Anisotropic Elastic Plates (eBook)

Preface 5
Contents 9
1 Linear Anisotropic Elastic Materials 17
1.1 Theory of Elasticity for Anisotropic Bodies 17
1.1.1 State of Stress 17
1.1.2 Deformation 19
1.1.3 Constitutive Laws 21
1.1.4 Boundary Conditions 21
1.2 Three-Dimensional Constitutive Relations 23
1.2.1 Generalized Hooke's Law 23
1.2.2 Material Symmetry 25
1.2.3 Engineering Constants 28
1.3 Two-Dimensional Constitutive Relations 30
1.3.1 Isotropic Materials 30
1.3.2 Anisotropic Materials 32
1.3.3 Monoclinic Materials 34
1.3.4 Orthotropic Materials 34
1.4 Laminate Constitutive Relations 35
1.4.1 Specially Orthotropic Lamina 36
1.4.2 Generally Orthotropic Lamina 37
1.4.3 Classical Lamination Theory 39
2 Lekhnitskii Formalism 44
2.1 Governing Differential Equations 44
2.2 General Solutions 49
2.3 Boundary Conditions 52
2.3.1 Lateral Surface Conditions 52
2.3.2 End Conditions 54
2.4 Special Cases 58
2.4.1 Generalized Plane Deformation 58
2.4.2 Plane Deformation 59
2.4.3 Generalized Plane Stress 62
2.4.4 Anisotropic Rod by Bending and Twisting 62
2.5 Anisotropic Cantilever Under Transverse Force 66
3 Stroh Formalism 68
3.1 General Solutions 68
3.2 Boundary Conditions 72
3.3 Material Eigenrelation 74
3.3.1 Sextic Eigenrelation 76
3.3.2 Generalized Sextic Eigenrelation 79
3.3.3 The Matrix Differential Equation 82
3.4 Some Identities 83
3.4.1 Explicit Expression of Fundamental Elasticity Matrix N 83
3.4.2 Explicit Expressions of Barnett--Lothe Tensors S, H, and L 84
3.4.3 Identities Relating N, N( ), S, H, L 86
3.4.4 Identities Converting Complex Form to Real Form 92
3.5 Degenerate Materials 99
4 Infinite Space, Half-Space, and Bimaterials 102
4.1 Infinite Space 102
4.1.1 Uniform Loading 102
4.1.2 Pure In-Plane Bending 106
4.1.3 Concentrated Forces 107
4.1.4 Couple Moments 110
4.1.5 Dislocations 114
4.2 Half-Space 115
4.2.1 Green's Function 115
4.2.2 Surface Green's Function 120
4.2.3 Distributed Load Along the Half-Space Surface 121
4.2.4 Couple Moments 122
4.2.5 Dislocations 122
4.3 Bimaterials 123
4.3.1 Green's Function 123
4.3.2 Interfacial Green's Function 126
5 Wedges and Interface Corners 129
5.1 Uniform Tractions on the Wedge Sides 129
5.1.1 Non-critical Wedge Angles 131
5.1.2 Critical Wedge Angles 136
5.1.3 Summary 137
5.2 Forces at the Wedge Apex 138
5.2.1 A Single Wedge -- Under a Concentrated Force 138
5.2.2 A Single Wedge -- Under a Concentrated Couple 139
5.2.3 Multi-material Wedge Spaces 142
5.2.4 Multi-material Wedges 144
5.3 Stress Singularities 145
5.3.1 Multi-material Wedge Spaces 148
5.3.2 Multi-material Wedges 149
5.3.3 Eigenfunctions 150
5.3.4 Special Cases 151
5.4 Stress Intensity Factors of Interface Corners 153
5.4.1 Near-Tip Field Solutions 155
5.4.2 A Unified Definition 158
5.4.3 H-Integral for Two-Dimensional Interface Corners 161
5.4.4 H-Integral for Three-Dimensional Interface Corners 166
5.4.5 Numerical Examples 169
6 Holes 173
6.1 Elliptical Holes 173
6.1.1 Uniform Loading at Infinity 175
6.1.2 In-Plane Bending at Infinity 179
