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.