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

Michel  Potier-Ferry; Foudil  Mohri; Fan  Xu; Noureddine  Damil; Bouazza  Braikat; Khadija  Mhada; Heng  Hu; Qun  Huang; Saeid  Nezamabadi

doi:10.3934/dcdss.2016013

Article Contents

2016, Volume 9, Issue 2: 585-597. Doi: 10.3934/dcdss.2016013

This issue Previous Article Stress gradient effects on the nucleation and propagation of cohesive cracks Next Article On estimation of internal state by an optimal control approach for elastoplastic material

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

1.
LEM3, Laboratoire d'Etudes des Microstructures et de Mécanique des Matériaux, UMR CNRS 7239, Université de Lorraine, Ile du Saulcy, 57045 Metz Cedex 01
2.
Department of Mechanics and Engineering Science, Fudan University, 220 Handan Road, 200433 Shanghai
3.
Laboratoire d'Ingénierie et Matériaux LIMAT, Faculté des Sciences Ben M'Sik, Université Hassan II de Casablanca, Sidi Othman, Casablanca
4.
School of Civil Engineering, Wuhan University, 8 South Road of East Lake, 430072 Wuhan
5.
Université de Montpellier, Laboratoire de Mécanique et Génie Civil, UMR CNRS 5508, CC048 Place Eugène Bataillon, 34095 Montpellier Cedex 05

Received: April 2015

Revised: October 2015

Published: April 2016

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Abstract

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.

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Mathematics Subject Classification: 34E13, 4K18, 35B36, 42B05, 74G60.

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