Advances in Materials Science and Engineering

Volume 2015, Article ID 134975, 8 pages

http://dx.doi.org/10.1155/2015/134975

## Surface Effect on Diffractions of Elastic Waves and Stress Concentration near a Cluster of Cylindrical Nanoholes Arranged as Quadrate Shape

Department of Engineering Mechanics, Xi’an University of Technology, Xi’an 710048, China

Received 18 September 2014; Accepted 27 November 2014

Academic Editor: Xing Chen

Copyright © 2015 Ru Yan. 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

We consider the multiple scattering of elastic waves (P-wave and SV-wave) by a cluster of nanosized cylindrical holes arranged as quadrate shape. When the radius of the holes shrinks to nanometers, the surface elasticity theory is adopted in analysis. Using the displacement potential method and wave functions expansion method, we obtain that the multiple scattering fields induced by incident P- and SV-waves around the holes are derived. The dynamic stress concentration around the holes is calculated to illustrate the effect of surface effects on the multiple scattering of P- and SV-waves.

#### 1. Introduction

The diffraction of elastic waves by a single inhomogeneity embedded in an elastic medium was discussed in detail by Pao and Mow [1]. With the development of composite materials, there is an increasing demand for understanding the dynamic behavior of composite materials, and much attention has been directed toward the multiple scattering of elastic waves. Fang et al. obtained the multiple diffraction fields by two-particle reinforced composite system [2]. Twersky investigated the scattering of acoustic or electromagnetic wave by an arbitrary configuration of parallel cylinders [3]. Lakhtakia et al. [4] observed the reflection and transmission of incident waves by an array of circular cylinders in an elastic slab. Wang and Sudak obtained the scattering field of elastic waves by a cluster of circular cylinders with imperfect interfaces [5].

Nanomaterials have different physical, optical, and mechanical properties distinct from their macroscopic counterparts [6]. At nanoscale, surface has significant effects on the physical and mechanical properties of solids, due to the increasing ratio of surface area to volume [7, 8]. To account for the surface effects, Gurtin et al. [9] developed a continuum model of surface elasticity. Based on the surface elasticity theory, Wang et al. [10] analyzed the diffraction of plane compressional wave (P-wave) by a nanosized circular hole. Ru et al. [11] obtained the scattering field of P- and SV-waves by a nanosized inhomogeneity. Wang [12] and Zhang et al. [13] analyze the diffraction of elastic waves by a pair and an array of nanosized inhomogeneities. Recently, Ru et al. discussed the surface effect on the scattering field induced by nanosized cylindrical holes [14]. Those investigations illustrated the importance of surface effects on the diffraction of elastic waves. In this paper, we discussed the multiple scattering of plane P- and SV-waves by a cluster of nanosized cylindrical holes arranged as quadrate shape.

#### 2. Basic Equations

At nanoscale, we consider the problem in the framework of surface elasticity theory because of the surface effect [9]. According to the surface elasticity theory, a surface is regarded as a negligibly thin membrane adhered to the bulk without slipping and possesses material constants different from the bulk material. On the surface, the surface stress leads to a set of nonclassical boundary conditions. In the bulk, however, the classical theory of elasticity is still applicable.

The surface stress tensor is related to the surface energy density as [9]where is the Kronecker delta and is the second-rank tensor of surface strain. In this paper, Einstein’s summation convention is adopted for all repeated Latin indices (1, 2, 3) and Greek indices (1, 2).

Without residual surface tension, for an isotropic surface, the relationship between the surface stresses and the surface strains is [9]where and are two material constants on surface.

Assume that the surface adheres perfectly to the bulk material without slipping, and then the equilibrium equations on the surface are [15]where is the tangential component of the traction in the -direction, is the normal vector of the surface, is the curvature of the surface, and is stress tensor of the surface. Generally, the surface inertia force can be neglected for dynamic problems.

In the bulk solid, the equilibrium and constitutive equations are the same as those in the classical theory of elasticity:where is the mass density of the material, is the time, and are shear modulus and Poisson’s ratio, respectively, and and are stress tensor and strain tensor in the bulk material, respectively.

The strain tensor is related to the displacement vector by

Based on surface elasticity theory, we derive the solutions for elastic fields near a cluster of cylindrical nanoholes arranged as quadrate shape induced by incident P-wave and SV-wave, respectively.

#### 3. Diffraction of Elastic Waves by Cylindrical Nanoholes

We consider the diffraction of elastic waves by a cluster of ( approaching infinity) identical cylindrical holes with radius of in an infinite elastic matrix, as shown in Figure 1. The holes are arranged as quadrate shape and the distance between the centers and adjacent holes is , as shown in Figure 2. The global polar coordinate system is set up at the center of the middle hole. For convenience, at the center of th hole (), the local polar coordinate system is set up. The plane strain condition () is assumed; thus, .