Korn-Poincar\'e-type inequalities and energy density for a subspace of $SBD^p$

Specialized functions of bounded deformation are established as an setting for elasticity problems in which deformations are allowed to jump on some $(n-1)$-dimensional set.

We are interested in a specific subspace of these functions (in 2d), in which we fix the possible direction of the jump set for each component. In this setting we are able to provide different Korn-Poincaré-type estimates, independent of the size of the jump set.

We apply these estimates to approximate such functions with functions that are smooth up to a jump set of finitely many segments and that satisfy the same directionally constraint.