IMPACT RESISTANCE OF
GRAPHITE / EPOXY SANDWICH PANELS
by
PUI HO WILSON TSANG
B.Eng., Imperial College of Science and Technology, London.
(August 1987)
Submitted to the Department of Aeronautics and Astronautics
in partial fulfillment of
the requirements for the degree of
Master of Science
in Aeronautics and Astronautics
at the
Massachusetts Institute of Technology
August 1989
@ Massachusetts Institute of Technology
Signature of Author
Department of Aeronautics and Astro utics
Certified by
Professor John Dugun i
i
.I
Theasis
,.gnyisor
Accepted by
(rofessor Harold Y. Wantnan
Chairman, Departmental Graduate Committee
A|ASSACHUSETTS INSTT1
OF rF:'-.,
SLP 2 9 1989
,rE
WITHDRAAWNf
M.I.T.
LBRARIES
LIBAeIitb
Aero
Impact Resistance
Graphite / Epoxy Sandwich Panels
by
Pui Ho Wilson Tsang
Submitted to the Department of Aeronautcs and Astonautics on August 11,
1989, in partial fulfillment of the requirements for the degree of Master of
Science in Aeronautics and Astronautics.
Abstract
The impact behaviour of sandwich panels is studied at 2 levels: global
bending and local indentation. A theoretical model is developed to tackle
each of these two aspects of the problem. The global model is based on plate
theory including shear deformation and is used to find the force-time (F-T)
history and the acceleration-time (A-T) history at the point of impact. The
local model is an axisymmetric elasticity model and is used to find the
stress and strain fields in the panel near the point of impact.
The global model predicts that a sandwich panel with a thicker core
experiences a higher peak force than a thinner core panel for a given
impact velocity. The local model shows that intense and localized stress and
strain fields are set up near the point of impact and that these stresses and
strains are "diffused" away as they are transmitted through the core.
Static indentation tests and impact tests were done on sandwich
panels with AS4/3501-6 [±45/0]s facesheets and Rohacell core of three
thicknesses (3.175mm, 6.35mm, and 12.7mm). The local model fails to
predict well the static indentation test results because the compressive
strength of the Rohacell core is exceeded at a very low load level. Using the
experimentally determined "contact stiffness" and assuming a linear
"contact spring", the global model predicts well the F-T history measured
from the impact tests until the front facesheet (i.e. the impacted face) fails.
Thesis supervisor : John Dugundji
Title : Professor of Aeronautics and Astronautics
Acknowledgements
The most valuable assets of a thesis supervisor are knowledge,
patience, and time for his or her students. I am ever so thankful to my
thesis supervisor, Prof. John Dugundji, for his ample supplies of all these
assets. I also owe my gratitude to Al Supple for his indispensable
assistance in the laboratory. His ability and willingness to help are much
appreciated. I would also like to thank Prof. Paul Lagace for providing me
with the opportunity to work in Telac.
The most treasurable part of working in Telac is the companionship
of other fellow graduate students. Thanks to Kevin, Pierre, Simon,
Kiernan, Narendra, Chris, Peter, Tom, and Ken for their advices (both
technical
and personal)
and
friendships.
Also,
thanks
to my
undergraduate helpers, Joe, Monte, and Simone for their dedicated
assistance with the experiments.
Finally, I would like to thank members of my family for their
continuous moral and financial support during these past two years.
Special thanks to my parents who foot all the long distance telephone bills.
This research was partially supported by the Navy contract no.
N62269-89-M-3192.
Table of Contents
List of Tables
7
List of Figures
8
List of Symbols
12
Chapter I : Introduction
1.1
Background
14
1.2
Research Objectives
16
Chapter II : Theoretical Analysis
2.1
Overview
17
2.2
Global Model
2.2.1 Assumptions
2.2.2 Equations of Motion
2.2.3 Solution Method
18
18
18
28
2.3
Local
2.3.1
2.3.2
2.3.3
2.3.4
30
30
30
37
40
Model
Assumptions
Equations of Motion
Solution Method
Bending Moment Correction and Inertia Loading
2.3.5 Strains relative to Ply Principal Axes
43
Chapter III : Experiments
3.1
Manufacturing
46
3.1.1 Specimens
3.1.2 Specimen Holder
46
50
3.2
Static Indentation Tests
56
3.3
Impact Tests
56
3.4
Damage Inspection
Chapter IV : Results and Discussion
4.1
4.2
Global Model
62
4.1.1
4.1.2
4.1.3
4.1.4
4.1.5
62
66
66
69
69
Local Model
4.2.1
4.2.2
4.2.3
4.2.4
4.2.5
4.2.6
4.2.7
4.2.8
4.2.9
4.3
4.4
Inputs
Symmetric Modes
Convergence
Effect of Core Thickness
Effect of Impact Velocity
Inputs
Convergence
Effect of Core Thickness
Stress Distributions in r
Strain Distributions in z
Stress Distributions in z
Bending Moment Correction
Inertia Loading
Strains relative to Ply Principal Axes
72
72
72
76
76
84
89
89
94
101
Comparison between Experiments and Analysis
104
4.3.1 Static Indentation Tests
104
4.3.2 Impact Tests
110
Damage Inspection
117
Chapter V : Conclusions and Recommendations
5.1
Conclusions
122
5.1.1
5.1.2
5.1.3
5.1.4
122
123
Global Model
Local Model
Experiment
Comparison between Experiments and Analysis
124
124
5.2
Recommendations
125
References
126
Appendix A Generalized Beam Functions (GBF's)
128
Appendix B
Elements of the Mass Matrix and the Stiffness Matrix
130
Appendix C
Displacements u and w of the Local Model
132
Appendix D
Bending Moment Correction
135
Appendix E
Inertia Loading
137
Appendix F
Impulsive Force on a Rigid Wall
141
Appendix G Interpretation of the X-ray Photograph of an ImpactDamaged Panel
144
List of Tables
Table 2.1
Numbering system for beam functions
Table 3.1
AS4/3501-6 material properties
Table 3.2
Mechanical properties of Rohacell 71WF
Table 3.3
Test matrix for static indentation tests and
impact tests
Table 4.1
D-matrices of the sandwich panels
Table 4.2
Transverse shear parameters of the core
Table 4.3
Input parameters for the global model
Table 4.4
Equivalent engineering properties of [±45/0]
s
AS4/3501-6 facesheets
Table 4.5
Curve-fit results of the analytical F vs. a curves
Table 4.6
Linear curve-fit results of the experimental
Table A.1
F vs. a curves
108
Shape parameters for the GBF's
129
-7-
List of Figures
Figure 2.1
Panel geometry and coordinate system
for the global model
Figure 2.2
Dynamic system of the global model
Figure 2.3
Geometry and coordinate system for the
local model
Figure 2.4
Boundary conditions for the local model
Figure 2.5
Indentation of the local model
Figure 2.6
Using the global model to find Mrr
Figure 2.7
Bending moment correction
Figure 2.8
Strain transformation to ply principal
axes
Figure 3.1
Assembly for facesheet cure
Figure 3.2
Facesheet cure cycle
Figure 3.3a
Sandwich panel assembly
Figure 3.3b
Alignment of facesheets
Figure 3.4
Assembly for bond cure
Figure 3.5
Bond cure cycle
Figure 3.6
Specimen holder
Figure 3.7
Static indentation test setup
Figure 3.8
Impact machine "Fred"
Figure 3.9
Clamping stand for impact tests
Figure 4.1
Convergence of F-T history and A-T
history
Figure 4.2
Effect of integration time step on F-T
history and A-T history
-8-
Figure 4.3
Effect of core thickness on F-T history
and A-T history
Figure 4.4
Effect of impact velocity on F-T history
and A-T history
Figure 4.5
Determination of equivalent engineering
properties for the facesheets
Figure 4.6
Convergence of the analytical F-a curve
Figure 4.7
Effect of core thickness on the analytical
F-a curve
Figure 4.8a
orr distribution in r due to a static load
Figure 4.8b
(00 distribution in r due to a static load
Trz distribution in r due to a static load
Figure 4.8c
Figure 4.8d
Figure 4.9a
ozz distribution in r due to a static load
Err distribution in z due to a static load
for the 6.35mm-core panel
Figure 4.9b
E0 distribution in z due to a static load
for the 6.35mm-core panel
Figure 4.9c
Trz distribution in z due to a static load
for the 6.35mm-core panel
Figure 4.9d
Ezz distribution in z due to a static load
for the 6.35mm-core panel
Figure 4.10a
orr distribution in z due to a static load
for the 6.35rmm-core panel
Figure 4.10b
Goe distribution in z due to a static load
for the 6.35mm-core panel
Figure 4.10c
Trz distribution in z due to a static load
for the 6.35mm-core panel
Figure 4.10d
ozz distribution in z due to a static load
for the 6.35mm-core panel
Figure 4.11
Strains
due
to
bending
moment
correction
Figure 4.12
Comparison between Err distribution in z
of the global model and of the local model
at the boundary of the local region
Figure 4.13a
Effect of inertia
loading on
arr
distribution in r for the 6.35mm-core
panel
Figure 4.13b
Effect of inertia loading on a 0
distribution in r for the 6.35mm-core
panel
Figure 4.13c
Effect
of inertia
loading
on
crz
distribution in r for the 6.35mm-core
panel
Figure 4.13d
Effect of inertia
loading
on a
distribution in r for the 6.35mm-core
panel
Figure 4.14a
Strain, Ell
100
,
contours relative to ply
principal axes
Figure 4.14b
102
Strain, E1 3 , contours relative to ply
principal axes
Figure 4.15a
103
Static indentation test result for a
3.175mm-core panel
Figure 4.15b
105
Static indentation test result for a
6.35mm-core panel
-10-
106
Figure 4.15c
Static indentation test result for a
12.7mm-core panel
Figure 4.16
,zz distribution in z for the 6.35mm-core
panel indicating core crushing
Figure 4.17a
113
Experimental and analytical F-T history
for 12.7mm-core panels
Figure 4.18
111-112
Experimental and analytical F-T history
for 6.35mm-core panels
Figure 4.17c
109
Experimental and analytical F-T history
for 3.175mm-core panels
Figure 4.17b
107
114-115
Experimental and analytical rebound
velocities
116
Figure 4.19
Core crushing
118
Figure 4.20
X-ray photograph of an impact-damaged
panel
118
Figure 4.21
Delamination length vs. impact velocity
120
Figure 4.22
Depth of crushed core vs. impact velocity
121
Figure F.1
Impact on a rigid wall
143
Figure G.1
X-ray photograph of a typical impactdamaged panel
Figure G.2
145
X-ray photograph of an impact-damaged
panel with extensive facesheet damage
-11-
146
List of Symbols
a
dimension of the panel along x-axis of the global model
b
dimension of the panel along y-axis of the global model
Yx
rotation about y-axis
Yy
rotation about x-axis
w
lateral displacement of the panel
tf
thickness of the facesheets
tc
thickness of the core
h
total thickness of the panel
m
mass of the impactor
u
displacement of the impactor
F
magnitude of the contact force between the impactor and
the panel
Kx, Ky, Kxy
curvatures of the panel
Ub
Us
bending strain energy
We
work done by the impact force
shear strain energy
coordinates of the point of impact
T
kinetic energy of the panel
Ai , Bi , Ci
modal amplitudes of the global model
fi, gi, hi,...
beam functions and their derivatives
normalized x-coordinate
normalized y-coordinate
V
total strain energy of the panel
K
stiffness matrix of the panel
M
mass matrix of the panel
-12-
R
generalized force vector of the global model
k
contact stiffness of the panel
indentation of the panel
a
index of the constitutive equation of the contact spring
modal amplitudes of the global model
p(r)
pressure loading of the local model
Rc
radius of contact between the impactor and the panel
Rp
radius of the local region
¢i
stress functions of the local model
Jo
Bessel function of order zero
Pm
roots of JO
Ami, Bmi,...
modal amplitudes of the local model
coefficients of the Fourier-Bessel expansion of p(r)
yi
inertia loading
-13-
Chapter I
Introduction
1.1 Back~eround
Advanced composites have been receiving much attention in
engineering research and development for the past two decades. Current
evidences suggest that the trend will continue to intensify rather than
diminish. Compared with metals, advanced composites offer some superior
properties for engineering applications. For example, composites have
relatively high strength-to-weight and stiffness-to-weight ratios which
render them indispensable in aerospace industries. Moreover, composites
have practically unlimited tailorability which makes them very attractive to
engineering design.
Sandwich panels, in the present context, refer to flat panels which
consist of two composite laminates (facesheets) with a sheet of different
material (core) embedded in between. Usually the facesheets are thin
compared with the core. Although, by definition, sandwich structure is a
subset of composite materials, they exhibit some distinctive properties
different from those of a composite laminate, e.g. local buckling of
facesheets (facesheet wrinkling). From the application point of view,
sandwich panels offer greater weight saving than conventional laminates
of the same thickness because the core is usually made from light-weight
materials. They also provide a better alternative for conventional skinstiffener
constuctions
because
sanwich panels
are
more easily
manufactured.
During daily applications, sandwich structures might experience
impact-type loading, e.g., tool drop during maintanance and runway
-14-
kickups on the fuselage. Like composite laminates, sandwich panels with
composite facesheets can sustain internal damage under impact loading
without visible surface damage. The presence of the core complicates the
damage mechanisms of sandwich panels because the core has its own
damage mechanisms
and provides damage interactions
with the
facesheets. The study of impact problem of structural components
comprises two different aspects: damage resistance and damage tolerance.
Damage resistance concerns how much and what types of damage the
structure will suffer due to impact. Damage tolerance, on the other hand,
involves how much load the structural component can carry after it is
damaged by impact. A clear understanding of both aspects of the problem
for sandwich panels will help engineers design safer and more efficient
structures.
The literature on impact study of sandwich structures is scarce.
Latest studies (within last ten years) include work by t'Hart [1] who found
that delamination of the facesheet could occur at relatively low impact
velocity. Koller [2] performed impact experiments on panels with fiber glass
facesheets and polyurethanes core. He calculated the displacement
underneath the point of impact and obtained good agreement with the
experiments. Van Veggel [3] conducted a parametric study on the impact
and damage tolerance properties of sandwich panels. Research has also
been done in TELAC on the impact resistance and damage tolerance of
sandwich structures. Bernard [4] impacted sandwich panels of three
different core materials and three different core thicknesses. He then
characterized the damages in the facesheet and in the core by means of Xray and cross-sectioning. Once the types of damage were determined, they
were modelled by various techniques and artificially implanted in
-15-
specimens which were subsequently tested in compression. No theoretical
analysis was done by Bernard. Lie [5], on the other hand, studied the
impact resistance and damage tolerance of sandwich panels both
experimentally and analytically. All Lie's sandwich panels had Nomex
honeycomb core and "thin" facesheets made from a plain weave fabric.
