Wave Momentum And Quasi-particles In Physical Acoustics

by Gerard A Maugin, Martine Rousseau

Overview

This unique volume presents an original approach to physical acoustics with additional emphasis on the most useful surface acoustic waves on solids. The study is based on foundational work of Léon Brillouin, and application of the celebrated invariance theorem of Emmy Noether to an element of volume that is representative of the wave motion.This approach provides an easy interpretation of typical wave motions of physical acoustics in bulk, at surfaces, and across interfaces, in the form of the motion of associated quasi-particles. This type of motion, Newtonian or not, depends on the wave motion considered, and on the original modeling of the continuum that supports it. After a thoughtful review of Brillouin's fundamental ideas related to radiative stresses, wave momentum and action, and the necessary reminder on modern nonlinear continuum thermomechanics, invariance theory and techniques of asymptotics, a variety of situations and models illustrates the power and richness of the approach and its strong potential in applications. Elasticity, piezoelectricity and new models of continua with nonlinearity, viscosity and some generalized features (microstructure, weak or strong nonlocality) or unusual situations (bounding surface with energy, elastic thin film glued on a surface waveguide), are considered, exhibiting thus the versatility of the approach.This original book offers an innovative vision and treatment of the problems of wave propagation in deformable solids. It opens up new horizons in the theoretical and applied facets of physical acoustics.

Product Details

ISBN-13:	9789814663786
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	05/18/2015
Series:	World Scientific Series On Nonlinear Science Series A , #88
Pages:	252
Product dimensions:	5.90(w) x 9.10(h) x 0.70(d)

