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MODELING DYNAMIC BEHAVIOUR INCLUDING SHOCKWAVE
PROPAGATION AND SPALL FAILURE IN ORTHOTROPIC MATERIALS
NORZARINA BINTI MA’AT
A project submitted in partial
fulfilment of the requirements for the award of the
Degree of Master of Mechanical Engineering
Faculty of Mechanical and Manufacturing Engineering
Universiti Tun Hussein Onn Malaysia
AUGUST 2017
iii
DEDICATION
To my beloved family, my husband and family in-law
iv
ACKNOWLEDGEMENT
“By the Name of Allah, Most Gracious, Most Merciful”
Alhamdulillah, praise be to Allah, finally I managed to complete this study.
I would like to express my sincere gratitude to my advisor Dr. Mohd Khir Bin
Mohd Nor for the continuous support during of my Master study and research, for his
patience, motivation, enthusiasm, and immense knowledge. His guidance helped me
in all the time of research and writing of this thesis.
I would like to acknowledge the Department of Mechanical Engineering and
manufacturing at University Tun Hussein Onn Malysia (UTHM). My graduate
experience benefitted greatly from the courses I took, the opportunities I had under Dr
Mohd Khir Bin Mohd Nor to serve as a graduate research assistant (GRA) and the
seminars that the department organized.
Last but not the least, I would like to thank my family: my parents En. Ma’at
Bin Abdul Ghani and Pn. Siti Khalijah Binti Rauna, for giving birth to me and
supporting me throughout my life. Thanks to my husband that always not give up to
me and supporting me. Also to my sibling and my family in-law that always give me
words of encouragement. I love you all.
.
v
ABSTRACT
In practice, most of the engineering materials manufactured using sheet metal forming
processes, are orthotropic. The technological demands on such materials are coming
from various manufacturing processes, aerospace structures, car crashworthiness and
defence. Much research has been performed involving analytical, experimental and
computational methods, it is generally accepted that there is still a need for improved
constitutive models. Moreover, there are numerous mechanics of materials issues that
have yet to be solved, related to finite strain deformation and failure of elastic and
plastic of material orthotropic. Based on this motivation, a constitutive model is
developed to predict a complex elastoplastic deformation behaviour involving
shockwaves and spall failure in orthotropic materials at high pressure. The important
feature of the proposed hyperelastic-plastic constitutive model formulated in this
research project is a Mandel stress tensor combined with the new generalised
orthotropic pressure. The formulation is developed in the isoclinic configuration and
allows for a unique treatment for elastic and plastic part. The stress tensor
decomposition of the new generalised pressure and Hill’s yield criterion aligned
uniquely within the principal stress space is adopted to characterize elastic and plastic
orthotropy. An isotropic hardening is adopted to define the evolution of plastic
orthotropy. The formulation is further combined with a shock equation of state (EOS)
and Grady spall failure model to predict shockwave propagation and spall failure in
the materials, respectively. The algorithm of the proposed constitutive model is
implemented as a new material model in the Lawrence Livermore National Laboratory
(LLNL)-DYNA3D code of UTHM’s version, named Material Type 92 (Mat92). The
𝝍 tensor used to define the alignment of the new yield surface is first validated in this
work. This is continued with an internal validation related to elastic isotropic, elastic
orthotropic and elastic-plastic orthotropic of the proposed formulation before a
comparison against range of Plate Impact Test data at 234ms−1, 450ms−1 and
895ms−1 impact velocities is performed. A good agreement is obtained in each test.
vi
ABSTRAK
Secara praktikal, kebanyakan pembuatan bahan kejuruteraan menggunakan proses
pembentukan kepingan logam. Permintaan teknologi terhadap bahan tersebut datang
dari pelbagai proses pembuatan, struktur aeroangkasa, kecacatan kemalangan dan
pertahanan. Banyak penyelidikan telah dilakukan melibatkan kaedah analitik,
eksperimen dan pengiraan, secara amnya diterima bahawa masih terdapat keperluan
untuk menambah baik model juzuk. Selain itu, terdapat banyak masalah mekanik
bahan yang masih belum dapat diselesaikan, yang berkaitan dengan perubahan bentuk
terikan yang terhingga dan kegagalan elastik dan plastik bahan ortotropik. Berdasarkan
motivasi ini, model juzuk dibangunkan untuk meramalkan tingkah laku ubah bentuk
elastoplastik kompleks yang melibatkan gelombang kejutan dan kegagalan spall dalam
bahan ortotropik pada tekanan tinggi. Ciri penting dalam model juzuk plastik yang
dicadangkan dalam projek penyelidikan ini ialah tensor tekanan Mandel yang
digabungkan dengan tekanan orthotropik yang baru. Perumusan dibangunkan dalam
konfigurasi isoklinik dan membolehkan rawatan untuk elastik dan plastik. Penguraian
tensor tegasan tekanan umum yang baru dan kriteria hasil Hill sejajar dengan unik di
dalam ruang tekanan utama yang digunakan untuk mencirikan orthotropi elastik dan
plastik. Pengerasan isotropik digunakan untuk menentukan evolusi orthotropi plastik.
Perumusan digabungkan dengan persamaan kejutan (EOS) dan model kegagalan
Grady untuk meramalkan penyebaran shockwave dan kegagalan spall dalam bahan.
Algoritma model yang dicadangkan ini dilaksanakan sebagai model bahan baru dalam
kod versi UTHM makmal Lawrence Livermore (LLNL) -DYNA3D, yang dinamakan
Material Type 92 (Mat92). 𝝍 tensor yang digunakan untuk menentukan penjajaran
permukaan hasil baru disahkan untuk peringkat pertama. Seterusnya pengesahan
dalaman yang berkaitan dengan formulasi isotopik elastik, ortotropik elastik dan
ortotropik elastik plastik yang dicadangkan sebelum perbandingan terhadap data ujian
plat impak pada halaju impak 234ms−1, 450ms−1 dan 895ms−1 dilakukan.
Perjanjian yang baik diperolehi dalam setiap ujian.
vii
TABLE OF CONTENTS
TITLE i
DECLARATION ii
DEDICATION iii
ACKNOWLEDGEMENT iv
ABSTRACT v
ABSTRAK vi
TABLE OF CONTENTS vii
LIST OF FIGURES xi
LIST OF TABLES xiv
LIST OF APPENDICES xv
CHAPTER 1 INTRODUCTION 1
1.1 Background of The Study 1
1.2 Problem Statement 2
1.3 Objective 2
1.4 Scope 3
1.5 Significant 4
CHAPTER 2 LITERATURE REVIEW 6
2.1 Introduction 6
2.2 Overview of Orthotropic Aluminium Alloys 6
2.3 Summary of Previous Research Work 7
2.4 Insight into Elasticity 16
2.5 Insight into Plasticity 20
2.6 Overview of Isotropic Plasticity 20
2.7 Insight into Anisotropic Plasticity 24
2.7.1 Yield Criteria for Plastic Anisotropy 24
i. Hill’s Orthotropic Yield Criterion 27
ii. Advantages of Hills Yield Criterion 29
viii
2.8 Suitable Decomposition Approach for
Orthotropic Materials
30
2.9 Formulation Related to Multiplicative
Decomposition
31
2.9.1 Rate of Deformation of the
Multiplicative Decomposition
34
2.9.2 Structural Tensor for the Principal
Directions of Materials Orthotropy
35
2.10 New Stress Tensor Decomposition for
Orthotropic Materials
36
2.10.1 New Alignment of Generalized
Orthotropic Pressure
39
2.11 Shockwave Propagation in Materials 40
2.12 Spall 43
2.12.1 Advantages of Grady Failure Model 45
2.13 Plate Impact Test 46
2.13.1 Insight into Plate Impact Test 47
2.14 Continuum Thermodynamic 50
2.14.1 The First Law of Thermodynamics 50
2.14.2 The Second Law of Thermodynamics 51
2.14.3 Helmholtz Free Energy 52
2.14.4 The Elastic and Plastic Parts of Free
Energy Function
53
2.15 Finite Element Code of LLNL-DYNA3D 54
2.15.1 Structure of LLNL-DYNA3D 54
2.15.2 Data Handling 55
2.15.3 Input 56
2.15.4 Initialisation 57
2.15.5 Restart 57
2.15.6 Solution 58
2.15.7 Output 59
ix
CHAPTER 3 METHODOLOGY 61
3.1 Introduction 61
3.2 Research Methodology 63
3.3 New Constitutive Formulation 66
3.3.1 Kinematic Assumptions 66
3.3.2 New Definition of Mandel stress tensor 69
i. Coupling with Equation of State 71
3.3.3 Elastic Free Energy Function 72
3.3.4 Orthotropic Yield Criterion 72
3.3.5 Evolution Equations 73
3.3.6 Grady Failure Model 75
3.4 Implementation of Constitutive Formulation in
DYNA3D Finite Element Code
77
4.3 3.4.1 Initial Implementation of New
Constitutive Model
78
3.4.2 Implementation of Equation of State
(EOS)
