Piecewise rigidity

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 a $SBV$ deformation of $\text{Om}$ whose associated crack $J_u$ has finite energy in the sense of Griffith's theory (i.e., $H^{N-1}(J_u)<\mathrm{\infty }$), and whose approximate gradient $\mathrm{\nabla }u$ 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.