A Study of the Effect of Delamination Size on the Critical Sublaminate
Buckling Load in a Composite Plate Using the Ritz Method
By
Christopher Klobedanz
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING
Major Subject: MECHANICAL ENGINEERING
Approved:
_________________________________________
Professor Ernesto Gutierrez-Miravete, Project Advisor
Rensselaer Polytechnic Institute
Hartford, Connecticut
December, 2014
© Copyright 2014
by
Christopher Klobedanz
All Rights Reserved
ii
CONTENTS
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
LIST OF SYMBOLS ....................................................................................................... vii
LIST OF KEYWORDS .................................................................................................. viii
ACKNOWLEDGMENT................................................................................................... ix
ABSTRACT ....................................................................................................................... x
1. Introduction .................................................................................................................. 1
2. Background .................................................................................................................. 3
3. Theory/Methodology ................................................................................................... 5
3.1
Analytical-Numerical Method............................................................................ 5
3.1.1
Assumptions ........................................................................................... 5
3.1.2
Classical Laminate Plate Behavior ......................................................... 6
3.1.3
Sublaminate Behavior ............................................................................ 9
3.1.4
Potential Energy Minimization ............................................................ 12
3.1.5
Newton-Raphson Method .................................................................... 13
3.1.6
Evaluating the Critical Buckling Load ................................................. 14
4. Results/Discussion ..................................................................................................... 15
4.1
4.2
Analytical Results ............................................................................................ 15
4.1.1
Inputs .................................................................................................... 15
4.1.2
Load-Strain Maps ................................................................................. 16
4.1.3
Buckling Load vs. Delamination Size .................................................. 17
Validation of Maple Code ................................................................................ 18
4.2.1
Inputs .................................................................................................... 19
4.2.2
Comparison of Analysis Tools ............................................................. 19
5. Conclusions ................................................................................................................ 21
iii
REFERENCES ................................................................................................................ 22
Appendix A: Maple Code .............................................................................................. A-1
Appendix B: Reference (61) Validation - Maple Code ................................................. B-1
Appendix C: Plate Displacement Equation Derivation ................................................. C-1
Appendix D: Sublaminate Displacement Equation Derivation ..................................... D-1
Appendix E: Load-Strain Chart Plotpoint Data ............................................................. E-1
iv
LIST OF TABLES
Table 1: Laminate Material Properties ......................................................................................... 15
Table 2: Buckling Load vs. Delamination Size ............................................................................ 18
Table 3: Validation Input Additional Properties ........................................................................... 19
Table 4: Validation Output Results............................................................................................... 20
Table 5: 0.5" Delamination - Critical Buckling Load ..................................................................E-1
Table 6: 0.6" Delamination - Critical Buckling Load ..................................................................E-1
Table 7: 0.7" Delamination - Critical Buckling Load ..................................................................E-2
Table 8: 0.8” Delamination – Critical Buckling Load .................................................................E-2
Table 9: 0.9" Delamination - Critical Buckling Load ..................................................................E-3
Table 10: 1.0" Delamination - Critical Buckling Load ................................................................E-3
v
LIST OF FIGURES
Figure 1: Laminate Composite Structure [2] .................................................................................. 1
Figure 2: Delaminated Composite Loading Condition [1] ............................................................. 2
Figure 3: Buckling Modes [8] ......................................................................................................... 6
Figure 4: Material and Composite Axis Orientations [19] ............................................................. 7
Figure 5: Ply Transverse Dimension Number System [19] ............................................................ 8
Figure 6: Sublaminate Boundary Conditions [1] .......................................................................... 11
Figure 7: Plate Geometry .............................................................................................................. 15
Figure 8: Load vs. Strain Curves as a Function of Delamination Size ......................................... 17
Figure 9: Buckling Load vs. Delamination Size Chart ................................................................. 18
Figure 10: Plate Geometry (Validation Input) .............................................................................. 19
Figure 11: Validation Output Results ........................................................................................... 20
vi
LIST OF SYMBOLS
Axes
Symbol
Variable
x,y,z
Material Coordinate System
x1,x2,x3
𝑥̇ 1, 𝑥̇ 2
Variables (continued)
Symbol
Variable
Unit
a
Ellipse Major Semi-Axis
in
Composite Plate Coordinate System
b
Ellipse Minor Semi-Axis
in
Adjusted Plate Coordinate System
A
Sublaminate Area
in2
E
Modulus of Elasticity
psi
ν
Poisson’s Ratio
-
G
Shear Modulus
psi
Variable Subscripts
Symbol
Variable
sl
Pertaining to the Sublaminate
pl
Pertaining to the Composite Plate
tot
Pertaining to the Composite
mid
Pertaining to the Mid-Plane
123
Pertaining to the Full 1-6 Matrix
126
Pertaining to In-Plane Matrix Components
α
45 Pertaining to Transverse Matrix Components
Thermal Expansion Coefficientin/(in*°F)
Q
Stiffness Component
psi
ϴ
Fiber Angle
°
c
cos(ϴ)
-
s
sin(ϴ)
-
ε
Strain
in/in
γ
Shear Strain
rad
C1,C2
Constant Terms
-
k
Composite Layer
u
Displacement
in
i,j
Unspecified Index
σ
Stress
psi
n
Nth Number
τ
Shear Stress
psi
κ
Curvatures
1/in
Variables
upoly Polynomial Displacement Function
-
Symbol
Variable
Unit
ψpoly
Polynomial Rotation Function
-
Ni
In-Plane Load
lbf/in
pci
Polynomial Coefficients
-
K
Foundation Modulus
lbf/in3
z
Ply Transverse Coordinate
in
ΔT
Temperature Change
°F
ΔP
Transverse Pressure Load
psi
N
Ply Number (Top to Bottom)
-
in
ψ
Rotation Angle
°
h
Thickness
A,B,D,E,F,HGlobal Stiffness Matrices lb/in,lb,lb*in,...
lcomp
Composite Plate Length
in
ζ,ϕ
Higher Order Rotation Angles
°
wcomp
Composite Plate Width
in
Π
Total Potential Energy
in*lb
rdel
Circular Delamination Radius
in
eqi
Energy Minimization Equation
-
vii
LIST OF KEYWORDS

Composite

Delamination

Sublaminate

Buckling

Ritz

Galerkin

Newton-Raphson
viii
ACKNOWLEDGMENT
A lot of work went into completing this project and even more went into getting through
the graduate program as a whole. None of it could have been accomplished without the support
of my friends and family, the help from my professors, and the patience and guidance of my
advisor, Ernesto.
The strongest recognition goes to my fiancé, Megan. She sacrificed right along with me
when I had deadlines to meet and had no time for anything else. She kept me fed when I was
consumed with work, she always offered encouragement, and she was the light at the end of
every semester that kept me motivated and sane.
My parents, who were especially influential to my success throughout the graduate
program, have always been supportive and proud of my work. Their confidence in me has
shaped the expectations I hold myself to and the lessons they taught me gave me the tools I
needed to accomplish this goal.
Every semester, the Rensselaer staff has been able and willing to help whenever called
upon. Each class has pushed forward my education and further prepared me for future challenges
in my career.
Finally, I would not have had the means to start working towards this degree so soon
without the support of Electric Boat.
Thank you all!
ix
ABSTRACT
This project analyzes the effect of delamination size on the localized critical buckling
load of a partially delaminated composite plate sublaminate under uniform, uniaxial
compression. The plate being analyzed is a square, 24-layered, symmetric graphite/epoxy
laminate composite of uniform thickness, with a centrally-located circular delamination between
the fourth and fifth layers. A model, based on the code outlined in Reference (1), was built in the
symbolic computation program Maple to conduct this analysis.
The model first applies Classical Laminate Plate Theory to define the composite
behavioral response to compression. It then applies the Ritz variational method to minimize the
total potential energy of the system and create an expression from which the critical buckling
load of the composite sublaminate can be predicted. The potential energy term is initially
expressed in terms of known geometric dimensions and material properties, and unknown
polynomial coefficients. The Newton-Raphson method solves for the values of the polynomial
coefficients, which can then be substituted back into the sublaminate stress, strain, and
displacement equations to fully describe the sublaminate behavior. By conducting this analysis at
various loads, a load-strain plot illustrates the critical buckling load as a maximum point in the
resulting curve. This process was repeated for delaminations of various diameters to understand
the relationship between delamination size and the compressive-tolerance capabilities of
composites.
This analysis was validated by recreating the results of a model created in the Reference
(1) study. After verifying the Maple model, the initial study was able to be expanded to support
the goals of this project. Ultimately the project determined that as the delamination size
increases, the strength of the composite can be severely reduced.
x
1. Introduction
Composite materials have become increasingly important in industrial applications such
as civil engineering, aerospace and aeronautical structures, and a variety of other areas.
