Explore constitutive modeling from fundamental theory through cutting-edge AI applications
Constitutive Models of Solid Materials: From Mechanical Principles to Engineering Applications provides researchers and engineers with comprehensive methods to predict material responses under complex loading conditions. Written by internationally recognized experts in materials mechanics and computational methods, this systematic treatment connects rigorous theoretical foundations to practical engineering applications, demonstrating the accurate modeling of material behavior across multiple scales.
The text progresses methodically from tensor analysis and continuum mechanics through elasticity, plasticity, and damage mechanics to micromechanics and numerical implementation strategies. Coverage extends to artificial intelligence integration in constitutive research, featuring Physics-Informed Neural Networks for constitutive parameter prediction. Four detailed case studies examine sintered nano-silver, high-strength steel, solder alloys, rock modeling, and high-temperature concrete performance.
The book offers:
- Comprehensive coverage from mathematical foundations through elastoplastic theory, damage mechanics, and micromechanics to AI-enhanced modeling approaches
- Numerical implementation strategies including time-stepping schemes, Newton-Raphson iteration, and elastic predictor plastic corrector methods for simulations
- Detailed case studies on sintered nano-silver, high-strength steels, solder alloys, rocks, and concrete under extreme conditions
- Integration of machine learning including Artificial Neural Networks, XGBoost, and Physics-Informed Neural Networks with example programs
- Multiscale frameworks combining Eshelby’s theory, Hill’s method, and homogenization techniques linking microstructure to macroscopic behavior
This comprehensive resource serves materials scientists, mechanical engineers, civil engineers, aerospace professionals, and graduate students seeking to master constitutive modeling. By combining rigorous mathematical formulation with computational methods and practical case studies, it provides an essential foundation for advanced materials research and engineering practice.
Table of Contents:
Preface xi
1 Overview of Continuum Mechanics 1
1.1 Definition of Tensor 1
1.1.1 Vectors and Tensors 1
1.1.2 Definition of Tensors 2
1.2 Coordinate Transformations 4
1.2.1 Summation Convention 4
1.2.2 Kronecker Increment 6
1.2.3 Coordinate Transformations 7
1.2.4 Permutation Symbols 8
1.3 Basic Operations of Tensors 9
1.3.1 Vector Operations 9
1.3.2 Operations of Tensors 11
1.3.3 Isotropic Tensors 13
1.3.4 Tensor Functions and Calculus Operations 14
1.3.5 Gradient, Divergence, and Curl 16
1.3.6 Green’s Theorem and Stokes’ Theorem 18
1.4 Application of Tensors in Mechanics 20
1.5 Fundamental Laws of Continuum Mechanics 21
1.5.1 Law of Conservation of Mass 21
1.5.2 Law of Conservation of Momentum 22
1.5.3 Law of Conservation of Moment of Momentum 23
1.5.4 Law of Conservation of Energy 23
1.5.5 Second Law of Thermodynamics 25
1.6 Constitutive Model Construction Principle 26
References 29
2 Fundamentals of Elasticity Theory 31
2.1 Fundamental Assumptions of Elasticity Theory 31
2.1.1 Continuity Assumption 32
2.1.2 Perfect Elasticity Assumption 32
2.1.3 Homogeneity Assumption 32
2.1.4 Isotropy Assumption 33
2.1.5 Small Deformation Assumption 33
2.2 Three Fundamental Equations 34
2.2.1 Equilibrium Equations 34
2.2.2 Geometric Equations 35
2.2.3 Physical Equations 37
2.3 Deformation and Strain 38
2.3.1 Reference Configuration and Transient Configuration 38
2.3.2 Strain Tensor 40
2.3.3 Principal Strain and Volumetric Strain 41
2.3.4 Strain Coordination Equation 44
2.4 Stress Analysis 45
2.4.1 Cauchy Principle of Stress 45
2.4.2 Cauchy Stress Formula 47
2.4.3 Equilibrium Different Equations and Stress Tensor Symmetry 48
2.4.4 Principal Stress and Stress Invariants 49
2.5 Foundation of the Elastic Constitutive 51
2.5.1 Elastic Constitutive Theory 51
2.5.2 Generalized Hooke’s Law 51
