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1 Introduction1

1.1 Elasto-Plastic Finite Elements1

1.2 Bounds and Region of the Convex Yield Surface3

1.3 Unified Strength Theory and its Implementati on in Computer Codes4

1.4 The Effect of Yield Criteria on the Numerical Analysis Results7

1.5 Historical Review:With Emphasis on the Implementation and Application of Unified Strength Theory12

1.6 Brief Summary17

References19

2 Stress and Strain29

2.1 Introduction29

2.2 Stress at a Point,Stress Invariants29

2.3 Deviatoric Stress Tensor and its Invariants31

2.4 Stresses on the Oblique Plane33

2.4.1 Stresses on the Oblique Plane33

2.4.2 Principal Shear Stresses33

2.4.3 Octahedral Shear Stress35

2.5 From Single-Shear Element to Twin-Shear Element37

2.6 Stress Space38

2.7 Stress State Parameters42

2.8 Strain Components45

2.9 Equations of Equilibrium46

2.10 Generalized Hooke's Law46

2.11 Compatibility Equations48

2.12 Governing Equations for Plane Stress Problems49

2.13 Governing Equations in Polar Coordinates50

2.14 Brief Summary51

References52

3 Material Models in Computational Plasticity53

3.1 Introduction53

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

3.2.1 Hydrostatic Stress Independence55

3.2.2 The Tensile Yield Stress Equals the Compressive Yield Stress56

3.2.3 Sixfold Symmetry of the Yield Function56

3.2.4 Convexity of the Yield Function57

3.2.5 Bounds of the Yield Function for Non-SD Materials58

3.3 Material Models for SD Materials66

3.3.1 General Behavior of Yield Function for SD Materials66

3.3.1.1 Six Basic Experimental Points for SD Materials66

3.3.1.2 Threefold Symmetry of the Yield Function66

3.3.1.3 Convexity of the Yield Function67

3.3.2 Three Basic Models for SD Materials67

3.4 Multi-Parameter Criteria for Geomaterials70

3.4.1 Multi-Parameter Single-Shear Failure Criterion70

3.4.2 Multi-Parameter Three-Shear Failure Criterion71

3.4.3 Multi-Parameter Twin-Shear Failure Criterion74

3.5 Bounds and the Region of the Convex Yield Function75

3.6 Brief Summary77

References78

4 Unified Strength Theory and its Material Parameters81

4.1 Introduction81

4.2 Mechanical Model of Unified Strength Theory82

4.3 Mathematical Modelling and the Determination of the Material Parameters of the Unified Strength Theory85

4.4 Mathematical Expression of the Unified Strength Theory86

4.5 Special Cases of the Unified Strength Theory87

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 Parameters92

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 Theory95

4.7.1 Material Parameters β and C are Determined by Experimental Results of Uniaxial Tension Strength σt and Shear Strength τ096

4.7.2 Material Parameters β and C are Determined by Experimental Results of Uniaxial Compressive Strength σc and Shear Strength τ096

4.7.3 Material Parameters β and C are Determined by Experimental Results of Uniaxial Compressive Strength σc and Biaxial Compressive Strength σcc97

4.7.4 Material Parameters β and C are Determined by Experimental Results of Uniaxial Compressive Strength σc and Biaxial Compressive Strength σcc97

4.7.5 Material Parameters β and C are Determined by Experimental Results of Uniaxial Compressive Strength σc and Biaxial Compressive Strength σcc97