81
i. Generalized Orthotropic Pressure
Implementation
82
3.4.3 Elastoplastic with Hardening
Implementation
84
3.4.4 Grady Spall Model Implementation 85
CHAPTER 4 RESULTS AND ANALYSIS 87
4.0 Introduction 87
4.1 Validation Framework 89
4.2 New Yield Surface Alignment 89
4.3 Internal Validation 93
4.3.1 Finite Element (FE) Model for a Single
Element Analysis
93
4.3.2 Finite Element (FE) Model for Multiple
Element Analysis
94
4.3.3 Analysis of Elastic Isotropy
Formulation
95
x
i. Uniaxial Stress Analysis 96
ii. Uniaxial Strain Analysis 97
iii. Multiple Element Analysis 99
4.3.4 Analysis of Elastic Orthotropy
Formulation
101
i. Uniaxial Stress Analysis 103
ii. Uniaxial Strain Analysis 104
iii. Multiple Element Analysis 106
4.3.5 Analysis of Elastic-Plastic Orthotropy
Formulation
108
i. Isotropic Elastic-Perfectly Plastic
Analysis
109
ii. Orthotropic Elastic-Plastic with
Linear Hardening Analysis
112
4.4 Validation against Experimental Data of Plate
Impact Test
114
4.5 Summary of Result and Analysis 121
CHAPTER 5 CONCLUSION 123
5.0 Introduction 123
5.1 Summary and Conclusion 124
5.2 Recommendations for Future Research 125
REFERENCES 127
APPENDICES 138
xi
LIST OF FIGURES
2.1 Configuration of a multiplicative decomposition of the
deformation gradient
31
2.2 The conventional stress tensor representation 39
2.3 𝝍 and 𝜹 as a vector in a principal stress space. 40
2.4 Examples of incipient spall, intermediate and spall fracture 43
2.5 Schematic of a Plate Impact Test apparatus 46
2.6 Distance-time diagram 47
2.7 Ideal free surface velocity profile showing features corresponding
to various phenomena
48
2.8 Characteristics for plate impact test 49
2.9 Basic structure of DYNA3D code 55
2.10 Structure of main solution subroutine 60
2.11 Main subroutines of hexahedron element section, for strength
model requiring an equation of state
61
3.1 Research Phases 63
3.2 General framework for the research project methodology 64
3.3 Implementation algorithm in DYNA3D of UTHM’s Version 65
3.4 Definition of the chosen isoclinic configuration ��𝒊. 68
3.5 Common block for n3a pointer 79
3.6 Call statement for subroutine matin 79
3.7 Section for UTHM’s Material Model in subroutine matin 80
3.8 Additional data blocks for UTHM’s material models 80
3.9 Section to initialize parameters of UTHM’s material models 81
3.10 Section for UTHM Material Models in subroutine solde 82
3.11 Algorithm For Grady Spall implementation 87
4.1 Summary of the proposed validation method 90
xii
4.2 Values of 𝝍 tensor calculated in subroutine f3dm92 91
4.3 Values of 𝝍 tensor calculated in checking spreadsheet.xls 92
4.4 Identification of a unique yield surface using 𝝍 tensor 92
4.5 FE model for single element analysis 93
4.6 Solid elements used in the single element analysis 93
4.7 Configuration of the Plate Impact test simulation 95
4.8 Stress strain curves in x-direction of uniaxial stress, Mat92 vs.
Mat10
96
4.9 Stress strain curves in y-direction of uniaxial stress, Mat92 vs.
Mat10
97
4.10 Stress strain curves in z-direction of uniaxial stress, Mat92 vs.
Mat10
97
4.11 Stress strain curves in x-direction of uniaxial strain, Mat92 vs.
Mat10
98
4.12 Stress strain curves in y-direction of uniaxial strain, Mat92 vs.
Mat10
98
4.13 Stress strain curves in z-direction of uniaxial strain, Mat92 vs.
Mat10
99
4.14 Z-Stress vs. Time of Material Type 92 at 1ms−1 impact velocity 100
4.15 Z-Stress vs. Time of Material Type 10 at 1ms−1 impact velocity 100
4.16 Z-Stress vs. Time of Material Type 92 at 10ms−1 impact velocity 101
4.17 Z-Stress vs. Time of Material Type 10 at 10ms−1 impact velocity 101
4.18 Orthotropic material axes type 2 (AOPT 2), Lin (2004) 102
4.19 Stress strain curves of uniaxial stress in x-direction, Mat92 vs.
Mat22
103
4.20 Stress strain curves of uniaxial stress in x-direction, Mat92 vs.
Mat22,
104
4.21 Stress strain curves of uniaxial stress in z-direction, Mat92 vs.
Mat22
104
4.22 Stress strain curves of uniaxial strain in x-direction, Mat92 vs.
Mat22,
105
4.23 Stress strain curves of uniaxial strain in y-direction, Mat92 vs.
Mat22,
105
xiii
4.24 Stress strain curves of uniaxial strain in z-direction, Mat92 vs.
Mat22,
106
4.25 Z-Stress vs. Time of Material Type 22 at 1ms−1 impact velocity -
Elastic Orthotropy
107
4.26 Z-Stress vs. Time of Material Type 92 at 1ms−1 impact velocity -
Elastic Orthotropy
107
4.27 Stress vs. Time of Material Type 22 at 10ms−1 impact velocity -
Elastic Orthotropy
108
4.28 Z-Stress vs. Time of Material Type 92 at 10ms−1 impact velocity
- Elastic Orthotropy
108
4.29 Stress vs. strain curve, elastic-perfectly plastic in x-direction of
Mat92
110
4.30 Stress vs. strain curve, elastic-perfectly plastic in y-direction of
Mat92
110
4.31 Stress vs. strain curve, elastic-perfectly plastic in z-direction of
Mat92
111
4.32 Stress strain curve in x direction, Mat92 vs. Mat33 113
4.33 Stress strain curve in y direction, Mat92 vs. Mat33 113
4.34 Stress strain curve in z direction, Mat92 vs. Mat33 114
4.35 Longitudinal Stress at 234ms−1 impact in longitudinal direction 116
4.36 Longitudinal Stress at 234ms−1 impact in transverse direction 116
4.37 Longitudinal Stress at 450ms−1impact in longitudinal direction 117
4.38 Longitudinal Stress at 450ms−1 impact in transverse direction 117
4.39 Longitudinal stress at 895ms−1impact in longitudinal direction 118
4.40 Longitudinal stress at 895ms−1impact in transverse direction 118
4.41 The Advantages of New Material Model 112
xiv
LIST OF TABLES
2.1 Summary of Previous Research work in Modelling Constitutive
Model
7
3.1 Six subroutines involved in Material Type 33 78
4.1 Elastic Properties of AA7010 91
4.2 Boundary conditions for a single element analysis in 𝑥 direction 94
4.3 Aluminium material properties 96
4.4 Elastic orthotropy properties for aluminium alloy 102
4.5 Aluminium alloy properties for isotropic elastic-perfectly plastic
analysis
109
4.6 Summary of elastic-perfectly plastic analysis 111
4.7 Tantalum material properties 112
4.8 Material properties used in the Plate Impact test analysis 115
4.9 Mat92 vs. Plate Impact Test Data 119
xv
LIST OF APPENDICES
APPENDIX TITLE
PAGE
A The Clausius Plank Inequality 139
B Result of Elastic Orthotropy in uniaxial stress
analyses
142
C Gantt Chart 143
1
CHAPTER 1
INTRODUCTION
1.1 Background of The Study
Orthotropic material is common in many engineering applications. They are a subset
of anisotropic materials where the properties change as measured in three mutually
orthogonal symmetry planes and behave in far more complex compared to isotropic
materials. The example of familiar orthotropic materials is sheet of aluminium alloy
formed by squeezing thick sections of metal between heavy rollers. The strain rate
dependent and mechanical behaviour can be observed as important for applications
involving impact and dynamic loading in aerospace structures, car crashworthiness
and defence. For instance, aluminium alloy 7010 in temper conditions T7651 has been
extensively used as structure components in aero industry due to its high strength and
high resistance to stress corrosion cracking and good fracture toughness. The ability
to predict the deformation behaviour for many aluminium alloys undergoing finite
strain deformation therefore becoming more and more important and has attracted
attention designer and the user of metal structures for many years. This has brought
great challenges in understanding the behaviour of these materials from quasi-static to
high strain rate regimes as adopted in broad engineering applications (Chen et al.,
2009). To develop models capable of modelling deformation behaviour of aluminium
alloys, it is necessary to understand the formation and propagation of shock waves.
The constitutive models intended to represent plastic behaviour are of great importance
2
in the current design and analysis of forming processes due to their broad engineering
application (Barlat et al., 2003). The ability to appropriately capture the behaviour of
deformation processes in these applications is becoming more important (Mohd Nor,
2015).
1.2 Problem Statement
Much research has been carried out in respect to complex materials behaviour of
orthotropic materials under dynamic loading conditions, leading to results in various
technologies involving analytical, experimental and computational methods. Despite
of this current status, it is generally agreed that there is a need for improved constitutive
formulation as well as the corresponding procedures to identify the parameters for
these models. Modelling finite strain deformation and failure in such materials requires
an appropriate mathematical description. This is a real deal since an appropriate
formulation can be very complex, specifically to deal with the orientations of materials
orthotropy (Sitnikova et al., 2015). Moreover, there are numerous mechanics of
materials issues that have yet to be solved, related to orthotropic elastic and plastic
behaviour. For example, shape changes resulting from a deformation process on a
continuum level can be very complex when dealing with orthotropic materials since
the co-linearity of the principal axis of the stress and strain tensors is no longer in
place. Based on this motivation, this research project is conducted to develop a
constitutive formulation to predict the behaviour of commercial aluminium alloys
undergoing large deformation including failure.
1.3 Objective
The objective of this research project is:
1. To formulate an orthotropic constitutive model to predict the dynamic
deformation behaviour including shockwave propagation and spall failure of
orthotropic materials
3
2. To establish the DYNA3D finite element code of UTHM’s version as a good
simulation tool with a proper implementation guideline for a new constitutive
relation.
3. To develop a simplified validation approach for any constitutive model
formulated for orthotropic materials.
1.4 Scope
The scope of this research project is:
1. The development of a new constitutive formulation is defined within a
consistent thermodynamic framework adopting a multiplicative decomposition
of deformation gradient F.
2. The description of finite strain deformation within elastic regime is defined
using the new stress tensor decomposition generalized for orthotropic
materials.
3. A yield function is used to capture the initial plastic orthotropy and the
subsequent evolution in a unique alignment within the principal stress space.
