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ОбложкаSlawinski M.A. Waves and rays in elastic continua. - 3rd ed. - Singapore: World scientific, 2015. - xxxi, 621 p.: ill., portr. - Bibliogr. in the notes and p.: 567-589. - Ind.: p.591-619. - ISBN 978-981-4641-75-3
Шифр: (И/Д21-S67) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
Dedication .................................................... vii
Acknowledgments ................................................ ix
Preface ........................................................ xi
Changes from First Edition ..................................... xv
Changes from Second Edition ................................... xix
List of Figures .............................................. xxxi

Part 1. Elastic continua ........................................ 1
Introduction to Part 1 .......................................... 3

Chapter 1. Deformations ......................................... 7
Preliminary Remarks ............................................. 7
1.1  Notion of Continuum ........................................ 8
1.2  Rudiments of Continuum Mechanics .......................... 10
     1.2.1  Axiomatic format ................................... 10
     1.2.2  Primitive concepts of continuum mechanics .......... 11
1.3  Material and Spatial Descriptions ......................... 14
     1.3.1  Fundamental concepts ............................... 14
     1.3.2  Material time derivative ........................... 16
     1.3.3  Conditions of linearized theory .................... 18
1.4  Strain .................................................... 23
     1.4.1  Introductory comments .............................. 23
     1.4.2  Derivation of strain tensor ........................ 23
     1.4.3  Physical meaning of strain tensor .................. 27
1.5  Rotation Tensor and Rotation Vector ....................... 33
     Closing Remarks ........................................... 34
1.6  Exercises ................................................. 35

Chapter 2. Forces and Balance Principles ....................... 43
     Preliminary Remarks ....................................... 43
2.1  Conservation of Mass ...................................... 44
     2.1.1  Introductory comments .............................. 44
     2.1.2  Integral equation .................................. 44
     2.1.3  Equation of continuity ............................. 46
2.2  Time Derivative of Volume Integral ........................ 47
2.3  Stress .................................................... 49
     2.3.1  Stress as description of surface forces ............ 49
     2.3.2  Traction ........................................... 50
2.4  Balance of Linear Momentum ................................ 51
2.5  Stress Tensor ............................................. 54
     2.5.1  Traction on coordinate planes ...................... 54
     2.5.2  Traction on arbitrary planes ....................... 56
2.6  Cauchy's Equations of Motion .............................. 61
     2.6.1  General formulation ................................ 61
     2.6.2  Surface-forces formulation ......................... 64
2.7  Balance of Angular Momentum ............................... 67
     2.7.1  Introductory comments .............................. 67
     2.7.2  Integral equation .................................. 69
     2.7.3  Symmetry of stress tensor .......................... 70
2.8  Fundamental Equations ..................................... 74
     Closing Remarks ........................................... 75
2.9  Exercises ................................................. 76

Chapter 3. Stress-Strain Equations ............................. 87
     Preliminary Remarks ....................................... 87
3.1  Rudiments of Constitutive Equations ....................... 88
3.2  Formulation of Stress-Strain Equations: Hookean Solid ..... 90
     3.2.1  Introductory comments .............................. 90
     3.2.2  Tensor form ........................................ 91
     3.2.3  Matrix form ........................................ 95
3.3  Determined System ......................................... 97
3.4  Anelasticity .............................................. 98
     3.4.1  Introductory comments .............................. 98
     3.4.2  Viscosity: Stokesian fluid ......................... 98
     3.4.3  Viscoelasticity: Kelvin-Voigt model ................ 99
     3.4.4  Viscoelasticity: Maxwell model .................... 104
     Closing Remarks .......................................... 105
3.5  Exercises ................................................ 106

Chapter 4. Strain Energy ...................................... 113
     Preliminary remarks ...................................... 113
4.1  Strain-energy Function ................................... 114
4.2  Strain-energy Function and Elasticity-tensor Symmetry .... 116
     4.2.1  Fundamental considerations ........................ 116
     4.2.2  Elasticity parameters ............................. 118
     4.2.3  Matrix form of stress-strain equations ............ 118
     4.2.4  Coordinate transformations ........................ 119
4.3  Stability Conditions ..................................... 120
     4.3.1  Physical justification ............................ 120
     4.3.2  Mathematical formulation .......................... 120
     4.3.3  Constraints on elasticity parameters .............. 121
4.4  System of Equations for Elastic Continua ................. 122
     4.4.1  Elastic continua .................................. 122
     4.4.2  Governing equations ............................... 123
     Closing Remarks .......................................... 125
4.5  Exercises ................................................ 126

