Non-local approximation of free-discontinuity problems in linear elasticity and application to stochastic homogenisation

We analyse the $\mathrm{\Gamma }$-convergence of general non-local convolution type functionals with varying densities depending on the space variable and on the symmetrized gradient. The limit is a local free-discontinuity functional, where the bulk term can be completely characterized in terms of an asymptotic cell formula. From that, we can deduce an homogenisation result in the stochastic setting.