Bonding position (effect), 66, 68–69
Bonding area (effect), 68–69
Boundary element method, 305–323
Boundary element method (formulation),
305–306
Brittle adhesives, 116, 142, 145, 162, 172, 296
Brittle failure, 116
Buckling, 187, 265, 269–271
Bulk adhesive, 91, 95–96, 98–100, 102,
109–111, 112, 114, 124, 156, 165,
179, 219
Butt joints, 53–74, 108, 115–117, 138, 144,
244, 246, 293, 298–299
C
Carbon fibre reinforced plastic, 109–110, 138,
193, 201, 204–205, 211, 219, 228–229,
234, 244, 268
Carrier, 95, 97, 107, 109, 124, 189
Castigliano theorem, 167, 171
Castro-Macosko model (conversion
effect), 125
Cathodic delamination, 237–238
Cavitation, 116, 189
327
328
Characteristic length (continuum damage
model), 178
Charpy test, 280
Chemical reactions, 229
Chemorheology, 125
Circumferential stresses, 259
Clamped-clamped, 37–38
Cleavage loading (butt joint), 60–64
Coarse meshes, 136, 140, 146, 148
Coefficient of thermal expansion, 99, 109–110,
245–246, 257
Coffin and Manson law, 202
Cohesive parameters, 158, 165–176
Cohesive strength, 162–164
Cohesive zone model, 47, 143, 150, 155, 161,
162, 189, 232, 234, 236
Collocation error, 312
Complementary energy method, 117
Compliance (creep), 101–102, 113
Compliance (fracture mechanics), 161,
166, 170
Compliance based beam method, 166,
167–169, 171–172
Compliance calibration method, 166, 170
Composite delamination, 26, 28
Composite laminates, 33, 43, 50, 135, 144, 263
Composite materials, 14, 15, 23, 95–96,
109–112, 111, 132–133, 134–135,
139–140, 156–157, 169, 179, 184, 211,
234, 243–244, 246, 259, 266–268
Computation memory, 147, 291–292
Computation time, 133, 140, 145, 150
Conductive adhesives, 109, 120, 125
Constant-life diagram, 193–194
Constitutive equation of a linear elastic
solid, 282
Constitutive models (adherend), 29, 30,
34, 285
Constitutive models (adhesive), 95–126,
230, 290
Constrained matrix, 147
Continuity conditions, 29, 34, 117, 309, 312
Continuity error, 312
Continuum damage model, 155, 160, 177–179,
189, 218–219, 232, 234–236
Corner joints, 133, 245
Corner radius, 108
Corrected beam theory, 166
Coupled analyses (environmental degradation),
230, 235, 239
Crack acceleration effect (fatigue), 204, 214
Crack arrest point (fatigue), 212
Crack equivalent concept, 166–167
Crack front, 158, 238–239
Index
Crack growth rate, 190, 208–209
Crack growth, 136, 158–159, 161–162,
164–167, 170, 181, 183, 186, 192, 195,
203, 206–207, 209–213, 228
Crack initiation, 136, 183, 236
Crack length correction, 165, 168, 170
Crack path, 220
Crack root plasticity (overload), 199
Cracked lap shear (CLS) joints, 25–50
Cracking, 189
Crash analysis, 292–293
Creep, 101, 103–104, 108, 112, 184, 187,
187–188, 202, 214–215, 218
Creep-fatigue, 214–216
Critical dimension (failure prediction), 157
Cross-head rate, 112–113
Crystalline polymers, 111
Cumulative fatigue damage model, 122, 190,
195, 203, 219
Cure, 95–96, 124–125
Curing stresses, 229
Cyano-acrylic adhesive, 298
Cycle mix method, 204–205
Cyclic strain hardening, 201
Cyclic strain softening, 201
Cyclic symmetry, 146
Cylindrical joint, 257, 293
D
D’Alembert’s solution, 283, 291
Damage, 189
Damage equivalent stress, 219
Damage growth, 155, 160, 161–162, 165, 172,