Composite materials are more expensive to use than traditional metallic materials, but have the
cost-benefit advantage of exhibiting high strength-to-weight ratios, increased toughness, and
increased corrosion resistance. They also have the advantage of being customizable for their
intended applications.
A standard composite laminate material is composed of individual plies, bound together
to form a multi-layered structure. Each ply is made up of strong fibrous materials, oriented in a
specified direction, and held together by an enveloping matrix material. Figure 1 illustrates a
typical composite laminate with the layout of individual laminate layers oriented at various
angles with respect to the direction of the fiber.
Figure 1: Laminate Composite Structure [2]
The bonding of the laminate layers to one another creates the overall body of the
composite structure. In a composite, the fibers are responsible for the majority of the loadbearing properties, while the matrix distributes the load to the fibers, bonds the composite
together, and adds transverse strength to the structure.
Composites are subject to a wide range of defects capable of causing strength and
stiffness reductions, or even critical failure, of the structure. One of the main modes of failure in
a layered composite is the delamination of adjacent layers [3]. Delamination is the separation of
laminates along their interfacial boundary and can result from several factors including poor
manufacturing processes, point impacts, free edge effects, structural discontinuities, drilling,
moisture and temperature effects, and internal failure mechanisms such as matrix cracking [3].
Delaminations are crucial defects to understand because they lie beneath the surface of the
material and can go unnoticed during inspection, but can cause premature failure under certain
loading conditions.
1
This project analyzes the effect of an interlaminar circular delamination in a clamped,
symmetric, graphite/epoxy composite plate under uniaxial compressive loading. The localized
critical buckling load of the sublaminate, the thin subsection of the composite above the
delamination, is determined analytically and numerically in Maple [4] based on the code outlined
in Reference (1). In this project, it is assumed that once the sublaminate buckles, the strength
contribution of that section of the composite is negligible. Therefore, the critical buckling load of
the sublaminate can be considered a significant failure criterion for the composite as a whole. By
varying the diameter of the initial delamination, the critical buckling loads for multiple
delamination sizes are determined and compared. Figure 2 illustrates the loading condition of the
delaminated composite in this project.
Figure 2: Delaminated Composite Loading Condition [1]
Delaminations have a tendency to propagate during cyclic loading and can potentially
grow unstably until failure [5]. This project did not examine post-buckling growth of
delaminations; only the static buckling failure mode was analyzed.
2
2. Background
As composites have become increasingly useful in current industries, the amount of
research in the field has grown significantly. There are numerous analytical models, numerical
models, simplified models, and specific models regarding composite behavior. However,
because of the complexity of the structure of a composite, there are no all-encompassing theories
that can fully describe the behavioral responses of composites in all loading scenarios. The
complexity of the problem is due to the multitude of variables that are needed to define a given
composite system. Even simple systems have numerous layers, all of which may carry different
material properties or orientations; the defects in a composite can be one or many, can exist
between layers or within a single laminate, can be oriented in any direction, and can be of simple
or complex shapes. Many studies in the field restrict the scope of their problem and document
trends that exist under the controlled conditions of the study. Universal properties are then
extrapolated based on the findings, but are often less accurate once outside the scope of the
study. Some studies which have contributed to knowledge in the field consist of the following:
Reference (6), which analyzes delamination behavior in response to bending, and
Reference (7), which analyzes a delaminated composite plate experiencing torsional forces, are
examples of studies which analyze responses due to various loading and geometric conditions.
References (8), (9), and (10) are examples of studies which model buckling effects of
composite plates using numerical methods; Reference (8) was also able to experimentally
validate the results of the numerical model with correlating data.
Reference (11) analyzed the vibrational response of a Timoshenko beam with an internal
delamination using the Rayleigh-Ritz method.
Composite defects other than just delaminations, are thoroughly discussed in Reference
(2). Liu and Nairn also contributed a variational technique for predicting microcrack (cracks
between fibers in a laminate) propagation based on fracture mechanic methods in Reference (12).
Similar studies exist for interlaminar delamination in References (13), (14), and (15). Many of
these studies also examine post-buckling crack growth.
Through-width delamination models were analyzed in References (16), (17), and (18).
Both References (17) and (18) expanded the scope of the study to account for multiple
delaminations within a single composite.
3
The studies mentioned above only represent a fraction of the research that has been
conducted regarding composite materials. These studies are meant to highlight the variety of
topics that can be analyzed in the field, and the variety of methods with which analysis can be
conducted. Many of the previous studies that analyzed circular delaminations in a composite
plate relied on numerical methods or finite element models to analyze the system. This project
aims to provide a tool which will be able to predict the critical buckling load (and subsequently
assess the strength of the composite as a whole) using simple Classical Laminate Plate Theory
(CLPT) equations. Reference (1) provides a comprehensive methodology for analyzing such a
problem, but draws different conclusions than those sought out by this project (delamination
depth vs. buckling load, delamination orientation vs. buckling load, etc.). The conclusions that
were found in Reference (1) are expanded upon in this project to illustrate the relationship
between delamination size and the critical sublaminate buckling load. The method, and results
are provided below and the Maple code which incorporates the methodology is attached in
Appendix A.
4
3. Theory/Methodology
3.1 Analytical-Numerical Method
The analytical results are primarily derived by following the system of equations outlined
in chapters 3 and 4 of Reference (1). These equations are useful for more complicated systems
than the one being analyzed in this project; for example, they can be applied to systems with
elliptical delaminations in various orientations, systems with various in-plane loading conditions,
they can account for hygrothermal (temperature and moisture) effects, and can take into account
minor contact between the sublaminate and the base plate. Incorporation of the hygrothermal
effects, contact force effects, and elliptical geometry effects into the Maple code was necessary
in order to replicate trials from Reference (1) for validation of the Maple code (see Appendix B),
but those contributions are not reflected in this section nor in the results produced by this project.
The approach used in this project can be broken into three primary steps from which the
critical buckling load of a given composite arrangement is ultimately determined. First, basic
CLPT equations [19] are used to calculate the strains, displacements, and stresses within the
composite as if no delamination was present. Second, the sublaminate strains, displacements, and
stresses are calculated based on fitting polynomial coefficients and functions to represent them.
The polynomial equations are selected based on satisfying boundary conditions at the edge of the
delaminated region. Third, the total potential energy of the system is minimized with respect to
the unknown polynomial coefficients and a system of nonlinear algebraic equations is obtained.
The Newton-Raphson method solves for the coefficients and by substituting the coefficients back
into the sublaminate strain equation for various loads, the critical buckling load can be
determined by producing and analyzing a load-strain curve.
The assumptions listed in the next section are made in order to satisfy the laws that
govern the CLPT equations.
3.1.1 Assumptions
1. The laminate properties are treated as linear elastic.
2. Each laminate is orthotropic (or specially orthotropic if at a 0° orientation angle).
3. The laminate thickness is constant for all layers and the composite as a whole. The
thickness is small relative to the other dimensions of the plate.
4. The strains within the plate and the sublaminate are relatively small.
5
5. There is no slippage between adjacent layers within the composite except at the location
of the delamination.
6. The method is only valid for local buckling (as in Figure 3(a)), so the sublaminate must
be fairly small relative to the base plate.
Figure 3: Buckling Modes [8]
7. The base plate is assumed to experiences no transverse displacements or rotations.
8. All loading is in-plane with respect to the composite orientation. Multiaxial loading,
shear loading, and combined loading are all possible with the given method, however this
project is restricted to uniaxial compressive loading.
3.1.2 Classical Laminate Plate Behavior
The first step in the analysis of the delaminated plate is to use basic CLPT equations to
evaluate the composite plate as if no delamination were present whatsoever. Using these
equations, the pre-delamination strains, displacements, and stresses within the plate can be
derived.
