Calculus of Variations and Geometric Measure Theory

A. F. Donnarumma - M. Friedrich

Stochastic homogenisation for functionals defined on asymptotically piecewise rigid functions

created by donnarumma on 20 Dec 2023



Inserted: 20 dec 2023

Year: 2023

ArXiv: 2312.12082 PDF


We study stochastic homogenisation of free-discontinuity surface functionals defined on piecewise rigid functions which arise in the study of fracture in brittle materials. In particular, under standard assumptions on the density, we show that there exists a $\Gamma$-limit almost surely and that it can be represented by a surface integral. In addition, the effective density can be characterised via a suitable cell formula and is deterministic under an ergodicity assumption. We also show via $\Gamma$-convergence that the homogenised functional defined on piecewise rigid functions can be recovered from a Griffith-type model by passing to the limit of vanishing elastic deformations.