Null-Field Approach for the Multi-inclusion Problem Under Antiplane Shears

Abstract

In this paper, we derive the null-field integral equation for an infinite medium containing circular holes and/or inclusions with arbitrary radii and positions under the remote antiplane shear. To fully capture the circular geometries, separable expressions of fundamental solutions in the polar coordinate for field and source points and Fourier series for boundary densities are adopted to ensure the exponential convergence. By moving the null-field point to the boundary, singular and hypersingular integrals are transformed to series sums after introducing the concept of degenerate kernels. Not only the singularity but also the sense of principle values are novelly avoided. For the calculation of boundary stress, the Hadamard principal value for hypersingularity is not required and can be easily calculated by using series sums. Besides, the boundary-layer effect is eliminated owing to the introduction of degenerate kernels. The solution is formulated in a manner of semi-analytical form since error purely attributes to the truncation of Fourier series. The method is basically a numerical method, and because of its semi-analytical nature, it possesses certain advantages over the conventional boundary element method. The exact solution for a single inclusion is derived using the present formulation and matches well with the Honein et al.’s solution by using the complex-variable formulation (Honein, E., Honein, T., and Hermann, G., 1992, Appl. Math., 50, pp. 479–499). Several problems of two holes, two inclusions, one cavity surrounded by two inclusions and three inclusions are revisited to demonstrate the validity of our method. The convergence test and boundary-layer effect are also addressed. The proposed formulation can be generalized to multiple circular inclusions and cavities in a straightforward way without any difficulty.