Constitutive Models of Solid Materials

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.

• Preface xi1 Overview of Continuum Mechanics 11.1 Definition of Tensor 11.1.1 Vectors and Tensors 11.1.2 Definition of Tensors 21.2 Coordinate Transformations 41.2.1 Summation Convention 41.2.2 Kronecker Increment 61.2.3 Coordinate Transformations 71.2.4 Permutation Symbols 81.3 Basic Operations of Tensors 91.3.1 Vector Operations 91.3.2 Operations of Tensors 111.3.3 Isotropic Tensors 131.3.4 Tensor Functions and Calculus Operations 141.3.5 Gradient, Divergence, and Curl 161.3.6 Green’s Theorem and Stokes’ Theorem 181.4 Application of Tensors in Mechanics 201.5 Fundamental Laws of Continuum Mechanics 211.5.1 Law of Conservation of Mass 211.5.2 Law of Conservation of Momentum 221.5.3 Law of Conservation of Moment of Momentum 231.5.4 Law of Conservation of Energy 231.5.5 Second Law of Thermodynamics 251.6 Constitutive Model Construction Principle 26References 292 Fundamentals of Elasticity Theory 312.1 Fundamental Assumptions of Elasticity Theory 312.1.1 Continuity Assumption 322.1.2 Perfect Elasticity Assumption 322.1.3 Homogeneity Assumption 322.1.4 Isotropy Assumption 332.1.5 Small Deformation Assumption 332.2 Three Fundamental Equations 342.2.1 Equilibrium Equations 342.2.2 Geometric Equations 352.2.3 Physical Equations 372.3 Deformation and Strain 382.3.1 Reference Configuration and Transient Configuration 382.3.2 Strain Tensor 402.3.3 Principal Strain and Volumetric Strain 412.3.4 Strain Coordination Equation 442.4 Stress Analysis 452.4.1 Cauchy Principle of Stress 452.4.2 Cauchy Stress Formula 472.4.3 Equilibrium Different Equations and Stress Tensor Symmetry 482.4.4 Principal Stress and Stress Invariants 492.5 Foundation of the Elastic Constitutive 512.5.1 Elastic Constitutive Theory 512.5.2 Generalized Hooke’s Law 51References 583 Plastic Constitutive Theory 593.1 Plastic Behavior of Solid Materials 593.2 Variable Definitions 613.2.1 Decomposition of the Strain Tensor 613.2.2 Strain Rate and Strain Increment 623.3 Drucker Postulate 643.3.1 Important corollary of the Drucker axioms 683.4 Incremental Plastic Flow Theory 713.5 Total Plastic Deformation Theory 793.6 Characteristics of Constitutive Relations in Plastic Deformation 82References 844 Damage Constitutive Principles and Methods 874.1 Overview of Material Damage 874.1.1 Continuum Damage Mechanics 874.1.2 Relationship Between Material Damage and Microstructure 884.1.3 Types of Material Damage 894.2 Research Methods of Damage Mechanics 904.3 Definition and Basic Assumptions of Damage 914.3.1 Variable Selection 914.3.2 Definition of Damage 934.4 Damage Evolution Equation 954.4.1 State Variables 954.4.2 Damage Local State Principle 964.4.3 Basic Principles of Irreversible Thermodynamics 964.4.4 Basis of Thermodynamic Damage 974.5 Helmholtz Free Energy (HFE) 994.5.1 Physical Significance of HFE 994.5.2 Tensor Decomposition of HFE 994.5.3 Damage Criteria Based on Energy Dissipation 100References 1015 Basics of Micromechanics 1035.1 Basic Concepts of Micromechanics 1035.1.1 Introduction to Representative Volume Elements 1045.1.2 Localization 1055.2 Eshelby Eigenstrain Theory and Equivalent Inclusion Theory 1075.2.1 Eigenstrain Theory 1075.2.2 Equivalent Inclusion Theory 1125.3 Hill Theorem 1145.4 Mean Field Method Based on Eshelby Equivalent Inclusion Theory 1155.4.1 Basic Process of Homogenization Method 1155.4.2 Sparse Methods 1185.4.3 Mori–Tanaka Method 1195.4.4 Self-consistent Method for Polycrystalline Materials 121References 1216 Numerical Implementation of Constitutive Relations 1236.1 Basic Concepts 1236.1.1 Numerical Solution of Differential Equations 1236.1.2 Newton’s Iterative Method 1256.1.3 One-dimensional Elastoplastic Constitutive Model 1276.1.4 Rate Form of the Stress–Strain Relationship 1286.1.5 Stress–Strain Relationship with Kinematic Hardening in Rate Form 1296.1.6 Incremental Form of the Governing Equations 1306.1.7 Elastic Predictor/Plastic Corrector Solution Algorithm 1316.1.8 Consistent Elastoplastic Modulus 1346.2 Three-dimensional Elastoplastic Model and Numerical Solution Framework 1366.2.1 Model Overview 1366.2.2 Stress–Strain Rate Relations 1386.2.3 Elastic Predictor/Plastic Corrector Solution Algorithm 1396.2.4 Consistency Elastoplastic Modulus 1426.3 Three-dimensional Elastoviscoplastic