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 of damage and worsening of the elastic properties 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 linear (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.

中文翻译：

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脆性损伤中的浓度与振荡效应

这项工作涉及在 $\Gamma$-收敛的意义上，对线性化弹性背景下的一系列脆性损伤变分模型进行渐近分析。该研究是在损坏区域集中到一组零体积时进行的，同时以相同的顺序 $\varepsilon$，损坏材料的刚度变小。三个主要特征使分析非常重要：在 $\varepsilon$ 固定时，最小化每个脆性损伤模型的序列振荡和发展微观结构；当 $\varepsilon\to 0$ 时，损伤集中和弹性变差是有利的；并且这些现象的竞争转化为弹性能量增长的退化，其从二次型（在 $\varepsilon$ 固定）变为线性（在极限内）。因此，同质化效应以一种非平凡的方式与奇点形成相互作用，这需要新的分析方法。特别是，线性弹性框架中的均质化与奇点形成的相互作用目前似乎尚未在文献中考虑过。我们明确地确定了各向同性胡克张量的二维和三维的 $\Gamma$-limit。极限有效能的表达式结果是 Hencky 塑性类型。我们进一步考虑散度在极限内保持平方可积的状态，这导致了 Tresca 型模型。迄今为止，文献中似乎尚未考虑线性化弹性框架中均质化与奇点形成的相互作用。我们明确地确定了各向同性胡克张量的二维和三维的 $\Gamma$-limit。极限有效能的表达式结果是 Hencky 塑性类型。我们进一步考虑散度在极限内保持平方可积的状态，这导致了 Tresca 型模型。迄今为止，文献中似乎尚未考虑线性弹性框架中的均质化与奇点形成的相互作用。我们明确地确定了各向同性胡克张量的二维和三维的 $\Gamma$-limit。极限有效能的表达式结果是 Hencky 塑性类型。我们进一步考虑散度在极限内保持平方可积的状态，这导致了 Tresca 型模型。