N-Phase Elliptical Inhomogeneities With Internal Uniform Stresses in Plane Elasticity

Abstract

We investigate the problem of an N-phase elliptical inhomogeneity in plane elasticity. The elliptical inhomogeneity is bonded to the unbounded matrix through the intermediate (N−2) interphases, and the matrix is subjected to remote uniform stresses. We observe that the stress field inside the elliptical inhomogeneity is still uniform when the following two conditions are satisfied: (i) The formed interfaces are (N−1) confocal ellipses, and (ii) the interphases and the matrix possess the same shear modulus but different Poisson’s ratios. In Appendixes , we also discuss an arbitrary number of interacting arbitrary shaped inhomogeneities embedded in an infinite matrix, and an N-phase inhomogeneity with (N−1) interfaces of arbitrary shape. Here all the phases comprising the composite possess the same shear modulus but different Poisson’s ratios. The results in the main body and in Appendixes are further extended in Appendix to finite plane strain deformations of compressible hyperelastic harmonic materials.