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1 Introduction 1

1.1 Elasto-Plastic Finite Elements 1

1.2 Bounds and Region of the Convex Yield Surface 3

1.3 Unified Strength Theory and its Implementati on in Computer Codes 4

1.4 The Effect of Yield Criteria on the Numerical Analysis Results 7

1.5 Historical Review:With Emphasis on the Implementation and Application of Unified Strength Theory 12

1.6 Brief Summary 17

References 19

2 Stress and Strain 29

2.1 Introduction 29

2.2 Stress at a Point,Stress Invariants 29

2.3 Deviatoric Stress Tensor and its Invariants 31

2.4 Stresses on the Oblique Plane 33

2.4.1 Stresses on the Oblique Plane 33

2.4.2 Principal Shear Stresses 33

2.4.3 Octahedral Shear Stress 35

2.5 From Single-Shear Element to Twin-Shear Element 37

2.6 Stress Space 38

2.7 Stress State Parameters 42

2.8 Strain Components 45

2.9 Equations of Equilibrium 46

2.10 Generalized Hooke's Law 46

2.11 Compatibility Equations 48

2.12 Governing Equations for Plane Stress Problems 49

2.13 Governing Equations in Polar Coordinates 50

2.14 Brief Summary 51

References 52

3 Material Models in Computational Plasticity 53

3.1 Introduction 53

3.2 Material Models for Non-SD Materials(Metallic Materials) 55

3.2.1 Hydrostatic Stress Independence 55

3.2.2 The Tensile Yield Stress Equals the Compressive Yield Stress 56

3.2.3 Sixfold Symmetry of the Yield Function 56

3.2.4 Convexity of the Yield Function 57

3.2.5 Bounds of the Yield Function for Non-SD Materials 58

3.3 Material Models for SD Materials 66

3.3.1 General Behavior of Yield Function for SD Materials 66

3.3.1.1 Six Basic Experimental Points for SD Materials 66

3.3.1.2 Threefold Symmetry of the Yield Function 66

3.3.1.3 Convexity of the Yield Function 67

3.3.2 Three Basic Models for SD Materials 67

3.4 Multi-Parameter Criteria for Geomaterials 70

3.4.1 Multi-Parameter Single-Shear Failure Criterion 70

3.4.2 Multi-Parameter Three-Shear Failure Criterion 71

3.4.3 Multi-Parameter Twin-Shear Failure Criterion 74

3.5 Bounds and the Region of the Convex Yield Function 75

3.6 Brief Summary 77

References 78

4 Unified Strength Theory and its Material Parameters 81

4.1 Introduction 81

4.2 Mechanical Model of Unified Strength Theory 82

4.3 Mathematical Modelling and the Determination of the Material Parameters of the Unified Strength Theory 85

4.4 Mathematical Expression of the Unified Strength Theory 86

4.5 Special Cases of the Unified Strength Theory 87

4.5.1 Special Cases of the Unified Strength Theory(Varying b) 87

4.5.2 Special Cases of the Unified Strength Theory(Varyinga) 89

4.6 Other Formulations of the UST and Material Parameters 92

4.6.1 UST with Principal Stress and Compressive Strength(σ1,σ2,σ3,a,σc) 92

4.6.2 UST with Stress Invariant and Tensile Strength F(I1,J2,θ,σt,a) 93

4.6.3 UST with Stress Invariant and Compressive Strength F(I1,J2,θ,a,σc） 94

4.6.4 UST with Principal Stress and Cohes ive Parameter F(σ1,σ2,σ3,C0,？) 94

4.6.5 UST with Stress Invariant and Cohesive Parameter F(I,J2,θ,C0,？) 95

4.7 Other Material Parameters ofthe Unified Strength Theory 95

4.7.1 Material Parameters β and C are Determined by Experimental Results of Uniaxial Tension Strength σt and Shear Strength τ0 96

4.7.2 Material Parameters β and C are Determined by Experimental Results of Uniaxial Compressive Strength σc and Shear Strength τ0 96

4.7.3 Material Parameters β and C are Determined by Experimental Results of Uniaxial Compressive Strength σc and Biaxial Compressive Strength σcc 97

4.7.4 Material Parameters β and C are Determined by Experimental Results of Uniaxial Compressive Strength σc and Biaxial Compressive Strength σcc 97

4.7.5 Material Parameters β and C are Determined by Experimental Results of Uniaxial Compressive Strength σc and Biaxial Compressive Strength σcc 97

