A numerical solution to the dynamic response of a compressible Blatz–Ko sphere with spherically symmetric eigenstrains

Abstract

This paper investigates nonlinear spherical wave propagation in a compressible Blatz–Ko ball with a spherically symmetric inclusion. First, the nonlinear dynamic equilibrium equations governing both the inclusion region and the surrounding matrix are derived using a known geometric approach. The finite difference method together with the Richardson and Newton methods are employed to obtain the nonlinear response. Numerical evaluations are conducted to examine the response under sudden loading and unloading conditions for different inclusion sizes and eigenstrain values. Results indicate that inside the inclusion region, equal eigenstrains lead to a uniform and hydrostatic stress field that is consistent with the static response. Moreover, it is shown that the convergence time of the deformation depends on inclusion size and eigenstrain value for the loading and unloading case studies. Larger inclusions slow convergence when the eigenstrains are less than unity, but quicken it when the eigenstrains are greater than unity.