Hwu Anisotropic Elastic Plates

1. Auflage 2010
ISBN: 978-1-4419-5915-7
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 673 Seiten

As structural elements, anisotropic elastic plates find wide applications in modern technology. The plates here are considered to be subjected to not only inplane load but also transverse load. In other words, both plane and plate bending problems as well as the stretching-bending coupling problems are all explained in this book. In addition to the introduction of the theory of anisotropic elasticity, several important subjects have are discussed in this book such as interfaces, cracks, holes, inclusions, contact problems, piezoelectric materials, thermoelastic problems and boundary element analysis.

Autoren/Hrsg.

Hwu, Chyanbin

Inhaltsverzeichnis

1;Preface;5
2;Contents;9
3;1 Linear Anisotropic Elastic Materials;17
3.1;1.1 Theory of Elasticity for Anisotropic Bodies;17
3.1.1;1.1.1 State of Stress;17
3.1.2;1.1.2 Deformation;19
3.1.3;1.1.3 Constitutive Laws;21
3.1.4;1.1.4 Boundary Conditions;21
3.2;1.2 Three-Dimensional Constitutive Relations;23
3.2.1;1.2.1 Generalized Hooke's Law;23
3.2.2;1.2.2 Material Symmetry;25
3.2.3;1.2.3 Engineering Constants;28
3.3;1.3 Two-Dimensional Constitutive Relations;30
3.3.1;1.3.1 Isotropic Materials;30
3.3.2;1.3.2 Anisotropic Materials;32
3.3.3;1.3.3 Monoclinic Materials;34
3.3.4;1.3.4 Orthotropic Materials;34
3.4;1.4 Laminate Constitutive Relations;35
3.4.1;1.4.1 Specially Orthotropic Lamina;36
3.4.2;1.4.2 Generally Orthotropic Lamina;37
3.4.3;1.4.3 Classical Lamination Theory;39
4;2 Lekhnitskii Formalism;44
4.1;2.1 Governing Differential Equations;44
4.2;2.2 General Solutions;49
4.3;2.3 Boundary Conditions;52
4.3.1;2.3.1 Lateral Surface Conditions;52
4.3.2;2.3.2 End Conditions;54
4.4;2.4 Special Cases;58
4.4.1;2.4.1 Generalized Plane Deformation;58
4.4.2;2.4.2 Plane Deformation;59
4.4.3;2.4.3 Generalized Plane Stress;62
4.4.4;2.4.4 Anisotropic Rod by Bending and Twisting;62
4.5;2.5 Anisotropic Cantilever Under Transverse Force;66
5;3 Stroh Formalism;68
5.1;3.1 General Solutions;68
5.2;3.2 Boundary Conditions;72
5.3;3.3 Material Eigenrelation;74
5.3.1;3.3.1 Sextic Eigenrelation;76
5.3.2;3.3.2 Generalized Sextic Eigenrelation;79
5.3.3;3.3.3 The Matrix Differential Equation;82
5.4;3.4 Some Identities;83
5.4.1;3.4.1 Explicit Expression of Fundamental Elasticity Matrix N;83
5.4.2;3.4.2 Explicit Expressions of Barnett--Lothe Tensors S, H, and L;84
5.4.3;3.4.3 Identities Relating N, N( ), S, H, L;86
5.4.4;3.4.4 Identities Converting Complex Form to Real Form;92
5.5;3.5 Degenerate Materials;99
6;4 Infinite Space, Half-Space, and Bimaterials;102
6.1;4.1 Infinite Space;102
6.1.1;4.1.1 Uniform Loading;102
6.1.2;4.1.2 Pure In-Plane Bending;106
6.1.3;4.1.3 Concentrated Forces;107
6.1.4;4.1.4 Couple Moments;110
6.1.5;4.1.5 Dislocations;114
6.2;4.2 Half-Space;115
6.2.1;4.2.1 Green's Function;115
6.2.2;4.2.2 Surface Green's Function;120