178, 189, 195, 220
Damage initiation, 151, 160, 162, 164, 172,
184–185, 226–227
Damage mechanics models, 95–96, 109, 120,
155–179, 185
Damage parameter, 120, 163, 217
Damage partition method, 215, 217
Damage shift model (fatigue), 212
Data reduction scheme (fracture energy),
165–166, 169–170, 171, 175
Debond length, 26
Debonding (adhesive), 25, 28, 161
Debonding (filler), 189
Deflection wave, 281
Deformation gradient, 248
Deformation gradient tensor, 248
Deformation theory, 96–99
De-ionised water, 236
Delamination, 26
Density, 101
Index
Density functions (boundary element method),
309, 310–311, 312–313
Deviatoric strain, 119
Deviatoric stress tensor, 96
Diagonal matrix, 162–163
Diameter of the rings (effect), 78, 84
Diameter ratio (effect), 74, 78
Differential geometry (surface theory),
117–119
Diffusion, 111, 227–228
Diffusion coefficient, 227–228
Dilatational waves, 281, 283
Direct boundary element method, 308–309,
315
Discretization, 305–306, 310–314, 317
Displacement vector, 247, 290
Distortion energy criterion, 101
Dissimilar adherends, 142, 156, 245
Dissimilar meshes, 147–149
Double cantilever beam, 162, 166–168,
173–174, 179, 190–191, 209–212,
214–215
Double lap joint, 19, 20, 104–106, 108,
133–134, 138, 140, 142, 156–157, 190,
193, 201, 204–205, 210, 219, 244,
319, 321
Dual bilinear fit, 98
Dual boundary element method, 309
Ductile adhesives, 116, 155, 161–162,
173, 179
Ductile failure, 116
Dynamic analysis, 280–281, 285, 289,
291, 292
Dynamic loadings, 244
Dynamic stress concentration, 289
Dynamic Volkersen model, 287–288, 293, 295
E
Edge moment, 29, 38–39, 44–45
Edge shear force, 31
Effective plastic strain criterion, 156
Effective stress tensor, 120
Elastic limit strain, 104, 107
Elastic limit stress, 97–98, 105, 107, 121
Elastomer modified epoxy adhesive, 97
Elasto-plastic fracture mechanics, 208, 211
Embedded process zone, 161
End loaded split, 169–170
End notched flexure, 161–162, 165,
169–175, 179
Energetic criterion, 158, 159
Energy barrier (viscoelasticity), 106
Engagement length (effect), 78, 84
Entropy, 103
329
Environmental degradation, 11, 21, 179, 218,
225–240
Epoxy adhesive, 97, 107, 124, 135, 137, 145,
193, 195, 204–205, 211, 219, 244, 253,
256, 296, 298–299
Equilibrium equations, 29
Equivalent composite modulus, 117
Equivalent crack length, 168–169, 183
Equivalent elastic modulus, 114
Equivalent mixed-mode displacement, 164
Equivalent plastic displacement (damage
model), 234
Error (numerical), 305–306, 307
Euler beam theory, 30–32, 35, 285
Excimer pulse lasers, 111
Explicit solver (dynamic analysis), 279,
292–293
Eyring theorem, 105–106
F
Failure criterion, 22, 28, 47, 100
Failure criterion (stress/strain based), 47, 136,
156, 161–162, 179, 196
Failure criterion (fracture mechanics), 48,
157–160, 179
Fatigue crack growth curve, 209, 211
Fatigue failure, 184–185
Fatigue initiation, 184, 189–190, 192, 195,
207, 217–218
Fatigue life (limit), 120–124, 155, 192,
196–198, 205–206, 210–211, 218, 256
Fatigue loading, 186–189
Fatigue (modelling), 183–220
Fatigue propagation, 184, 189, 190, 192, 195,
217–218
Fatigue spectrum, 186, 199
Fatigue threshold, 208, 211–212
Fibre pull-out, 162
Fickian diffusion, 14, 21, 227–228
