Classical continuum models close the motion of a defect - a phase boundary, a twin boundary, a dislocation - with a kinetic RELATION: an instantaneous algebraic law giving the defect velocity as a function of the configurational (driving) force acting on it at that instant. This concept replaces that closure with a kinetic EQUATION: a differential relation in which the velocity obeys its own evolution law, so the present velocity depends on the history of the driving force rather than on its current value alone. The motivation is a discrete overdamped bistable chain - a lattice of masses with a double-well nearest-neighbour interaction plus linear next-nearest-neighbour coupling - whose exact travelling-wave solutions show that no single-valued velocity-versus-force function reproduces the lattice response once the drive varies in time. Reducing the chain to the few atoms that are actually switching (the active points) yields a small system of ordinary differential equations for the velocity: a first-order (K=1) reduction already captures rate effects and the pinning threshold, while a second-order (K=2) reduction reproduces the non-monotone, memory-carrying response. The result is a way to carry discrete lattice information - the Peierls-Nabarro landscape, the depinning threshold, the overshoot at fast drive - into a continuum-level constitutive law without abandoning the continuum description.
A kinetic relation is the algebraic closure that continuum mechanics uses to make the motion of a defect - a phase boundary, a dislocation, a crack - well posed: it assigns a velocity to each instant…