6.2.3;4.2.3 Distributed Load Along the Half-Space Surface;121
6.2.4;4.2.4 Couple Moments;122
6.2.5;4.2.5 Dislocations;122
6.3;4.3 Bimaterials;123
6.3.1;4.3.1 Green's Function;123
6.3.2;4.3.2 Interfacial Green's Function;126
7;5 Wedges and Interface Corners;129
7.1;5.1 Uniform Tractions on the Wedge Sides;129
7.1.1;5.1.1 Non-critical Wedge Angles;131
7.1.2;5.1.2 Critical Wedge Angles;136
7.1.3;5.1.3 Summary;137
7.2;5.2 Forces at the Wedge Apex;138
7.2.1;5.2.1 A Single Wedge -- Under a Concentrated Force;138
7.2.2;5.2.2 A Single Wedge -- Under a Concentrated Couple;139
7.2.3;5.2.3 Multi-material Wedge Spaces;142
7.2.4;5.2.4 Multi-material Wedges;144
7.3;5.3 Stress Singularities;145
7.3.1;5.3.1 Multi-material Wedge Spaces;148
7.3.2;5.3.2 Multi-material Wedges;149
7.3.3;5.3.3 Eigenfunctions;150
7.3.4;5.3.4 Special Cases;151
7.4;5.4 Stress Intensity Factors of Interface Corners;153
7.4.1;5.4.1 Near-Tip Field Solutions;155
7.4.2;5.4.2 A Unified Definition;158
7.4.3;5.4.3 H-Integral for Two-Dimensional Interface Corners;161
7.4.4;5.4.4 H-Integral for Three-Dimensional Interface Corners;166
7.4.5;5.4.5 Numerical Examples;169
8;6 Holes;173
8.1;6.1 Elliptical Holes;173
8.1.1;6.1.1 Uniform Loading at Infinity;175
8.1.2;6.1.2 In-Plane Bending at Infinity;179
8.1.3;6.1.3 Arbitrary Loading Along the Hole Boundary;182
8.1.4;6.1.4 Concentrated Force at Arbitrary Location;187
8.1.5;6.1.5 Dislocation at Arbitrary Location;190
8.2;6.2 Polygon-Like Holes;190
8.2.1;6.2.1 Transformation Function;192
8.2.2;6.2.2 Uniform Loading at Infinity;196
8.2.3;6.2.3 Pure In-Plane Bending at Infinity;198
8.2.4;6.2.4 Discussions;200
9;7 Cracks;201
9.1;7.1 Singular Characteristics of Cracks;201
9.1.1;7.1.1 Cracks in Homogeneous Materials;202
9.1.2;7.1.2 Interfacial Cracks;204
9.1.3;7.1.3 Cracks Terminating at the Interfaces;205
9.2;7.2 A Finite Straight Crack;205
9.2.1;7.2.1 Uniform Loading at Infinity;206
9.2.2;7.2.2 In-plane Bending at Infinity;207
9.2.3;7.2.3 Arbitrary Loading on the Crack Surfaces;208
9.2.4;7.2.4 Concentrated Force at Arbitrary Location;208
9.2.5;7.2.5 Dislocation at Arbitrary Location;209
9.3;7.3 Collinear Cracks;209
9.3.1;7.3.1 General Solutions;210
9.3.2;7.3.2 Two Collinear Cracks;212
9.3.3;7.3.3 Collinear Periodic Cracks;214
9.3.4;7.3.4 Fracture Parameters;215
9.3.4.1;7.3.4.1 Two Collinear Cracks;216
9.3.4.2;7.3.4.2 Collinear Periodic Cracks;218
9.3.4.3;7.3.4.3 Discussions;219
9.4;7.4 Collinear Interface Cracks;220
9.4.1;7.4.1 General Solutions;221
9.4.2;7.4.2 A Semi-infinite Interface Crack;224
9.4.3;7.4.3 A Finite Interface Crack;225
9.4.3.1;7.4.3.1 Point Load;225
9.4.3.2;7.4.3.2 Uniform Load;226
9.4.4;7.4.4 Two Collinear Interface Cracks;228
9.4.5;7.4.5 Fracture Parameters;230
9.4.5.1;7.4.5.1 Proper Definition for Bimaterial Stress Intensity Factors;230
9.4.5.2;7.4.5.2 Some Explicit Expressions;235
9.4.5.3;7.4.5.3 A Semi-infinite Interfacial Crack Subjected to Point Load;237
