Transition Matrix Method for Determining Stress Distribution of an Inhomogeneity in Elastostatic Medium

The transition matrix method has been extensively utilized to solve scattering in elastodynamic media. It is based on the reciprocity theorem, continuity of the interface boundary conditions, and applicable to arbitrary shape of inhomogeneity in systematic matrix multiplication. However, the transition matrix method has never been applied to determine stress distribution in elastostatic media. One important reason is the problem of the shortage of the basis functions of the elastostatic media that must be used to develop the transition matrix. This study investigates the required basis functions, and finds a set thereof that include Love’s special solutions of three dimensional elastostatics and three vector functions that are applicable to elastic waves. The proposed basis functions also can be adopted to derive the three significant orthogonality conditions for reciprocity at the surface of the inhomogeneity, which are useful in developing the transition matrix. The novel basis functions make the process of derivation of the T-matrix in elastostatics similar to that in elastodynamics. This process is illustrated for a spherical inhomogeneity that embedded in an elastic medium and stress patterns are compared with Goodier’s solutions, demonstrating high accuracy.