Cellular instabilities analyzed by multi-scale Fourier series: A review

Michel Potier-Ferry (1) , Foudil Mohri (1) , Fan Xu (2) , Noureddine Damil (3) , Bouazza Braikat (3) , Khadija Mhada (3) , Heng Hu (4) , Qun Huang (4) , Saeid Nezamabadi (5)

1 LEM3 - Laboratoire d'Etude des Microstructures et de Mécanique des Matériaux
2 Department of Mechanics and Engineering Science
3 LIMAT - Laboratoire d'Ingénierie et Matériaux [Casablanca]
4 School of Civil Engineering
5 PMMD - Physique et Mécanique des Milieux Divisés

The paper is concerned by multi-scale methods to describe instability pattern formation, especially the method of Fourier series with variable coefficients. In this respect, various numerical tools are available. For instance in the case of membrane models, shell finite element codes can predict the details of the wrinkles, but with difficulties due to the large number of unknowns and the existence of many solutions. Macroscopic models are also available, but they account only for the effect of wrinkling on membrane behavior. A Fourier-related method has been introduced in order to modelize the main features of the wrinkles, but by using partial differential equations only at a macroscopic level. Within this method, the solution is sought in the form of few terms of Fourier series whose coefficients vary more slowly than the oscillations. The recent progresses about this Fourier-related method are reviewed and discussed.

Mots clés: Multi-scale models; wrinkling; bifurcation; slowly variable Fourier coefficients

Domaines: Mécanique des solides [physics.class-ph]; Mécanique des structures [physics.class-ph]

Michel Potier-Ferry, Foudil Mohri, Fan Xu, Noureddine Damil, Bouazza Braikat, et al.. Cellular instabilities analyzed by multi-scale Fourier series: A review. Discrete and Continuous Dynamical Systems - Series A, 2016, 9 (2), pp.585-597. ⟨10.3934/dcdss.2016013⟩. ⟨hal-01314358⟩