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

TL;DR

This paper establishes a piecewise Korn inequality for $GSBD^2$ functions in 2D, enabling control of functions' deviations from rigid motions, and applies it to embedding, density, and Korn-Poincaré inequalities.

Contribution

It introduces a novel piecewise Korn inequality for $GSBD^2$ functions, extending elasticity results to functions with jump discontinuities and deriving new embedding and density results.

Findings

$GSBD^2$ functions have bounded variation after subtracting a piecewise rigid motion.

Embedding of $GSBD^2$ into $GBV$ and $SBV$ spaces is established.

A Korn-Poincaré inequality for functions with small jump sets is proved.

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-Poincar\'e inequality for functions with small jump sets in arbitrary space dimension.