Null-field approach for the antiplane problem with elliptical holes and/or inclusions

Abstract

In this paper, we extend the successful experience of solving an infinite medium containing circular holes and/or inclusions subject to remote shears to deal with the problem containing elliptical holes and/or inclusions. Arbitrary location, different orientation, various size and any number of elliptical holes and/or inclusions can be considered. By fully employing the elliptical geometry, fundamental solutions were expanded into the degenerate kernel by using an addition theorem in terms of the elliptic coordinates and boundary densities are described by using the eigenfunction expansion. The difference between the proposed method and the conventional boundary integral equation method is that the location point can be exactly distributed on the real boundary without facing the singular integral and calculating principal value. Besides, the boundary stress can be easily calculated free of the Hadamard principal values. It is worthy of noting that the Jacobian terms exist in the degenerate kernel, boundary density and contour integral; however, these Jacobian terms would cancel each other out and the orthogonal property is preserved in the process of contour integral. This method belongs to one kind of meshless methods since only collocation points on the real boundary are required. In addition, the solution is regarded as semi-analytical form because error purely attributes to the number of truncation term of eigenfunction. An exact solution for a single elliptical inclusion is also derived by using the proposed approach and the results agree well with Smith’s solutions by using the method of complex variables. Several examples are revisited to demonstrate the validity of our method.