Self-Consistent Methods for Composites

Table of Contents

1. Frontmatter

2. Chapter 1. Introduction

   Sergey Kanaun, Valery Levin

   Abstract

   Specific features of the homogenization problem for wave fields in heterogeneous materials are discussed. The place and advantages of self-consistent methods over other methods of solution of the homogenization problem are indicated. The structure of the book is reviewed. The homogenization problem for time-harmonic wave fields is formulated in detail.

3. Chapter 2. Scalar Waves in Heterogeneous Media

   Sergey Kanaun, Valery Levin

   Abstract

   Scalar wave fields in homogeneous host media with a set of isolated inclusions are considered. The problem of calculation of these fields is reduced to the integral equation for the fields in the region occupied by the inclusions. For an isolated spherical inclusion subjected to a plane monochromatic incident wave (the one-particle problem), the solution is obtained in the form of series of spherical harmonics. The total scattering cross section of a spherical inclusion is calculated.

4. Chapter 3. Self-consistent Methods for Solution of the Homogenization Problem in the Case of Scalar Waves

   Sergey Kanaun, Valery Levin

   Abstract

   The homogenization problem for monochromatic scalar wave fields in the medium with a random set of spherical inclusions is solved by two self-consistent methods: the effective medium and effective field methods. Predictions of the methods for phase velocities and attenuation factors of the mean wave fields in the composites are compared in the long, middle, and short-wave regions of the propagating waves.

5. Chapter 4. Propagation of Electromagnetic Waves in Matrix Composites and Polycrystals

   Sergey Kanaun, Valery Levin

   Abstract

   The homogenization problems for electromagnetic wave fields in the composites with spherical inclusions and polycrystals with quasispherical grains are considered. The dispersion equations for the wave numbers of the mean wave fields are derived by the effective field and effective medium methods. The asymptotic solutions of these equations in the long- and short-wave regions are obtained. Predictions of the methods are compared in a wide regions of frequencies and dielectric permittivities of the host medium and the inclusions.

6. Chapter 5. Propagation of Axial Elastic Shear Waves in Fiber Reinforced Composites

   Sergey Kanaun, Valery Levin

   Abstract

   Time-harmonic axial shear waves in the elastic composites reinforced with cylindrical infinite fibers of the same orientations are considered. The wave problem is reduced to solution of a system of two integral equations for displacement and strain fields in the composite medium. The effective medium and effective field methods are used to derive the dispersion equations for the wave numbers of the mean wave fields propagating in the composite. Asymptotic solutions of the dispersion equations in the long and short-wave regions are obtained. The numerical solutions of the dispersion equations are compared in a wide region of frequencies of the incident field and elastic properties of the host medium and the fibers.

7. Chapter 6. Diffraction of Long Elastic Waves on Isolated Inclusions

   Sergey Kanaun, Valery Levin

   Abstract

   The problem of diffraction of long elastic waves on an isolated inclusion in an infinite homogeneous medium (the one-particle problem) is considered. For long incident waves, explicit solutions of the one-particle problems are obtained for inclusions widely used as fillers for composite materials. Ellipsoidal inclusions, elliptical cracks, stiff elliptical flakes and ribbons, and stiff axisymmetric fibers are considered. For such inclusions, efficient algorithms for calculating the total scattering cross sections for longitudinal and transverse waves of arbitrary directions are presented.

8. Chapter 7. Effective Wave Operator for the Medium with a Random Set of Isolated Inclusions

   Sergey Kanaun, Valery Levin

   Abstract

   A homogeneous elastic medium containing random sets of isolated inclusions is considered. The medium is subjected to a plane monochromatic incident wave whose length is more than the inclusion’s linear sizes. The effective field method is applied to the construction of the effective wave operator governing the mean wave fields propagating in the composite. The equations for the velocities and attenuation factors of the mean wave fields are obtained for the inclusions widely used as fillers for particulate composites.

9. Chapter 8. Propagation of Elastic Waves in the Medium with Spherical Inclusions. The Effective Medium Method

   Sergey Kanaun, Valery Levin

   Abstract

   The effective medium method is applied to the solution of the homogenization problem for elastic wave propagation in the composites with random sets of spherical inclusions. For spherical isotropic inclusions in an isotropic host medium, the solutions of the one-particle problem of the EMM can be obtained for arbitrary frequency of the incident field. As a result, the method predictions for the velocities and attenuation factors of the mean wave fields can be obtained in long, middle and short-wave regions. The dispersion equation of the EMM and its long and short-wave asymptotic solutions are obtained. Comparisons of predictions of the method for velocities and attenuation factors of the mean wave fields with experimental data are presented.

10. Chapter 9. Propagation of Elastic Waves in the Medium with Spherical Inclusions: The Effective Field Method

    Sergey Kanaun, Valery Levin

    Abstract

    The effective field medium is applied to solution of the homogenization problem for elastic wave propagation in the composites with a random set of spherical inclusions. For isotropic inclusions and isotropic host medium, the solutions of the one-particle problem of the EFM can be found in the form of series of spherical harmonics for any frequency of the incident field. It allows us to obtain the solution of the homogenization problem in the long, middle and short-wave regions. Predictions of the effective field and effective medium methods are compared with available in the literature experimental data.

11. Chapter 10. Propagation of Elastic Waves Through Polycrystalline Materials

    Sergey Kanaun, Valery Levin

    Abstract

    The problem of elastic wave propagation in polycrystals with quasispherical grains is solved by the effective medium method. The general scheme of the method in application to polycrystals is developed. The approximate solution of the homogenization problem based on the perturbation theory (Born approximation) is presented. The phase velocities and attenuation factors of the longitudinal and transversal waves in polycrystals of Al and Zn are obtained. The method predicts physically correct behavior of the mean wave fields in the long wave, stochastic, and diffusive regions. Comparison of the predictions of the effective medium method with experimental data and the Born approximation is presented.

12. Backmatter