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;

The book is dedicated to the application of self-consistent methods to the solution of static and dynamic problems of the mechanics and physics of composite materials. The effective elastic, electric, dielectric, thermo-conductive and other properties of composite materials reinforced by ellipsoidal, spherical multi-layered inclusions, thin hard and soft inclusions, short fibers and unidirected multi-layered fibers are considered. Explicit formulas and efficient computational algorithms for the calculation of the effective properties of the composites are presented and analyzed. The method of the effective medium and the method of the effective field are developed for the calculation of the phase velocities and attenuation of the mean (coherent) wave fields propagating in the composites. The predictions of the methods are compared with experimental data and exact solutions for the composites with periodical microstructures. The book may be useful for material engineers creating new composite materials and scholars who work on the theory of composite and non-homogeneous media.