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UMI
DYNAMIC PENETRATION OF METAL/FIBER LAMINATES
Thesis by
Wei Li
Department of Mechanical Engineering
McGill University
Montreal, Quebec, Canada
August 2003
A Thesis Submitted to the Faculty of Graduate Studies and Research in Partial Fulfillment of the Requirements for the Degree of Master of
Engineering
©Wei Li, 2003
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ABSTRACT
Laminates composed of alternating layers of metal and fiber reinforced polymers
(FRP's) exhibit a number of properties, which are preferable to either metals or FRP's
alone, making them attractive materials for a number of industries, particularly aerospace.
A number of questions persist, however, before these new composites can be widely
accepted and utilized; one of which is their response to impact, which may occur over a
wide range ofvelocities. Numerical methods, especially the FEA method, have been widely
used to simulate the impact response because they can reduce the cost and save time
comparing with the experiment. In this work, a continuum damage based model (CDM) is
developed and implemented into FEA commercial software ABAQUS. Using a rate
dependent plasticity model for the constitutive behavior of Aluminum and the CDM for the
behavior of fiberglass laminates, the dynamic penetration is simulated using ABAQUS.
Force vs. displacement results compare weIl with those obtained from the experiments. In
addition, the computed damage region is in close agreement with that se en in sectioned
specimens of the tested material. Simulations are also performed for ballistic experiments
conducted on 150mm x 150mm clamped panels of the same laminates. Ballistic
experiments involve both the local penetration response as weIl as the global deformation
behavior, particularly at velocities near the ballistic limit, where significant flexural
deformation takes place. Results from the simulation agree weIl with the ballistic
experiment results. Given the validity of the modeling approach, the high velocity impact
response of the other metal/fi ber systems can be examined minimizing the need for trial and
error fabrication.
Résumé
Les stratifiés composés de couches alternantes de polymères renforcés de métals et de
fibres (PRF's) démontrent plusieurs propriétés, caractérisant les metaux ou les PRF's. Cela
fait d'eux des matériaux intéresants pour un grand nombre d'industries, particulièrement
l'aérospatiale. Plusieurs questions restent sans réponse avant que ses nouveaux composites
puissent être utilisés et acceptés. L'une d'elles est leur réponse à l'impact, variant selon la
vitesse. Les méthodes numériques, en particulier, l'analyse par éléments finis, ont été
utilisées à plusieurs reprises pour simuler l'impact parce qu'elles réduisent le coût et le
temps d'expérimentation. Un modèle de dommages continues (MDC) a été développé et
implémenté dans le programme ABAQUS. Utilisant un modèle de plasticité dépendant de
la cinétique pour le comportement constitutif de l'aluminium et le MDC pour le
comportement de stratifié en fibre de verre, la pénétration dynamique a été simulé par
ABAQUS. Les résultats de la force vs. le déplacement ressemblent à ceux obtenus
expérimentalement. De plus, la région de dommages calculés a été très similaire avec les
spécimens des matériaux testés. Les simulations ont également été effectuées pour les tests
de ballistique utilisant des stratifiés encastrés de même type mesurant 150mm x 150mm.
Les tests de ballistique comportent la réponse de pénétration locale ainsi que le
comportement de déformation globale, en particulier dans la région de vitesse maximale de
ballistique, où la déformation par flexion est considérable. Les résultats de simulations
concordent avec ceux de ballistique. Avec la validité de la modélisation, la réponse à
l'impact à haute vitesse pour d'autres systèmes métals/fibres peuvent être examinés,
diminuant la fabrication par essai et erreur.
li
ACKNOWLEDGMENTS
1 would first like to express the deepest gratitude to Profess James A. Nemes for his
financial support, supervision, encouragement, suggestions and especially the creative ideas
while supervising this thesis work. 1 have learned a great deal from his teaching and his
help in writing papers.
1 would also like to thank my vice supervisor, post Doctor Ben Yahia Faycal. His
help with ABAQUS/CAE and Unix increasing the efficiency of my work and without his
help this thesis could not have been completed.
1 am really thankful to Christine EI-Lahham for their careful proofreading of the
thesis and Robert Glenns who translated the abstract of the thesis into French.
1 am deeply indebted to my wife Hongfang Wu and my whole family, whose love,
support and encouragement will always sustain me.
Finally 1 would like to extend my sincere gratitude to everybody in McGill University
who helped me complete this study.
1ll
TABLE OF CONTENTS
Abstract ...................................................................................................................................... i
Resume ..................................................................................................................................... ii
Acknowledgments ................................................................................................................... iii
Table of Contents ..................................................................................................................... iv
List of Figures .......................................................................................................................... vi
List of Tables ......................................................................................................................... viii
Glossary ................................................................................................................................... ix
Chapter 1 Introduction .......................................................................................................... 1
1.1 What Are Fiber-reinforced Metal Laminates (FML)? .......................................... 1
1.2 The Advantages of FML ...................................................................................... 2
1.3 Applications ofFML ............................................................................................ 3
1.4 Introduction to Experiments ................................................................................. 4
1.4.1 Dynamic Punch Experiments .................................................................. .4
1.4.2 Ballistic Experiments ............................................................................... 5
1.5 Previous Work ..................................................................................................... 5
1.5.1 Experimental Research of Penetration of Composite Materials ............... 6
1.5.2 N umerical Simulation of Ballistic Penetration of Composite Materials ... 6
1.6 Objective ofthis Project. ...................................................................................... 8
Chapter 2 Review of Material Models .................................................................................. 9
2.1 Johnson-Cook Model ........................................................................................... 9
2.2 Review of Composites Material's Failure Criteria ............................................. 10
2.2.1 The Maximum Stress and Strain Criteria ............................................... 11
2.2.2 Tsai-Wu Criterion .................................................................................. 12
2.2.3 Hashin Failure Criterion ......................................................................... 13
2.2.4 Tsai-Hill and Azzi-Tsai-Hill Criteria ..................................................... 14
2.2.5 Envelope of Different Type Failure Criteria ........................................... 15
2.3 Review of CDM for Fiber Composites .............................................................. 15
Chapter 3 Constitutive Model for Transversely Isotropie Material and Its Subroutine in ABAQUS ........................................................................................................... 18
3.1 Constitutive Equation of Transversely Isotropie Material .................................. 18
3.2 User Subroutine ................................................................................................. 24
IV
3.2.1 Overview of User Subroutine: VUMAT ................................................ 24
3.2.2 Governing Equations and Flow Chart .................................................... 24
3.2.3 Test ofSubroutine Using only one Element ........................................... 25
Chapter 4 Simulation and Results ....................................................................................... 27
4.1 Material Paralneters ........................................................................................... 27
4.2 Simulation ofDynamic Punch and Ballistic Experiments with ABAQUS ......... 35
Chapter 5
5.1
4.2.1 Punch Experiments ................................................................................ 35
4.2.2 Ballistic Experiments ............................................................................. 38
Parameters Research .......................................................................................... 47
The Effect of Parameters on Punch experiment ................................................ .47
5.1.1 The Effect of Ms ..................................................................................... 47
5.1.2 The Effect ofEd ..................................................................................... 49
5.1.3 The Effect of Mf and Mm ........................................................................ 50
5.2 The Effect ofParameters on Ballistic experiment .............................................. 51
5.2.1 The Effect ofEd ..................................................................................... 52
5.2.2 The Effect of Ms .................................................................................... 53
5.2.3 The Effect ofMfand Mm ........................................................................ 54
Chapter 6 Conclusions and Recommended Future Studies ................................................. 56
References ........................................................................................................................... 58
Appendix A Subroutine .......................................................................................................... 62
Appendix B Flowchart ........................................................................................................... 66
v
LIST OF FIGURES
Number Page Title
Figure 1.1 2 Configuration oftypical FML material (2.54 mm, GLARE 5)
Figure 1.2 3 Advanced composite, including carbon FiberglasslEpoxy (CFRP), FiberglasslThermoplastic and GLARE, are used extensively in the A380's primary and secondary structures
Figure 1.3 4 Schematic of the dynamic punch experiment
Figure 2.1 11 "on-axis" coordinates system with 1-axis along the fiber direction and 2-axis along matrix direction
Figure 2.2 15 Tsai-Hill versus maximum stress failure envelope (IF = 1.0)
Figure 2.3 15 Tsai-Hill versus Tsai-Wu failure envelope (IF = 1.0, F'.2 = 1.0)
Figure 2.4 15 Tsai-Hill versus Azzi-Tsai-Hill failure envelope (IF = 1.0)
Figure 2.5 17 The transformation ofthel-D damaged bar to the I-D effective homogenized bar
Figure 3.1 24 Strain/Stress response in uniaxial strain for different value Mf
Figure 3.2 26 Predicted response for the S2-GlasslEpoxy
Figure 4.1 29 Stress/Strain transformation between on-axis and off-axis
Figure 4.2 34 In-plane modulus of [0/90]s fiberglass prepreg ply
Figure 4.3 36 Configuration of punch experiment used in simulations
Figure 4.4 37 Predicted and measured force vs. displacement response in dynamic punch of Glare
Figure 4.5 37 The comparison of energy absorbed in dynamic punch of Glare
Figure 4.6 38 Section through the punched region
Figure 4.7 39 Configuration ofballistic experiment used in simulations
Figure 4.8 41 Ballistic limit vs. areal weight density of2024-T3 Aluminum at room temperature
vi
Figure 4.9 43 Ballistic limit vs. areal weight density of GLARE panels at room temperature
Figure 4.10 44 The effect of modeling assumption on the response of two layer GLARE
Figure 4.11 45 Sequence of penetration of the 2/1 GLARE-5 at impact velocity of 112 mis showing contours of plastic strain in the Aluminum and failure in the composite layers
Figure 4.12 46 Ballistic penetration, showing contours of plastic strain in the Aluminum. Failed elements in composite layers have been deleted
Figure 5.1 48 Load-displacement response of2.54 mm GLARE for different Ms
Figure 5.2 48 Load-displacement response of 1.9304 mm GLARE for different Ms
Figure 5.3 49 Load-displacement response of2.54 mm GLARE for different Ed
Figure 5.4 50 Load-displacement response of 1.9304 mm GLARE for different Ed
Figure 5.5 51 Load-displacement response of2.54 mm GLARE for different Mf and Mm
Figure 5.6 51 Load-displacement response of 1.9304 mm GLARE for different Mf and Mm
Figure 5.7 53 Ballistic limit ofGLARE for different Ed (Mt=Mm=Ms=0.2)
Figure 5.8 54 Ballistic limit ofGLARE for different Ms (Mt=Mm=O.2, Ed=1.1)
Figure 5.9 55 Ballistic limit ofGLARE for different Mfand Mm (Ms=O.2, Ed=1.1)
vü
LIST OF TABLES
Number Page Title
Table 4.1 32 Mechanical properties of the unidirectional S2 GlasslEpoxy prepreg
Parameters of the quasi-transverse isotropic lamina Table 4.2 35
([ 0/90]s S2-Glass /Epoxy )
Table 4.3 35 Parameters describing the behavior of Aluminum
Table 4.4 40 Experimental and computed ballistic limits of Aluminum
Table 4.5 41 Description of GLARE used in ballistic experiments
Table 4.6 42 Experimental and computed ballistic limits of GLARE
Table 5.1 52 Ballistic limit ofGLARE for different Ed (MFMm=Ms=0.2)
Table 5.2 53 Ballistic limit of GLARE for different Ms (MFMm=0.2, Ed= 1.1)
Table 5.3 54 Ballistic limit of GLARE for different Mf and Mm (Ms=O .2, Ed= 1.1)
vüi
s
r-O-z
Â
GLOSSARY
The equivalent plastic strain rate
Material parameter in Johnson-Cook model
The transition temperature
The non-dimensional temperature
The melt temperature of material
Tsai-Wu coefficients, where i, j = 1,2,6
The equivalent stress
The equivalent strain
The equivalent plastic strain
The ultimate strength in fiber direction, where i = I,e
The ultimate strain in fiber direction, where i = I,e
The ultimate strength in matrix direction, where i = l, e
The ultimate strain in matrix direction, where i = I,e
The ultimate shear stress
The ultimate shear strain
The equi-biaxial stress at failure
Constant between 0 and 1
Effective stress
Cylindrical coordinate system
Effective cross-sectional area
Stress components in cylindrical coordinate system, where i,j=r,O,z
ix
h
A,B,n,c,m
D
M(D)
The components of the stiffness matrix, where i,} = 1,2,3,4
Displacement, where i = r,z,B
Scale factor in CDM model, where i = 1,2,3
The element deletion coefficient
Parameters used in CDM mode!, where i = f,m,s
Young's modulus, where i=r,B,z orp,t
Poisson's ratio, where i,} = r,B,z or p,t
Laminate resultant in direction i , where i = 1,2,3
The components of stiffuess matrix, where i,} = 1,2,6
The total thickness of laminate
Material parameters in Johnson-Cook model
Damage variable
Index of failure
Damage tensor
x
Chapter 1
INTRODUCTION
1.1 What Are Fiber-reinforced Metal Laminates (FML)?
With the advancement of the aircraft industry, the load in the fuselage skin has
increased many times, so the design of a modern pressurized fuselage shell structure
demands more refined techniques and improved materials (Vogelesang, et al. 1990).
