Analysis of membrane instability with a two-parameter extended system

Membrane instability typically exhibits small wavelength compared to the structural size, which often leads to numerical difficulties in computational efficiency and convergence problem. Recently, the Fourier-based reduced technique that is similar to the famous Ginzburg–Landau equation has shown the potential to overcome these difficulties. However, the wrinkling wavelength, an internal parameter, should be defined a priori, and how to determine it is still questionable. In this paper, we propose a two-parameter extended system attempting to lift this restriction. In this system, a Fourier-based reduced model with the wavelength as the second path-control parameter is firstly established, and then augmented by appending a constraint equation that characterizes the critical state (i.e., bifurcation point). The resulting critical equilibrium path, where the bifurcation points and the corresponding buckling modes depend on the wavelength, is tracked by the pseudo-arclength algorithm. Numerical results show that the proposed system is able to correctly and efficiently predict the wrinkling wavelength and buckling behavior of the membrane. This study could be extended to other similar models (e.g., amplitude equations) that characterize instability phenomena with internal parameters.