本书系统地介绍了作者们在非稳态环境条件下多物理岩土塑性理论、模型及数值算法方面的研究工作。在本书的前半部分，首先介绍了一种严密的塑性建模方法，包括无需构造塑性势函数的非正交流动规则、确定非稳态环境下塑性应变增量和加/卸载准则、增量型应力-应变环境本构关系。基于该理论框架，作者们建立了一系列稳态和非稳态环境下的土和混凝土的非正交弹塑性模型，这些模型的性能得到了大量实验数据的充分验证。在本书的后半部分，作者基于前沿的数值方法，如数值优化算法、数值微分方法、正则化技术等，提出了一些高效、稳健的应力更新算法，用于解决弹塑性模型数值实现中常见的计算困难，这些算法在岩土材料弹塑性模型的数值应用中表现出了良好的性能。

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(1)2001-09至2006-11,北京航空航天大学,道路与铁道工程,博士(硕博连读) (2)1996-09至2000-07,南京航空航天大学,飞行器环境控制,学士2006-11-29至2008-09-01 北京工业大学 助理研究员 2008-09-01至2010-12-31 北京工业大学 副研究员 2011-01-01至今 北京工业大学 教授（1）“非正交岩土塑性理论及模型”，中国岩石力学与工程学会自然科学一等奖，2023年，排名第1（本书依托项目）。 （2）“城市大型地下结构抗震设计理论与方法及工程应用”，国家科技进步二等奖，2017年，排名第2（本书依托项目）。路德春教授为国家杰出青年科学基金获得者、北京市学术创新团队带头人。现任北京工业大学城市建设学部副主任、岩土与地下工程研究所所长，兼任《岩土工程学报》、《岩石力学与工程学报》、《岩土力学》等期刊编委。

Contents
Chapter 1 Introduction 1
1.1 Geotechnical Plasticity Modeling 2
1.1.1 Magnitude of Plastic Strain Increment 3
1.1.2 Direction of Plastic Strain Increment 6
1.2 Stress Update Algorithms for Elastoplastic Models 8
1.2.1 Numerical Techniques for Non-smooth Problems 8
1.2.2 Numerical Methods for Nonlinear Problems 10
1.2.3 Regularization Methods for Mesh Dependence Problem 11
1.3 Main Research Work 11
References 13
Chapter 2 Multi-physical and Non-orthogonal Geotechnical Plasticity Theory 19
2.1 Magnitude of Plastic Strain Increment 20
2.2 Direction of Plastic Strain Increment 23
2.2.1 Non-orthogonal Phenomena in Numerical Experiments 24
2.2.2 Non-orthogonal Plastic Flow Rule 27
2.3 Multi-physical Loading/Unloading Criterion 29
2.4 Constitutive Equations of Multi-physical and Non-orthogonal Plasticity Theory 33
2.5 Determination Method for Model Parameters 34
2.6 Summary 36
References 37
Chapter 3 Non-orthogonal Elastoplastic Model for Soil in Steady Environments 39
3.1 Characteristic Stress Method 40
3.2 Non-orthogonal Elastoplastic Model for Clay 44
3.2.1 Yield Function 44
3.2.2 Direction of Plastic Strain Increment 45
3.2.3 Magnitude of Plastic Strain Increment 53
3.2.4 Stress-Strain Relationship 55
3.2.5 Parameter Calibration and Model Validation 55
3.3 Non-orthogonal Elastoplastic Model for Sand with Dilatancy 62
3.3.1 Direction of Plastic Strain Increment 62
3.3.2 Magnitude of Plastic Strain Increment 65
3.3.3 Stress-Strain Relationship 72
3.3.4 Parameter Calibration and Model Validation 73
3.4 Summary 76
References 78
Chapter 4 Non-orthogonal Elastoplastic Model for Soil in Unsteady Water Environment 81
4.1 Modeling Approach for Soil in Unsteady Water Environment 82
4.2 Establishment of Constitutive Model 86
4.2.1 Yield Function in Steady Water Environment 86
4.2.2 Influence of Water Environment on Yield Function 88
4.2.3 Description of Unsteady Water Environment 91
4.3 Constitutive Equations 94
4.4 Model Validation 97
4.4.1 Silt Sand 97
4.4.2 Pearl Clay 99
4.4.3 Mixed Soil 104
4.5 Model Performance Analysis 106
4.5.1 Isotropic Compression Path with Constant Suction 107
4.5.2 Wetting Path with Constant Net Stress 108
4.5.3 Conventional Triaxial Drained Shearing Path 112
4.6 Summary 113
References 114
Chapter 5 Non-orthogonal Plasticity Damage Model for Concrete 117
5.1 Theoretical Framework of Plasticity Damage Model 118
5.1.1 Damage Formulation 118
5.1.2 Plasticity Formulation 119
5.2 Specifics Adopted for Concrete Model 121
5.2.1 Damage Variable 121
5.2.2 Effective Hardening Function 123
5.2.3 Yield Function 125
5.2.4 Non-orthogonal Flow Rule 128
5.3 Parameter Calibration and Model Validation 133
5.3.1 Parameter Calibration 133
5.3.2 Model Validation for Monotonic Loading 135
5.3.3 Model Validation for Cyclic Loading 136
5.4 Summary 139
References 141
Chapter 6 Non-orthogonal Elastoplastic Model for Concrete in Variable Strain Rate Environment 143
6.1 Specifics Adopted for Concrete Model 144
6.1.1 Yield Function 144
6.1.2 Hardening Parameter 146
