Beyond kinetic relations

In this paper we discuss a new concept of a constitutive kinetic equation which represents a quasicontinuum analog of the conventional continuum mechanical concept of a kinetic relation. Kinetic relations link the velocity of the defect with the driving force and are widely used to model dynamical response of dislocations, cracks and phase boundaries. To illustrate the difference between kinetic relations and kinetic equations we consider a prototypical model of an overdamped defect in a one-dimensional lattice where all computations can be done explicitly. We show that in the strongly discrete limit the minimal kinetic equation is in fact a system of two ordinary differential equations involving the coordinate of the defect and another internal parameter describing the configuration of the core region.