# Concentration versus oscillation effects in brittle damage

## TL;DR

This paper analyzes the asymptotic behavior of brittle damage models in linear elasticity as damage concentrates and stiffness diminishes, revealing a transition from quadratic to linear growth energy and identifying the limit energy as Hencky plasticity or Tresca-type.

## Contribution

It introduces a novel analysis of the interaction between homogenization and damage concentration, deriving the $ ext{Gamma}$-limit for brittle damage models with microstructures.

## Key findings

- The $ ext{Gamma}$-limit of the damage models is explicitly identified in 2D and 3D.
- The limit energy is of Hencky plasticity type under general conditions.
- A Tresca-type model emerges when divergence remains square-integrable.

## Abstract

This work is concerned with an asymptotic analysis, in the sense of $\Gamma$-convergence, of a sequence of variational models of brittle damage in the context of linearized elasticity. The study is performed as the damaged zone concentrates into a set of zero volume and, at the same time and to the same order $\varepsilon$, the stiffness of the damaged material becomes small. Three main features make the analysis highly nontrivial: at $\varepsilon$ fixed, minimizing sequences of each brittle damage model oscillate and develop microstructures; as $\varepsilon\to 0$, concentration and saturation of damage are favoured; and the competition of these phenomena translates into a degeneration of the growth of the elastic energy, which passes from being quadratic (at $\varepsilon$ fixed) to being of linear-growth type (in the limit). Consequently, homogenization effects interact with singularity formation in a nontrivial way, which requires new methods of analysis. In particular, the interaction of homogenization with singularity formation in the framework of linearized elasticity appears to not have been considered in the literature so far. We explicitly identify the $\Gamma$-limit in two and three dimensions for isotropic Hooke tensors. The expression of the limit effective energy turns out to be of Hencky plasticity type. We further consider the regime where the divergence remains square-integrable in the limit, which leads to a Tresca-type model.