On the dynamics of moving singularities in solids under the use of the level-set method and the configurational force concept

In this paper we examine the motion of a moving singularity within a continuum body under the use of the level-set method and the material force concept. The level set method introduces an implicit representation of a singular surface where it transforms into a thin transition layer of non-zero thickness where all quantities take inhomogeneous expressions within the body. The existence of an inhomogeneous energy of the material predicts inhomogeneity forces which drive the singularity. The driving force is in fact a material force entering the canonical momentum equation (pseudo-momentum) in a natural way. The kinetic relation is produced by invoking relations that can be considered as the regularized versions of the Rankine–Hugoniot jump conditions combined with non-equilibrium jump relations at the singularity. The effectiveness of the method is illustrated by the use of two examples: (1) propagating phase-transition front that corresponds to a stress-induced martensitic transformation in an elastic bar and (2) propagation of a straight brittle crack.