4.8 Three-Parameter Unified Strength Theory98

4.9 Stress Space and Yield Loci ofthe UST98

4.10 Yield Surfaces of the UST in Principal Stress Space102

4.11 Extend of UST from Convex to Non-Convex107

4.12 Yield Loci of the UST in Plane Stress State108

4.13 Unified Strength Theory in Meridian Plane112

4.14 Extend of UST from Linear to Non-Linear UST114

4.15 Equivalent Stress of the Unified Strength Theory116

4.15.1 Equivalent Stresses for Non-SD Materials117

4.15.2 Equivalent Stresses for SD Materials117

4.15.3 Equivalent Stresses of the Unified Yield Criterion117

4.15.4 Equivalent Stress of the Unified Strength Theory118

4.16 Examples119

4.17 Summary122

References125

5 Non-Smooth Multi-Surface Plasticity129

5.1 Introduction129

5.2 Plastic Deformation in Uniaxial Stress State130

5.3 Three-Dimensional Elastic Stress-Strain Relation132

5.4 Plastic Work Hardening and Strain Hardening133

5.5 Plastic Flow Rule136

5.6 Drucker's Postulate-Convexity of the Loading Surface137

5.7 Incremental Constitutive Equations in Matrix Formulation141

5.8 Determination of Flow Vector for Different Yield Functions144

5.9 Singularity of Piecewise-Linear Yield Functions146

5.10 Process of Singularity of the Plastic Flow Vector151

5.11 Suggested Methods153

5.12 Unified Process of the Corner Singularity156

5.12.1 Tresca Yield Criterion156

5.12.2 Mohr-Coulomb Yield Criterion157

5.12.3 Twin-Shear Yield Criterion157

5.12.4 Generalized Twin-Shear Yield Criterion157

5.13 BriefSummary159

References160

6 Implementation of the Unified Strength Theory into FEM Codes163

6.1 Introduction163

6.2 Bounds of the Single Criteria for Non-SD Materials165

6.3 Bounds of the Failure Criteria for SD Materials166

6.4 Unification of the Yield Criteria for Non-SD Materials and SD Materials168

6.5 Material Models170

6.6 Program Structure and its Subroutines Relating to the Unified Strength Theory:INVARY,YIELDY,FLOWVP172

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 Summary178

References178

7 Examples of the Application of Unified Elasto-Plastic Constitutive Relations183

7.1 Introduction183

7.2 Plane Stress Problems184

7.2.1 Elasto-Plastic Analysis of a Cantilever Beam184

7.2.2 Elasto-Plastic Analysis of a Trapezoid Structure under Uniform Load187

7.3 Plane Strain Problems188

7.4 Spatial Axisymmetric Problems190

7.4.1 Analysis of Plastic Zone for Thick-Walled Cylinder190

7.4.2 Analysis for Limit-Bearing Capacity of a Circular Plate193

7.4.3 Truncated Cone under the Uniform Load on the Top195

7.5 Brief Summary197

References198

8 Strip with a Circular Hole under Tension and Compression199

8.1 Introduction199

8.2 Plastic Analysis of a Strip with a Circular Hole for Non-SD Material200

8.3 Elasto-Plastic Analysis of a Strip with a Circular Hole for SD Material under Tension203

8.4 Plastic Zone of a Strip with a Circular Hole for SD Material under Compression204

8.5 Comparison of Numerical Analysis with Experiments205

8.6 Elasto-Plastic Analysis of a Strip with a Circular Hole for a Special SD Material:Concrete207

8.7 Brief Summary208

References211

9 Plastic Analysis of Footing Foundation Based on the Unified Strenghth Theory213

9.1 Introduction213

9.2 Effect of Yield Criterion on the Limit Analysis of Footing216

9.3 Elasto-Plastic Analysis of Foundation Using UST218

9.4 Plastic Analysis of Strip Foundation Using UST220

9.5 Plastic Analysis of Circular Foundation Using UST226

9.5.1 Unified Characteristics Line Field of Spatial Axisymmetric Problem226

9.5.2 Numerical Simulation of Spatial Axisymmetric Problem227

9.5.3 Effect of UST Parameter ？ on the Spread ofShear Strain230

9.6 Effect of UST Parameter b and ？ on the Spread ofShear Strain232

9.7 Brief Summary233

References234

10 Underground Caves,Tunnels and Excavation of Hydraulic Power Station239

10.1 Introduction239

10.2 Effect of Yield Criterion on the Plastic Zone for a Circular Cave241

10.3 Plastic Zone for Underground Circular Cave under Two Direction Compressions242

