Properties of the self-similarly expanding Eshelby inclusions and application to the dynamic “Hill Jump Conditions”

Abstract

For a self-similarly subsonically dynamically expanding Eshelby inclusion, we show by an analytic argument (based on the analyticity of the coefficients of the ensuing elliptic system and the Cauchy–Kowalevska theorem) that the particle velocity vanishes in the whole interior domain of the expanding inclusion. Since the acceleration term is thus zero in the interior domain in the Navier equations of elastodynamics, this reduces to an Eshelby problem. The classical Hill jump conditions across the interface of a region with transformation strain are expanded here to dynamics when the interface is moving with inertia satisfying the Hadamard jump conditions. The validity of the Eshelby property and the determination of the constrained strain from the dynamic Eshelby tensor in the interior domain allow one to fully determine from the Hill jump conditions the stress across the moving phase boundary of a self-similarly expanding ellipsoidal Eshelby inhomogeneous inclusion. The driving force can then be obtained. Self-similar motion grasps the early response of the system.