The expanding spherical inhomogeneity with transformation strain

Within the context of linear elastodynamics, the radiated fields (including inertia) for a spherical inhomogeneous (of different elastic constants) inclusion with dilatational transformation strain (or eigenstrain), expanding in general motion under applied loading, can been obtained on the basis of Eshelby’s equivalent inclusion method by using the strain field of the expanding homogeneous spherical inclusion (as a function of the eigenstrain) to determine the equivalent eigenstrain. With the equivalent dynamic eigenstrain (which is dependent on the boundary motion), the radiated fields for the inhomogeneous spherical expanding inclusion can be obtained. Based on them, the “driving force” (self-force) on the moving boundary can be computed, and this is the rate of mechanical work (with inertia) required to create an incremental region of inhomogeneity with eigenstrain, i.e with the elastic constants changing as the region of the eigenstrain expands. The self-force depends on the history of the motion, and, in the presence of external loading the driving force yields a Peach-Koehler type force, which exhibits coupling of the applied loading to the history of the motion of the boundary velocity.