4. The description for non-linear behaviour at high pressures is determined using
shock equation of state (EOS) of new generalized pressure and spall failure.
5. The implementation of the proposed formulation involves modification of
several subroutines in the DYNA3D finite element code of UTHM’s version.
6. The selected code is further enhanced to deal with more complicated analysis.
7. The implementation method is also developed as a guideline for other
constitutive models development in the code.
8. The numerical simulation results obtained using the new constitutive model is
internally validated using the newly simplified approach of single and multiple
element analyses.
9. New yield surface alignment is conducted to confirm the compliance and
stiffness matrices including the value of 𝝍 tensor is correctly calculated
4
10. Single and multiple element analysis is adopted to confirm the capability of
proposed constitutive model to predict the behaviour of orthotropic materials
under consideration and provide an efficient book keeping.
11. The final stage of validation process is a validation against the aluminium
alloys AA7010 experimental data of Plate Impact test.
12. Aluminium Alloy (AA7010) is used in this research work due to its strength,
light weight, good corrosion resistant and good toughness.
13. Plate impact test is used to investigate dynamic deformation and failure modes
of materials that require impacting a flyer plate at high strain rate against a
specimen (target).
14. The limitations of this research work are:
i. It is impossible to capture softening and to ensure the distributions of
plastic strains including the initiation and evolution of damage
parameters.
ii. Unable to capture strain rates and temperatures sensitivities of the
materials under consideration.
iii. Not capable to describe the Bauschinger effect of the materials.
iv. Unable to analyse a complex behaviour of orthotropic materials in three
dimensional mode including damage characteristics, and further produce
the corresponding input parameters.
1.5 Significant
The goal of this research project is to develop a constitutive model capable of
demonstrating the dynamic behaviour in commercial aluminium alloys including spall
failure at high pressures, which applicable in various engineering applications.
To develop new constitutive model, the required criteria is:
a) Able to demonstrate the main characteristic of orthotropic metals
b) Simple in mathematical, thus computationally uncomplicated.
c) Requires measurable input parameters.
5
The achievement of this research project signals a good guidance for appropriate
material models for commercial materials that can help towards better comprehension
of the materials behaviour which undergoing finite strain deformation.
The implementation of the first ever UTHM’s constitutive formulation in this
work is really important to guide the other researchers to develop more constitutive
formulation. This achievement contributes for a better prediction provided by the
DYNA3D of UTHM’s version. Further the finite element code can be commercialized
to meets real industrial applications.
In addition, the validation method developed in this work can be regarded as a
simplified approach for validation of any constitutive model formulation and its
implementation in any finite element simulation code.
6
CHAPTER 2
LITERATURE REVIEW
2.1 Introduction
This chapter is very important since a comprehensive literature review related to the
modelling deformation behaviour of orthotropic materials including the description of
a finite element code of LLNL-DYNA3D is concisely discussed. The chapter starts by
a discussion of orthotropic aluminium alloys. This is followed by a discussion of the
fundamental theory associated to the elasticity and plasticity. Further, the theory of
plasticity of isotropic and anisotropic materials is presented. Subsequently, the
decomposition suitable for orthotropic materials is discussed, before the multiplicative
decomposition of the deformation gradient 𝐅 is further considered. The chapter then
rigorously review the literature related to structural tensors, the new generalized
pressure for orthotropic materials, shockwave propagation in materials, spall criterion
including the related theory of thermodynamic framework. The chapter finally review
the background and the basic structure of the LLNL-DYNA3D code of UTHM’s
version.
2.2 Overview of Orthotropic Aluminium Alloys
Aluminium alloys with a high specific strength and excellent combination of
mechanical properties are widely used in many engineering applications such
automotive, aerospace and others. Its advantages are mainly due to its strength, light
7
weight, good thermal conductivity, high corrosion resistance and much easier to
machine and form than steel. It can be observed that such materials exhibit orthotropic
behaviour while undergoing large elasto-plastic deformation at unit-cell level due to
the preferred orientation as a result of various manufacturing processes (Mohd Nor,
2016). Sheet of aluminium alloys formed by squeezing thick sections of metal between
heavy rollers is an example of orthotropic materials produced in manufacturing
industry. There are many other examples such as Advanced High Strength Steel
(AHSS) and fibre-reinforced elastomers.
The prediction of aluminium alloys behaviour undergoing finite strain
deformation including shock wave propagation has attracted attention due to its broad
engineering applications. Therefore, the constitutive model that influence the
prediction capability in many finite element code specifically the behaviour within
plastic regime is very crucial (Anagnostopoulos et al., 2005). The formulation to
describe complex materials responses at high strain rate and pressures should be one
of integration between elasticity, plasticity and equation of state (EOS) including spall
failure. It is difficult for engineer and the user of metal structures to ignore the realm
of this topics, regardless of his particular field of interest. Lack of this knowledge can
impose limitations on engineering design.
2.3 Summary of Related Previous Works
Table 2.1 shows the summary of the previous research works performed by other
researchers.
Table 2.1 Summary of the Previous Research works related Modelling Constitutive
Model
Author Summary Result
Vili Panov,
(2005)
The aim of the work presented in this
thesis was to produce the improvement
of the existing simulation tools used for
the analysis of materials and structures,
which are dynamically loaded and
The new material model
has been implemented in
DYNA3D, using for this
purpose developed elastic
(predictor or plastic),
8
subjected to the different levels of
temperatures and strain rates. The main
objective of this work was
development of tools for modelling of
strain rate and temperature dependant
behaviour of aluminium alloys, typical
for aerospace structures with
pronounced orthotropic properties, and
their implementation in computer
codes. Explicit finite element code
DYNA3D has been chosen as
numerical test-bed for implementation
of new material models. Constitutive
model with an orthotropic yield
criterion, damage growth and failure
mechanism has been developed and
implemented into DYNA3D.
corrector or damage
mapping, integration
algorithm. Numerical
simulations of Taylor
impact cylinder test for
AA7010, have been
carried out to validate
implemented model and
simulation results are
given to illustrate the
potential applicability of
the proposed model.
Good agreement with
experimental results was
obtained. Comparison of
numerical results for
purely isotropic models
like as Johnson-Cook,
Mechanical Threshold
Stress and anisotropic
elastoplastic model with
proposed anisotropic
elastoplastic damage
model illustrates
significant differences in
material response. It
could be concluded
following: proposed
model is capable to
capture more accurately
major and minor
distributions of plastic
strains, and furthermore,
9
developed model can
describe evolution of
damage adequately.
Vignjevic et
al., (2008)
A constitutive relationship for
modeling of shock wave propagation in
orthotropic materials is proposed for
nonlinear explicit transient large
deformation computer codes
(hydrocodes). A procedure for
separation of material volumetric
compression (compressibility effects
equation of state) from deviatoric strain
effects is formulated, which allows for
the consistent calculation of stresses in
the elastic regime as well as in the
presence of shock waves. According to
this procedure the pressure is defined as
the state of stress that results in only
volumetric deformation, and
consequently is a diagonal second
order tensor. As reported by Anderson
et al. Comput. Mech.15,201,1994 the
shock response of an orthotropic
material cannot be accurately predicted
using the conventional decomposition
of the stress tensor into isotropic and
deviatoric parts. This paper presents
two different stress decompositions
based on the assumption that the stress
tensor is split into two components: one
component is due to volumetric strain
and the other is due to deviatoric strain.
Both decompositions are rigorously
The simulation results
showed a good agreement
with the experimental
data. Differences between
the experimental traces
and numerical results in
these test cases could be a
consequence of the
orientation and position
of the gauges related to
the layup of the
composite material, as
well as the gauge
averaging the recorded
stress across its area.
Thus an important feature
of further validation
would be that such
simulations should
improve understanding of
experimental averaging
occurring in the gauge so
that it can be replicated in
the simulation.
10
derived. In order to test their ability to
describe shock propagation in
orthotropic materials, both algorithms
were implemented in a hydrocode and
their predictions were compared to
experimental plate impact data. The
material considered was a carbon fiber
reinforced epoxy material, which was
tested in both the through-thickness
and longitudinal directions.
Czarnotaa et
al., (2008)
Model of void nucleation and growth is
proposed and used to characterize
dynamic damage in metals subjected to
extreme loading conditions. The
material is elastic–visco plastic and
initially void free (but the presence of
initial voids can be as well considered).
Micro voids are generated at potential
nucleation sites when the nucleation
pressure is overcome by the local
pressure loading. Owing to
microstructural heterogeneities and to
the presence of residual stresses, the
nucleation pressure varies from site to
site. A Weibull probability density
function is adopted to describe the
fluctuation of nucleation pressures
within the material. When the local
pressure increases, more voids are
nucleated. The growth of a void is
described by using a hollow sphere
model where micro-inertia effects are
accounted for. The matrix weakening
it is generally deduced in
the literature (acoustic
assumption) that for
aluminium, titanium and
tantalum, the maximum
stress sustained by the
material (spall stress) is
not affected by the impact
velocity. In this model,
the maximum stress in the
spall plane strongly
increases with the impact
velocity. Nevertheless,
the velocity pullback at
the target free surface is
shown, as in experiments,
to be independent of the
impact velocity for a
given flyer geometry. In
addition, the effect of the
flyer size on the velocity
pullback is exactly
captured. These results
11
due to void growth is also included.
Dynamic damage model has been
implemented in ABAQUS/ Explicit.
All the model parameters have been
identified from independent
experiments.
bring some confidence on
the physical mechanisms
introduced in the present
dynamic damage model.
The stabilizing effect of
micro-inertia is of
particular importance, as
it controls the level of the
maximum stress and the
development of damage
during stress relaxation in
the spall plane.
