Effects of oscillation scales in discrete brittle damage models
Elise Bonhomme
TL;DR
The paper studies the asymptotic behavior of discrete brittle damage models in two dimensions under simultaneous scaling of damage strength and discretization. It combines a finite-element–type discretization with a variational Brittle damage energy that penalizes damaged volume and degenerates stiffness, introducing scaling parameters $\alpha$ and $\beta$ to describe the relative decay rates $\eta_\varepsilon/\varepsilon$ and $h_\varepsilon/\varepsilon$. The main contribution is a $\Gamma$-convergence analysis of the discrete energies $\mathcal{F}_{\varepsilon}$, showing that concentration phenomena are only recovered in regimes where the mesh-induced oscillation scale and the damaged-region scale align (i.e., $\alpha$ and $\beta$ are compatible). This clarifies how spatial discretization influences effective fracture models and connects to prior continuous and numerical studies, highlighting when discrete approximations can reproduce fracture-like limits or fail to do so depending on the discretization scale.
Abstract
This paper is concerned with the asymptotic analysis of a sequence of variational models of brittle damage in the context of linearized elasticity in the two-dimensional discrete setting. We consider a discrete version of Francfort and Marigo's brittle damage model, where the total energy is restricted to continuous and piecewise affine displacements; within different regimes where the damaged regions concentrate on vanishingly small sets while the stiffness of the damaged material degenerates to $0$. In this setting, the convergence of the space discretization, the concentration of the damaged regions, and the decay of the elastic properties of the damaged phase all compete simultaneously in non-trivial ways according to the scaling law under consideration. The mesh size turns out to be a crucial feature of the analysis, as it induces a minimal scale of spatial oscillations for admissible displacements. This study was motivated by the numerical investigations performed in [2] on the one hand, where Allaire-Jouve-Van Goethem have shown that forcing the stiffness decay on sets of arbitrarily small measure seems to lead to concentrations phenomena such as in brittle fracture; and by the static analysis performed in [10] on the other hand, where Babadjian-Iurlano-Rindler addressed the rigorous asymptotic analysis of this observation in terms of the $Γ$-convergence of the total energies, in the continuous setting in space. Surprisingly, they showed that fracture-type models were not obtained asymptotically, thus raising the question of the dependence of the effective models with respect to the scaling of a spatial discretization in Francfort and Marigo's model. This is the content of the present work, where we show that concentrations phenomena are only captured asymptotically in regimes where the mesh size and the concentration of damaged regions are of same order.