1.2 Research Objectives
The ultimate goal of the present work is to develop theoretical models
which can predict through-the-thickness damages in a sandwich panel due
to impact. Ideally, the model should be able to predict damages in the
facesheets, in the core, and at the interfaces between the core and the
facesheets. Facesheet damages include fibre breakage, matrix cracks, and
delaminations while core damages are typified by core crushing and
transverse cracks. Some experiments will be done to verify the analytical
prediction.
The present work only targets at the damage resistance aspect of the
impact problem. No damage tolerance analysis is attempted.
-16-
Chapter II
Theoretical Analysis
2.1 Overview
When a sandwich panel is impacted by a foreign object, its response
can be viewed as a combination of two different phenomena: (i) the local
indentation of the panel at the point of impact and (ii) the global bending of
the panel as a whole. Koller [2] argues that part of the kinetic energy of the
impactor is transformed into (i) the potential energy stored in the local
stress field around the point of impact, and (ii) the bending energy of the
panel. Based on similar arguments Cairns [6] developed two theoretical
models to study the impact resistance of composite laminates. The idea is to
use a plate model (global model) to find the force and acceleration, as
functions of time, experienced by the laminate due to impact. Then an
elasticity model (local model) is used to find the local stress and strain fields
near the point of impact due to the peak force experienced by the laminate.
The dynamic effect of the impact is included as an inertia loading in the
local model.
In the present work, Cairns' global and local models are modified to
study the impact resistance of sandwich panels. Briefly speaking, the
current global model is essentially the same as Cairns' with the core of the
panel modelled as a "ply" made of different material. The current local
model uses three stress functions to represent the facesheets and the core
respectively while Cairns' local model uses one stress function to represent
a laminate.
-17-
2.2.1 Assumptions
The global panel model assumes that:
(i)
through-the-thickness strain, Ezz, is negligible and, hence, w (out-ofplane displacement) is a function of x and y (in-plane coordinates)
only;
(ii)
the panel deforms both in shear and in bending;
(iii)
the core is a "ply" made of different material;
(iv)
only the core contributes to the transverse shear stiffness of the
panel;
(v)
the contact force between the impactor and the panel is a point load;
(vi)
the local indentation of the panel can be accounted for by a contact
spring;
(vii)
no damage and no damping are present.
2.2.2 Equations of Motion
The geometry and coordinate system of the panel are shown in
Fig.2.1. The deformation of the panel is described by three displacement
variables, Tx (mid-plane rotation about y), Ty (mid-plane rotation about x),
and w (midplane displacement in z) which are functions of x and y only.
The energy method is used to derive the governing equation of the
system which contains the impactor and the panel coupled together by a
contact spring (Fig.2.2). Firstly, the potential energy and kinetic energy of
the panel are expressed in terms of the displacement variables. Secondly,
the displacement variables are written as a summation of mode shapes.
-18-
--
y4
P
--
Y
L
a
4
L
..-
I
core
facesheet "
Il
z , w
14
Z, W
tc
tc
--tf
X
x
Figure 2.1
Panel
geometry
and
coordinate
system for the global model
-19-
-h
-h
impactor
TU q
F
contact
spring
sandwich panel
Figure 2.2 Dynamic system of the global model
-20-
Thirdly, Lagrange's equation of motion is applied to obtain the governing
equation of the panel which is then combined with the equation of motion of
the impactor. Finally, the resultant equation of motion is solved together
with the constitutive equation of the contact spring.
According to the Reissner-Mindlin plate theory, the kinematic
relationships for the panel can be written as,
SKx
,x
Kx
x ,y
y ,x
Tyz
jy + W
Sxz
+
x
(2.1a)
(2.l
W,x
(2.1b)
where,
,x
ax
,y -
y
The potential energy of the panel consists of the bending energy, Ub, and
the shear energy, U s , which can be written as surface integrals over the
area of the panel,
x 1112
D16x
UUb= 1 ff K
K
DY
D DD Ky
D
D22
D
16 D26
S
1 =yz
"
SYz
I
K
66
GT
45
Yyz
45
55
x
dxdy
(2.2a)
dxdy
(2.2b)
When the impactor touches the panel, it exerts a force on the panel and
gives rise to an external work term which is given by,
We=~-
F 8(xc,y,) w dxdy
-21-
(2.3)
where,
F
magnitude of the contact force;
6 the Kronecker Delta function;
Xc, Yc a coordinates of the point of impact.
The corresponding kinetic energy, T, of the panel is given by,
00
T = 1
I
x
I 0
0P
y
ýv
dx dy
(2.4)
where,
h
2
P= I pdz
h
2
h
2
I= f pz 2dz
h
( )at
;p
density of the panel
The panel is discretized by writing the displacement variables as,
Ix = 11 Ars(t) fr()
gs( 1)
rs
T y =
(2.5a)
1 B rs(t)
rs
hr() l s(l)
w = 11 Crs(t) mr ()
rs
where,
-22-
(2.5b)
ns(T1)
(2.5c)
fr()=
mr()
d
(2.6a)
h r()=mr()
g s()
Is( )
(2.6b)
= n s(T)
(2.6c)
d
-d n s(l)
dl
(2.6d)
where,
x
=a ; 1' yb
Ars, Brs, and Crs are modal amplitudes to be found from the
analysis.
mr(A) and ns(T
l ) are beam functions satisfying the geometric boundary
conditions of the panel. These beam functions are given in Appendix A.
Cairns [6] rewrites the expressions for the three displacement
variables (Eq.2.5) in the form,
T x=
i
Y y=
i
A i(t) fr(•) gs(1
)
(2.7a)
B i(t) h,(A) l s(1)
(2.7b)
w= I Ci(t) mr()
i
ns( 1)
(2.7c)
where r and s are related to i through some organized scheme. This is
merely a renumbering of the modal amplitudes. The renumbering has two
advantages: (i) mathematically it gets rid of a summation sign from Eq.2.5
and, as a result of which, the squares of the three displacement variables
-23-
will contain a double summation instead of a quadruple summation; (ii)
physically each value of "i" in Eq.2.7 represents a "plate function" which is
a product of two beam functions. The exact renumbering scheme is not
explicitly given in Cairns' thesis. In the present work the scheme shown in
Table 2.1 is used.
Substituting Eq.2.7 into Eq.2.1 and then into Eq.2.2 gives Ub and U s in
terms of the modal amplitudes, Ai , Bi, and Ci . The next step is to apply
Lagrange's equation in the form,
dd (T
dt
)
+
i
DA.
ai
(2.8a)
av
d_( IT
dt f13 .
bi
MB. I
d
(2.8b)
av =P.
ci
aC.I
dt
(2.8c)
where V=Ub+Us.
Note that Eq.2.8 are three matrix equations which, after considerable
amount of algebra, can be written in the form,
A
Ix
0
o
y
aa
IF
B.
KT
B
1
ab
K
ac
bb
K
bc
R
A
B
K
K
ab
ac
=-F Rai
bc
K
CC
bi
Ci
ci
(-9
The elements of the mass matrix, the stiffness matrix, and the
generalized force vector are given in Appendix B. The rotary inertia Ix and
Iy can be condensed out and the resultant equation becomes,
-24-
Table 2.1
Numbering system for beam functions
Considering r=s=1,2,3 as an example:
r (x-direction)
s (y-direction)
i
1
1
1
1
2
2
1
3
3
2
1
4
2
2
5
2
3
6
3
1
7
3
2
8
3
3
9
-25-
M + Kq=- F
(2.10a)
where,
C 1
(2.10b)
-1
K
K=
c -
T KT
ac
b
ab
gaa
KT
K
ab
R=R
i
K ac
K
bb
bc
(2.10c)
ci
(2.10d)
It is insightful to examine the dimensions of the matrix equation
2.10. Considering a 3x3 mode analysis (i.e. 3 modes are used along the xdirection and 3 modes are used along the y-direction), r=s=1,2,3 in Eq.2.5
and i=1,...,9 in Eq.2.7. Hence, the dimensions of Eq.2.10a will be 9x9. One
immediately realizes that Eq.2.10a expands very quickly as the number of
modes increases. The nomenclature could be deceiving because going from
a 3x3 mode to a 4x4 mode analysis means Eq.2.10a expanding from 9x9 to
16x16.
So far only the motion of the panel has been considered. The impactor
is modelled as a point mass whose equation of motion is simply,
mii = - F
(2.11)
where u is the displacement of the impactor as shown in Fig.2.2. Eq.2.10a
and Eq.2.11 can be combined to give,
m
fi
_0+ 0
-26-
u= -
1
(2.12)
The impactor and the panel are coupled together by a contact spring
(Fig.2.2) whose constitutive equation can be written as,
F = kap
(2.13)
where k is the contact stiffness of the panel and 0 is the parameter
controlling the stiffening (P>1) or softening (3<1) property of the contact
spring. The physical indentation, cx, of the panel is modelled as the
compression of the contact spring and is given by,
cc = u +
(2.14a)
where wc is the panel displacement at the point of impact. In terms of the
generalized coordinates q, this gives,
[R
1
a=
Tu
(2.14b)
Eqs.2.12, 2.13, and 2.14b are solved together to give .q, u, F, and a as
functions of time. Note that there are i+3 algebraic equations (Eqs.2.12, 2.13,
and 2.14b) for i+3 unknowns. The initial conditions are,
t=o
0
[q ] 0
u t=o 0o
(2.15a)
(2.15b)
where u 0 is the impact velocity. Also one has,
F
which, together with Eq.2.12, gives,
-27-
t=o
=0
(2.15c)
t=0
(2.15d)
2.2.3 Solution Method
Following Cairns' formulation, the Newmark constant-averageacceleration integration scheme [7] is adopted to solve Eq.2.12 with Eq.2.13
and Eq.2.14b. Consider a general second order differential equation in time,
A
+i B x =- FS
(2.16)
where _A, B, and S are given and do not vary with time. Since we are going
to use a numerical step-by-step integration method in time, it is necessary
to rewrite Eq.2.16 in the form,
..(+)
(j+1)
Ax
F (j+1)
+B x
=-F
S
(2.17)
where the superscript (j+1) represents the j+lth integration time step. The
Newmark method assumes,
.(j+l1)
. (j)
x
=x
(j+1)
At
(j+l) +
+L
+x
2
(j)
(j+ )
At
(j)]
(2.18a)
(j)
2
(2.18b)
where At is the integration time step. Rearranging gives,
.(j+l
)
4 [x(j+l)
(j)
(J)
At
(j+1)
X
2
A
. (J)
(2.19a)
(-)1
]
(j+l)
At -(2.19b)
-28-
_X
(i)
X
Substituting Eq.2.19a into Eq.2.17 and rearranging give,
11- (j+1)_ -x
_(
+B
4
(j)
(AAt
SAt- (
J
(j+l
An additional relationship between F(j+ l ) and _x
(2.20)
)
exists for this
problem. From Eqs.2.13 and 2.14b, one has,
F j+= k
SX
(2.21)
where ST is the transpose of S while k and 1 are given.
Substituting Eq.2.20 into Eq.2.21 gives an equation in F0(j+ 1 ) because
all information at the jth time step is known. When 3 is equal to 1 Eq.2.21 is
linear and can be solved directly. When 3 is not equal to 1, Newton-Rapson
method will be used to solve Eq.2.21. Once F(j+ l ) is solved, it can be
backsubstituted into Eq.2.20 to givex_(j+ 1)which, in turn, is used to evaluate
0(j + 1) and "_xj+l)via Eq.2.19. The same procedures are then repeated for the
next time step.
Eq.2.12 in the previous section can readily be solved using the
Newmark method by comparing Eq.2.12 with Eq.2.16, and recognizing that,
A
B
[0
K
0
1
-29-
m
(2.22a)
0
(2.22b)
2
(2.22c)
q
u
(2.22d)
and that Eq.2.21 is the same as Eq.2.13.
2.3 Local Model
2.3.1 Assumptions
Imagine a circular "plug" being cut out of the sandwich panel
around the point of impact, as shown in Fig.2.3. The local model is
developed to describe the stress and strain fields in this plug due to a
certain axisymmetric pressure loading p(r) at the point of impact. The local
model assumes that:
(i)
the facesheets can be treated as two homogeneous transversely
isotropic "plates", and the core as a homogeneous isotropic "plate";
(ii)
the deformation of the panel is axisymmetric;
(iii)
linear stress-strain relationship and linear strain-displacement
relationship are obeyed;
(iv)
the contact pressure due to impact is Hertzian;
(v)
the part of the panel surrounding the local region can be accounted
for by imposing a constant bending moment at the lateral boundary of
the local region;
(vi)
the dynamic effect of the impact can be accounted for by a uniform
inertia loading on the local region.
2.3.2 Equations of Motion
The sandwich panel comprises three components: the two facesheets
and the core. By using separate through-the-thickness coordinate, zi, as
-30-
r,u
p(r)
W2'
R
zl, w
1
-2
2
Z3' W 2
Rp ---
Figure 2.3
Geometry and coordinate system for the local model
-31-
shown in Fig.2.3, they can be described by the same set of equations derived
by Lekhnitskii [8] and subsequently used by Cairns [6]. In the following
equations the subscript "i" is used to distinguish the three components of
the panel with i=1,2,3 for the top facesheet, the core, and the bottom
facesheet respectively.
For a transversely
isotropic material
under axisymmetric
deformation, there are 4 stress components and 4 strain components
related by 5 independent compliances,
I 1r
all
0_
a12
a13
O rr
0
a 12 a 1 a13 0
o 0l
Ezz
a 113
~ zz
Yrz
0
a
0
33
0
0
a44
rz(2.23)
For convenience the subscript "i" is dropped for the compliances, ars , and
for the parameters, s 1 and s2 (see below) of each of the 3 components of the
panel.The governing equations of the problem are given by,
2+
Dr2
+
1 a +
2
2
r ar
rd2r
a0
S2
=
12
2
1
(2.24a)
where,
1/2
12
2d.i
a.13(a
11a33a
11 -
11 33
(2.24b)
a12)
-a13
13
-32-
(2.24c)
(2.24c)
b
11 33
al3(allC.
