An Equal Area Rule for Dissipative Kinetics of Propagating Strain Discontinuities

Abstract

Travelling phase boundaries and shock waves in elastic bars with fairly general nonlinear stress-strain relations are investigated. The stress can be either a nonmonotone function of strain with a local maximum followed by a local minimum, or it may be monotone, but its curvature is always assumed to change sign. The effects of adding linear strain gradient but nonlinear strain rate effects to the stress are considered, with emphasis on the kinetics of travelling strain discontinuities. For a special nonlinearity in the viscosity this leads to an explicit additional jump condition for phase boundaries, valid for a general class of stress-strain laws. This is interpreted as a generalized equal area rule for dynamical phase transition, which reduces to the Maxwell rule in the static case. As a consequence, the thermodynamic force acting on phase boundaries (driving traction) is delivered through a kinetic relation as a function of their speed. Shock waves are restricted by physically meaningful inequalities.