Uniform fields inside two non-elliptical inclusions

The problem of two non-elliptical inclusions with internal uniform fields embedded in an infinite matrix, subjected at infinity to a uniform stress field, is discussed in detail by means of the conformal mapping technique. The introduced conformal mapping function can map the matrix region (excluding the two inclusions) onto an annulus. The problem is completely solved for anti-plane isotropic elasticity, anti-plane piezoelectricity, anti-plane anisotropic elasticity, plane elasticity and finite plane elasticity. The correctness of the solution is verified by comparison with existing solutions and by checking an extreme situation. Our results indicate that it is permissible for the two inclusions to have different material properties and different shapes. Finally, two interesting applications of the obtained results are given, and we find that when the two inclusions have different material properties, the elastic polarization tensor associated with the two non-elliptical inclusions does not lie on the lower Hashin-Shtrikman bound.