Chapter 5.   Material Symmetry ................................ 133
     Preliminary remarks ...................................... 133
5.1  Orthogonal Transformations ............................... 134
     5.1.1  Transformation matrix ............................. 134
     5.1.2  Symmetry group .................................... 134
5.2  Transformation of Coordinates ............................ 135
     5.2.1  Introductory comments ............................. 135
     5.2.2  Transformation of stress-tensor components ........ 135
     5.2.3  Transformation of strain-tensor components ........ 140
     5.2.4  Stress-strain equations in transformed
            coordinates ....................................... 141
     5.2.5  On matrix forms ................................... 142
5.3  Condition for Material Symmetry .......................... 144
5.4  Point Symmetry ........................................... 147
5.5  Generally Anisotropic Continuum .......................... 148
5.6  Monoclinic Continuum ..................................... 149
     5.6.1  Elasticity matrix ................................. 149
     5.6.2  Vanishing of tensor components .................... 151
     5.6.3  Natural coordinate system ......................... 152
5.7  Orthotropic Continuum .................................... 154
5.8  Trigonal Continuum ....................................... 157
     5.8.1  Elasticity matrix ................................. 157
     5.8.2  Natural coordinate system ......................... 158
5.9  Tetragonal Continuum ..................................... 159
     5.9.1  Elasticity matrix ................................. 159
     5.9.2  Natural coordinate system ......................... 161
5.10 Transversely Isotropic Continuum ......................... 162
     5.10.1 Elasticity matrix ................................. 162
     5.10.2 Rotation invariance ............................... 163
5.11 Cubic Continuum .......................................... 168
5.12 Isotropic Continuum ...................................... 170
     5.12.1 Elasticity matrix ................................. 170
     5.12.2 Lame's parameters ................................. 171
     5.12.3 Tensor formulation ................................ 172
     5.12.4 Physical meaning of Lame's parameters ............. 174
5.13 Relations Among Symmetry Classes ......................... 175
     Closing Remarks .......................................... 177
5.14 Exercises ................................................ 178

Part 2. Waves and rays ........................................ 203
     Introduction to Part 2 ................................... 205

Chapter 6. Equations of Motion: Isotropic Homogeneous
Continua ...................................................... 209
     Preliminary Remarks ...................................... 209
6.1  Wave Equations ........................................... 210
     6.1.1  Equation of motion ................................ 210
     6.1.2  Wave equation for P waves ......................... 213
     6.1.3  Wave equation for S waves ......................... 214
     6.1.4  Physical interpretation ........................... 216
6.2  Plane Waves .............................................. 217
6.3  Displacement Potentials .................................. 220
     6.3.1  Helmholtz's decomposition ......................... 220
     6.3.2  Gauge transformation .............................. 221
     6.3.3  Equation of motion ................................ 222
     6.3.4  P and S waves ..................................... 223
6.4  P and S Waves in Terms of Displacements .................. 226
6.5  Solutions of Wave Equation for Single Spatial Dimension .. 228
     6.5.1  d'Alembert's approach ............................. 228
     6.5.2  Directional derivative ............................ 234
     6.5.3  Well-posed problem ................................ 235
     6.5.4  Causality, finite propagation speed and
            sharpness of signals .............................. 240
6.6  Solution of Wave Equation for Two and Three Spatial
     Dimensions ............................................... 242
     6.6.1  Introductory comments ............................. 242
     6.6.2  Three spatial dimensions .......................... 243
     6.6.3  Two spatial dimensions ............................ 245
6.7  On Evolution Equation .................................... 247
6.8  Solutions of Wave Equation for One-Dimensional
     Scattering ............................................... 250
6.9  On Weak Solutions of Wave Equation ....................... 258
     6.9.1  Introductory comments ............................. 258
     6.9.2  Weak derivatives .................................. 260
     6.9.3  Weak solution of wave equation .................... 260
6.10 Reduced Wave Equation .................................... 262
     6.10.1 Harmonic-wave trial solution ...................... 262
     6.10.2 Fourier's transform of wave equation .............. 264
6.11 Extensions of Wave Equation .............................. 266
     6.11.1 Introductory comments ............................. 266
     6.11.2 Standard wave equation ............................ 266
     6.11.3 Wave equation and elliptical velocity dependence .. 267
     6.11.4 Wave equation and weak inhomogeneity .............. 271
     Closing Remarks .......................................... 278
6.12 Exercises ................................................ 279

