$Γ$-convergence for free-discontinuity problems in linear elasticity: Homogenization and relaxation

Abstract

We analyze the Γ -convergence of sequences of free-discontinuity functionals arising in the modeling of linear elastic solids with surface discontinuities, including phenomena as fracture, damage, or material voids. We prove compactness with respect to Γ -convergence and represent the Γ -limit in an integral form defined on the space of generalized special functions of bounded deformation (GSBD p ). We identify the integrands in terms of asymptotic cell formulas and prove a non-interaction property between bulk and surface contributions. Eventually, we investigate sequences of corresponding boundary value problems and show convergence of minimum values and minimizers. In particular, our techniques allow to characterize relaxations of functionals on GSBD p , and cover the classical case of periodic homogenization.

Investigations on the influence of the boundary conditions when computing the effective crack energy of random heterogeneous materials using fast marching methods

“…In a recent extension of the work of Braides et al [43], Friedrich et al [58] lift the restriction to the anti-plane shear case for their periodic homogenization result.…”

“…In this work we studied the influence of the boundary conditions on the effective crack energy of heterogeneous materials. Based on homogenization result [43,48,58] for the Francfort-Marigo model of brittle fracture [14] in a quasi-static setting and without crack irreversibility, we investigated a method for computing the effective crack energy using the fast marching method [49]. We validated our approach and compared it to recent FFT-based methods using periodic boundary conditions [44,45].…”

From atomistic systems to linearized continuum models for elastic materials with voids

“…At this point, a further step consists in proving the weak convergence of the (symmetrized) discrete gradients in the setting of GSBD functions. As the analogous result for Sobolev functions [43] is not directly applicable, beforehand we need to perform a delicate approximation of GSBD functions by Sobolev functions, using recent blow-up techniques for GSBD functions [15,16,18,28] and generalizing them to a discrete setting.…”

Computing the effective crack energy of microstructures via quadratic cone solvers

“…Homogenization methods enable to compute the mechanical behavior of heterogeneous materials. Recent theoretical works established a homogenization result [1][2][3] for the Francfort-Marigo [4] model of brittle fracture. For a fixed time discretization, the heterogeneous model converges to the effective model…”