Advances in Acoustics and Vibration

Volume 2015, Article ID 515263, 7 pages

http://dx.doi.org/10.1155/2015/515263

## On Peculiarities of Propagation of a Plane Elastic Wave through a Gradient Anisotropic Layer

Kazan Federal University, 18 Kremlyovskaya Street, Kazan 420008, Russia

Received 15 October 2015; Accepted 3 December 2015

Academic Editor: Kim M. Liew

Copyright © 2015 Anastasiia Anufrieva et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The problem of diffraction of a plane elastic wave by an anisotropic layer is studied. The diffraction problem is reduced to a boundary value problem for the layer. The grid method is used for solving the resulting boundary value problem. The diffraction of a plane longitudinal wave by the layer is considered. Some peculiarities of the gain-frequency and the gain-angle characteristics of a normal component of an energy flow of a passed longitudinal wave are numerically studied.

#### 1. Introduction

Compared to homogenous and isotropic materials, anisotropic materials represent an enormous interest from the practical point of view. Their properties vary in different spatial directions. Anisotropic materials and media are encountered in nature and they are often used in various industries.

During the last 50 years, interest to anisotropic materials has become strong, especially from research groups working in the areas of geology [1–4] and seismology [5–7]. It is well recognized that anisotropy of geological formations is caused by stratification of the prevalent direction of microcracks as well as by orientation of elements of geological formations possessing initial anisotropy. It is worth noting that this type of anisotropy originates from crystal texture of the materials. Experimental investigations [8] indicate that the anisotropy is caused by stresses induced by the rock masses. All of this allowed geologists to gain a better understanding of structure of the Earth’s crust [9] and motivated further studies in the area of anisotropic materials.

One more research direction of the area of anisotropic materials is acoustic metamaterials and phononic crystals [10]. These materials are artificial composite materials possessing a rather complicated structure [11, 12]. Investigation of these kinds of materials allowed making a huge progress in such engineering areas as developing good screening, noise insulation, and reaching full control over the ratio of energy of the transmitted wave to that of the reflected wave [13].

It is recognized that elastic properties of materials can often vary in the point-to-point manner. Sometimes, variation of elastic properties of geological layers with the depth can be caused by stresses in the surrounding rock masses. In industrial production of layered structures, one can create continuous-type variations of the ratios of the included constituent matters and, thereby, obtain a sufficiently smooth variation of the resulting material properties. By doing this way, various materials possessing a continuous (gradient) distribution of elastic characteristics in the transverse direction and homogeneous characteristics in the longitudinal direction of the layer can be produced.

Aside from variation of elastic properties in the point-to-point manner, another type of variation across the material is also quite often encountered both in nature and in industries. The variation is in the abrupt changing of the material’s elastic properties from one layer to another. One of the approaches to modeling such layered structures is associated with applying of a stratified model to the anisotropic material [14].

One of the first and simplest approaches to solving the problem was related to analysis of diffraction of elastic waves by isotropic layers possessing a continuous distribution of parameters. However, it is noteworthy that a more precise approach to an anisotropic model is in treating it as a transversely isotropic model. This approach was applied by our research group in the past to investigate the problem of elastic wave propagation through gradient isotropic materials [15] and gradient transversely isotropic materials [16, 17]. In [18], diffraction of plane acoustic wave by inhomogeneous layer was investigated. Sonic wave transmission through inhomogeneous anisotropic layer adjacent to a viscous fluid was investigated in [19].

In the present work, diffraction of elastic waves by an inhomogeneous anisotropic layer is chosen to be a subject of our investigation. Differential equations governing the diffraction problem for half-planes are considered separately from those for the layer. The problems for half-planes are overdetermined (using the overdetermination is, indeed, the essence of our proprietary approach) allowing establishing a relationship between traces of desired functions at the media’s interface [20]. Thus, the original problem reduces to the boundary value problem for the partial differential equations system with boundary conditions of the third type. After that, we applied the Fourier transform with respect to the variable, for which uniformity is preserved, to the resulting boundary value problem. The obtained system of ordinary differential equations was solved using the grid method.

Let us note that the algorithm was applied by our research group in the past not only in the works cited above but also in a number of other works of our group as well. For example, the problem of elastic wave diffraction by the layer with the fractal distribution of density was considered in [21].

Numerical results of the elastic wave propagation through a gradient layer with a linear variation of parameters, which are similar to properties of sandstone, are presented in the present work. Peculiarities of the wave propagation through such layers are shown. The transmission range for the gradient anisotropic layer at angle values close to the tangent angle is given. In addition, an example of passage of an elastic wave through a layer having parameters varying in the continuous manner from amazonite to labradorite is considered.

#### 2. Statement of the Problem

Let an elastic harmonic wave of type fall on anisotropic layer of thickness (medium 2 ), which is a nonuniform in the transverse direction, with continuous density and tensor of elasticity modulus from medium 1 under the angle with respect to the axis (see Figure 1). The diffraction result is a wave reflected toward medium 1 and a wave propagating toward medium 3 and field in the layer. We need to investigate response frequency characteristics of the longitudinal part of the wave . Media 1 and 3 are assumed to be uniform and isotropic. Medium 2 is assumed to be of the gradient type and anisotropic.