Mohd Nor
et al., (2013)
a finite strain constitutive model for
orthotropic materials was developed
within a consistent thermodynamic
framework of irreversible process in
this paper. The important features of
this material model used the
multiplicative decomposition of the
deformation gradient and a Mandel
stress tensor combined with the new
stress tensor decomposition
generalised for orthotropic materials.
The elastic free energy function and the
yield function are defined within an
invariant theory by means of the
introduction of the structural tensors.
The plastic behaviour is characterised
within the associative plasticity
framework using the Hill’s yield
criterion. The complexity was further
extended by coupling the formulation
with the equation of state (EOS) to
The results generated by
the proposed material
model were compared
against the experimental
data of Plate Impact test
of the Aluminium Alloy
7010. The results were
satisfactory with respect
to the experimental data.
12
control the response of the material to
shock loading. This material model
which was developed and integrated in
the isoclinic configuration provides a
unique treatment for elastic and plastic
anisotropy. The effects of elastic
anisotropy are taken into account
through the stress tensor decomposition
and plastic anisotropy through yield
surface defined in the generalized
deviatoric plane perpendicular to the
generalised pressure. To test its ability
to describe shockwave propagation, the
new material model was implemented
into the LLNL-DYNA3D code.
Mohd Nor,
(2015)
A unique orthogonal rotation tensor
was defined and implemented into the
Lawrence Livermore National
Laboratory-DYNA3D code in this
paper. This tensor is vital for the
implementation work of a newly
formulated constitutive model
proposed by M. K. Mohd Nor, to
ensure the analysis was precisely
integrated in the isoclinic
configuration. The implementation of
this unique orthogonal rotation tensor
into the LLNL-DYNA3D was
performed by referring to three
theorems; the deformation gradient F is
invertible the plastic stretch 𝑈𝑝 is
symmetric and positive definite and
finally the rotation tensor �� is assumed
Several theorems were
proved and applied to
ensure a unique rotation
tensor was successfully
implemented into this
code. Accordingly, a new
subroutine was added to
examine the accuracy of
the newly implemented
rotation tensor algorithm.
The results obtained for
uniaxial stress test of
reversed loading and the
Plate Impact test proved
the integration were
precisely performed in
isoclinic configuration
throughout the analysis
13
orthogonal hence ��−1 = ��𝑇. The
subroutine chkrot93 was adopted to
check the accuracy of the proposed
algorithm to calculate a proper rotation
tensor ��. The accuracy of the proposed
algorithm to define a unique orthogonal
rotation tensor was tested and validated
with the uniaxial tensile test of reversed
loading and Plate Impact test of
orthotropic materials that have so much
application in real world practices.
using the proposed
algorithm and provide a
satisfactory results with
respect to the reference
data.
Mohd Nor,
(2016)
Proposed constitutive model used a
new Mandel stress tensor that was
combined with the new stress tensor
decomposition. The Mechanical
Threshold Model (MTS) was adopted
as a referential curve to control the
yield surface expansion that accounts
for isotropic plastic hardening. The
formulation was developed in the
isoclinic configuration by using a
multiplicative decomposition of the
deformation gradient framework. The
complexity was increased by
combining the proposed formulation
with equation of states (EOS). The
proposed constitutive model was
implemented into the LLNL-
DYNA3D. To validate the proposed
formulation, the final radius and length
of the deformed cylinder profile
obtained experimentally were
From the analysis
performed, it is noticed
that the proposed
formulation of the new
constitutive model,
Material Type 93 is
capable of producing a
good agreement with
respect to the three
dimensional stress-state
of commercial aluminium
alloy AA7010 using
Taylor Impact test data.
14
compared with the results generated by
new material model.
Mohd Nor,
(2016)
A finite strain constitutive model to
predict the deformation behaviour of
orthotropic metals is developed in this
paper. The important features of this
constitutive model are the
multiplicative decomposition of the
deformation gradient and a new
Mandel stress tensor combined with the
new stress tensor decomposition
generalized into deviatoric and
spherical parts. The elastic free energy
function and the yield function are
defined within an invariant theory by
means of the structural tensors. The
Hill’s yield criterion is adopted to
characterize plastic orthotropy, and the
thermally micromechanical-based
model, Mechanical Threshold Model
(MTS) is used as a referential curve to
control the yield surface expansion
using an isotropic plastic hardening
assumption. The model complexity is
further extended by coupling the
formulation with the shock equation of
state (EOS). The proposed formulation
is integrated in the isoclinic
configuration and allows for a unique
treatment for elastic and plastic
anisotropy. The effects of elastic
anisotropy are taken into account
through the stress tensor decomposition
It can be observed the
proposed formulation of
the new constitutive
model integrated in the
isoclinic configuration
and updated in a unique
alignment of deviatoric
plane within the stress
space is capable of
producing a good
agreement with respect to
the Taylor Cylinder
Impact test data of
orthotropic metals. At this
point, it can be concluded
that the validation process
is completed since the
capability of the proposed
formulation of the new
constitutive model to
simulate the deformation
behaviour of orthotropic
metals at high strain rates
within three-dimensional
stress state has finally
been validated.
15
and plastic anisotropy through yield
surface defined in the generalized
deviatoric plane perpendicular to the
generalized pressure. The proposed
formulation of this work is
implemented into the Lawrence
Livermore National Laboratory-
DYNA3D code by the modification of
several subroutines in the code. The
capability of the new constitutive
model to capture strain rate and
temperature sensitivity is then
validated. The final part of this process
is a comparison of the results generated
by the proposed constitutive model
against the available experimental data
from both the Plate Impact test and
Taylor Cylinder Impact test
The above summary proves the needs of improved hyperelastic-plastic constitutive
formulation using the new stress tensor decomposition. The chosen framework
primarily distinguishes the chosen approach from hypoelastic-plastic formulation. To
predict a complex deformation behaviour involves very high pressures and
shockwaves, the shock equation of state (EOS) and spall failure must be adopted.
2.4 Insight into Elasticity
The theory of elasticity is a branch of continuum mechanics which is dealing with
deformable solid bodies and having the physical property. Elastic deformation may be
defined as a reversible deformation, means the body returns to its original shape and
recovers all stored energy after removal of the applied load. Most metals at low
stresses and ceramics are elastic.
16
To understand the behaviour of an elastic body, a mathematical formulation
can be developed to generalize theory concerning elastic solids. The continuum theory
of elasticity is first correctly introduced in Cauchy’s works. Stress and strain are related
to each other by Hooke's Law where the strain is assumed to be sufficiently small. For
small deformations, it can be assumed that this relationship is linear. This observation
is well supported by experimental evidence provided.
This linear elastic stress–strain relationship can be expressed as
𝜎𝑖𝑗 = 𝐶𝑖𝑗𝑘𝑙휀𝑘𝑙 (2.1)
where 𝜎𝑖𝑗, 𝐶𝑖𝑗𝑘𝑙 and 휀𝑘𝑙 refer to Cauchy stress tensor, fourth-order stiffness tensor and
the infinitesimal strain tensor respectively. Alternatively, the relationship can be
written as
휀𝑖 = 𝑆𝑖𝑗𝑘𝑙𝜎𝑗 (2.2)
where 𝑆𝑖𝑗 refers to elastic compliance. It should be noted that both 𝐶𝑖𝑗𝑘𝑙 and 𝑆𝑖𝑗𝑘𝑙 are
proportionally constants where 𝑖 and 𝑗 can have any value from 1 to 3.
Equation (2.1) also can be expressed in tensor notation as
𝝈 = 𝑪 ∶ 𝜺 (2.3)
The indices are avoided in tensor notation. It is important to note that both 𝜎𝑖𝑗 and 휀𝑖𝑗
are symmetric tensors. Therefore, stress tensor can be written in matrix form as:
(
𝜎11 𝜎12 𝜎13
𝜎21 𝜎22 𝜎23
𝜎31 𝜎32 𝜎33
) = (
𝜎11 𝜎21 𝜎31
𝜎12 𝜎22 𝜎32
𝜎13 𝜎23 𝜎33
) (2.4)
Similarly, the strain tensor can be expressed as:
17
(
휀11 휀12 휀13
휀21 휀22 휀23
휀31 휀32 휀33
) = (
휀11 휀21 휀31
휀12 휀22 휀32
휀13 휀23 휀33
) (2.5)
Symmetry effect leads to a significant simplification of the stress-strain relationship of
Equations (2.1) and (2.2) as follows:
휀𝑖𝑗 = 𝑆𝑖𝑗𝑘𝑙𝜎𝑘𝑙 or 휀𝑖𝑗 = 𝑆𝑖𝑗𝑘𝑙𝜎𝑙𝑘 (2.6)
and since
𝑆𝑖𝑗𝑘𝑙𝜎𝑘𝑙 = 𝑆𝑖𝑗𝑘𝑙𝜎𝑙𝑘 ; 𝜎𝑘𝑙 = 𝜎𝑙𝑘 ; 𝑎𝑛𝑑 𝑆𝑖𝑗𝑘𝑙 = 𝑆𝑖𝑗𝑙𝑘 (2.7)
Further, Equation (2.6) also can be expressed as
휀𝑗𝑖 = 𝑆𝑗𝑖𝑘𝑙𝜎𝑘𝑙 using 𝑆𝑖𝑗𝑘𝑙 = 𝑆𝑗𝑖𝑘𝑙 (2.8)
Hooke’s tensor can be written with respect to a Cartesian coordinate system in
tensorial notation as
𝑪 = 𝐶𝑖𝑗𝑘𝑙𝑒𝑖 ⊗ 𝑒𝑗 ⊗ 𝑒𝑘 ⊗ 𝑒𝑙 (2.9)
where 𝑒𝑖,𝑗,𝑘,𝑙 represents the principal strain. Due to the symmetry of the strain tensor
휀, 휀𝑘𝑙 = 휀𝑙𝑘, it can be simplified that
𝐶𝑖𝑗𝑘𝑙 = 𝐶𝑖𝑗𝑙𝑘 and 𝐶𝑖𝑗𝑘𝑙 = 𝐶𝑗𝑖𝑙𝑘 (2.10)
The direct consequence of the symmetry in the stress and strain tensors is that only 36
components of the compliance and stiffness tensors are independent and distinct terms.
