Wrinkles on Tori

Wrinkling patterns in soft materials have been extensively studied due to their important roles in determining surface morphologies in biological structures and developing multifunctional devices. Most existing work focuses on relatively simple geometries, such as flat structures and curved structures with constant curvature such as the cylinder and 2-sphere. In this talk we discuss wrinkling patterns on a torus, the Gaussian and mean curvatures of which vary along the poloidal direction. We observe eight different wrinkling patterns from large-scale finite element simulations and construct a phase diagram for these patterns. We further show that the non-uniform curvature and anisotropic deformation play critical roles in determining the formation and evolution of these wrinkling patterns. The anisotropic deformation along the toroidal and poloidal directions controls pattern transitions from stripes to hexagons and the non-uniform curvatures determine the nucleation sites of the wrinkling patterns. Our results show that global deformations of a torus lead to strong coupling between elasticity and curvature which may enlarge the design space as well as the dynamically control of wrinkling patterns.