Young-measure approximations for elastodynamics with non-monotone stress-strain relations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 3, p. 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.

DOI : https://doi.org/10.1051/m2an:2004019
Classification:  35G25,  47J35,  65P25
Keywords: non-monotone evolution, nonlinear elastodynamics, Young-measure approximation, nonlinear wave equation
@article{M2AN_2004__38_3_397_0,
author = {Carstensen, Carsten and Rieger, Marc Oliver},
title = {Young-measure approximations for elastodynamics with non-monotone stress-strain relations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {38},
number = {3},
year = {2004},
pages = {397-418},
doi = {10.1051/m2an:2004019},
zbl = {1130.74383},
mrnumber = {2075752},
language = {en},
url = {http://www.numdam.org/item/M2AN_2004__38_3_397_0}
}

Carstensen, Carsten; Rieger, Marc Oliver. Young-measure approximations for elastodynamics with non-monotone stress-strain relations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 3, pp. 397-418. doi : 10.1051/m2an:2004019. http://www.numdam.org/item/M2AN_2004__38_3_397_0/

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