Unified Strength Theory and Its Applications (eBook)

Yu Mao-Hong is a Professor at Xi'an Jiao Tong University, China. He has been studying the strength theory of materials and structures for a long time since he graduated from Zhejiang University in 1955. He proposed the Twin-shear Yield Criterion for metals, the Twin-shear Strength Theory for geo-materials and the Unified Strength Theory for different kinds of materials for metals(1961)/for geo-materials(1985)/for different kinds of materials(1991). The twin shear unified strength theory has been written into more than 300 monographs and textbooks.Yu Mao-Hong is the winner of the National Natural Science Price of China and the winner of the Ho Leung Ho Lee Foundation Mechanics Prize. His research areas focus on Mechanics of Materials, Plasticity, Structural Mechanics, non-linear FEM, Soil Mechanics and Rock Mechanics.Yu Mao-Hong has already published the followed monographs with Springer:Unified Strength Theory and Its Applications (2004)Generalized Plasticity (2006)Structural Plasticity: Limit, Shakedown and Dynamic Plastic Analyses of Structures (2009)Computational Plasticity: With Emphasis on the Application of the Unified Strength Theory (2012)

Preface to the Second Edition 5
Preface to the First Edition 7
Review of the “Unified Strength Theory and Its Applications” Petre P. Teodorescu 11
Comment on the “Unified Strength Theory” Kolupaev V A and Altenbach H 12
Contents 13
Notations 21
1 Introduction 25
1.1 Strength of Materials and Structures 25
1.2 Strength of Materials Under Complex Stress State 26
1.3 Definition of Strength Theory 28
1.4 Significance and Development of Strength Theory 29
1.5 Shape of the Limit Surface of Strength Theory 31
1.6 Summary 32
1.7 Readings 34
2 Stress State and Elements 37
2.1 Elements 37
2.2 Stress at a Points: Stress Invariants 38
2.3 Deviatoric Stress Tensor, Deviatoric Tensor Invariants 40
2.4 Stresses on the Oblique Plane 41
2.4.1 Stresses on the Oblique Plane 41
2.4.2 Principal Shear Stresses 41
2.4.3 Octahedral Shear Stress 42
2.5 Hexahedron, Octahedron, Dodecahedron 43
2.5.1 Principal Stress Element (?1, ?2, ?3) 43
2.5.2 Isoclinal Octahedron Element (?8, ?8) 43
2.5.3 Single-Shear Element (?13??13, ?12??12 ?23??23)
2.5.4 Twin-Shear Element (?13??13, ?12??12) (?13??13, ?23??23)
2.5.5 Three-Shear Element (?13, ?12, ?23 ?13, ?12, ?23)
2.5.6 Twin-Shear Stress State and Twin-Shear Element 47
2.6 Stress Space 49
2.6.1 Relation Between ( /sigma_{{{/bf 1}}} ,/,/sigma_{{{/bf 2}}} ,/,/sigma_{{{/bf 3}}} ) and (X, Y, Z) 52
2.6.2 Relation ( /sigma_{{{/bf 1}}} ,/,/sigma_{{{/bf 2}}} ,/,/sigma_{{{/bf 3}}} ) and ( {{/varvec /upxi}}, {{/varvec r}},/,{{/varvec /uptheta}} ) or ( {{/varvec J}}_{{{/bf 2}}},/,{{/varvec /uptau}}_{{{/bf m}}} ,/,{{/varvec /theta}} ) 52
2.7 Stress State Parameters 54
2.8 Summary 56
2.9 Readings 57
3 Unified Yield Criterion 60
3.1 Introduction 60
3.2 General Behavior of the Yield Function 61
3.2.1 Hydrostatic Stress Independence 62
3.2.2 The Tensile Yield Stress Equals the Compressive Yield Stress 63
3.2.3 Symmetry of the Yield Function 63
3.3 Yield Surface 64
3.4 Mechanical Model of the Unified Yield Criterion 65
3.5 Unified Yield Criterion 66
3.6 Other Forms of the Unified Yield Criterion 68