Damage tolerance and fatigue properties of aircraft structures have been the main attention
of design engineer.
Many years ago the idea of using two materials to form a hybrid structural material to
overcome most of the disadvantages of both materials was born. In the mid 1970's
researchers at the Delft University of Technology found that putting fibers in the bond layer
between metal layers can improve the metal' s fatigue properties by bridging cracks to keep
them from growing. Since then Delft has done a great de al of research work with this kind
of material. In the late 1970's Delft first made an APALL laminate which is composed of
alternating layers of unidirectional aramid/epoxy laminate and Aluminum-alloy sheets, and
in the early 1980's switched to fiberglass (called GLARE - Glass fiber-reinforced
Aluminum). Now FML (fiber-reinforced metal laminates) have been widely accepted and
already have been used in many areas, especially in the aerospace industry.
Generally speaking, FML are hybrid composites consisting of alternating layers of
metal-alloy sheets and fiber-reinforced epoxy prepreg, which is usually regarded as a family
ofhighly damage tolerant materials with a high weight saving potential.
For example, Figure 1.1 shows one typical FML, which is built up by bonding three
Aluminum alloy sheets and two S2-GlasslEpoxy layers laminates alternately. The laminates
can be applied in various thicknesses, e.g. a 3/2 lay-up [Al/pre-preg/AI/pre-preg/AI], that is
a laminate with three Aluminum layers and two intermediate Fiberglass/Epoxy layers. The
FiberglasslEpoxy layers of a 3/2 lay-up can be multiple cross-plied 0/90 layers or can be
unidirectional, e.g.: [Al/oo/90o/Al/oo/90o/AI] is a cross-plied lay-up.
1
Aluminum S2-
Glass/Epoxy [0/90]s
Figure 1.1 Configuration oftypical FML material (2.54 mm, GLARE 5)
1.2 The Advantages of FML
Fiber metal laminates were primarily developed for fatigue prone areas of modern
civilian aircraft and are available commercially under a few different trade names.
However, after many years' research, Delft has found several grades of this material offer
additional advantages such as damage tolerance, fire resistance and impact resistance in
addition to high strength per weight. According to Vohrlrdsang, et al. (1995), the following
conclusions are acquired:
• Fatigue Behavior: FML exhibits crack growth rates 10-100 times slower than their
monolithic Aluminum constituents.
• Impact resistance: For the complete range of thickness, FML shows higher
resistance to cracking than non-clad 2024-T3 in a standard drop weight set-up. This
impact performance of FML is attributed to a favorable high strain rate
strengthening phenomenon which occurs in the glass fibers, combined with their
relatively high failure strain.
• Corrosion: FML has very high anti-corrosion properties. Due to the barrier role
played by the fiber-epoxy layers, while monolithic metal is fully penetrated, the
laminate is merely pitted to the first fiber-epoxy interface.
• Flame resistance: The flame resistance is a very important property for aircrafts.
Higher flame resistance can provide the passengers more time to escape in case of
2
fire. Experiments prove that the flame resistance of FML is much better th an
monolithic Aluminum alloys due to the fact that the glass fibers have high melting
point, which can protect the second Aluminum layer from melting for a much
longer time period.
1.3 Applications of FML
FML material will play a big role in the success of the world's largest commercial
aircraft due to its advantages. In fact, FML has already been used in production such as on
the C-17 aft cargo door and sorne transport aircraft flooring applications. And according to
Airbus' current plans, GLARE has been chosen for the upper fuselage shell of the A380
(now the largest aircraft in the world). It will carry 30 metric tons of structural composites,
primarily of carbon-Fiberglass/Epoxy or 16 percent of its airframe weight and results in a
weight saving of around 800 kg (Figure 1.2).
Figure 1.2 Advanced composites, including carbon FiberglasslEpoxy (CFRP),
Fiberglassrrhermoplastic and GLARE, are used extensively in the A380's primary and
secondary structures
3
1.4 Introduction to Experiments
This research work is mainly based on the simulation of two experiments, one is the
punch experiment which was performed by Nemes and Asamoah-Attiah (2001) at McGill
University, and the other is the ballistic impact experiment which was performed at NASA
Glenn Research Center (Hoo-Fatt and Lin, 2003).
Punch experiments were performed on two different types of GLARE-5 metal/fiber
laminates having a 3/2 layup, one of which has a thickness of 2.54 mm (.1 00 in.) and the
other has a thickness of 1.93 mm (.076 in.). The 2.54 mm thick panels consist of3 layers of
Aluminum 2024-T3, with thickness of .508 mm, alternating with 2 layers of S2-
Glass/Epoxy with thickness of .508 mm. The 1.93 mm panels are identical with the
exception of the thickness of the Aluminum layers, which are .305 mm. Each S2-
GlasslEpoxy layer has a layup of [0/90] s. The 1.524 mm thick panels also are used for
ballistic impact experiments in addition to the above two types of GLARE-5 laminates and
4 different of thickness of 2024-T3 Aluminum.
1.4.1 Dynamic Punch Experiments
Dynamic punch experiments were performed using a modified Hopkinson bar, in
which the incident bar serves as the punch and the transmitted bar serves as the die. Square
specimens cut to 25.4 mm x 25.4 mm are fit between the punch and die as shown
schematically in Figure 1.3. The punch has an outer diameter of 9.47 mm. The die has an
inner diameter of 9.70 mm and outer diameter of 12.70 mm. The narrow gap between the
punch bar and die tube results in very large strains and correspondingly large strain rates,
making the experiment ideal for simulating impacts, which may occur at much higher
velocities.
Die (Tœnsmitted)
Punch (tncident)
1 2
Figure 1.3 Schematic of the dynamic punch experiment
4
A striker bar of length .406 m and the same diameter as that of the incident bar, is
launched by agas gun, creating a compressive wave, which travels down the length of the
incident bar. At the specimen interface, a portion of the wave is reflected and a portion is
transmitted into the die tube. Gages on the incident and transmitted bars record the strain
history. Considering the bars to remain elastic and in a state of uniaxial stress, the force
history and displacement history in the specimen can be obtained using classical methods to
obtain the force vs. displacement response.
1.4.2 Ballistic Experiments
The ballistic experiments on the GLARE-5 composites were performed in the
Ballistic Impact Lab at NASA Glenn as described by Hoo-Fatt and Lin (2003). Panels of
17.80 cm x 17.80 cm were clamped on all sides in a fixture having a square aperture of
15.24 cm. Projectiles were flat-faced Ti-6AI-4V cylinders, 25.4 mm long with 12.7 mm
diameter. Ballistic limits were determined by performing tests at different velocities to
determine the lowest velocity at which penetration occurs. That velocity is defined as the
ballistic limit. In addition to the GLARE-5 laminates described ab ove, a 2/1 laminate
consisting oftwo .508 mm thick Aluminum and a .508 mm thick S2 glass was tested, along
with two panels constructed by bonding 2 identical 3/2 GLARE-5 panels together,
producing overall laminates twice the thickness. Five different thicknesses of the
Aluminum 2024-T3 were also tested as a baseline.
1.5 Previous Work
The response of composite materials to dynamic impact and ballistic penetration is
very important since it is used in aircraft structures and engine components where impact
loads can occur. The response of composite materials has been extensively investigated
both in experiment and simulation in the past two decades. However the study of
penetration mechanics for composite materials is still in its infancy. Early efforts mainly
concentrate on two areas, i.e. experimental research and numerical simulation. Numerical
methods have the potential to provide a good understanding of the whole penetration
process compared with traditional experimental methods, but it still must he validated by
experiments.
5
1.5.1 Experimental Research of Penetration of Composite Materials
One of the earliest experiments about impact response of composite materials is that
of Ross, et al. (1976) who investigated the effect of several important parameters including
fiber type, fiber orientation, ply-arrangement and matrix-fiber interaction on impact failure
mechanisms of composites. Jang, et al. (1989) investigated the response of hybrid
composites to low-velocity impact; results show that the stacking sequences in hybrid
laminates play a critical role controlling plastic deformation and delamination.
In the 1990's, more and more researchers became interested in impact experiments of
the composite material. Riendeau and Nemes (1996) performed the high rate experiment by
using a punch shear version of the split Hopkinson bar apparatus and investigated the rate
dependent response of AS4/3501-6. The result shows the peak punch load and displacement
at the peak are relatively insensitive to the punching speed. Nemes, et al. (1998)
investigated the effect of deformation rate on the penetration of graphite/epoxy, quasi
isotropic laminates with different thickness and stacking sequences. Results indicate that
stacking sequence has a slight effect on the load vs. displacement response, but the effect of
specimen thickness and loading rate are quite significant. Vlot, et al. (1998) investigated the
impact characteristics of fiber metal laminates (ARALL and GLARE) and characterize the
impact properties and dynamic behavior of FML in comparison with other high
performance aerospace structural materials. The results demonstrate the superior behavior
of GLARE compared with the other materials.
1.5.2 Numerical Simulation of Ballistic Penetrations of Composite Materials
For penetration problems, it is preferable to use numerical methods due to the
complexity of many impact events. However, numerical studies involving impact and
penetration in composite are relatively few in the Iiterature before the 1990's. With the
development and maturity of FEA technology in recent decades, more and more researchers
use numerical methods to study the impact response of ballistic penetration of composite
materials.
Royalance and Wang (1981) are two of the earliest researchers to use computer
simulation methods to study the ballistic penetration of the composite materiaI under impact
loads. Lee and Sun (1993) developed a model, which was incorporated into an FE code to
predict the ballistic Iimit. Good agreement between experimental data and computational
results were obtained. Sun, et al. (1995,1996,1997) used a punch curve, which was based on
6
the results of the experiments, as the "structural constitutive mode l" that captures the highly
nonlinear behavior of the laminate in the penetration process. This model was used in
conjunction with a special two-noded ring element to model damage processes during static
and dynamic penetration. The model predicts the residual velocity of the projectile at the
end of the penetration process. Chen, et al. (1997,1998) proposed the smoothed particle
hydrodynamics (SPH) method and by using it, the development of damage in laminate
composite materials subjected to high-velocity impact has been investigated. The numerical
results show that damage in laminates due to high velocity impact is caused not only by
perforation but also by a great deal of delamination.
Currently there is a trend to use the Continuum Damage Mechanics (CDM) concepts
to simulate the dynamic penetration of composite materials. Early researchers in this field
were Talreja (1985) and Dumont, et al. (1987) who first used the concepts of CDM to
predict the mechanical behavior of composite materials. Talreja used two damage variables
to represent fibre and matrix damage separately. However Dumont, et al. introduced two
scalar variables D and !1 (both of them vary from 1 to 0) to represent Y oung's modulus and
shear modulus reduction for tridirectional composite materials.
In 1992 both Li, et al. and Renard, et al. use the CDM to research the effect of cracks
in the fiber-reinforced composite materials. Li's model is used where the matrix cracks are
assumed to be an array of parallel cracks along fibers in the plies whereas Renard
considered the case where the cracks are transverse to fiber direction.
