A Density Result in GSBDp with Applications to the Approximation of Brittle Fracture Energies

We prove that any function in GSBDp(), with a n-dimensional open bounded set with finite perimeter, is approximated by functions ukSBV(;Rn)L(;Rn) whose jump is a finite union of C-1 hypersurfaces. The approximation takes place in the sense of Griffith-type energies W(e(u))dx+Hn-1(Ju), e(u) and J(u) being the approximate symmetric gradient and the jump set of u, and W a nonnegative function with p-growth, p>1. The difference between u(k) and u is small in L-p outside a sequence of sets Ek< subset of> whose measure tends to 0 and if |u|rL1() with r(0,p], then |uk-u|r0 in L1(). Moreover, an approximation property for the (truncation of the) amplitude of the jump holds. We apply the density result to deduce -convergence approximation a la Ambrosio-Tortorelli for Griffith-type energies with either Dirichlet boundary condition or a mild fidelity term, such that minimisers are a priori not even in L1(;Rn).