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

Carsten Carstensen; Marc Oliver Rieger

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 38, Issue: 3, page 397-418
  • ISSN: 0764-583X

Abstract

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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 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.

How to cite

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Carstensen, Carsten, and Rieger, Marc Oliver. "Young-Measure approximations for elastodynamics with non-monotone stress-strain relations." ESAIM: Mathematical Modelling and Numerical Analysis 38.3 (2010): 397-418. <http://eudml.org/doc/194220>.

@article{Carstensen2010,
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. },
author = {Carstensen, Carsten, Rieger, Marc Oliver},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Non-monotone evolution; nonlinear elastodynamics; Young-measure approximation; nonlinear wave equation.},
language = {eng},
month = {3},
number = {3},
pages = {397-418},
publisher = {EDP Sciences},
title = {Young-Measure approximations for elastodynamics with non-monotone stress-strain relations},
url = {http://eudml.org/doc/194220},
volume = {38},
year = {2010},
}

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
DA - 2010/3//
PB - EDP Sciences
VL - 38
IS - 3
SP - 397
EP - 418
AB - 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.
LA - eng
KW - Non-monotone evolution; nonlinear elastodynamics; Young-measure approximation; nonlinear wave equation.
UR - http://eudml.org/doc/194220
ER -

References

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