Calculus of Variations and Geometric Measure Theory

A. Chambolle - A. Giacomini - M. Ponsiglione

Piecewise rigidity

created by ponsiglio on 22 Feb 2006
modified by giacomini on 25 Sep 2008


Published Paper

Inserted: 22 feb 2006
Last Updated: 25 sep 2008

Journal: J. Funct. Anal.
Volume: 244
Pages: 134-153
Year: 2007


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,RN)$ is a deformation of $A$ whose associated crack $Ju$ 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.$$