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

Published online by Cambridge University Press:  15 June 2004

Carsten Carstensen
Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany.
Marc Oliver Rieger
Scuola Normale Superiore, Piazza dei Cavalieri 7, 56100 Pisa, Italy.
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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 $u_{tt}=\mbox{div}\:S(Du)$ 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.

Research Article
© EDP Sciences, SMAI, 2004

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