  We present a Korn-type inequality in a planar setting for special functions
of bounded deformation. We prove that for each function in SBD$^2$ with
sufficiently small jump set the distance of the function and its derivative
from an infinitesimal rigid motion can be controlled in terms of the linearized
elastic strain outside of a small exceptional set of finite perimeter.
Particularly the result shows that each function in SBD$^2$ has bounded
variation away from an arbitrarily small part of the domain.
