The mathematics of thin structures

Babadjian, Jean-François; Di Fratta, Giovanni; Fonseca, Irene; Francfort, Gilles; Lewicka, Marta; Muratov, Cyrill

doi:10.1090/qam/1628

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The mathematics of thin structures

Authors: Jean-François Babadjian, Giovanni Di Fratta, Irene Fonseca, Gilles A. Francfort, Marta Lewicka and Cyrill B. Muratov
Journal: Quart. Appl. Math. 81 (2023), 1-64
MSC (2020): Primary 49J45, 74B20, 74K20, 74K35, 53A35
DOI: https://doi.org/10.1090/qam/1628
Published electronically: September 1, 2022
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Abstract: This article offers various mathematical contributions to the behavior of thin films. The common thread is to view thin film behavior as the variational limit of a three-dimensional domain with a related behavior when the thickness of that domain vanishes. After a short review in Section 1 of the various regimes that can arise when such an asymptotic process is performed in the classical elastic case, giving rise to various well-known models in plate theory (membrane, bending, Von Karmann, etc…), the other sections address various extensions of those initial results. Section 2 adds brittleness and delamination and investigates the brittle membrane regime. Sections 4 and 5 focus on micromagnetics, rather than elasticity, this once again in the membrane regime and discuss magnetic skyrmions and domain walls, respectively. Finally, Section 3 revisits the classical setting in a non-Euclidean setting induced by the presence of a pre-strain in the model.

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