n matrix format, the stress-strain relation showing the 36 (6 x 6) independent
components of stiffness can be represented as:
18
[ 𝜎1
𝜎2
𝜎3𝜎4
𝜎5
𝜎6]
=
[ 𝐶11 𝐶12
𝐶21 𝐶22
𝐶13 𝐶14
𝐶23 𝐶24
𝐶15 𝐶16
𝐶25 𝐶26
𝐶31 𝐶32
𝐶41 𝐶42
𝐶33 𝐶34
𝐶43 𝐶44
𝐶35 𝐶36
𝐶45 𝐶46
𝐶51 𝐶52
𝐶61 𝐶62
𝐶53 𝐶54
𝐶63 𝐶64
𝐶55 𝐶56
𝐶65 𝐶66]
[ 휀1
휀2
휀3휀4
휀5
휀6]
(2.12)
The above matrix expression can be written in indicial notation as:
𝜎𝑖 = 𝐶𝑖𝑗휀𝑗 𝑎𝑛𝑑 휀𝑖 = 𝑆𝑖𝑗𝜎𝑖 (2.13)
Further reductions in the number of independent constants are possible by employing
other symmetry considerations to Equation 2.13 to remark that in the most general case
C may have the maximum number of 21 independent components. The 21 independent
elastic constants can be reduced still further by considering the symmetry conditions
found in different crystal structures. In isotropic case, the elastic constants are reduced
from 21 to 2. This symmetry requirement ensures the existence of an elastic energy
potential.
A great number of materials can be treated as isotropic, although they are not
microscopically. Individual grains exhibit the crystalline anisotropy and symmetry, but
when they form a poly-crystalline aggregate and are randomly oriented, the material
is microscopically isotropic. If the grains forming the poly-crystalline aggregate have
preferred orientation, the material is microscopically anisotropic. Often, material is not
completely isotropic; if the elastic modulus E is different along three perpendicular
directions, the material is orthotropic. It is accepted within the linear elastic framework
that the strains and stresses are connected through a one-to-one relation. This tensor
actually links the deformation of a medium to an applied stress (Khan et al., 2009).
Referring to Equation (2.2), it can be seen that the elastic compliance tensor S is the
inverse tensor of C such that
S = 𝑪−1.
(2.14)
As shown above, the similar symmetry requirements can be imposed on the
compliance tensor to give
19
𝑆𝑖𝑗𝑘𝑙= 𝑆𝑘𝑙𝑖𝑗 = 𝑆𝑗𝑖𝑘𝑙 (2.15)
As briefly defined above, materials with three mutually perpendicular planes of
symmetry is also known as orthotropic materials. In the case of elastic orthotropic
materials, nine parameters are required to define the stress strain relationship using
stiffness and compliance tensors.
The stiffness and compliance tensors can be written in matrix form as follows
𝑪𝑒 =
[
𝐸1(1 − 𝜐32𝜐23)
Δ𝑒
𝐸2(𝜐12 + 𝜐13𝜐32)
Δ𝑒
𝐸3(𝜐13 + 𝜐12𝜐23)
Δ𝑒0 0 0
𝐸1(𝜐21 + 𝜐23𝜐31)
Δ𝑒
𝐸2(1 − 𝜐13𝜐31)
Δ𝑒
𝐸3(𝜐23 + 𝜐21𝜐13)
Δ𝑒0 0 0
𝐸1(𝜐31 + 𝜐21𝜐32)
Δ𝑒
000
𝐸2(𝜐32 + 𝜐32𝜐23)
Δ𝑒
000
𝐸3(1 + 𝜐21𝜐12)
Δ𝑒
000
0G23
0 0
00
G31
0
0 0 0G12]
(2.16)
where:
∆𝑐= 1 − 𝑣21𝑣12 − 𝑣31𝑣13 − 𝑣32𝑣23 − 𝑣12𝑣23𝑣31 − 𝑣21𝑣13𝑣32
𝑉12
𝐸1=
𝑉21
𝐸2,𝑉31
𝐸3=
𝑉13
𝐸1,𝑉32
𝐸3=
𝑉23
𝐸2
𝑪𝑒 =
[ 1𝐸𝑥
𝑣𝑦𝑧
𝐸𝑦
𝑣𝑧𝑥
𝐸𝑧 0 0 0
𝑣𝑧𝑦
𝐸𝑥
1𝐸𝑦
𝑣𝑧𝑦
𝐸𝑧0 0 0
𝑣𝑥𝑧
𝐸𝑥
000
𝑣𝑦𝑧
𝐸𝑦
000
1𝐸𝑧
000
01
G𝑥𝑦
00
001
G𝑥𝑧
0
0001
G𝑦𝑧 ]
(2.17)
where E, 𝜈 and G is refer to Young’s modulus, Poisson’s ratio and shear modulus
respectively.
20
2.5 Insight into Plasticity
Permanent deformation is known as plastic deformation which the behaviour can be
very complex in general. After the applied load is removed the deformation body is
remains. The physical process permanently changing their relative positions and
involves sliding of atoms past each other. Since the normal stress does not affect
significantly the sliding process in metals., it may be stated that only shear stress can
and does induce plastic deformation in metals.
There are two approaches can be used at microscopic and macroscopic levels
in the study of plasticity. At microscopic scale, it can be characterised as a theory of
dislocation. Which is effective at the level of atomic and essential to appropriately
describe the accurate phenomenon of materials physically. While, at macroscopic
scale, its lead to plasticity theory.
2.6 Overview of Isotropic Plasticity
According to the theory of plasticity, the behaviour of material undergoing plastic
deformation is characterised by a yield function, a flow rule and a strain-hardening
law. The yield function determines the stress state when yielding occurs. The direction
of plastic strain rate is obtained by the gradient (from normal to yield surface at the
loading point). The yield surface described the limitation of the elastic regime of
material and specify the point of material when start to yield (Khan et al., 2009). The
flow rule describes the increment of plastic strain when yielding occurs and define the
relationship between the deviatoric stress and the strain rate tensor.
The Levy–Mises equation described the relationship between the ratio of strain
and the ratio of stress that specifies the increment of total strain which basically
suggested by Levy and further enhanced by Von Mises. The theory of Levy-Mises is
based on two assumptions. First, the elastic strain 𝛆𝑒 is assumed small, and ignored.
Further, the theory assume the increment of strain dε or correspondingly the strain rate
�� is coaxial with stress, σ. The same assumption applies to the deviatoric stress S,
hence the coaxiality between dε and S can be written as:
𝑑𝛆 = 𝐒 ∙ 𝑑λ (2.18)
21
where dλ or λ is a scalar factor of proportionality that determined from the yield
criterion. This theory is later extended to allow for elastic and plastic strains takes the
following form:
d𝜺𝑖𝑗 = 𝑑𝛆𝑒𝑖𝑗+ 𝑑𝛆𝑝𝑖𝑗
(2.19)
The theory is known as the Prandtl–Reuss equation where the total strain increment
𝑑𝜺 is the sum of the elastic 𝑑𝛆𝑒𝑖𝑗 and the plastic 𝑑𝛆𝑝𝑖𝑗
increments. Plastic deformation
is considered to be constant volume process, while elastic deformation causes volume
changes and shape changes.
Basically during metal forming process, the elastic increment strain is ignored
compare to the plastic increment fully developed plastic region. However, when
collaborate with elastoplastic deformation the elastic component of the total increment
is become essential and it is needed to accompanist to Prandtl–Reuss assumption.
Otherwise, the material is assumed as rigid-perfectly plastic if the elastic incremental
strain is ignored and this condition is well-suited with the Levy-Mises assumption.
𝑑𝛆𝑝𝑖𝑗= 𝑑λS (2.20)
An important assumption has been made in Equation (2.20) that the principal axes of
plastic strain increment and deviatoric stress are coincident. This theory matches with
the behaviour in most metals where the plastic strain 𝜺𝑝 is much larger than elastic
strain 𝜺𝑒.
Von Mises has made a significant contribution by proposing a simplified
constitutive equation for plastic deformation. In the theory of elasticity, Von Mises
relates the strain tensor to the stress tensor through an elastic potential function, as a
complementary to elastic strain energy 𝛹𝑒 as
𝜺𝑖𝑗 =∙𝑑𝛹𝑒
𝑑𝜎𝑖𝑗 (2.21)
22
This idea is further generalized to the theory of plasticity by assuming an existence of
plastic potential function 𝛹(𝝈) and the plastic strain rate ��𝑝 to define the following
formulation
��𝑝 = λ ∙𝑑𝛹(𝛔)
𝑑𝜎 (2.22)
Yield criterion can be used to determine λ in the above equation. It should be noted
that a plasticity potential theory is defined based on a flow rule assumption, focusing
on a potential, 𝛹(𝝈) identification. The plastic potential function 𝛹(𝝈) is normally
assumed identical to the yield function 𝑓(𝛔) in the associative plasticity theory, such
that
𝛹(𝝈) = 𝑓(𝛔) (2.23)
Subsequently Equation (2.21) can be re-expressed as
��𝑝 = λ ∙𝑑𝑓(𝛔)
𝑑σ (2.24)
In this equation, the plastic strain rate ��𝑝 is set normal to the yield surface, called an
associated flow rule. This theory is supported by experimental evidences as observed
in the plastic deformation of numerous metals. Contrarily, a non-associated flow rule
is defined as 𝛹(𝛔) ≠ 𝑓(𝛔).
