We present an approximation result for functions u : Ω → R n belonging to the space GSBD(Ω) ∩ L 2 (Ω, R n) with e(u) square integrable and H n−1 (Ju) finite. The approximating functions u k are piecewise continuous functions such that u k → u in L 2 (Ω, R n), e(u k) → e(u) in L 2 (Ω, M n×n sym), H n−1 (Ju k Ju) → 0, and´J u k ∪Ju |u ± k − u ± | ∧ 1dH n−1 → 0. As an application, we provide the extension to the vector-valued case of the Γ-convergence result in SBV (Ω) proved by Ambrosio and Tortorelli in [4, 5].

We prove an integral representation result for functionals with growth conditions which give coercivity on the spaceThe space SBD p of functions whose distributional strain is the sum of an L p part and a bounded measure supported on a set of finite H 1 -dimensional measure appears naturally in the study of fracture and damage models. Our result is based on the construction of a local approximation by W 1,p functions. We also obtain a generalization of Korn's inequality in the SBD p setting.

We obtain a cohesive fracture model as -limit, as ε → 0, of scalar damage models in which the elastic coefficient is computed from the damage variable v through a function f ε of the form f ε (v) = min{1, ε 1 2 f (v)}, with f diverging for v close to the value describing undamaged material. The resulting fracture energy can be determined by solving a one-dimensional vectorial optimal profile problem. It is linear in the opening s at small values of s and has a finite limit as s → ∞. If in addition the function f is allowed to depend on the parameter ε, for specific choices we recover in the limit Dugdale's and Griffith's fracture models, and models with surface energy density having a power-law growth at small openings.

Abstract.We provide an approximation result in the sense of Γ-convergence for energies of the form Ω Q 1 (e(u)) dx, where Ω ⊂ R n is a bounded open set with Lipschitz boundary, Q 0 and Q 1 are coercive quadratic forms on M n×n sym , a, b are positive constants, and u runs in the space of fields SBD 2 (Ω); i.e., it's a special field with bounded deformation such that its symmetric gradient e(u) is square integrable, and its jump set Ju has finite (n − 1)-Hausdorff measure in R n . The approximation is performed by means of Ambrosio-Tortorellitype elliptic regularizations, the prototype example being where (u, v) ∈ H 1 (Ω, R n )×H 1 (Ω), ε ≤ v ≤ 1, and γ > 0.

We prove that special functions of bounded deformation with small jump set are close in energy to functions which are smooth in a slightly smaller domain. This permits to generalize the decay estimate by De Giorgi, Carriero, and Leaci to the linearized context in dimension n and to establish the closedness of the jump set for local minimizers of the Griffith energy.