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Wave propagation in elastic solids
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CONTENTS Preface vii Introduction 1 The propagation of mechanical disturbances 1 Continuum mechanics 3 Outline of contents 4 Historical sketch 7 Bibliography 8 1. One-dimensional motion of an elastic continuum 10 1.1. Introduction 10 1.2. Nonlinear continuum mechanics in one dimension 11 1.2.1. Motion 11 1.2.2. Deformation 11 1.2.3. Time-rates of change 12 1.2.4. Conservation of mass 14 1.2.5. Balance of momentum 15 1.2.6. Balance of energy 16 1.2.7. Linearized theory 17 1.2.8. Notation for the linearized theory 20 1.3. Half-space subjected to uniform surface tractions 21 1.4. Reflection and transmission 26 1.5. Waves in one-dimensional longitudinal stress 29 1.6. Harmonic waves 30 1.6.1. Traveling waves 30 1.6.2. Complex notation 32 1.6.3. Standing waves 32 1.6.4. Modes of free vibration 32 1.7. Flux of energy in time-harmonic waves 33 1.7.1. Time-average power per unit area 34 1.7.2. Velocity of energy flux 35 1.7.3. Energy transmission for standing waves 36 1.8. Fourier series and Fourier integrals 37 1.8.1. Fourier series 37 1.8.2. Fourier integrals 39 1.9. The use of Fourier integrals 41 1.10. Problems 42 2. The linearized theory of elasticity 46 2.1. Introduction 46 2.2. Notation and mathematical preliminaries 47 2.2.1. Indicial notation 47 2.2.2. Vector operators 48 2.2.3. Gauss' theorem 49 2.2.4. Notation 50 2.3. Kinematics and dynamics 50 2.3.1. Deformation 50 2.3.2. Linear momentum and the stress tensor 51 2.3.3. Balance of moment of momentum 52 2.4. The homogeneous, isotropic, linearly elastic solid 52 2.4.1. Stress-strain relations 52 2.4.2. Stress and strain deviators 54 2.4.3. Strain energy 55 2.5. Problem statement in dynamic elasticity 55 2.6. One-dimensional problems 57 2.7. Two-dimensional problems 58 2.7.1. Antiplane shear 58 2.7.2. In-plane motions 58 2.8. The energy identity 59 2.9. Hamilton's principle 61 2.9.1. Statement of the principle 61 2.9.2. Variational equation of motion 63 2.9.3. Derivation of Hamilton's principle 64 2.10. Displacement potentials 65 2.11. Summary of equations in rectangular coordinates 66 2.12. Orthogonal curvilinear coordinates 68 2.13. Summary of equations in cylindrical coordinates 73 2.14. Summary of equations in spherical coordinates 75 2.15. The ideal fluid 78 3. Elastodynamic theory 79 3.1. Introduction 79 3.2. Uniqueness of solution 80 3.3. The dynamic reciprocal identity 82 3.4. Scalar and vector potentials for the displacement field 85 3.4.1. Displacement representation 85 3.4.2. Completeness theorem 85 3.5. The Helmholtz decomposition of a vector 88 3.6. Wave motion generated by body forces 89 3.6.1. Radiation 89 3.6.2. Elastodynamic solution 93 3.7. Radiation in two dimensions 93 3.8. The basic singular solution of elastodynamics 96 3.8.1. Point load 96 3.8.2. Center of compression 101 3.9. Three-dimensional integral representation 102 3.9.1. Kirchhoff's formula 103 3.9.2. Elastodynamic representation theorem 104 3.10. Two-dimensional integral representations 105 3.10.1. Basic singular solutions 106 3.10.2. Antiplane line load 107 3.10.3. In-plane line load 108 3.10.4. Integral representations 109 3.11. Boundary-value problems 110 3.12. Steady-state time-harmonic response 115 3.12.1. Time-harmonic source 115 3.12.2. Helmholtz's equation 116 3.12.3. Helmholtz's first (interior) formula 117 3.12.4. Helmholtz's second (exterior) formula 117 3.12.5. Steady-state solutions in two dimensions 118 3.13. Problems 119 4. Elastic waves in an unbounded medium 122 4.1. Plane waves 122 4.2. Time-harmonic plane waves 124 4.2.1. Inhomogeneous plane waves 125 4.2.2. Slowness diagrams 127 4.3. Wave motions with polar symmetry 128 4.3.1. Governing equations 128 4.3.2. Pressurization of a spherical cavity 129 4.3.3. Superposition of harmonic waves 132 4.4. Two-dimensional wave motions with axial symmetry 135 4.4.1. Governing equations 135 4.4.2. Harmonic waves 136 4.5. Propagation of wavefronts 138 4.5.1. Propagating discontinuities 138 4.5.2. Dynamical conditions at the wavefront 140 4.5.3. Kinematical conditions at the wavefront 141 4.5.4. Wavefronts and rays 142 4.6. Expansions behind the wavefront 144 4.7. Axial shear waves by the method of characteristics 148 4.8. Radial motions 152 4.9. Homogeneous solutions of the wave equation 154 4.9.1. Chaplygin's transformation 154 4.9.2. Line load 156 4.9.3. Shear waves in an elastic wedge 157 4.10. Problems 160 Plane harmonic waves in elastic half-spaces 165 5.1. Reflection and refraction at a plane interface 165 5.2. Plane harmonic waves 166 5.3. Flux of energy in time-harmonic waves 166 5.4. Joined half-spaces 168 5.5. Reflection of SH-waves 170 5.6. Reflection of P-waves 172 5.7. Reflection of SV-waves 177 5.8. Reflection