In 1995, Matzenmiller, et al., based on the work of Talreja (1985), proposed a
constitutive model (ML T model) for anisotropic damage in fiber-composites. In their model
there are three damage parameters, two of them are associated with the in-plane principal
lamina directions and one represents the effect of damage in shear. ML Ts growth law for
damage parameters wi (i = 1,2, S) is assumed to follow a Weibull distribution. Since then,
Nandlall, et al. (1998) and Williams, et al. (2001) used the MLT model to simulate the
ballistic response of S2-glass-fibre-reinforced plastic (GRP) laminates and with good
success.
Recently Johnson, et al. (2001) developed a continuum damage-mechanics (CDM)
model for fabric-reinforced composites as a framework within which both in-ply and
delamination failure may be modeled during impact loading. Damage-development
equations are derived and appropriate material parameters are determined from
experiments. This model allows the inter-laminar layers to be modeled and strength
7
reduction to be represented due to delamination; it also provides a computationally efficient
method for the analysis of large-scale structural parts.
1.6 Objective ofthis Project
As a new hybrid material, GLARE consists of two totally different materials, one is a
typical elastic-plastic material - Aluminum and the other is a typical brittle material -
Fiberglass/Epoxy. Two different materials couple together giving GLARE a better
performance than either alone. For composite materials, as mentioned in section 1.5, many
researchers already have done a lot of work and succeeded in using numerical methods to
simulate the dynamic response. However, when two different materials couple together,
how GLARE behaves under the impact load is not clear. In order to clearly understand the
failure mechanism and penetration process, it is very necessary to study it using numerical
methods and so far no one has simulated the impact response of hybrid materials like
GLARE.
The main objective of this project focuses on the development and implementation of
a composite material model based on continuum damage mechanics (CDM) into the
commercial FE code ABAQUS. Once the code was verified, this model together with the
Johnson-Cook model is used to simulate the Force vs. Displacement response under the
punch load and to predict the ballistic limits under the high-velocity impact for GLARE
laminates.
8
Chapter2
REVIEW OF MATERIAL MODELS
2.1 Johnson-Cook Model
The well-known Johnson-Cook Model (1983) has been widely used in metal plastic
deformation especially where it is subjected to large strains, high strain rates and high
temperatures. The Johnson-Cook model has the following characters:
• It is a particular type of Mises plasticity model with analytical forms of the
hardening law and rate dependence;
• It is suitable for high-stain-rate deformation of many materials, including most
metals;
• Parameters used in Johnson-Cook are sensitive to the computational algorithm, it is
very important to use proper parameters in order to get good results;
• It is typically used in adiabatic transient dynamic simulations.
The Johnson-Cook model can be described by equation (2.1):
CT = (A + BB/)(l +Cln ~p)(l-êm) Bo
(2.1)
where A, B, C, n, m and 1;0 are material parameters measured at or below the transition
temperature 8( ; ê is the non-dimensional temperature; (J is the equivalent stress, B p is the
equivalent plastic strain and 1; pis the equivalent plastic strain rate. They are defined by the
following equations:
9
Îi = {fo-o, )/( 0, -Omo<)
for B<BI
for Bt ::;; B::;; Bmell
for B> Bmell
(2.4)
Considering the Aluminum to be a rate-dependent solid and assuming a von Mises
yield criterion with isotropic hardening, the evolution of the yield surface is described in
terms of a single scalar variable, which is taken to be the equivalent stress a. In this work,
the effect oftemperature was not considered, so equation (2.1) becomes the following one:
(2.5)
In addition to the constitutive equation, it is necessary to include a failure model to
describe the ductile failure of the Aluminum. Although, numerous ductile failure models
have been proposed in recent decades, in this work a relatively simple model of failure
based on equivalent plastic strain is used. Material failure is said to occur when the
equivalent plastic strain reaches a critical value. In general, this critical value can be taken
as a function of strain rate, temperature, and state of stress. However, for simplicity a
constant value 8 1 is chosen. In ABAQUS when the Johnson-Cook model is used, the user
must specify the value of 8 f . When the equivalent plastic strain reaches the user given
critical value 8 1 , ABAQUS considers failure to occur in the material and the element will
be deleted.
2.2 Review of Composite Material's Failure Criteria
There are two types of failure criteria for unidirectional composites, stress based and
strain based criteria, but usually the strain-based criterion is preferable because strain is
easier to measure in practice.
AIl components in the following criteria use the so called "on-axis" coordinate system
defined in Figure 2.1, where the 1-2 plane is the laminate plane, the I-direction is the fiber
direction and the 2-direction is the matrix direction. Under this coordinate system, the stress
10
For simplicity, each criterion below uses the uniform symbol IF (Index of Failure)
to express whether material failure has occurred or not, i.e. if IF ;::: 1.0, failure occurs.
~1 Figure 2.1 "on-axis" coordinates system with 1-axis along the fiber direction and 2-axis
along matrix direction
2.2.1 The Maximum Stress and Strain Criteria
Maximum Stress Criterion
Assume that XI and Xc are the tension strength and the compression strength in the
1-direction; ~ and 1';. are the tension stress strength and the compression stress strength in
the 2-direction; S is the in 1-2 plane shear stress strength.
The failure occurs when any of the following inequalities is met:
For 0'11>0, if 0'11 ;::: XI' fiber tension failure occurs
For O'll <0, if 0'11 ~ Xc' fiber compression fai/ure occurs
For 0'22>0, if 0'22 ;:::~, matrix tension failure occurs (2.6)
For 0'22 <0, if 0'22 ~ 1';., matrix compression fai/ure occurs
if 10'121;::: S, in plane shear fai/ure occurs
The maximum stress failure criterion requires that:
1, = max(_11 -B.. ---R);::: 1.0 0' 0' 10' 1 F X' Y , S (2.7)
where
11
and
{1'; y-J:.
Maximum Strain Criterion
when 0"11>0
when 0"1l<0
when 0"22>0
when 0"22<0
(2.8)
(2.9)
The maximum strain Criterion is almost same as the maximum stress Criterion if the
stress components are substituted by strain components. Here XI" and X cc are the ultimate
tension strain and the ultimate compression strain in the I-direction; 1';" and J:." are the
ultimate tension strain and the ultimate compression strain in the 2-direction; S" is the in 1-
2 plane ultimate shear strain. The failure occurs when any of the following inequalities is
met:
For 8 11 >0, if 8 11 ;::: Xie' fiber tension failure occurs
For 8 11 <0, if 8 11 S; X cc ' fiber compression fai/ure occurs
For 822 >0, if 8 22 ;:::1';", matrix tension failure occurs
For 822 <0, if 8 22 s; J:.", matrix compression fai/ure occurs
if 18 121;::: S", in plane shear fai/ure occurs
The maximum stress failure criterion requires that:
where
and
2.2.2 Tsai-Wu Criterion
8 8 8 1 = max(_I_1 ---R --.ll..);::: 1.0 F X'y'S
& b' B
when 8 11 >0
when 8 11 <0
when 8 22 >0
when 8 22 <0
The Tsai-Wu failure criterion requires that
12
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
Where {1<;, F2' 1<; l' 1<;2' F22' F66} are Tsai-Wu coefficients, which are defined as follows:
Il Il 1 1 1 F. =-+- F =-+- F. =--- F =- F =-
1 X X' 2 Y y' 11 X X '22 yy' 66 8 2 t,' 1 c J c 1 c
(2.15)
O"bias is the equibiaxial stress at failure. If it is known, then
(2.16)
Otherwise,
Where -1.0::; j* ::; 1.0. The default value of j* is zero.
2.2.3 Hashin Failure Criterion
The Hashin failure criterion has sorne advantages over other criteria; one is that it can
predict the different modes of failure (i.e. fiber and matrix failure)
Tensile Fiber Mode (when 0"11 > 0 ):
or (2.17)
Compressive Fiber Mode (when 0"11 < 0 ):
(2.18)
Tensile Matrix Mode (wh en 0"22 + 0"33 > 0 ):
(2.19)
13
Compressive Matrix Mode «(722 + (733 < 0):
(2.20)
Where Sr is the transverse shear strength and S is the axial shear strength.
2.2.4 Tsai-Hill and Azzi-Tsai-Hill Criteria
The Tsai-Hill failure Criterion requires that:
where
and
y= {Y, i'c
when (711)0
when (711<0
when (722)0
when (722<0
(2.21)
(2.22)
(2.23)
The Azzi-Tsai-Hill failure Criterion is the same as the Tsai-Hill Criterion, except that
the absolute value of the cross product term is taken as:
(2.24)
The difference between the two failure criteria shows up only when lTll and lT22 have
opposite signs.
14
2.2.5 Envelope of Different Type Failure Criteria:
Figure 2.2 Tsai-Hill versus maximum stress failure envelope ( IF = 1.0)
Figure 2.3 Tsai-Hill versus Tsai-Wu failure envelope (IF = 1.0, F;2 = 1.0)
Figure 2.4 Tsai-Hill versus Azzi-Tsai-Hill failure envelope ( IF = 1.0).
From the failure envelopes (HKS inc., 2001) in the Œil -Œ22 plane, shown in Figures
2.2-2.4, we can see the difference for the different criteria.
2.3 Review of CDM for Fiber Composites
In section 2.2 a few of the most popular failure criteria for composite materials were
introduced. Generally different criteria have different advantages and disadvantages. For
15
example, the maximum stress and strain criterion is the roughest one but it is easy to use
due to ignoring the coupling of the stress/strain tensor. On the contrary, the Hashin
criterion can accurately predict failure and has the ability to distinguish between modes of
failure, but it is too complicated.
Whatever criteria are used, they don't consider the behavior after the failure occurs
and only simply assume that material is ideally brittle. This means that once the failure
criteria are satisfied, the material can't carry any more loads and the dominant stiffness and
stress components reduced to zero instantaneously. Generally speaking, this kind of criteria
is useful for anticipating the maximum load for product design under static loads, but for
post-failure analysis, for example in the impact response, or dynamic penetration, where the
post failure behavior is critical; the ideaIly brittle model is obviously unreasonable.
In 1958 Kachanov first introduced the concept of effective stress. Since then many
researchers such as Rabotnov (1963), Krajcinovic (1981,1984,1989) and Lemaitre (1984),
have devoted a lot of work to it in the past two decades, and founded the basic framework
of continuum damage mechanics (CDM). The key concept in CDM is the assumption that a
micromechanical process (micro-crack growth) can be treated as a macro level
homogenized continuum, irrespective of the damage state. Although CDM has succeeded in
being used in fields ranging from metals' elastic and plastic deformation to brittle fracture,
there are a few difficulties for the application ofCDM in composite materials. One ofthem
is first-, second-, even third- and fourth-order damage tensors are needed to account for the
anisotropy of the damage due to composite materials' anisotropic property. Even for first
order damage tensors, the model is very difficult to solve if coupling is considered between
the damage tensor and the stress tensor, and it is almost impossible to measure the damage
tensor in practice through experimental methods. That is the reason why most work in this
field was based on the scalar damage variables.
First, let's consider the I-D case where isotropic damage is assumed, as shown in
Figure 2.5. The left configuration is the I-D bar which contains aIl kinds of defects such as
voids and cracks, where 0', A, T are stress, cross-sectional area and tensile force. The
configuration b) is the configuration that is obtained by homogenized aIl defects in the bar,
where Ô', Â, T are effective stress, effective cross-sectional area and tensile force. Assume
that the original configuration and the effective homogenized configuration are subjected to
the same tensile force T, i.e.:
16
T=o-A=â (2.25)
000 ---r--.
}T ~ (TA c=::::> 0 0
~ (J" 0 4 La &
Figure 2.5 The transformation ofthel-D damaged bar to the I-D effective homogenized bar
In CDM, the damage variable D can be defined as in the following manner:
A
A-A D=-
A
Putting equation (2.26) into (2.25), we get:
A A 1 0- = 0--;::- = --0-
A I-D
(2.26)
(2.27)
Since the effective stress is the stress in the homogenized configuration, then:
(2.28)
Combining equations (2.27) and (2.28):
(2.29)
where M(D) is called the damage tensor. For a I-D isotropic damage case it is the scalar variable which can be defined as:
M(D) =_1_ I-D
(2.30)
Usually, for composite materials, M(D) is more complicated such that it can be a
second-order or even a fourth-order tensor.
