An approximation result for special functions with bounded deformation

A "special displacement with bounded deformation" is a function u: Omega subset of R-N --> R-N whose symmetrized gradient is a bounded measure which coincides, outside a (N - 1)-dimensional rectifiable "jump set" J(u), 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 H-1 (J(u)) is finite, then such a displacement is approximated with a sequence (u(n))(ngreater than or equal to1) of piecewise continuous displacements whose jump sets J(un) are (relatively) closed, with u(n) and e(u(n)) converging strongly in L-2, respectively to u and e(u), and the lengths H-1(J(un)) converging to H-1 (J(u)). As an application, we approximate with a sequence of elliptic functionals a functional which appears in the theory of brittle fracture in linearized elasticity. (C) 2004 Elsevier SAS. All rights reserved.