Plane deformations of a composite with interacting harmonic inclusions in the presence of non-uniform loading

Abstract

We consider plane deformations of a composite material containing two interacting elastic inclusions of irregular shape which do not disturb the first invariant of an applied non-uniform stress field at infinity (so-called ‘harmonic inclusions’). Our analysis is based on a conformal map which maps the matrix region outside the two inclusions onto an annulus in the mapped plane. The unknown coefficients in the mapping function are uniquely determined via the continuity conditions of traction and displacement across the two inclusion-matrix interfaces. In our design, the two inclusions may have different and arbitrary elastic constants and the hoop stress remains constant along each of the two inclusion-matrix interfaces from the matrix side. The results are illustrated graphically. The set of permissible parameters leading to a one-to-one mapping is also discussed.