9.4.5.4;7.4.5.4 A Finite Interface Crack Subjected to Point Load;237
9.4.5.5;7.4.5.5 A Finite Interface Crack Subjected to Uniform Load;238
9.4.5.6;7.4.5.6 Two Collinear Interface Cracks Subjected to Uniform Load at Infinity;238
9.5;7.5 Delamination Fracture Criteria;239
9.5.1;7.5.1 Stress Intensity Factors and Energy Release Rates;240
9.5.2;7.5.2 Experimental Details;240
9.5.2.1;7.5.2.1 Materials and Specimen Fabrication;242
9.5.2.2;7.5.2.2 Testing Procedure;243
9.5.3;7.5.3 Delamination Fracture Toughness;243
9.5.4;7.5.4 Mixed-Mode Fracture Criteria;246
9.5.5;7.5.5 Prediction of Delamination Fracture;248
9.5.5.1;7.5.5.1 The Onset of Delamination in a Perfect Composite Laminate;249
9.5.5.2;7.5.5.2 The Onset of Delamination in a Delaminated Composite;251
10;8 Inclusions;252
10.1;8.1 Elliptical Elastic Inclusions;252
10.1.1;8.1.1 Uniform Loading at Infinity;258
10.1.2;8.1.2 Concentrated Forces at the Matrix;259
10.2;8.2 Rigid Inclusions;261
10.2.1;8.2.1 Elliptical Rigid Inclusions;263
10.2.2;8.2.2 Rigid Line Inclusions;267
10.2.3;8.2.3 Polygon-Like Rigid Inclusions;268
10.3;8.3 Interactions Between Inclusions and Dislocations;269
10.3.1;8.3.1 Dislocations Outside the Inclusions;270
10.3.2;8.3.2 Dislocations Inside the Inclusions;271
10.3.3;8.3.3 Dislocations on the Interfaces;276
10.3.4;8.3.4 Interaction Energy;279
10.4;8.4 Interactions Between Inclusions and Cracks;281
10.4.1;8.4.1 Cracks Outside the Inclusions;281
10.4.2;8.4.2 Cracks Inside the Inclusions;284
10.4.3;8.4.3 Cracks Penetrating the Inclusions;285
10.4.4;8.4.4 Curvilinear Cracks Lying Along the Interfaces;286
11;9 Contact Problems;289
11.1;9.1 Rigid Punches on a Half-Plane;290
11.1.1;9.1.1 General Solution;290
11.1.2;9.1.2 Indentation by a Flat-Ended Punch;295
11.1.3;9.1.3 A Flat-Ended Punch Tilted by a Couple;297
11.1.4;9.1.4 Indentation by a Parabolic Punch;299
11.1.5;9.1.5 Analogy with the Interface Crack Problems;300
11.2;9.2 Rigid Stamp Indentation on a Curvilinear Hole Boundary;302
11.2.1;9.2.1 General Solution;302
11.2.2;9.2.2 Elliptical Hole Boundaries;305
11.2.3;9.2.3 Polygonal Hole Boundaries;307
11.2.4;9.2.4 Numerical Calculation;309
11.3;9.3 Rigid Punches on a Perturbed Surface;310
11.3.1;9.3.1 Straight Boundary Perturbation;312
11.3.2;9.3.2 Elliptical Boundary Perturbation;314
11.3.3;9.3.3 Illustrative Examples;318
11.4;9.4 Sliding Punches With or Without Friction;320
11.4.1;9.4.1 General Solution;321
11.4.2;9.4.2 A Sliding Wedge-Shaped Punch;324
11.4.3;9.4.3 A Sliding Parabolic Punch;326
11.4.4;9.4.4 Two Sliding Flat-Ended Punches;330
11.5;9.5 Contact Between Two Elastic Bodies;333
11.5.1;9.5.1 Contact in the Presence of Friction;338
11.5.2;9.5.2 Contact in the Absence of Friction;340
11.5.3;9.5.3 Contact in Complete Adhesion;343
12;10 Thermoelastic Problems;345
12.1;10.1 Extended Stroh Formalism;345
12.2;10.2 Holes and Cracks;349
12.2.1;10.2.1 Elliptical Holes Under Uniform Heat Flow;350