Filler, 95, 109, 125, 189, 245–246
Fine meshes, 136, 140, 145, 293, 318, 322
Finite difference method, 293, 305
Finite element analysis, 18, 19
Finite element method (butt joint), 72–73
Finite element method (equation), 289
First law of thermodynamics, 251
Flexible adhesive, 44
Fluid dynamics, 135
Forward elimination, 291
Fourier series, 18, 54, 60
Fourier’s law, 251
Four-Point End Notched Flexure, 169–170
330
Fracture mechanics, 28, 96, 104, 136, 155,
157–160, 185, 189–190, 192, 207–217,
227, 307, 314
Fracture process zone, 163, 166–167, 168,
169, 178
Fracture toughness, 26–27, 48, 208
Free end reflection (stress wave), 284
Fixed end reflection (stress wave), 284
Free-fixed, 38, 41
Free surface, 16–17, 138, 245–247, 252–254,
256–257, 260–263, 268, 270
Free volume, 110–111
Frequency (fatigue), 186
Functionally gradient joints, 110–111, 142
G
Gaps (adhesive), 138, 244
Gas constant, 106
Gauss point, 178
Gauss quadrature, 311
Generalised plane strain analysis, 139
Generalized stress intensity factors, 108,
136, 217
Genetic algorithm, 172
Gerber equation, 194
Gibbs free energy, 102
Glass fibre reinforced plastic, 162
Glass transition temperature, 101, 190
Global yielding, 14
Goland and Reissner, 8, 27, 42–44, 47,
131, 293
Goodman equation, 194
Green deformation tensor, 248
Griffith, 208
H
Heat transfer, 228, 243
Heat transfer coefficients, 243, 246,
252–253, 261
Heat transfer (conductive), 243, 246–247, 253,
257, 260, 261
Heat transfer (convective), 243, 246–247, 253,
254, 257, 260, 261
Helmholtz free energy, 103
Hemming technique, 279
High cycle fatigue, 192–193, 203
High strain rate, 280, 297, 298
High stress amplitudes, 201–202
Higher order theories, 43
Highly deformable adhesives, 116
Hilber-Hughes-Tailor method, 292
Hooke’s law, 115, 266
h-type meshing, 145, 314
hp -type meshing, 314
Index
hpr -type meshing, 314, 322
hr -type meshing, 314, 317
Humidity (effect), 123
Hybrid joints, 140–141
Hydrostatic component, 137, 156
Hygro-mechanical finite element analysis, 211
Hygroscopic expansion, 227
Hygroscopic strains, 229
Hygroscopic stresses, 229
Hygro-thermal stresses, 230–232
Hygro-thermo-mechanical stress, 233
Hyperbolic tangent model, 20–21
I
Identity matrix, 163
Impact dynamics (fundamentals), 281–285
Impact energy, 296
Impact modeling, 279–299
Impact strength, 279–281, 295, 299
Impact wedge-peel, 296–297
Implicit solver (dynamic analysis), 280,
291–292
Incremental finite element method, 245
Incremental theory of plasticity, 119
Indirect boundary element method, 308–309
Inertia force, 291
Inertia wheel test, 299
Initial elastic modulus, 110
Initial stiffness, 163
Instantaneous compliance (creep), 104
Instantaneous strain (creep), 104
Integration error, 311–312
Interface (displacement), 75–76, 80–82
Interface (failure), 58, 161–162, 177, 190, 232,
236, 238
Interface stiffnesses (cohesive zone
model), 162
Interface (stress), 25, 57–59, 64–86, 90,
115–116, 133, 256–257, 260, 269, 274,
307, 313–314
Interface (traction), 18
Interface crack theory, 29
Interface crack, 25
Interface elements, 148, 161–162, 177
Interfacial strength, 107, 111–113
Interface (stress wave reflection), 284
Interlaminar fracture, 169
Internal strain energy, 158
