Moving singularities in thermoelastic solids

The solution of the evolution problem of a discontinuity requires the formulation of a kinetic law of the progress relating the driving force and the velocity of the singularity. In the case of a crack, the energy-release rate can be computed (in quasi-statics and in the absence of thermal and intrinsic dissipations) by means of the celebrated J-integral of fracture that is known to be path-independent and, therefore, provides a very convenient estimation of the driving force once the field solution is known. However, the velocity at the crack tip remains undetermined. A similar situation holds for a displacive phase-transition front propagation. The driving force acting on the phase boundary can be determined, but not the velocity of the displacive phase-transition front. From the thermodynamic point of view, both the phase transition and the crack propagation are non-equilibrium processes; entropy is produced at the evolving discontinuity. Therefore, stress jumps are determined by means of non-equilibrium jump relations at the discontinuity. Then the kinetic relations can be obtained depending on the choice of excess stress behavior. The procedure is illustrated on the example of a phase-transition front propagation in a shape-memory alloy bar.