An approximation result for special functions with bounded deformation

AbstractA “special displacement with bounded deformation” is a function u:Ω⊂RN→RN whose symmetrized gradient is a bounded measure which coincides, outside a (N−1)-dimensional rectifiable “jump set” Ju, with a summable function e(u). We show that in dimension N=2, when u and e(u) are square integrable, and the total length H1(Ju) is finite, then such a displacement is approximated with a sequence (un)n⩾1 of piecewise continuous displacements whose jump sets Jun are (relatively) closed, with un and e(un) converging strongly in L2, respectively to u and e(u), and the lengths H1(Jun) converging to H1(Ju). As an application, we approximate with a sequence of elliptic functionals a functional which appears in the theory of brittle fracture in linearized elasticity