Interphase, 95–96, 111–117, 125–126, 246
Interpolation error, 310–311
Inverse method, 162, 172, 175, 176
Irwin-Kies equation, 166, 171
Isostress condition (interphase), 114
Isothermal chemoviscosity model, 125
Index
Isothermal theory of separation, 120
Iterative finite element analysis, 138
Izod test, 280
J
J-integral, 29, 48–49, 208, 215
Joint design, 13–14, 116, 133, 185, 240, 244,
247, 259, 280, 296, 305, 323
Joint rotation, 10, 132
Joint strength, 4, 14, 20, 66, 68, 77–78, 85,
91, 104, 110, 116, 133–137, 138–139,
141–142, 143–144, 173, 244, 270
K
Kelvin model, 100
Kinetic equation of solids, 281–282
Kinetic model (impact loads), 285–286
Kolsky bar, 297–298
L
Lagrangian formulation, 247, 293
Lamé constant, 309
Lap joints, 3–22, 116, 120, 132, 133, 137, 192,
217, 245, 279, 296, 314–315
Lap-strap joint, 190–191, 199, 212, 219
Large deflection (overlap), 33, 34
Lifetime prediction, 226
Linear elastic fracture mechanics, 208,
215, 232
Linear fracture energetic criterion, 165, 175
Liu model (filler volume fraction effect), 125
L-joint, 234
Load eccentricity, 244
Load distribution (effect), 72–73, 77
Loading history, 119, 220
Load sequencing effects (fatigue),
188–189, 201
Log-log method (creep), 103–104
Longitudinal stress (adhesive), 14, 16–17
Longitudinal wave, 281, 283, 288
Low cycle fatigue, 193, 201–203
Ludwik’s equation, 107
Lumped weight matrix, 292
M
Mass matrix (dynamic analysis), 291
Master curve, 101
Mat, 98
Matrix (composite material), 266
Matrix conditioning error, 313–314
Maximum cyclic load, 120–123, 124
Maximum peel stress (adhesive), 41–43,
45, 320
331
Maximum shear stress (adhesive), 43, 104,
113, 289, 320
Maximum stress, 97, 120
Maximum strain criterion, 47, 156
Maximum stress criterion, 47, 70, 97, 156–157
Maximum principal stress (adhesive), 135
Maxwell model, 96, 104, 106, 120–121
Mean stress (fatigue), 186, 192–193
Mean stress jumps (fatigue), 204
Mechanical adhesion, 117
Mechanical interlocking, 113, 117
Mechanical-hygro-thermal stress-strain
state, 225
Mechanical stresses, 229
Mesh dependency, 150, 156, 160
Mesh insensitive, 136
Mesh convergence, 156
Mesh error, 314
Mesh refinement, 156, 305, 306, 314, 319
Metal adherend, 94, 111, 135, 138, 156, 211,
243–244, 268
Michell’s stress functions, 70, 74, 88–89, 91
Micro-cracking, 163, 189
Micromechanics (composite materials),
266–268
Microscopy measurement (crack), 212
Miner’s law, 198–203, 205
Mixed-adhesive joint, 111, 141–142
Mixed-mode, 155, 158–159, 161,
177–179, 210
Mixed mode flexure, 234, 236
Mixed-mode ratio, 164
Mixed-resin joint, 111
Mode I, 46, 158–159, 161, 165, 166, 168, 170,
172–173, 190
Mode II, 46, 158–159, 161, 162, 169–175, 177
Mode III, 158, 160, 175
Modified Bingham model, 120–121
Modified distortion energy criterion, 47
Moire interferometry, 135
Moisture, 13, 21, 184, 230, 234–235, 239
Moisture absorption, 226, 228, 230
Moisture concentration, 225, 227–229, 234
Moisture transport, 225, 227–229
Mixed-mode (loading), 26, 49
Multi-physics finite element analysis, 220
Multivariable nonlinear regression
analysis, 125
N
Navier equation, 282
Newmark-β method, 292
Newton quadratic method, 117
Newton’s law of cooling, 251
332
Newton-Raphson method, 250