4.8 Three-Parameter Unified Strength Theory 98

4.9 Stress Space and Yield Loci ofthe UST 98

4.10 Yield Surfaces of the UST in Principal Stress Space 102

4.11 Extend of UST from Convex to Non-Convex 107

4.12 Yield Loci of the UST in Plane Stress State 108

4.13 Unified Strength Theory in Meridian Plane 112

4.14 Extend of UST from Linear to Non-Linear UST 114

4.15 Equivalent Stress of the Unified Strength Theory 116

4.15.1 Equivalent Stresses for Non-SD Materials 117

4.15.2 Equivalent Stresses for SD Materials 117

4.15.3 Equivalent Stresses of the Unified Yield Criterion 117

4.15.4 Equivalent Stress of the Unified Strength Theory 118

4.16 Examples 119

4.17 Summary 122

References 125

5 Non-Smooth Multi-Surface Plasticity 129

5.1 Introduction 129

5.2 Plastic Deformation in Uniaxial Stress State 130

5.3 Three-Dimensional Elastic Stress-Strain Relation 132

5.4 Plastic Work Hardening and Strain Hardening 133

5.5 Plastic Flow Rule 136

5.6 Drucker's Postulate-Convexity of the Loading Surface 137

5.7 Incremental Constitutive Equations in Matrix Formulation 141

5.8 Determination of Flow Vector for Different Yield Functions 144

5.9 Singularity of Piecewise-Linear Yield Functions 146

5.10 Process of Singularity of the Plastic Flow Vector 151

5.11 Suggested Methods 153

5.12 Unified Process of the Corner Singularity 156

5.12.1 Tresca Yield Criterion 156

5.12.2 Mohr-Coulomb Yield Criterion 157

5.12.3 Twin-Shear Yield Criterion 157

5.12.4 Generalized Twin-Shear Yield Criterion 157

5.13 BriefSummary 159

References 160

6 Implementation of the Unified Strength Theory into FEM Codes 163

6.1 Introduction 163

6.2 Bounds of the Single Criteria for Non-SD Materials 165

6.3 Bounds of the Failure Criteria for SD Materials 166

6.4 Unification of the Yield Criteria for Non-SD Materials and SD Materials 168

6.5 Material Models 170

6.6 Program Structure and its Subroutines Relating to the Unified Strength Theory:INVARY,YIELDY,FLOWVP 172

6.6.1 Subroutine"Invar" 172

6.6.2 Subroutine"Invary" 174

6.6.3 Subroutine"Yieldy" 175

6.6.4 Subroutine"Criten" 176

6.7 Brief Summary 178

References 178

7 Examples of the Application of Unified Elasto-Plastic Constitutive Relations 183

7.1 Introduction 183

7.2 Plane Stress Problems 184

7.2.1 Elasto-Plastic Analysis of a Cantilever Beam 184

7.2.2 Elasto-Plastic Analysis of a Trapezoid Structure under Uniform Load 187

7.3 Plane Strain Problems 188

7.4 Spatial Axisymmetric Problems 190

7.4.1 Analysis of Plastic Zone for Thick-Walled Cylinder 190

7.4.2 Analysis for Limit-Bearing Capacity of a Circular Plate 193

7.4.3 Truncated Cone under the Uniform Load on the Top 195

7.5 Brief Summary 197

References 198

8 Strip with a Circular Hole under Tension and Compression 199

8.1 Introduction 199

8.2 Plastic Analysis of a Strip with a Circular Hole for Non-SD Material 200

8.3 Elasto-Plastic Analysis of a Strip with a Circular Hole for SD Material under Tension 203

8.4 Plastic Zone of a Strip with a Circular Hole for SD Material under Compression 204

8.5 Comparison of Numerical Analysis with Experiments 205

8.6 Elasto-Plastic Analysis of a Strip with a Circular Hole for a Special SD Material:Concrete 207

8.7 Brief Summary 208

References 211

9 Plastic Analysis of Footing Foundation Based on the Unified Strenghth Theory 213

9.1 Introduction 213

9.2 Effect of Yield Criterion on the Limit Analysis of Footing 216

9.3 Elasto-Plastic Analysis of Foundation Using UST 218

9.4 Plastic Analysis of Strip Foundation Using UST 220

9.5 Plastic Analysis of Circular Foundation Using UST 226

9.5.1 Unified Characteristics Line Field of Spatial Axisymmetric Problem 226

9.5.2 Numerical Simulation of Spatial Axisymmetric Problem 227

9.5.3 Effect of UST Parameter ？ on the Spread ofShear Strain 230

9.6 Effect of UST Parameter b and ？ on the Spread ofShear Strain 232

9.7 Brief Summary 233

References 234

10 Underground Caves,Tunnels and Excavation of Hydraulic Power Station 239

10.1 Introduction 239

10.2 Effect of Yield Criterion on the Plastic Zone for a Circular Cave 241

10.3 Plastic Zone for Underground Circular Cave under Two Direction Compressions 242

10.3.1 Material Model 243