17
Chapter 3
CONSTITUTIVE MODEL FOR TRANSVERSELY ISOTROPIC MATERIALAND ITS SUBROUTINE IN
ABAQUS
3.1 Constitutive Equation of Transversely Isotropie Material
In order to capture the stress and damage evolution during the penetration process in
fiber ply composites of GLARE, 3-D elements are more advantageous than 2-D plate/shell
elements. However an accurate 3D continuum analysis of a structure is a computationally
expensive proposition. Even with the advantage of finite element methods, most
engineering problems become extremely cumbersome to simulate with brick elements. It is
a good idea to reduce 3D continuum to 2D continuum analyses by using geometry and load
properties in the FEA method.
In this work, an axi-symmetric model is used in order to reduce computation time.
However there are two conditions that must be satisfied for the case of using the axi
symmetric mode!. One is that the material must be transversely isotropie, and the other one
is that the configuration of the model (including the geometry, the load and boundary
condition) also must be axi-symmetric. ln this work, the following assumptions are made in
order to use the axi-symmetric model:
Firstly, it is assumed that the Fiberglass/Epoxy layers can be considered as a
transversely isotropie material. Of course, the material properties require sorne changes for
this assumption. Chapter 4 will give more detai\ed information about how to transform the
unidirectional properties to the equivalent transversely isotropic properties, since GLARE
consists of alternating layers of aluminum-alloy sheets and fiberglass-reinforced epoxy
prepreg, whose fiberglass ply is cross-plied 0/90 layer. Strictly speaking, it is orthotropic
and not axi-symmetric, so considering the GLARE as a transversely isotropic material is
only an approximation.
The geometric configuration of the axi-symmetric model must be circular. However
the specimens are square in both the punch experiment experiment and the impact
experiment. The second assumption is to consider the specimens as circular rather than
square. In this work, the penetration area is concentrated in the center of specimen and it is
18
very small compared with the ove raIl size of the model, so it is reasonable to neglect the
effect ofthe edge area and to consider the model as axi-symmetric.
Through the above assumptions, the transversely isotropic model can be used. By
definition, the transversely isotropic symmetry is orthotropic plus symmetric with respect to
ail directions in a given plane. Usually for the axi-symmetric model it is more convenient to
use the r - e - z cylindrical coordinate system. Here we con si der symmetry in aIl
directions about the z- axis. Under this coordinate system, the stress and strain components
of the transversely isotropic material have the following relation:
(l-V~t ~)rEp (vI' +v~t ~)rEp (1 + vI' )vptr Et 0 0 0 El' El'
ur,
(1-V~t ~)rEp E 2 lirr
U eo (1 +Vp)Vpt _t r 0 0 0 lioo El' El'
U zz (l-v~)rE, 0 0 0 lizz (3.1) =
Ure El'
lirO
U rz 0 0 lirz
U ez 2(1+vp)
liOz Gt 0
S 0 Gt
where 1 1
and the r=1 2 2 / 2 /
= (1 +vp )(l-vp -2v/ Et/El') -vI' -2vpt Et El' -2vpvpt Et El'
elasticity "stiffness" matrix has only 5 independent elastic coefficients, which are
{Ep,E"vp,vp"G,} . Here the subscript p means in-plane, the subscript t means transverse
plane.
By considering the axi-symmetric condition, it is easy to show that not aIl strain/stress
tensor components are necessary; 6 stress and strain components can be reduced to 4.
Under the assumption ofaxi-symmetry, the displacements only are a function of {r,z},
illustrated by the following equation:
(3.2)
Therefore,
19
(3.3)
So:
(3.4)
For simplicity, the subscript notation {1,2,3,4} are used to substitute {rr,ee,zz,rz} :
(JI
(J2
(J3
(J4
and let:
(Jrr &1
<=> (Jzz
and &2
<=> (J(J(J &3
(Jrz &4
C11 = (l-vp/ E,/ Ep )rEp
C22 = ( 1-v / ) rEl
CI2 = (l+vp )vp,rE,
Cl3 = (vp +vpl2 EJ Ep )rEp
C44 =GI
&rr
&zz
&(J(J
&rz
Finally equation (3.1) can be reduced to equation (3.7) :
(JI CIl C I2 C 13 0 &1
(J2 C I2 C22 C I2 0 &2 =
(J3 C 13 C l2 Cil 0 &3
(J4 0 0 0 C44 &4
(3.5)
(3.6)
(3.7)
According to the continuum damage mechanics (CDM) theory (see equation (2.29))
the stress-strain relation can be rewritten as:
20
0"] Cil C]Z C13 0 8]
O"z (M(D)t
CIZ C22 CI2 0 8Z = (3.8) 0"3 C13 CIZ Cil 0 8 3
0"4 0 0 0 C44 8 4
For a fiber-reinforced composite material, the damage evolution is very complicated,
as mentioned before, the damage tensor M(D) could be a third- even higher tensor. For
simplicity, the fiberglass sub-Iayers, which are idealized as initially transversely isotropic
(in the r - e plane), are described using a continuum damage model, based in part on a
model used by Matzenmiller, et al. (1995) and Williams and Vaziri (2001). Assuming the
damage to be decoupled, the damage tensor is taken to have the form:
1 0 0 0
I-DI
0 0 0
M(D) = I-Dz
(3.9)
0 0 0 I-D3
0 0 0 I-D s
Where the phenomenological damage parameters, Dl, D2, D3 and Ds vary from 0 to 1
and represent modulus reduction under different loading conditions due to progressive
damage in the material. According to the basic concepts of the MLT constitutive model
(Matzenmiller, et al. 1995) adopted for this study, by physical meaning Dl and D2 represent
the effects of damage on the rand 8 direction (Both of them are fiber direction), D3
represents the effect of damage on z direction (Le. matrix direction) and Ds represents the
effect of damage on r-8 direction (i.e. shear direction).
Putting equation (3.9) into equation (3.8) gives:
0"1 I-DI 0 0 0 Cil CI2 CI3 0 8 1
O"z 0 I-Dz 0 0 CI2 C22 CI2 0 8 2 = (3.10) 0"3 0 0 I-D3 0 C13 CI2 Cil 0 8 3
0"4 0 0 0 I-D, 0 0 0 C44 8 4
21
Damage evolution is assumed to begin once a threshold value (elastic limit) of strain
is reached, as summarized in the expressions below:
o Xc .. <8( <XI ..
8( sXco
(3.11 )
X I8 s 8( < (Ed *XIJ
[ ( JMfJ 1 SF; * 8( l-SF; * exp ---
Mfe X('6
[ 1 (SF * JMIJ l-SF; * exp ___ (8(
Mfe X I8
1 8( ? (Ed *XI .. )
o 1';'8 < 8 2 < Y, ..
8 2 S 1'; ... (3.12)
8 2 ? Y, ..
o X",. < 8 3 <XI ..
8 3 S X co
(3.13)
Xlt.· S 8 3 S (Ed *X16 )
8 3 ? (Ed *XI .. )
8 4 < Is .. 1
8 4 ?IS .. I (3.14)
...L ...L ...L
where SF; =eM1 ,SF2 =eMm,SF; =eMs . X'c'Xcc,J';c,t:c and Sc are the in-plane, through-
thickness, and shear strain limits respective1y. Ed is the element deletion coefficient used to
describe the material capacity to carry load and should be larger than 1. Once the strain
reaches Ed*Xte in the in-plane direction, we assume that the material cannot carry load and
the element is deleted. Element deletion does not occur in the case of compressive strain.
22
The through thickness and shear components are reduced as described in equation (3.14)
when the damage threshold is reached without element deletion, since the material is still
able to carry load in the in-plane direction. Parameters Mf, Mm and Ms are used to control
the rate of modulus reduction in the post-elastic region. The larger the parameters Mf, Mm
and Ms are, the faster the material modulus decreases. They are a1so the material properties,
however their values are very hard to obtain from experiments. In this study, the numerical
experimental method (i.e. iteration) is used to ob tain those values, which satisfy the simulation
result both in punch and ballistic experiment. e is exponent function constant and equal to
2.7183 ...
For the special case - the uniaxial strain where:
(3.15)
The equation(3.10) becomes:
al (I-D1)Cll&1
a 2 (1-D2)Cl2&1 =
a 3 (1-D3)CI3 &1 (3.16)
a 4 0
Since &2 = &3 = 0, then D2 = D3 = 0, so equation (3.16) becomes:
al (1- D1)Cll&1
a 2 Cl2 &1 (3.17) =
a3 Cl3&1
a 4 0
Putting equation (3.11) into (3.17) we obtain the stress/strain relation in the r
direction as described by equation (3.18):
23
(_~( SrÎ Xli, )Mf J MIe Xlt:
0"1 = SF., xe Cil 8 1
0"1 =0
1 .. +09
/ 8 .. +08
<a-
I CL 6e+08 ~
~ (7.j
4e+08
2 .. +08
-0.04'" -O. '.
\ \ , , , ,
\ , , \ 1
\ \ \ , '. , , , "
8 1 :::; XCii
XCii <81 < X'ii
X, •. :::; 8 1 < EdX'1i 8 1 ~EdX'1i
Mf -1
Mf -5
- M:f --15
(3.18)
0.1
Figure 3.1 Strain/Stress response in uniaxial strain for different value Mf
The stress/strain response of an uniaxial strain case are shown in Figure 3.1, from it
we see we can control the post failure behavior by changing the parameter Mf, i.e. if we
want to increase the degradation speed after material failure, we can increase the values of
Mf. When Mf equals 15, once the strain exceeds (XT&) the stress decreases so fast that its
behavior is like a brittle material.
3.2 User Subroutine
3.2.1 Overview of User Subroutine: VUMAT
The ab ove model is implemented into the FE code ABAQUSlExplicit by using a
material subroutine referred to as VUMAT.
User subroutine VUMAT in ABAQUS (HKS Inc, 2001):
• is used when none of the existing material models included in the ABAQUS
materiallibrary accurately represents the behavior of the material to be modeled;
• is used to define the mechanical constitutive behavior of a material;
• can use and update solution-dependent state variables;
24
• can use any field variables that are passed in;
• cannot be used in an adiabatic analysis.
3.2.2 Governing Equations and Flow Chart
It is required that the stress - strain relations must be written in incremental form in
ABAQUSlExplicit. Differentiating equation (3.10) gives:
=
Le.
-dDI
o o o
I-DI
o o o
o -dD2
o o o
I-D2
o o
o o
-dD3
o o o
I-D3
o
o Cll Cl2 C13 0
o CI2 C22 CI2 0
o C13 Cl2 CIl 0
-dD,. 0 0 0 C44
o Cil CI2 Cn 0
o CI2 C22 Cl2 0
o CI3 Cl2 CIl 0
I-D, 0 0 0 C44
8 3
8 4
d81
d82
d83
d84
dal = (1- DI)(Clld81 + CI2d82 + Cl3 d83 ) - dDI (C1l81 + C12d82 + C13d83 )
da2 = (1- D2 )(C12d81 + C22 d82 +C12d83 ) -dD2 (CI281 + C22d82 + C12d83 )
da3 = (l-D3 )(C13d81 +C12d82 +Clld83)-dD3(C1381 +C12d82 +Clld83)
da4 = (1- Ds )C44d84 - dD.\.C4484
(3.19)
(3.20)
The code and the flow chart of the subroutine are shown in Appendix A and B.
3.2.3 Test of Subroutine Using only One Element
We must be very cautious that the implementation of any realistic constitutive model
requires extensive development and testing. Initial testing on a single element model with
prescribed traction loading is strongly recommended by ABAQUS. In order to make sure
our subroutine works properly; five different tests are done using one element in each test.
Those tests are pure tension in the r direction, pure compression in the r direction, pure
tension in the z direction, pure compression in the z direction and pure shear load in r-z
plane separately.
Of particular importance in this work are the tensile response of the fiberglass
material in the in-plane direction and the response in shear. The predicted response under
25
uniaxial in-plane stress for different values of the parameter Ed and the shear response for
different values of Ms are shown in Figure 3.2.
[1<10')
1.60
..... ..... CIJ 1.00
li! !
0.50
0.00 0.02 0.04 0.06 0.08
Straln E11
a)
[xIO'] 80.00 r--t--r-,---,--,-,-,-,.--,-,--r-"
60.00
N .... (f.)