Before analyzing the composite as a whole, the stiffness of each ply must be accounted for. The
following is the constitutive relationship between stress and strain in a single, orthotropic
composite ply, in the material coordinate system of the ply (Reference (19), equation 2.26):
6
𝑸𝒙𝒙
𝝈𝒙
𝑸𝒙𝒚
𝝈𝒚
𝝉𝒚𝒛 = 𝟎
𝝉𝒛𝒙
𝟎
[𝝉𝒙𝒚 ] [ 𝟎
𝑸𝒙𝒚
𝑸𝒚𝒚
𝟎
𝟎
𝟎
𝟎
𝟎
𝑸𝒚𝒛
𝟎
𝟎
𝟎
𝟎
𝟎
𝑸𝒛𝒙
𝟎
𝟎
𝜺𝒙
𝟎
𝜺𝒚
𝟎 𝜸𝒚𝒛
𝟎 𝜸𝒛𝒙
𝑸𝒚𝒛 ] [𝜸𝒙𝒚 ]
[1]
The engineering constants that make up the compliance matrix in equation [1] are defined below
(Reference (19), equation 2.27):
𝐸𝑥
𝑄𝑥𝑥 = 1−𝜈
𝑥𝑦 𝜈𝑦𝑥
𝑄𝑥 = 𝐺𝑦𝑧
𝐸𝑦
𝑄𝑦𝑦 = 1−𝜈
𝑥𝑦 𝜈𝑦𝑥
𝑄𝑦 = 𝐺𝑧𝑥
𝐸𝑥
𝑄𝑥𝑦 = 1−𝜈
𝑥𝑦 𝜈𝑦𝑥
𝑄𝑧 = 𝐺𝑥𝑦
[2]
The following equation shows the transformation matrix from which the compliance matrix
components from the material coordinate system can be converted to the composite plate
coordinate system (Reference (19), equation 2.36); Figure 4 illustrates how these axes are
defined:
𝑸𝟏𝟏
𝒄𝟒
𝒔𝟒
𝑸𝟐𝟐
𝟐
𝒄 ∗ 𝒔𝟐
𝑸𝟏𝟐
𝒄𝟐 ∗ 𝒔𝟐
𝑸𝟔𝟔
𝑸𝟏𝟔 = 𝒄𝟑 ∗ 𝒔
𝑸𝟐𝟔
𝒄 ∗ 𝒔𝟑
𝑸𝟒𝟒
𝟎
𝑸𝟒𝟓
𝟎
[𝑸𝟓𝟓 ] [ 𝟎
𝒔𝟒
𝒄𝒄𝟒
𝟐
𝒄 ∗ 𝒔𝟐
𝒄𝟐 ∗ 𝒔𝟐
−𝒄 ∗ 𝒔𝟑
−𝒄𝟑 ∗ 𝒔
𝟎
𝟎
𝟎
𝟐𝒄𝟐 ∗ 𝒔𝟐
𝟎
𝟎
𝟒𝒄𝟐 ∗ 𝒔𝟐
𝟐𝒄𝟐 ∗ 𝒔𝟐
𝟎
𝟎
𝟒𝒄𝟐 ∗ 𝒔𝟐
𝑸𝒙𝒙
𝟒
𝟒
𝒄 +𝒔
𝟎
𝟎
−𝟒𝒄𝟐 ∗ 𝒔𝟐
𝑸𝒚𝒚
(𝒄𝟐 − 𝒔𝟐 )𝟐
−𝟐𝒄𝟐 ∗ 𝒔𝟐
𝟎
𝟎
𝑸𝒙𝒚
𝒄 ∗ 𝒔𝟑 − 𝒄𝟑 ∗ 𝒔
𝟎
𝟎
𝟐(𝒄 ∗ 𝒔𝟑 − 𝒄𝟑 ∗ 𝒔)
𝑸𝒙
𝒄𝟑 ∗ 𝒔 − 𝒄 ∗ 𝒔𝟑
𝟎
𝟎
𝟐(𝒄𝟑 ∗ 𝒔 − 𝒄 ∗ 𝒔𝟑 ) 𝑸
𝒚
𝟎
𝒄𝟐
𝒔
𝟎
[ 𝑸𝒛 ]
𝟎
−𝒄 ∗ 𝒔 𝒄 ∗ 𝒔
𝟎
]
𝟎
𝒔𝟐
𝒄
𝟎
𝒄 = 𝐜𝐨𝐬 𝜽
𝒘𝒉𝒆𝒓𝒆
𝒔 = 𝐬𝐢𝐧 𝜽
Figure 4: Material and Composite Axis Orientations [19]
7
[3]
Having defined the compliance matrix components in the composite coordinate system, the
constitutive relationship between stress and strain in the composite coordinate system can be
written (Reference (19), equation 2.35):
𝜎1
𝑄11
𝜎2
𝑄12
𝜎4 = 0
𝜎5
0
[𝜎6 ] [𝑄16
𝑄12
𝑄22
0
0
𝑄26
0
0
𝑄44
𝑄45
0
0
0
𝑄45
𝑄55
0
𝑄16 𝜀1
𝑄26 𝜀2
0 𝜀4
0 𝜀5
𝑄66 ] [𝜀6 ]
𝑤ℎ𝑒𝑟𝑒
𝜎4 = 𝜏23
𝜎5 = 𝜏31
𝜎6 = 𝜏12
𝑎𝑛𝑑
𝜀4 = 𝛾23
𝜀5 = 𝛾31
𝜀6 = 𝛾12
[4]
The next step is to calculate the global stiffness matrix of the composite, which is a function of
the stiffness of each ply. Since this project only accounts for in-plane loading and transverse
displacements/rotations of the plate are assumed to be zero (see assumption 7), the total matrix
has been reduced to eliminate coupling and bending stiffness terms (Reference (19), equation
7.38):
ℎ𝑡𝑜𝑡
2
ℎ
− 𝑡𝑜𝑡
2
𝐴𝑝𝑙,126 = ∫
𝑛
𝑡𝑜𝑡
(𝑄𝑖𝑗 )𝑑𝑧 = ∑𝑘=1
(𝑄𝑖𝑗 )𝑘 (𝑧𝑘 − 𝑧𝑘−1 )
𝑓𝑜𝑟
𝑖, 𝑗 = 1,2,6
[5]
Where the (Qij)k term is the compliance matrix for the kth ply of the composite (top to bottom),
and the zn terms are defined by Figure 5:
Figure 5: Ply Transverse Dimension Number System [19]
Multiplying the inverse of the global stiffness matrix (equation [5]) by the in-plane loading
matrix yields the mid-plane strains in the composite (Reference (19), equation 7.46). Since only
in-plane loads are applied to this system, the pre-buckling strains throughout the composite are
equivalent to the pre-buckling mid-plane strains:
8
𝐴𝑝𝑙,11
𝜀𝑝𝑙,1
𝜀
[ 𝑝𝑙,2 ] = [𝐴𝑝𝑙,12
𝜀𝑝𝑙,6
𝐴𝑝𝑙,16
𝐴𝑝𝑙,11 −1 𝑁1
𝐴𝑝𝑙,26 ] [ 𝑁2 ]
𝑁12
𝐴𝑝𝑙,66
𝐴𝑝𝑙,12
𝐴𝑝𝑙,22
𝐴𝑝𝑙,26
[6]
The following are the strain-displacement relationships for in-plane strains, from which the
displacements throughout the composite can be derived (Reference (19), equation 7.28):
𝜀𝑝𝑙,1 =
𝜀𝑝𝑙,2 =
𝜀𝑝𝑙,6 =
𝛿𝑢𝑝𝑙,1
𝛿𝑥1
𝛿𝑢𝑝𝑙,2
[7]
𝛿𝑥2
𝛿𝑢𝑝𝑙,1
𝛿𝑢𝑝𝑙,2
𝛿𝑥2
+
𝛿𝑥1
Appendix C derives the following displacement equations by substituting the strains from
equation [6] into the equation [7] expressions:
1
𝑢𝑝𝑙,1 = 𝜀𝑝𝑙,1 𝑥1 + 2 𝜀𝑝𝑙,6 𝑥2
[8a]
1
𝑢𝑝𝑙,2 = 𝜀𝑝𝑙,2 𝑥2 + 2 𝜀𝑝𝑙,6 𝑥1
And from assumption 7:
𝑢𝑝𝑙,3 = 0
[8b]
Finally, by substituting the equation [7] plate strains back into equation [4], the stresses in each
ply can be calculated as follows:
𝜎𝑝𝑙,1
𝑄11
[𝜎𝑝𝑙,2 ] = [𝑄12
𝜎𝑝𝑙,6 𝑘
𝑄16
𝑄12
𝑄22
𝑄26
𝑄16 𝜀𝑝𝑙,1
𝑄26 ] [𝜀𝑝𝑙,2 ]
𝑄66 𝑘 𝜀𝑝𝑙,6
𝑓𝑜𝑟
𝑘 = 1 𝑡𝑜 𝑛𝑡𝑜𝑡
[9]
3.1.3 Sublaminate Behavior
The second step in the analysis of the delaminated plate is to calculate the strains,
displacements, and stresses in the sublaminate. In this section, they are solved for in terms of
unknown polynomial coefficients. The coefficients will be solved for in section 3.1.4.
