YOUNG-MEASURE APPROXIMATIONS FOR ELASTODYNAMICS WITH NON-MONOTONE STRESS-STRAIN RELATIONS

Abstract. Microstructures in phase-transitions of alloys are mod-eled by the energy minimization of a nonconvex energy density φ. Their time-evolution leads to a nonlinear wave equation utt = div S(Du) with the non-monotone stress-strain relation S = Dφ plus proper boundary and initial conditions. This hyperbolic-elliptic initial-boundary value problem of changing types allows, in general, solely Young measure solutions. This paper introduces a fully-numerical time-space discretization of this equation in a corresponding very weak sense. It is shown that discrete solutions exist and generate weakly convergent subsequences whose limit is a Young measure solution. Numerical examples in one space dimension illustrate the time-evolving phase transitions and mi-crostructures of a nonlinearly vibrating string. 1