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We present a piecewise Korn inequality for generalized special functions ofbounded deformation ($GSBD^2$) in a planar setting generalizing the classicalresult in elasticity theory to the setting of functions with jumpdiscontinuities. We show that for every configuration there is a partition ofthe domain such that on each component of the cracked body the distance of thefunction from an infinitesimal rigid motion can be controlled solely in termsof the linear elastic strain. In particular, the result implies that $GSBD^2$functions have bounded variation after subtraction of a piecewise infinitesimalrigid motion. As an application we prove a density result in $GSBD^2$.Moreover, for all $d \ge 2$ we show $GSBD^2(\Omega) \subset (GBV(\Omega;{\BbbR}))^d$ and the embedding $SBD^2(\Omega) \cap L^\infty(\Omega;{\Bbb R}^d)\hookrightarrow SBV(\Omega;{\Bbb R}^d)$ into the space of special functions ofbounded variation ($SBV$). Finally, we present a Korn-Poincar\'e inequality forfunctions with small jump sets in arbitrary space dimension.

A piecewise Korn inequality in SBD and applications to embedding and density results