I
a
d
o
a
12 )
a
-
11 33
a 2 -a2
11
--
22
13
-a
a
a
(2.24d)
+ a l l a44
a12
13
(2.24e)
2
12
alla 33
(2.24f)
The stress functions, Pi, take the form,
mr)
Zf m i(zi)J0P
.= m
'
m
(2.25a)
where,
fmi(zi) = A
+Cmisinh(
m
em
=RP
-
+ B micosh(sl
1 mZi)
isinh(sCmzi)
s
2
m z i)
+ D icosh (
2
Omzi)
S m -roots of Bessel function Jo
(2.25b)
(2.25c)
Ami, Bmi, Cmi, and Dmi are coefficients to be found.
The four stress components are given by,
2
G rrri.=
00i
z.
1r
az.b
ar
2
b
a
+ r
+ a.
ar
i.
1.
+ r Dr +a a
i
(2.26a)
i
(2.26b)
aa
zzi= az-i
e8
a
+ cirar
r ar +d
-33-
2
82
i
(2.26c)
ar ar 2
rzi
r ar
+
i za i
i
(2.26d)
The two displacements are obtained by integrating the corresponding strain
components (Appendix C),
u i(r,z i) = (- a11 -b. a 1
S(rz=a 44
,rr
+
c.a 3)i,rz
1r
2a13a
(2.27a)
+ a3d)
33d i•,.zz
(2.27b)
where,
,r
ar '
i,z-
etc.
1
At this point all the basic equations for the local model have been
derived. As in most elasticity problems the next step is to find the right
number of boundary conditions which describe the physical problem shown
in Fig.2.3. Examining Eq.2.25 shows that there are 4 coefficients, Ami,
B mi Cmi, and Dmi for each of the 3 stress functions Oi. Hence, a total of 12
boundary conditions is required. These boundary conditions are given in
Eq.2.28 and depicted in Fig.2.4,
a
zz
-
tft
2j
p
(2.28a)
Iz
rj r,- 22 0
(2.28b)
-2
zzi
2
zz2(
-34-
2
0(2.28c)
=
Qz1z
Ozzl= azz2
-p(r)
trzl =
trzl = Trz2
U1 =
u2
w
w2
1
-
azz2
0zz3
Trz2= trz3
u
0 zz3
=
2
w2
= 0
Irz3 = 0
Figure 2.4 Boundary conditions for the local model
-35-
=
u3
w3
2
Srzlr
t-=
2 0
1r rz2( r,-
ul(r,& -.
tc0
2)
wir, ) -f
w2 r,-
azz2( r,L
2
c
c
tr,c)2(r
t2
(2.28e)
2SC= 0
(2.28f)
2
(2.28g)
zz3
t
rz(r '2
rz3
u
(2.28d)
=
r,
(2.28h)
tf
(r,-2
u3
(2.28i)
3
2f t~) 0
(2.28j)
(2.28k)
r
r,
-=
0
(2.281)
So far nothing has been said about the contact loading p(r).
Timoshenko [9] shows that for two isotropic bodies in contact the pressure
distribution is of the form,
1/2
p(r) = p
11 r
-36-
where p 0 is the peak pressure and Rc is the radius of contact. This is
known as the Hertzian pressure loading. Assuming that po and Rc are
given, p(r) can be expanded as a Fourier-Bessel series,
00
J
p(r)=
om
(C mr)
m=1
(2.30a)
R
Pm
2
2 Jp(r)rJ (cmr) dr
2
o
(p m)R Po,(2.30b)
Eq.2.30a is used in the boundary condition given by Eq.2.28a.
2.3.3 Solution Method
Eqs.2.26c, 2.26d, 2.27a, and 2.27b basically express
0
zzi' trzi, ui, and
w i respectively in terms of Ami, Bmi, Cmi, and Dmi. Imposing the
boundary conditions shown in Eq.2.28 gives rise to a 12x12 matrix equation,
12 x12
[
m4
0
(2.31)
which will be solved by the Gauss-Jordan elimination [10].
The analysis of the local model described so far solves the following
problem: for a given p 0 and Rc the stress functions Oi can be found and,
hence, the stresses, strains, and displacements in the three components of
the panel. However, our problem is posed slightly differently. We would like
to accomplish the following: (i) to find the total load, F, required to produce
a given amount of indentation, cx, which in practical terms is to find the
-37-
parameter k and p in Eq.2.13; (ii) to find the stress and strain fields, which
are necessary for damage prediction, due to a certain load, F.
In order to do that, we need more information. As shown in Fig.2.5 a
and Rc are related geometrically by,
1/2
a=R.- (R-R)
c
i
(2.32)
Alternatively, a can be defined as the relative displacement between the top
surface and the bottom surface of the panel at the point of impact, i.e.,
a = w11 0,
2
- w3 0,-
3(0 1t2
(2.33)
(2.33)
The total load, F, is obtained by integrating p(r) over the contact area, which
yields,
2
2cR CP0
F=
(2.34)
The solution procedures are as follows:
(i)
for given a find Rc from Eq.2.32;
(ii)
assume a unit load F and calculate p 0 from Eq.2.34;
(iii)
expand p(r) as in Eq.2.30;
(iv)
solve for Ami, Bmi, Cmi, and Dmi by matchir 9g the boundary
conditions given by Eq.2.28;
(v)
calculate ca from Eq.2.33 and compare with the inpu t value;
(vi)
linearly scale the unit load F by the ratio of the inpu.t value of a to the
calculated value of a.
-38-
nc
Figure 2.5 Indentation of the local model
-39-
By repeating these procedures for several values of a a curve of F vs. a can
be plotted and a subsquent curve fit will give the values of k and 0 in Eq.2.13.
Once k and 1 are found, we can attempt to predict damage due to a certain
applied load, F, as follows:
(i)
for given F find a by Eq.2.13;
(ii)
find Rc by Eq.2.32;
(iii)
expand p(r) as in Eq.2.30;
(iv)
solve for Ami, Bmi, Cmi, and Dmi as before;
(v)
calculate stress and strain fields in the facesheets and the core;
(vi)
rotate strains or stresses to ply principal axes (see later discussion);
(vii)
apply appropriate failure criteria on a ply-by-ply basis.
2.3.4 Bending Moment Correction and Inertia Loading
In the local model described in the previous section, no boundary
condition is imposed on the lateral surface of the local region. In order to
account for the presence of the surrounding part of the sandwich panel, a
uniform bending moment is superimposed on the local region at the lateral
boundary.
Firstly, use the global model to find the average bending moment,
Mrr, at r=Rp due to a static load, F (Fig.2.6, Appendix D). Then apply the
bending moment, Mrr, to the local region using standard lamination
theory with the facesheets and the core being isotropic (in-plane) as shown
in Fig.2.7. This will give the through-the-thickness strain and stress
distributions as follows,
-40-
Figure 2.6 Using the global model to find Mrr
Mrr
C
[facesheet/core]
Figure 2.7 Bending moment correction
-41-
-1
Krr
D11
12
16
Mrr
K0 = D12 D22 D26
0
r
K
D 16
D 26
err
-
E00
=
Sr0
D 66
(2.35)
Krr
00 I
K rO
-
(2.36)
Note that the D-matrix in Eq.2.35 is isotropic and hence is different from
that in Eq.2.2a. err and e00 given by Eq.2.36 are added to the corresponding
axisymmetric strains calculated in the previous section. Note that yr0 in
Eq.2.36 is zero because the local model assumes axisymmetric deformation.
When the panel is impacted, it accelerates. This dynamic effect could
be important when the impact velocity is high. Cairns applies a uniform
inertia loading on the local model to account for this effect. For simplicity
the same technique is used for the present local model. However, it should
be pointed out that this is a crude approximation because the acceleration
experienced by the panel varies over the area of the local region as well as
through the thickness of the panel. Cairns shows that in order to include
the inertia loading, ozz i has to be rewritten as (Appendix E),
Gz-z z
z
c
i
r2
+---+dr
jJ
2Z2
@
i
i
+Y.z.
i i
(2.37)
As a result of this change, u i and wi become,
ui(r,zi)= (- a1 1 - ba 1+ca 13 )irz
-42-
+.a13ri
(2.38a)
wi(r,zi) =a4M(.i
+
i (a
2
+
+ (- 2a
z2 - a
33 1
13
+ a33 d i)
r 2)
(2.38b)
where,
yi = PiCc
Pi - density of the 3 components of the panel
;
2.3.5 Strains relative to Ply Principal Axes
In order to predict through-the-thickness damage in the panel, the
axisymmetric strain components need to be rotated to the principal axes of
individual plies. Consider a general ply (k th ply, say), making an angle Ok
with the principal axes of the facesheets (Fig.2.8). It is assumed that the
strains do not vary through the thickness of the ply and their values are
equal to those at the midplane of the ply. The axisymmetric strains are
evaluated at a number of points, hereon referred to as the grid points, and
transformed to the ply axes by,
PE
11
err
E 00
C3 3
C
= [T]
23
zz
0
C rz
13
0
]12
ply axes
axisymmetric
(2.39a)
whereE2 3 ,E1 3 ,E2 3 arethetensorstraincomponents.
-43-
fibre direction
91
,·1
IN
panel axis
Figure 2.8
Strain transformation to ply
principal axes
-44-
where,
[T] =
cos20
sin28
0
0
sin 2 0
2
COs' O
0
0
0
2cos OsinO
0
0
0
- 2 cos 0 sin 0
0
1
0
0
0
0 cosO -sinO
0
0 sin 0 cos 0
- cos OsinO cosOsinO
0
0
S= Ok - g
0
0
0
cos 20 - sin20
(2.39b)
(2.39c)
As shown by Eq.2.39 and Fig.2.8, the transformation of the axisymmetric
strains depends on the position of the grid point.
-45-
Chapter III
Experiments
3.1 Manufacturing
3.1.1 Specimens
The facesheets of the sandwich panels are made from Hercules
AS4/3501-6 graphite-epoxy which is supplied in 305mm wide impregnated
tape ("prepreg"). The material properties of AS4/3501-6 are shown in Table
3.1. The core of the sandwich panels is made from a closed-cell rigid
polymethacrylimide foam material called "Rohacell". The type of Rohacell
used for the present work is 71WF manufactured by Rohm Tech Inc. The
Rohacell is supplied in the form of rectangular sheets in 3 thicknesses
(3.175 mm, 6.35 mm, and 12.7 mm). The mechanical properties of the
Rohacell are given in Table 3.2. The core and the facesheets are bonded
together by 2 layers of FM123-2 modified nitrile-epoxy film adhesive (0.06
lb/sq.ft.) supplied by American Cyanamid.
A [±45/0] s layup was chosen for the facesheets to provide continuity
in the data base established by previous work in Telac [4,11] on sandwich
panels. The facesheets were manufactured by standard procedures
developed in Telac over the years. The procedures are briefly described here
while detailed documentation can be found in Ref.[12]. The prepreg is first
cut into 305mm x 350mm plies. The plies are then stacked together in the
correct order and orientation to form a laminate. Several laminates and
other supplementary cure materials are put on an aluminum cure plate as
shown in Fig.3.1. The cure assembly is rolled into the autoclave in which
the laminates undergo a 2-stage cure cycle. The whole cure cycle lasts for
-46-
Table 3.1 AS4/3501-6 material properties
XT :
2150 MPa
XC :
1550 MPa
YT :
54 MPa
YC :
221 MPa
105 MPa
S
EL :
139.3 GPa
ET :
GLT:
11.1 GPa
vLT:
0.3
Density
1540 kg/m 3
6.0 GPa
-47-
Table 3.2 Mechanical properties of Rohacell 71WF
Compressive strength
1.7 MPa
Tensile strength
2.2 MPa
Flexural strength
2.9 MPa
Shear strength
1.3 MPa
Modulus of elasticity
105 MPa
Shear modulus
29 MPa
Elongation at break
3%
Gross density
75 kg/m 3
-48-
Vacuum Bag
Fiberglass Air
Breather
Aluminum
Top Plate "
:,
r "":':"
.
Laminate
Laminate .
f
Non-porous Teflon
-
Porous Teflo.-
-- -- ----
- Peel Ply
Aluminum
T-Dam
Cork"
Bleeder paper
..-
-- -
Peel Ply
-
,-
-Yiiiiii// ii///////////////
/////'
,,d
i
- -
'i
Aluminum Cure Plate
Figure 3.1 Assembly for facesheet cure
-49-
Non-porous Teflon
...
Vacuum tape
about 5 hours with a full vacuum maintained within the vacuum bag. The
pressure and temperature as functions of time during the cure are depicted
in Fig.3.2. The cured laminates are then trimmed along all 4 edges by about
5 mm (to remove any epoxy ridges) with a milling machine equipped with
diamond grit blade and water cooling. Exact dimensions of the laminates
are not important at this stage.
The Rohacell core
and the film adhesive are cut into 305mm x
350mm sheets. The facesheets, the core, and the film adhesive are stacked
together as shown in Fig.3.3a to form a sandwich panel. Extreme care is
taken to align the facesheets relative to the core and to each other. Firstly,
two 450 lines are marked on both surfaces of the core. Then the facesheets
are attached to the core with the fiber direction of the +45' ply aligned with
the two 450 marks on the core as shown in Fig.3.3b. A bond cure assemby is
set up as shown in Fig.3.4 and then put into the autoclave to undergo a bond
cure cycle (Fig.3.5) with the vaccum bag vented to atmospheric pressure.
The cured panels are finally cut into squares of 279.4 mm x 279.4 mm with
a tolerance of ± 3.0 mm using the aforementioned milling machine.
3.1.2 Specimen Holder
The specimen holder provides the required boundary conditions for
the panel during the static indentation tests and the impact tests. The test
procedures will be described in the next two sections. The holder is made of
aluminum and consists of two clamping plates, each with a square cut-out.
The dimensions of the holder are given in Fig.3.6. The rectangular rods can
be replaced by circular rods to provide simply-supported boundary condition
or they can be removed to provide free boundary condition. For the present
work, only clamped boundary condition will be used.
