Quasistatic damage evolution with spatial $\mathrm{BV}$-regularization

Marita Thomas

doi:10.3934/dcdss.2013.6.235

Article Contents

2013, Volume 6, Issue 1: 235-255. Doi: 10.3934/dcdss.2013.6.235

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Quasistatic damage evolution with spatial $\mathrm{BV}$-regularization

Marita Thomas^1,

1.
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin

Received: April 30, 2011

Revised: June 30, 2011

Published: January 2013

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Abstract

An existence result for energetic solutions of rate-independent damage processes is established. We consider a body consisting of a physically linearly elastic material undergoing infinitesimally small deformations and partial damage. In [23] an existence result in the small strain setting was obtained under the assumption that the damage variable $z$ satisfies $z\in W^{1,r}(\Omega)$ with $r\in(1,\infty)$ for $\Omega⊂ \mathbb{R}^d.$ We now cover the case $r=1$. The lack of compactness in $W^{1,1}(\Omega)$ requires to do the analysis in $\mathrm{BV}(\Omega)$. This setting allows it to consider damage variables with values in {0,1}. We show that such a brittle damage model is obtained as the $\Gamma$-limit of functionals of Modica-Mortola type.

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Mathematics Subject Classification: Primary: 74C05, 74R05, 49J45; Secondary: 49S05, 74R20.

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