The hardening law is used in this theory to define the evolution of the yield
surface and describes how the material is strain-hardened as the plastic strain increases
(Khan et al., 2009). There are two hypotheses can be adopted to analyze the hardening
effects. The first hypothesis known as the plastic strain hardening that states the
amount of hardening is controlled by the effective plastic strain as shown below
��𝑝 = ∫𝑑��𝑝 (2.25)
The second hypothesis is initiated in Hill (1948) where the hardening is assumed
depends on the total plastic work done, 𝑊𝑝 to give
23
𝑊𝑝 = ∫𝛔: 𝑑𝛆𝑝 (2.26)
The yield surface evolution can be characterized as isotropic, kinematic, anisotropic,
or a combination of the three, regardless of the initial shape developed when the
materials start to yield. Using an isotropic hardening assumption, the yield expansion
is assumed take place without any distortion of the initial curve. The isotropic
hardening can be generalized for isotropic yield function in terms of the second and
third invariants of the deviatoric stress 𝐒, J2′ and J3
′ respectively.
𝑓 = 𝑓(J2′ , J3
′ ) − 𝑓(α) = 0 (2.27)
It can be seen in the above equation, parameter 𝛼 is necessary to represent the isotropic
hardening which refer to the yield surface radius. Subsequently, parameter 𝛼 can be
expressed as
𝑑α = 𝑑ε𝑝 (2.28)
Using the above equation, the evolution of the yield surface radius is proportional to
plastic deformation. To consider the Bauschinger effect in the uniaxial tension-
compression such as the case of cyclic loading, the kinematic hardening must be
included by observing the centre of the yield surface. An internal variable 𝞪 known as
the backstress is introduced in this concept to determine the yield surface centre
position in the stress space which the changes is controlled by hardening of plastic. If
the initial yield surface is determined by following equation,
𝑓 = 𝑓(𝛔) − 𝑓(α) = 0 (2.29)
the kinematic hardening can be further embedded to the formulation to define the
centre of the yield surface during the plastic deformation as
𝑓 = 𝑓(𝛔 − 𝞪) − 𝑓(α) (2.30)
24
In addition to the above concepts, an equation of state (EOS) must be included in the
constitutive model to model the behaviour of materials undergoing very high pressures
and shockwaves. The deformation behaviour of metals can be closely predicted within
elastic and plastic regimes by introducing damage and failure models (Gotoh, 1971).
2.7 Insight into Anisotropic Plasticity
It can be observed from the discussion in the previous sections, the theory related to
isotropic materials is not very complex since the response is not affected by the
materials orientations. A similar characteristic however, does not experienced by
anisotropic materials. In such materials, the mechanical properties will start to changes
(be distorted) as the materials starts to rotate during plastic deformation.
Generally, anisotropic materials exhibit different response in different
direction due to its unique magnitude and orientation in each direction. The initial
shape of the yield surface is influenced by elastic anisotropic that represents the initial
anisotropy of materials.
2.7.1 Yield Criteria for Plastic Anisotropy
In order to simulate forming processes, the mechanics of continua offers several
general equations that apply to any medium. However, the constitutive equations are
valid only for a particular class of materials. These equations account for the intrinsic
nature of the materials and can be applied only under given conditions. Often, in sheet
metal forming problems at room temperature, the classical flow theory of plasticity is
employed. This theory assumes the existence of a plastic potential, usually identified
with the yield surface, which must be convex. Therefore, strains are related to stresses
through the normality rule. However, the yield surface of sheet metals can be quite
different according to the nature of the material and its degree of anisotropy, (Hill,
1979).
Hill (1948) introduced the anisotropic theory of plasticity and described a
theory of yielding and plastic flow of anisotropic material, (Chin Wu, 2003). He also
the first one who proposed the minimal framework of orthotropic plastic response of
rolled sheet which commonly modelled by a homogeneous yield function of degree
127
REFERENCES
Anderson, C. E., Cox, P. A., Johnson, G.R. and Maudlin, P. J., A Constitutive
Formulation for Anisotropic Materials Suitable for Wave Propagation
Computer Program-II, Computer Mechanic, Vol. 15, 201–223, 1994.
Anagnostopoulos, K. A., Charalambopoulos, A. and Massalas, C. V., On The
Investigation of Elasticity Equation for Orthotropic Materials and The
Solution of the Associated Scattering Problem, International Journal of
Solids and Structures, Vol. 42, pp. 6376–6408, 2005.
Aravas, N., Finite-Strain Anisotropic Plasticity and The Plastic Spin, Modelling
Simulation Material Science, Vol. 2, 483–504, 1994.
Asaro, R. J., Micromechanics of Crystals and Polycrystals, Advances in Applied
Mechanics, Vol. 23, 1–115, 1983.
Asay, J. R. and Shahinpoor, M., High-Pressure Shock Compression of Solids,
Springer, New York, 1993.
Banabic, D., Balan T. and Comsa, D. S., A New Yield Criterion for Orthotropic
Sheet Metals Under Plane-Stress Conditions, Processing of 7th Cold Metal
Forming Conference, Cluj Napoca, Romania, pp. 217 -224, 2000.
Banabic, D., Kuwabara, T., Balan, T., Comsa, D. S. and Julean, D., Non-Quadratic
Yield Criterion for Orthotropic Sheet Metals Under Plane-Stress
Conditions, International Journal of Mechanical Sciences, Vol. 45, pp.
797–811, 2003.
Barlat, F. and Brem, J. C., Plane Stress Yield Function for Aluminum Alloy Sheets
- Part 1. Theory. International Journal of Plasticity, Vol. 19. 1297–1319,
2003.
Barlat, F., Lege, D. J. and Brem, J. C., A Six-Component Yield Function for
Anisotropic Materials, International Journal of Plasticity, Vol. 7, pp. 693-
712, 1991.
128
Barlat, F., Maeda, Y., Chung, K., Yanagawa, M., Brem, J. C., Hayashida, Y. D.,
Cege, J. K. and Murtha S. J., Yield Function Development for Aluminium
Alloy Sheet, PII:S0022-5096, pp. 200034-3, 1997.
Barlat, F. and Lian, J., Plastic Behavior and Stretchability of Sheet Metals. Part 1:
A Yield Function for Orthotropic Sheets Under Plane Stress Conditions,
International Journal of Plasticity, Vol. 5, pp. 51-66, 1989.
Barlat, F., Crystallographic Texture, Anisotropic Yield Surfaces and Forming
Limits of Sheet Metals, Materials Science and Engineering, Vol. 91, pp.
55-72, 1987.
Belytschko, T., Guo, Y., Liu W. K. and Xiao, S. P., A Unified Stability Analysis
of Meshless Particle Methods, International Journal Numerical method in
Engineering, Vol. 48, Issue 9, 2000.
Bohlke, T. and Bertram, A., The Evolution of Hookes Law Due to Texture
Development in FCC Polycrystal, International Journal of Solid and
Structures, Vol. 38, pp. 9437-9459, 2001.
Boehler, J. P., On irreducible Representations for Isotropic Scalar Functions,
ZAMM 57, pp. 323–327, 1977.
Bronkhorst, C. A., Cerreta, E. K., Xue, Q., Maudlin, P. J., Mason, T. A. and Gray,
G. T., An Experimental and Numerical Study of the Localization Behavior
of Tantalum and Stainless Steel, International Journal of Plasticity, Vol.
22, pp. 1304–1335, 2006.
Callen H. B., Thermodynamics and Introduction to Thermostatics, 2nd Edition,
Wiley, New York 493, 1985.
Campbell, J., Lagrangian Hydrocode Modelling of Hypervelocity Impact On
Spacecraft, PhD Dissertation, Cranfield University, 1998.
Cazacu, O., Plunkett, B. and Barlat, F., Orthotropic Yield Criterion for Hexagonal
Closed Packed Metals, International Journal of Plasticity, Vol. 22, pp.
1171–1194, 2006.
Cazacu, O. and Barlat, F., Application of The Theory of Representation to
Describe Yielding of Anisotropic Aluminium Alloys, International
Journal of Engineering Science, Vol. 41, pp. 1367–1385, 2003.
Chen, Y., Asay, J. R., Dwivedi, S. K. and Field, D. P., Spall Behaviour of
Aluminum with Varying Microstructures, Journal Applied Physic, Vol. 99,
pp. 1–13, 2006.
129
Chen, Y., Pedersen, K., Clausen, A., Hopperstad, O. and Langseth, M., An
Experimental Study on the Dynamic Fracture of Extruded AA6xxx and
AA7xxx Aluminium Alloys. Materials Science and Engineering, Vol. 523,
pp. 253-262, 2009.
Chung, K. and Shah, K., Finite Element Simulation of Sheet Metal Forming for
Planar Anisotropic Metals, International Journal of Plasticity, Vol. 8, pp.
453-476, 1992.
Costas, G. F., George, A. G. and Bryan, A. C., Computational Modelling of
Tungsten Carbide Sphere Impact and Penetration into High-Strength-Low-
Alloy (HSLA)-100 Steel Targets, Journal Mechanics of Material and
Structure, Vol. 2 (10), pp. 1965–1979, 2007.
Czarnota, C., Jacques, N., Mercier, S. and Molinari, A., Modelling of Dynamic
Ductile Fracture and Application to The Simulation of Plate Impact Tests
On Tantalum, Journal of the Mechanics and Physics of Solids, Vol. 56, pp.
1624–1650, 2008
Dafalias, Y. F. and Rashid, M. M., The Effect of Plastic Spin On Anisotropic
Material Behaviour, International Journal of Plasticity, Vol. 5, pp. 227-
246, 1989.
Davison, L. and Graham, R. A., Shock Compression of Solids, Journal Applied
Physic, Vol. 55 pp. 255–379, 1979.
