A generalized continuum approach to predict local buckling patterns of thin structures
Keywords:
Ginzburg-Landau equation, multi-scale analysis, local instability, local-global coupling, buckling, wrinklingAbstract
Macroscopic descriptions of instability pattern formation can be predicted by generic amplitude equations of Ginzburg-Landau type. A variant of this approach is presented, that permits 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 macroscopic stresses are Fourier coefficients of the microscopic stresses. This new double scale description of cellular instabilities is applied to beam on an elastic foundation and to 3D nonlinear elasticity. We shall also discuss the behaviour of these new continuum models after a finite element discretisation.
Downloads
References
Cochelin B., Damil N., Potier-Ferry M., Méthode Asymptotique Numérique, Hermès Science
Publishing, Paris, Londres, 2007.
Damil N., Potier-Ferry M., “Wavelength selection in the postbuckling of a long rectangular
plate”, International Journal of Solids and Structures, vol. 22, 1986, p. 511-526.
Damil N., Potier-Ferry M. “Amplitude equations for cellular instabilities”, Dynamics and
Stability of Systems, vol. 7, 1992, p. 1-34.
Damil N., Potier-Ferry M., “A generalized continuum approach to describe instability pattern
formation by a multiple scale analysis”, Comptes Rendus Mécanique, vol. 334, 2006,
p. 674-678.
Diaby A., Le Van A., Wielgosz C., « Modélisation du plissage dans les structures
membranaires », Revue Européenne de Mécanique Numérique, vol. 13, n° 1-2-3 (Giens
, 2006, p. 143-154.
Iooss G., Mielke A., Demay Y. “Theory of steady Ginzburg-Landau equation in
hydrodynamic stability problems”, European Journal of Mechanics B/Fluids, vol. 8, 1989,
p. 229-268.
Léotoing L., Drapier S., Vautrin A., “Nonlinear interaction of geometrical and material
properties in sandwich beam instabilities”, International Journal of Solids and Structures,
vol. 39, 2002, p. 3717-3739.
Newell, A., Whitehead J., “Finite band width, finite amplitude convection”, Journal of Fluid
Mechanics, vol. 38, 1969, p. 279-303.
Segel L., “Distant side walls cause slow amplitude modulation of cellular convection”,
Journal of Fluid Mechanics, vol. 38, 1969, p. 203-224.
Sridharan S., Zeggane M., “Stiffened plates and cylindrical shells under interactive buckling”,
Finite Elements in Analysis and Design, vol. 38, 2001, p. 155-178.
Wesfreid J.E., Zaleski S. (Ed.), Cellular Structures in instabilities, Lecture Notes in Physics
, Springer-Verlag, Heidelberg, 1984.
Wong Y.W., Pellegrino S., “Wrinkled membranes-Part1: experiments”, Journal of Mechanics
of Materials and Structures, vol. 1, 2006, p. 3-25.