10.3.1 Material Model243

10.3.2 Elastic Bearing Capacity244

10.3.3 Lasto-Plastic Analysis245

10.3.4 Comparison of Different Criteria246

10.4 Laxiwa Hydraulic Power Plant on the Yellow River249

10.5 Plastic Analysis for Underground Excavation at Laxiwa Hydraulic Power Station252

10.5.1 Strength of the Laxiwa Granite252

10.5.2 Plastic Zones Around the Underground Excavation Using the Single-Shear and Twin-Shear Theories254

10.5.3 Plastic Zones Around the Underground Excavation with Four Yield Cone Criteria255

10.6 The Effect of Failure Criterion on the Plastic Zone of the Underground Excavation256

10.7 Three Dimension Numerical Modeling of Underground Excavation for a Pumped-Storage Power Station257

10.8 Dynamic Response and Blast-Resistance Analysis of a Tunnel Subjected to Blast Loading262

10.9 Brief Summary264

References266

11 Implementation of the Unified Strength Theory into ABAQUS and its Application269

11.1 Introduction269

11.2 Basic Theory270

11.2.1 Expression of the Unified Strength Theory270

11.2.2 The General Expression of Elastic-Plastic Increment Theory271

11.3 ABAQUS UMAT(User Material)272

11.3.1 General Introduction of UMAT272

11.3.2 Interface and Algorithm of UMAT273

11.3.3 Elastic and Plastic State273

11.3.4 Constitutive Relationship Integration(Stress Update Method)275

11.3.5 Tangent Stiffness Method277

11.3.6 Treatment of the Singular Points on the Yield Surface277

11.4 Typical Numerical Example277

11.4.1 Model Conditions277

11.4.2 Comparison of 2D and 3D Solution from ABAQUS278

11.4.3 Results from UMAT of the United Strength Theory278

11.5 Engineering Applications281

11.5.1 Project Background and Material Parameters281

11.5.2 FEM Mesh and Boundary Condition282

11.5.3 Results of Analysis282

11.6 Conclusions286

References287

12 2D Simulation of Normal Penetration Using the Unified Strength Theory289

12.1 Introduction289

12.2 Penetration and Perforation291

12.3 Constitutive Model of Concrete293

12.4 Penetration and Perforation of Reinforced Concrete Slab301

12.5 Perforation of Fibre Reinforced Concrete Slab305

12.6 High Velocity Impact on Concrete Slabs Using UST and SPH Method309

12.6.1 Material Model for the Concrete Slab310

12.6.2 The Failure Surface310

12.6.3 The Elastic Limit Surface312

12.6.4 Strain Hardening313

12.6.5 Residual Failure Surface313

12.6.6 Damage Model313

12.7 Numerical Example314

12.8 Brief Summary317

References318

13 3D Simulation of Normal and Oblique Penetration and Perforation321

13.1 Introduction321

13.2 Simulation of Normal Impact Process321

13.3 Simulation of Oblique Impact Process325

13.4 Conclusions330

References331

14 Underground Mining333

14.1 Introduction333

14.2 Elastic-Brittle Damage Model Based on Twin-Shear Theory336

14.2.1 Damage Model336

14.2.2 Three-Dimensional Damage Model336

14.3 Non-Equilibrium Iteration for Dynamic Evolution338

14.4 Numerical Simulation of Caving Process Zone340

14.4.1 Introduction to Block Cave Mining340

14.4.2 Geometry and Undercut Scheme340

14.4.3 Result of Numerical Simulation341

14.5 Numerical Simulation for Crack Field Evolution in Long Wall Mining344

14.5.1 Geometry and FEM Model344

14.5.2 Evolution of Crack Field in the Roof345

14.5.3 Results of Displacement and Stress346

References348

15 Reinforced Concrete Beam and Plate349

15.1 Introduction349

15.2 Elasto-Plastic Analysis for Reinforced Concrete Beams350

15.2.1 Material Modelling350

15.2.2 Material Modeling of Concrete352

15.2.3 Reinforcing Steel353

15.2.4 Structural Modeling353

15.2.5 Simply Supported Beams353

15.3 Punching Shear Failure Analysis of Flat Slabs by UST355

15.3.1 Slab-Column Connections355

15.3.2 Conclusions356

15.4 Elasto-Plastic Analysis for an Ordinary RC Beam357

15.5 Elasto-Plastic Analysis of an RC Deep Beam359

15.6 Elasto-Plastic Analysis of an RC Box Sectional Beam361

15.7 Summary365

References366

16 Stability Analysis of Underground Caverns Based on the Unified Strength Theory369

16.1 Introduction369

