Calculus of Variations and Geometric Measure Theory

R. Marziani - F. Solombrino

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

created by marziani on 27 Dec 2022
modified by solombrino on 08 Jun 2023


Published Paper

Inserted: 27 dec 2022
Last Updated: 8 jun 2023

Journal: Proceedings of the Royal Society Edinburgh Sect. A
Pages: 35
Year: 2023
Doi: 10.1017/prm.2023.51

ArXiv: 2212.13759 PDF


We analyse the $\Gamma$-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.

Keywords: Non-local approximation, variational fracture, free discontinuity problems, deterministic and stochastic homogenisation, functions of bounded deformations, Gamma-convergence.