Shock and Vibration

Volume 2019, Article ID 6978305, 7 pages

https://doi.org/10.1155/2019/6978305

## Explicit Solutions to Single Scattering of SH Waves with a Radially Gradient Interphase Layer

^{1}School of Mechanical and Electrical Engineering, Guilin University of Electronic Technology, Guilin 541004, China^{2}College of Aerospace Engineering, Chongqing University, Chongqing 400044, China^{3}Chongqing Key Laboratory of Heterogeneous Material Mechanics, Chongqing University, Chongqing 400044, China

Correspondence should be addressed to Jun Zhang; nc.ude.uqc@gnahzjem

Received 22 October 2018; Accepted 10 December 2018; Published 14 January 2019

Academic Editor: Jean-Mathieu Mencik

Copyright © 2019 Jiading Bao 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

In this work, analytical solutions to the single scattering of horizontally polarized shear waves (SH) by cylindrical fibers with two specific radially gradient interphase layers are presented. In the first case, the shear modulus and the square of wave number is a linear function of ; in the second case and is a linear function of . As an example, we solve the single scattering of SH waves by a SiC fiber with the two interphase layers in an aluminum matrix. The calculated scattering cross sections are compared to values obtained by an approximate method (dividing the continuous varying layer into multiple homogeneous sublayers). The results indicate the current approach gives excellent performance.

#### 1. Introduction

The multiple scattering of waves occurs throughout our daily lives, for example, the scattering of sound by water drops in fog and the scattering of light by fine particles in the air. Study on the multiple scattering of waves has a lengthy history by virtue of its ubiquity [1]. To date, multiple scattering of elastic waves in composites is still an active topic since more and more new composites appear. Even if the composite’s matrix and inclusions are both elastic, elastic waves in such a composite are still attenuated due to the multiple scattering effects; this is called attenuation caused by scattering. There have been many theoretical models [2] proposed to evaluate this attenuation because its accurate evaluation facilitates the dynamic characterization and nondestructive assessment [3] of composite materials.

The extant research can be roughly split into two groups. The first group consists of methods based on the wave function expansion that employ rational averaging techniques, which was pioneered by Foldy [4–8]. In these methods, the exciting waves and scattered waves of each inclusion in the composite were expressed as a series of wave functions. The expansion coefficients of the scattered waves by each inclusion are related through a matrix **T** to those of the exciting waves. The components of the matrix **T** are determined by the boundary conditions of the corresponding single scattering problem [9]. Note that the matrix **T** becomes a scalar for SH waves since no mode conversion happens. After a sequence of mathematical operations, a homogeneous system for the ensemble-averaged expansion coefficients of the exciting waves or scattered waves can be obtained. Solving this homogeneous system yields the attenuation coefficient. The second group consists of studies of the self-consistent method [10–13], in which each inclusion is assumed to behave as an isolated inclusion in a medium with the effective properties of the composite. In addition, the wave acting on each inclusion is a coherent wave. The attenuation coefficient was then obtained through various consistency conditions; for example, the mean wave field equals the coherent wave field [13] or the total scattering by the inclusions embedded in the effective medium effectually vanishes [12].

Both types of theoretical models necessitate solving a corresponding single scattering problem [9], that is, the scattering of a plane incident wave by a single inclusion embedded in an infinite matrix. In the theoretical models proposed by Waterman and Truell [4] and Norris and Conoir [14], for example, the single scattering problem was solved to obtain the far-field scattering amplitude, , which was further used to evaluate the scattering cross section . In most of the existing theoretical models, the fibers/particulates were usually assumed to be perfectly bonded to the matrix where the interphase layer between the fibers and matrix was neglected. The interphase layer should be accounted for, however, because it features changes in gradient properties between the fibers and matrix that are generated via chemical reaction and atom diffusion during the manufacturing process (or created artificially to improve compatibility between the fibers and the matrix).

Until now the effect of the functional gradient interphase layer on the attenuation of elastic waves in composite materials has been meagerly covered in the literatures [15–19]. In order to solve the corresponding single scattering problem in these studies, the inhomogeneous interphase layer was usually divided into multilayers, each with a homogeneous property to approximate a continuously varying layer. The wave field in the fibers, intermediate layers, and matrix was then still expressed as a series of wave functions. The continuity conditions of displacements and stresses at all interfaces were listed, and the resulting coupled linear equations for the expansion coefficients were solved. When the intermediate layers are sufficiently thin, this yielded exact solutions [16]. This treatment is straightforward and can be used to deal with any type of radial gradient profiles, but the coefficient matrix of the resulting linear system is sometimes ill-conditioned. Additionally, for composites with a high contrast of properties between the fibers and matrix, the interphase layer must be divided into a large number of sublayers to obtain convergent results. This increases the computational cost as well as the condition number of the coefficient matrix [20].

The transfer matrix method has also been applied to the gradient interphase problem [19]. In this method the displacements and stresses at the inner surface are related through a transfer matrix **M** to the corresponding values at the outer interface for each intermediate sublayer, the components of which are functions of the material properties and geometries, uniquely defined by the sublayer. By using the transfer matrix for each intermediate sublayer, the displacements and stresses on the outer surface of the interphase layer are related to the inner surface. Assuming the fiber, interphase layer, and matrix are perfectly bonded, a linear system for the expansion coefficients of waves in the fiber and matrix is established and solved. For SH waves, the transfer matrix **M** is of size . Therefore, the final linear system to be solved is of size 2, which is much smaller than that in the method described above. Similarly, the final coefficient matrix of the linear system can be ill-conditioned if the sublayers are thin.

Though several approximate methods to solve the single scattering problem with a radially gradient interphase layer have been proposed, analytical solutions still remain elusive. In this work, analytical solutions to the single scattering with a radially gradient interphase layer of several specific profiles were provided for SH waves. The derivation process followed the work of Martin [21], in which general solutions to the single scattering of acoustic waves by an inhomogeneous sphere with spherically symmetrical properties were investigated. The two transformations used in our derivation process differed from Martin’s in that the governing equations for distinct waves are different. We demonstrated that the proposed method of how to make the transformations is also applicable to other waves. Additionally, the detailed expressions of solutions were presented and a specific example was calculated.

The remainder of this paper was organized as follows: Section 2 derived the governing equations for SH waves in a radially gradient medium. The general solutions for two specific radially gradient materials were then derived in Section 3. Section 4 presented analytical solutions for the single scattering problem of the SH wave by a cylindrical fiber with the two specific interphase layers and a detailed example. Section 5 provided a brief summary and conclusion.

#### 2. Description of the Problem and Governing Equations

In this work, as shown in Figure 1, the single scattering of SH waves by a fiber with a radially gradient interphase layer was considered [17]. Here *a* denotes the radius of the fiber and *b* the outer radius of the interphase layer. The incident wave is a plane SH wave propagating in the positive *x* direction with a unit magnitude. Throughout the work, *a* = 71 *μ*m and *b* = 1.1*a*.