6.1.3 Arbitrary Loading Along the Hole Boundary 182
6.1.4 Concentrated Force at Arbitrary Location 187
6.1.5 Dislocation at Arbitrary Location 190
6.2 Polygon-Like Holes 190
6.2.1 Transformation Function 192
6.2.2 Uniform Loading at Infinity 196
6.2.3 Pure In-Plane Bending at Infinity 198
6.2.4 Discussions 200
7 Cracks 201
7.1 Singular Characteristics of Cracks 201
7.1.1 Cracks in Homogeneous Materials 202
7.1.2 Interfacial Cracks 204
7.1.3 Cracks Terminating at the Interfaces 205
7.2 A Finite Straight Crack 205
7.2.1 Uniform Loading at Infinity 206
7.2.2 In-plane Bending at Infinity 207
7.2.3 Arbitrary Loading on the Crack Surfaces 208
7.2.4 Concentrated Force at Arbitrary Location 208
7.2.5 Dislocation at Arbitrary Location 209
7.3 Collinear Cracks 209
7.3.1 General Solutions 210
7.3.2 Two Collinear Cracks 212
7.3.3 Collinear Periodic Cracks 214
7.3.4 Fracture Parameters 215
7.3.4.1 Two Collinear Cracks 216
7.3.4.2 Collinear Periodic Cracks 218
7.3.4.3 Discussions 219
7.4 Collinear Interface Cracks 220
7.4.1 General Solutions 221
7.4.2 A Semi-infinite Interface Crack 224
7.4.3 A Finite Interface Crack 225
7.4.3.1 Point Load 225
7.4.3.2 Uniform Load 226
7.4.4 Two Collinear Interface Cracks 228
7.4.5 Fracture Parameters 230
7.4.5.1 Proper Definition for Bimaterial Stress Intensity Factors 230
7.4.5.2 Some Explicit Expressions 235
7.4.5.3 A Semi-infinite Interfacial Crack Subjected to Point Load 237
7.4.5.4 A Finite Interface Crack Subjected to Point Load 237
7.4.5.5 A Finite Interface Crack Subjected to Uniform Load 238
7.4.5.6 Two Collinear Interface Cracks Subjected to Uniform Load at Infinity 238
7.5 Delamination Fracture Criteria 239
7.5.1 Stress Intensity Factors and Energy Release Rates 240
7.5.2 Experimental Details 240
7.5.2.1 Materials and Specimen Fabrication 242
7.5.2.2 Testing Procedure 243
7.5.3 Delamination Fracture Toughness 243
7.5.4 Mixed-Mode Fracture Criteria 246
7.5.5 Prediction of Delamination Fracture 248
7.5.5.1 The Onset of Delamination in a Perfect Composite Laminate 249
7.5.5.2 The Onset of Delamination in a Delaminated Composite 251
8 Inclusions 252
8.1 Elliptical Elastic Inclusions 252
8.1.1 Uniform Loading at Infinity 258
8.1.2 Concentrated Forces at the Matrix 259
8.2 Rigid Inclusions 261
8.2.1 Elliptical Rigid Inclusions 263
8.2.2 Rigid Line Inclusions 267
8.2.3 Polygon-Like Rigid Inclusions 268
8.3 Interactions Between Inclusions and Dislocations 269
8.3.1 Dislocations Outside the Inclusions 270
8.3.2 Dislocations Inside the Inclusions 271
8.3.3 Dislocations on the Interfaces 276
8.3.4 Interaction Energy 279
8.4 Interactions Between Inclusions and Cracks 281
8.4.1 Cracks Outside the Inclusions 281
8.4.2 Cracks Inside the Inclusions 284
8.4.3 Cracks Penetrating the Inclusions 285
8.4.4 Curvilinear Cracks Lying Along the Interfaces 286
9 Contact Problems 289
9.1 Rigid Punches on a Half-Plane 290
9.1.1 General Solution 290
9.1.2 Indentation by a Flat-Ended Punch 295
9.1.3 A Flat-Ended Punch Tilted by a Couple 297
9.1.4 Indentation by a Parabolic Punch 299