; 40.00
~ ............ lis_02
.. '1 ....... _~ Ms" 15 ............ Ifs .. S
0.00 L..J--.l-..1........L.-J......J.........J.-...J.-J.-.l.......J-...I.-J..J
0.00 0.C2 0.04 0.08 0.08 0.10 0.12
Strain E12
b)
Figure 3.2 Predicted response for the S2-GlasslEpoxy a) in-plane tension (MFMs=Mm=O.2) b) transverse shear (Ed=!.1)
From Figure 3.2 a) we know that parameter Ed determines the limit strain of the
material to which it can deform without breaking. The larger Ed is, the further the element
can stretch without being deleted. However from Figure 3.2 b) we know the parameter Ms
(same as Mf and Mm) controls the degradation speed after the material reach the post failure
phase. i.e. if we want to increase the degradation speed after material failure, we just
increase the value of Ms.
26
Chapter 4
SIMULATION AND RESULTS
4.1 Material Parameters
For orthotropic elasticity, the strain-stress relation under the r - e - z coordinate
system is defined by equation( 4.1):
-VOr -Vzr 0 0 0 Err Eoo Ezz
-VrO -Vzo 0 0 0 &rr Err Eoo Ezz CYrr
&00 -Vrz -voz 1 0 0 0
CYoo
&zz Err Eoo Ezz CYzz (4.1) = &rO
0 0 0 0 0 CYrO
&rz G rO CYrz
&Oz 0 0 0 0 0 CYOz
G rz
0 0 0 0 0 1
G Oz
Transverse isotropy is a special subclass of orthotropy, which is characterized by a
plane of isotropy at every point in the material. Assuming the r - e plane to be the plane of
isotropy at every point, transverse isotropy requires:
Ep = Err = Eoo
El =Ezz
VIP = vzr = vzo
G, =Grz =Goz
27
(4.2)
where p and t stand for "in-plane" and "transverse" respeetively. Henee, while V'P has the
physieal interpretation of the Poisson's ratio whieh eharaeterizes the strain in the plane of
isotropy resulting from stress normal to it, v pl eharaeterizes the transverse strain in the
direction normal to the plane of isotropy resulting from stress in the plane of isotropy.
Generally the quantities VIp and VI'I are not equal but are related by Vip / El = V pl / E p .
For transversely isotropie materials if the engineering constants { El" El' vI" V pl' GI }
are known, the elastie stiffness matrix parameters {Ctp C22 , C12 , C13 , C44 } ean be obtained
using equation (3.6). However the fiberglass layers of GLARE are not transversely isotropie
sinee they consist of [0/ 90 l~ laminates, so the model eannot direetly be used. In order to
use the model deseribed in Chapter 3, we have assumed there exists a transverse isotropy
layer, whieh has the effective properties of the orthotropie fiberglass layer. Aeeording to
this assumption, 9 parameters {Err,Ezz,Eoo,vrz,vrO,vzO,Grz,GrO,GzO} of the orthotropie
material ean be reduced to 5 equivalent parameters {E p' El' vI" V pl' GI } of transversely
isotropie material, following the steps deseribed:
First we consider a sm aIl pieee of square [0/9oL laminate, as shown in Figure 4.1.
Assume the laminate plane is the r-O plane; for eonvenienee, here we use two eoordinate
systems known as "off-axis" (the 1-2 eoordinate system) and "on-axis" (the x-y coordinate
system) rather than r-O coordinate system. Stresses and strains can be expressed in
different coordinates, for example, the strain {cp C2' c6 }, {c: , c: ' c~ } and stress
{O""0"2' 0"6} ,{O": ,0": ,O":y} , where subscript {1,2,6} means the 1, 2 and shear direction in the off
axis coordinate, subscript {x,y,xy} means the 1, 2 and shear direction in the on-axis
coordinate system and superscript {0,90+0} means the 0 and 90+0 ply of laminates.
However whatever coordinates systems are used, the aetual "state of stress or strain"
remains unchanged.
28
ur. ~ ........ a!
~
• :2
~l
~DtCI)
T~
~ x
Ca) 9nss Re5U.Ùl1l:S (Il) ln-Plane bh (c) Dn-.ll..JŒ P:tf::tmh (d) Qn..ll..JŒ P:tf Stress
Figure 4.1 Stress/Strain transformation between on-axis and off-axis
Since the different layers of [0/90]s are bonded together and are relatively thin, we
assume the different layers have the same strain in any direction. According to this
assumption and through the transformation of stresses and strains from one coordinate
system to another, the relation between the in-plane stress resultants {Np N2 , N6 } and the in-
plane strain {8p 8 2,86} is given by equation (4.3). The detailed information is described by
Lessard (1999).
(4.3)
29
defined by equation (4.4)
~ = U l + ~*U2 + V2'U3 h
~2 = U l - ~'U2 + V2*U3
AI2 -U -V:U h - 4 2 3
A66 - U -V'U h - 5 2 3
Al6 = .LV' + V' h 2 3 4
A26 =.L V' - V' h 2 3 4
(4.4)
Where h is the total thickness of the laminate, {~* ,v2*'~', V4*} is defined by equation (4.5)
~. = h cos(281)+ fz cos(282 )+ h cos(283 )+··· V2* = h cos ( 481 ) + fz cos ( 482 ) + h cos ( 483 ) + ...
~. = h sin(281)+ fz sin(282 )+ h sin(283 )+'" V4* = h sin( 481)+ 12 sin (482 )+ h sin (483 ) + ...
(4.5)
where {1;, i = 1,2,3 .. ·} is volume fraction of plies with 8i orientation, and
h+fz+h+"·=l.
30
(4.6)
For the GLARE material, since fiber plies of prepreg consist of [0/901,<
direction B .
~* = cos (2B] ) + cos ( 2 ( B] + 90) ) = 0
V; = cos ( 4B] ) + cos ( 4 ( B] + 90) ) = cos ( 4B] )
V;* = sin (2BI )+ sin (2( BI + 90)) = 0
V4* = sin (4B]) + sin (4( B] + 90)) = sin (4BI)
(4.7)
We have assumed there exists an equivalent transverse isotropy layer, which has the
effective properties of the orthotropic fiberglass layer. Here we use the arithmetic average
of the modulus of general laminates as the equivalent modulus of the transverse isotropy
layer. . In fact such an equivalent transverse isotropy layer doesn't exist, and strictly
speaking it should be called the quasi-transverse isotropy layer, so the equivalent modulus
obtained here by use the arithmetic average are only approximations.
31
tr tr tr
[:' AI2 ~" ]
f AlJde f A\2de f AI6de tr tr
Az2 Az6 = f Az2de f Az6
de (4.8)
~6 tr
S f AlJde
Inverting the modulus matrix (4.8) we can get the compliance matrix:
(4.9)
From the compliance in equation(4.9), we can calculate the effective engineering
constants:
P (kg/m)
1980
In-plane longitudinal modulus
In-plane transverse modulus
In-plane shear modulus
(4.10) ° a\2 In-plane Poisson's ratio = V 21 =--
aIl
° a61 In-plane shear coupling coefficient = V61 =-ail
In-plane normal coupling coefficient = V106 = ~ a66
Table 4.1 Mechanical properties of the unidirectional S2 Glass/Epoxy prepreg
(Hoo Fatt and Lin, 2003)
E" E2r E)3 G'2"'Ü13"'Ü2) Xl Xc YI Y, VI]=V12 V23
(GPa) (GPa) (GPa) (MPa) (MPa) (MPa) (MPa)
52 17 7 0.25 0.32 1779 1040 93.5 274.5
S
(MPa)
77
Table 4.1 contains the mechanical properties of unidirectional S2 GlasslEpoxy
prepreg in the GLARE material. Substituting these into equation (4.4), we get:
32
A _II = 30.9994 + 4.22025 cos( 48) h
~2 = 30.9994 + 4.22025 cos( 48) h
A12 = 8.5589-4.22025cos(48) h
A66 = 15.5589-4.22025 cos(48) h
AI6 = 4.22025 sine 48) h
~6 = 4.22025 sine 48) h
and Figure 4.2 illustrates the {AIl' ~2' A12' A66' A16' ~6} change with ().
A11 ~30~----~~~----+---~~~~----+-~ CIl
P-I ~ '-"'
~20r-----r----+--~+=~~-----r----+-~ o<t: .. I.D
~ A12 ~10~----~--~~--+---~~=-~----+-~
,E .. .....
....;.;' A16 ~ O~~--~---+----~~~~--~----~~
o 0.25 0.5 0.75 1 1.25 1.5 e (rad)
Figure 4.2 In-plane modulus of [0/90]s fiberglass prepreg ply
Putting equation (4.11) into (4.8), we get:
lA!! ~!2 ~!6] _[30.9994 8.5589 A22 ~6 - 30.9994
S A66 S
o ] o h
15.5589
Inverting (4.12) and using (4.9), we get:
33
(4.11 )
(4.12)
a12 a161_ 1 [0.0349207 -0.00964057 a22 a26 - - 0.0349207
h a62 a66 S
So the effective engineering constants of fiber ply in GLARE are:
EIO = 1/(allh) = 1/0.0349207 = 28.6363
Eg = 1/(a22h) = 1/0.0349207 = 28.6363
E~ = 1/(a66h) = 1/0.0642719 = 15.5589
vgl = -a12 /all = -(-0.00964057)/0.0349207 = 0.276099
V~I = a61 /all = 0
V?6 = a16 /a66 = 0
(4.13)
(4.14)
Now the effective engineering constants satisfy equation (4.2), we can use the
following parameters as the equivalent transverse isotropie constants:
Ep = 28.64 GPa
El =17 GPa
vp = 0.276
GI =7 GPa
vpl =0.3
Putting equation (4.15) into equation (3.6), we get:
Cil = 34.4 GPa
C22 = 19.94 GPa
Cl2 = 7.89 GPa
CI3 = 11.98 GPa
C44 = 7.0 GPa
(4.15)
(4.16)
Since the glass/epoxy is a rate-sensitive material, we must consider the strain rate
effect. According to Armenakas and Sciammarella (1973), the stiffness of unidirectional
glass/epoxy laminates increases by about 50% when loaded to strain rates of about 500 S·l.
Hoo-Fatt and Lin (2003) also assume that Eü and Gü are 1.5 times that of the static values.
34
By considering the global strain rate in our case, which is lower than 500 S-I, we use 1.2
times values of equation (4.16), illustrated in Table 4.2:
Cil C22
(GPa) (GPa)
41.31 23.93
Table 4.2 Parameters of the quasi-transverse isotropie lamina
( [0/ 90 l~ S2-Glass/Epoxy )
C]2 Cn C44
(GPa) (GPa) (GPa) X'E XCE Y'E YCE SE ED
9.47 14.37 8.4 0.05 -0.02 0.055 -0.078 0.012 1.1
Mr=Mm =Ms
0.2
For the Aluminum layers in GLARE, the Johnson-Cook model is used to simulate its
behavior and the following parameters (Table 4.3) are used:
Table 4.3 Parameters describing the behavior of Aluminum
p (kg/m3) E (GPa) v A (MPa) B (MPa) C n "a Er
2780 72.4 0.33 350 140 0.022 0.1 1.0 1.0
4.2 Simulation of Dynamic Punch and Ballistic Experiments with
ABAQUS
The dynamic punch and ballistic experiments are simulated using the commercial
finite element code ABAQUS/Explicit with the CDM description for the S2-GlasslEpoxy
implemented as a user subroutine. The material parameters used for the two material
descriptions are shown in Tables 4.2 and 4.3. Four-node bilinear reduced integration
elements are utilized. A minimum of 4 elements is used through the thickness of each sub
layer and aspects ratios in the penetration area are maintained close to 1.
4.2.1 Punch Experiments
For punch experiments, the two GLARE-5 samples with 3/2 layups are tested using
the Hopkinson bar. The results of the punch experiments indicate the punch penetrates the
35
GLARE at a velocity of approximately 20 rn/s, remaining essentially constant throughout
the duration of punching. So when the punch experiments are simulated, we use this
velocity as a boundary condition of the punch and a fixed boundary condition for the die.
The configuration for the punch experiment is shown in Figure 4.3.