3.7 Special Cases of the UYC (Unified Yield Criterion) 68
3.7.1 Single-Shear Yield Criterion (b = 0) 69
3.7.2 New Yield Criterion (b = 1/4) 70
3.7.3 New Yield Criterion (b = 1/2) 71
3.7.4 New Yield Criterion (b = 3/4) 75
3.7.5 Twin-Shear Yield Criterion (b = 1) 75
3.8 Determination of the UYC Parameter b 77
3.9 Unified Yield Criterion in the Plane Stress State 78
3.9.1 ?1 ? ?2  greaterthan  0, ?3 = 0 79
3.9.2 ?1 ? 0, ?2 = 0, ?3  lessthan  0 79
3.9.3 ?1 = 0, ?2 ? ?3  lessthan  0 79
3.10 Unified Yield Criterion in the ?–? Stress State 82
3.11 Examples 83
3.11.1 Example 3.1 84
3.11.2 Example 3.2 87
3.12 Summary 89
4 Verification of the Yield Criterion 93
4.1 Introduction 93
4.2 Comparison of the Unified Yield Criterion with the General Behavior of the Yield Criterion 93
4.2.1 Hydrostatic Stress Independence 94
4.2.2 The Tensile Yield Stress Equals the Compressive Yield Stress 94
4.2.3 Symmetry of the Yield Function 95
4.3 Comparison of the Unified Yield Criterion with Experimental Data 95
4.4 Comparison of the Yield Criteria with the Tests of Taylor and Quinney 98
4.5 Comparison of the Yield Criteria with the Tests of Ivey 99
4.6 Comparison of the Yield Criteria with the Tests of Winstone 100
4.7 Comparison of the Yield Criteria with the Experimental Results of Ellyin 102
4.8 Summary 105
5 Extended Unified Yield Criterion 109
5.1 Introduction 109
5.2 Extended Unified Yield Criterion 110
5.3 Special Cases of the Extended Unified Yield Criterion 111
5.3.1 Extended Single-Shear Yield Criterion (Extended Tresca Yield Criterion) 111
5.3.2 New Extended Yield Criterion (b = 1/4) 112
5.3.3 New Extended Yield Criterion (b = 1/2, Linear Drucker–Prager Criterion) 113
5.3.4 New Extended Yield Criterion (b = 3/4) 114
5.3.5 New Extended Yield Criterion (b = 1, Extended Twin-Shear Yield Criterion) 115
5.4 Yield Loci of the Extended Yield Criterion in the Meridian and Deviatoric Planes 116
5.5 Quadratic Extended Unified Yield Criterion 119
5.6 Summary 120
6 Basic Characteristics of Strength of Materials Under Complex Stress 123
6.1 Introduction 123
6.2 Strength Difference Effect in Tension and Compression (SD Effect) 124
6.3 Effect of Hydrostatic Stress 125
6.4 Effect of Normal Stress 131
6.5 Effect of Stress Angle 133
6.6 Research on the Effect of Intermediate Principal Stress 134
6.7 Effects of the Intermediate Principal Stress in Metals 135
6.8 Effects of the Intermediate Principal Stress in Rock 138
6.9 Characteristics of the Effect of Intermediate Principal Stress in Rock 146
6.10 Effects of the Intermediate Principal Stress in Concrete 147
6.11 Engineering Applications of the Effect of Intermediate Principal Stress in Concrete 151
6.12 Summary 154
6.13 Readings 154
7 Principles for Comment, Formulation and Choice of the Strength Theory Function 159
7.1 Introduction 159
7.2 Principle 1: A Strength Theory Function Must Contain All the Three Variables Both in Principal Stress Coordinate and Stress Invariant Coordinate 159
7.3 Principle 2: Three-fold Symmetry and Six-fold Symmetry 160
7.4 Principle 3: Drucker Postulate and Convexity of the Limit Surface 162
7.5 Principle 4: Two Boundaries of the Limit Surface 164
7.6 Principle 5: The Strength Theory Function Should Be Fitted to Test Results 166
7.7 Applications of the Principles 167