9.1.5 Analogy with the Interface Crack Problems 300
9.2 Rigid Stamp Indentation on a Curvilinear Hole Boundary 302
9.2.1 General Solution 302
9.2.2 Elliptical Hole Boundaries 305
9.2.3 Polygonal Hole Boundaries 307
9.2.4 Numerical Calculation 309
9.3 Rigid Punches on a Perturbed Surface 310
9.3.1 Straight Boundary Perturbation 312
9.3.2 Elliptical Boundary Perturbation 314
9.3.3 Illustrative Examples 318
9.4 Sliding Punches With or Without Friction 320
9.4.1 General Solution 321
9.4.2 A Sliding Wedge-Shaped Punch 324
9.4.3 A Sliding Parabolic Punch 326
9.4.4 Two Sliding Flat-Ended Punches 330
9.5 Contact Between Two Elastic Bodies 333
9.5.1 Contact in the Presence of Friction 338
9.5.2 Contact in the Absence of Friction 340
9.5.3 Contact in Complete Adhesion 343
10 Thermoelastic Problems 345
10.1 Extended Stroh Formalism 345
10.2 Holes and Cracks 349
10.2.1 Elliptical Holes Under Uniform Heat Flow 350
10.2.2 Cracks Under Uniform Heat Flow 355
10.3 Rigid Inclusions 356
10.3.1 Elliptical Rigid Inclusions Under Uniform Heat Flow 357
10.3.2 Rigid Line Inclusions Under Uniform Heat Flow 359
10.4 Collinear Interface Cracks 360
10.4.1 General Solutions 361
10.4.2 Uniform Heat Flow 365
10.5 Multi-material Wedges 368
10.5.1 Stress and Heat Flux Singularity 368
10.5.2 Near-Tip Solutions 376
10.5.3 Special Cases 377
11 Piezoelectric Materials 380
11.1 Constitutive Laws 381
11.1.1 Three-Dimensional State 381
11.1.2 Two-Dimensional State 383
11.2 Expanded Stroh Formalism 385
11.2.1 General Solutions 385
11.2.2 Boundary Conditions 388
11.3 Explicit Expressions 389
11.3.1 Fundamental Matrix N 390
11.3.2 Material Eigenvector Matrices A and B 391
11.3.3 Barnett--Lothe Tensors S, H, and L 400
11.3.4 Bimaterial Matrices D and W 403
11.4 Multi-material Wedges 404
11.4.1 Orders of Stress/Electric Singularity 405
11.4.2 Near-Tip Solutions 405
11.4.3 Stress/Electric Intensity Factors 407
11.4.4 Corner Opening Displacement/Electric Potential 409
11.5 Singular Characteristics of Cracks 409
11.5.1 Cracks in Homogeneous Piezoelectric Materials 410
11.5.2 Interface Cracks Between Two Dissimilar Piezoelectric Materials 412
11.6 Some Crack Problems 415
11.6.1 Cracks 416
11.6.2 Interface Cracks 417
12 Plate Bending Analysis 422
12.1 Bending Theory of Anisotropic Plates 423
12.2 Lekhnitskii Bending Formalism 426
12.2.1 General Solutions 426
12.2.2 Boundary Conditions 427
12.2.3 Degenerate Materials 431
12.3 Stroh-Like Bending Formalism 431
12.3.1 General Solutions 432
12.3.2 Material Eigenrelation and Its Explicit Expressions 435
12.3.3 Explicit Expressions of S, H, and L 437
12.4 Holes/Inclusions/Cracks 439
12.4.1 Elliptical Holes 439
12.4.2 Elliptical Rigid Inclusions 442
12.4.3 Cracks 444
13 Coupled StretchingBending Analysis 446
13.1 Coupled StretchingBending Theory of Laminates 447
13.2 Complex Variable Formulation 450
13.2.1 Displacement Formalism 451
13.2.2 Mixed Formalism 457
13.2.3 Explicit Expressions of N, A, and B 462
13.2.4 Reduction to Symmetric Laminates 465
13.2.5 Comparison and Discussion 468
