The mathematics of thin structures

Jean-François Babadjian (1) , Giovanni Di Fratta (2) , Irene Fonseca (3) , Gilles A Francfort (4) , Marta Lewicka (5) , Cyrill B Muratov (6)

1 LMO - Laboratoire de Mathématiques d'Orsay
2 Universita di Napoli Frederico II
3 CMU - Carnegie Mellon University [Pittsburgh]
4 UP13 - Université Paris 13
5 Department of Mathematics [Pittsburgh]
6 NJIT - New Jersey Institute of Technology [Newark]

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.

Domaines: Equations aux dérivées partielles [math.AP]

Jean-François Babadjian, Giovanni Di Fratta, Irene Fonseca, Gilles A Francfort, Marta Lewicka, et al.. The mathematics of thin structures. Quarterly of Applied Mathematics, 2023, 81 (1), pp.1-64. ⟨10.1090/qam/1628⟩. ⟨hal-03773118⟩