Skip to main content
main-content

Inhaltsverzeichnis

Frontmatter

1. Introduction

The problem of wave propagation in composite materials has many important applications. The solution of this problem allows us to predict the response of composite materials to various types of dynamic loadings; this problem forms the theoretical background for non-destructive ultrasonic evaluation of microstructures of composites. The main objectives of the theory in this problem are the dependencies of the phase velocity and attenuation coefficient of the mean (coherent) wave field propagating in the composite on the frequency of the incident field (dispersion curves) and on the details of the composite microstructure. For composite materials with random microstructures, this problem cannot be solved exactly, and only approximate solutions are available. Self-consistent methods are widely used for the construction of such approximate solutions.
In self-consistent methods, complex actual wave fields propagating in heterogeneous media are approximated by simple ones using physically reasonable hypotheses. All the self-consistent methods are based on two types of such hypotheses. The first one reduces the problem of interactions between many inclusions in the composite to a problem for one inclusion (the one particle problem). The second hypothesis is the condition of self-consistency. For application of these methods, the heterogenous medium should have specific features: a typical element (particle) should exist in the medium. Such a particle may be an inclusion in the matrix-inclusion composites, a grain in random polycrystalline materials, a crack in materials with defects, etc.

2. Self-consistent methods for scalar waves in composites

In this chapter, we consider self-consistent methods in application to a simple model problem: propagation of scalar waves in a medium with isolated inclusions. The main hypotheses of the effective field method and various versions of the effective medium method are introduced. For every method, the algorithm of development of dispersion equations for the mean (coherent) wave field propagating in the composites is presented.

3. Electromagnetic waves in composites and polycrystals

In this chapter, the self-consistent methods are applied to the solution of the problem of electromagnetic wave propagation in the composites with spherical inclusions and polycrystals with quasispherical grains. For every method, the dispersion equation for the wave number of the mean wave field propagating in the composites is derived. The asymptotic solutions of these equations in the long and short-wave regions are obtained in closed analytical forms. Phase velocities and attenuation coefficients of the mean wave fields are calculated in wide regions of the parameters of the composites and frequencies of the incident waves. Predictions of various self-consistent methods are compared and analyzed.

4. Axial elastic shear waves in fiber-reinforced composites

The self-consistent methods developed in Chapters 2 and 3 may be applied to the analysis of elastic wave propagation in composites without essential modifications. Nevertheless, elastic waves introduce specific difficulties. First, two types of elastic waves (longitudinal and transverse waves of various polarizations) may propagate in the composites, and the dispersion equations for each wave should be derived by the methods. Secondly, elastic waves oblige us to consider a system of two integral equations for the displacement and strain fields, and this makes the analysis more cumbersome than that for scalar or electromagnetic waves.
In this Chapter we consider a relatively simple case: propagation of axial elastic shear waves through composites reinforced with long unidirectional fibers. The wave vector of these waves is orthogonal to the fiber axes, and the polarization vector coincides with the fiber directions. In this case, there is only one nonzero component of the displacement field in the composite, and only one type of wave propagates in the composite. This makes the algorithm of the self-consistent methods more transparent than this for other composites in which a wave of one type generates waves of other types. The structure of this chapter is as follows.
In Section 4.1, the integral equations of the axial shear wave propagation problem are considered. In Section 4.2, the general scheme of the EMM is developed for construction of the dispersion equation for the mean wave field in the composite. In Section 4.3, the EFM is applied to the solution of the same problem. Section 4.4 is devoted to the solutions of the one-particle problems of both methods. In Sections 4.5 and 4.6, the solutions of the dispersion equations in the long and short-wave regions are constructed. In Section 4.7, the results of numerical solutions of the dispersion equations of both methods are compared in a wide region of frequencies of the incident field. Section 4.8 is devoted to wave propagation in composites with periodic arrangements of cylindrical fibers. We show that the EFM predicts the existence of pass and stop bands in the frequency region for the propagating waves.

5. Diffraction of long elastic waves by an isolated inclusion in a homogeneous medium

We consider the one-particle problem for self-consistent methods. The solution of this problem is constructed for incident waves longer than characteristic sizes of the inclusion (long-wave approximation). Explicit equations for the elastic field inside an ellipsoidal inclusion and its limiting forms (oblate and prolate spheroids) are obtained in Sections 5.1–5.3. Thin soft and hard inclusions, and hard axisymmetric fibers are considered in Section 5.4. The final Section 5.5 is devoted to the proof of the optical theorem for diffraction of elastic waves, and the calculation of the total scattering cross-sections of inclusions of various forms.

6. Effective wave operator for a medium with random isolated inclusions

In this chapter, we consider a homogeneous medium containing a random set of isolated inclusions. The effective field method is applied to the solution of the homogenization problem for wave propagation. The dispersion equation for the mean wave field in the composite is derived using the long-wave solutions of the one-particle problem. We show that this dispersion equation corresponds to a homogeneous medium (effective medium) with attenuation and dispersion. The Green function of the wave operator for the effective medium is constructed and analyzed. The velocities and attenuation coefficients of the long waves propagating in the composites with inclusions of various forms are calculated in the framework of the EFM.

7. Elastic waves in a medium with spherical inclusions

In this chapter, the effective medium and effective field methods are applied to elastic wave propagation in composites with a set of spherical inclusions. for spherical isotropic inclusions in an isotropic matrix, the series solutions of the one-particle problems of the self-consistent methods may be obtained for any frequency of the incident field. As a result, the predictions of the methods may be analyzed and compared in a wide region of frequencies of the incident fields that covers long, medium and short waves. The contents of the chapter is as follows.
In Section 7.1, version I of the EMM is developed in a form that may be applied for propagating waves of any length. The solutions of the one-particle problems are considered in Section 7.2, and the final forms of the dispersion equations of the EMM are presented in Section 7.3. The long- and short-wave asymptotic solutions of the EMM dispersion equations are obtained in this section. In Section 7.4, versions II and III of the EMM are considered in the long-wave region. Numerical solutions of the EMM dispersion equations, and comparison of predictions of the three versions of the EMM in a wide region of frequencies are presented in Section 7.5. In Section 7.6, the effective field method is developed for the problem. Specific features of the EFM for longitudinal wave propagation are indicated. It is shown that in this case, the local exciting field acting on each particle is not a plane wave but a sum of plane and radial waves. Solutions of the one-particle problems of the EFM are presented in Section 7.7. The dispersion equations of the EFM in the long- and short-wave region are considered in Section 7.8. Numerical solutions of the dispersion equations of the EFM, and comparison of predictions of the EMM and EFM with experimental data for epoxy-lead composites are considered in Section 7.9.

8. Elastic waves in polycrystals

The problem of wave propagation through polycrystalline materials has attracted much attention due to its important theoretical and practical aspects. for instance, the solution of this problem gives a theoretical foundation for a nondestructive analysis of the microstructure of real metals.
In this chapter, the problem of elastic wave propagation in polycrystals is solved by the effective medium method. The general scheme of the method is developed in Sections 8.1-8.3. A special basis of four-rank tensors is proposed in Section 8.4 in order to perform the method for polycrystals with orthorhombic symmetry of the monocrystals.

Backmatter

Weitere Informationen

Premium Partner

    Marktübersichten

    Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen. 

    Bildnachweise