Table of Contents

Preface v

1 Prolegomena: wave momentum and radiative stresses in 1D in the line of Brillouin 1

1.1 Introduction 1

1.2 One-dimensional motion in the Euleriandescription 3

1.2.1 Basic equations 3

1.2.2 Method of perturbations 5

1.2.3 First-order approximation 5

1.2.4 Second-order approximation 6

1.2.5 Example of momentum and radiative stress in a thin rod 7

1.3 One-dimensional motion in the Lagrangian description 11

1.3.1 Basic equations 11

1.3.2 Perturbation analysis at the first-order of approximation 12

1.3.3 Perturbation analysis at the second order of approximation 13

1.4 Summary and concluding remarks 14

2 Elements of continuum tliermomcchanics 19

2.1 Material body 19

2.2 Balance laws of the thermomechanics of continua 23

2.2.1 Global balance laws in the Euler-Cauchy format 23

2.2.2 Euler-Cauchy format of the local balance laws of thermomechanics 25

2.2.3 Global balance laws in the Piola-Kirchhoff format 27

2.2.4 Piola-Kirchhoff format of the local balance laws of thermomechanics 27

2.3 General theorems of thermodynamics 29

2.3.1 Thermodynamic hypotheses 29

2.3.2 Local expression of the general theorems of thermomechanics 29

2.4 Finite-strain elasticity 30

2.4.1 Measures of finite strains 31

2.4.2 Time rates of finite strains 31

2.4.3 Rigid-body motions 32

2.5 Strains in small-strain elasticity 32

2.6 Constitutive equations for finite-strain elasticity 33

2.7 Constitutive equations for small-strain elasticity 35

3 Pseudornornentum and Eshelby stress 37

3.1 Introduction 37

3.2 Pseudornomentuin in hyperelastic materials 39

3.3 Field-theoretical formulation in the case of elasticity 41

3.4 The case of small strains 45

3.5 Peculiarity of a one-dimensional motion 46

3.6 Small strains in the presence of dissipation 48

4 Action, phonons and wave mechanics 51

4.1 Wave-particle dualism and phonons 52

4.2 Action in continuum mechanics 53

4.3 Wave kinematics and wave action 56

4.4 Evolution equation for the wave amplitude 59

4.5 Hamiltonian formulation 60

4.6 Further analytical mechanics 61

4.7 The case of inhomogeneous waves 63

5 Transmission-reflection problem 67

5.1 Introduction 67

5.2 Reminder on the wavelike picture 68

5.2.1 One-dimensional case 68

5.2.2 Transmission-reflection problem for a perfect interface 69

5.2.3 Transmission-reflection problem for an interface with delamination 69

5.3 Associated quasi-particle picture 70

5.3.1 Basic equations 70

5.3.2 Transmission-reflection problem (perfect interface) 72

5.3.3 Case of an imperfect interface for an interface with delaminating 74

5.4 Case of a sandwiched slab 74

5.5 Conclusion 76

6 Application to dynamic materials 79

6.1 Reminder on the notion of dynamic materials 79

6.2 General properties of linear wave propagation 82

6.3 Case of a fixed material interface or transition layer 84

6.4 Case of a time-line or thin time-like interface layer 85

6.5 Quasi-particle re-interpretation at a time-like interface layer 86

6.6 Waves along a rod of finite length 89

6.7 Space-time homogenization of dynamic materials 91

6.7.1 So-called "slow" time-like configuration 92

6.7.2 So-called "fast" space-like configuration 93

6.7.3 Space-time homogenization for a long time 94

6.8 Generalization to moving interfaces 95

6.9 Conclusion 96

7 Elastic surface waves in terms of quasi-particles 99

7.1 The notion of surface wave 99

7.2 The Rayleigh surface wave in isotropic linear elasticity 102

7.2.1 Definition of Rayleigh waves 102

7.2.2 Conservation of wave momentum for Rayleigh surface waves 104

7.2.3 Quasi-particles associated with Rayleigh surface waves 105

7.2.4 The influence of surface energy 108

7.2.5 The case of leaky surface wavesp 108

7.3 The case of Love waves 115

7.3.1 The Love SAW solution 115

7.3.2 Conservation of wave momentum and energy 117

7.3.3 Mass and energy of the associated quasi-particle 118

7.3.4 Summary of this section 122

7.4 The case of Murdoch waves 123

7.4.1 Definition of Murdoch waves 123

7.4.2 Murdoch SAW linear solution 125

7.4.3 Canonical conservation laws for Murdoch linear SAWs 126

7.4.4 Associated quasi-particle 128

7.4.5 Consideration on the Lagrangian of the wave system 130

7.4.6 Murdoch case as a limit of the Love case 131

7.5 Conclusion 132

8 Electroelastic surface waves in terms of quasi-particles 135

8.1 The notion of electroelastic surface wave 135

8.2 Basic equations of piezoelectricity 137

8.3 Conservation laws of energy and wave momentum in electroelasticity 139

8.4 The Bleustein Gulyaev surface wave 140

8.4.1 The general surface wave problem in piezoelectric materials 140

8.4.2 The Bleustein Gulyaev surface wave problem per se 142

8.4.3 Dynamics of the associated quasi-particle 144

8.4.4 Another case of electric boundary condition 147

8.5 Perturbation by elastic nonlinearitics 150

8.5.1 Basic equations 150

8.5.2 Surface wave solution 151

8.5.3 Quasi-particle associated with the wave solution 153

8.6 Perturbation by viscosity 157

8.6.1 Some general words 157

8.6.2 Reminder of the Bleustein--Gulyaev surface wave problem in presence of weak viscous losses 157

8.6.3 Global equations of wave momentum and energy 159

9 Waves in generalized elastic continua 171

9.1 The notion of generalized continuum 171

9.2 Weak nonlocality and gradient model of elasticity 173

9.2.1 Summary of strain-gradient elasticity 173

9.2.2 Wave solution for a simplified problem 174

9.2.3 Wave momentum 174

9.3 The case of Cosserat continua 175

9.3.1 Summary of the general linear theory 175

9.3.2 Wave solution for a simplified problem 177

9.3.3 Wave momentum and quasi-particles 178

9.4 The case of strong nonlocality 179

9.4.1 Summary of nonlocal elasticity 179

9.4.2 Wave solution for a simplified problem 180

9.4.3 Wave momentum and quasi-particles 182

9.5 Conclusion 185

10 Examples of solitonic systems 187

10.1 Introduction: The notion of solitolin 187

10.2 Reminder: Some standard cases 188

10.2.1 The Boussinesq model in elastic crystals 188

10.2.2 The Korteweg-De Vries equation 190

10.2.3 The sine-Gordon equation 195

10.2.4 The nonlinear Schrödinger model 197

10.2.5 Comments 198

10.3 The generalized Boussinesq model (gradient elasticity) 198

10.4 Surface elastic solitons 201

10.4.1 The basic equations 201

10.4.2 Reminder: Linear harmonic approximation 204

10.4.3 Solitary-wave solutions for envelope signals 205

Appendix A Reminder on Noether's theorem 211

Appendix B Justification of (4.33)?(4.34) by a two-timing method 217

Bibliography 221

Index 233