Non linear pattern selection in a problem of elasticity

2014 The buckling of long rectangular elastic plates offers the possibility of testing a recent proposal. The wavenumber of cellular structures in slightly supercritical conditions is determined by the boundary conditions. In the present case a supercritical decrease of the wavelength is predicted. Tome 42 ? 1 ler JANVIER 1981 LE JOURNAL DE PHYSIQUE LETTRES J. Physique LETTRES 42 (1981) LI L-4 1 er JANVIER 1981, Classification Physics Abstracts 03.40D Having in mind the wavelength selection in cellular flows, as in Rayleigh-Benard convection in large horizontal layers, S. Zaleski and the author [1] solved the question of non linear pattern selection in slightly supercritical conditions for one dimensional models. This sort of problem is formulated as follows : space dependent fluctuations with a (horizontal) fixed wavenumber, say qo, become linearly unstable around a homogeneous rest state whenever a control parameter, say 8, exceeds some critical value, which can be taken at s = 0. The growth of these fluctuations is limited by non linear effects and a new steady state is reached via a supercritical (or normal) bifurcation. In a large class of problems, for slightly positive values of 8 linearly unstable fluctuations grow from the homogeneous state whenever their wavenumber belongs to a band of width of order 81/2 near the threshold value qo. However, owing to the boundary conditions limiting the lateral extent of the structure, the supercritical steady pattern has its wavenumber in a much narrower band of width of order 8 near qo. The applicability of this sort of consideration to the Rayleigh-Benard problem is not obvious, since long rolls, when parallel to a lateral boundary, are unstable against a cross roll instability localized near this boundary [1]. One should account for the structure of these boundary rolls parallel to their axis. This structure is due either to the cross roll modulation or to lateral boundaries inhibiting their growth. A realistic treatment of this problem is not an easy task. Therefore it is of interest to look at a physical situation involving a non linear selection of the wavelength, but nevertheless permitting quantitative predictions from simple ab initio calculations. That is why I have considered the following version of the von Karman problem in elasticity of thin plates [2]. This is the buckling of long rectangular elastic plates [3] submitted to a load along their long axis (see Fig. 1). According to the general considerations of reference [1], whenever the length of the long axis, say L, is much larger than a quantity of order E-1, one may limit oneself to the consideration of a half infinite problem. The possible buckling patterns for a large (but finite) L are obtained by gluing together two half infinite solutions in a convenient way. We shall not consider this specific problem here ; it is treated in reference [1]. Fig. 1. At the centre of the figure the plate is represented from above; its cuts along the short (A) and long (B) are represented on the left and right. A similar figure is in reference [4]. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019810042010100 L-2 JOURNAL DE PHYSIQUE LETTRES The von Karman [2] equations for the static buckling are [4, 5] :