The Application of the Boundary Element Method to the Problem of Wave Diffraction from a Diamond Shaped Inclusion

We investigate the scattering of in-plane compressional (P) and shear (SV) waves by diamond shaped inclusions using a boundary element method. Special case that the shape of the diamond becomes square is also considered. Numerical results for some diamond shapes of the inclusion are obtained, and the effects of frequency on the scattering cross sections are discussed in detail. The response simplifies for the limited cases of a hole and a rigid inclusion, but with quite different behavior for each. Calculations for a diamond shaped SiC-fiber-reinforced Al composite are also carried out, and the results are shown graphically. INTRODUCTION Elasticity solutions involving the static and dynamic behavior of inclusions embedded in dissimilar materials have received a good deal of attention, partly because of the growing applications for composites. The problem of a single polygonal inclusion in an elastic matrix has been examined by several authors. Chen [1] studied the singular stress fields created by an antiplane deformation at a diamond inclusion corner. Reedy and Guess [2] considered a rigid square inclusion embedded within an elastic disk, and discussed the stress state generated by the inclusion. Recently, Pan and Jiang [3] investigated the stress singularity at vertices of regular polygonal inclusions with various numbers of sides, and revealed the effects of material orientation and corner angle at the vertex on the elastic singularity. The evaluation of elastic waves propagating in composites is fundamental to the investigation of microstructure. This plays a significant role in design, development, processing and in-service inspection of the composites. The scattering of elastic waves by inclusions has been studied by several authors [4, 5]. In this paper, we study the effect of inclusion shape on the scattering of time-harmonic in-plane compressional (P) and shear (SV) waves by a diamond shaped inclusions. Under the plane strain assumption, the solutions are obtained using the boundary element method. Numerical results are given as a function of frequency, and the effect of cross sectional shape of the diamond shaped inclusion on the scattering cross sections are discussed in detail. Numerical calculations are also examined for a diamond shaped SiC-fiberreinforced Al composite. *Address correspondence to this author at the Department of Materials Processing, Graduate School of Engineering, Tohoku University, Aobayama 6-6-02, Sendai 980-8579, Japan; Tel/Fax: +81-22-795-7341; E-mail: [email protected] PROBLEM STATEMENT AND SCATTERED FAR FIELD Consider a diamond shaped inclusion inscribed into a circle of radius a embedded in an infinite matrix. The Cartesian coordinate system ( 1 2 3 x x x , , ) with origin at center of the circle will be used. Let μ , , , be the Lam é constants, mass density, Poisson’s ratio of the matrix, and 0 0 0 0 μ , , , those of the inclusions. The geometry is depicted in Fig. (1), where is the angle defining the orientation of inclusions, is the corner angle, is the surface of inclusions, and D and C are the domains outside and inside inclusions, respectively. Fig. (1). Diamond shaped inclusion and coordinate systems. The components in the 1 x and 2 x directions of the displacement vector u are 1 u and 2 u , while the component 3 u is absent because the problem is plane strain. For the same reason, derivatives with respect to 3 x are zero. The time factor exp( ) i t , which is common to all field variThe Application of the Boundary Element Method to the Problem The Open Mechanics Journal, 2008, Volume 2 63 ables in a steady-state regime, is omitted throughout this paper; is the circular frequency and t is the time. The stress equations of motion are given by 2 0 ji j i u , + = (1) where a comma followed by an index denotes partial differentiation with respect to the space coordinate i x , indices can assume the values 1 and 2 only, and ij define the components of stress tensor. We have introduced the summation convention for repeated tensor indices. The stress components are related to the displacement gradients by Hooke’s law ( ) ij ij k k i j j i u u u μ , , , = + + (2)