Preface |
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1 Prolegomena: wave momentum and radiative stresses in 1D in the line of Brillouin |
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1 | (18) |
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1 | (2) |
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1.2 One-dimensional motion in the Eulerian description |
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3 | (8) |
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3 | (2) |
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1.2.2 Method of perturbations |
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5 | (1) |
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1.2.3 First-order approximation |
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5 | (1) |
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1.2.4 Second-order approximation |
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6 | (1) |
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1.2.5 Example of momentum and radiative stress in a thin rod |
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7 | (4) |
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1.3 One-dimensional motion in the Lagrangian description |
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11 | (3) |
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11 | (1) |
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1.3.2 Perturbation analysis at the first-order of approximation |
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12 | (1) |
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1.3.3 Perturbation analysis at the second order of approximation |
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13 | (1) |
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1.4 Summary and concluding remarks |
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14 | (5) |
2 Elements of continuum thermomechanics |
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19 | (18) |
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19 | (4) |
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2.2 Balance laws of the thermomechanics of continua |
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23 | (6) |
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2.2.1 Global balance laws in the Euler—Cauchy format |
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23 | (2) |
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2.2.2 Euler—Cauchy format of the local balance laws of thermomechanics |
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25 | (2) |
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2.2.3 Global balance laws in the Piola—Kirchhoff format |
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27 | (1) |
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2.2.4 Piola—Kirchhoff format of the local balance laws of thermomechanics |
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27 | (2) |
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2.3 General theorems of thermodynamics |
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29 | (1) |
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2.3.1 Thermodynamic hypotheses |
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29 | (1) |
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2.3.2 Local expression of the general theorems of thermomechanics |
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29 | (1) |
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2.4 Finite-strain elasticity |
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30 | (2) |
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2.4.1 Measures of finite strains |
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31 | (1) |
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2.4.2 Time rates of finite strains |
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31 | (1) |
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32 | (1) |
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2.5 Strains in small-strain elasticity |
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32 | (1) |
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2.6 Constitutive equations for finite-strain elasticity |
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33 | (2) |
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2.7 Constitutive equations for small-strain elasticity |
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35 | (2) |
3 Pseudomomentum and Eshelby stress |
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37 | (14) |
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37 | (2) |
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3.2 Pseudomomentum in hyperelastic materials |
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39 | (2) |
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3.3 Field-theoretical formulation in the case of elasticity |
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41 | (4) |
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3.4 The case of small strains |
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45 | (1) |
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3.5 Peculiarity of a one-dimensional motion |
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46 | (2) |
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3.6 Small strains in the presence of dissipation |
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48 | (3) |
4 Action, phonons and wave mechanics |
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51 | (16) |
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4.1 Wave-particle dualism and phonons |
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52 | (1) |
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4.2 Action in continuum mechanics |
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53 | (3) |
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4.3 Wave kinematics and wave action |
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56 | (3) |
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4.4 Evolution equation for the wave amplitude |
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59 | (1) |
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4.5 Hamiltonian formulation |
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60 | (1) |
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4.6 Further analytical mechanics |
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61 | (2) |
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4.7 The case of inhomogeneous waves |
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63 | (4) |
5 Transmission-reflection problem |
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67 | (12) |
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67 | (1) |
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5.2 Reminder on the wavelike picture |
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68 | (2) |
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5.2.1 One-dimensional case |
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68 | (1) |
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5.2.2 Transmission-reflection problem for a perfect interface |
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69 | (1) |
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5.2.3 Transmission-reflection problem for an interface with delamination |
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69 | (1) |
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5.3 Associated quasi-particle picture |
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70 | (4) |
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70 | (2) |
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5.3.2 Transmission-reflection problem (perfect interface) |
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72 | (2) |
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5.3.3 Case of an imperfect interface for an interface with delaminating |
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74 | (1) |
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5.4 Case of a sandwiched slab |
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74 | (2) |
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76 | (3) |
6 Application to dynamic materials |
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79 | (20) |
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6.1 Reminder on the notion of dynamic materials |
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79 | (3) |
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6.2 General properties of linear wave propagation |
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82 | (2) |
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6.3 Case of a fixed material interface or transition layer |
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84 | (1) |
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6.4 Case of a time-line or thin time-like interface layer |
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85 | (1) |
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6.5 Quasi-particle re-interpretation at a time-like interface layer |
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86 | (3) |
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6.6 Waves along a rod of finite length |
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89 | (2) |
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6.7 Space-time homogenization of dynamic materials |
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91 | (4) |
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6.7.1 So-called "slow" time-like configuration |
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92 | (1) |
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6.7.2 So-called "fast" space-like configuration |
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93 | (1) |
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6.7.3 Space-time homogenization for a long time |
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94 | (1) |
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6.8 Generalization to moving interfaces |
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95 | (1) |
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96 | (3) |
7 Elastic surface waves in terms of quasi-particles |
