Microstructures in phase-transitions of alloys are modeled by the energy minimization of a nonconvex energy density . Their time-evolution leads to a nonlinear wave equation with the non-monotone stress-strain relation 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 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 },
pages = {397--418},
year = {2004},
publisher = {EDP-Sciences},
volume = {38},
number = {3},
doi = {10.1051/m2an:2004019},
mrnumber = {2075752},
zbl = {1130.74383},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an:2004019/}
}
TY - JOUR AU - Carstensen, Carsten AU - Rieger, Marc Oliver TI - Young-measure approximations for elastodynamics with non-monotone stress-strain relations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2004 SP - 397 EP - 418 VL - 38 IS - 3 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an:2004019/ DO - 10.1051/m2an:2004019 LA - en ID - M2AN_2004__38_3_397_0 ER -
%0 Journal Article %A Carstensen, Carsten %A Rieger, Marc Oliver %T Young-measure approximations for elastodynamics with non-monotone stress-strain relations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2004 %P 397-418 %V 38 %N 3 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an:2004019/ %R 10.1051/m2an:2004019 %G en %F 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 , Tome 38 (2004) no. 3, pp. 397-418. doi: 10.1051/m2an:2004019
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