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

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 A x B, 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 A x B are further extended in Appendix C to finite plane strain deformations of compressible hyperelastic harmonic materials.