A quantitative geometric rigidity result in SBD

Abstract: We present a quantitative geometric rigidity estimate for special functions of bounded deformation in a planar setting generalizing the result by Friesecke, James, M\"uller obtained in nonlinear elasticity theory and the piecewise rigidity result by Chambolle, Giacomini, Ponsiglione for brittle materials which do not store elastic energy. We show that for each deformation there is an associated triple consisting of a partition of the domain, a corresponding piecewise rigid motion being constant on each connected component of the cracked body and a displacement field measuring the distance of the deformation from the piecewise rigid motion. We also present a related estimate in the geometrically linear setting which can be interpreted as a `piecewise Korn-Poincar\'e inequality'.