Density in $SBD$ and approximation of fracture energies

Abstract

We prove three density theorems, in the strong $BD$ topology, for the three subspaces of $SBD$ functions: $SBD$; $SB{D}_{\infty }^p$, where the absolutely continuous part of the symmetric gradient is in $L^p$, with $p>1$; $SB{D}^p$, whose functions are in $SB{D}_{\infty }^p$ and the jump set has finite ${\mathcal{H}}^{n-1}$-measure. We compare them with existing results, discussing related approximation of fracture energies.