10.3.2 Elastic Bearing Capacity 244

10.3.3 Lasto-Plastic Analysis 245

10.3.4 Comparison of Different Criteria 246

10.4 Laxiwa Hydraulic Power Plant on the Yellow River 249

10.5 Plastic Analysis for Underground Excavation at Laxiwa Hydraulic Power Station 252

10.5.1 Strength of the Laxiwa Granite 252

10.5.2 Plastic Zones Around the Underground Excavation Using the Single-Shear and Twin-Shear Theories 254

10.5.3 Plastic Zones Around the Underground Excavation with Four Yield Cone Criteria 255

10.6 The Effect of Failure Criterion on the Plastic Zone of the Underground Excavation 256

10.7 Three Dimension Numerical Modeling of Underground Excavation for a Pumped-Storage Power Station 257

10.8 Dynamic Response and Blast-Resistance Analysis of a Tunnel Subjected to Blast Loading 262

10.9 Brief Summary 264

References 266

11 Implementation of the Unified Strength Theory into ABAQUS and its Application 269

11.1 Introduction 269

11.2 Basic Theory 270

11.2.1 Expression of the Unified Strength Theory 270

11.2.2 The General Expression of Elastic-Plastic Increment Theory 271

11.3 ABAQUS UMAT(User Material) 272

11.3.1 General Introduction of UMAT 272

11.3.2 Interface and Algorithm of UMAT 273

11.3.3 Elastic and Plastic State 273

11.3.4 Constitutive Relationship Integration(Stress Update Method) 275

11.3.5 Tangent Stiffness Method 277

11.3.6 Treatment of the Singular Points on the Yield Surface 277

11.4 Typical Numerical Example 277

11.4.1 Model Conditions 277

11.4.2 Comparison of 2D and 3D Solution from ABAQUS 278

11.4.3 Results from UMAT of the United Strength Theory 278

11.5 Engineering Applications 281

11.5.1 Project Background and Material Parameters 281

11.5.2 FEM Mesh and Boundary Condition 282

11.5.3 Results of Analysis 282

11.6 Conclusions 286

References 287

12 2D Simulation of Normal Penetration Using the Unified Strength Theory 289

12.1 Introduction 289

12.2 Penetration and Perforation 291

12.3 Constitutive Model of Concrete 293

12.4 Penetration and Perforation of Reinforced Concrete Slab 301

12.5 Perforation of Fibre Reinforced Concrete Slab 305

12.6 High Velocity Impact on Concrete Slabs Using UST and SPH Method 309

12.6.1 Material Model for the Concrete Slab 310

12.6.2 The Failure Surface 310

12.6.3 The Elastic Limit Surface 312

12.6.4 Strain Hardening 313

12.6.5 Residual Failure Surface 313

12.6.6 Damage Model 313

12.7 Numerical Example 314

12.8 Brief Summary 317

References 318

13 3D Simulation of Normal and Oblique Penetration and Perforation 321

13.1 Introduction 321

13.2 Simulation of Normal Impact Process 321

13.3 Simulation of Oblique Impact Process 325

13.4 Conclusions 330

References 331

14 Underground Mining 333

14.1 Introduction 333

14.2 Elastic-Brittle Damage Model Based on Twin-Shear Theory 336

14.2.1 Damage Model 336

14.2.2 Three-Dimensional Damage Model 336

14.3 Non-Equilibrium Iteration for Dynamic Evolution 338

14.4 Numerical Simulation of Caving Process Zone 340

14.4.1 Introduction to Block Cave Mining 340

14.4.2 Geometry and Undercut Scheme 340

14.4.3 Result of Numerical Simulation 341

14.5 Numerical Simulation for Crack Field Evolution in Long Wall Mining 344

14.5.1 Geometry and FEM Model 344

14.5.2 Evolution of Crack Field in the Roof 345

14.5.3 Results of Displacement and Stress 346

References 348

15 Reinforced Concrete Beam and Plate 349

15.1 Introduction 349

15.2 Elasto-Plastic Analysis for Reinforced Concrete Beams 350

15.2.1 Material Modelling 350

15.2.2 Material Modeling of Concrete 352

15.2.3 Reinforcing Steel 353

15.2.4 Structural Modeling 353

15.2.5 Simply Supported Beams 353

15.3 Punching Shear Failure Analysis of Flat Slabs by UST 355

15.3.1 Slab-Column Connections 355

15.3.2 Conclusions 356

15.4 Elasto-Plastic Analysis for an Ordinary RC Beam 357

15.5 Elasto-Plastic Analysis of an RC Deep Beam 359

15.6 Elasto-Plastic Analysis of an RC Box Sectional Beam 361

15.7 Summary 365

References 366

16 Stability Analysis of Underground Caverns Based on the Unified Strength Theory 369

16.1 Introduction 369

16.2 Huanren Pumped-Storage Powerhouse and Geology 370

16.2.1 The Powerhouse Region 370

