Comptes Rendus
A generalized continuum approach to describe instability pattern formation by a multiple scale analysis
Comptes Rendus. Mécanique, Volume 334 (2006) no. 11, pp. 674-678.

Macroscopic descriptions of instability pattern formation can be obtained by the generic amplitude equations of Ginzburg–Landau type. In the simple example of beam buckling, a variant of this approach is established, that permits one to account for the coupling between local and global instabilities. The mean field and the amplitude of the fluctuations are governed by similar equations. The resulting model is a generalized continuum, where the generalized stresses are Fourier coefficients of the microscopic stress.

L'évolution des instabilités spatio-temporelles peut se décrire macroscopiquement par des équations d'amplitude génériques de type Ginzburg–Landau. Dans l'exemple élémentaire du flambage d'une poutre, on établit une variante de cette approche, qui permet de prendre en compte des couplages entre instabilités locales et globales et qui traite de la même manière le champ moyen et la fluctuation. Le modèle final est un milieu continu généralisé, où les contraintes généralisées sont des coefficients de Fourier de la contrainte microscopique.

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Accepted:
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DOI: 10.1016/j.crme.2006.09.002
Keywords: Continuum mechanics, Ginzburg–Landau equation, Multiple scale, Local instability, Local–global coupling, Buckling
Mot clés : Milieux continus, Équation de Ginzburg–Landau, Échelles multiples, Instabilités locales, Couplage local–global, Flambage

Noureddine Damil 1; Michel Potier-Ferry 2

1 Laboratoire de calcul scientifique en mécanique, faculté des sciences Ben M'Sik, université Hassan II – Mohammedia, BP 7955, Sidi Othman, Casablanca, Morocco
2 Laboratoire de physique et mécanique des matériaux, UMR CNRS 7554, université Paul-Verlaine Metz, île du Saulcy, 57045 Metz, France
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Noureddine Damil; Michel Potier-Ferry. A generalized continuum approach to describe instability pattern formation by a multiple scale analysis. Comptes Rendus. Mécanique, Volume 334 (2006) no. 11, pp. 674-678. doi : 10.1016/j.crme.2006.09.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2006.09.002/

[1] Cellular Structures in Instabilities (J.E. Wesfreid; S. Zaleski, eds.), Lecture Notes in Physics, vol. 210, Springer-Verlag, Heidelberg, 1984

[2] M.C. Cross; P.C. Hohenberg Pattern formation out of equilibrium, Reviews of Modern Physics, Volume 65 (1993), pp. 851-1112

[3] N. Damil; M. Potier-Ferry Wavelength selection in the postbuckling of a long rectangular plate, International Journal of Solids and Structures, Volume 22 (1986), pp. 511-526

[4] A. Newell; J. Whitehead Finite band width, finite amplitude convection, Journal of Fluid Mechanics, Volume 38 (1969), pp. 279-303

[5] L. Segel Distant side walls cause slow amplitude modulation of cellular convection, Journal of Fluid Mechanics, Volume 38 (1969), pp. 203-224

[6] G. Iooss; A. Mielke; Y. Demay Theory of steady Ginzburg–Landau equation in hydrodynamic stability problems, European J. Mech. B/Fluids, Volume 8 (1989), pp. 229-268

[7] S. Forest; K. Sab Cosserat overall modeling of heterogeneous materials, Mechanics Research Communications, Volume 25 (1998), pp. 449-454

[8] V. Kouznetsova; M.G.D. Geers; W.A.M. Brekelmans Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy, Computer Methods in Applied Mechanics and Engineering, Volume 193 (2004), pp. 5525-5550

[9] S. Sridharan; M. Zeggane Stiffened plates and cylindrical shells under interactive buckling, Finite Elements in Analysis and Design, Volume 38 (2001), pp. 155-178

[10] L. Léotoing; S. Drapier; A. Vautrin Nonlinear interaction of geometrical and material properties in sandwich beam instabilities, International Journal of Solids and Structures, Volume 39 (2002), pp. 3717-3739

[11] Y.W. Wong; S. Pellegrino Winkled membranes—Part 1: experiments, Journal of Mechanics of Materials and Structures, Volume 1 (2006), pp. 3-25

[12] E. Sanchez-Palencia Non-Homogeneous Media and Vibration Theory, Lecture Notes in Physics, vol. 127, Springer-Verlag, Heidelberg, 1980

[13] M. Potier-Ferry Foundations of elastic postbuckling theory, Lecture Notes in Physics, vol. 288, Springer-Verlag, Heidelberg, 1987, pp. 1-82

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