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