13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- To the authors’ knowledge, Noda and Matsuo [5] have used the Cauchy-type singular integral equations to solve an interaction problem of elliptical inclusions distributed in an infinite medium under a longitudinal shear loading. They discussed different outlet of two elliptical inclusions as well as different ratios of shear moduli. Later, Lee and Kim [6] also revisited the problem of Noda and Matsuo by using the volume integral equation method. Lee and Chen [7] also successfully used the null-field boundary integral equation in conjunction with degenerate kernels to solve the problem. Besides, we don’t find other works to discuss on this issue containing more than two inclusions. For the piezoelectricity problems with circular inclusions, many researchers [8-12] made much contribution on this issue. However, for containing elliptical inhomogenieties, Meguid and Zhong [13] used the complex-variable method to study the problem of a piezoelectric elliptical inhomogeneity. They derived the analytical solution in their works. Pak [14] used the conformal mapping technique to obtain a closed-form solution. The previous works were very similar. The main difference is that Meguid and Zhong provided a general series solution, but Pak derived an explicit closed-form solution. Besides, numerous researchers have successfully solved similar problems with an elliptical inclusion. However, to the authors’ best knowledge, we don’t find any work on dealing with anti-plane piezoelectric problems containing two or more than two elliptical inclusions in the literature. This is our main concern. In this paper, we extend the successful experience of solving piezoelectricity with circular inclusions to deal with the problem containing elliptical holes and/or inclusions. By fully employing the elliptical geometry, fundamental solutions were expanded into the degenerate kernel by using an addition theorem in terms of the elliptical coordinates, and boundary densities are approximated by the eigenfunction expansion. The proposed approach can be seen as one kind of meshless and semi-analytical methods because only collocation points on the real boundary are required and the error purely attributes to the number of truncation terms. In order to verify the accuracy for solving two or more than two elliptical inclusions, the available result of two circular-inclusion case is used to compare with the solution of present approach by numerically approaching the length of the major axis to be equal to the minor axis. 2. Problem statement and formulation 2.1. Problem statement The problem to be considered here is an infinite piezoelectric medium with multiple elliptical inclusions under the remote anti-plane shears ( zx and zy ) and the in-plane far-field electric field Figure 1 Sketch of the problem
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