The sublaminate displacements are defined using a higher-order lamination theory, which
defines them as nonlinear functions of the transverse coordinate, x3 (Reference (19), equation
7.25):
𝑢𝑠𝑙,1 = 𝑢𝑚𝑖𝑑,𝑠𝑙,1 + 𝑥3 (𝜓𝑠𝑙,1 ) + 𝑥32 (𝜁𝑠𝑙,1 ) + 𝑥33 (𝜙𝑠𝑙,1 )
𝑢𝑠𝑙,2 = 𝑢𝑚𝑖𝑑,𝑠𝑙,2 + 𝑥3 (𝜓𝑠𝑙,2 ) + 𝑥32 (𝜁𝑠𝑙,2 ) + 𝑥33 (𝜙𝑠𝑙,2 )
𝑢𝑠𝑙,3 = 𝑢𝑚𝑖𝑑,𝑠𝑙,3
9
[10]
The appropriate three dimensional strain-displacement relationships are defined by the GreenLagrange strain tensor (Reference (1), equation C.14):
1 𝜕𝑢
𝜕𝑢
𝜀𝑠𝑙,𝑖𝑗 = 2 𝜕𝑥 𝑖 + 𝜕𝑥𝑗 +
𝑗
𝑖
𝜕𝑢3 𝜕𝑢3
[11]
𝜕𝑥𝑖 𝜕𝑥𝑗
Appendix D derives the following displacement equations by substituting the equation [10]
displacements into equation [11]:
𝜕𝑢𝑚𝑖𝑑,𝑠𝑙,3
4
𝑢𝑠𝑙,1 = 𝑢𝑚𝑖𝑑,𝑠𝑙,1 + 𝑥3 (𝜓𝑠𝑙,1 ) + 𝑥33 (− 3(ℎ
𝑠𝑙
4
𝑢𝑠𝑙,2 = 𝑢𝑚𝑖𝑑,𝑠𝑙,2 + 𝑥3 (𝜓𝑠𝑙,2 ) + 𝑥33 (− 3(ℎ
𝑠𝑙
)2
(
𝜕𝑥1
𝜕𝑢𝑚𝑖𝑑,𝑠𝑙,3
(
)2
𝜕𝑥2
+ 𝜓𝑠𝑙,1 ))
+ 𝜓𝑠𝑙,2 ))
[12]
𝑢𝑠𝑙,3 = 𝑢𝑚𝑖𝑑,𝑠𝑙,3
The sublaminate displacements in equation [12] are now defined by three displacement
functions, 𝑢𝑚𝑖𝑑,𝑠𝑙,1 , 𝑢𝑚𝑖𝑑,𝑠𝑙,2 , 𝑢𝑚𝑖𝑑,𝑠𝑙,3 , and two rotation functions, 𝜓𝑠𝑙,1 , 𝜓𝑠𝑙,2 . Using the
Reference (1) methodology, those functions are approximated by a linear combination of the
plate displacements (equations [8a] and [8b]) and polynomial coordinate functions (Reference
(1), equation 3.24):
𝑢𝑚𝑖𝑑,𝑠𝑙,1 = 𝑢𝑝𝑙,1 + 𝑢𝑝𝑜𝑙𝑦𝑚𝑖𝑑,𝑠𝑙,1
𝑢𝑚𝑖𝑑,𝑠𝑙,2 = 𝑢𝑝𝑙,2 + 𝑢𝑝𝑜𝑙𝑦𝑚𝑖𝑑,𝑠𝑙,2
𝑢𝑚𝑖𝑑,𝑠𝑙,3 = 𝑢𝑝𝑜𝑙𝑦𝑚𝑖𝑑,𝑠𝑙,3
𝜓𝑠𝑙,1 = 𝜓𝑝𝑜𝑙𝑦𝑚𝑖𝑑,𝑠𝑙,1
𝜓𝑠𝑙,2 = 𝜓𝑝𝑜𝑙𝑦𝑚𝑖𝑑,𝑠𝑙,2
[13]
Where the polynomial coordinate functions are defined below (Reference (1), equation 3.27):
𝑢𝑝𝑜𝑙𝑦𝑚𝑖𝑑,𝑠𝑙,1 = (1 − 𝑥̇ 12 − 𝑥̇ 22 )(𝑝𝑐1 𝑥̇ 1 + 𝑝𝑐2 𝑥̇ 2 + 𝑝𝑐3 𝑥̇ 13 + 𝑝𝑐4 𝑥̇ 12 𝑥̇ 2 + 𝑝𝑐5 𝑥̇ 1 𝑥̇ 22 + 𝑝𝑐6 𝑥̇ 23
+ 𝑝𝑐7 𝑥̇ 15 + 𝑝𝑐8 𝑥̇ 14 𝑥̇ 2 + 𝑝9 𝑥̇ 13 𝑥̇ 22 + 𝑝10 𝑥̇ 12 𝑥̇ 23 + 𝑝𝑐11 𝑥̇ 1 𝑥̇ 24 + 𝑝𝑐12 𝑥̇ 25 + 𝑝𝑐13 𝑥̇ 17
+ 𝑝𝑐14 𝑥̇ 16 𝑥̇ 2 + 𝑝𝑐15 𝑥̇ 15 𝑥̇ 22 + 𝑝𝑐16 𝑥̇ 14 𝑥̇ 23 + 𝑝𝑐17 𝑥̇ 13 𝑥̇ 24 + 𝑝𝑐18 𝑥̇ 12 𝑥̇ 25 + 𝑝𝑐19 𝑥̇ 1 𝑥̇ 26
+ 𝑝𝑐20 𝑥̇ 27 )
𝑢𝑝𝑜𝑙𝑦𝑚𝑖𝑑,𝑠𝑙,2 = (1 − 𝑥̇ 12 − 𝑥̇ 22 )(𝑝𝑐21 𝑥̇ 1 + 𝑝𝑐22 𝑥̇ 2 + 𝑝𝑐23 𝑥̇ 13 + 𝑝𝑐24 𝑥̇ 12 𝑥̇ 2 + 𝑝𝑐25 𝑥̇ 1 𝑥̇ 22 + 𝑝𝑐26 𝑥̇ 23
+ 𝑝𝑐27 𝑥̇ 15 + 𝑝𝑐28 𝑥̇ 14 𝑥̇ 2 + 𝑝29 𝑥̇ 13 𝑥̇ 22 + 𝑝30 𝑥̇ 12 𝑥̇ 23 + 𝑝𝑐31 𝑥̇ 1 𝑥̇ 24 + 𝑝𝑐32 𝑥̇ 25 + 𝑝𝑐33 𝑥̇ 17
+ 𝑝𝑐34 𝑥̇ 16 𝑥̇ 2 + 𝑝𝑐35 𝑥̇ 15 𝑥̇ 22 + 𝑝𝑐36 𝑥̇ 14 𝑥̇ 23 + 𝑝𝑐37 𝑥̇ 13 𝑥̇ 24 + 𝑝𝑐38 𝑥̇ 12 𝑥̇ 25 + 𝑝𝑐39 𝑥̇ 1 𝑥̇ 26
+ 𝑝𝑐40 𝑥̇ 27 )
𝑢𝑝𝑜𝑙𝑦𝑚𝑖𝑑,𝑠𝑙,3 = (1 − 𝑥̇ 12 − 𝑥̇ 22 )(𝑝𝑐41 + 𝑝𝑐42 𝑥̇ 12 + 𝑝𝑐43 𝑥̇ 22 + 𝑝𝑐44 𝑥̇ 1 𝑥̇ 2 )
𝜓𝑝𝑜𝑙𝑦𝑠𝑙,1 = (1 − 𝑥̇ 12 − 𝑥̇ 22 )(𝑝𝑐45 𝑥̇ 1 + 𝑝𝑐46 𝑥̇ 2 + 𝑝𝑐47 𝑥̇ 13 + 𝑝𝑐48 𝑥̇ 12 𝑥̇ 2 + 𝑝𝑐49 𝑥̇ 1 𝑥̇ 22 + 𝑝𝑐50 𝑥̇ 23 )
𝜓𝑝𝑜𝑙𝑦𝑠𝑙,2 = (1 − 𝑥̇ 12 − 𝑥̇ 22 )(𝑝𝑐51 𝑥̇ 1 + 𝑝𝑐52 𝑥̇ 2 + 𝑝𝑐53 𝑥̇ 13 + 𝑝𝑐54 𝑥̇ 12 𝑥̇ 2 + 𝑝𝑐55 𝑥̇ 1 𝑥̇ 22 + 𝑝𝑐56 𝑥̇ 23 )
10
𝑤ℎ𝑒𝑟𝑒
𝑥1
𝑥̇ 1 = 𝑟
𝑑𝑒𝑙
𝑎𝑛𝑑
𝑥2
𝑥̇ 2 = 𝑟
𝑑𝑒𝑙
[13]
The terms that describe the equation [13] polynomial functions are able to very closely model the
complicated behavior of the sublaminate. The functions also fully satisfy imposed boundary
conditions where the edges of the sublaminate meet the plate. The boundary conditions require
that the sublaminate displacements, 𝑢𝑠𝑙,1 , 𝑢𝑠𝑙,1 , and 𝑢𝑠𝑙,1 , are set equal to the plate displacements,
𝑢𝑝𝑙,1 , 𝑢𝑝𝑙,1 , and 𝑢𝑝𝑙,1, the sublaminate rotations, 𝜓𝑠𝑙,1 and 𝜓𝑠𝑙,2 , are set equal to zero, and the
sublaminate is modeled as clamped to the plate. The boundary conditions are illustrated by
Figure 6:
Figure 6: Sublaminate Boundary Conditions [1]
By substituting the equation [13] displacements into the equation [11] expression, the following
sublaminate strains are derived. In these expressions, the midplane strains and curvatures have
been separated out as the coefficients to x30, x31, x32, and x33,as applicable. The midplane strains