12.2.2;10.2.2 Cracks Under Uniform Heat Flow;355
12.3;10.3 Rigid Inclusions;356
12.3.1;10.3.1 Elliptical Rigid Inclusions Under Uniform Heat Flow;357
12.3.2;10.3.2 Rigid Line Inclusions Under Uniform Heat Flow;359
12.4;10.4 Collinear Interface Cracks;360
12.4.1;10.4.1 General Solutions;361
12.4.2;10.4.2 Uniform Heat Flow;365
12.5;10.5 Multi-material Wedges;368
12.5.1;10.5.1 Stress and Heat Flux Singularity;368
12.5.2;10.5.2 Near-Tip Solutions;376
12.5.3;10.5.3 Special Cases;377
13;11 Piezoelectric Materials;380
13.1;11.1 Constitutive Laws;381
13.1.1;11.1.1 Three-Dimensional State;381
13.1.2;11.1.2 Two-Dimensional State;383
13.2;11.2 Expanded Stroh Formalism;385
13.2.1;11.2.1 General Solutions;385
13.2.2;11.2.2 Boundary Conditions;388
13.3;11.3 Explicit Expressions;389
13.3.1;11.3.1 Fundamental Matrix N;390
13.3.2;11.3.2 Material Eigenvector Matrices A and B;391
13.3.3;11.3.3 Barnett--Lothe Tensors S, H, and L;400
13.3.4;11.3.4 Bimaterial Matrices D and W;403
13.4;11.4 Multi-material Wedges;404
13.4.1;11.4.1 Orders of Stress/Electric Singularity;405
13.4.2;11.4.2 Near-Tip Solutions;405
13.4.3;11.4.3 Stress/Electric Intensity Factors;407
13.4.4;11.4.4 Corner Opening Displacement/Electric Potential;409
13.5;11.5 Singular Characteristics of Cracks;409
13.5.1;11.5.1 Cracks in Homogeneous Piezoelectric Materials;410
13.5.2;11.5.2 Interface Cracks Between Two Dissimilar Piezoelectric Materials;412
13.6;11.6 Some Crack Problems;415
13.6.1;11.6.1 Cracks;416
13.6.2;11.6.2 Interface Cracks;417
14;12 Plate Bending Analysis;422
14.1;12.1 Bending Theory of Anisotropic Plates;423
14.2;12.2 Lekhnitskii Bending Formalism;426
14.2.1;12.2.1 General Solutions;426
14.2.2;12.2.2 Boundary Conditions;427
14.2.3;12.2.3 Degenerate Materials;431
14.3;12.3 Stroh-Like Bending Formalism;431
14.3.1;12.3.1 General Solutions;432
14.3.2;12.3.2 Material Eigenrelation and Its Explicit Expressions;435
14.3.3;12.3.3 Explicit Expressions of S, H, and L;437
14.4;12.4 Holes/Inclusions/Cracks;439
14.4.1;12.4.1 Elliptical Holes;439
14.4.2;12.4.2 Elliptical Rigid Inclusions;442
14.4.3;12.4.3 Cracks;444
15;13 Coupled StretchingBending Analysis;446
15.1;13.1 Coupled StretchingBending Theory of Laminates;447
15.2;13.2 Complex Variable Formulation;450
15.2.1;13.2.1 Displacement Formalism;451
15.2.2;13.2.2 Mixed Formalism;457
15.2.3;13.2.3 Explicit Expressions of N, A, and B;462
15.2.4;13.2.4 Reduction to Symmetric Laminates;465
15.2.5;13.2.5 Comparison and Discussion;468
15.3;13.3 Stroh-Like Formalism;469
15.3.1;13.3.1 General Solutions;469
15.3.2;13.3.2 Material Eigenrelation;470
15.3.3;13.3.3 Stress Functions;472
15.3.4;13.3.4 Explicit Expressions of N;476
15.3.5;13.3.5 Explicit Expressions of A and B;478
15.3.6;13.3.6 Explicit Expressions of ;480
15.3.7;13.3.7 Explicit Expressions of S, H, and L;483
15.4;13.4 Hygrothermal Stresses;484
15.4.1;13.4.1 Basic Equations;485
15.4.2;13.4.2 Extended Stroh-Like Formalism;487
15.5;13.5 Electro-elastic Composite Laminates;492
