A separation-of-variable method for the wrinkling problems of orthotropic rectangular stretched sheets

Film structures have been extensively employed in space and soft materials. The flexural rigidity of a sheet is so small that it can hardly withstand pressure, causing instability and forming wrinkles. This work develops a separation-of-variable (SOV) method to the stretched-induced wrinkling analysis of orthotropic rectangular sheets. In this SOV method, wrinkling mode functions are in a separation-of-variable form, and two ordinary characteristic differential equations of wrinkling mode functions are derived through the minimum principle of the potential energy, then the mode functions are expressed in terms of the eigenvalues of the characteristic differential equations. Finally, closed-form mode functions and explicit equations for critical tensile strains are achieved with the boundary conditions of rectangular sheets. In this work, both stretched edges are clamped or simply supported, and another pair of edges are free, and the closed form solutions for the rectangular sheets with clamped stretched edges are achieved for the first time. The present results agree well with the analytical and numerical results in literature. In addition, the stretching process is described and several observations are found through theoretical and numerical analyses.