16.2.2 In Situ Stress Measurement in Huanren Pumped Storage Powerhouse 371

16.3 Comparison of Failure Criteria for Geomaterials 371

16.4 Determination of Rock Mass Strength Parameters 373

16.5 Constitutive Formulation of Unified Strength Theory Used for Fast Lagrangian Analysis 374

16.6 Development of Unified Strength Theory Model in Flac-3D 379

16.7 Test of User-Defined Unified Strength Theory Constitutive Model in Flac-3D 379

16.8 Stability Analysis of Underground Powerhouse 382

16.8.1 Generation of Numerical Model and Selection of Parameters 382

16.8.2 Simulations for Different Excavation Schemes 383

16.9 Excavation and Support Modeling 390

16.10 Comparison of the Stabilities in these Models with Different b Values 393

16.11 Conclusions 397

References 398

17 Stability of Slope 399

17.1 Introduction 399

17.2 Effect of Yield Criterion on the Analysis of a Slope 402

17.3 Stability of Three Gorges High Slope 407

17.4 Stability of a Vertical Cut 410

17.5 Stability for a Slope of a Highway 411

References 415

18 Unified Strength Theory and FLAC 417

18.1 Introduction 417

18.2 Unified Strength Theory Constitutive Model 419

18.3 Governing Equation 420

18.3.1 Balance Equation 420

18.3.2 Explicit Numerical Procedure 422

18.3.3 Constitutive Equation 422

18.4 Unified Elasto-Plastic Constitutive Model 425

18.4.1 Unified Elasto-Plastic Constitutive Model 425

18.4.2 The Key to Implementation of the Constitutive Model 428

18.5 Calculation and Analysis 428

18.5.1 Slope Stability Analysis 428

18.5.1.1 Associated Flow Rule 429

18.5.1.2 Non-associated Flow Rule 431

18.5.2 Thick-Walled Cylinder under Internal Pressure 432

18.5.3 Bearing Capacity of Strip Footings 434

18.6 Three Dimensional Simulation of a Large Landslide 439

18.7 Conclusions 444

References 445

19 Mesomechanics and Multiscale Modelling for Yield Surface 447

19.1 Introduction 447

19.2 Interaction Yield Surface of Structures 450

19.3 Models in Mesomechanics and Macromechanics 451

19.3.1 RVE and HEM Model 451

19.3.2 Equivalent Inclusion Model 451

19.3.3 CSA and CCA Models 451

19.3.4 Gurson Homogenized Model 452

19.3.5 Periodic Distribution Model 452

19.3.6 PHA Model and 3-Fold Axissymmetrical Model 452

19.3.7 A Unit Cell of Masonry 452

19.3.8 Topological Disorder Models 452

19.3.9 Random Field Models of Heterogeneous Materials 453

19.4 Failure Surface for Cellular Materials under Multiaxial Loads and Damage Surfaces ofa Spheroidized Graphite Cast Iron 453

19.5 Mesomechanics Analysis of Composite Using UST 455

19.6 Multiscale Analysis of Yield Criterion of Metallic Glass Based on Atomistic Basis(Schuh and Lund,2003) 457

19.7 Multiscale Analysis of Yield Criterion of Molybdenum and Tungsten Based on Atomistic Basis (Groger et al,2008) 459

19.8 Phase Transformation Yield Criterion of Shape-Memory Alloys 459

19.9 Atomic-Scale Study of Yield Criterion in Nanocrystalline CU 461

19.10 A General Yield Criteria for Unit Cell in Multiscale Plasticity 463

19.11 Virtual Material Testing Based on Crystal Plasticity Finite Element Simulations 468

19.12 Meso-Mechanical Analysis of Failure Criterion for Concrete 469

19.13 Brief Summary 472

References 473

20 Miscellaneous Issues:Ancient Structures,Propellant of Solid Rocket,Parts of Rocket and Generator 481

20.1 Introduction 481

20.2 Stability of Ancient City Wall in Xi'an 484

20.3 Stability of the Foundation of Ancient Pagoda 487

20.3.1 Structure of Foundation of Ancient Pagoda 487

20.3.2 The Effect of Yield Criterion on Plastic Zone of Soil Foundation of Pagoda 489

20.4 Plastic Analysis of Thick-Walled Cylinder 492

20.5 Plastic Analysis of the Structural Part of a Rocket 494

20.6 Numerical Analysis of Rocket Motor Grain 496

20.7 3D Numerical Simulation for a Solid Rocket Motor 499

20.8 Structural Part of the Generator of Nuclear Power Station 503

20.9 The Effect of Yield Criterion on the Spread of the Shear Strain of Structure 504

20.10 About the Unified Strength Theory:Reviews and Comments 505

20.11 Signification and Determination of the UST Parameterb 510

20.11.1 Signification of the UST Parameter b 510

20.11.2 Determination of the UST Parameter b 512

20.12 BriefSummary 514

References 517

Index 521