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99 | (36) |
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7.1 The notion of surface wave |
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99 | (3) |
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7.2 The Rayleigh surface wave in isotropic linear elasticity |
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102 | (13) |
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7.2.1 Definition of Rayleigh waves |
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102 | (2) |
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7.2.2 Conservation of wave momentum for Rayleigh surface waves |
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104 | (1) |
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7.2.3 Quasi-particles associated with Rayleigh surface waves |
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105 | (3) |
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7.2.4 The influence of surface energy |
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108 | (1) |
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7.2.5 The case of leaky surface waves |
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108 | (7) |
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7.3 The case of Love waves |
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115 | (8) |
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7.3.1 The Love SAW solution |
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115 | (2) |
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7.3.2 Conservation of wave momentum and energy |
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117 | (1) |
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7.3.3 Mass and energy of the associated quasi-particle |
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118 | (4) |
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7.3.4 Summary of this section |
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122 | (1) |
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7.4 The case of Murdoch waves |
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123 | (9) |
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7.4.1 Definition of Murdoch waves |
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123 | (2) |
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7.4.2 Murdoch SAW linear solution |
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125 | (1) |
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7.4.3 Canonical conservation laws for Murdoch linear SAWs |
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126 | (2) |
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7.4.4 Associated quasi-particle |
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128 | (2) |
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7.4.5 Consideration on the Lagrangian of the wave system |
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130 | (1) |
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7.4.6 Murdoch case as a limit of the Love case |
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131 | (1) |
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132 | (3) |
8 Electroelastic surface waves in terms of quasi-particles |
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135 | (36) |
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8.1 The notion of electroelastic surface wave |
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135 | (2) |
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8.2 Basic equations of piezoelectricity |
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137 | (2) |
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8.3 Conservation laws of energy and wave momentum in electroelasticity |
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139 | (1) |
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8.4 The Bleustein—Gulyaev surface wave |
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140 | (10) |
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8.4.1 The general surface wave problem in piezoelectric materials |
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140 | (2) |
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8.4.2 The Bleustein—Gulyaev surface wave problem per se |
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142 | (2) |
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8.4.3 Dynamics of the associated quasi-particle |
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144 | (3) |
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8.4.4 Another case of electric boundary condition |
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147 | (3) |
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8.5 Perturbation by elastic nonlinearities |
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150 | (7) |
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150 | (1) |
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8.5.2 Surface wave solution |
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151 | (2) |
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8.5.3 Quasi-particle associated with the wave solution |
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153 | (4) |
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8.6 Perturbation by viscosity |
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157 | (14) |
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157 | (1) |
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8.6.2 Reminder of the Bleustein—Gulyaev surface wave problem in presence of weak viscous losses |
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157 | (2) |
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8.6.3 Global equations of wave momentum and energy |
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159 | (12) |
9 Waves in generalized elastic continua |
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171 | (16) |
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9.1 The notion of generalized continuum |
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171 | (2) |
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9.2 Weak nonlocality and gradient model of elasticity |
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173 | (2) |
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9.2.1 Summary of strain-gradient elasticity |
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173 | (1) |
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9.2.2 Wave solution for a simplified problem |
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174 | (1) |
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174 | (1) |
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9.3 The case of Cosserat continua |
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175 | (4) |
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9.3.1 Summary of the general linear theory |
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175 | (2) |
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9.3.2 Wave solution for a simplified problem |
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177 | (1) |
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9.3.3 Wave momentum and quasi-particles |
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178 | (1) |
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9.4 The case of strong nonlocality |
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179 | (6) |
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9.4.1 Summary of nonlocal elasticity |
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179 | (1) |
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9.4.2 Wave solution for a simplified problem |
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180 | (2) |
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9.4.3 Wave momentum and quasi-particles |
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182 | (3) |
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185 | (2) |
10 Examples of solitonic systems |
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187 | (24) |
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10.1 Introduction: The notion of soliton |
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187 | (1) |
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10.2 Reminder: Some standard cases |
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188 | (10) |
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10.2.1 The Bousginesq model in elastic crystals |
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188 | (2) |
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10.2.2 The Korteweg—De Vries equation |
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190 | (5) |
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10.2.3 The sine-Gordon equation |
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195 | (2) |
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10.2.4 The nonlinear Schrodinger model |
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197 | (1) |
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198 | (1) |
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10.3 The generalized Boussinesq model (gradient elasticity) |
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198 | (3) |
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10.4 Surface elastic solitons |
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201 | (10) |
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10.4.1 The basic equations |
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201 | (3) |
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10.4.2 Reminder: Linear harmonic approximation |
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204 | (1) |
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10.4.3 Solitary-wave solutions for envelope signals |
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205 | (6) |
Appendix A Reminder on Noether's theorem |
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211 | (6) |
Appendix B Justification of (4.33)—(4.34) by a two-timing method |
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217 | (4) |
Bibliography |
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221 | (12) |
Index |
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