Punch (Velocity = 20 mis)
Aluminum
Axial Symmetry FibergiassJEpoxy (Red)
Die (Fixed)
Figure 4.3 Configuration of punch experiment used in simulations
The force vs. displacement response curve of the punch experiment is one of the best
ways to understand the material's dynamic behavior and it is has been obtained by Nemes
and Asamoah (2001). In ABAQUS/CAE the force vs. displacement response curve can be
easily acquired by plotting the reaction force vs. displacement of punch. The results are
compared with those obtained from the experiments in Figure 4.4. Another thing to better
understand the dynamic behavior is comparison of the energy absorbed in the penetration
process, shown in Figure 4.5. As can be seen in the Figure, the following conclusion can be
obtained:
• Both peak force and punch energy agree quite weIl for the two specimens.
36
• There are discrepancies in the displacements at which peak force occurs, Le. the
peak force of simulation occurs earlier than the experimental one. One reason is
that the punch and the die are considered as rigid bodies in the model and
therefore the elastic deformation in the punch and die is neglected.
[x10') 20.00 .---r--cl\-r-----r-,--.----r-.,..---,
.. r~ .. ~' .aAQus l'''V i~ Bxpariment
16.00 1 'I~~\ /1 ! ~r~\
[x10') 15.00 .---r----.-.,..--,--.--.--,---,
Bxperiment
"0 10.00 ~ ,/ ~v ~\tl\'
,1 \ Il \,,-, 5.00 : '\ i\ " .' .. ! n r\ ",
1: V ~V'lJ 1 \ 0.00 ;.---'-----'---'---'------"\1'-----'-1--'----'
-1
0.00 DAO 0.80 1.20 1.60 [x10i 0.00 OAO 0.80 1.20 1.60 [X10'"!
Displacement [ml Displacement [ml
a) b)
Figure 4.4 Predicted and measured force vs. displacement response in dynamic punch
of Glare a) 2.54 mm thick b) 1.93 mm thick
Enera;y(J)
1&.00 ...---....---r. __ .,-.......---r-."'~-'"''''1'-.''''''''1'"'I
10.00
5.00
./ ,," /'
/ 8.00 ABAQUS
~ /
/"" ! Experiment !
J ,/ è /
li //'
ABAQUS ,
~" .. " 6.00
Experiment 4.00
2.00
0.00 !..";: ... ",,,.,....L_ ..... __ .I.. ...... J.._'"",,,J,,,,,,,,,,,,,,,k,,",,,_,U Time (s·l) 0.00 .-=--'-----'---'_-'----'-_'----'-----'-' Time (s·l)
0.00 \UO D.SC 1.20 1.60 [X10i 0.00 DAO 0.80 1.20 1.60 1)c10"l
a) b)
Figure 4.5 The comparison of energy absorbed in dynamic punch of Glare a) 2.54 mm
thick b) 1.93 mm thick
37
A section of the punched region, showing contours of plastic strain is compared with
the sectioned specimen in Figure 4.5. The deformation shape is very similar to that obtained
from experiment.
a) b)
Figure 4.6 Section through the punched region. a) Computed contours of plastic strain in
the Aluminum b) micrograph of sectioned sample
4.2.2 Ballistic Experiments
Ballistic experiments are simulated in a similar manner as the punch experiments
with the exception that the projectile is given a specified initial velocity and the outside of
the specimen is taken fixed as the boundary condition, as shown in Figure 4.7. For the
axisymmetric analysis the plate is modeled with a diameter of 15.24 cm rather than 15.24 x
15.24mm square.
38
-
,--~i9id body(lnltial Velocity)
FlxedEndl
Figure 4.7 Configuration of ballistic experiment used in simulations
The ballistic penetration is more complicated than the punch since it involves both
global deformation and the local penetration. It is difficult to obtain the force vs.
displacement response curve for the ballistic penetration experiment, so we can't compare
the simulation results with experimental ones. However of principal interest in the ballistic
penetration is the determination of the V 50 ballistic limit. The V 50 ballistic limit test is a
statistical test, originally developed by the U.S. military to evaluate hard armor. The V 50
ballistic limit velocity for a material is defined as that velocity for which the probability of
penetration of the chosen projectiles is exactly 0.5. In order to get the V 50 ballistic limit of
GLARE for different thickness GLARE panels, NASA Glenn Research Center has done a
large number of tests. To verify the validation of our subroutine, the simulation results are
compared with the test results from NASA Glenn.
For the simulation the V 50 ballistic limit is determined by performing a sequence of
analyses, which begin with a projectile velocity higher than the ballistic limit, then
decreasing the initial projectile velocity until the case where complete penetration does not
occur and projectile velocity decreases to zero.
In fact the ballistic limit is very sensitive and the exact value is very difficult to
determine numerically. However we can use the way illustrated as following to determine
39
the approximate ballistic limit with an error less than 1 mis (For the case there the ballistic
limit V50 is larger than 100 mis).
Assume the penetration energy of the material is constant in the experiments when
the same projectile with the different initial velocity is used, so
where m is the mass ofthe projectile,
Ep is the penetration energy,
V 50 is the ballistic limit;
I:!. V 50 is the error of ballistic limit,
Vrem is the velocity after the penetration.
Since vrcm = 5 and Vso > 100 , so
From equation (4.18) get:
25 AVso < - < 0.125m/ s
2Vso
(4.17)
(4.18)
(4.19)
Although the exact value of the ballistic limit is difficult to determine, the error of the
ballistic limit can be obtained within 0.125m/s by using the above method. In other word, if
we take the initial velocity of the projectile where the velocity after the penetration is less
than 5 mis as the ballistic limit V50, then the accuracy of the result is acceptable. One
advantage of using this method is that this takes fewer trials so it can reduce the
computation time.
4.2.2.1 Simulations for pure Aluminum panels
Prior to simulating ballistic penetration of GLARE, the experiments performed on
Aluminum panels are simulated. The thicknesses of Aluminum panels simulated are .508
mm, 1.6 mm, 3.175 mm and 6.35 mm separately. The resulting ballistic Iimit for each
40
thickness is shown in Table 4.4 and Figure 4.8. As can be seen in the Table 4.4, good
agreement is obtained over the range of thickness considered, which provides confidence in
the modeling approach.
Table 4.4 Experimental and computed ballistic limits of Aluminum
Thickness Areal weight Ballistic limit (mis) Material Error
(mm) density(N/m2) Experimental ABAQUS results
0.508 13.84 68 66 -2.94%
Aluminum 1.6002 43.60 130 137 5.38%
(2024-T3) 3.175 86.50 197 213 8.12%
6.350 173.00 215 220 2.33%
.. "if'sàlilstic limit v. lIreal wei~hl dell$~yoi2Ol4.J3 Aluminum . . .
250 ' .• ::.: "i':':': 'i" .... '[' , .... ·r·· .. · "1'''' :"1'":" "j'" ····1· .. ····]
~j,: ••••••• [ ••••••• !;I •• t··i-·~el~~l •• l .• u·' ""., l , .' ".,., <
J!r};I;~100 ----mt--· mrm----r--m--r---m.)Exp.eflmenti---m-j--.----j 1: : : : :ABA~US ;"': :"
50 -·-----f-·-----f-------f-------f-------!-------!------_! _______ ! _______ ~Z~;&~f:,i ! : l l l l l l L /'~,c l [1 l l l 1 L.;;:'
2D 40.,', SQ 80 ,100· ·120 140 ';J80 180 '·\At.al Wei9IitD~~~!y!t;t{~~)
Figure 4.8 Ballistic limit vs. areal weight density of2024-T3 Aluminum at room temperature
4.2.2.2 Simulations resuIts for GLARE-5 panels
There are five GLARE-5 samples modeled using the constitutive models described
previously. They are the two with 3/2 layups, the 2/1 GLARE-5, and bonded GLARE-5
41
samples with 3/2 layups as illustrated in Table 4.5. With the exception of the 1.9304 mm
thick GLARE, all of the panels had Aluminum and S2-Glass/Epoxy layers which were
0.508 mm thick. The 1.9304 mm thick GLARE had 0.508 mm thick S2-Glass/Epoxy layers
but 0.3048 mm thick Aluminum layers.
Table 4.5 Description of GLARE used in ballistic experiments
(Hoo Fatt and Lin, 2003)
Material AI/Glass-Epoxy Total Panel Thickness AI Thickness Layup (mm) (mm)
GLARE-5 AI/GE/AI 1.524 0.0508
GLARE-5 AI/GE/ AI/GE/AI 1.9304 0.03048
GLARE-5 AI/GE/ AI/GE/AI 2.54 0.0508
GLARE-5, 2-Layer (1.9304 mm) (AI/GE/AI/GE/AI)2 4.064 0.03048
GLARE-5, 2-Layer (2.54 mm) (AI/GE/AI/GE/Alh 5.2832 0.0508
Table 4.6 and Figure 4.9 show the comparison of the ballistic limit for the GLARE
specimens between ABAQUS results and experimental ones. From the results the following
conclusions can be obtained:
• The V 50 ballistic limits from simulation agree fairly well with the experimental
one, with the biggest error being 22.18% for 1.93mm thick GLARE. For
ballistic limit problems the se results are reasonable considering the nature of the
ballistic problems.
• The errors are negative for aIl cases Le. the simulations underestimate the actual
ballistic limit. One reason why the simulations may underestimate the energy
absorbed by the FiberglasslEpoxy layer in GLARE maybe that the delamination
energy is not considered. As mentioned by Hoo Fatt and Lin (2003), the energy
dissipated in delamination represented up to 9% of the total absorbed energy.
Table 4.6 Experimental and Computed Ballistic Limits of GLARE
Thickness Areal weight Ballistic Iimit (m/s) Material Errors
(mm) density(N/m2) Experimental
1 ABAQUS results
42
GLARE-5
GLARE-5
(2-1ayer)
1.524 37.54 137 112
1.9304 44.63 151 117.5
2.540 61.23 157 140
1.9304*2 89.26 186 160
2.54*2 122.46 211 190
, ',' ,. " ' , , ; ~'
Bam$t{~,!rfuUl:Y:i;~raill •. ............... ;.. ...... ;..,:' .. '..- ..... '..-'.;i,;;.· .. " .. ' ............. ... , , , ,
----·-i-- .... --~· ........ ·_~ ............ -........ .. , ,
, ........
----~ ~- ---- --:- ------~------1 1 1 1
1 : ~ .... r- : : ~~~~': : 1 .... """ .... 1 1 1
-~... : : : 1 1 1 1
-----t------i------~-------~------1 1 1 1 1 1 1 1 1 1 1 1 , ,
1 1 1 1 .......... T ............ ,- ...... -- -.- ........... -ï ........ --, , , , , , , ,
... ~ J--~""'~:
:.,. .................. : + ........ t'",w .... : 1
1 1 l , .,e ............ ~ ...... -- .. ~ ...... - .... -~- .. ----~--- .. --~·.P .
, , , ,
1\;' 1 (.",'
1 1 1 1 1·····:
------i------~-------~------~------t~:i. A;BAQIJS +.: ~ ..
: : : : ~~'i:,: E~perif11ent +--:
1 1 1 • i'
------~------~-------~------~------~~~ 1 1 1 1 1
1 : 1 : ... n~'· ,
-18.25%
-22.18%
-10.83%
-13.95%
-10.05%
Figure 4.9 Ballistic limit vs. areal weight density of GLARE panels at room temperature
4.2.2.3 The comparison of simulations results for two modeling assumptions
Since the strength of the bond between the two layers of GLARE is unknown the se
cases have been simulated using two different conditions. Under the first condition perfect
continuity is assumed whereas under the second condition only contact between the two
layers with friction is assumed. The difference in response for the two modeling
assumptions is shown in Figure 4.10.
43
Step Tlme' 9,7501E-05 Step Tlme' 9,7501E-05
a) b)
Figure 4.10 The effect ofmodeling assumption on the response oftwo layer GLARE a) Fully continuous b) Unbonded with friction on contact surface
Figure 4.10 shows that the unbonded model involves more global deformation than
the fully continuous one since it was easier for them to bend and stretch before failure, so it
can absorb more energy and has a higher ballistic limit. The other phenomenon is the
separation that occurs between two unbonded panels, similar to the delamination observed
in the experiment.
