A three-phase circular inhomogeneity with imperfect interface under thermomechanical loadings in plane elasticity

Summary The solution for a homogeneous circular inhomogeneity embedded in an infinite elastic matrix with a single interphase layer plays a fundamental role in many practical and theoretical applications. In particular, it serves as the basis for the solution of the generalized self-consistent method in the mechanics of composite materials. Thus, the study of three-phase problems is of great interest.A general method is presented for the rigorous solution of a three-phase circular inhomogeneity under thermomechanical loadings in plane elasticity. The bonding at the inhomogeneity-interphase interface is considered to be inperfect with the assumption that the interface imperfections are constant. On the remaining boundary, that being the interphase-matrix interface, the bonding is considered to be perfect. Although the problem of a three-phase circular inhomogeneity with imperfect bonding has previously been studied, it seems that the explicit expressions for the complete solutions cannot be located in the literature. In this paper, it is found that stress field within the inhomogeneity is determined by three, in general, complex coefficients while the stress field in the matrix is controlled by three other, in general, complex coefficients. The role of the interphase layer as well as the influence of the imperfect bonding condition, on the stress fields, is manifested by their effect on the six, in general, complex coefficients.The exact closed-form solutions are applied to the design of a three-phase circular inhomogeneity. In particular, for specific thermomechanical loadings, it is shown that a uniform stress state within the inhomogeneity can be achieved with the imperfect interface model provided the imperfect interface parameters are suitably chosen.