$N$-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 xB, 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 xB are further extended in Appendix  C to finite plane strain deformations of compressible hyperelastic harmonic materials.