13.3 Stroh-Like Formalism 469
13.3.1 General Solutions 469
13.3.2 Material Eigenrelation 470
13.3.3 Stress Functions 472
13.3.4 Explicit Expressions of N 476
13.3.5 Explicit Expressions of A and B 478
13.3.6 Explicit Expressions of 480
13.3.7 Explicit Expressions of S, H, and L 483
13.4 Hygrothermal Stresses 484
13.4.1 Basic Equations 485
13.4.2 Extended Stroh-Like Formalism 487
13.5 Electro-elastic Composite Laminates 492
13.5.1 Basic Equations 492
13.5.2 Expanded Stroh-Like Formalism 495
14 Holes/Cracks/Inclusions in Laminates 503
14.1 Holes in Laminates Under Uniform Stretching and Bending Moments 504
14.1.1 Field Solutions 506
14.1.2 Stress Resultants and Moments Along the Hole Boundary 507
14.2 Holes in Laminates Under Uniform Heat Flow and Moisture Transfer 508
14.2.1 Uniform Heat Flow and Moisture Transfer in x 1 x 2 -Plane 508
14.2.2 Uniform Heat Flow and Moisture Transfer in x 3 -Direction 511
14.3 Holes in Electro-Elastic Laminates 513
14.4 Greens Functions for Laminates 515
14.4.1 Concentrated In-Plane Forces and Out-of-Plane Moments ( ) 516
14.4.2 Concentrated Transverse Force ( ) 518
14.4.3 Concentrated In-Plane Torsion ( ) 520
14.4.4 Explicit Real-Form Solutions 522
14.5 Greens Functions for Laminates with Holes/Cracks 525
14.5.1 Field Solutions 526
14.5.2 Stress Resultants and Moments Along the Hole Boundary 532
14.5.3 Verification and Discussions 533
14.5.4 Cracks 534
14.6 Greens Functions for Laminates with Elastic Inclusions 536
14.6.1 Concentrated Forces/Moments Outside the Inclusions 539
14.6.2 Concentrated Forces/Moments Inside the Inclusions 544
14.6.3 Verification and Discussions 549
15 Boundary Element Analysis 554
15.1 Two-Dimensional Elastic Analysis 554
15.1.1 Boundary Integral Equations 554
15.1.2 Fundamental Solutions 555
15.1.3 Boundary Element Formulation 561
15.1.4 Stresses and Displacements at Internal Points 563
15.1.5 Stress Intensity Factors for Crack Problems 564
15.1.6 Subregion Technique 566
15.1.7 Numerical Implementation 568
15.2 Two-Dimensional Electro-Elastic Analysis 569
15.2.1 Boundary Element Formulation 569
15.2.2 Numerical Examples 570
15.3 Coupled StretchingBending Analysis 572
15.3.1 Boundary Integral Equations -- Internal Points 574
15.3.2 Fundamental Solutions 579
15.3.3 Boundary Integral Equations -- Boundary Points 584
15.3.4 Free-Term Coefficients 589
Appendix A Symbols, Sign Convention, and Units 598
A.1 Common Symbols 598
A.2 Extended Symbols 603
A.3 Sign Convention 608
A.4 Units 609
Appendix B Hilbert Problem 615
B.1 Solution to the Hilbert Problem in Scalar Form 615
B.2 Solution to the Hilbert Problem in Matrix Form 616
B.3 Evaluation of a Line Integral in Scalar Form 618
B.4 Evaluation of a Line Integral in Matrix Form 620
Appendix C Summary of Stroh Formalism 622
C.1 Two-Dimensional Problems: 622
C.2 Coupled Stretching–Bending ding Problems 625
C.3 Dimensions of Matrices Used in Stroh Formalism 631
Appendix D Collection of the Problem Solutions 632
References 661
Author Index 669
Subject Index 672