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

Carstensen, Carsten; Rieger, Marc Oliver

doi:10.1051/m2an:2004019

Geodesic

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

Carstensen, Carsten ; Rieger, Marc Oliver

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 38 (2004) no. 3, pp. 397-418

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Résumé

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 $u_{t t} = div S (D u)$ with the non-monotone stress-strain relation $S = D φ$ 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.

MR Zbl

DOI : 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 },
     pages = {397--418},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {3},
     year = {2004},
     doi = {10.1051/m2an:2004019},
     mrnumber = {2075752},
     zbl = {1130.74383},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an:2004019/}
}

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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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