Chapter 7. Equations of Motion: Anisotropic Inhomogeneous
Continua ...................................................... 307
     Preliminary Remarks ...................................... 307
7.1  Formulation of Equations ................................. 308
7.2  Formulation of Solutions ................................. 309
     7.2.1  Introductory comments ............................. 309
     7.2.2  Trial-solution formulation: General wave .......... 309
     7.2.3  Trial-solution formulation: Harmonic wave ......... 312
     7.2.4  Asymptotic-series formulation ..................... 315
7.3  Eikonal Equation ......................................... 321
     Closing Remarks .......................................... 324
7.4  Exercises ................................................ 325

Chapter 8. Hamilton's Ray Equations ........................... 337
     Preliminary Remarks ...................................... 337
8.1  Method of Characteristics ................................ 338
     8.1.1  Level-set functions ............................... 338
     8.1.2  Characteristic equations .......................... 339
     8.1.3  Consistency of formulation ........................ 343
8.2  Time Parametrization of Char ............................. 344
     8.2.1  General formulation ............................... 344
     8.2.2  Equations with variable scaling factor ............ 345
     8.2.3  Equations with constant scaling factor ............ 346
     8.2.4  Formulation of Hamilton's ray equations ........... 347
8.3  Physical Interpretation of Hamilton's Ray Equations
     and Solutions ............................................ 348
     8.3.1  Equations ......................................... 348
     8.3.2  Solutions ......................................... 349
8.4  Relation between p and ẋ ................................. 350
     8.4.1  General formulation ............................... 350
     8.4.2  Phase and ray velocities .......................... 350
     8.4.3  Phase and ray angles .............................. 353
     8.4.4  Geometrical illustration .......................... 355
8.5  Example: Elliptical Anisotropy and Linear Inhomogeneity .. 356
     8.5.1  Introductory comments ............................. 356
     8.5.2  Eikonal equation .................................. 357
     8.5.3  Hamilton's ray equations .......................... 359
     8.5.4  Initial conditions ................................ 360
     8.5.5  Physical interpretation of equations and
            conditions ........................................ 360
     8.5.6  Solution of Hamilton's ray equations .............. 362
     8.5.7  Solution of eikonal equation ...................... 367
     8.5.8  Physical interpretation of solutions .............. 368
8.6  Example: Isotropy and Inhomogeneity ...................... 368
     8.6.1  Parametric form ................................... 368
     8.6.2  Explicit form ..................................... 370
     Closing Remarks .......................................... 371
8.7  Exercises ................................................ 372

Chapter 9. Christoffel's Equations ............................ 387
     Preliminary Remarks ...................................... 387
9.1  Explicit form of Christoffel's Equations ................. 388
9.2  Christoffel's Equations and Anisotropic Continua ......... 393
     9.2.1  Introductory comments ............................. 393
     9.2.2  Monoclinic continua ............................... 394
     9.2.3  Transversely isotropic continua ................... 399
9.3  Phase-slowness Surfaces .................................. 407
     9.3.1  Introductory comments ............................. 407
     9.3.2  Convexity of innermost sheet ...................... 407
     9.3.3  Intersection points ............................... 408
     Closing Remarks .......................................... 411
9.4  Exercises ................................................ 411

Chapter 10. Reflection and Transmission ....................... 421
     Preliminary Remarks ...................................... 421
10.1 Angles at Interface ...................................... 422
     10.1.1 Phase angles ...................................... 422
     10.1.2 Ray angles ........................................ 424
     10.1.3 Example: Elliptical velocity dependence ........... 425
10.2 Amplitudes at Interface .................................. 428
     10.2.1 Kinematic and dynamic boundary conditions ......... 428
     10.2.2 Reflection and transmission amplitudes ............ 434
     Closing Remarks .......................................... 440
10.3 Exercises ................................................ 442