Nodal displacement vector, 250, 290–291
Nodal temperature vector, 251
Non-destructive inspection, 189
Non-linearity (geometry), 26, 34, 44, 47,
102–103, 127, 131, 243, 244–245, 259,
260, 260–261, 268, 271, 274
Non-linearity (adhesive), 20–21, 98, 131, 230,
236, 243, 244, 260
Non-linearity (thermal), 243–274
Non-self-similar crack growth, 159
Non-uniform temperature distributions, 239,
243, 245, 254, 256, 257, 263–264,
269–270, 273–274
Normal stress (adhesive), 116
Notches (adherend), 138
Number of cycles to failure, 209
Number of cycles to fatigue initiation, 217
Numerical crack growth integration, 209–210
Nusselt number, 253
O
Optical measurement (crack), 190
Origin symmetry, 146
Orthotropic material, 167
Overlap length, 5, 10, 13, 48, 134–135, 155,
176–177, 190, 256, 259, 265, 295
Overload (fatigue), 212
Over-stress, 119–120
P
Parametric studies, 133, 144, 150,
306–307, 322
Particles, 109, 125, 142, 189
Patches, 138, 142, 144
Parameters (fatigue spectrum), 186
Paris’ law, 208, 211
Peak stress (adhesive), 7, 12, 15, 44, 122, 135,
142, 143, 200, 246, 255–257, 256, 259,
261, 265, 269, 271, 273–274, 316
Peel joint, 133, 156
Peel stress (adhesive), 6, 9, 11, 14–19, 29,
31–33, 35, 41, 42–43, 135, 319
Penalty function, 117–119
Pendulum hammer, 280, 295–296
Pendulum test, 295–297
Perturbation technique (nonlinear viscoelasticity), 103
Photoelasticity, 133
Piezo-electric loadcell, 296
Pin-and–collar joint, 298
Plane strain problem, 145
Plastic adherend, 131
Plastic energy density, 137
Index
Plastic shakedown (fatigue), 201
Plastic strain, 97
Plastic strain amplitude, 202
Plasticisation, 227
Plasticity (adherend), 134, 285
Plasticity (adhesive), 92, 131–132, 134, 161,
179, 198, 199, 281
Point symmetry, 146–147
Poisson’s ratio effects, 139–140
Post-failure examination, 190
Potential barrier (viscoelasticity), 105
Power-law response (nonlinear viscoelasticity),
108, 125
Prandtl number, 253
Pre-crack, 153, 160, 171, 207
Pressure sensitive tape, 116, 120, 280
Primer, 111
Principal of virtual work, 249, 289
Processing conditions, 95–96, 124–126
Progressive wave, 283–284
p-type meshing, 145, 314
P –δ curve, 171–175, 179
Q
Quadratic stress criterion, 164, 178, 298–299
R
Radial stresses, 258
Ramberg and Osgood equation, 97
Random access memory, 147
Randomly oriented short fibers, 110
Rapid fracture (fatigue), 187, 187, 195, 207
Rate dependent material behaviour, 97, 98,
102–103, 108, 189
Ratio between fatigue limit and quasi-static
strength, 193
Rayleigh waves, 281
R-curve, 172, 173–174
R-ratio (fatigue), 193–194, 208
Reflective symmetry, 146
Reinforcement (joint), 259–260, 268, 270
Reinforcement (composite material), 266
Relaxation modulus, 101–102
Relaxation time, 108
Repairs, 138, 142
Residual, 250
Residual failure load (fatigue), 205–206
Residual strains, 230, 246
Residual strength, 144, 199, 203, 225, 227,
234–238
Residual stress, 70, 229–230, 245–246, 257
Retained (“master”) node, 147
Retrograde wave, 283–284
Reversibility of the ageing processes, 240
Index
Reverse bent joints, 133
Reynolds number, 253
Rice, J.R., 208, 215
Rivet-bonded joints, 132, 142–143
Riveted joints, 184
Rizzo’s method (boundary element
method), 309
Roller-clamped, 38–39
Roller-roller, 38–39, 41–46
