Wrinkled and wrinkle-free membranes

Abstract

The extreme out-of-plane flexibility makes membranes vulnerable of wrinkling, which limits the applications with stringent-accuracy demands and high-frequency requirements. In this study, a theoretical analysis is proposed to predict the minimum of min-principal stress min(σmin), representing the wrinkling capability of the stretched membrane with an arbitrary aspect ratio and random curved edges, while the curved edges are represented by a Fourier series function. The Marguerre function is adopted to calculate the stress solutions and the sigma-approximation approach is employed to eliminate the Gibbs phenomenon. The determination of numbers of truncated terms of Marguerre function and the normalization strategy are helpful to solve the ill-posed problem. The accuracy and generality of our analytical solutions are verified by the finite element results. Moreover, the wrinkling criterion indicates that the entire membrane is taut and wrinkle-free if min(σmin) is positive. With maximizing the membrane area as the objective function, the positive min(σmin) as constraint, and the Fourier coefficients tuning edge profiles as design variables, the structural design approach gives the optimal configurations, whose wrinkle-free performance is verified by the finite element analyses and physical experiments.