The self-similarly expanding Eshelby ellipsoidal inclusion: II. The Dynamic Eshelby Tensor for the expanding sphere

Abstract

The field solution of a self-similarly (subsonically) expanding Eshelby ellipsoidal inclusion obtained in Part I is evaluated for the case of the expanding spherical inclusion under general uniform eigenstrain ϵij⁎ in self-similar motion R=υt, starting from zero dimension. The particle velocity in the interior domain vanishes and the displacement gradient is constant exhibiting the Eshelby property in the self-similar dynamic case. All components of the interior and exterior Dynamic Eshelby Tensor are obtained for the sphere, with the interior ones depending on the wave speeds and the expansion speed of the inclusion, while the exterior ones depend, in addition, on the variable of self-similarity r/t and the direction of the field point. By a limiting procedure the static Eshelby tensor both interior and exterior is retrieved, thus making the static inclusion a special limit of the dynamic self-similarly expanding one. The jump of the particle velocity across the moving inclusion boundary is obtained, and it depends only on the wave and expansion speeds and the direction of the normal.