Asymptotic Analysis of Ambrosio-Tortorelli Energies in Linearized Elasticity

We provide an approximation result in the sense of Γ-convergence for energies of the form ˆ Ω Q1(e(u)) dx+ aH(Ju) + b ˆ Ju Q 1/2 0 ([u] νu) dH n−1, where Ω ⊂ Rn is a bounded open set with Lipschitz boundary, Q0 and Q1 are coercive quadratic forms on Mn×n sym , a, b are positive constants, and u runs in the space of fields SBD2(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 Rn. The approximation is performed by means of Ambrosio-Tortorelli type elliptic regularizations, the prototype example ˆ Ω ( v|e(u)| + (1− v)2 ε + γ ε|∇v| ) dx, where (u, v) ∈ H1(Ω,Rn)×H1(Ω), ε ≤ v ≤ 1 and γ > 0.