"Special functions with bounded deformation" and p-integrable strain arise naturally in the study of geometrically linear fracture models. They have a jump set of finite (n − 1)-dimensional measure and, away from the jump set, a symmetrized gradient e(u) in Lp , p ≥ 1. We show that if the measure of the jump set is sufficiently small with respect to the size of the domain, then the function u can be approximated by an affine function away from a small exceptional set, with an error which depends solely on e(u). We also derive a corresponding trace statement.