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