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2010-10-18 - Colloque/Présentation - communication orale - Anglais - 65 page(s)

Troestler Christophe , "Wrinkling of thin membranes on fluid substrates." in Workshop on Variational Methods in Nonlinear Differential Equations, Oaxaca, Mexique, 2010

  • Codes CREF : Calcul des variations (DI1128), Analyse numérique (DI1123), Equations différentielles et aux dérivées partielles (DI1127)
  • Unités de recherche UMONS : Analyse numérique (S835)
  • Instituts UMONS : Institut de Recherche sur les Systèmes Complexes (Complexys)
  • Centres UMONS : Modélisation mathématique et informatique (CREMMI)
Texte intégral :

Abstract(s) :

(Anglais) In this talk, we will examine the shape of a clamped thin membrane resting on a substrate such as water and compressed horizontally. Physical experiments show that, for small compressions, wrinkles appear. We assume that the shape of the membrane is constant in the direction orthogonal to the compression so that it can be described by a curve $[0,L] \to \mathbb{R}^2 : s \mapsto \bigl(x(s), y(s)\bigr)$. The balance of energy between the curvature of the membrane and the displacement of the substrate implies that the curve $(x,y)$ must minimize the functional \begin{equation*} (x,y) \mapsto \tfrac{1}{2} \int_0^L \kappa^2(s) \,\mathrm{d}s + \tfrac{1}{2} K \int_0^{L} \bigl(y(s)\bigr)^2 \, \partial_sx(s) \,\mathrm{d}s , \end{equation*} where $\kappa$ is the curvature of $(x,y)$, and $K$ is a constant depending on the substrate. The set of configurations $(x,y)$ is constrained to reflect the fact that the membrane is clamped and compressed. We will describe the symmetry and nodal properties of the solutions when the compression is small.