and curvatures will be incorporated into the potential energy equation in section 3.1.4:
𝜀𝑠𝑙,11 = 𝜀𝑠𝑙,1 = 𝜀𝑚𝑖𝑑,𝑠𝑙,1 + 𝑥3 𝜅1𝑚𝑖𝑑,𝑠𝑙,1 + 𝑥33 𝜅2𝑚𝑖𝑑,𝑠𝑙,1
𝜀𝑠𝑙,22 = 𝜀𝑠𝑙,2 = 𝜀𝑚𝑖𝑑,𝑠𝑙,2 + 𝑥3 𝜅1𝑚𝑖𝑑,𝑠𝑙,2 + 𝑥33 𝜅2𝑚𝑖𝑑,𝑠𝑙,2
𝜀𝑠𝑙,33 = 𝜀𝑠𝑙,3 = 0
𝜀𝑠𝑙,23 = 𝜀𝑠𝑙,4 = 𝜀𝑚𝑖𝑑,𝑠𝑙,4 + 𝑥32 𝜅2𝑚𝑖𝑑,𝑠𝑙,4
𝜀𝑠𝑙,31 = 𝜀𝑠𝑙,5 = 𝜀𝑚𝑖𝑑,𝑠𝑙,5 + 𝑥32 𝜅2𝑚𝑖𝑑,𝑠𝑙,5
𝜀𝑠𝑙,12 = 𝜀𝑠𝑙,6 = 𝜀𝑚𝑖𝑑,𝑠𝑙,6 + 𝑥3 𝜅1𝑚𝑖𝑑,𝑠𝑙,6 + 𝑥33 𝜅2𝑚𝑖𝑑,𝑠𝑙,6
[14]
The sublaminate stresses for each laminate are then determined by plugging the equation [14]
strains back into equation [4] as follows:
11
𝜎1
𝑄11
𝜎2
𝑄12
𝜎4 = 0
𝜎5
0
[𝜎6 ]𝑘 [𝑄16
𝑄12
𝑄22
0
0
𝑄26
0
0
𝑄44
𝑄45
0
0
0
𝑄45
𝑄55
0
𝑄16 𝜀1
𝑄26 𝜀2
𝜀4
0
𝜀5
0
𝑄66 ]𝑘 [𝜀6 ]
𝑓𝑜𝑟
𝑘 = 1 𝑡𝑜 𝑛𝑠𝑙
[15]
3.1.4 Potential Energy Minimization
The third step in the analysis of the delaminated plate is to determine the values for the
unknown polynomial coefficients, so that the system can be quantitatively evaluated and the
critical buckling load of the sublaminate can be derived. To solve for the coefficients, the total
potential energy of the system must be determined, and then minimized with respect to each of
the unknowns. This process creates a system of nonlinear equations which can be solved via the
Newton-Raphson Method.
The potential energy equation is composed of the following (Reference (1), equation 3.33):
[16]
In which the sublaminate global stiffness terms are defined by the following (Reference (1),
equation 3.34):
ℎ𝑠𝑙
(𝐴𝑖𝑗 , 𝐵𝑖𝑗 , 𝐷𝑖𝑗 , 𝐸𝑖𝑗 , 𝐹𝑖𝑗 , 𝐻𝑖𝑗 ) = ∫ 2ℎ𝑠𝑙 𝑄𝑖𝑗 (1, 𝑥3 , 𝑥32 , 𝑥33 , 𝑥34 , 𝑥36 )𝑑𝑥3
−
ℎ𝑠𝑙
2
ℎ
− 𝑠𝑙
2
(𝐴𝑖𝑗 , 𝐷𝑖𝑗 , 𝐹𝑖𝑗 ) = ∫
𝑓𝑜𝑟
2
𝑖, 𝑗 = 1,2,6
[17]
𝑄𝑖𝑗 (1, 𝑥32 , 𝑥34 )𝑑𝑥3
𝑓𝑜𝑟
𝑖, 𝑗 = 4,5
And the limits of integration over the sublaminate area are defined by (Reference (1), equation
3.36):
12
2
𝑥
𝑟𝑑𝑒𝑙 √1−( 1 )
𝑟𝑑𝑒𝑙
∫−𝑟
𝑑𝑒𝑙
𝑟𝑑𝑒𝑙
∫
2
𝑥
−𝑟𝑑𝑒𝑙 √1−( 1 )
𝑟𝑑𝑒𝑙
[… ] 𝑑𝑥2 𝑑𝑥1
[18]
The stable solution for the values of the coefficients is obtained by minimizing equation [16] as
follows (Reference (1), equation 3.38):
𝑑𝛱
𝑒𝑞𝑖 = 𝑑𝑝𝑐𝑠𝑙 = 0
𝑖
𝑓𝑜𝑟
𝑖 = 1 𝑡𝑜 56
[19]
Equation [19] produces 56 nonlinear equations which are solved in the next section. These
equations are functions of the unknown polynomial coefficients.
3.1.5 Newton-Raphson Method
The Newton-Raphson Method is an efficient tool used to solve complex systems of nonlinear
equations like the following (Reference (20), equation 3):
𝑓1 (𝑥1 , 𝑥2 , … , 𝑥𝑛 ) = 0
𝑓2 (𝑥1 , 𝑥2 , … , 𝑥𝑛 ) = 0
⋮
𝑓𝑛 (𝑥1 , 𝑥2 , … , 𝑥𝑛 ) = 0
[20]
This is the form that the equation [19] expressions take, where pcn is the unknown in lieu of xn.
The unknowns are solved by inputting a reasonable initial guess for every unknown (pci,0 = 0 for
i = 1 to 56), and then iteratively solving the following equation to derive pci,m+1 (Reference (20),
equation 5):
𝜕𝑒𝑞1 (𝑝𝑐1,𝑚 ,𝑝𝑐2,𝑚 ,…,𝑝𝑐56,𝑚 )
𝑝𝑐1,𝑚+1
𝑝𝑐2,𝑚+1
[
]=
⋮
𝑝𝑐56,𝑚+1
𝜕𝑒𝑞1 (𝑝𝑐1,𝑚 ,𝑝𝑐2,𝑚 ,…,𝑝𝑐56,𝑚 )
𝜕𝑝𝑐1,𝑚
𝜕𝑝𝑐2,𝑚
𝜕𝑒𝑞2 (𝑝𝑐1,𝑚 ,𝑝𝑐2,𝑚 ,…,𝑝𝑐56,𝑚 )
𝜕𝑒𝑞2 (𝑝𝑐1,𝑚 ,𝑝𝑐2,𝑚 ,…,𝑝𝑐56,𝑚 )
𝜕𝑝𝑐1,𝑚
𝜕𝑝𝑐2,𝑚
⋮
⋮
𝜕𝑒𝑞56 (𝑝𝑐1,𝑚 ,𝑝𝑐2,𝑚 ,…,𝑝𝑐𝑛,𝑚 )
𝜕𝑒𝑞56 (𝑝𝑐1,𝑚 ,𝑝𝑐2,𝑚 ,…,𝑝𝑐𝑛,𝑚 )
([
𝜕𝑝𝑐1,𝑚
𝜕𝑝𝑐2,𝑚
−𝑒𝑞1 (𝑝𝑐1,𝑚 , 𝑝𝑐2,𝑚 , … , 𝑝𝑐56,𝑚 )
−𝑒𝑞2 (𝑝𝑐1,𝑚 , 𝑝𝑐2,𝑚 , … , 𝑝𝑐56,𝑚 )
⋮
[−𝑒𝑞56 (𝑝𝑐1,𝑚 , 𝑝𝑐2,𝑚 , … , 𝑝𝑐56,𝑚 )]
⋯
⋯
⋱
⋯
−1
𝜕𝑒𝑞1 (𝑝𝑐1,𝑚 ,𝑝𝑐2,𝑚 ,…,𝑝𝑐56,𝑚 )
𝜕𝑝𝑐56,𝑚
𝜕𝑒𝑞2 (𝑝𝑐1,𝑚 ,𝑝𝑐2,𝑚 ,…,𝑝𝑐56,𝑚 )
∗
𝜕𝑝𝑐56,𝑚
⋮
𝜕𝑒𝑞56 (𝑝𝑐1,𝑚 ,𝑝𝑐2,𝑚 ,…,𝑝𝑐𝑛,𝑚 )
𝜕𝑝𝑐56,𝑚
]
𝑝𝑐1,𝑚
𝑝𝑐2,𝑚
+[ ⋮ ]
𝑝𝑐56,𝑚
[21]
)
Where m is the indicator for the number of iterations that have been completed. In the Maple
code, the equation [21] is iterated 40 times to produce a stable solution. Equation [21] should be
13
further iterated until pci,m – pci,m+1 approaches zero for all terms if 40 is not enough. The final
iteration yields the solution to the polynomial coefficients.
