#### Vol. 3, No. 1, 2015

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On the approximation theorem for structured deformations from $\operatorname{\mathit{BV}}(\Omega)$

### Miroslav Šilhavý

Vol. 3 (2015), No. 1, 83–100
##### Abstract

This note deals with structured deformations introduced by Del Piero and Owen. As treated in the present paper, a structured deformation is a pair $\left(g,G\right)$ where $g$ is a macroscopic deformation giving the position of points of the body and $G$ represents deformations without disarrangements. Here $g$ is a map of bounded variation on the reference region $\Omega$, and $G$ is a Lebesgue-integrable tensor-valued map. For structured deformations of this level of generality, an approximating sequence ${g}_{k}$ of simple deformations is constructed from the space of maps of special bounded variation on $\Omega$, which converges in the ${L}^{\mathfrak{1}}\left(\Omega \right)$ sense to $\left(g,G\right)$ and for which the sequence of total variations of ${g}_{k}$ is bounded. The condition is optimal. Further, in the second part of this note, the limit relation of Del Piero and Owen is established on the above level of generality. This relation allows one to reconstruct the disarrangement tensor $M$ of the structured deformation $\left(g,G\right)$ from the information on the approximating sequence.

##### Keywords
structured deformation, fracture, approximations, maps of bounded variation, maps of special bounded variation
##### Physics and Astronomy Classification Scheme 2010
Primary: 81.40.Lm
Primary: 74R99
Secondary: 74A05