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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
27/2006

Piecewise rigidity

Antonin Chambolle, Alessandro Giacomini and Marcello Ponsiglione

Abstract

In this paper we provide a Liouville type theorem in the framework of fracture mechanics, and more precisely in the theory of $SBV$ deformations for cracked bodies.

We prove the following rigidity result: if $u\in SBV(\Omega,\mathbb R^N)$ is a deformation of $\Omega$ whose associated crack $J_u$ has finite energy in the sense of Griffith's theory (i.e., $\mathcal H^{N-1}(J_u)<\infty$), and whose approximate gradient $\nabla u$ is almost everywhere a rotation, then $u$ is a collection of an at most countable family of rigid motions. In other words, the cracked body does not store elastic energy if and only if all its connected components are deformed through rigid motions. In particular, global rigidity can fail only if the crack disconnects the body.

Received:
Mar 14, 2006
Published:
Mar 14, 2006

Related publications

inJournal
2007 Repository Open Access
Antonin Chambolle, Alessandro Giacomini and Marcello Ponsiglione

Piecewise rigidity

In: Journal of functional analysis, 244 (2007) 1, pp. 134-153