Inserted: 26 jun 2015
Last Updated: 26 jun 2015
We present a quantitative geometric rigidity estimate for special functions of bounded deformation in a planar setting generalizing a result by Friesecke, James and Müller for Sobolev functions obtained in nonlinear elasticity theory and a qualitative piecewise rigidity result by Chambolle, Giacomini and 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-Poincare inequality'.
Keywords: free discontinuity problems, Functions of Bounded Deformation, Brittle materials, variational fracture , Geometric rigidity, piecewise rigidity