New nonlinear multi-scale models for wrinkled membranes

A new macroscopic approach to the modelling of membrane wrinkling is presented. Most of the studies of the literature about membrane behaviour are macroscopic and phenomenological, the influence of wrinkles being accounted for by nonlinear constitutive laws without compressive stiffness. The present method is multi-scale and it permits to predict the wavelength and the spatial distribution of wrinkling amplitude. It belongs to the family of Landau–Ginzburg bifurcation equations and especially relies on the technique of Fourier series with slowly varying coefficients. The result is a new family of macroscopic membrane models that are deduced from Föppl–von Kármán plate equations. Numerical solutions are presented, giving the size of the wrinkles as a function of the applied compressive and tensile stresses.