3.1.6 Evaluating the Critical Buckling Load
The stability of the solution from equation [21] should be tested by applying the following
equation (Reference (1), equation 3.39):
𝑑2 𝛱𝑠𝑙 (𝑝𝑐1,𝑚+1 ,𝑝𝑐2,𝑚+1 ,⋯,𝑝𝑐56,𝑚+1 )
|
𝑑𝑝𝑐𝑖 𝑑𝑝𝑐𝑗
|>0
𝑓𝑜𝑟
𝑖, 𝑗 = 1 𝑡𝑜 56
[22]
If equation [22] is true, then the solution is stable. Once stability has been verified, the
polynomial coefficients can be substituted back into equations [12], [14], and [15] to determine
the sublaminate displacements, strains, and stresses, respectively. A check to verify that the
polynomials are correct can be completed by ensuring the boundary conditions where the
sublaminate meets the plate hold true; for example, upl,1(a,0,0) = usl,1(a,0,0).
By varying the initial compressive load, Nx, and comparing it to the sublaminate strain (in this
project, 𝜀𝑠𝑙,1 (0,0,0) was used for comparison), the resulting load vs. strain curve illustrates the
critical buckling load of the sublaminate for the given system as a peak in the curve. The critical
buckling loads for delaminations of various sizes can then be compared.
14
4. Results/Discussion
4.1 Analytical Results
The methodology described in section 3 was coded into the Maple worksheet in
Appendix A. By adjusting the inputs (laminate thicknesses, laminate orientation angles, number
of plies, location of delamination, and size of delamination) to model the desired system, the
worksheet can be used to calculate the behavioral response of the composite, and can be used as
a tool to derive the buckling load of the sublaminate. By plotting the strain in the sublaminate
against varying loads along a load-strain curve, the critical buckling load is easily illustrated as a
peak in the chart.
4.1.1 Inputs
The individual composite laminate plies in the context of this project are assumed to have
the following material properties:
Table 1: Laminate Material Properties
Ex (psi)
Ey (psi)
νxy
Gyz (psi)
Gzx (psi)
Gxy (psi)
19.5E6
1.32E6
0.3
1.01E6
1.01E6
0.5E6
The composite is assembled with n = 24 plies, where the circular delamination lies
centrally between the 4th and 5th ply. All plies are of equal, constant thickness (hi = 0.005 in), and
are oriented in a [(02/902)3]s arrangement.
The geometric parameters of the plate are shown in Figure 7 and consist of the following:
lcomp
Where:
lcomp = wcomp = 6.0 in
rdel = {0.5, 0.6, 0.7, 0.8, 0.9, 1.0} in
Nx
Nz = Varied
rdel
wcomp
Figure 7: Plate Geometry
15
The in-plane loads placed on the composite are given as Ny = Nxy = 0 lbf/in, and Nx is
varied around the critical buckling load of the sublaminate to converge on a maximum in the
load-strain plot.
4.1.2 Load-Strain Maps
The curves generated by iteratively running the Maple program at various delamination
sizes are shown in Figures 8(a-f), below. The peaks of each plot indicate the load at which the
sublaminate buckles. The data points for each of the curves are included in Tables 5-10 in
Appendix E.
Load vs. Strain (0.5 in)
Load vs. Strain (0.6 in)
1.600E-03
1.200E-03
-Strain (in/in)
-Strain (in/in)
1.400E-03
1.000E-03
8.000E-04
6.000E-04
4.000E-04
2.000E-04
0.000E+00
0
500
1000
1500
2000
1.000E-03
9.000E-04
8.000E-04
7.000E-04
6.000E-04
5.000E-04
4.000E-04
3.000E-04
2.000E-04
1.000E-04
0.000E+00
0
500
-Load (lbf/in)
1000
-Load (lbf/in)
(a)
(b)
16
1500
Load vs. Strain (0.7 in)
Load vs. Strain (0.8 in)
8.000E-04
6.000E-04
7.000E-04
5.000E-04
-Strain (in/in)
-Strain (in/in)
6.000E-04
5.000E-04
4.000E-04
3.000E-04
2.000E-04
4.000E-04
3.000E-04
2.000E-04
1.000E-04
1.000E-04
0.000E+00
-300
0.000E+00
200
700
1200
0
200
-Load (lbf/in)
600
800
1000
-Load (lbf/in)
(c)
(d)
Load vs. Strain (0.9 in)
Load vs. Strain (1.0 in)
4.500E-04
4.000E-04
3.500E-04
3.000E-04
2.500E-04
2.000E-04
1.500E-04
1.000E-04
5.000E-05
0.000E+00
4.000E-04
3.500E-04
-Strain (in/in)
-Strain (in/in)
400
3.000E-04
2.500E-04
2.000E-04
1.500E-04
1.000E-04
5.000E-05
0.000E+00
0
200
400
600
0
800
200
400
600
-Load (lbf/in)
-Load (lbf/in)
(e)
(f)
Figure 8: Load vs. Strain Curves as a Function of Delamination Size
4.1.3 Buckling Load vs. Delamination Size
Table 2 lists the dimension of the delaminations against their respective sublaminate
critical buckling loads:
17
Table 2: Buckling Load vs. Delamination Size
Load vs. Delamination Size
Del. Radius
-Load
(in)
(lbf/in)
0.5
1705
0.6
1185
0.7
875
0.8
670
0.9
530
1
430
Figure 9 illustrates the correlation between buckling load and delamination size. Figure 9 shows
a clear trend in the data.
Load vs. Delamination Size
-Critical Buckling Load (lbf/in)
1800
1600
1400
1200
1000
800
600
400
200
0
0.4
0.5
0.6
0.7
0.8
0.9
1
Delamination Radius (in)
Figure 9: Buckling Load vs. Delamination Size Chart
4.2 Validation of Maple Code
In order to ensure the reliability of the Appendix A Maple Code, a series of trials were run
using the inputs from Reference (1), Appendix G. The Maple code that was developed for this
project had to be expanded (see Appendix B) to incorporate hygrothermal, contact force, and
elliptical geometry effects, which were input parameters in the Reference (1) code. The full
system is described in the following section.
18
4.2.1 Inputs
The Reference (1), Appendix G system analyzes a 16-ply, rectangular graphite/epoxy
composite laminate with a centrally located elliptical delamination between the 4th and 5th ply.
The material properties are the same as those used in Table 1. All plies are of equal, constant
thickness (hi = 0.00556”), and are oriented in a [(02/902)2]s arrangement.
The geometric parameters of the plate are shown in Figure 10 and consist of the following:
wcomp
Where:
b
a
Nx
lcomp
lcomp = 6.0 in
wcomp = 3.0 in
a = 1.0 in
b = 0.75 in
Nz = Varied
Figure 10: Plate Geometry (Validation Input)
The in-plane loads placed on the composite are given as Ny = Nxy = 0 lbf/in, and Nx is varied
around the critical buckling load of the sublaminate.
The thermal and contact properties, along with atmospheric conditions are displayed in Table 3,
below:
Table 3: Validation Input Additional Properties
αx (in/(in-°F))
αy (in/(in-°F))
ΔT (°F)
K (lbf/in3)
ΔP (psi)
0.50E-6
18.0E-6
-180
1.0E6
3
4.2.2 Comparison of Analysis Tools
Table 4 and Figure 11 Illustrate the output data points of the Appendix B system, run at
various compressive loads.
19
Table 4: Validation Output Results
Load vs. Strain
-Load
-Strain
(lbf/in)
(in/in)
200
5.551E-04
300
6.626E-04
400
7.700E-04
500
8.775E-04
600
9.850E-04
677.92
1.073E-03
700
9.188E-04
800
1.733E-04
900
5.993E-04
1000
1.415E-03
1100
4.402E-04
1200
3.166E-04
The figure 11 plot shows that the strain initially varies linearly with the load until reaching the
critical buckling load of the sublaminate. At that point, the strain begins to fluctuate between low
points and peaks. It should be noted that the minimal peak is the one at which buckling occurs.
Load vs. Strain
-Strain (in/in)
1.500E-03
1.000E-03
5.000E-04
0.000E+00
0
200
400
600
800
1000
1200
1400
-Load (lbf/in)
Figure 11: Validation Output Results
By analyzing the first peak in the Figure 11 table, the critical compressive buckling load
of the sublaminate from the Reference (1), Appendix G system is found to be about 680 lbf/in.
