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Concentration versus oscillation effects in brittle damage
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Babadjian, Jean-Francois, Iurlano, Flaviana and Rindler, Filip (2021) Concentration versus oscillation effects in brittle damage. Communications on Pure and Applied Mathematics, 74 (9). pp. 1803-1854. doi:10.1002/cpa.21953 ISSN 0010-3640.
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Official URL: https://doi.org/10.1002/cpa.21953
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.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Library of Congress Subject Headings (LCSH): | Mathematical analysis, Homogenization (Differential equations), Asymptotic expansions, Plasticity, Brittleness -- Mathematical models | ||||||||
Journal or Publication Title: | Communications on Pure and Applied Mathematics | ||||||||
Publisher: | John Wiley & Sons | ||||||||
ISSN: | 0010-3640 | ||||||||
Official Date: | September 2021 | ||||||||
Dates: |
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Volume: | 74 | ||||||||
Number: | 9 | ||||||||
Page Range: | pp. 1803-1854 | ||||||||
DOI: | 10.1002/cpa.21953 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Reuse Statement (publisher, data, author rights): | "This is the peer reviewed version of the following article: Babadjian, J.‐F., Iurlano, F. and Rindler, F. (2021), Concentration versus Oscillation Effects in Brittle Damage. Comm. Pure Appl. Math., which has been published in final form at https://doi.org/10.1002/cpa.21953. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Date of first compliant deposit: | 13 January 2020 | ||||||||
Date of first compliant Open Access: | 11 October 2021 | ||||||||
RIOXX Funder/Project Grant: |
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