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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

REFERENCES

Armenakas AE and Sciammarella CA, Response of Glass-fiber-reinforced Epoxy

Specimens to High Rates of Tensile Loading, Experimental Mechanics 13 (1973) 433-440

Chaboche JL and Maire JF, A New Micromechanics Based CDM Model and Its

Application to CMC' s, Aerospace Science and Technology 6 (2002) 131-145

Chen JK and Medina DF, The Effects of Projectile Shape on Laminated Composite

Perforation, Composite Science and Technology 58 (1998) 1629-1639

Chen JK, Allahdadi FA and Carney TC, High Velocity Impact of Graphite/Epoxy

Composite Laminates, Composite Science and Technology 57 (1997) 1369-1379

Dumont JP, Ladeveze P, Poss M and Remond Y, Damage Mechanics for 3-D Composites,

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

for Publication (2003)

Jang BZ, Chen LC, Wang CZ, Lin HT and Zee RH, Impact Resistance and Energy

Absorption Mechanisms in Hybrid Composites, Composites Science and Technology 34

(1989) 305-335

Johnson AF, Pickett AK and Rozycki P, Computational Methods for Predicting Impact

Damage in Composite Structures, Composites Science and Technology 61 (2001) 2183-

2192

58

Johnson GR and Cook WH, A Constitutive Model and Data for Metals Subjected to Large

Strains, High Strain Rates and High Temperatures, Proceedings ofthe Seventh International

Symposium on Ballistic, The Hague, The Netherlands (1983) 541-547

Kachanov LM, On the Creep Rupture Time, Izv AN SSSR Otd Tekhn Nauk 8 (1958) 26-31

Krajcinovic D and Fonseka GU, The Continuous Damage Theory of Brittle Materials, J

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

Lee SWR and Sun CT, Dynamic Penetration of Graphite/Epoxy Laminates Impacted by A

Blunt-Ended Projectile, Composites Science and Technology 49 (1993) 369-380

Lemaitre J, How to Use Damage Mechanics, Nuclear Engineering and Design (1984) 233-

245

Lessard L, Mechanics of Composite Materials (Course Notes), Department of Mechanical

Engineering, McGill University (1999)

Li SX, Jiang CR and Han SL, Modeling of the Characteristics of Fiber-Reinforced

Composite Materials Damaged by Matrix-Cracking, Composites Science and Technology

43 (1992) 185-195

Matzenmiller A, Lubliner J and Taylor RL, A Constitutive Model for Anisotropie Damage

in Fiber-Composites, Mechanics ofMaterials 20 (1995) 125-152

NandlaIl D, Williams K and Vaziri R, Numerical Simulation of the Ballistic Response

GRP-Plates, Composites Science and Technology 58 (1998) 1463-1469

Nemes JA and Asamoah-Attiah S, Punch Testing of AluminumlFiberglass Laminates (July

2001).

59

Nemes JA, Eskandari H and Rakitch L, Effect of Laminate Parameters On Penetration of

GraphitelEpoxy Composites, International Journal Impact Engineering Vol. 21 (1998) 97-

112

Potti SV and Sun CT, Prediction of Impact Induced Penetration and Delamination in Thick

Composite Laminates, International Journal Impact Engineering, Vol. 19, No. 1 (1997) 31-

48

Rabotnov YN. On the Equations of State for Creep, Progress in Applied Mechanics, Prager

Anniversary Volume, New York: Macmillan (1963)

Renard J, Favre JP and Jeggy T, Influence of Transverse Cracking on Ply Behavior:

Introduction of A Characteristic, Composites Science and Technology 46 (1993) 29-37

Riendeau Sand Nemes JA, Dynamic Punch Behavior of AS4/3501-6, Journal of Composite

Materials, 30(13) (1996) 1494-1512

Ross CA, Cristescu N and Sierakowski RL, Experimental Studies On Failure Mechanisms

of Impacted Composite Plates, Fiber Science and Technology (9), Applied Science

Publishers Ud, England (1976)

Roylance D and Wang SS, Influence of Fiber Properties On Ballistic Penetration of Textile

Panels, Fiber Science and Technology 14 (1981) 183-190

Sun CT and Potti SV, A Simple Model to Predict Residual Velocities ofThick Composite

Laminates Subjected to High Velocity Impact, International Journal, Impact Engineering

Vol. 18. No. 3 (1996) 339-353

Sun CT and Potti SV, Modeling Dynamic Penetration Of Thick Section Composite

Laminates, AlAA/ ASME/ ASCE/ AHS Structures, Structural Dynamics and Materials

Conf., Collection ofTechnical Papers (1995) 383-393

Talreja RA, Continuum Mechanics Characterization of Damage in Composite Materials,

Proc. Royal Soc. London A399 (1985) 195-216.

60

Varna J, Joffe Rand Talreja R, A Synergistic Damage Mechanics Analysis of Transverse

Cracking in [±B / 904 ]s Laminates, Composites Science and Technology 61 (2001) 657 -

665

Vlot A, Kroon E and La Rocca G, Impact Response of Fiber Metal Laminates, Key

Engineering Material, Vol. 141-143 (1998) 235-276

Vogelesang LB, Gunnink JW, Roebroeks JHJJ, Muller RPG, Toward the Supportable and

Durable Aircraft Fuselage Structure, In: Grandage JM, Jost(Eds.) GS, Proceedings of the

18th Symposium of the International Committee on Aeronautical Fatigue, Melbourne,

Australia, 3-5 (May 1995) 257-272.

Vohrlrdsang LB and Schijve J, Fibre Metal Laminates: Damage Tolerant Aerospace

Materials, In: Case Studied in Manufacturing with Advanced Materials, Vol. 2, Elsevier,

ISBN: 0-444-88934-5 (1995) 259-260.

Williams KV and Vaziri R, Application of A Damage Mechanics Model for Predicting the

Impact Response Composite Materials, Computer and Structures 79 (2001) 997-1011

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