Chapter 11.  Lagrange's Ray Equations ......................... 449
     Preliminary Remarks ...................................... 449
11.1 Legendre's Transformation of Hamiltonian ................. 450
11.2 Formulation of Lagrange's Ray Equations .................. 450
11.3 Beltrami's Identity ...................................... 453
     Closing Remarks .......................................... 453
11.4 Exercises ................................................ 454

Part 3. Variational formulation of rays ....................... 459
     Introduction to Part 3 ................................... 461

Chapter 12.  Euler's Equations ................................ 465
     Preliminary Remarks ...................................... 465
12.1 Mathematical Background .................................. 466
12.2 Formulation of Euler's Equation .......................... 467
12.3 Beltrami's Identity ...................................... 470
12.4 Generalizations of Euler's Equation ...................... 471
     12.4.1 Introductory comments ............................. 471
     12.4.2 Case of several variables ......................... 471
     12.4.3 Case of several functions ......................... 472
     12.4.4 Higher-order derivatives .......................... 472
12.5 Special Cases of Euler's Equation ........................ 473
     12.5.1 Introductory comments ............................. 473
     12.5.2 Independence of z ................................. 474
     12.5.3 Independence of x and z ........................... 474
     12.5.4 Independence of x ................................. 475
     12.5.5 Total derivative .................................. 475
     12.5.6 Function of x and z ............................... 476
12.6 First Integrals .......................................... 479
12.7 Lagrange's Ray Equations as Euler's Equations ............ 480
     Closing Remarks .......................................... 481
12.8 Exercises ................................................ 482

Chapter 13.  Variational Principles ........................... 491
     Preliminary Remarks ...................................... 491
13.1 Fermat's Principle ....................................... 492
     13.1.1 Statement of Fermat's principle ................... 492
     13.1.2 Properties of Hamiltonian fig.4 ...................... 493
     13.1.3 Variational equivalent of Hamilton's ray
            equations ......................................... 494
     13.1.4 Properties of Lagrangian fig.3 ....................... 494
     13.1.5 Parameter-independent Lagrange's ray equations .... 497
     13.1.6 Ray velocity ...................................... 498
     13.1.7 Proof of Fermat's principle ....................... 498
13.2 Hamilton's Principle: Example ............................ 500
     13.2.1 Introductory comments ............................. 500
     13.2.2 Action ............................................ 500
     13.2.3 Lagrange's equations of motion .................... 503
     13.2.4 Wave equation ..................................... 505
     Closing Remarks .......................................... 509
13.3 Exercises ................................................ 509

Chapter 14.  Ray Parameters ................................... 519
     Preliminary Remarks ...................................... 519
14.1 Traveltime Integrals ..................................... 520
14.2 Ray Parameters as First Integrals ........................ 521
14.3 Example: Elliptical Anisotropy and Linear Inhomogeneity .. 522
     14.3.1 Introductory comments ............................. 522
     14.3.2 Rays .............................................. 523
     14.3.3 Traveltimes ....................................... 526
14.4 Rays in Isotropic Continua ............................... 529
14.5 Lagrange's Ray Equations in xz-Plane ..................... 530
14.6 Conserved Quantities and Hamilton's Ray Equations ........ 531
     Closing Remarks .......................................... 533
14.7 Exercises ................................................ 534

Part 4. Appendices ............................................ 545
     Introduction to Part 4 ................................... 547

Appendix A.  Euler's Homogeneous-Function Theorem ............. 549
     Preliminary Remarks ...................................... 549
A.l  Homogeneous Functions .................................... 550
A.2  Homogeneous-Function Theorem ............................. 551
     Closing Remarks .......................................... 553
Appendix В.  Legendre's Transformation ........................ 555
     Preliminary Remarks ...................................... 555
B.l Geometrical Context ....................................... 556
     В.1.1  Surface and its tangent planes .................... 556
     B l.2  Single-variable case .............................. 556
B.2  Duality of Transformation ................................ 558
B.3  Transformation between Lagrangian fig.3 and Hamiltonian 
     fig.4 ....................................................... 559
B 4  Transformation and Ray Equations ......................... 560
     Closing Remarks .......................................... 562
Appendix C. List of Symbols ................................... 563
C 1  Mathematical Relations and Operations .................... 563
C.2  Physical Quantities ...................................... 565
     C 2.1  Greek letters ..................................... 565
     C.2.2  Roman letters ..................................... 566

Bibliography .................................................. 567
Index ......................................................... 591
About the Author .............................................. 621

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