This is the same result that Reference (1) calculated (see page 124), so the Appendix A code is
validated as being a reliable tool for determining outputs.
20
5. Conclusions
In this project, the Reference (1) code was successfully translated into a Maple code that
can analyze multi-layered composite laminates with a circular delamination between two plies to
determine the critical buckling load. The code can support composites made of plies with
different materials, thicknesses, and orientations. The validity of the code was verified by
replicating the Reference (1), Appendix G results with the Maple code in Appendix B of this
report, and outputting the same critical buckling load.
The goal of this project was to determine the relationship between the size of a circular
delamination and the critical buckling load of the sublaminate. Figure 9 shows that as the
delamination size gets smaller and smaller, the structure is able to handle larger loads without
buckling. This result is further supported by literature in the field which suggests that the
buckling load of the sublaminate has an inverse relationship with the square of the lateral
dimension of the delamination [3].
With this code proven, further research could determine the effects of any other one of
the controlled variables on buckling load. Reference (1) examines several of these trends.
Reference (1) also examined the post-buckling response of composites and the affinity for
delaminations to grow when subject to high stresses. In future, it could be possible to grow the
code even further to be able to account for multiple delaminations.
Ultimately, the understanding that can be taken away from this project is that composites
can quickly become compromised due to delaminations within the structure. All measures should
be taken in the manufacturing process to ensure strict adherence to quality, and techniques for
inspecting in-service composites should be advanced so as to prevent premature critical failures
of structures.
21
REFERENCES
1. Peck, S. and Springer, G., Compression Behavior of Delaminated Composite Plates.
NASA Ames Research Center NCC2-304, Department of Aeronautics and Astronautics,
Stanford University. October 1989.
2. Talreja, R. and Singh, C., Damage and Failure of Composite Materials. Cambridge
University Press. Chapter 3. 2012.
3. Sridharan, S., Delamination Behaviour of Composites.Woodhead Publishing Limited.
Chapters 1-3.October 2008.
4. http://www.maplesoft.com/
5. Kardomateas, G., Pelegri, A., and Malik, B., Growth of Internal Delaminations under
Cyclic Compression in Composite Plates. Journal of the Mechanics and Physics of
Solids, Volume 43, No. 6, Pages 847-868. May 1994.
6. Kardomateas, G., Snap Buckling of Delaminated Composites Under Pure Bending.
Composites Science and Technology, Volume 39, Pages 63-74. June 1990.
7. Shan, L., Explicit Buckling Analysis of Fiber-Reinforced Plastic (FRP) Composite
Structures. Washington State University Thesis. May 2007.
8. Bisagni, C., Buckling Analyses of Composite Laminated Panels with Delaminations.
Politecnico Di Milano Thesis. 2010.
9. Remmers, J. and Borst, R., Delamination Buckling of Fibre-Metal Laminates.
Composites Science and Technology, Volume 61, Pages 2207-2213. June 2001.
10. Keshava, K., Ranjan, G., and Dineshkumar, H., Partial Delamination Modeling in
Composite Beams Using Finite Element Method. Department of Aerospace Engineering,
Indian Institute of Science. July 2013.
11. Hobeck, J. and Obenchain, M., A Rayleigh-Ritz Model for Dynamic Response and
Buckling Analysis of Delaminated Composite Timoshenko Beams. Department of
Aerospace Engineering, University of Michigan. April 2013.
12. Liu, S. and Nairn, J., Fracture Mechanics Analysis of Composite Microcracking:
Experimental Results in Fatigue. Materials Science and Engineering Department,
University of Utah. June 1990.
13. Pelegri, A., Kardomateas, G., and Malik, B., The Fatigue Growth of Internal
Delaminations under Compressive Loading of Cross-Ply Composite Plates. Composite
Materials: Fatigue and Fracture (Sixth Volume), Pages 143-161. 1997.
22
14. Szekrenyes, A., Interlaminar Stresses and Energy Release Rates in Delaminated
Orthotropic Composite Plates. International Journal of Solids and Structures, Volume 49,
Pages 2460-2470. February 2012.
15. Krueger, R., Fracture Mechanics for Composites: State of the Art and Challenges.
National Institute of Aerospace Presentation, Nordic Seminar. June 2006.
16. Gillespie J. and Pipes, R., Compressive Strength of Composite Laminates with
Interlaminar Defects. Composite Structures, Volume 2, Pages 49-69. 1984.
17. Kharazi, M. and Ovesy, H., Compressional Stability Behavior of Composite Plates with
Multiple Through-the-Width Delaminations. Journal of Aerospace Science and
Technology, Volume 5, No. 1, Pages 13-22. March 2008.
18. Huang, H. and Kardomateas, G., Buckling of Orthotropic Beam-Plates with Multiple
Central Delaminations. International Journal of Solids and Structures, Volume 35, No.
13, Pages 1355-1362. March 1997.
19. Gibson, R., Principles of Composite Material Mechanics. McGraw Hill, Inc. Chapters 13, 5-8. 1994.
20. Skaflestad, B., Newton’s Method for Systems of Non-Linear Equations.
http://www.math.ntnu.no. Pages 3-7. October 2006.
23
Appendix A: Maple Code
A-1
