# 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

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topCarstensen, 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 -

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