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Young-Measure approximations for elastodynamicswith non-monotone stress-strain relations

Published online by Cambridge University Press:  15 June 2004

Carsten Carstensen and
Marc Oliver Rieger
Carsten Carstensen
Affiliation:
Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany. cc@math.hu-berlin.de.
Marc Oliver Rieger
Affiliation:
Scuola Normale Superiore, Piazza dei Cavalieri 7, 56100 Pisa, Italy. rieger@sns.it.
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Abstract

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.

Keywords

Non-monotone evolutionnonlinear elastodynamicsYoung-measure approximationnonlinear wave equation.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 3 , May 2004, pp. 397 - 418
Copyright
© EDP Sciences, SMAI, 2004

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