De Vuyst, T. A., Hydrocode Modelling of Water Impact, Phd Thesis Cranfield
University, Cranfield, UK, 2003.
Drumheller, D. S., Introduction to Wave Propagation in Nonlinear Fluids and
Solids, Cambridge University Press, Cambridge, UK, 1998.
Eidel, B. and Gruttmann, F., Elastoplastic Orthotropy at Finite Strains:
Multiplicative Formulation and Numerical Implementation,
Computational Materials Science, Vol. 28, pp. 732–742, 2003.
Eliezer, S., Ghatak, A., Hora, H. and Teller, E., An Introduction to Equations of
State, Theory and Applications, Cambridge University Press, Cambridge,
UK, 1986.
Elmarakbi, A. M., Hub, N. and Fukunaga, Finite Element Simulation of
Delamination Growth in Composite Materials Using LS-DYNA.
Composites Science and Technology, Vol. 69, pp. 2383–2391, 2009.
130
Feigenbaum, H., Modeling Yield Surface Evolution with Distortional Hardening,
ECI 285, June 15, 2004
Furnish, M. D. and Chhabildas, L. C., Alumina Strength Degradation in The
Elastic Regime, in: AIP Conference Proceedings, Vol. 429 (1), pp. 501–
504073502, 1988.
Fountzoulas, C. G., Gazonas, G. A. and Cheeseman, B. A., Computational
Modeling of Tungsten Carbide Sphere Impact and Penetration into High-
Strength-Low-Alloy (HSLA)-100 Steel Targets, Journal Mechanics of
Material and Structure, Vol. 2 (10), pp. 1965, 2007
Gotoh, K., Equation of State of Simple Liquids from Coordination Number in
Random Assemblage. Nature Physical Science, Vol. 234, pp. 138-139,
1971.
Gotoh, K., A Theory of Plastic Anisotropy Based On Yield Function of Fourth
Order (Plane Stress State)-II, International Mechanic Science, Vol. 19, pp.
513-520. Pergamon Press 1977.
Grady, D., Incipient Spall, Crack Branching, and Fragmentation Statistics in the
Spall Process, Journal de Physique Colloques, Vol. 49 (C3), pp.C3-175-
C3-182, 1988.
Gray, G. T., Bourne, N. K. and Millett, J. C. F., Shock Response of Tantalum:
Lateral Stress and Shear Strength Through the Front, Journal Applied
Physic, Vol. 94 (10) pp. 6430–643, 2003.
Gruneisen, E., The state of solid body, NASA R19542, 1959.
Hallquist, J. O., Wainscotta B. and Schweizerhof, K., Improved Simulation of
Thin-Sheet Metalforming Using LS-DYNA3D on Parallel Computers,
Journal of Materials Processing Technology, Vol. 50, pp. 144-157, 1995.
Han, C. W., The Evolution of Yield Surface in The Finite Strain Range,
Department of Civil and Environmental Engineering, The University of
Iowa, Iowa City, IA 52242, U.S.A., 2004.
Han, C. W., Anisotropic Plasticity for Sheet Metals Using the Concept of
Combined Isotropic Kinematic Hardening, International Journal of
Plasticity, Vol. 18, pp. 1661–1682, 2002.
Hansen, N. R. and Schreyer, H. L., A Thermodynamically Consistent Framework
for Theories of Elastoplasticity Coupled with Damage, International
Journal Solids Structure, Vol. 31, pp. 359–389. 1994.
131
Hill, R., A Theory of the Yielding and Plastic Flow of Anisotropic Metals,
Processing Research Society, pp. 281–297, 1948.
Hill, R., Theoretical Plasticity of Textured Aggregates, Mathematic Processing
Cambridge Philosophy Society, Vol. 85, pp. 179–191, 1979.
Holzapfel, G. A., Nonlinear Solid Mechanics, A Continuum Approach for
Engineering, John Wiley & Sons Ltd, Chichester, 2007.
Inal, K., Wu, P. D., Neale, K. W., Simulation of earing in textured aluminum
sheets’, International Journal of Plasticity, Vol. 16, pp. 635-648, 2000.
Itskov, M. and Aksel, N., A Constitutive Model for Orthotropic Elastoplasticity at
Large Strains, Archive of Applied Mechanics, Vol. 74, pp. 75 – 91, 2004.
Jeffrey, D. C., Roger, W. M. and Daniel, H. K., A Model for Plasticity Kinetics
and Its Role in Simulating the Dynamic Behavior of Fe at High Strain
Rates, International Journal of Plasticity, Vol. 25, pp. 603–611, 2009.
Joshi K. D. and Gupta S., Measurement, of Spall Strength of Al-2024-T4 and
SS304 in Plate Impact Experiment, Conference High Pressure Result, Vol.
27, pp. 259, 2007
Joshi, K. D., Amit S. R., Gupta, S. and Benerjee, S., Measurement Spall Strength
of Al 2024-T4 and SS304 in Plate Impact Test, Journal of Physic:
Conference Series, Vol. 215, pp. 012149, 2010.
Kanel G. I., Razorenov S. V., Bogatch, A. V. and D. E. Grady., Simulation of Spall
Fracture of Aluminium and Magnesium Over a Wide Range of Load
Duration and Temperature, International Journal Impact Engineering,
Vol. 20, pp. 467–478, 1997.
Kanel, G. I., Zaretsky, E. B., Rajendran, A. M., Razorenov, S. V. Savinykh, A. S.
and Paris, V., Search for Conditions of Compressive Fracture of Hard
Brittle Ceramics at Impact Loading, International Journal of Plasticity,
Vol. 25, pp. 649–670, 2009.
Karafillis A. P. and Boyce J. M. C., A General Anisotropic Yield Criterion Using
Bound and A Transformation Weighting Tensor, Journal Mechanic Physic
Solids, Vol. 41 (12), pp.1859-1886, 1993.
Khan, A. S. and Meredith, C. S., Thermo-Mechanical Response of Al 6061 with
and without Equal Channel Angular Pressing (ECAP), International
Journal of Plasticity, Vol. 26, pp. 189–203, 2010.
132
Khan, A. S., Kazmi, R., Pandey, A. and Stoughton, T., Evolution of Subsequent
Yield Surfaces and Elastic Constants with Finite Plastic Deformation. Part-
I: A Very Low Work Hardening Aluminium Alloy (Al6061-T6511),
International Journal of Plasticity, Vol. 25, pp. 1611–1625, 2009.
Khan, A. S., Kazmi, R., Farrokh, B. and Zupan M., Effect of Oxygen Content and
Microstructure On the Thermo-Mechanical Response of Three Ti–6Al–4V
alloys: Experiments and Modelling Over a Wide Range of Strain-Rates and
Temperatures, International Journal of Plasticity, Vol. 23, pp. 1105–1125,
2007.
Kruger, L., Meyeer, L.W., Razorenov S. V. and Kanel, G. I., Investigation of
Dynamic Flow and Strength Properties of Ti-6-22-22S at Normal and
Elevated Temperature, International Journal of Impact Engineering, Vol.
28, pp. 877-890, 2003.
Lee, D. and Zaverl, F., A Generalized Strain Rate Dependent Constitutive Equation
for Anisotropic Metals, Acta Metallurgica. Vol. 26, pp. 1771-1780.
Pergamon Press 1978
Lin, J. I., DYNA3D: A Nonlinear, Explicit, Three-Dimensional Finite Element
Code for Solid and Structural Mechanics User Manual, Lawrence
Livermore National Laboratory, 2004.
Logan R. W. and William F. H., Upper-Bound Anisotropic Yield Locus
Calculations Assuming Lid-Pencil Glide, International Journal Mechanic
Science, Vol. 22, pp. 419-430, 1980.
Ls-Dyna Keyword User's Manual, Volume II, Material Models, Version 971
R6.1.0 Models, August 2012
LS DYNA3D Theoretical Manual, LSTC Report 1018 Revision 2, 2006.
Lubarda V. A. and Krajcinovic D., Some Fundamental Issues in Rate Theory of
Damage-Elastoplasticity, International Journal Plastic, Vol. 11, pp. 763-
797, 1995.
Man, C., On The Correlation of Elastic and Plastic Anisotropy in Sheet Metals,
Journal Elastic-Plastic, Vol. 39 (2), pp. 165–173, 1995.
Mandel, J., Plasticité Classiqueet Viscoplastiécism Lecture Notes, Springer-
Verlag, Wien, 1972.
133
Martin H. Sadd, Elasticity Theory, Application and Numeric, Second Edition,
ISBN: 978-0-12-374446-3, Academic Press Elsevier’s Science &
Technology, Department in Oxford, 2009.
Maudlin, P. J., Bingert, J. F., House J. W. and Chen, S. R., On The Modeling of
the Taylor Cylinder Impact Test for Orthotropic Textured Materials:
Experiments and Simulations, International Journal of Plasticity, Vol. 15,
pp. 139-166, 1999.
Meredith, C. S. and Khan, A. S., Texture Evolution and Anisotropy in The
Thermo-Mechanical Response of UFG Ti Processed Via Equal Channel
Angular Pressing, International Journal of Plasticity, Vol. 30–31, pp. 202–
217, 2012.
Meyers, M. A., Dynamic Behaviour of Materials, Wiley, Inc, New York, 1994.
Minich, R., Cazamias, J., Kumar, M. and Schwartz, A., Effect of Microstructural
Length Scales On Spall Behaviour of Copper, Metallurgical and Materials
Transactions, Vol. 35, pp. 2663-2673, 2004.
Mohd Nor, M. K., The Development of Unique Orthogonal Rotation Tensor
Algorithm in the LLNL DYNA3D for Orthotropic Materials Constitutive
Model. Australian Journal of Basic and Applied Sciences, Vol. 9 (37). pp.