-50-
AUTOCLAVE
0.59
0
275 280
10
TIME (mins)
AUTOCLAVE
350
225
150
RT
-70
0
10
95
35
115
235
275 280
TIME (mins)
VACUUM (mmHg)
760
I
L
w
A•
A
280
TIME (mins)
Figure 3.2 Facesheet cure cycle
-51-
Core
Facesheet
Core
Film
Adhesive
II
Figure 3.3a Sandwich panel assembly
a----
450 mark on
the core
/45
Facesheet
Core
Figure 3.3b Alignment of facesheets
-52-
Vacuum Bag
erglass Air
ather
Ste
Top Plat
rr
'
~I~
"
'
"
~-r-"
"
*
---*
**
*
,,,*
--
**
Aluminun
Edge Bai
d-
. Non-porous
• Teflon
Assemble(
Pane
////
//
*2,%,7,7////
Aluminum Cure Plate
Figure 3.4 Assembly for bond cure
-53-
-9
-
-
Vacuum
tape
AUTOCLAVE
PRESSURE (MPa)
0.14-
0-
__
155 170
TIME (mins)
AUTOCLAVE
STEMPERATURE
(OF)
225150
-
RT
-70
J- --I
010
35
155 170
TIME (mins)
Figure 3.5 Bond cure cycle
-54-
25.4
j
-*1 I4-
25.4
O 63.5 © 3O
o
50.8
o
0
203.2 --
0
38.1
203.2
381.0
T
I
317.5
]
"
0
0
0
O0
1--0
0
0
I
355.6
9.5
I
0
[4
-
"-
241.3
"-1
I I
I I
fI
-~
4
I --l
I
rectangular rods
for clamped
boundary condition
all dimensions in mm.
12.7mm diameter
o
3.2mm diameter
Figure 3.6 Specimen holder
-55-
sandwich
panel
12.7
3.2 Static Indentation Tests
The purpose of the static indentation tests is to find the contact
stiffness of the sandwich panels. The setup for the test is shown in Fig.3.7.
Essentially, the specimen holder is supported by a holding jig at the four
corners. The jig, in the form of an inverted table, is designed to accomodate
the Trans-Tek Model 354 Linear Variable Differential Transformer (LVDT).
During the tests the LVDT and the indentor remain stationary while the
test jig moves upwards. The LVDT, thus, measures the relative
displacement between the top surface and the bottom surface of the panel
i.e. the indentation. This method is adopted from Tan and Sun [13].
The tests were conducted in a MTS-810 uniaxial testing machine.
Data were collected by a DEC PDP-11/34 computer. They include force data
from the load cell of the testing machine and the displacement data from
the LVDT. The tests were run under stroke control with a stroke rate of
0.0564 mm/s. Two panels of each thickness were tested (Table 3.3).
3.3 Impact Tests
The purpose of the impact test is to verify the global model prediction
of the loading history. The tests were done using the impact machine
"Fred" constructed by Lie [5]. The essential components of Fred are shown
in Fig.3.8. The operation of Fred is briefly as follows. The main spring is
compressed by cranking up the winch manually with the handle located at
the end of the machine. When the electrical circuit to the magnet is
disconnected, the striker is released. The striker hits the impacting rod
(impactor) which in turn hits the panel. The impact velocity is measured by
the light gate (which also activates the data aquisition when the timing flag
-56-
(stationary)
ndentor
lose radius
- 6.35 mm
Specime
holder
Sandwich
Panel
LVDT
holding jig
12.7m
(moving up)
Figure 3.7 Static indentation test setup
-57-
Table 3.3 Test matrix for static indentation tests and impact tests
Static indentation tests
core thickness
3.175 mm
6.35 mm
12.7 mm
no. of panels
Impact tests
core thickness
3.175 mm
no. of panels
-58-
6.35 mm
12.7 mm
55-
C
CL;
c;
0
o
W
6Q
dr "
(U5
C,
0
.i..;
IC-
Ca
O
0
E
0
E
0
u
.p.
a
- -·
L
-
I
Figure 3.8 Impact machine "Fred"
-59-
passes throught it) and the impact force is measured by the force
transducer (PCB Model 208A05). The data were collected by a DEC PDP 1123 computer equipped with a Data Translation DT-3382-G-32DI A/D
converter. A sampling frequency of 15 kHz. was used.
The range of impact velocities used was found from preliminary tests
on a 6.35mm-core panel. It was found that the velocity range from about
0.8m/s to 3.5 m/s produces a reasonable damage range from invisible front
face (i.e. the impacted face) damage to front face puncture. For the impact
tests the specimen holder was mounted on a clamping stand which was
also used by Lie [5] and Bernard [6]. The stand is modified from an old
drilling table and hold the specimen holder at the same level as the
impactor (Fig.3.9). Seven panels of each of the 3 thicknesses were tested
(Table 3.3).
3.4 Damage Inspection
Two different techniques were employed to examine the damaged
panels: x-ray photography and cross-sectioning. For panels impacted at low
speed and sustained no surface fracture, a 0.8 mm diameter hole was
drilled in the center of the impacted region. The hole was drilled to a depth
as close to the film adhesive as possible. A x-ray opaque dye (diiodobutane)
was injected into the front facesheet through the hole. The injected panels
were allowed to sit for about an hour before the x-ray photographs were
taken using the Scanray Torrex 150D X-ray Inspection System.
After the x-ray photographs were taken the panels were sectioned
through the damaged area by the aforementioned millng machine. The
cross-sections were examined under an Olympus SZ-Tr microscope.
-60-
Specimen Holder
/
[Lz
D ii
Clamping Stand
Figure 3.9
Clamping stand for impact tests
-61-
Chapter IV
Results and Discussion
4.1 GlobalModel
4.1.1 Inputs
Standard lamination theory is used to work out the D-matrix of the
sandwich panel using the material properties given in Table 3.1 and Table
3.2, except for the shear modulus, G, of the Rohacell core. This is because
the manufacturer's values for E (105 MPa) and G (29 MPa) give an isotropic
Poisson ratio of v = 0.81. Since the positive definiteness of the stiffness
matrix requires v < 0.5, a value of v = 0.3 is chosen, and G is calculated from
E by G = E/2(1+v), which gives G = 40.4 MPa. It can be shown that the local
model results are insensitive to this change in G.
The sandwich panel is treated as a [±45/0/Rohacell] s laminate ,i.e.,
the core is modelled as 2 "plies" of Rohacell for this symmetric layup. The
resultant D-matrices for the 3 core thicknesses are shown in Table 4.1. The
global model also requires the transverse shear parameters used in Eq.2.2b,
G 4 4 , G4 5 , and G55 of the panels. It is assumed that only the core
contributes to the shear stiffness of the panel. Hence,
GG
G
core
panel
A A°
core
The values of A4 5 , A4 4 , and A5 5 for the Rohacell core are given in Table 4.2.
For the present discussion, a linear contact spring is assumed, i.e. 0
equals 1 in Eq.2.13. Strictly speaking, f and k in Eq.2.13 should be
calculated by the local model and subsequently verified by the static
-62-
Table 4.1 D-matrices of the sandwich panels
3.175 mm
6.35 mm
12.7 mm
D11
501.5
1613.0
5746.7
D12
152.1
485.7
1726.4
D16
1.28
1.28
1.28
D22
218.9
700.3
2495.8
D26
1.28
1.28
1.28
D66
171.7
548.6
1950.6
all values in Nm
-63-
Table 4.2
Transverse shear parameters of the core
3.175 mm
6.35 mm
12.7 mm
A4 4
0.128
0.256
0.513
A4 5
0
0
0
ASS
0.128
0.256
0.513
all values in MNm - 1
-64-
Table 4.3 Input parameters for the global model
boundary
all sides clamped
condition
241.3 mm x 241.3 mm
dimensions of panel (test area)
facesheet
0.804 mm
thickness
1.534 kg
impactor mass
facesheet
1540 kg/m 3
density
75 kg/m 3
core density
-65-
indentation tests. However, the value of
P does
not affect the results of the
global model in a qualitative sense and later on it will be seen that 3 = 1 is
not an unreasonable value to use. A contact stiffness, k= 0.5 MNm- 1 is
used. This value is of the same order of magnitude as that determined by
the static indentation tests (as will be seen later). Other input parameters,
which correspond to the actual experimental condition, are summarized in
Table 4.3.
4.1.2 Symmetric Modes
In the experiments all panels were impacted at the center. Under
this condition anti-symmetric modes of the beam functions (Appendix A)
are excited only if D16 and D26 are nonzero. Table 4.1 shows that D16 and
D26 terms are negligible compared with the other D-terms. Therefore, only
symmetric modes are used in the global analysis.
4.1.3 Convergence
In the global analysis we are interested in the convergence of the
force-time (F-T) history and the acceleration-time (A-T) history at the center
of the panel. Fig.4.1 shows that a 7x7 mode analysis gives a fairly converged
F-T history but a not-so-good A-T history. In fact, a converged A-T history
requires a fairly large number of modes which renders the global analysis
inefficient. However, it can be argued that in low velocity impact (impact
velocity < 4 ms-1), the stresses caused by the acceleration are negligible
compared with that cause by the "static" load (i.e. the peak force indicated
by the F-T history). Hence, we can afford to use a less than converged A-T
history. It can also be shown that the higher the contact stiffness is, the
more modes are required to give a converged solution.
-66-
6.35mm-rCORE PANEL
V- 3.6 m/r
r,
cu
Figure 4.1a
3x3 modes
5x5 modes
7x7 modes
08.
2.5
5.0
7.5
10.0
12.5
15.8
17.5
20.0
17.5
20.8
Time [ms]
6.35mm-CORE PANEL
V- 3.0 m/r
Ge
1
3x3 modes
Figure 4.1b
S
0
5
4'
5x5 modes
I
o
4'
7x7 modes
0.0
2.5
5.0
7.5
10.0
12.5
15.8
Time [me]
Figure 4.1 Convergence of F-T history and A-T history
-67-
6.35mmn-CORE PANEL
0
,•
V- 3.0 m/s
(N
Figure 4.2a
delt- 0.1 me
dolt- 0.05 me
delt- 0.02 me
9.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
17.5
20.8
Time [me]
6.35mm--CORE PANEL
V- 3.0 m/r
1
Figure 4.2b
*
d*lt- 0.05
me
4.ur
~P
I.--LAJu AI U..
l6 JMIl
ldM
k16l
0.0
2.5
5.0
7.5
10.0
12.5
15.0
Time [me]
Figure 4.2 Effect of integration time step on F-T history and A-T history
-68-
The effect of the integration time step (delt) is shown in Fig.4.2. It can
be seen that the A-T history is more sensitive to the variation in the time
step than the F-T history.
All subsequent global analyses use 7x7 modes and 0.1 ms time step.
4.1.4 Effect of Core Thickness
Fig.4.3a shows that the thicker core panels experience a higher peak
load than the thinner core ones. A plausible explanation is that the thicker
core panel has greater resistance against bending and hence most of the
kinetic energy of the impactor is transformed into potential energy stored in
the contact spring which gives a higher contact force. It should be pointed
out that the same contact stiffness, k= 0.5 MNm-1, is used for all three core
thicknesses in Fig.4.3. This agrees with the experimental observation that
the contact stiffness is independent of the core thickness (as will be seen
later) but disagrees with the result of the local analysis (see later
discussion). The impact duration is shorter for the thicker core panels.
Fig.4.3b shows that the thicker core panel, which has greater resistance to
bending, experiences less acceleration. The F-T history for the case of a
rigid wall (Appendix F) is also shown in Fig.4.2a. This represents the
limiting case when the thickness of the core becomes infinite.
4.1.5 Effect of Impact Velocity
Fig.4.4a shows that the peak force increases as the impact velocity
increases. The impact duration remains constant within the velocity range
shown. It can be shown that the peak force and the acceleration (Fig.4.4b)
vary linearly with the impact velocity, for this case of a linear contact
spring, 13 = 1.
-69-
EFFECT OF CORE THICKNESS
V- 3.6 m/s
rigid wall
Figure 4.3a
12.7 mm
S6.5
m
3.175 mm
0.0
2.5
5.0
7.5
18.6
12.5
15.0
17.5
20.0
17.5
26.0
Time [me]
EFFECT OF CORE THICKNESS
V- 3.0 m/s
3.175 m
Figure 4.3b
C
4
eJ
aU6
u
12.7 m
6.0
2.5
5.0
7.5
10.0
12.5
15.6
Time [me]
Figure 4.3 Effect of core thickness on F-T history and A-T history
-70-
EFFECT OF IMPACT VELOCITY
6.35ram-CORE PANEL
Figure 4.4a
V- 3.0 m/s
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
17.5
29.8
Time [ms]
EFFECT OF IMPACT VELOCITY
-6.35mm-CME PANEL~PANEL
5sam--CORE
3.8
1
3.
m/
Figure 4.4b
e4
C
gs
V- 2.e m/9
V- 1.0 n/
0.0
2.5
5.0
7.5
10.0
12.5
15.0
Time [me]
Figure 4.4 Effect of impact velocity on F-T history and A-T history
-71-
4.2.1 Inputs
The local model assumes the facesheets to be homogeneous
transversely isotropic. Cairns [6] developed a program to calculate the
equivalent (3D) engineering properties of a laminate given the material type
and layup (Fig.4.5). For a [±45/0] s laminate made from AS4/3501-6 the
equivalent properties for the laminate are given in Table 4.4.
It is noticed from Table 4.3 that the laminate is not transversely
isotropic because Exx # Eyy, Vxz # Vyz , and Gxz # Gyz. Cairns' strategy is to
use Exx, vxz, Gxz as the "major" analysis and Eyy, Vyz , Gy z as the "minor"
analysis to bound the problem. It is found that the results of the local model
are quite insensitive to the difference between the 2 cases. All local analyses
in the present work use major axis properties.
4.2.2 Convergence
One of the goals of the local model is to find the constitutive relation of
the contact spring, i.e. to find k and P in Eq.2.13. This is done by fitting
Eq.2.13 to the analytical prediction of load vs. indentation (F-a) curve. The
convergence of the F-a curve depends on the value of Rp (radius of the local
region). The convergence for two values of Rp is shown in Fig.4.6. It is seen
that convergence is quicker for the smaller Rp and that the converged F-a
curves for the two values of Rp shown are practically the same.
The criteria for selecting the value of Rp are: (a) Rp should be large
enough to include the local damages around the point of impact;
(b) Rp
should be large enough so that at r=Rp the distribution of trz is symmetric
-72-
(uniply) material
properties
layup
N1J
equivalent engineering properties
of the laminate
Figure 4.5
Determination
of
equivalent
properties for the facesheets
-73-
engineering
Table 4.4 Equivalent engineering properties of [±45/0] s AS4/3501-6 facesheet
moduli are in GPa
-74-
3.175mm-CORE PANEL
Rpm 31.75 mn
G
16 modes
Figure 4.68a
e0.0
.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
9.7
6.8
Indentation [mn]
3.175mr-CORE PANEL
Rpm. 63.5 -m
G
10 modes
G
20 modes
)
30 modes
Figure 4.6b
0.0
6.1
0.2
0.3
0.4
9.5
9.6
Indentation [mm]
Figure 4.6 Convergence of the analytical F-a curve
-75-
about the midplane of the panel (so-called plate solution); (c) Rp should be
small so that the solution method is reasonably "economical". It is found
that Rp/R i = 5 (where Ri is the radius of the impactor or indentor) gives a
practical value for the analysis. Since Ri=6.35mm is used in the
experiments, subsequent analyses will use a value of Rp=31.75mm.
