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 ( σ min ) , 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 ( σ min ) is positive. With maximizing the membrane area as the objective function, the positive min ( σ min ) 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.