15.5.1;13.5.1 Basic Equations;492
15.5.2;13.5.2 Expanded Stroh-Like Formalism;495
16;14 Holes/Cracks/Inclusions in Laminates;503
16.1;14.1 Holes in Laminates Under Uniform Stretching and Bending Moments;504
16.1.1;14.1.1 Field Solutions;506
16.1.2;14.1.2 Stress Resultants and Moments Along the Hole Boundary;507
16.2;14.2 Holes in Laminates Under Uniform Heat Flow and Moisture Transfer;508
16.2.1;14.2.1 Uniform Heat Flow and Moisture Transfer in x 1 x 2 -Plane;508
16.2.2;14.2.2 Uniform Heat Flow and Moisture Transfer in x 3 -Direction;511
16.3;14.3 Holes in Electro-Elastic Laminates;513
16.4;14.4 Greens Functions for Laminates;515
16.4.1;14.4.1 Concentrated In-Plane Forces and Out-of-Plane Moments ( );516
16.4.2;14.4.2 Concentrated Transverse Force ( );518
16.4.3;14.4.3 Concentrated In-Plane Torsion ( );520
16.4.4;14.4.4 Explicit Real-Form Solutions;522
16.5;14.5 Greens Functions for Laminates with Holes/Cracks;525
16.5.1;14.5.1 Field Solutions;526
16.5.2;14.5.2 Stress Resultants and Moments Along the Hole Boundary;532
16.5.3;14.5.3 Verification and Discussions;533
16.5.4;14.5.4 Cracks;534
16.6;14.6 Greens Functions for Laminates with Elastic Inclusions;536
16.6.1;14.6.1 Concentrated Forces/Moments Outside the Inclusions;539
16.6.2;14.6.2 Concentrated Forces/Moments Inside the Inclusions;544
16.6.3;14.6.3 Verification and Discussions;549
17;15 Boundary Element Analysis;554
17.1;15.1 Two-Dimensional Elastic Analysis;554
17.1.1;15.1.1 Boundary Integral Equations;554
17.1.2;15.1.2 Fundamental Solutions;555
17.1.3;15.1.3 Boundary Element Formulation;561
17.1.4;15.1.4 Stresses and Displacements at Internal Points;563
17.1.5;15.1.5 Stress Intensity Factors for Crack Problems;564
17.1.6;15.1.6 Subregion Technique;566
17.1.7;15.1.7 Numerical Implementation;568
17.2;15.2 Two-Dimensional Electro-Elastic Analysis;569
17.2.1;15.2.1 Boundary Element Formulation;569
17.2.2;15.2.2 Numerical Examples;570
17.3;15.3 Coupled StretchingBending Analysis;572
17.3.1;15.3.1 Boundary Integral Equations -- Internal Points;574
17.3.2;15.3.2 Fundamental Solutions;579
17.3.3;15.3.3 Boundary Integral Equations -- Boundary Points;584
17.3.4;15.3.4 Free-Term Coefficients;589
18;Appendix A Symbols, Sign Convention, and Units;598
18.1;A.1 Common Symbols;598
18.2;A.2 Extended Symbols;603
18.3;A.3 Sign Convention;608
18.4;A.4 Units;609
19;Appendix B Hilbert Problem;615
19.1;B.1 Solution to the Hilbert Problem in Scalar Form;615
19.2;B.2 Solution to the Hilbert Problem in Matrix Form;616
19.3;B.3 Evaluation of a Line Integral in Scalar Form;618
19.4;B.4 Evaluation of a Line Integral in Matrix Form;620
20;Appendix C Summary of Stroh Formalism;622
20.1;C.1 Two-Dimensional Problems:;622
20.2;C.2 Coupled Stretching–Bending ding Problems;625
20.3;C.3 Dimensions of Matrices Used in Stroh Formalism;631
21;Appendix D Collection of the Problem Solutions;632
22;References;661
23;Author Index;669
24;Subject Index;672