7.8 Summary 169
8 Unified Strength Theory (UST) 172
8.1 Introduction 172
8.2 Voigt-Timoshenko Conundrum 173
8.3 Mechanical Model of the Unified Strength Theory 174
8.4 Mathematical Modelling of the Unified Strength Theory 176
8.5 Experimental Determination of Material Parameters 176
8.6 Mathematical Expression of the Unified Strength Theory 177
8.7 Other Formulations of the Unified Strength Theory 177
8.7.1 In Terms of Stress Invariant {{/bi F(I}}_{{{/bf 1}}} {{,/,/bi J}}_{{{/bf 2}}} {{/bf ,/,}}{{/varvec /theta}}{{/bf ,/,}}{{/varvec /upsigma}}_{{{/bf t}}} {{/bf ,/,}}{{/varvec /upalpha}}{{/bf )}} 177
8.7.2 In Terms of Principal Stress and Cohesive Parameter {{/bi F(}}{{/varvec /upsigma}}_{{{/bf 1}}} {{/bf ,}}/,{{/varvec /upsigma}}_{{{/bf 2}}} {{/bf ,}}/,{{/varvec /upsigma}}_{{{/bf 3}}} {{/bi ,C}}_{{{/bf 0}}} {{/bf ,}}/,/varphi {{/bf )}} 178
8.7.3 In Terms of Stress Invariant and Cohesive Parameter {{/bi F(I}}_{{{/bf 1}}} {{/bi ,/,J}}_{{{/bf 2}}} {{/bf ,}}/,{{/varvec /theta}}{{/bi ,C}}_{{{/bf 0}}} {{/bf ,}}/,/varphi ) 178
8.7.4 In Terms of Principal Stresses and Compressive Strength Parameter {{/bi F(}}{{/varvec /upsigma}}_{{{/bf 1}}} {{/bf ,}}/,{{/varvec /upsigma}}_{{{/bf 2}}} {{/bf ,/,}}{{/varvec /upsigma}}_{{{/bf 3}}} {{/bf ,}}/,{{/varvec /upalpha}}{{/bf ,}}/,{{/varvec /upsigma}}_{{{/bf c}}} {{/bf )}} 179
8.7.5 In Terms of Stress Invariant and Compressive Strength Parameter {{/bi F(}}{{/bi I}}_{{{/bf 1}}} {{/bf ,}}/,{{/bi J}}_{{{/bf 2}}} {{/bf ,}}/,{{/varvec /theta}}{{/bf ,}}/,{{/varvec /upalpha}}{{/bf ,}}/,{{/varvec /upsigma}}_{{{/bf c}}} {{/bf )}} 179
8.8 Relation Among the Parameters of the UST 180
8.9 Special Cases of the UST for Different Parameter b 180
8.10 Special Cases of the UST by Varying Parameter ? 182
8.11 Limit Loci of the UST by Varying Parameter b in the ?-Plane 183
8.12 Variation of Limit Loci of the UST When ? =1/2 186
8.13 Limit Surfaces of the Unified Strength Theory in Principal Stress Space 189
8.14 Limit Loci of the Unified Strength Theory in the Plane Stress State 192
8.14.1 Variation of the Unified Strength Theory with b 193
8.14.2 Limit Locus of the Unified Strength Theory by Varying {{/varvec /upalpha}} 194
8.15 Limit Loci of the Unified Strength Theory Under the {{/varvec /upsigma}} - {{/varvec /uptau}} Combined Stress State 196
8.16 Unified Strength Theory in Meridian Plane 197
8.17 Generalizations of the UST 200
8.18 Effective Stress UST for Saturated and Unsaturated Soils 202
8.19 Significance of the UST 203
8.20 Summary 205
8.21 Readings 207
9 Experimental Verification of Strength Theory 212
9.1 Introduction 212
9.2 Equipment for Complex Stress State Experiments 212
9.2.1 Experimental Equipment for Tension (Compression)–Torsion Stress States 213
9.2.2 Biaxial Plane Experimental Equipment 213
9.2.3 Equipment for Axisymmetric Triaxial Experiments 214
9.2.4 Equipment for True Triaxial Experiments 215
9.3 Axial–Loading and Torsion Experiments 219
9.4 Experimental Verification of Strength Theory for Rock 220
9.5 Experiments on Rock Under True Triaxial Stress 223
9.5.1 Strength of Rock Under High Pressure 224
9.5.2 The Effect of Intermediate Principal Stress 224
9.5.3 The Effect of Stress Angle 225
9.5.4 Limit Meridian Loci 226
9.5.5 The Limit Loci on the ?–Plane 227
