ASYMPTOTIC ANALYSIS OF AMBROSIO-TORTORELLI ENERGIES IN LINEARIZED ELASTICITY

We provide an approximation result in the sense of Gamma-convergence for energies of the form integral(Omega) L-1 (e(u)) dx + a Hn-1 (J(u)) + b integral (Ju) L-0(1/2) ([u] circle dot nu(u)) dH(n-1) where Omega subset of R-n is a bounded open set with Lipschitz boundary, L-0 and L-1 are coercive quadratic forms on M-sym(nxn), a, b are positive constants, and u runs in the space of fields SBD2(Omega); i.e., it's a special field with bounded deformation such that its symmetric gradient e(u) is square integrable, and its jump set J(u) has finite (n-1)-Hausdorff measure in R-n. The approximation is performed by means of Ambrosio-Tortorellitype elliptic regularizations, the prototype example being integral(Omega) (v vertical bar e(u)vertical bar(2) + gamma epsilon vertical bar del v vertical bar(2)) dx, where (u, v) is an element of H-1(Omega, R-n) x H-1(Omega), epsilon <= v <= 1, and gamma > 0.