4.2.2.4 The penetration sequence
To better understand the sequence leading to penetration, the evolution of plastic
strain and damage in the composite layers during ballistic penetration is shown in Figure
4.11 for the case of the 2/1 GLARE-5 impacted at 112 rn/s. Because of the low flexural
rigidity of panel, the specimen separates from the projectile, except for contact at the
projectile edge. The edge contact results in significant plastic strain in the upper layer of
Aluminum. As the penetration proceeds, damage progresses to the composite layers, as
shown at 40 I-ls. Progressive failure of the composite seen at 80 I-ls leads to large plastic
strain in both the upper and lower Aluminum layers and resulting failure and penetration.
In addition, plastic strain also extends away from the penetration, which leads to the
permanent deformation ofthe panel, which was noted by Hoo-Fatt and Lin (2003).
44
Step TIme = 2.o001E..()5
a) c)
b) cl)
Figure 4.11 Sequence of penetration of the 2/1 GLARE-5 at impact velocity of 112 mis
showing contours of plastic strain in the Aluminum and failure in the composite layers.
4.2.2.5 Comparison of two different thicknesses
The simulations also show the difference in response depending on the thickness of
the specimen. The response of thinner specimens is dominated much more by flexural
behavior, whereas the thick specimens are dominated by the local shear, similar to that
observed in the high rate punch experiments. This can be clearly seen in Figure 4.12, which
shows a 2/1 layup of GLARE-5 and a bonded GLARE-5, both near penetration at velocities
close to their ballistic limit.
45
L, L, a) b)
Figure 4.12 Ballistic penetration, showing contours of plastic strain in the Aluminum.
Failed elements in composite layers have been deleted a) 2/1 layup GLARE at initial
velocity of 115 mis b) Bonded GLARE at initial velocity of 180 mis
46
Chapter 5
PARAMETERS RESEARCH
Parameters in the model including Ed, Mf, Mm and Ms which should be determined
from experiments, are very difficult to measure directly. The effect on the predicted stress
vs. strain response on the material was shown in Chapter 4. Here we can investigate how
much effect each of these parameters has on the punch and ballistic penetration results by
numerical methods. For example, in order to investigate the effect of Ed, we keep aIl other
parameters constants and vary Ed with different values, and then we can know how much Ed
affects material response by comparing the simulation result. The foIlowing sections mainly
discuss the effect of the parameters Ed, Ms, Mf and Mm on the simulation results.
5.1 The Effect of Parameters on Punch Experiment
5.1.1 The Effect of Ms
In order to investigate the effect of Ms, we fix MFMm=0.2 and Ed=l.l, and let Ms =
0.1, 0.2 and 2.0 separately. Figure 5.1 and Figure 5.2 are the force vs. displacement
simulation response curve for 2.54 mm and 1.9304 mm GLARE. From them we can see
that Ms has a large effect on the LoadlDisplacement response curve; the higher Ms is, the
lower the peak value of the force vs. displacement is. It is reasonable since for punch
experiment, the material mainly carries a shear stress in the narrow gap between the punch
and die, and Ms represents the shear failure degradation parameter where higher values
means that the degradation rate is quicker, so the peak value of the force vs. displacement
should be lower.
47
[x10')
20.00
16.00
Z 12.00 -"0
.§ 8.00
4.00
0.00 0.00 0.50 1.00
Displaœment lm] (Mf=Mm=0.2 Ed=1.1)
M,,=O.l M,,=O.2 M,,=2
1.50 [x10~
Figure 5.1 Load-displacement response of2.54 mm GLARE for different Ms
15.00 M3=O.2 M3=O.1 M,,=2
~ 10.00 "0
.§ 5.00
\\
Il \\W\~\ ' '\~i r~~
\JIU1. tJYy l! 0.00 '-----'-_--'-_-'----"'-'-----'-_-"-_...L.....I
0.00 0.50 1.00 1.50 [xi 0 ~ Displaœment lm]
(Ed=1.1 Mf=Mm=O.2)
Figure 5.2 Load-displacement response of 1.9304 mm GLARE for different Ms
48
5.1.2 The Effect of Ed
In the same manner, this time we fix Mf, Mm and Ms (aIl ofthem equal 0.2) and vary
Ed from 1.0, 1.1 and 1.3. The simulation results are illustrated in Figure 5.3 and Figure 5.4.
From the curves we can see the Ed has almost no effect on the peak value of the force vs.
displacement; but it increases slightly the width of the force/displacement response curve
when we increase Ed. In our model, Ed is used to describe the strain when the element is
deleted, i.e. when El = Ed X El f' the element is deleted, which means the larger Ed is, the later
the element is deleted, so the element can absorb more energy. From the Figures we see the
punch energy (areas under the curve) absorbed by the plate is increased since the width of
the curve is Ïncreased when we increase the value of Ed.
Z -"0
.3
[x101 1
20.00
15.00
10.00
5.00
0.00 '--_..1-.._-'-_-'
0.00 0.50 1.00
Displac::ement lm] (Mf=Mm=Ms=O.2)
Bd=1. 0 Bd=1. 1 Bd=1. 3
1.50 [x10"l
Figure 5.3 Load-displacement response of2.54 mm GLARE for different Ed
49
[x101)
15.00
j
/ 10.00 i\J z - r
~ i 5.00
0.00 .'------'-_ .... I_-'--=:.w "--'-'-_-'----'--'
0.00 0.50 1.00 [x10~ Displaœment [ml
(Mf=Mm=Ms=O.2)
Figure 5.4 Load-Displacement response of 1.9304 mm GLARE for different Ed
5.1.3 The Effect of Mf and Mm
Finally in order to investigate the effect of Mf and Mn, we keep Ms =0.2 and Ed = 1.1
as constants, and each time we let MFMm, the simulation results are illustrated in Figure
5.5 and Figure 5.6 for 2.54 mm and 1.9304 mm thickness GLARE. From the curves we see
Mf and Mm only have a small effect on the peak value of the force vs. displacement curve;
the higher Mf and Mm are, the lower peak value of the curve, but the difference is not so big.
Concerning the width of the response curve, the effect of Mf and Mm is not very apparent.
Because Mf and Mm are parameters used to describe the degradation for the fiber and matrix,
as we have said before, for the punch experiment, the main load case is shear load which is
mainly carried by matrix, so it is clear why Mf and Mm only have little effect on the force vs.
displacement curve.
50
:[
[x1011
20.00 ,---,----.-,----.------.---,----,
15.00
Mf=Mm=O.2 Mf=Mm=2 Mf=Mm=10
~ 10.00
5.00
Q.oo :,-. _---'-_---L_--l.-"'-.lL...!OII...I-~"--"-_____' 0.00 0.50 1.00 1.50 [x10j
Displaœment lm] (Ed=1.1 Ms=O.2)
Figure 5.5 Load-displacement response of2.54 mm GLARE for different Mf and Mm
[x10' 1 .--..-~-.--.--r-.-~~
15.00
;
1Q.00 ;\ f
:[ "l; "0
.3 5.00
0.00 '-----L_--1-_"'--""--'l~---'-_l.L..J.L-.:!.Wd 0.00 0.50 1.00 1.50 [x1 0 j
Displaœment lm] (Ed=1.1 Ms=O.2)
Figure 5.6 Load-displacement response of 1.9304 mm GLARE for different Mf and Mm
5.2 The Effect of Parameters on Ballistic Experiment
We know for the punch experiments, the plates mainly carry shear load in a very
narrow area, but for the ballistic experiments, there are different effects, which involve both
51
the local penetration response as weIl as the global deformation behavior, particularly at
velocities near the ballistic limit, where significant flexural deformation takes place. So the
effects of parameters Ed, Mf, Mm and Ms on the ballistic experiment may be somewhat
different from the punch experiment, and also we can only compare the ballistic limit rather
than comparing the force vs. displacement curve.
5.2.1 The Effeet ofEd
Table 5.1 Ballistic limit [mIs] ofGLARE for different Ed (MFMm=Ms=O.2)
Areal weight (N/m2) Ed=1.0 Ed=1.1 Ed=1.2 Ed=1.3
37.54 112 112 113 115
44.63 117 118 119 120
61.23 139 140 142 144
Table 5.1 contains the simulation results for different Ed with fixed MFMm=Ms=O.2.
We can see the trend more clearly in Figure 5.7. Comparing the force vs. displacement
response of the punch experiment where the Ed almost has no effect, the effect of Ed on
ballistic limit is obvious; GeneraIly, the higher value of Ed is, the higher ballistic limit is.
The reason is that for ballistic experiments the penetration area mainly involves the tension
due to the fixed ends of plate rather than shear; but for punch experiments the penetration
area has almost no tension; the failure is caused by shear.
52
45 ····so .......... 55;
.. ~l~~~I'!~~tDen.~I~lrdliJf) ..
Figure 5.7 Ballistic limit ofGLARE for different Ed (Mt=Mm=Ms=O.2)
5.2.2 The Effeet of Ms
For the punch experiments, the parameter Ms is a very important factor in the force vs.
displacement response curve, however for the ballistic experiment, as shown in Table 5.2
and illustrated in Figure 5.8, the effect of Ms is very small (less than 2% for ballistic limit).
It is reasonable since the ballistic penetration is mainly caused by fiber/matrix tension
failure rather than shear failure.
Table 5.2 Ballistic limit [mIs] ofGLARE for different Ms (Mt=Mm=O.2, Ed=l.l)
Area) weight (N/m2) Ms=O.l Ms=O.2 Ms=2 Ms=10
37.54 113 112 111 111
44.63 118 117 116 115
61.23 141 140 /39 139
53
Figure 5.8 Ballistic limit ofGLARE for different Ms (MFMm=O.2, Ed=l.l)
5.2.3 The Effect ofMrand Mm
The simulation results for different Mf and Mm with fixed Ed=l.l and Ms=O.2 are
listed in Table 5.3 and illustrated by Figure 5.9. Again comparing with the force vs.
displacement response ofthe punch experiment where the Mf and Mm have very little effect,
the effect of Mf and Mm on ballistic limit is much bigger. Generally, the higher value of Mf
and Mm are, the lower ballistic limit is. The reason is the same as the effect of Ed, because
for ballistic experiments the penetration area mainly involves fiber tension and matrix
compression rather than shear so Mf and Mm have very big effects on ballistic limits; but for
punch experiments the penetration area almost have no tension, the failure is caused by
shear, so the effect of Mf and Mm on force vs. displacement is very small.
Table 5.3 Ballistic limit of GLARE for different Mf and Mm (Ms=O.2, Ed= 1.1)
Areal weight (N/m2) MFMm=O.l MFMm=O.2 MFMm=2 MFMm=lO
37.54 112 112 111 109
44.63 118 117.5 116 114
61.23 141 140 133 132
54
, ~ ______ ~~:;~~~~~~~~~_c_t .~~ ~tl~M~~JfJ:~~_ ~~_I~i~~~c_~~{~t1\{L~~~_;~. ~ __ ._ o 0 o o
Mf= Mri1 = 0.1 1 1 1 l ,
- -- - -- - - -~ - - -- -- - - --~-- -- ---- - -~- - - --- -- - -~- -- - - - - - __ 0 " Mf:::" Mm-=n.:z 1 1 1 1 ., ,
, 1 1 l " ,
1 1 1 ,,,;: :
i' 0 0
- -- -- •• - -~ - - -- .-. - --~ -- - - ---- __ L. - - --- -- - -~- - ,~- - - --:-- - -- --- ---:- - - -- • ." i"
: ,/ : ,.' Mf= Mo/! = 2 '. ~7. ."'~o,,· Mf= Mrln = 10
________ -l- ----------:- ----------1- -----rI"~· -:- - - - :,;"<-"::,.:.;t:_ ----------1- -----"" 1 1 1 il ,' ... II' 1 1 1 1 ," 1 .4' ,., ,
: ,'~ ~4~ ,!,~.;': : - -- - -- -- -~ -- - --- - -- -~-- -- --- - 7-- ---,:,>-",,: -~~-- - - -- -- --:--- ----- ---:- - ----.'
,,': ,,~~~.~:., .. " : : :' ~~ ,"~"',. 1 1 1
___ ._. ___ ~ __ . _______ :.._ '! __ r!f~~~!~: _______ ~ __________ : ___________ : ______ _ l ,;;;" <".~ .... , 1 1
1 .' ," 1 1 1
l ""'~i+'''''' .".';: 1 :
..",~ ... ~~·~r .. "..: : : : - - -- -- -- -~".. :-::; ... lI:~-~:~...'- ---- - ---~-- -- --- - --i- - -- -- -- __ ,_w •••• __ • --•••• -.-~. ~>:<~\.