References 58
3 Plastic Constitutive Theory 59
3.1 Plastic Behavior of Solid Materials 59
3.2 Variable Definitions 61
3.2.1 Decomposition of the Strain Tensor 61
3.2.2 Strain Rate and Strain Increment 62
3.3 Drucker Postulate 64
3.3.1 Important corollary of the Drucker axioms 68
3.4 Incremental Plastic Flow Theory 71
3.5 Total Plastic Deformation Theory 79
3.6 Characteristics of Constitutive Relations in Plastic Deformation 82
References 84
4 Damage Constitutive Principles and Methods 87
4.1 Overview of Material Damage 87
4.1.1 Continuum Damage Mechanics 87
4.1.2 Relationship Between Material Damage and Microstructure 88
4.1.3 Types of Material Damage 89
4.2 Research Methods of Damage Mechanics 90
4.3 Definition and Basic Assumptions of Damage 91
4.3.1 Variable Selection 91
4.3.2 Definition of Damage 93
4.4 Damage Evolution Equation 95
4.4.1 State Variables 95
4.4.2 Damage Local State Principle 96
4.4.3 Basic Principles of Irreversible Thermodynamics 96
4.4.4 Basis of Thermodynamic Damage 97
4.5 Helmholtz Free Energy (HFE) 99
4.5.1 Physical Significance of HFE 99
4.5.2 Tensor Decomposition of HFE 99
4.5.3 Damage Criteria Based on Energy Dissipation 100
References 101
5 Basics of Micromechanics 103
5.1 Basic Concepts of Micromechanics 103
5.1.1 Introduction to Representative Volume Elements 104
5.1.2 Localization 105
5.2 Eshelby Eigenstrain Theory and Equivalent Inclusion Theory 107
5.2.1 Eigenstrain Theory 107
5.2.2 Equivalent Inclusion Theory 112
5.3 Hill Theorem 114
5.4 Mean Field Method Based on Eshelby Equivalent Inclusion Theory 115
5.4.1 Basic Process of Homogenization Method 115
5.4.2 Sparse Methods 118
5.4.3 Mori–Tanaka Method 119
5.4.4 Self-consistent Method for Polycrystalline Materials 121
References 121
6 Numerical Implementation of Constitutive Relations 123
6.1 Basic Concepts 123
6.1.1 Numerical Solution of Differential Equations 123
6.1.2 Newton’s Iterative Method 125
6.1.3 One-dimensional Elastoplastic Constitutive Model 127
6.1.4 Rate Form of the Stress–Strain Relationship 128
6.1.5 Stress–Strain Relationship with Kinematic Hardening in Rate Form 129
6.1.6 Incremental Form of the Governing Equations 130
6.1.7 Elastic Predictor/Plastic Corrector Solution Algorithm 131
6.1.8 Consistent Elastoplastic Modulus 134
6.2 Three-dimensional Elastoplastic Model and Numerical Solution Framework 136
6.2.1 Model Overview 136
6.2.2 Stress–Strain Rate Relations 138
6.2.3 Elastic Predictor/Plastic Corrector Solution Algorithm 139
6.2.4 Consistency Elastoplastic Modulus 142
6.3 Three-dimensional Elastoviscoplastic Model with Yield Surface and Solution Framework 142
6.3.1 Model Overview 142
6.3.2 The Definition of the Plastic Multiplier 143
6.3.3 Elastic Predictor/Plastic Corrector Solution Algorithm 144
6.4 Three-dimensional Elastoplastic Damage Model and Solution Framework 147
6.4.1 Model Overview 147
6.4.2 Elastic Prediction/Plastic Correction-based Solution Algorithm 148
6.4.3 Numerical Issues 149
6.5 Custom Constitutive Models Based on ABAQUS 150
6.6 Conclusion 152
References 152
7 Artificial Intelligence in Constitutive Research 153
7.1 Tensor Bases of Machine Learning 153
7.1.1 Function of Tensor in Machine Learning Algorithms 153
7.1.2 Key Operational Rule of Tensor Application in Machine Learning 154
7.2 Basic Mathematical Principles of Artificial Intelligence 156
7.2.1 MLP-based Machine Learning Algorithms 156
7.2.2 CART-based Machine Learning Algorithms 160
7.2.2.1 Classification and Regression Tree 161
7.2.2.2 Ensemble CART-based Algorithm 162
7.3 Current State of Machine Learning in Research of Constative Models 163
7.3.1 Data-driven Research of the Material Constitutive Models 164
7.3.2 Machine-learning Based Prediction of Life Cycle Material Properties 165
7.3.3 Physic-informed Research of the Solid Material 166
7.3.4 Practices to Improve the Applicability of Machine Learning Models 167
7.4 Example: Predicting Constitutive Parameters of Concrete 168
7.5 Conclusion 173
References 174
8 Tensile Creep Failure Mechanism and Theoretical Model of Sintered Nano-silver 179
