Esaim: Mathematical Modelling and Numerical Analysis Young-measure Approximations for Elastodynamics with Non-monotone Stress-strain Relations

Microstructures in phase-transitions of alloys are modeled by the energy minimization of a nonconvex energy density φ. Their time-evolution leads to a nonlinear wave equation utt =d ivS(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 microstructures of a nonlinearly vibrating string. Mathematics Subject Classification. 35G25, 47J35, 65P25.

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