Scale-invariant initial value problems with applications to the dynamical theory of stress-induced phase transformations

We deal with the 1D non-stationary wave propagation in an elastic phase-transforming bar. The stress in the bar σ is assumed to be a piecewise linear function of the strain ε containing a “negative slope segment,” thus, the strain energy is a non-convex function of the strain. It is known that the problem of elastostatics for such a material can have solutions with discontinuous deformation gradients. In the framework of the model of stress-induced phase transitions, the surfaces of the strain discontinuity are considered as the phase boundaries, and the domains of continuity are considered as zones occupied by different phases of the material. The solution of both statical and dynamical problems are generally non-unique; therefore, an additional thermodynamic boundary condition at the phase boundary is required. The comparative analysis for two types of problems is under consideration. The first problem is concerned with a new phase nucleation in a phase-transforming bar caused by a collision of two non-stationary waves [1-3]. The second one is a new phase nucleation caused by an impact loading applied at the end of a semi-infinite phase-transforming bar [4-6]. Both of the problem can be formulated as a scale-invariant initial value problem with additional restrictions in the form of several inequalities involving the problem parameters. The aim of the investigation is to determine the domains of existence of the solution in the parameter space.