Appendix B: Reference (61) Validation - Maple Code
B-1
Appendix C: Plate Displacement Equation Derivation
To derive the composite plate displacement equations in equation [8a], start by defining the
strain-displacement relationship (see equation [7]):
𝜀𝑝𝑙,1 =
𝜀𝑝𝑙,2 =
𝜀𝑝𝑙,6 =
𝛿𝑢𝑝𝑙,1
𝛿𝑥1
𝛿𝑢𝑝𝑙,2
𝛿𝑥2
𝛿𝑢𝑝𝑙,1
𝛿𝑢𝑝𝑙,2
𝛿𝑥2
+
[C-1]
𝛿𝑥1
By substituting the strains from equation [6] into the equation [7] expressions and rearranging,
the following expressions are derived:
𝑢𝑝𝑙,1 = ∫ 𝜀𝑝𝑙,1 𝛿𝑥1 = 𝜀𝑝𝑙,1 𝑥1 + 𝐶1 𝑥2
𝑢𝑝𝑙,2 = ∫ 𝜀𝑝𝑙,2 𝛿𝑥2 = 𝜀𝑝𝑙,2 𝑥2 + 𝐶2 𝑥2
[C-2]
The loads acting on the system are all in-plane (and for this project are only uniaxially
compressive), so the strains are constant throughout the plate. This means that C1 and C2 must be
constant terms. To solve for C1 and C2 the assumption that there is no rigid body motion is
applied, yielding the following:
𝛿𝑢𝑝𝑙,1
𝛿𝑥2
−
𝛿𝑢𝑝𝑙,2
𝛿𝑥1
= 𝐶1 − 𝐶2 = 0
[C-3]
The third expression in equation [C-1] gives us the following:
𝛿𝑢𝑝𝑙,1
𝛿𝑥2
−
𝛿𝑢𝑝𝑙,2
𝛿𝑥1
= 𝐶1 − 𝐶2 = 𝜀𝑝𝑙,6
[C-4]
By solving equations [C-3] and [C-4] as a linear system of equations, the nontrivial solution is
determined to be:
1
𝐶1 = 2 𝜀𝑝𝑙,6
1
𝐶2 = 2 𝜀𝑝𝑙,6
[C-5]
Substituting equation [C-5] back into equation [C-1], the composite plate displacements are
derived:
1
𝑢𝑝𝑙,1 = 𝜀𝑝𝑙,1 𝑥1 + 2 𝜀𝑝𝑙,6 𝑥2
1
𝑢𝑝𝑙,2 = 𝜀𝑝𝑙,2 𝑥2 + 2 𝜀𝑝𝑙,6 𝑥1
[C-6a]
And from assumption 7:
𝑢𝑝𝑙,3 = 0
C-1
[C-6b]
Appendix D: Sublaminate Displacement Equation Derivation
To derive the simplified sublaminate displacement equations in equation [12], the displacements
are initially defined using higher-order lamination theory (see equation [10]):
𝑢𝑠𝑙,1 (𝑥1 , 𝑥2 , 𝑥3 ) = 𝑢𝑚𝑖𝑑,𝑠𝑙,1 (𝑥1 , 𝑥2 ) + 𝑥3 𝜓𝑠𝑙,1 (𝑥1 , 𝑥2 ) + 𝑥32 𝜁𝑠𝑙,1 (𝑥1 , 𝑥2 ) + 𝑥33 𝜙𝑠𝑙,1 (𝑥1 , 𝑥2 )
𝑢𝑠𝑙,2 (𝑥1 , 𝑥2 , 𝑥3 ) = 𝑢𝑚𝑖𝑑,𝑠𝑙,2 (𝑥1 , 𝑥2 ) + 𝑥3 𝜓𝑠𝑙,2 (𝑥1 , 𝑥2 ) + 𝑥32 𝜁𝑠𝑙,2 (𝑥1 , 𝑥2 ) + 𝑥33 𝜙𝑠𝑙,2 (𝑥1 , 𝑥2 )
𝑢𝑠𝑙,3 (𝑥1 , 𝑥2 , 𝑥3 ) = 𝑢𝑚𝑖𝑑,𝑠𝑙,3 (𝑥1 , 𝑥2 )
[D-1]
The appropriate three dimensional strain-displacement relationships are defined by the GreenLagrange strain tensor (see equation [11]):
𝜀𝑠𝑙,𝑖𝑗 =
1 𝜕𝑢𝑖
2 𝜕𝑥𝑗
+
𝜕𝑢𝑗
𝜕𝑥𝑖
+
𝜕𝑢3 𝜕𝑢3
[D-2]
𝜕𝑥𝑖 𝜕𝑥𝑗
From assumption 5, there are no shear forces acting at the outer surfaces of the sublaminate, so
the following boundary conditions can be applied:
𝜀4 = 𝜀23 = 0
𝜀5 = 𝜀31 = 0
𝑎𝑡
𝑥3 = ±
ℎ𝑠𝑙
[D-3]
2
Incorporating the equation [D-3] boundary conditions into the equation [D-2] expressions yields
the following:
1
(𝜓𝑠𝑙,1 (𝑥1 , 𝑥2 ) + ℎ𝑠𝑙 𝜁𝑠𝑙,1 (𝑥1, 𝑥2 ) +
2
1
(𝜓𝑠𝑙,1 (𝑥1 , 𝑥2 ) − ℎ𝑠𝑙 𝜁𝑠𝑙,1 (𝑥1, 𝑥2 ) +
2
1
(𝜓𝑠𝑙,2 (𝑥1 , 𝑥2 ) + ℎ𝑠𝑙 𝜁𝑠𝑙,2 (𝑥1, 𝑥2 ) +
2
1
(𝜓𝑠𝑙,2 (𝑥1 , 𝑥2 ) − ℎ𝑠𝑙 𝜁𝑠𝑙,2 (𝑥1, 𝑥2 ) +
2
3ℎ2𝑠𝑙
4
3ℎ2𝑠𝑙
4
3ℎ2𝑠𝑙
4
3ℎ2𝑠𝑙
4
𝜙𝑠𝑙,1 (𝑥1 , 𝑥2 ) +
𝜙𝑠𝑙,1 (𝑥1 , 𝑥2 ) +
𝜙𝑠𝑙,2 (𝑥1 , 𝑥2 ) +
𝜙𝑠𝑙,2 (𝑥1 , 𝑥2 ) +
𝜕𝑢𝑚𝑖𝑑,𝑠𝑙,3 (𝑥1 ,𝑥2 )
𝜕𝑥1
)=0
𝜕𝑢𝑚𝑖𝑑,𝑠𝑙,3 (𝑥1 ,𝑥2 )
𝜕𝑥1
)=0
𝜕𝑢𝑚𝑖𝑑,𝑠𝑙,3 (𝑥1 ,𝑥2 )
𝜕𝑥2
)=0
𝜕𝑢𝑚𝑖𝑑,𝑠𝑙,3 (𝑥1 ,𝑥2 )
𝜕𝑥2
)=0
[D-4a]
[D-4b]
[D-4c]
[D-4d]
Doing basic arithmetic with the equation [D-4] expressions yields the following:
EQ[𝐃 − 𝟒𝐚] − 𝐄𝐐[𝐃 − 𝟒𝐛] 𝑦𝑖𝑒𝑙𝑑𝑠
𝜁𝑠𝑙,1 = 0
[D-5a]
𝐄𝐐[𝐃 − 𝟒𝐜] − 𝐄𝐐[𝐃 − 𝟒𝐝] 𝑦𝑖𝑒𝑙𝑑𝑠
𝜁𝑠𝑙,2 = 0
[D-5b]
4
𝐄𝐐[𝐃 − 𝟒𝐚] + 𝐄𝐐[𝐃 − 𝟒𝐛] 𝑦𝑖𝑒𝑙𝑑𝑠 𝜙𝑠𝑙,1 = − 3ℎ2 (
𝑠𝑙
4
𝐄𝐐[𝐃 − 𝟒𝐜] + 𝐄𝐐[𝐃 − 𝟒𝐝] 𝑦𝑖𝑒𝑙𝑑𝑠 𝜙𝑠𝑙,2 = − 3ℎ2 (
𝑠𝑙
𝜕𝑢𝑚𝑖𝑑,𝑠𝑙,3 (𝑥1 ,𝑥2 )
𝜕𝑥1
𝜕𝑢𝑚𝑖𝑑,𝑠𝑙,3 (𝑥1 ,𝑥2 )
𝜕𝑥2
+ 𝜓𝑠𝑙,1 (𝑥1 , 𝑥2 )) [D-5c]
+ 𝜓𝑠𝑙,2 (𝑥1 , 𝑥2 )) [D-5d]
By substituting the expressions in equation [D-5] back into equation [D-1], the simplified
sublaminate displacement equations are derived:
D-1
𝜕𝑢𝑚𝑖𝑑,𝑠𝑙,3
4
𝑢𝑠𝑙,1 = 𝑢𝑚𝑖𝑑,𝑠𝑙,1 + 𝑥3 (𝜓𝑠𝑙,1 ) + 𝑥33 (− 3(ℎ
2
𝑠𝑙 )
𝑠𝑙
𝑢𝑠𝑙,3 = 𝑢𝑚𝑖𝑑,𝑠𝑙,3
D-2
𝜕𝑥1
𝜕𝑢𝑚𝑖𝑑,𝑠𝑙,3
4
𝑢𝑠𝑙,2 = 𝑢𝑚𝑖𝑑,𝑠𝑙,2 + 𝑥3 (𝜓𝑠𝑙,2 ) + 𝑥33 (− 3(ℎ
(
)2
(
𝜕𝑥2
+ 𝜓𝑠𝑙,1 ))
+ 𝜓𝑠𝑙,2 ))
[D-6]
Appendix E: Load-Strain Chart Plotpoint Data
Table 5: 0.5" Delamination - Critical Buckling Load
Load vs. Strain (0.5 in)
-Load
-Strain
(lbf/in)
(in/in)
300
2.390E-04
600
4.781E-04
800
6.374E-04
1000
7.968E-04
1200
9.561E-04
1400
1.116E-03
1600
1.275E-03
1690
1.347E-03
1700
1.355E-03
1705
0.001421084
1710
1.243E-03
1750
1.035E-03
1800
8.920E-04
2000
5.452E-04
Table 6: 0.6" Delamination - Critical Buckling Load
Load vs. Strain (0.6 in)
-Load
-Strain
(lbf/in)
(in/in)
300
2.390E-04
600
4.781E-04
800
6.374E-04
1000
7.968E-04
1100
8.765E-04
1150
9.163E-04
1170
9.322E-04
1180
9.402E-04
1185
9.442E-04
1190
8.969E-04
1200
8.108E-04
1300
5.284E-04
E-1
Table 7: 0.7" Delamination - Critical Buckling Load
Load vs. Strain (0.7 in)
-Load
-Strain
(lbf/in)
(in/in)
200
1.594E-04
400
3.187E-04
600
4.781E-04
800
6.374E-04
850
6.773E-04
870
6.932E-04
875
6.972E-04
880
6.234E-04
900
5.273E-04
1000
3.189E-04
1100
1.904E-04
Table 8: 0.8” Delamination – Critical Buckling Load
Load vs. Strain (0.8 in)
-Load
-Strain
(lbf/in)
(in/in)
100
6.374E-05
200
1.594E-04
300
2.390E-04
400
3.187E-04
500
3.984E-04
600
4.781E-04
650
5.179E-04
665
5.299E-04
670
5.338E-04
675
4.771E-04
680
4.462E-04
700
3.744E-04
800
1.986E-04
E-2
Table 9: 0.9" Delamination - Critical Buckling Load
Load vs. Strain (0.9 in)
-Load
-Strain
(lbf/in)
(in/in)
100
7.968E-05
200
1.594E-04
300
2.390E-04
400
3.187E-04
500
3.984E-04
600
2.036E-04
700
8.048E-05
800
1.146E-05
525
4.183E-04
530
4.223E-04
535
3.697E-04
540
3.429E-04
550
3.069E-04
Table 10: 1.0" Delamination - Critical Buckling Load
Load vs. Strain (1.0 in)
-Load
-Strain
(lbf/in)
(in/in)
100
7.968E-05
200
1.594E-04
300
2.390E-04
400
3.187E-04
420
3.347E-04
425
3.386E-04
430
3.426E-04
435
2.917E-04
450
2.370E-04
500
1.449E-04
600
3.355E-05
E-3