Self-Consistent Methods for Composites: Vol.2: Wave Propagation in Heterogeneous Materials

Contents......Page 7
Preface......Page 11
Notations......Page 12
1. Introduction......Page 14
2.1 Integral equations for scalar waves in a medium with isolated inclusions......Page 18
2.2 The effective field method......Page 20
2.3.1 Version I of the EMM......Page 27
2.3.2 Version II of the EMM......Page 29
2.3.3 Version III and IV of the EMM......Page 30
2.4 Notes......Page 32
3.1 Integral equations for electromagnetic waves......Page 33
3.2 Version I of EMM for matrix composites......Page 36
3.3 One-particle EMM problems for spherical inclusions......Page 39
3.4 Asymptotic solutions of the EMM dispersion equation......Page 42
3.5 Numerical solution of the EMM dispersion equation......Page 45
3.6 Versions II and III of the EMM......Page 48
3.7 The effective field method......Page 54
3.8 One-particle EFM problems for spherical inclusions......Page 59
3.9.1 Long-wave asymptotics......Page 60
3.9.2 Short-wave asymptotics......Page 62
3.10 Numerical solution......Page 63
3.11 Comparison of version I of the EMM and the EFM......Page 66
3.12 Versions I, II, and III of EMM......Page 70
3.13.1 Variational formulation of the diffraction problem for an isolated inclusion......Page 71
3.13.2 Plane wave approximation......Page 72
3.14 The EFM for composites with regular lattices of spherical inclusions......Page 74
3.15 Versions I and IV of EMM for polycrystals and granular materials......Page 83
3.16 Conclusion......Page 87
3.17 Notes......Page 88
4. Axial elastic shear waves in fiber-reinforced composites......Page 89
4.1 Integral equations of the problem......Page 90
4.2 The effective medium method......Page 92
4.3.1 Integral equations for the local exciting fields......Page 96
4.3.2 The hypotheses of the EFM......Page 97
4.3.3 The dispersion equation of the EFM......Page 99
4.4.1 The one-particle problem of the EMM......Page 101
4.4.2 The one-particle problem of the EFM......Page 103
4.4.3 The scattering cross-section of a cylindrical fiber......Page 104
4.4.4 Approximate solution of the one-particle problem in the long-wave region......Page 107
4.5 Solutions of the dispersion equations in the long-wave region......Page 109
4.5.1 Long-wave asymptotic solution for EMM......Page 110
4.5.2 Long-wave asymptotic solution for EFM......Page 111
4.6 Short-wave asymptotics......Page 115
4.7 Numerical solutions of the dispersion equations......Page 117
4.8 Composites with regular lattices of cylindrical fibers......Page 120
4.9 Conclusion......Page 126
4.10 Notes......Page 127
5.1 The dynamic Green tensor for a homogeneous anisotropic medium......Page 128
5.2 Integral equations for elastic wave diffraction by an isolated inclusion......Page 133
5.3 Diffraction of long elastic waves by an isolated inclusion......Page 134
5.4 Diffraction of long elastic waves by a thin inclusion......Page 139
5.4.1 Thin soft inclusion......Page 140
5.4.2 Thin hard inclusion......Page 142
5.5 Diffraction of long elastic waves by a short axisymmetric fiber......Page 144
5.6 Total scattering cross-sections of inclusions......Page 149
5.6.1 An isolated inclusion......Page 150
5.6.2 Long-range scattering cross-sections......Page 155
5.7 Notes......Page 164
6.1 Diffraction of elastic waves by a random set of ellipsoidal inclusions......Page 166
6.2 The Green function of the effective wave operator......Page 173
6.3 Velocities and attenuations of long elastic waves in matrix composites......Page 177
6.4.1 Isotropic elastic medium with random crack-like inclusions......Page 179
6.4.2 Isotropic elastic medium with a random set of hard disks......Page 188
6.5 Long elastic waves in composites with short hard fibers......Page 193
6.5.1 Random sets of fibers homogeneously distributed over orientations......Page 194
6.5.2 Random set of fibers of the same orientation......Page 195
6.6 Notes......Page 198
7.1 Version I of the EMM for elastic waves......Page 199
7.2.1 Diffraction of a plane monochromatic wave by an isolated spherical inclusion......Page 204
7.2.2 An approximate solution of the one-particle problems in the long-wave region......Page 208
7.3 The dispersion equations of the EMM......Page 210
7.3.1 The EMM dispersion equation for longitudinal waves......Page 212
7.3.2 The EMM dispersion equation for transverse waves......Page 214
7.3.3 Total scattering cross-sections of a spherical inclusion......Page 215
7.3.4 The EMM dispersion equations in the short-wave region......Page 217
7.4 Versions II and III of EMM for long waves......Page 218
7.5 Numerical solution of the EMM dispersion equations......Page 223
7.6 The effective field method......Page 227
7.6.1 The hypotheses of the EFM......Page 228
7.6.2 The effective field for transverse waves......Page 230
7.6.3 The effective field equations for longitudinal waves......Page 233
7.7.1 Transverse waves......Page 238
7.7.2 Longitudinal waves......Page 242
7.8.1 Long transverse waves......Page 245
7.8.2 Short transverse waves......Page 248
7.8.3 Long longitudinal waves......Page 249
7.8.4 Short longitudinal waves......Page 250
7.9.1 Transverse waves......Page 254
7.9.2 Longitudinal waves......Page 257
7.9.3 Longitudinal waves in epoxy-lead composites......Page 260
7.10 Conclusion......Page 264
7.11 Notes......Page 266
8. Elastic waves in polycrystals......Page 267
8.1 General consideration......Page 268
8.2 The effective medium method......Page 270
8.3 The one-particle problem of EMM......Page 272
8.4 Polycrystals with orthorhombic grains......Page 275
8.5 The Born approximation......Page 280
8.6 Numerical results......Page 282
8.8 Notes......Page 286
A.1 E-basis......Page 288
A.2 P-basis......Page 289
A.3 Averaging the elements of the E- and P-bases......Page 291
A.4 Tensor bases of four-rank tensors in 2D-space......Page 292
B. The Percus-Yevick correlation function......Page 294
References......Page 296
O......Page 302
W......Page 303