9.6 Experimental Verification of Strength Theory for Concrete 228
9.7 Experimental on Clay and Loess Under Complex Stress 232
9.8 Experiments on Sand Under Complex Stress 233
9.9 The Ultimate Dynamic Strength of Sand Under Complex Stress 235
9.10 Summary 237
10 Visualization of the Unified Strength Theory 240
10.1 Introduction 240
10.2 Visualization of the Unified Strength Theory 241
10.2.1 The Visualization of the Twin-Shear Strength Theory 241
10.2.2 Limit Surfaces of Unified Strength Theory 242
10.2.3 Limit Surfaces of Unified Yield Criterion 245
10.3 Other Forms of Graphic Expression of UST 248
10.3.1 Limit Loci of UST in the Plane Stress State 248
10.3.2 Limit Loci of UST in the {{/varvec /uppi}} -Plane 249
10.4 Kolupaev Figure 250
10.5 Summary 251
11 Equivalent Stress of the Unified Strength Theory and Comparisons with Other Theories 254
11.1 Introduction 254
11.2 Equivalent Stress 255
11.2.1 Equivalent Stresses for Non-SD Materials 255
11.2.2 Equivalent Stresses for SD Materials 255
11.2.2.1 Equivalent Stresses of the Unified Yield Criterion 256
11.2.2.2 Equivalent Stress of the Unified Strength Theory 256
11.3 A Comparison of Limit Loci in the Deviatoric Plane 257
11.3.1 A Comparison with Drucker–Prager Criterion 259
11.3.2 A Comparison with Matsuoka–Nakai Criterion 260
11.3.3 A Comparison with Gudehus–Argyris Criterion 261
11.3.4 A Comparison with Willam–Warnke Criterion 262
11.4 Summary 263
12 Economic Signification of the Unified Strength Theory 266
12.1 Introduction 266
12.2 A Trapezoidal Structure 267
12.3 A Spatial Axisymmetric Problem 269
12.4 Thin-Walled Pressure Vessel Design 270
12.5 Elastic Limit Pressure of Thick-Walled Cylinders 272
12.6 Summary 279
13 Rhombicuboctahedron Stress Strength Theory 282
13.1 Introduction 282
13.2 Rhombicuboctahedron Model 283
13.3 Rhombicuboctahedron Stress Strength Theory 286
13.4 Application of the Rhombicuboctahedron Stress Strength Theory 288
13.5 Summary 288
14 The Beauty of Strength Theories 291
14.1 Introduction 291
14.2 The Beauty of Science 291
14.3 Garden of Strength Theories 292
14.4 Beauty of the Huber-Mises Theory 293
14.5 Beauty of the Unified Strength Theory 293
14.5.1 Simplicity 296
14.5.2 Unification 296
14.5.3 Clarity and Extension 297
14.5.4 Symmetry 298
14.5.5 Analogy 298
14.5.6 Diversity and Innovation 298
14.6 Summary 299
15 Applications of the Unified Strength Theory 302
15.1 Introduction 302
15.2 Application of UST on the Shape and Size of the Crack Tip Plastic Zone 304
15.2.1 Mode I Crack in Plane Stress State 304
15.2.2 Mode I Crack in Plane Strain 306
15.2.3 Mode II Crack in Plane Stress 307
15.2.4 Mode II Crack in Plane Strain State 308
15.3 Application of UYC on FEM Analysis for Limit-Bearing Capacity of Plate 309
15.4 Application of UYC on FEM Analysis of Plastic Zones for Thick-Walled Cylinders 311
15.5 Application of UYC on FEM Analysis of Plastic Zone for a Strip with a Hole 313
15.6 Application of UST on FEM Analysis of Plastic Zone for Circular Cave 315
15.7 FEM Analysis of Composite Using UYC 317
15.8 Application of UST on FEM Analysis for Underground Caves 320
15.9 Summary 323
16 Historical Reviews 327
16.1 Introduction 327
16.2 Strength Theories Before the Twentieth Century 328
16.2.1 Early Work 328
