An Eshelby inclusion of arbitrary shape in a degenerate orthotropic elastic plane

Abstract

Using Suo's complex variable formulation, we first derive a general solution to the plane problem of an infinite homogeneous degenerate orthotropic elastic plane containing an Eshelby inclusion of arbitrary shape undergoing uniform in-plane eigenstrains. The elastic field within the Eshelby inclusion is identified once the two polynomials representing the principal parts of the remote asymptotic behaviors of two auxiliary functions are determined. We next derive an explicit solution to the problem of an inclusion having an (n+1)-fold axis of quasi-symmetry (with n ≥ 1) in an infinite degenerate orthotropic elastic material. The inclusion boundary has an (n+1)-fold axis of symmetry in the z-plane, where z is the single complex variable appearing in Suo's formulation, and is described by a four-term mapping function. The non-uniform distributions of the total strains and rigid body rotation within the quasi-symmetric inclusion are completely determined. We further prove that when n ≥ 2, $n\ne 3$ the arithmetic mean of the Eshelby tensors at n+1 rotational symmetric points within the inclusion in the z-plane is equal to the constant Eshelby tensor within a special elliptical inclusion, the boundary of which is circular in the z-plane, and that it is independent of the rotation of the inclusion boundary in the z-plane.