8.1 Introduction 179
8.2 Molecular Dynamics Model of Sintered Neck 181
8.3 Damage Model 183
8.4 Creep Life Model 185
8.5 Parameter Determination 185
8.6 Damage Evolution Analysis of Sintered Nano-silver 186
8.7 Theoretical Analysis of Sintered Nano-silver Creep 187
8.8 Conclusion 187
References 189
9 Unified Creep-plasticity Model for High-strength Steel and Solder Alloys 193
9.1 Introduction 193
9.2 Viscoplastic Constitutive Framework 194
9.3 Application of the Proposed Theory 199
9.3.1 Applied to High-strength Steel 200
9.3.2 Applied to Sn–3.0Ag–0.5Cu Solder Alloy 203
9.4 Conclusion 205
References 205
10 A Multiscale Framework for the Constitutive Modeling of Rock 209
10.1 Introduction 209
10.2 Fundamentals of Rock Behavior 210
10.2.1 Heterogeneity and Anisotropy of Rock 210
10.2.2 Mechanical Behavior Across Scales 210
10.2.3 Role of Microstructure in Governing Macroscopic Behavior 211
10.3 Multiscale Modeling Framework 211
10.3.1 Concept of Scale Separation and Homogenization 211
10.3.2 Eshelby’s Equivalent Inclusion Theory and Effective Properties 212
10.3.3 Hill’s Incremental Method 213
10.4 Pressure Dependent Plasticity Model 215
10.4.1 Thermodynamic Formulation 215
10.4.2 Elastoplastic Constitutive Relations 215
10.5 Application of the Multiscale Constitutive Modeling to the COx Argillite 216
10.5.1 Mineralogy and the Mechanical Response of the COx Argillite 216
10.5.2 Modelling of the Porous Clay Matrix by the extended GTN Model 217
10.6 Computational Aspects 220
10.6.1 Local Integration Algorithm and the Consistent Tangent Moduli 220
10.6.2 Numerical Implementation of the Homogenization Procedure 222
10.7 Numerical Validations 224
10.7.1 Comparison Against FE Analysis on a Two-phase Unit Cell 224
10.7.2 Experimental Validation 226
10.8 Conclusion 232
References 233
11 Development of a High Temperature Constitutive Model for Concrete Based on Elastoplastic Theory 235
11.1 Introduction 235
11.2 Constitutive Theory of Concrete Based on Thermodynamic Framework 235
11.2.1 Constitutive Theory of Concrete 235
11.2.2 Thermodynamic Equations 242
11.2.3 Yield Criterion and Hardening Law 245
11.3 Damage Model of Concrete at High Temperature 250
11.3.1 External Load Damage Variables of UHPC 250
11.3.2 Heat Damage 251
11.3.3 Determination of parameters 254
11.4 Concrete High Temperature Transient Creep Model 258
11.4.1 High Temperature Transient Creep Strain Model 258
11.4.2 Drying Deformation 260
11.4.3 Dehydration Deformation 261
11.4.4 Chemical Decomposition 263
11.5 Model Validation 263
11.5.1 Numerical Implementation of the Constitutive Model 263
11.5.2 Numerical Model Establishment and Verification 263
11.6 Conclusion 269
References 270
Appendix 275
A.1 Machine Learning Case Studies (for Chapter 7) 275
A.2 Tensorial Notations and Operations (for Chapter 10) 281
A.3 Basic Framework for Calculation of Elastic-plastic Damage Constitutive Model of Concrete (for Chapter 11) 282
Index 287
About the Author :
Yao Yao is a Professor researching mechanical properties of materials and disaster prevention of engineering structures under extreme loads. His research is also focused on engineering material construction, fatigue, and damage under multi-physical fields.
Hu Fang, PhD, develops high-temperature damage constitutive models for solid materials. His research is also focused on material fracture and damage theory, as well as the research on the mechanical properties of composite material interfaces.
Hongcun Guo, PhD, researches the mechanical behavior of green building materials and the high-temperature mechanical properties of ultra-high performance concrete.
Tao Zeng, PhD, is a council member of the Shaanxi Society for Rock Mechanics and Engineering. His research is focused on underground engineering, including rock mechanics and the development of micromechanical models of rock materials.
Xu He, PhD, specializes in multi-scale modeling of mechanical properties, constitute behavior, and failure processes of advanced materials under extreme loads.
