*Published Paper*

**Inserted:** 22 feb 2006

**Last Updated:** 25 sep 2008

**Journal:** J. Funct. Anal.

**Volume:** 244

**Pages:** 134-153

**Year:** 2007

**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 $u$ is in $SBV$ (A,R^{N)$ is a deformation of $A$ whose associated crack $J}_{u$ has finite energy in the sense of
Griffith theory (i.e., with finite ($N}-1$)-area), 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.$$

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