A Korn-Poincar\'e-type inequality for special functions of bounded deformation

We present a Korn-Poincar\'e-type inequality in a planar setting which is in the spirit of the Poincar\'e inequality in SBV due to De Giorgi, Carriero, Leaci. We show that for each function in SBD$^2$ one can find a modification which differs from the original displacement field only on a small set such that the distance of the modification from a suitable infinitesimal rigid motion can be controlled by an appropriate combination of the elastic and the surface energy. In particular, the result can be used to obtain compactness estimates for functions of bounded deformation.