4.2.3 Effect of Core Thickness
Fig.4.7 and Table 4.5 show that the thicker the panel is, the lower is
the contact stiffness, k. Moreover, the value of p varies between 1.08 and 1.15
but remains close to unity. This justifies the assumption of a linear contact
spring in the global analysis. For the subsequent local analyses, the values
for k and p given in Table 4.5 were used.
4.2.4 Stress Distributions in r
Fig.4.8 shows the stress distribution as function of r for the 3
thicknesses under the same static loading. Some interesting observations
can be made from these plots. The distribution of arr and
o00 are very
similar. They are both discontinuous at the two interfaces between the core
and the facesheets. This is because in the local model, the in-plane
displacement, u, is matched at the interfaces while the core and the
facesheets have very different compliances. Both Fig. 4.8a and Fig. 4.8b
show a very intense and localized stress developed in the top facesheet (i.e.
the facesheet in contact with the indentor) around the center of the panel
(point of indentation) as shown by line 1 and line 2F. The stress dies away
rapidly as it moves towards the outer boundary. The bottom facesheet, on
the other hand, shows a more "spread-out" stress distribution pattern. It
looks like the stresses are "diffused" away as they are transferred from the
-76-
Table 4.5 Curve-fit of the analytical F vs. a
k
3.175 mm
14.92
1.15
6.35 mm
7.91
1.11
12.7 mm
4.68
1.08
k in MNm-P
-77-
EFFECT OF CORE THICKNESS
Rp - 31.75 mm
6.9
G
3.175mm core
G
6.35mm core
r
12.7m core
6.1
6.2
6.3
0.4
0.5
9.6
0.7
0.8
Indentation [mm]
Figure 4.7 Effect of core thickness on the analytical F vs. a curve
-78-
top facesheet to the bottom facesheet through the core. Also, both top and
bottom facesheets appear to be bending independently about their respective
midpoints. Negligibly low stress develops in the core because of its low
stiffness. The variation in the thickness of the core does not change the
stress distribution qualitatively. However, it can be seen that the thicker
core panel experiences lower stresses in general than the thinner core one.
Fig.4.8c shows that the transverse shear stress Trz is continuous
across the interfaces 2 and 3, and is zero at the top and bottom surfaces of
the panel as stipulated by the boundary conditions. Since the local model
assumes axisymmetric deformation, trz is also zero at the center of the
panel. Again an intense and localized stress is developed around the center
and decays towards the boundary. The bottom interface (3) develops a
higher stress than the top interface (2) near the center for the 3.175mm-core
and 6.35mm-core panels but the opposite is true for the 12.7mm-core panel.
In all cases the shear stresses at interfaces 2 and 3 approach each other as
they approach the outer boundary of the local region. Remember that one of
the criteria used to determine the value of Rp is that plate solution (i.e., a
symmetric rrz distribution about the midplane of the panel) should be
recovered at the boundary of the local region. As can be seen in Fig.4.8c, the
thickness of the core affects the recovery of a symmetric trz distribution.
Hence, strictly speaking, the thickness of the core should be taken into
account in determining
Rp.
The ozz distribution shown in Fig.4.8d indicates, once again, the
"diffusing" action of the core whereby the high local stress in the top
facesheet is dissipated as it is transferred through the core to the bottom
facesheet. Note that ozz is continuous at interfaces 2 and 3.
-79-
AXISYMMETRIC STRESS
FOR LOAD=-100.0 N
fŽLr
,A
2C,.3C
1
2C
2F
!
I
2C
I
3F
3F
-I
3.175nm-core
_·
-40.
-30.
-20.
6.
-10.
10.
20.
1
30.
40.
r [mm]
a
AXISYMMETRIC STRESS
AXISYIMETRIC STRESS
FOR LOAD-1000.6N
FOR LOAD-e1000.
N
I')
0
2F 0
a.
~I
2C.3C
2C, 3C
0L
o
u)
U)
3F
-4-.
-38.
·
·
·
1
·
·
1
-26.
-16.
e.
16.
20.
30.
48.
12.7mw-core
12.7m-ca
1
-core
-40.
-30.
-2e.
-10.
r [mm]
Figure 4.8a
rr distribution in r due to a static load
-80-
e.
r [mm]
10.
20.
36.
48
AXISYMMETRIC STRESS
FOR LOAD-1eee.e N
|,,
4ffý
.
.
.
r-,
1
.
2F
ir
2C
4
-40.
-30.
-29.
-10.
0.
10.
26.
30.
r [m]
AXISYMMETRIC STRESS
AXISYMMETRIC STRESS
FOR LOAD-10e0.o N
FOR LOAD-1000.0 N
2C,3C
3F
1.-7
12.7rm-core
-49.
-30.
-29.
-10.
0.
10.
20.
· · · · ·
30.
40e.
r [mm]
-48.
-30.
-20.
-10.
O. 10.
r [mm]
Figure 4.8b a0e distribution in r due to a static load
-81-
20.
30.
48e.
AXISYMMETRIC STRESS
FOR LOAD-1000.0 N
1
*.
m
4
ý-core
a
-4e.
-39.
-
-29.
10.
-10.
20.
30.
40.
r [mm]
AXISYMMETRIC STRESS
FOR LOAD-0le0.0 N
AXISYMMETRIC STRESS
FOR LOAD-100le.
N
3
12.7mm-core
6.35mm-core
-40.
-30.
-20.
-10.
0.
10.
20.
30.
40e.
r [-m]
Figure 4.8c
-40.
-30. -20.
10.
-18.
r [(m]
trz distribution in r due to a static load
-82-
20.
30.
40.
AXISYMMETRIC STRESS
FOR LOAD-1000.0 N
1
-W,
w!I
|
i
-Core
-40.
-3•. -26.
0.
-10.
10.
28.
30.
re
40.
r [mm]
AXISYMMETRIC STRESS
FOR LOAD-1eee.e N
AXISYMMETRIC STRESS
FOR LOAD-100le. N
3
4
3
4
6.35mm-core
12.7rm-core
·
-40. -30. -29. -10.
0.
10.
20.
30.
4e.
-40.
-36.
-20.
-10.
r [rm]
Figure 4.8d
6.
r [(m]
ozz distribution in r due to a static load
-83-
10.
20.
39.
4e.
4.2.5 Strain Distribution in z
Fig.4.9 shows the through-the-thickness strain distributions at 3
radial locations of the local region. It shows the general trend that the
magnitude of the strains decreases as one moves away from the center.
This is expected after seeing the stress distributions in r from the previous
section. Fig.4.9a and Fig.4.9b show that err and Eoe vary nonlinearly in the
top facesheet and in the core but they both have a linear distribution in the
bottom facesheet ("plate bending") at the center of the local region (r=Omm).
As we move away from the center the nonlinearity in the strain distribution
fades away. Eventually at the boundary of the local region (r=31.75mm) the
strain distributions in both the facesheets and the core are linear. In fact
the strain distributions at the boundary of the local region suggest that the
facesheets bend as two "independent" plates while the core serves as a
coupling medium between them. Both err and E0o
are continuous at the
interfaces 2 and 3.
Fig.4.9c shows that the yrz distribution is discontinuous at the
interfaces 2 and 3 because the facesheets and the core have different shear
moduli. It shows that the core deforms more than the facesheets because
the core is "softer". Fig.4.9d shows that ezz is practically zero at the
boundary of the local region. This is another indication of the recovery of the
plate solution at the boundary of the local region.
-84-
THROUGH-THE-THICKNESS STRAIN DISTRIBUTION
AT r- 0.00 mm; FOR LOAD- 10M.0 N
Co
•esheet
T
core
N
4.
00
(.4
·
·
otton
acesheet
·
-3.0 -2.0 -1.0 6.6
1.0 2.0
x10**-3
Epsilon rr
3.0
4.8
1
5.8
THROUGH-THE-THICKNESS STRAIN DISTRIBUTION
THROUGH-ThE-THICKNESS STRAIN DISTRIBUTION
AT r- 15.87 mm; FOR LOAD- 100.0 N
AT r- 31.75 mm; FOR LOAD- 10.60 N
U
T
Cl
aSesheet
$.,heet
s/
N
I
I
r-
I
f"
core
core
N
N
ao
4.
botto
acesheet
·
-3.0
-2.0
-1.8
xl10*-3
6.0
1.0
2.0
3.0
4.0
I
5.6
Epsilon rr
Figure 4.9a
-0.3
·
·
-0.2
-0.1
xle1,-3
j
6.0
bottor
facesheet
6.1
0.2
Epsilon rr
err distribution in z due to a static load for the
6.35mm-core panel
-85-
9.3
9.4
0.5
THROUGH-THE-THICKNESS STRAIN DISTRIBUTION
04
AT r- 0.00 ma; FOR LOAD- 100.0 N
----~I
uesheet
C14
N
T
core
1.
3
N4
1
-3.0
1
-2.0
·
·
1
-1.0 0.0 1.8 2.0
xOle0-3
Epsilon tt
·facesheet
·
bottoc
3.0
4.0
1
5.0
THROUGH-THE-THICKNESS STRAIN DISTRIBUTION
AT r- 15.87 m; FOR LOAD- 100.0 N
THROUGH-THE-THICKNESS STRAIN DISTRIBUTION
AT r- 31.75 mm; FOR LOAD- 188.e N
N!
T
1°esheet
1°•esheet
0u
N
4.
7
COre
e-"
N
b
ottom
acesheet
-3.0
-2.0
-1.0
x10lo-3
0.8
1.0
2.0
3.0
4.0
foesheet
5.0
Epsilon tt
Figure 4.9b
-0.3 -0.2 -0.1
xleo.-3
0.0
0.1
0.2
Epsillon tt
ege distribution in z due to a static load for the
6.35mm-core panel
-86-
0.3
9.4
0.5
THROUGH-THE-THICKNESS STRAIN DISTRIBUTION
THROUGH-THE-THICKNESS STRAIN DISTRIBUTION
AT r- 15.87 mm; FOR LOAD- 1ee.e N
AT r- 31.75 rm; FOR LOAD= 180.0 N
I
I
-3.5
-3.0
-2.5
xle**-3
-2.0
-1.5
-1.0
-6.5
0.0
0.5
Epsilon rz
Figure 4.9c
-3.5
-3.0
-2.5
x16e0-3
-2.0
-1.5
-1.8
Epsilon rz
yrz distribution in z due to a static load for the
6.35mm-core panel
-87-
-8.5
.e8 8.5
THROUGH-THE-THICKNESS STRAIN DISTRIBUTION
AT r- 0.00 mm; FOR LOAD- 100.e N
T
-14.
-12.
-10.
-8.
xl**S-3
C.
-6.
-4.
-2.
6.
2.
Epsilon zz
THROUGH-THE-THICKNESS STRAIN DISTRIBUTION
THROUGH-THE-THICKNESS STRAIN DISTRIBUTION
AT r- 15.87 mm; FOR LOAD- 160.9 N
AT r- 31.75 -m; FOR LOAD- 100.0 N
T
r,,
a esheet
o
?esheet
N
core
core
N
bottom
facesheet
-14.
-12.
-16.
xe160-3
-8.
-6.
-4.
-2.
bottom
0.
2.
Epsilon zz
Figure 4.9d
-14.
I
·
·
·
-12.
-19.
-8.
-6.
xl1e0-3
_acesheet
1
-4.
Epsilon zz
Ezz distribution in z due to a static load for the
6.35mm-core panel
-88-
-2.
1
·
0.
2.
4.2.6 Stress Distribution in z
The throught-the-thickness stress distributions in Fig.4.10 reveal
features similar to those shown by the strain distributions in z: (i) the core
"diffuses" away the intense and localized stresses (or strains) in the top
facesheet and as a result of which a more "spread-out" stress distribution is
observed in the bottom facesheet; (ii) the localized stresses (or strain) decays
very rapidly towards the boundary of the local region. Due to the difference
in the compliances between the facesheets and the core, a continuous stress
distribution corresponds to a discontinuous strain distribution at the 2
interfaces between the core and the facesheets. By the same token, the core
experiences much larger strains than the facesheets but much smaller
stresses compared with the facesheets. Note that Fig.4.10c shows an antisymmetric trz distribution at r = 15.87 mm which evolves into a symmetric
distribution at r = 31.75 mm.
4.2.7 Bending Moment Correction
Consider a load of 100 N being applied to a 6.35mm-core panel. The
average bending moment Mrr at the boundary of the local region
(r=31.75mm) as calculated by the global model is -10.4 Nm/m. Using Eq.2.35
gives rise to a curvature vector,
rr
oo
- 8.13
=
re
1.36
0
p~/mm
(4.1)
which in turn (Eq.2.35) gives,
-89-
THROUGH-THE-THICKNESS STRESS DISTRIBUTION
Nu
AT r- 0.0 mm; FOR LOAD- 100e.
N
top facesheet
I
I
!
T
°C
T1
core
00
NN
,4
ql,
Ylom facesheet
I
-60088.
-40088.
-200. 8.
288.
1eee.
Eee. 80see.
488.
Sigma rr [MPa]
THROUGH-THE-THICKNESS STRESS DISTRIBUTION
AT r- 31.75 mm; FOR LOAD- 180.8 N
THROUGH-THE-THICKNESS STRESS DISTRIBUTION
15.87mm; FOR LOAD- 188.8N
AT r-
N
*o
top facesheet
0u
top facesheet
I0
1
i-
N
-68.
-40.
-20.
8.
core
core
ltam facesheet
bottom facesheet
28.
40.
68.
88.
1ee.
Sigma rr [MPa]
-60.
-48.
-28.
8.
28.
40.
68.
Sigma rr [MPa]
Figure 4.10a crr distribution in z due to a static load for the
6.35mm-core panel
-90-
88.
1ee.
THROUGH-THE-THICKNESS STRESS DISTRIBUTION
CýAT
r- 0.e mm; FOR LOAD- 100.0 N
t
09
C4
*I
04
N
T
IC
core
°o
4.
N
cu
N
'4.
hoheet-600. -466. -266. 6.
206.
4e8.
606.
