Young-measure approximations for elastodynamics with non-monotone stress-strain relations

Carstensen, Carsten; Rieger, Marc Oliver

ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 3, pp. 397-418.

Microstructures in phase-transitions of alloys are modeled by the energy minimization of a nonconvex energy density $\phi$. Their time-evolution leads to a nonlinear wave equation $u_{tt}=divS\left(Du\right)$ with the non-monotone stress-strain relation $S=D\phi$ 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.

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