The Broad Wrinkling Landscape of Hyperelastic Parallelogram-Shaped Membranes: From Wrinkle Migration to Restabilization and their Subsequent Reappearance Elsewhere

Wrinkling is a commonly observed out-of-plane instability in membrane structures due to their extremely low bending-to-stretching stiffness ratio. It has been extensively investigated for symmetric membrane geometries and boundary conditions that induce planar non-uniform stress states by preventing the lateral contraction at the edges, and is also known to potentially display self-restabilization. This study investigates an initially flat, parallelogram-shaped hyperelastic membrane, focusing on the effect of the inclination angle that defines its deviation from rectangular geometry. It is shown that wrinkling can occur either centrally or at the two opposite obtuse-angled corners–even for small inclination angles–during stretching with unconstrained lateral contraction, a condition under which the flat configuration for the rectangular counterpart remains always stable. Three distinct evolutions of the wrinkling pattern are numerically identified, all ultimately leading to corner-localized wrinkles. This final state may arise (i) directly, without a prior bifurcation, or after the appearance of central wrinkling that either (ii) restabilizes or (iii) separates and migrates toward the corners. A closed-form expression for the critical wrinkling condition is derived by combining a perturbation approach with an energy-based method in the framework of linear elasticity. This provides an accurate estimate of the onset and pattern of central wrinkling. The present findings reveal new pathways in wrinkling pattern evolution and introduce a novel approach to unconventional boundary-value problems, with potential applications ranging from lightweight structural systems to flexible electronics.