""'~~'i~~'~ ~ .. .".,.. : 1 1 ~:,> ~,.,., , ..... "" 1 1 1 l , ~:; ,< '
.,.J"" , 1 1 l , __ >\ .. -.; ... "'- -: ----_. ----~ ----------: ----------: ----------;- ----------;- ------, ~:i;,
Figure 5.9 Ballistic limit ofGLARE for different Mfand Mm (Ms=O.2, Ed=l.l)
55
Chapter 6
CONCLUSIONS AND RECOMMENDED FUTURE STUDIES
6.1 Conclusions
Modeling the GLARE material by combining a continuum damage model (CDM) for
the fiberglass and the Johnson-Cook model for the Aluminum into an explicit finite-element
code, quite reasonable simulations results of both the punch and the ballistic experiment are
obtained.
The results of the high rate punch experiment have been shown to exhibit very similar
deformation behavior to the ballistic experiment, particularly for the thicker materials.
For the punch experiment simulation, in addition to the peak value of the force vs.
displacement curve and punch energy, the deformation shape of simulation is very similar
to that obtained from the experiment. Similarly for the ballistic simulation, the Vso baIlistic
limits agree weIl with the experiment ones.
In order to better understand the behavior of the model and provide more instruction
about using of the mode l, in this work we have also examined the effect of different model
parameters, namely Mf, Mm, Ms and Ed, both in the punch experiment and the ballistic
experiment. The results seems reasonable for the effect of different parameters in both
cases.
However, the predictions for the GLARE laminates are not as good as the Aluminum
al one, perhaps indicating deficiencies in the modeling of the composite layers. A number
of assumptions were made in the way the composite material was considered, ail of which
require closer examination in future studies.
6.2 Recommended Future Studies
For simplicity, a number of assumptions were made in the way the composite
material was considered, aH of which require closer examination. In order to better
understand the failure mechanism of the composite material under the dynamic penetration
and achieve better simulation results, the foIlowing areas of further investigation are
recommended. 56
First, as one of the most important failure mechanisms for composite materials, the
delamination failure should be considered in further work. According to Hoo-Fart and Lin
(2003), the energy dissipated in delamination represented 2-9% ofthe total absorbed energy
in the penetration of GLARE. Delamination not only directly absorbs energy but also
reduces the bending stiffness of the laminates, which can improve the ballistic performance.
Second, for computational expediency, the [0/90]s laminate has been considered as a
transversely isotropie solid, which requires further study. When delamination must be
considered, the [0/90]s laminate can't be considered as a transversely isotropie solid since a
transversely isotropie solid neglects the orientation property of different plies which is one
of the main factors for delamination.
Third" in this work we increase the stiffness values to 1.2 times the original ones
considering the effect ofthe strain-rate. However in the penetration pro cess the strain rate is
not same in the different areas of the model. For example the strain-rate of the penetration
area is much higher than that of the edge areas. Obviously this model (the rate-independent
model) is not completely reasonable, and the use of a rate-dependent model should be
considered.
The last recommendation is to consider the effect of temperature. In the penetration
process, a lot of dynamic energy changes to heat through plastic deformation especially for
Aluminum materials, which could strongly affect its behavior.
57
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Composite Structures 8 (1987) 119-141
Follansbee P, The Hopkinson Bar, Metals Handbook, 9th Ed., Vol. 8 (1985) 198-203
HKS Inc., ABAQUS Theory Manual, Version 5.8 (1998)
HKS Inc., ABAQUSlExplicit User's Manual, Version 6.2 (2001)
Hoo-Fatt M and Lin C, Ballistic Impact of GLARETM Fiber-Metal Laminates, Submitted
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Damage in Composite Structures, Composites Science and Technology 61 (2001) 2183-
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Johnson GR and Cook WH, A Constitutive Model and Data for Metals Subjected to Large
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Appl Mech 48 (1981) 809-824
Krajcinovic D, Continuum Damage Mechanics, Appl Mech Rev 37(1) (1984) 1-6
Krajcinovic D, Damage Mechanics, Mech Mater 8(22/3) (1989) 117-97
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Blunt-Ended Projectile, Composites Science and Technology 49 (1993) 369-380
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245
Lessard L, Mechanics of Composite Materials (Course Notes), Department of Mechanical
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Li SX, Jiang CR and Han SL, Modeling of the Characteristics of Fiber-Reinforced
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61
APPENDIX A: SUBROUTINE
C
C USER SUBROUTINE VUMA T
SUBROUTINE VUMAT (
CREADONLY -
* * * * * *
NBLOCK, NDIR, NSHR, NSTATEV, NFIELDV, NPROPS, LANNEAL,
STEPTIME, TOTALTIME, DT, CMNAME, COORDMP, CHARLENGTH,
PROPS, DENSITY, STRAININC, RELSPININC,
TEMPOLD, STRETCHOLD, DEFGRADOLD, FIELDOLD,
STRESSOLD, STATEOLD, ENERINTERNOLD, ENERINELASOLD,
TEMPNEW, STRETCHNEW, DEFGRADNEW, FIELDNEW,
C WRITE ONL y -
C
C
* STRESSNEW, STATENEW, ENERINTERNNEW, ENERINELASNEW)
INCLUDE 'VABA]ARAM.INC'
DIMENSION COORDMP(NBLOCK, *), CHARLENGTH(NBLOCK), PROPS(NPROPS),
DENSITY(NBLOCK), STRAININC(NBLOCK,NDIR+NSHR),
2 RELSPININC(NBLOCK,NSHR), TEMPOLD(NBLOCK),
3 STRETCHOLD(NBLOCK,NDIR+NSHR),
4 DEFGRADOLD(NBLOCK,NDIR+NSHR+NSHR),
5 FIELDOLD(NBLOCK,NFIELDV), STRESSOLD(NBLOCK,NDIR+NSHR),
6 STATEOLD(NBLOCK,NSTATEV), ENERINTERNOLD(NBLOCK),
7 ENERINELASOLD(NBLOCK), TEMPNEW(NBLOCK),
8 STRETCHNEW(NBLOCK,NDIR+NSHR),
9 DEFGRADNEW(NBLOCK,NDIR+NSHR+NSHR),
FIELDNEW(NBLOCK,NFIELDV),
2 STRESSNEW(NBLOCK,NDIR+NSHR), STATENEW(NBLOCK,NSTATEV),
3 ENERINTERNNEW(NBLOCK), ENERINELASNEW(NBLOCK)
CHARACTER * 80 CMNAME
PARAME TER (MAXBLK = 64, e = 2.718281828)
ClIO = PROPS(l)
C220 = PROPS(2)
62
Cl20 = PROPS(3)
Cl30 = PROPS(4)
C440 = PROPS(5)
ElIfl = PROPS(6)
ElIf2 = PROPS(7)
E22fl = PROPS(8)
E22f2 = PROPS(9)
El2fU = PROPS(IO)
Mf = PROPS(13)
Mm = PROPS(14)
Ms = PROPS(15)
SFI = e**(1.0/Mf)
SF2 = e**(1.0/Mm)
SF3 = e**(1.0/Ms)
DO K = l, NBLOCK
STATEOLD(k,l)= 1
El = STATEOLD(K, 2)*SFI
E2 = STATEOLD(K, 3)*SF2
E3 = STATEOLD(K, 4)*SFI
E4 = STATEOLD(K, 5)*SF3
ElD = STRAININC(K, 1)*SFl
E2D = STRAININC(K, 2)*SF2
E3D = STRAININC(K, 3)*SFI
E4D = STRAININC(K, 4)*SF3
IF (el .GE. 0) THEN
VI = EXP(-(el/Ellfl)**Mf/(Mf*e»
VlD = (ellEllfl)**(Mf-l)/(e*Ellfl)
GOTO 10
ENDIF
IF (el .LT. SFl*Ellf2) THEN
VI = EXP(-(ellEllf2)**Mf/(Mf*e»
Vld=O
C VlD = (el/Ellf2)**(Mf-l)/(e*Ellf2)
63
GOTO 10
ENDIF
V1=1
V1d= 0
10 IF (e2 .GE. 0) THEN
V2 = EXP(-(e2/E22fl)**Mm/(Mm*e))
V2D = (e2/E22fl)**(Mm-1)/(e*E22fl)
GO TO 20
ENDIF
IF (e2 .LT. SF2*E22f2) THEN
C V2 = EXP(-(e2/E22f2)**Mm/(Mm*e))
V2= 1
V2d= O.
C V2D = (e2/E22f2)**(Mm-1)/(e*E22f2)
GO TO 20
ENDIF
V2= 1
V2d= 0
20 IF (e3 .GE. 0) THEN
V3 = EXP(-(e3/E11fl)**Mf/(Mf'I'e))
V3D = (e3/E1Ifl)**(Mf-1)/(e*El1fl)
GOTO 30
ENDIF
IF (e3 .LT. SFI *E11f2) THEN
V3 = EXP(-(e3/Ellf2)**Mf/(Mf'I'e))
V3d= 0
C V3D = (e3/El1f2)**(Mf-1)/(e*E11f2)
GOTO 30
ENDIF
V3 = 1
V3d=0
30 Vs = EXP(-(ABS(e4)/E12fO)**Ms/(Ms*e))
IF (ABS(e4) .GT. SF3*E12fO) THEN
Vsd= 0
GOT035
64
ENDIF
Vsd = (ABS(e4)/E12fO)**(Ms-l)/(e*E12fO)
35 IF (el .GT. (1.5*Sfl *Ellfl» GOTO 40
C IF (e2 .GT. (1.5*Sf2*E22fl» GOTO 40
IF (e3 .GT. (1.5*Sfl *Ellfl» GOTO 40
STATENEW(k,l) = STATEOLD(K,l)
GOTO 50
40 STATENEW(K, 1) = 0
50 STATENEW(K, 2) = STATEOLD(K, 2) + STRAININC(K, 1)
STATENEW(K, 3) = STATEOLD(K, 3) + STRAININC(K, 2)
STATENEW(K, 4) = STATEOLD(K, 4) + STRAININC(K, 3)
STATENEW(K, 5) = STATEOLD(K, 5) + STRAININC(K, 4)
STRESSNEW(K,l) = STRESSOLD(K,l)+Vl *(CllO*ElD+C120*E2D+C130*E3D)
I-Vl *Vld*ElD*(CllO*El+C120*E2+C130*E3)
STRESSNEW(K,2) = STRESSOLD(K,2)+V2*(C220*E2D+C120*ElD+C120*E3D)
I-V2*V2d*E2D*(C120*El+C220*E2+C120*E3)
STRESSNEW(K,3) = STRESSOLD(K,3)+V3*(CIIO*E3D+C130*ElD+C120*E2D)
1-V3*V3d*E3D*(C130*El +C120*E2+CIIO*E3)
STRESSNEW(K,4) = STRESSOLD(K,4)+2. *Vs*E4D*C440*(1-Vsd*E4)
END DO
RETURN
END
65
APPENDIX B: FLOWCHART
START
Initialization 01 param eters:
C'I,C22,C'2,Cj],C44,M I,M m,Ms,e'tf,e'LI
() rel < el </
Yes
Post lailure and apply ML T m odet
O<D s;1
Yes
Post lailure and apply ML Tm odet
O<D S;1
If (!.1 > e .1//
(! 3 < (! J <'1
y e s
Post lailure and apply ML Tm odet
O<D,S;1
le 121 > e 12 f
y e s
Post lailure and apply ML T m odet
O<D s S;1
1 f el> 1 . 1 e 1 If
(}re~>l.le'l
y e s
Element i5 deleted
No
No
No
No
No
(1- D,)(C"dc, + C"d,', + C"dc,)- dD,(C"c, + e"c, + C"c,)
(1- D,)(C 12 dc, + C"dc, + C"dc,)- dD,(C"c, + C"E, + C"c,)
(1- D,)(C"dc, + C"dc, + C"dL',)- dD,(C.,c, + C,,&, + ClIC,)
= (1 - lJ s ) C 44d s 4 - d D se 44 & "
END
66
linear M odet
D = 0
Linear M odet
D 2 = 0
Linear M odet
D, = 0
linear M odet
D.I' = 0