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

created by friedrich on 06 May 2016

[BibTeX]

Preprint

Inserted: 6 may 2016
Last Updated: 6 may 2016

Year: 2016

Abstract:

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-Poincaré inequality for functions with small jump sets in arbitrary space dimension.