SOLUTION FOR A CRACK EMBEDDED IN MULTIPLY CONFOCALLY ELLIPTICAL LAYERS IN ANTIPLANE ELASTICITY

This paper provides a general solution for a crack embedded in multiply confocally elliptical layers in antiplane elasticity. In the problem, the elastic medium is composed of an inclusion, many confocally elliptical layers and the infinite matrix with different elastic properties. In addition, the remote loading is applied at infinity. The complex variable method and the conformal mapping technique are used. On the mapping plane, the complex potentials for the inclusion and many layers are assumed in a particular form with two undetermined coefficients. The continuity conditions for the displacement and traction along the interface between two adjacent layers are formulated and studied. By enforcing those conditions along the interface, the exact relation between two sets of two undetermined coefficients in the complex potentials for j-th layer and j + 1-th layer can be evaluated. From the traction free condition along the crack faces, the correct form of the complex potential for the cracked inclusion is obtained. Finally, many numerical results are provided.