16.2 Huanren Pumped-Storage Powerhouse and Geology370

16.2.1 The Powerhouse Region370

16.2.2 In Situ Stress Measurement in Huanren Pumped Storage Powerhouse371

16.3 Comparison of Failure Criteria for Geomaterials371

16.4 Determination of Rock Mass Strength Parameters373

16.5 Constitutive Formulation of Unified Strength Theory Used for Fast Lagrangian Analysis374

16.6 Development of Unified Strength Theory Model in Flac-3D379

16.7 Test of User-Defined Unified Strength Theory Constitutive Model in Flac-3D379

16.8 Stability Analysis of Underground Powerhouse382

16.8.1 Generation of Numerical Model and Selection of Parameters382

16.8.2 Simulations for Different Excavation Schemes383

16.9 Excavation and Support Modeling390

16.10 Comparison of the Stabilities in these Models with Different b Values393

16.11 Conclusions397

References398

17 Stability of Slope399

17.1 Introduction399

17.2 Effect of Yield Criterion on the Analysis of a Slope402

17.3 Stability of Three Gorges High Slope407

17.4 Stability of a Vertical Cut410

17.5 Stability for a Slope of a Highway411

References415

18 Unified Strength Theory and FLAC417

18.1 Introduction417

18.2 Unified Strength Theory Constitutive Model419

18.3 Governing Equation420

18.3.1 Balance Equation420

18.3.2 Explicit Numerical Procedure422

18.3.3 Constitutive Equation422

18.4 Unified Elasto-Plastic Constitutive Model425

18.4.1 Unified Elasto-Plastic Constitutive Model425

18.4.2 The Key to Implementation of the Constitutive Model428

18.5 Calculation and Analysis428

18.5.1 Slope Stability Analysis428

18.5.1.1 Associated Flow Rule429

18.5.1.2 Non-associated Flow Rule431

18.5.2 Thick-Walled Cylinder under Internal Pressure432

18.5.3 Bearing Capacity of Strip Footings434

18.6 Three Dimensional Simulation of a Large Landslide439

18.7 Conclusions444

References445

19 Mesomechanics and Multiscale Modelling for Yield Surface447

19.1 Introduction447

19.2 Interaction Yield Surface of Structures450

19.3 Models in Mesomechanics and Macromechanics451

19.3.1 RVE and HEM Model451

19.3.2 Equivalent Inclusion Model451

19.3.3 CSA and CCA Models451

19.3.4 Gurson Homogenized Model452

19.3.5 Periodic Distribution Model452

19.3.6 PHA Model and 3-Fold Axissymmetrical Model452

19.3.7 A Unit Cell of Masonry452

19.3.8 Topological Disorder Models452

19.3.9 Random Field Models of Heterogeneous Materials453

19.4 Failure Surface for Cellular Materials under Multiaxial Loads and Damage Surfaces ofa Spheroidized Graphite Cast Iron453

19.5 Mesomechanics Analysis of Composite Using UST455

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 Alloys459

19.9 Atomic-Scale Study of Yield Criterion in Nanocrystalline CU461

19.10 A General Yield Criteria for Unit Cell in Multiscale Plasticity463

19.11 Virtual Material Testing Based on Crystal Plasticity Finite Element Simulations468

19.12 Meso-Mechanical Analysis of Failure Criterion for Concrete469

19.13 Brief Summary472

References473

20 Miscellaneous Issues:Ancient Structures,Propellant of Solid Rocket,Parts of Rocket and Generator481

20.1 Introduction481

20.2 Stability of Ancient City Wall in Xi'an484

20.3 Stability of the Foundation of Ancient Pagoda487

20.3.1 Structure of Foundation of Ancient Pagoda487

20.3.2 The Effect of Yield Criterion on Plastic Zone of Soil Foundation of Pagoda489

20.4 Plastic Analysis of Thick-Walled Cylinder492

20.5 Plastic Analysis of the Structural Part of a Rocket494

20.6 Numerical Analysis of Rocket Motor Grain496

20.7 3D Numerical Simulation for a Solid Rocket Motor499

20.8 Structural Part of the Generator of Nuclear Power Station503

20.9 The Effect of Yield Criterion on the Spread of the Shear Strain of Structure504

20.10 About the Unified Strength Theory:Reviews and Comments505

20.11 Signification and Determination of the UST Parameterb510

20.11.1 Signification of the UST Parameter b510

20.11.2 Determination of the UST Parameter b512

20.12 BriefSummary514

References517

Index521