22-27, 2015
Mohd Nor, M. K., Modelling Inelastic Behaviour of Orthotropic Metals in a
Unique Alignment of Deviatoric Plane within The Stress Space,
International Journal of Non--Linear Mechanics, Vol. 87, pp. 43–57, 2016.
Mohd Nor, M. K., Modelling Rate Dependent Behaviour of Orthotropic Metals,
PhD Thesis Cranfield University, Cranfield, UK. 2012.
Mohd Nor, M. K., Modeling of Constitutive Model to Predict The Deformation
Behaviour of Commercial Aluminum Alloy AA7010 Subjected To High
Velocity Impacts, ARPN Journal of Engineering and Applied Sciences,
ISSN 1819-6608, 2016.
Mohd Nor, M. K., Vignjevic R. and Campbell J., Modelling of Shockwave
Propagation in Orthotropic Materials, Applied Mechanics and Materials,
Vol. 315, pp. 557-561, 2013.
Mohd Nor, M. K., Modelling Inelastic Behaviour of Orthotropic Metals in A
Unique Alignment of Deviatoric Plane Within The Stress Space,
International Journal of Non-Linear Mechanics, Vol. 87, pp. 43–57, 2016
134
Mohd Nor, M. K., Ma’at, N., Kamarudin, K. A. and Ismail, A. E. Implementation
of Finite Strain-Based Constitutive Formulation in LLLNL-DYNA3D to
Predict Shockwave Propagation in Commercial Aluminum Alloys
AA7010, IOP Conference Series: Material Science and Engineering, Vol.
160, 012023, 2016.
Mohd Nor, M. K., and Ma’at, N., Simplified Approach to Validate Constitutive
Model Formulation of Orthotropic Materials Undergoing Finite Strain
Deformation, Journal of Engineering Applied Science, Vol. 11, pp. 2146-
2154, 2016.
Nakamachi, E., Tam, N. N. and Morimoto, H., Multi-scale finite element analyses
of sheet metals by using SEM-EBSD measured crystallographic RVE
models’, International Journal of Plasticity, Vol. 23, pp. 450–489, 2007.
Ontario, R. N., Dubey and Waterloo, An Eulerian-Based Approach to Elastic-
Plastic Decomposition, Act Mechanic, Vol. 131, pp. 111 – 119, 1998.
Panov V., Modelling of Behaviour of Metals at High Strain Rates, PhD Thesis
Cranfield University, Cranfield, UK. 2005.
Pedrazas, N. A., Daniel L. W., Dalton, D. A., Sherek, P. A., Steuck, S. P., Quevedo,
H. J., Bernstein, A. C., Taleff, E. M. and Ditmire T., Effects of
Microstructure and Composition On Spall Fracture in Aluminum, Journal
Materials Science and Engineering, Vol. 536, 2012.
Peralta, P., Digiacomo, S., Hashemian, S., Luo, S. N., Paisley, D., Dickerson, R.,
Loomis, E., Byler, D., Maclellan K. J. and D'Armas, H., Characterization
of Incipient Spall Damage in Shocked Copper Multicrystals. International
Journal of Damage Mechanics, Vol. 18 (4), pp. 393-413. 2009.
Proud, W. G., Notes on Shock Compression, Wave propagation and Spall Strength,
Institute of Shock Physics, Blackett Laboratory, Imperial College London,
London, 2015.
Rajendran, A. M., Critical Measurements for Validation of Constitutive Equations
Under Shock and Impact Loading Conditions, Optics and Laser in
Engineering, Vol. 40, pp. 249-262, 2003.
Rosenberg, Z., Luttwak, G., Yeshurun Y. and Partom, Y., Spall Studies of
Differently Treated 2024A1 Specimens, Journal Applied Physic, Vol. 54,
pp. 2147–2152, 1983.
135
Rust, W. and Schweizerhof, K., Finite Element Limit Load Analysis of Thin-
Walled Structures by ANSYS (implicit), LS-DYNA (explicit) and in
Combination, Thin-Walled Structures, Vol. 41, pp. 227–244, 2003.
Sitnikova, E., Guan, Z. W., Schleyer, G. K. and Cantwell, W. J., Modelling of
Perforation Failure in Fibre Metal Laminates Subjected to High Impulsive
Blast Loading, International Journal of Solids and Structures, Vol. 51, pp.
3135–3146, 2014.
Steinberg, D. J., Equation of State and Strength Properties of Selected Materials,
Report No. UCRL-MA-106439, Lawrence Livermore National Laboratory,
Livermore, CA, 1991.
Sinha S. and Ghosh, S., Modelling Cyclic Ratcheting Based Fatigue Life of HSLA
Steels Using Crystal Plasticity FEM Simulations and Experiments,
International Journal Fatigue, Vol. 28 (12), pp. 1690–1704, 2006.
Sansour, C., Karsaj I. and Soric, J., Formulation of Anisotropic Continuum
Elastoplasticity at Finite Strains. Part I: Modelling, International Journal
of Plasticity, Vol. 22, pp. 2346–2365, 2006.
Schmidt, S. C., Shaner, J. W., Samara, G. A. and Ross, M., Investigation of the
dynamic behavior of laser-driven flyers, in High-Pressure Science and
Technology, AIP Conference Proceedings, Vol. 309, pp. 1655–1658, 1978.
Schroder, J., Gruttmann F. and L��blein, J., A Simple Orthotropic Finite Elasto–
Plasticity Model Based on Generalized Stress–Strain Measures,
Computational Mechanics, Vol. 30, pp. 48–64, 2002.
Sitko, M., Skoczeń, B. and Wróblewski, FCC-BCC Phase Transformation in
Rectangular Beams Subjected to Plastic Straining at Cryogenic
Temperatures, International Journal of Mechanical Sciences, Vol. 52 (7),
pp. 993-1007, 2010.
Stevens, A. L. and Tuler, F. R., Effect of Shock Precompression On the Dynamic
Fracture Strength of 1020 Steel and 6061-T6 Aluminum, Journal Applied
Physics, Vol. 42-13, pp. 5665–5670, 1971
Trană, E., Nicolae, A. R., Lixandru, P., Cristian, L. M., Enache, C. and Zecheru,
T., Experimental and Numerical Investigation on 6082 Temper Aluminium
Alloy Cartridge Tubes Drawing, Journal of Materials Processing
Technology, Vol. 216, pp. 59–70, 2015.
136
Thomas, M. A., Chitty, D. E., Gildea, M. L. and Kindt T., Constitutive Soil
Properties for Cuddeback, Applied Research Associates, Inc.,
Albuquerque, New Mexico, 2008.
Tugcu, P. and Neale, K.W. On the implementation of anisotropic yield functions
into FInite strain problems of sheet metal forming’, International Journal
of Plasticity, Vol. 15, pp. 1021-1040, 1999.
Tuler, F. R. and Butcher, B. M., A Criterion for the Time Dependence of Dynamic
Fracture, International Journal of Fracture Mechanics, Volume 4-4, pp
431–437, 1986.st
Vignjevic, R., Campbell, J. C., Bourne, N. K. and Djordjevic, N., Modeling Shock
Waves in Orthotropic Elastic Materials, Journal of Applied Physics, Vol.
104, pp. 044904, 2008.
Vignjevic, R., Bourne, N. K., Millett, C. F. and De Vuyst, T., Effects of
Orientation on the Strength of the Aluminum Alloy 7010-T6 During Shock
Loading: Experiment and simulation, Journal of Applied Physics, Vol. 92
(8), 2002.
Vignjevic, R., Djordjevic, N., Campbell, J. C. and Panov, V., Modelling of
Dynamic Damage and Failure in Aluminium Alloys’, International
Journal of Impact Engineering, Vol. 49, pp. 61-76, 2012.
Vladimirov, I. N., Pietryga, M. P. and Reese, S., On The Modelling of Non-Linear
Kinematic Hardening at Finite Strains with Application to Springback -
Comparison of Time Integration Algorithms, International Journal for
Numerical Methods in Engineering, International Journal Numerical
Mathematic Engineering, Vol. 75, pp. 1–28, 2008.
Vladimirov, I. N. and Reese, S., Anisotropic Finite Plasticity with Combined
Hardening and Application to Sheet Metal Forming, International Journal
Material Forming, pp. 293–296, 2008.
Worswick, M. J. and Finn M. J. The Numerical Simulation of Stretch Flange
Forming, International Journal of Plasticity, Vol. 16, pp. 701-720, 2000.
Wackerle, J., Shock-Wave Compression of Quartz, Journal Applied Physic, Vol.
33, pp. 922–937, 1962.
Wilson, L. T., Reedal, D. R., Kuhns, L. D., Grady, D. E. and Kipp, M. E., Using
A Numerical Fragmentation Model to Understand the Fracture and
Fragmentation of Naturally Fragmenting Munitions of Differing Materials
137
and Geometries, 19th International Symposium of Ballistics, pp. 7–11
Interlaken, Switzerland, 2001.
Xiao, H., Bruhns O. T. and Meyers, A., A consistent Finite Elastoplasticity Theory
Combining Additive and Multiplicative Decomposition of the Stretching
and The Deformation Gradient, International Journal of Plasticity, Vol. 16,
pp. 143-177, 2000.
Yuan F. and Prakash, V., Plate Impact Experiments to Investigate Shock-Induced
Inelasticity in Westerly Granite, International Journal of Rock Mechanics
& Mining Sciences, Vol. 60, pp. 277–287, 2013.
Zel’dovich, Y. B. and Raizer, Yu. P., Physics of Shock Waves and High-
Temperature Hydrodynamic Phenomena, Reprint of the Academic Press
Inc., New York, 1966–1967 editions
Zaretsky, E. B. and Kanel, G. I., Plastic Flow in Shock-Loaded Silver at Different
Strain Rates, Journal of Applied Physics, Vol. 110 (7), 2011.