860.
10eee.
Sigma tt [MPa]
Nu
THROUGH-THE-THICKNESS STRESS DISTRIBUTION
AT r- 15.87mm; FOR LOAD- 100.6 N
THROUGH-THE-THICKNESS STRESS DISTRIBUTION
AT r- 31.75 mm; FOR LOAD- 100.9 N
4.
top facesheet
top faceshoet
S
N
IO
I"4.
I*
N
core
N
4.
Tortom facesheoet
-60. -40. -26.
0.
29. 40. 60.
Sigma tt [MPa]
bottom facesheet
80.
10ee0.
-68.
-40.
-20.
0.
28. 40. 66.
Sigma tt [MPo]
Figure 4.10b a00 distribution in z due to a static load for the
6.35mm-core panel
-91-
8.
1ee0.
z
THROUGH-THE-THICKNESS STRESS DISTRIBUTION
THROUGH-THE-THICKNESS STRESS DISTRIBUTION
AT r- 15.87 mm; FOR LOAD- 108.8 N
AT r- 31.75 am; FOR LOAD- 188.8 N
top facesheet
top facesheet
1
T
N
Io
I
core
0*
1
·
·
·
·
-0.30 -0.25 -0.29 -4.15 -0.19 -4.05 8.o8
Sigma rz [MPa]
bottom
bottom facesheet
_
bottom faceshee
·
8.85 8.10
-0.38 -0.25 -0.28 -4.15 -0.18 -0.85 0.88 08.5 9.19
Sigma rz [MPo]
Figure 4.10c trz distribution in z due to a static load for the
6.35mm-core panel
-92-
cesh
THROUGH-THE-THICKNESS STRESS DISTRIBUTION
oN
AT r- 0e.
mm; FOR LOAD- 100.6 N
I
N
top facesheet
core
IN
1Z
bottom facesheet
-1ee. -80. -66.
-40. -26.
6.
2e.
40.
6e.
Sigma zz [MPo]
N
THROUGH-THE-THICKNESS STRESS DISTRIBUTION
THROUGH-THE-THICKNESS STRESS DISTRIBUTION
AT r- 15.87 mm; FOR LOAD- 18ee.eN
AT r- 31.75 mm; FOR LOAD- 100.0 N
I
top faceshe it
top faceshe
iD
Nl
T
core
o
core
N
N
*o
Sr
bottom face heet
·
-1e.
-8.
-6.
bottom face iheet
·
·
-4.
-2.
e.
2.
1
4.
6.
Sigma zz [MPa]
-16.
-8.
-6.
-4.
-2.
6.
2.
Sigma zz [MPa]
Figure 4.10d azz distribution in z due to a static load for the
6.35mm-core panel
-93-
4.
6.
E
rr
EO
=z
- 8.13
6rO
1.36
0
te
(4.2)
The resultant strain distribution is shown in Fig.4.11. Comparing the
magnitude of the strains in Fig.4.11 with that in Fig.4.9a and Fig.4.9b
shows that the bending moment correction has negligible effect on the local
strain field near the point of impact. Hence, in subsequent analyses the
bending moment correction is discarded.
Before we leave the present discussion on the bending moment
correction, it should be pointed out that, for the material properties given
here, the global model and the local model show two very different
phenomena: (i) the local model shows that the facesheets bend as 2
"independent" plates while (ii) the global model shows that the whole panel
(i.e. facesheets & the core) bends as a plate. This can be seen from the strain
distributions in Fig.4.9a and Fig.4.11 which are depicted qualitatively in
Fig.4.12.
4.2.8 Inertia Loading
As can be seen from Fig.4.1, a "static" load (i.e. peak force of the F-T
history) of -1800N is accompanied by a center acceleration of -104 ms- 2 . The
stresses due to the "static" loading and those due to the inertia loading are
calculated separately and compared in Fig.4.13. It can be seen from
Fig.4.13 that for an impact velocity of 3 m/s the inertia loading has
negligible effect on the local stress field of the panel. The maximum strain
due to inertia loading is less than 2% of the corresponding maximum
strain due to the "static" load.
-94-
-58.17te
9.73tE
top facesheet
-25.82te
4.32gtFe
core
T
Err
bottom facesheet
C00
Figure 4.11 Strains due to bending moment correction
-95-
T
top facesheet
core
b
Err (global model)
Err (local model)
bottom facesheet
Figure 4.12 Comparison between err distribution in z of the
global model and of the local model at the
boundary of the local region
-96-
0
'-
CU-
j
CrJ
",
g e.
e'8 e'9
C0
C,,
(D[J ]
.1
89e eO'-a"*-
DiWi1
L.
c~4
I
e
a's
'9 't
, e' e'e e'Z- e't-
"e£
"OZ
"l
O'le
DeO- "eL- *eO-
[OJcl] JJ oub61
[odo] JJ ouW61
Figure 4.13a Effect of inertia loading on arr in r for the
6.35mm-core panel
-97-
0
09
U
U
0LL
,
,-
e'8
9"9
e'*
e'z
O"le
,z- 't,-
[OdO] )1 oW618
e'g
0"9 0*~* er
[OdO] 1;
Oe
' er'-
"eO "Z *e0" e0''"L- "eG-
e'•-
[odfl]
DWSbl
4 OW861s
Figure 4.13b Effect of inertia loading on coe in r for the
6.35mm-core panel
-98-
*eO-
0
P.
LI)
Cl
C~j
'.t
*'L
'9* *'t-
G'*-
'£*- '*t-
[oEi] zJ DW6lS
4
·
e'•
'L
·
'*0 'L[ody] ZJ oW61
*-
-
e't-
V's
LI' '* V'e- Z'e[ocJ] zJ oW61s
Figure 4.13c Effect of inertia loading on trz in r for the
6.35mm-core panel
-99-
V'e- **e-
L2~
a
-J
U
cr.
(0j
U:
II
*Ot "
*0Og
*ee
' 9- *'Go- "CsL-
[oCC] zz D•bjS
"i
I
C4
0
7
wa -J
I,U
,.,
n,
..
0
N~
I'0
·
1
·
·
1
·
•O'9L "00L
' '6e-"0L-"a[DodY] zz oWb1s
1
·
g]- zz D-[DdM] zz DW618
S
Figure 4.13d Effect of inertia loading on ozz in r for the
6.35mm-core panel
-100-
·
-
4.2.9 Strains relative to Ply Principal Axes
As explained in Chapter II, the axisymmetric strain (or stress)
components can be rotated to the principal axes of individual plies
whereupon appropriate failure criteria (e.g. maximum strain criteria) can
then be applied. Fig. 4.14 shows a plot of strain contours (lines of constant
strains) as a result of the aforementioned strain transformation. If the
ultimate allowable strain levels of the material is given, the damage area
can be measured from these contour plots. Damage types can be inferred by
relating different strain components to different damage mechanisms.
Cairns [6] uses the empirical criteria which assumes that ll controls the
fiber breakage,
822 controls matrix cracking, and E31
delaminations.
-101-
controls
STRAINS IN PLY PRINCIPAL AXES FOR 6.35mm-CORE PANEL
1
EPSILON 11 OF PLY # 2 FOR LOAD-
lee N
77
fiber direction
I)0
-. eeee
·
-60.
-45.
-30.
-15.
6.
15.
39.
45.
6e.
9.
12.
i-AXIS [mm]
STRAINS IN PLY PRINCIPAL AXES FOR 6.35mm-CORE PANEL
EPSILON 11 OF PLY j 2 FOR LOAD- 106 N (blow-up)
10
-12.
-9.
-6.
-3.
0.
3.
--AXIS [nu]
6.
Figure 4.14a Strain, E11, contours relative to ply axes
-102-
STRAINS IN PLY PRINCIPAL AXES FOR 6.35.m-CORE PANEL
GAMMA 13 OF PLY I 2 FOR LOAD- 100 N
0
e
,
I
0I
Sn
e1.
0
Sn
I
-68.
-45.
-38.
-15.
8.
T--AXIS
15.
45.
38.
6e.
[m]
STRAINS IN PLY PRINCIPAL AXES FOR 6.3mm-CORE PANEL
GAMMA 13 OF PLY # 2 FOR LOAD-
18e N
(blow-up)
G
.008840
-4.6838
-0.8828
-0.9108
e.e6ee
-12.
-9.
-6.
-3.
8.
3.
6.
9.
X-AXIS [mm]
Figure 4.14b Strain, E1 3 , contours relative to ply axes
-103-
12.
4.3 ComDarison between Experiment and Analysis
4.3.1 Static Indentation Tests
The experimental load vs. indentation (F vs. ca) curves are shown in
Fig.4.15. The experimental curves shown are terminated at the point of
failure of the top facesheet (i.e. the facesheet in contact with the indentor).
"Failure" in the present context is defined as the loss of load carrying
capacity of the top facesheet. This failure is reflected as a sharp load drop in
the F vs. a curves. Since we are not interested in the F vs. a behaviour after
the point of failure, subsequent data points are not shown in Fig.4.15.
However, it can still be observed that the 12.7 mm-core panel failed at the
highest load, followed by the 6.35 mm-core panel, while the 3.175 mm-core
failed at the lowest load. All three core thicknesses show a softening
behavior at the beginning (load less than - 300 N) followed by a stiffening
behavior until failure. A curve fit of the form F = ka (i.e. a linear contact
spring) is applied on the experimental data. The values of k are shown in
Table 4.6. These values support the use of k = 0.5 MNm - 1 earlier in the
parametric study in Section 4.1. The results in Table 4.6 show no sign of the
effect of core thickness.
Fig.4.15 shows that the local model fails to predict well the load vs.
indentation (F-a)behaviour of the panel. At this point a serious limitation of
the local model is revealed by examining the throught-the-thickness stress
((Yzz) distribution in the panel due to a static load. It can be seen from
Fig.4.16 that the stress developed in the core is approaching its compressive
strength (1.7 MPa) for a static load of 100 N. The local model, which
assumes everything being elastic, obviously will not work beyond that point.
Fig.4.15 shows that the analytical curves are approximately tangential to
-104-
3.175-mm CORE PANEL
2000
1000
0.000
0.001
0.002
0.003
Indentation [m]
Figure 4.15a Static indentation test result for a 3.175mm-core panel
-105-
6.35mm-CORE PANEL
2000
1000
0.000
0.001
0.002
0.003
Indentation [m]
Figure 4.15b Static indentation test result for a 6.35mm-core panel
-106-
12.7mm-CORE PANEL
2000
1000
0
0.000
0.001
0.002
0.003
Indentation [m]
Figure 4.15c Static indentation test result for a 12.7mm-core panel
-107-
Table 4.6 Linear curve-fit results of the experimental F vs. a curves
core thickness
3.175 mm
6.35 mm
group (a)
0.716
0.787
0.774
0.862 *
0.729
0.748
group (b)
A
12.7 mm
values of k in MNm-1
*
all sides simply-supported
A
results shown in Fig.4.15 are those of group (a)
-108-
THROUGH-THE-THICKNESS STRESS DISTRIBUTION
Cr4
I
AT r- 0.0 mm; FOR LOAD- 100.0 N
top facesheet
-
4.
core
7
Figure 4.16a
I
bottom facesheet
I
-100.
-80.
i
-60.
·
/
-40.
-20.
1
0.
20.
40.
60.
Sigma zz [MPa]
THROUGH-THE-THICKNESS STRESS DISTRIBUTION
AT r- 0.0 mm; FOR LOAD- 100.0 N
top facesheet
I
core
Figure 4.16b
I
L
(blow-up of Figure 4.16a)
-3.0
-2.5
-2.0
-1.5
bottom facesheet
-1.0
-4.5
0.0
0.5
1.0
Sigma zz [MPa]
Figure 4.16
czz distribution in z for the 6.35mm-core panel
indicating core crushing
-109-
the corresponding experimental curves at the origin and start to deviate
from the experimental results around 100 N.
As a result of this limitation of the local model the experimentally
determined values of k for group (a) in Table 4.6 together with a linear
contact spring (P=1) were used in the global model to obtain the F-T history
shown in Fig.4.17. The difference in k between group (a) and group (b) has
negligible effect on the F-T history.
4.3.2 Impact Tests
During the impact tests, experimental data for one 3.175mm-core
panel and four 6.35mm-core panels were lost due to an equipment problem.
The remaining data are shown in Fig.4.17.
Fig.4.17 shows that the global model gives reasonably good
agreement with the experimental F-T history. The model over-predicts the
impact duration and under-estimates the peak load. The thicker core
panels show better agreement than the thinner core ones when no
significant damage is present (i.e. a smooth half-sinusoidal F-T history). A
plausible explanation is that the facesheets have more contribution to the
transverse stiffness of the panel when the core is thin. The ratio of the total
facesheet thickness to the panel thickness are 0.34, 0.20, 0.11 for the
3.175mm-core, 6.35mm-core, 12.7mm-core panels respectively. Therefore
the assumption that only the core contributes to the transverse stiffness of
the panel is less valid for the thinner panels.
The energy required to create impact damage comes from the kinetic
energy of the impactor. Therefore the rebound velocity of the impactor
provides an approximate measure of the extent of the damage suffered by
-110-
3.175mm-CORE PANEL
V- e.876 m/s
0
S
3.175mm-CORE PANEL
V- 1.277 m/s
experiment
e
,/V_
r
xpe iment
analysis
analysis
1
·
0.6
5.0
·
T
15.0
19.6
I
26.6
6.9
5.0
10.0
15.0
20.0
15.0
20.0
Time [ms]
Time [me]
3.175mm-CORE PANEL
3.175mm-CORE PANEL
V- 1.875 m/s
V- 2.353 m/s
experiment
experiment
0.6
5.0
analysis
A.