Rotational symmetry, 146
R-type meshing, 314
Rule of mixtures, 110
S
Safety factor (fatigue), 192
Sandwich (joint), 8
Saw tooth waveform (fatigue), 187–188
Scarf joint, 4, 116–117, 133, 143, 293
Secondary creep rate, 103
Secondary phase materials, 109
Shape function, 250–251, 290
Shear modulus (adhesive), 111, 125, 289
Shear modulus (effect), 75–76, 79–80, 82
Shear modulus (interphase), 114–115
Shear stress (adhesive), 3–4, 5, 12–13, 15–19,
29, 31–33, 35, 41, 43, 45–46, 100–101,
112, 131, 132, 284, 318
Shear stress (substrate), 15–17
Shear-lag analysis, 6, 12, 113, 131, 285
Shear waves, 281, 283
Shift factor (creep), 104–107, 113
Single lap joint, 4–22, 31, 120, 132–133,
136–137, 139, 143, 145, 146, 161–162,
173, 177, 185, 190, 210, 219, 228–231,
232–234, 235–236, 244, 253–254, 279,
285–287, 293–294, 296, 319
Singularity, 18, 43, 58–60, 64, 66–67, 70, 72,
76, 95, 96, 108, 115–116, 126, 133,
136, 145, 156–157, 160, 178, 198, 199,
217–218, 245, 254, 280, 293, 299, 308
Sinusoidal waveform (fatigue), 186
Sliding elements (viscoelasticity), 120
Small strain-large displacement theory, 243,
247–249, 254, 260, 269–270, 274
S-N curves, 122, 192–193, 196, 198
Soft adhesive, 50
Softening law (stress-displacement), 162–163,
173, 177–178
Solid cylinders (butt joint), 70–71, 88–91
Specific heat, 253
Spew fillet, 131, 132, 135, 140, 145, 150,
210–211, 232, 235, 244, 253–262,
263–265, 269–270, 272–274, 306–307,
312–313, 316, 319–320
333
Spike waveform (fatigue), 187
Split Hopkinson bar, 297–298
Spring element, 28
Spring model, 27
Square waveform (fatigue), 187
Statistical analysis of the data (fatigue), 192
Steel adherend, 133, 138, 189, 240, 253, 257,
261, 293, 299
Stepped joint, 4, 143
Stiffeners, 192
Stiffness degradation approach (fatigue), 205
Stiffness matrix (continuum damage
model), 178
Stiffness matrix (finite element method),
250, 290
Strain-displacement relation matrix, 290
Strain equivalence (damage), 120
Strain energy release rate, 26–29, 41, 48–49,
158–159, 190, 208, 210, 212
Strain-life approach (fatigue), 201–203
Strain matrix, 250
Strain rate, 97, 105, 112
Strain tensor, 247, 282, 289–290
Strap joints, 133
Strength degradation approach (fatigue),
203–204
Strength of materials, 156–157, 227
Strength of singularities, 108
Strength prediction, 47–49, 136, 143, 158, 165,
176, 299
Stress amplitude, 193–194, 201–203
Stress concentration, 95–96, 108, 116, 132,
135, 137, 142, 143, 150, 155, 167, 171,
184, 243, 244, 246, 254, 258, 263–264,
269, 285, 293, 305, 307, 323
Stress concentration factor curves, 323
Stress-free temperature, 229
Stress functions, 80–91, 117
Stress intensity factor, 28, 48, 157, 158,
208, 323
Stress jump (interface), 119
Stress-life approach (fatigue), 192–203, 220
Stress ratio, 124
Stress tensor, 250, 289–290
Stress wave, 279–280, 283–284, 287–288,
293, 295, 298
Stress wave propagation, 281, 289
Stress whitening, 97–99, 124
Stress-strain curve, 97
Submodelling, 140, 150
Superposition method (fatigue life), 123
Support (joint), 259–265, 268, 269–273
Surface acoustic wave, 281
Surface (adherend), 95
334
Surface roughness, 116, 138, 139, 256
Surface topography, 12, 96, 111, 115–117
Surface treatment, 111