Model with Yield Surface and Solution Framework 1426.3.1 Model Overview 1426.3.2 The Definition of the Plastic Multiplier 1436.3.3 Elastic Predictor/Plastic Corrector Solution Algorithm 1446.4 Three-dimensional Elastoplastic Damage Model and Solution Framework 1476.4.1 Model Overview 1476.4.2 Elastic Prediction/Plastic Correction-based Solution Algorithm 1486.4.3 Numerical Issues 1496.5 Custom Constitutive Models Based on ABAQUS 1506.6 Conclusion 152References 1527 Artificial Intelligence in Constitutive Research 1537.1 Tensor Bases of Machine Learning 1537.1.1 Function of Tensor in Machine Learning Algorithms 1537.1.2 Key Operational Rule of Tensor Application in Machine Learning 1547.2 Basic Mathematical Principles of Artificial Intelligence 1567.2.1 MLP-based Machine Learning Algorithms 1567.2.2 CART-based Machine Learning Algorithms 1607.2.2.1 Classification and Regression Tree 1617.2.2.2 Ensemble CART-based Algorithm 1627.3 Current State of Machine Learning in Research of Constative Models 1637.3.1 Data-driven Research of the Material Constitutive Models 1647.3.2 Machine-learning Based Prediction of Life Cycle Material Properties 1657.3.3 Physic-informed Research of the Solid Material 1667.3.4 Practices to Improve the Applicability of Machine Learning Models 1677.4 Example: Predicting Constitutive Parameters of Concrete 1687.5 Conclusion 173References 1748 Tensile Creep Failure Mechanism and Theoretical Model of Sintered Nano-silver 1798.1 Introduction 1798.2 Molecular Dynamics Model of Sintered Neck 1818.3 Damage Model 1838.4 Creep Life Model 1858.5 Parameter Determination 1858.6 Damage Evolution Analysis of Sintered Nano-silver 1868.7 Theoretical Analysis of Sintered Nano-silver Creep 1878.8 Conclusion 187References 1899 Unified Creep-plasticity Model for High-strength Steel and Solder Alloys 1939.1 Introduction 1939.2 Viscoplastic Constitutive Framework 1949.3 Application of the Proposed Theory 1999.3.1 Applied to High-strength Steel 2009.3.2 Applied to Sn–3.0Ag–0.5Cu Solder Alloy 2039.4 Conclusion 205References 20510 A Multiscale Framework for the Constitutive Modeling of Rock 20910.1 Introduction 20910.2 Fundamentals of Rock Behavior 21010.2.1 Heterogeneity and Anisotropy of Rock 21010.2.2 Mechanical Behavior Across Scales 21010.2.3 Role of Microstructure in Governing Macroscopic Behavior 21110.3 Multiscale Modeling Framework 21110.3.1 Concept of Scale Separation and Homogenization 21110.3.2 Eshelby’s Equivalent Inclusion Theory and Effective Properties 21210.3.3 Hill’s Incremental Method 21310.4 Pressure Dependent Plasticity Model 21510.4.1 Thermodynamic Formulation 21510.4.2 Elastoplastic Constitutive Relations 21510.5 Application of the Multiscale Constitutive Modeling to the COx Argillite 21610.5.1 Mineralogy and the Mechanical Response of the COx Argillite 21610.5.2 Modelling of the Porous Clay Matrix by the extended GTN Model 21710.6 Computational Aspects 22010.6.1 Local Integration Algorithm and the Consistent Tangent Moduli 22010.6.2 Numerical Implementation of the Homogenization Procedure 22210.7 Numerical Validations 22410.7.1 Comparison Against FE Analysis on a Two-phase Unit Cell 22410.7.2 Experimental Validation 22610.8 Conclusion 232References 23311 Development of a High Temperature Constitutive Model for Concrete Based on Elastoplastic Theory 23511.1 Introduction 23511.2 Constitutive Theory of Concrete Based on Thermodynamic Framework 23511.2.1 Constitutive Theory of Concrete 23511.2.2 Thermodynamic Equations 24211.2.3 Yield Criterion and Hardening Law 24511.3 Damage Model of Concrete at High Temperature 25011.3.1 External Load Damage Variables of UHPC 25011.3.2 Heat Damage 25111.3.3 Determination of parameters 25411.4 Concrete High Temperature Transient Creep Model 25811.4.1 High Temperature Transient Creep Strain Model 25811.4.2 Drying Deformation 26011.4.3 Dehydration Deformation 26111.4.4 Chemical Decomposition 26311.5 Model Validation 26311.5.1 Numerical Implementation of the Constitutive Model 26311.5.2 Numerical Model Establishment and Verification 26311.6 Conclusion 269References 270Appendix 275A.1 Machine Learning Case Studies (for Chapter 7) 275A.2 Tensorial Notations and Operations (for Chapter 10) 281A.3 Basic Framework for Calculation of Elastic-plastic Damage Constitutive Model of Concrete (for Chapter 11) 282Index 287