6.1.3 Flow Rule 148
6.1.4 Loading/Unloading Criterion 149
6.2 Test of Loading Under Constant- and Variable-Strain Rate 149
6.2.1 Testing Procedures and Apparatus 150
6.2.2 Preparation of Materials and Specimens 150
6.2.3 Experimental Results 151
6.3 Comparison of Model Predictions and Test Results 153
6.3.1 Constant-Strain Rate 153
6.3.2 Variable-Strain Rate 156
6.4 Analysis of Incremental Stress-Strain-Strain Rate Model 160
6.4.1 Loading Paths with Independent Variation in Strain Rate 160
6.4.2 Designed Variable-Strain Rate Loading Test 163
6.5 Summary 164
References 164
Chapter 7 Unconstrained Stress Update Algorithm for Elastoplastic Models 167
7.1 Implicit Stress Integral Scheme for Elastoplastic Models 168
7.2 Return Mapping Stress Update Strategy 169
7.3 Unconstrained Stress Update Strategy 170
7.4 Application to 1D Compression Problem 171
7.4.1 Return Mapping Stress Update Strategy Solution 172
7.4.2 Unconstrained Stress Update Strategy Solution 175
7.5 Application to Non-orthogonal Elastoplastic Model for Clay 176
7.5.1 Implicit Stress Update Implementation 176
7.5.2 Smoothing Consistent Tangent Operator 178
7.6 Numerical Examples 183
7.6.1 Cylindrical Cavity Expansion Problems 184
7.6.2 Cylinder under Cyclic Shear Loads 186
7.6.3 Foundation Bearing Capacity Test 187
7.7 Summary 191
References 194
Chapter 8 Stress Update Algorithm Based on Numerical Optimization Methods 197
8.1 Unconstrained Minimization Problem: Modified Cam-Clay Model Case 198
8.2 Stress Update Algorithm with Line Search Method 200
8.3 Stress Update Algorithm with Trust Region Method 203
8.3.1 Trust Region Radius Determination 205
8.3.2 Non-monotonic Trust Region Method 205
8.3.3 Double Dogleg Method for Subproblem Solving 207
8.4 Numerical Examples 210
8.4.1 Element Test 210
8.4.2 Strip Foundation under Inclined Load 213
8.4.3 Cylinder under Cyclic Shear Loads 214
8.5 Summary 219
References 221
Chapter 9 Stress Update Algorithm Based on Numerical Differentiation Methods 223
9.1 Numerical Differentiation Methods 224
9.2 Finite Difference Methods 224
9.2.1 Complex Step Derivative Approximation 226
9.2.2 Hyper-Dual Step Derivative Approximation 226
9.2.3 Numerical Evaluation 229
9.3 Implicit Stress Integral Scheme of Mohr-Coulomb Plasticity Model 229
9.3.1 Non-associated Mohr-Coulomb Plasticity Model 230
9.3.2 Implicit Stress Integration Scheme 233
9.4 Return Mapping Algorithm Using Line Search Method and Hyper-Dual Step Derivative Approximation 234
9.4.1 Solution of Nonlinear Equations Based on Line Search Method 234
9.4.2 Derivative Evaluation Based on Hyper-Dual Step Derivative Approximation 236
9.5 Numerical Evaluations 238
9.5.1 3D Notched Plate Under Tension 240
9.5.2 Cylinder Under Shear Loads 242
9.5.3 Strip Footing on Slope 244
9.5.4 Slope Stability Analysis 245
9.5.5 Tunnel Excavation Simulation 247
9.6 Summary 249
References 251
Chapter 10 Stress Update Algorithm Based on Integral Nonlocal Method 253
10.1 Integral Type Nonlocal Method 254
10.2 Nonlocal Implicit Stress Update Algorithm 256
10.2.1 Plastic Part 257
10.2.2 Damage Part 258
10.2.3 Consistent Tangent Operator 259
10.3 Application to Mohr-Coulomb Plasticity Damage Model 261
10.3.1 Basic Constitutive Elements 261
10.3.2 Nonlocal Numerical Implementation 265
10.4 Numerical Examples 267
10.4.1 2D Plate Plane Strain Compression Test 269
10.4.2 3D Notched Plate Compression Test 269
10.4.3 Concrete-Filled Steel-Tube Column under Central Compression 273
10.4.4 Reinforced Concrete Column under Eccentric Compression 277
10.4.5 3D Column Specimen Axial Compression Test 279
10.5 Summary 281
References 282
Appendix A Definitions and Properties of Fractional Derivatives 285
Appendix B Operation Rules and Numerical Implementation of Hyper-Dual Numbers 289