A new method for an inhomogeneity with stepwise graded interphase under thermomechanical loadings

The solution for a circular inhomogeneity embedded in an infinite elastic m atrix with a multilayered interphase plays a fundamental role in many pract ical and theoretical problems. Therefore, improved analysis methods for thi s problem are of great interest. In this paper, a new procedure is presente d to obtain the exact stress fields within the inhomogeneity and the matrix under thermomechanical loadings, without the need of solving the full mult iphase composite problem. With this short-cut method, the problem is reduce d to a single linear algebraic equation and two coupled linear algebraic eq uations which determine the only three real coefficients of the stress fiel d within the inhomogeneity. In particular, the average stresses within the inhomogeneity can be calculated directly from the three real coefficients. Further, the other three unknown real coefficients associated with the stre ss field in the matrix can be determined subsequently. Hence, the influence of the stepwise graded interphase on the stress fields is manifested by it s effect on the six real coefficients. All these results hold for stepwise graded interphase composed of any number of interphase layers. Several exam ples serve to illustrate the method and its advantages over other existing approaches. The explicit solutions are used to study the design of harmonic elastic inclusions, and the effect of a compliant interphase layer on ther mal-mismatch induced residual stresses.