Self-Consistent Methods for Composites

1. Introduction; Self-consistent methods for scalar waves in composites; 2.1 Integral equations for scalar waves in a medium with isolated inclusions; 2.2 The effective field method; 2.3 The effective medium method; 2.3.1 Version I of the EMM; 2.3.2 Version I1 of the EMM; 2.3.3 Version I11 and nT of the EMM; 2.4 Notes; Electromagnetic waves in composites and polycrystals; 3.1 Integral equations for electromagnetic waves; 3.2 Version I of EMM for matrix composites; 3.3 One-particle EMM problems for spherical inclusions; 3.4 Asymptotic solutions of the EMM dispersion equation; 3.5 Numerical solution of the EMM dispersion equation; 3.6 Versions I1 and I11 of the EMM; 3.7 The effective field method; 3.8 One-particle EFM problems for spherical inclusions; 3.9 Asymptotic solutions of the EFM dispersion equation; 3.9.1 Long-wave asymptotics; 3.9.2 Short-wave asymptotics; 3.10 Numerical solution; 3.11 Comparison of version I of the EMM and the EFM; 3.12 Versions I, 11, and I11 of EMM; 3.13 Approximate solutions of one-particle problems; 3.13.1 Variational formulation of the diffraction problem for an isolated inclusion; 3.13.2 Plane wave approximation; 3.14 The EFM for composites with regular lattices of spherical inclusions; 3.15 Versions I and IV of EMM for polycrystals and granular materials; 3.16 Conclusion; 3.17 Notes; 4. Axial elastic shear waves in fiber reinforced composites; 4.1 Integral equations of the problem;4.2 The effective medium method; 4.3 The effective field method; 4.3.1 Integral equations for the local exciting fields; 4.3.2 The hypotheses of the EFM; 4.3.3 The dispersion equation of the EFM; 4.4 One-particle problems of EMM and EFM; 4.4.1 The one-particle problem of the EMM; 4.4.2 The one-particle problem of the EFM; 4.4.3 The scattering cross-section of a cylindrical fiber; 4.4.4 Approximate solution of the one-particle problem in the long-wave region; 4.5 Solutions of the dispersion equations in the long-wave region; 4.5.1 Long-wave asymptotic solution for EMM; 4.5.2 Long-wave asymptotic solution for EFM; 4.6 Short-wave asymptotics; 4.7 Numerical solutions of the dispersion equations; 4.8 Composites with regular lattices of cylindrical fibers; 4.9 Conclusion; 4.10 Notes; 5. Diffraction of long elastic waves by an isolated inclusion in a homogeneous medium; 5.1 The dynamic Green tensor for a homogeneous anisotropic medium; 5.2 Integral equations for elastic wave diffraction by an isolated inclusion; 5.3 Diffraction of long elastic waves by an isolated inclusion; 5.4 Diffraction of long elastic waves by a thin inclusion; 5.4.1 Thin soft inclusion; 5.4.2 Thin hard inclusion; 5.5 Diffraction of long elastic waves by a short axisymmetric fiber; 5.6 Total scattering cross-sections of inclusions; 5.6.1 An isolated inclusion; 5.6.2 Long range scattering cross-sections; 5.7 Notes; 6. Effective wave operator for a medium with random isolated inclusions; 6.1 Diffraction of elastic waves by a random set of ellipsoidal inclusions; 6.2 The Green function of the effective wave operator; 6.3 Velocities and attenuations of long elastic waves in matrix composites; 6.4 Long elastic waves in composites with random thin inclusions; 6.4.1 Isotropic elastic medium with random crack-like inclusions; 6.4.2 Isotropic elastic medium with a random set of hard disks; 6.5 Long elastic waves in composites with short hard fibers; 6.5.1 Random sets of fibers homogeneously distributed over orientations; 6.5.2 Random set of fibers of the same orientation; 6.6 Notes; 7. Elastic waves in a medium with spherical inclusions; 7.1 Version I of the EMM for elastic waves; 7.2 The one-particle problems of EMM; 7.2.1 Diffraction of a plane monochromatic wave by an isolated spherical inclusion; 7.2.2 An approximate solution of the one-particle problems in the long-wave region; 7.3 The dispersion equations of the EMM; 7.3.1 The EMM dispersion equation for longitudinal waves; 7.3.2 The EMM dispersion equation for transverse waves; 7.3.3 Total scattering cross-sections of a spherical inclusion; 7.3.4 The EMM dispersion equations in the short-wave region; 7.4 Versions I1 and I11 of EMM for long waves; 7.5 Numerical solution of the EMM dispersion equations; 7.6 The effective field method; 7.6.1 The hypotheses of the EFM; 7.6.2 The effective field for transverse waves; 7.6.3 The effective field equations for longitudinal waves; 7.7 One-particle problems of EFM; 7.7.1 Transverse waves; 7.7.2 Longitudinal waves; 7.8 EFM dispersion equations; 7.8.1 Long transverse waves; 7.8.2 Short transverse waves; 7.8.3 Long longitudinal waves; 7.8.4 Short longitudinal waves; 7.9 Numerical solution of the EFM dispersion equations; 7.9.1 Transverse waves; 7.9.2 Longitudinal waves; 7.9.3 Longitudinal waves in epoxy-lead composites; 7.10 Conclusion; 7.11 Notes; 8 . Elastic waves in polycrystals; 8.1 General consideration; 8.2 The effective medium method; 8.3 The one-particle problem of EMM; 8.4 Polycrystals with orthorhombic grains; 8.5 The Born approximation; 8.6 Numerical results; 8.7 Conclusions; 8.8 Notes; A . Special tensor bases of four rank tensors; A.l E-basis; A.2 P-basis; A.3 Averaging the elements of the E and P-bases; A.4 Tensor bases of four-rank tensors in 2D-space; B . The Percus-Yevick correlation function; References