Superposition law (stress waves), 283
Swelling, 227, 232
Swelling coefficient, 230
Symmetrical lay-ups (composites), 43
T
Tailored interphases, 112
Taper, 132, 134, 138, 293, 306
Tap water (environmental degradation), 237
Taylor expansion, 291–292
Telegraph equation (dynamic Volkersen
model), 287
Temperature (effect), 99–100, 101–102, 112,
123, 124, 134, 135, 138, 214, 228,
229–230, 239, 245
Tensile loading (butt joint of solid cylinder),
70–71
Tensile loading (butt joint of thin plates),
54–55
Tensile loading (tubular joint), 74–75
Theory of elasticity, 54
Thermal analysis, 243, 245–246, 257,
260–269, 270
Thermal boundary conditions, 243, 245, 247,
252–253, 257, 263–264, 268–269, 270
Thermal conductivity, 228, 252–253
Thermal cycling, 246
Thermal loads, 133, 243, 245–246
Thermal model, 251–252
Thermal strains, 230, 243, 245, 250, 252, 254,
256, 258, 262, 263–264, 269–270, 273
Thermal strain tensor, 250
Thermal stresses, 19, 111–112, 135, 138, 229,
243–274
Thermodynamic type displacement (environmental degradation), 236–238
Thermoplastic polyimidesulfone, 98
Thermoplastics, 98, 110, 120
Thermosets, 97, 98, 110, 124
Thin plates (butt joint), 53–70, 80, 91
Three dimensional finite element analysis,
28, 123, 134, 140, 144, 195, 228–229,
230, 244
Three dimensional failure map (fatigue),
217–218
Three-parameter solid model, 100, 120
Tied (“slave”) node, 146
Time (effect), 100
Time dependent fracture mechanics, 215
Time shift factor, 101
Index
Time-temperature superposition principle,
101–102
Timoshenko beam, 43, 285
T-joints, 2, 134, 243, 259–274
Tongue and groove joints, 133
Torsional loading (tubular joint), 75–77, 79–80
Total-life methods (fatigue), 192–203
T-peel joint, 161–162
Traction-separation law, 161–162, 165,
173, 179
Transient strain (creep), 104
Transient temperature distribution, 243, 253
Transition elements, 147
Transverse stiffness (composites), 43
Transverse strength of composites, 137
Transverse stress (adhesive), 118
Trapezium waveform (fatigue), 187, 216–217
Tubular joints, 74–91, 134, 139, 156, 244, 245,
247, 256–259, 270, 299
U
Ultrasonics, 190
Unidirectional reinforced polymers, 109, 246
Uniform temperature distribution, 243–245,
253, 268
V
Variable amplitude fatigue, 211–214, 240
Variable frequency fatigue, 215–216
Variational principle, 19
Varying loading conditions, 122
Virtual crack closure technique, 28, 159–160
Viscoelasticity, 95–96, 99–101, 102, 113–114,
214, 252, 285, 293
Viscosity, 125
Viscosity coefficient, 101
Viscoelasticity (nonlinear), 99–102, 104, 108
Void, 96, 114, 138
Volkersen, 6, 13, 133, 279, 285, 287
Volume fractions, 109, 114, 125
Von Mises criterion, 47, 70, 156, 298–299
W
Wavy lap joints, 133
Weakening of the interface, 227
Weighted residue, 306
Weld-bonded joints, 143, 279
Wet joints, 235
Wetting, 125
Width effects, 133, 139–140
Williams-Landel-Ferry (WLF) equation, 101
Wöhler, 192
Work hardening, 97, 108
Index
X
x-radiography, 190, 212
Y
Yield criterion, 119
Yield point, 120
Yield stress, 97, 100
Young’s modulus (adhesive), 59, 91, 105
335
Young’s modulus (effect), 54–56, 57–58,
65–66, 70–71, 74–75, 83, 90, 91, 145,
245–246
Z
Zero adhesive shear stress, 19, 45
Zero volume element, 232
Zigzag joint, 144