16.2.2 Strength Theories Before the Twentieth Century 330
16.2.3 Strength Theories at the Beginning of the Twentieth Century 332
16.3 Three Series of Strength Theories 335
16.3.1 Single-Shear Strength Theory (SSS Theory) 335
16.3.1.1 Single-Shear Yield Criterion (Tresca 1864) 336
16.3.1.2 Single-Shear Strength Theory (Mohr–Coulomb 1900) 337
16.3.1.3 Multiparameter Solo-Shear Criterion 338
16.3.1.4 Application of the SSS Theory 339
16.3.2 Octahedral-Shear Strength Theory (OSS Theory) 340
16.3.2.1 Octahedral-Shear Stress Yield Criterion (Huber–Mises Yield Criterion) 340
16.3.2.2 Octahedral-Shear Failure Criterion (Drucker–Prager Criterion) 342
16.3.2.3 General Octahedral-Shear Failure Criterion 343
16.3.2.4 Application of the OSS Theory 346
16.3.3 Twin-Shear Strength Theory (TSS Theory) 346
16.3.3.1 Twin-Shear Yield Criterion (Yu 1961a, 1983) 347
16.3.3.2 Twin-Shear Strength Theory (Yu and Song 1983, Yu et al. 1985) 348
16.3.3.3 Twin-Shear Multiparameter Criterion (Yu and Liu 1988, 1990) 349
16.3.3.4 Applications of the TSS Theory 350
16.4 Establishment of the Unified Yield Criterion 351
16.4.1 Curved General Yield Criterion 351
16.4.1.1 Curved General Yield Criterion Between SSS and OSS Yield Criteria 351
16.4.1.2 Curved General Yield Criterion Between OSS and TSS Yield Criteria 351
16.4.1.3 Curved General Criterion Between SSS and TSS Yield Criteria 352
16.4.1.4 Drucker Criterion 352
16.4.1.5 Hosford Criterion 352
16.4.1.6 Simplification of Anisotropic Yield Criterion 352
16.4.2 Linear Unified Yield Criterion 353
16.4.2.1 Unified Yield Criterion (Yu and He 1991, 1992) 353
16.4.2.2 Concave Yield Criterion 354
16.4.2.3 Applications of the Unified Yield Criterion 355
16.5 Failure Criteria of Rock, Concrete, Soil, Iron, Polymer and Other Materials 355
16.5.1 Failure Criteria for Rock 356
16.5.2 Failure Criteria for Concrete 357
16.5.3 Failure Criteria for Soil 358
16.5.4 Failure Criteria of Iron 360
16.5.5 Failure Criteria for Ice 361
16.5.6 Failure Criteria of Wood 361
16.5.7 Failure Criteria of Polymers 362
16.5.8 Failure Criteria of Energetic Materials (TNT, RDX and Solid Rocket Propellant) 363
16.5.9 Failure Criteria of Ceramic and Glass 364
16.5.10 Failure Criteria of Other Materials 364
16.5.10.1 Cellular Material, Solid Foams 364
16.5.10.2 Brick Masonry 365
16.5.10.3 Smart Materials: Piezoelectric Solids, Shape Memory Alloys 366
16.5.10.4 Photoplastic Materials 366
16.5.10.5 Soft Rock and Coal 367
16.5.10.6 Powder 367
16.5.10.7 Coatings 367
16.5.10.8 Frozen Soil 368
16.5.10.9 Biomaterials 368
16.5.10.10 Other Materials 368
16.6 Unified Strength Theory 369
16.6.1 Octahedral-Shear Generalized Strength Theory 369
16.6.2 Unified Strength Theory (Yu and He 1991 Yu 1992, 1994)
16.6.3 Special Cases of the Unified Strength Theory 372
16.6.4 Comparison and Choice 375
16.6.5 Application of the Unified Strength Theory 375
16.7 Computational Implementation of the Strength Theory 377
16.8 Summary 383
17 References and Bibliography 389
17.1 Early Works (Before 1900) 389
17.2 Works from 1901 to 1950 390
17.3 Works from 1951 to 1960 395
17.4 Works from 1961 to 1970 400
17.5 Works from 1971 to 1980 409
17.6 Works from 1981 to 1990 418
17.7 Works from 1991 to 2000 429
17.8 Works from 2001 to 2010 446
17.9 Works from 2011 to 2017 465
Index 477