10.6
15.6
20.0
Time [me]
0.0
5.0
18.6
Time [ms]
Figure 4.17a Experimental and analytical F-T history for
3.175mm-core I
3.175mm-CORE PANEL
V- 2.791 m/s
3.175mm-CORE PANEL
V- 3.333 m/s
experiment
e
experiment
ý analysis
analysis
0
6.0
5.0
16.6
Time [ms]
15.0
6.6
26.6
5.0
10.0
Time [ms]
Figure 4.17a Experimental and analytical F-T history for
3.175mm-core panels
-112-
15.0
20.0
6.35mm-CORE PANEL
6.35mm-CORE PANEL
V- 0.863 m/s
V- 1.364 m/s
experiment
A
analysis
experiment
,
analysis
0.0
10.0
5.0
15.8
20.8
0.0
5.0
10.0
Time [ms]
Time [me]
6.35mm-CORE PANEL
V- 1.935 m/s
experiment
A' analysis
0.0
5.0
15.0
10.0
29.0
Time [me]
Figure 4.17b Experimental and analytical F-T history for
6.35mm-core panels
-113-
15.0
20.0
12.7mm-CORE PANEL
12.7mm-CORE PANEL
V- 1.143 m/s
V- 1.667 m/s
experiment
1
\ analysis
iment
\\analysis
6.e
5.6
15.6
16.0
20.0
0.9
5.0
10.0
15.0
20.0
15.0
20.0
Time [ms]
Time [me]
12.7m-CORE PANEL
V- 2.553 m/s
12.7mwe-CORE PANEL
V- 2.143 m/s
analysis
analysis
ex eriment
6.6
5.6
15.0
16.0
26.0
0.6
Time [mes]
Figure 4.17c
5.0
18.6
Time [ms]
Experimental and analytical F-T history for
12.7mm-core panels
-114-
12.7mm-CORE PANEL
12.7mm-CORE PANEL
V- 2.791 m/s
SV 3.429 m/s
analysis
analysis
iriment
0.0
15.0
10.0
5.0
0.0
20.0
5.0
10.0
Time [ms]
Time [ma]
12.7mm-CORE PANEL
V, 3.871 mr/s
analysis
I*ment
0.0
5.0
15.0
10.0
20.8
Time [me]
Figure 4.17c Experimental and analytical F-T history for
12.7mm-core panels
115-
15.0
20.0
6.35mm-CORE PANEL
3.175mm-CORE PANEL
0NF)
O
NY
.a CC
O,
0O
'I
S11xp1rim•nt
G
experiment
G experiment
0.
1.
2.
3.
4.
0.
1.
2.
12.7mm-CORE PANEL
0
4.
F)
C
analysis
experiment
CD
0.
1.
2.
3.
3.
Impact velocity [m/s]
Impact velocity [m/S]
4.
Impact velocity [m/8]
Figure 4.18 Experimental and analytical rebound velocities
-116-
4.
the panel. Fig.4.18 shows the rebound velocity measured in the experiment
compared with that calculated by the global model. The analytical results,
which do not take damage into account, monotonically increase while the
experimental curve show a decrease in rebound velocity, indicating
damage is created in the panel. It is also noted from Fig.4.18 that the
damage occurs at a lower impact velocity for the thicker core panels. This
agrees with the F-T history shown in Fig.4.17 where the occurrance of
damage is indicated by a sharp load drop.
4.4 Damage Inspection
The evolution of damage sustained by the panel can be divided into 3
stages, as the impact velocity increases: (i) core crushing near the top
interface between the front facesheet (i.e. the impacted side) and the core;
(ii) delamination of the interface between the 5th and the 6th plies (+45' and
-450) of the front facesheet (hereon referred to as the 5-6 delamination); (iii)
visible surface fiber damage. In the first stage, the region of crushed core
increases both in area and in depth (Fig.4.19) as the impact velocity
increases. This region of crushed core constitutes a void between the
facesheet, which has sprung back to its flat position and the core which
remains crushed after the impact. In the second stage, the 5-6
delamination area is elliptical with the major axis aligned with the +45'
direction (Fig.4.20). As the impact velocity increases the 5-6 delamination
area increases and additional delaminations start to appear between other
plies in the front facesheet. However, the 5-6 delamination always remains
as the largest of all the delaminations and, hence, only the 5-6 delamination
can be identified in the X-ray photographs. In the third stage, visible
surface fiber damage is accompanied by the "caving in" of the front
-117-
impact
front
facesheet
core
back
facesheet
region of
crushed core
expands as impact velocity
increases
Figure 4.19
Core crushing
7-·
J
-li·L·~·IL~LI~AI·
~c
Figure 4.20 X-ray photograph of an impact-damaged panel
118-
facesheet, extensive delaminations and matrix cracks throught out thefront
facesheet, and significant core crushing (cavity). No damage is observed in
the back facesheet for all cases.
Due to the lack of precise control of the impact velocity, it is
impossible to compare the initiaton of the aforementioned damage stages
for the 3 core thicknesses except for stage (iii). Visible surface fiber damage
is first observed at 2.79 m/s for the 12.7mm-core panel, 3.15 m/s for the
6.35mm-core panel, and 3.33 m/s for the 3.175mm-core panel. The major
axis (Appendix G) of the 5-6 delamination area (delamination length) was
measured from the X-ray photographs and plotted against the impact
velocity (Fig.4.21). Figure 4.21 shows that the thicker core panels suffer
larger 5-6 delamination than the thinner core panels at comparable impact
velocities. The figure also shows that the 5-6 delamination length tends to
level off at about 40 mm as impact velocity increases beyond 3 m/s. The
depth of the region of crushed core provides another measure of impact
damage of the panels. Fig.4.22 shows that the thicker core panels suffer
more severe core damage than the thinner core panels at comparable
velocities. However, it should be pointed out that the result of core crushing
shown in Fig.4.22 is not as accurate as that of the delamination length
shown in Fig.4.21. It is because the measured depth of crushed core
depends on where the panel is sectioned. The maximum depth should be
obtained if the panel is sectioned exactly at the major axis of the 5-6
delamination. Practically, it is very difficult to achieve this precise cutting
because the 5-6 delamination cannot be seen with naked eyes.
-119-
Facesheet Damage
50
0 3.175mm-core
+
+
40 *
6.35mm-core
+
12.7mm-core
*
+
+
*
0
*3
30 -
20
13
10
a
-,.
I
:66.
.I
Impact velocity [m/s]
Figure 4.21 Delamination length vs. impact velocity
-120-
Core Damage
0
2
1
3
Impact velocity [m/s]
Figure 4.22 Depth of crushed core vs. impact velocity
-121-
4
Chapter V
Conclusions and Recommendations
5.1 Conclusions
5.1.1 Global Model
The goal of the global model is to find the force-time (F-T) history and
the acceleration-time (A-T) history at the point of impact. The following
conclusions are drawn from the analytical results shown in Chapter IV:
1.
Anti-symmetric modes can be neglected in the global analysis when
the panel is impacted at the center, due to D16 and D26 - 0.
2.
The F-T history is practically converged at 7x7 modes while the A-T
history requires a much larger number of modes to get the same
degree of convergence.
3.
The convergence of the F-T history and the A-T history depends on
the contact stiffness: the higher the contact stiffness is, the slower is
the convergence.
4.
The A-T history is more sensitive to the variation of integration time
step than the F-T history.
5.
Assuming a linear contact spring, the peak force is linearly
proportional to the impact velocity.
6.
Thicker core panels experience higher peak force than thinner
panels at the same impact velocity.
-122-
5.1.2 Local Model
The local model is used to find the contact stiffness of the panel and
the localized stress and strain fields in the panel due to static indentation
loading or impact loading. The following conclusions are drawn from the
analytical results shown in Chapter IV:
1.
The convergence of the load vs. indentation (F vs. a) curve depends on
the value of the radius of the local region, Rp: the larger Rp is, the
slower is the convergence.
2.
The converged F vs. a curve for different values of Rp are practically
the same providing that "plate solution" is recovered at the boundary
of the local region.
3.
Thicker core panels have lower contact stiffness than thinner core
panels.
4.
The stress distributions in r show an intense and localized stress
field developed near the center of the panel (i.e. the point of impact or
the point of indentation). The magnitudes of the stresses decay very
rapidly away from the center.
5.
The strain and stress distributions in z show that the aforementioned
high localized stresses or strains are "diffused" away as they are
transmitted through the core.
6.
Stresses and strains due to the bending moment correction are
negligible compared with those due to the impact or the indentation
load.
7.
Inertia loading is unimportant for the impact velocity range
considered (< 4 m/s).
-123-
5.1.3 Experiments
The purpose of the static indentation tests is to verify the analytical F
vs. a curve obtained by the local model. The impact tests are used to verify
the analytical F-T history of the global model.
1.
Static indentation test results show that thicker core panels fail at
higher load than thinner core panels.
2.
The experimental F vs. a curve shows little difference among
different core thicknesses.
3.
The impact test results show that the front facesheet of thicker core
panels fails at lower impact velocity than that of thinner core panels
(This may sound contradictory to the first point above, but apparently
occurs because at the same impact velocity, the thicker core panel
experiences a higher peak load than the thinner core panel.)
4.
Damage inspection results shows the 5-6 delamination area ( the only
delamination that can be measured from the X-ray photographs) is
larger for thicker core panels than for thinner core panels at
comparable impact velocities.
5.
It is possible to have core damage (core crushing) without visible
facesheet damage at low impact velocities (< 2 m/s).
5.1.4 Comparison between Experiments and Analysis
1.
The analytical F-T history from the global model remains in good
agreement with the experimental F-T history as the impact velocity
increases until damage occurs.
-124-
2.
The thicker core panels show better agreement of the F-T history
than the thinner core panels when no significant damage is present
(i.e. a smooth half-sinusoidal F-T history).
3.
The local model fails to predict well the experimental F vs. a curve
because the compressive strength of the Rohacell core is exceeded
(core crushing) at very low load levels.
5.2 Recommendations
1.
Material non-linearity must be included in the local model to account
for the core crushing.
2.
A more accurate method, which takes into account the shear
property of the facesheets, could be used to calculate the transverse
shear property of the panel .
3.
Some static indentation tests should be run up to the peak load
indicated by the experimental F-T history. The statically damaged
panel should then be compared with the dynamically damaged panel
to verify that the inertia loading is unimportant for the impact
velocity range considered.
4.
Different impactor
or indentor
geometry
should
be used
experimentally to test the validity of the local model.
5.
Different layup of the facesheets should be used to test the validity of
the axisymmetric assumption in the local model.
6.
The modelling of the inertia loading in the local analysis could take into
account the variation of the acceleration in the plane of the panel and
through the thickness of the panel.
-125-
References
1.
t' Hart, W.G.J., "The Effect of Impact Damage on the TensionCompression Fatigue Properties of Sandwich Panels with Face
Sheets
of Carbon/Epoxy",
National
Aerospace
Laboratory,
Amsterdam (Netherlands), (December 1981).
2.
Koller, M.G., "Elastic Impact of Spheres on Sandwich PLates",
Journal of Applied Mathematics and Physics (ZAMP), Vol.37,
(March 1986).
3.
Van Veggel, L.H., "Impact and Damage Tolerance Properties of
CFRP Sandwich Panels - An Experimental Parameter Study for the
Fokker 100 CA - EP Flap", New Materials and Fatigue Resistant
Aircraft Design, The Proceedings of the 14th Symposium of the
ICAF, (June 1987).
4.
Bernard, M.L., "Impact Resistance and Damage Tolerance of
Composite Sandwich Plates", S.M. Thesis, Dept. of Aeronautics and
Astronautics, M.I.T., (May 1987).
5.
Lie, S.C., "Damage Resistance and Damage Tolerance of Thin
Composite Facesheet Honeycomb Panels", S.M. Thesis, Dept. of
Aeronautics and Astronautics, M.I.T., (March 1989).
6.
Cairns, D.S., "Impact and Post-Impact Response of Graphite/Epoxy
and Kevlar/Epoxy Structures", Ph.D. Thesis, Dept. of Aeronautics
and Astronautics, M.I.T., (August 1987).
7.
Bathe, K.-J., Numerical Methods in Finite Element Analysis,
Prentice-Hall, Inc. (1982).
8.
Lekhnitskii, S.G., Theory of Elasticity of an Anisotropic Body,
English translation, MIR Publishers, Moscow (1981).
-126-
9.
Timoshenko, S.P., Goodier, J.N., Theory of Elasticity, 3rd ed.,
McGraw-Hill Book Company (1987).
10.
Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.,
Numerical Recipes - The Art of Scientific Computing, Cambridge
University Press (1986).
11.
Minquet, P.J., "Buckling of Graphite/Epoxy Sandwich Plates", S.M.
Thesis, Dept. of Aeronautics and Astronautics, M.I.T., (May 1986).
12.
Lagace, P.A., Brewer, J.C. & Varnerin, C.F., Telac Manufacturing
Course Notes, Technology Laboratory for Advanced Composites,
TELAC Report 88-4, M.I.T., (1988).
13.
Tan, T.M. and Sun, C.T., "Use of Statical Indentation Laws for the
Impact of Composite Plates", Journal of Applied Mechanics, Vol.52,
(March 1985).
14.
Dugundji, J., "Simple Expression for Higher Vibration Modes of
Uniform Euler Beams", AIAA Journal, Vol.26, No.8 (August 1988).
-127-
Appendix A
Generalized Beam Functions (GBF's)
Dugundji [14] derives approximate beam functions for various
boundary conditions. Although these GBF's are approximations to the
traditional beam functions, the difference between the two becomes neglible
when the mode number is bigger than 2. The GBF's has the advantages
that they can be written in one single parametric form and that they and
their products can be integrated or differentiated analytically.
The GBF's are written in the form,
0n(x)=
2 sin(3n x + ) + Ae -
where the constants or shape parameters
nx
+ Be-On(1-x)
3n
,
(A.1)
0, A, and B are given in Table
A.1 for some common boundary conditions and x is the normalized
coordinate along the beam (O<x<l). For the present analysis, the CL-CL
boundary conditions were chosen.
The beam functions are sometimes catergorized into two classes:
symmetric modes and anti-symmetric modes. When expressed in the form
of Eq.A.1, a symmetric mode is symmetric about x = 1/2 while an antisymmetric mode is anti-symmetric about x = 1/2. It can easily be verified
that symmetric modes are given by n being odd and anti-symmetric modes
by n being even in Table A. 1.
-128-
Table A.1 Shape parameters for the GBF's
boundary condition
Pn
0
A
B
SS-SS
nic
0
0
0
CL-FR
[n-(1/2)]7
- 7r/4
1
(- 1 )n+1
CL-CL
[n+(1/2)]r
-
x/4
1
(- 1 )n+l
FR-FR
[n+(1/2)]x
37/4
1
(-1)n+ l
SS-CL
[n+(1/4)]7
0
0
(, 1 )n+1
SS-FR
[n+(1/4)]7r
0
0
(-1 )n
-129-
Appendix B
Elements of the Mass Matrix and the Stiffness Matrix
The following equations are rederived after Cairns' work (which
contains some typographical mistakes):
Stiffness Matrix
D
K aa(i, ) =iJ
igi)(
gj)
+ D 16( figi)ff j
+ G.(f ig
K
(i,j)=
ab
(B.1)
+D
16 (f
' gi)(
lj)
hjl + .(fi gi)(
II + D .(fig'
f
+ D 6(fgi i)(f j g ) dxdy
:
)(hjl
D,(f
+ D 16 (fig )(figi
dx dy
\(
+ G45figih
K ac(ij ) =
(B.2)
'I[
G45(figi)(mj
n
D 2 h.il')(hjl'.)