and partition of energy at a free surface 181 5.9. Reflection and refraction of SH-waves 182 5.10. Reflection and refraction of P-waves 185 5.11. Rayleigh surface waves 187 5.12. Stoneley waves 194 5.13. Slowness diagrams 196 5.14. Problems 198 6. Harmonic waves in waveguides 202 6.1. Introduction 202 6.2. Horizontally polarized shear waves in an elastic layer 203 6.3. The frequency spectrum of SH-modes 206 6.4. Energy transport by SH-waves in a layer 208 6.5. Energy propagation velocity and group velocity 211 6.6. Love waves 218 6.7. Waves in plane strain in an elastic layer 220 6.8. The Rayleigh-Lamb frequency spectrum 226 6.9. Waves in a rod of circular cross section 236 6.10. The frequency spectrum of the circular rod of solid cross section 240 6.10.1. Torsional waves 241 6.10.2. Longitudinal waves 242 6.10.3. Flexural waves 246 6.11. Approximate theories for rods 249 6.11.1. Extensional motions 250 6.11.2. Torsional motions 251 6.11.3. Flexural motions - Bernoulli-Euler model 251 6.11.4. Flexural motions - Timoshenko model 252 6.12. Approximate theories for plates 254 6.12.1. Flexural motions - classical theory 255 6.12.2. Effects of transverse shear and rotary inertia 256 6.12.3. Extensional motions 257 6.13. Problems 258 7. Forced motions of a half-space 262 7.1. Integral transform techniques 262 7.2. Exponential transforms 264 7.2.1. Exponential Fourier transform 265 7.2.2. Two-sided Laplace transform 267 7.2.3. One-sided Laplace transform 268 7.3. Other integral transforms 269 7.3.1. Fourier sine transform 270 7.3.2. Fourier cosine transform 270 7.3.3. Hankel transform 270 7.3.4. Mellin transform 271 7.4. Asymptotic expansions of integrals 271 7.4.1. General considerations 271 7.4.2. Watson's lemma 272 7.4.3. Fourier integrals 273 7.4.4. The saddle point method 273 7.5. The methods of stationary phase and steepest descent 274 7.5.1. Stationary-phase approximation 274 7.5.2. Steepest-descent approximation 278 7.6. Half-space subjected to antiplane surface disturbances 283 7.6.1. Exact solution 284 7.6.2. Asymptotic representation 288 7.6.3. Steepest-descent approximation 288 7.7. Lamb's problem for a time-harmonic line load 289 7.7.1. Equations governing a state of plane strain 290 7.7.2. Steady-state solution 291 7.8. Suddenly applied line load in an unbounded medium 295 7.9. The Cagniard-de Hoop method 298 7.10. Some observations on the solution for the line load 301 7.11. Transient waves in a half-space 303 7.12. Normal point load on a half-space 310 7.12.1. Method of solution 310 7.12.2. Normal displacement at z = 0 313 7.12.3. Special case X = ц 316 7.13. Surface waves generated by a normal point load 318 7.14. Problems 321 8. Transient waves in layers and rods 326 8.1. General considerations 326 8.2. Forced shear motions of a layer 327 8.2.1. Steady-state harmonic motions 328 8.2.2. Transient motions 330 8.3. Transient in-plane motion of a layer 331 8.3.1. Method of solution 332 8.3.2. Inversion of the transforms 335 8.3.3. Application of the method of stationary phase 337 8.4. The point load on a layer 342 8.5. Impact of a rod 344 8.5.1. Exact formulation 347 8.5.2. Inversion of the transforms 349 8.5.3. Evaluation of the particle velocity for large time 350 8.6. Problems 353 9. Diffraction of waves by a slit 357 9.1. Mixed boundary-value problems 357 9.2. Antiplane shear motions 358 9.2.1. Green's function 359 9.2.2. The mixed boundary-value problem 362 9.3. The Wiener-Hopf technique 365 9.4. The decomposition of a function 369 9.4.1. General procedure 369 9.4.2. Example: the Rayleigh function 371 9.5. Diffraction of a horizontally polarized shear wave 372 9.6. Diffraction of a longitudinal wave 380 9.6.1. Formulation 380 9.6.2. Application of the Wiener-Hopf technique 382 9.6.3. Inversion of transforms 385 9.7. Problems 388 10. Thermal and viscoelastic effects, and effects of anisotropy and non- linearity 391 10.1. Thermal effects 391 10.2. Coupled thermoelastic theory 392 10.2.1. Time-harmonic plane waves 392 10.2.2. Transverse waves 394 10.2.3. Longitudinal waves 394 10.2.4. Transient waves 396 10.2.5. Second sound 398 10.3. Uncoupled thermoelastic theory 399 10.4. The linearly viscoelastic solid 399 10.4.1. Viscoelastic behavior 399 10.4.2. Constitutive equations in three dimensions 401 10.4.3. Complex modulus 402 10.5. Waves in viscoelastic solids 403 10.5.1. Time-harmonic waves 403 10.5.2. Longitudinal waves 403 10.5.3. Transverse waves 404 10.5.4. Transient waves 404 10.5.5. Propagation of discontinuities 407 10.6. Waves in anisotropic materials 409 10.7. A problem of transient nonlinear wave propagation 412 10.8. Problems 417 Author index 420 Subject index 422 Download link：http://www.isload.com.cn/store/m4342vmiispjd [ Last edited by zhq025 on 2008-1-16 at 10:20 ] |
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