K (i,j)=
II+ D
+ G44(hl
+G55(fig im'nj)] dx dy
+ D 6( hil
i
h 1'i) + D
66(
fJ
h'l
.)
33J
h'iI)(h' j)
dx dy
.h.1.)
(B.4)
K b(i,j) = IIG 45(h.l i)(m'.nj) + G
Kcc ( i j, ) =
(B.3)
m n'j ldxdy
G J4m in') (m jn'.i
n' + G45(min')(
SG45(m'.n'.)m. )
\-.AJ
in.j)
(
j)+ G55(m'ini)
+5
-130-
(R
r%
(B5)'
J i/
J -
/
dx dy
(B.6)
Mass Matrix
Ix(ij)= I (figi
fgj)dx dy
I y(i,j)= IJJ(hil ihjij)dxdy
M(i,j) =PJ(m ni.)(m.nj)dxdy
(B.7)
(B.8)
(B.9)
where I, P, fi' gi' hi) li, mi, ni are defined in Chapter II (Eq.2.4) and in
Appendix I.
Generalized Force Vector
ai
bi
ci.
0
0
mi
c)nil c)
(B.10)
where,
(c ' Tlic) - normalized coordinates of the point of impact.
-131-
Appendix C
Displacements u and w of the Local Model
Before the expressions for u and w are derived, the following relationships
are introduced,
-
13 (1
+b i)
-a
12a-a
+
a 33ci = a44
(C.la)
1 1a.+a13 d i =0
(C.lb)
- a12 - b.ial + c a 13 = 0
a
(C. c)
a. + a 11 + a 12 b i -c .a
44 1
11
1 13
= - 2a 13a. 131
+ a 33 d33d
i
1
(C.1d)
Eq.C.1 can easily be verified by substituting in the expressions for ai , b i , ci ,
and di from Eq.2.24. The amn's refer to the appropriate component, i, of the
sandwich panel.
u is obtained as follows:
u.(r,z.) = re 00 i
= ra12C rri + a 11%00i + a 13 Zz).
12rn
a 12
=rz
I2.a
11 (b
b
Ji,rr+
S-
r 1 i,r
+ai. .
ai izz
1
i ,rr
c.
+ a13 Ci i,rr
r
i,r+d
izz
(C.2)
Using Eq.C.lb and Eq.C.lc,
U
(r ,z
i
) = (- a12b i - a 11
+
Alternatively,
-132-
c ia 13) 0 i,rz
(C.3)
ui(r,z i ) = f
=
.dr + g(z)
a
J---(a
zzi)dr
rr i + a 12 G00 i + al3c
1
+
g(z)
b.
- all
a
-a
= J
ir.
(b.
12
1
+ a4i. izz
r
. ,rr
O
+
ir a
C.o
izz
dr+ g(z)
c
Sa
(c i i,rr
+ d i. izz
+r
r
+ g(z)
= (- 1all
- bia12
+ c ia13)i,rz
11
1 12
13
irz
(C.4)
Comparing Eq.C.3 and Eq.C.4 gives,
g(z) - 0
(C.5)
Hence,
u.(r,z.) = (- a 1 1 -b.a
12
+
cia 13 )0 i,rz
(C.6)
w is obtained as follows:
+gi(r)
w.i(r,zi)== Je ZZ .dz.
1
= J(a1(~r. +a0)
a 33
+a a
.z)dz.+g.(r)
(13+bi) ( i ,rr
aI- a
a 33
i
i,rr
+ Ir i
+ 2a.i]o].
dz i +gi(r)
+
Oi,r+ di
r
iZZ
1
(C.7)
Using Eq.C.la,
w.(r,z.)=
44(
a
+rr.
1 i,r )+ (w1~ z)= 44
¢i,rr
2a 13 a.i +a
33 di)
i,zz + gi(r)
(C.8)
-133-
Alternatively,
wi(r,zi) =w.i,rdr+ gi(z i )
(
uiu.
=
J
i dr + gi(z i )
Yrz
rzi
- z Ui
r +g(zi
= a44 zi- au
a
44
447
f- z
b. irr + +a
(-
11
1
(
+ a.
i ,zz
dr +i(zi
-b.a
+c.a13)
1 1 13
1 12
+
b ia 12
)
r
i ,zz
r-O i, r
a 44(o i,rr
+(all +
i,r
)
ci a 13 )
o i ,z z
i(i)
(C.9)
Using Eq.C.ld,
a(1
w.(r,z.)=
wi
a
o J44
,rr + ri
J
+(- 2a ial3 + d ia 33)
i ,zz
+
gi(i)
(C.10)
Comparing Eq.C.8 and Eq.C.10 gives,
gi(r)= gi(zi ) = Ki
(C.11)
where Ki is a constant. Hence,
w(rz)=a(
S44=
,rr
1.)+,ri
r
+
(-2a 13 a.+ad.)
4.1,zz +K. 1
i
33
(C.12)
Since we are only interested in the indentation of the panel, which is the
relative displacement between the top and the bottom surfaces of the panel,
Ki can be discarded.
-134-
Appendix D
Bending Moment Correction
The global model can be used to calculate the bending moment at the
boundary of the local region. The same energy method as that described in
Section 2.2 is used to find the governing equation for a point load applied at
the center of the panel. The equation can be written as,
K
aa
K•
K
ab
K
K
ac
K
ab
bb
T
T
ac
be
Rai
A
T i
B
R
bc
bi
C
Rci
(D.la)
cc i
which is the same as Eq.2.9 without the "acceleration term" because Eq.D.1
is a static load case. Hence, the modal amplitudes are given by,
..
A.
B
K
C1
..
K K
aa
T
ab
KT
Sac
K
..-
ab
bb
KT
bc
K
K
-1
-1]R
ac
R
bc
ai
bi
R
K
(ci
(D~lb)
CC
Once the modal amplitudes are found, Kx , Ky, and Kxy can be calculated as
functions of x and y using Eq.2.1a and Eq.2.7. The bending moments in
polar coordinates are found by first obtaining Mx , My, and Mxy in
rectangular coordinates, and then rotating to get Mrr, M6o, and MrO
,
Mx
M
Mxy
=
11
D12
12
D22
16
D26
K
D16
D26
D66
Kxy -
-135-
x
(D.2a)
Mrr
[
cos 20
M ore
=
sin 2
sin 2o
- cos 6 sin 0
2cos sin0
-2cos0sin
cos 0 sin
cos 20 - sin 2 0
Mx
My
Mxy
(D.2b)
Mrr is calculated at 0 = 00, 100, ... , 360' and then the average is used as the
bending moment correction.
-136-
Appendix E
Inertia Loading
The equations of equilibrium for the local model are [8] (the subscript
"i" in Section 2.3 is omitted in the present discussion),
ayrr
+
rZ
+r
rz
rr -
+
O00
rrr
ZZ
rz
z
r
(E.la)
(E.lb)
The stress components given by Eq.2.26 satisfy Eq.E.la while Eq.E.1b gives
rise to the governing equation of the problem (Eq.2.24a). If inertia loading is
included as a body force. Eq.E.1b becomes,
rz
Dr
+
zz
rz
z
r
(E.2a)
where,
y = pw
(E.2b)
Correspondingly, azz is given by,
S
-z
zz
c a- +
ar 2
r ar
+d
+z
az2
J
(E.2c)
so that the stress components still satisfy Eq.E.la while Eq.E.2a leads to
Eq.2.24a as before.
As a result of the above change in ozz, the displacements u and w
will be different. The following derivation of u and w, which includes the
inertia loading, should be compared with that in Appendix C.
-137-
u is obtained as follows:
u.i(r,z
i)=
re0
i
= (a12
rri
a 1100i + a13a
13
11 Oi
rn
i)
zzi
b
a12 (i,rr+
=r
1a
1(b iO
azi
+ a 13 ( i
+ a.i i ,zz
r
1+i,r
,r+a
ai)
i,rr
T
r +di
irr
1,T
r
1
+
i,zz
13
i
z
1,Z
(E.3a)
Using Eq.C.lb and Eq.C.lc,
u.i(r,zi
= (- a 12 b -a 11 +cia 13 ) i rz +ra13iz
)
(E.3b)
Alternatively,
u.i(r,z i ) =
rri dr + g(z)
= I(all rri + a12 G00i
{
-a
13
b.zz
zzi) dr
+
g(z)
b
irr +
11
+ a i i.zz)
r
- a 12(b
-- Dz.i
+a
Ci
+ a13
= (- all - b ia
12 +
i,rr + 1,ir
T
C
1i,rr
r
C ia 13 )
+ aii
z
i
i1,izz)
+yzi a 131 dr+
g(z)
ir
IZz)
i
+ry.z.a
+ g(z)
(E.3c)
Comparing Eq.E.3b and Eq.E.3c gives,
g(z) = 0
(E.3d)
Hence,
-138-
ui(r,zi) = (- a 12b i - a 11+ c ia 13) 0 i,rz + ra l3i.z.
13 1 1
(E.3e)
w is obtained as follows:
w.i(r,zi) = E
.dz. + g.(r)
=f
zzi)dz i + gi( r)
+ a 33
ri +
= (a 13(
ia a13[(1 ++bi)(
a33[c
r
i,rr
O
i,r)
+ 2ai4 i,zz.
+ a33y.iz.dzi + gi(r)
c.
+dio i
io i,rr
zz]
(E.4a)
Using Eq.C.la,
2
a
wi(r,zi)= a 4
i,
rI0,r)+ (-
+
2a 13a
13 i1 +
a 33 d.)
0
i,zz
a 33Yi
2
z
+ gi(r)
(E.4b)
Alternatively,
w.(r,zi)=
Jw i,rdr +
gi ( z
i)
u.i)
7
=i rzi
=
z1
a 4 4 7rzi
-
jdr
+g.(z.)
i
auir
-
.- a
,r + a i. i,z
-bia2
1
)
+gi(z1
·
+
a44 -aIrr
r
=
1
+ cia 13)
z)
rz
-i
- ry .a
=a440 i,rr
)
i,rz dr + gi(zi
1+ .
+---
+a.i.
i ,zz
,r
r 2 y.a 13
+ (all+ bi
a 12 -
ci
a 13 )
-139-
,z -
2
+ g(z.
)
(E.4c)
Using Eq.C.ld,
w.(r,z i ) =a44
,
r
+ (- 2ai
+
r2y.a
1 13
ri
,r
a 13 +
di a
2
33)
S1 I )
Oi,zz +gi(z
(E.4d)
Comparing Eq.E.4b and Eq.E.4d gives,
2
gi(r)=-
2
(E.4e)
gi(r)=- y.iz2r2 a
-
1 33
)1
(E.4d)
Hence,
(z)
=
+ a 33 di) i,zz
+ rOi, r + (- 2a 13a.
13 1
(
+
2 -
a
z2
33 i
a1 r2
-140-
13
(E.4e)
Appendix F
Impulsive Force on a Rigid Wall
Consider a ball of mass m hitting a semi-infinite wall with velocity
V 0 . The contact stiffness of the wall is modelled by a contact spring with a
spring constant k as shown in Fig.F.1. The equation of motion of the ball is,
mil = - F
(F.1)
where u is the displacement of the ball, and F is the contact force between
the ball and the wall.
Assuming a linear contact spring, the constitutive equation of the
spring is given by,
F = ku
(F.2)
Combining Eq. F.1 and Eq. F.2 gives,
mii + ku = 0
(F.3)
The solution of Eq. F.3 is of the form,
u = A sinot + Bcos ot
(F.4)
where (2 = m
At t=0,
u=0
=:
B=0
V
iu=V
V
A=
0
-141-
o0
0
(F.5)
Hence,
V
U
=
0 sin
cot
(F.6)
(F.6)
kmV 0 sin cot
(F.7)
F =ku
=
Using the values (same as those used for the parametric study in Section
4.1),
k = 0.5 MNm
m = 1.543 kg
-1
VVo=3m
=3ms - 1
(F.8)
F = 2627 .4 sin 570.9 t
(F.9)
gives,
The F-T history given by Eq.F.9 is included in Fig.4.3a.
It should also be noted that for this linear model, the force developed,
F, and the impact duration, Tp, can be expressed as,
F=
2kE
Tp
c0 -
b
sin ct
t
(F.7a)
/m
k
(F.10)
where Eb is the kinetic energy of the ball, mV 0 2 / 2. Also, if the wall is nonrigid, i.e. replaced by an appropriate spring-mass system, the maximum
force developed decreases and the pulse duration increases, as shown in
Fig.4.3a.
-142-
m
rigid wall
Figure F.1 Impact on a rigid wall
-143-
Appendix G
Interpretation of the X-ray Photograph of an Impact-Damaged Panel
A typical X-ray photograph of a damaged panel shows three regions
of interest (Fig.G.1): (i) a black patch at the point of impact; (ii) a peanut
shaped light coloured region; and (iii) a dark coloured elliptical boundary
surrounding region (ii). Examination of the cross-sections of the damaged
panels identifies these 3 regions as: (i) fiber damage and matrix cracks; (ii)
"large" delaminations which appear as highly visible gaps between the 5th
and the 6th plies; and (iii) "small" delaminations which appear as barely
visible slits between the 5th and the 6th plies. At high impact velocity when
significant damage occurs in the front facesheet, a large black patch may
appear at the point of impact. Sometimes this black patch is so big that it
overshadows the delamination region (Fig.G.2). This is because large
amount of dye was absorbed by the core.
The difference in contrast between region (ii) and region (iii) is
caused by the capillary action which sucks the opaque dye from region (ii) to
region (iii). The major axis (delam. length in Fig.4.21) is measured as the
distance between the end points of region (iii).
-144-
)n (ii)
region (iii)
0001
"large"
delamina
B B
B•
"I
delamination
7.
,i
-,~
cvl
i!
i
i
1.
--
1
.....
-
·.
,~-·i~.·I-
Figure G.1
;·`·
,.
J ··
9;s~s!r,:..-:.
LLI-~~uu~o~arl,
--
4rr.
(r
LI
II
,ij
r,
X-ray photograph of a typical impact-damaged panel
-145-
.7
Figure G.2
X-ray photograph of an impact-damaged panel
with extensive facesheet damage
-146-