On Topological Solitons and Geometry in Thin Shells.

The intimate connection between geometry and mechanics in thin shells gives rise to all sorts of phenomena, such as wrinkling, folding, and crumpling. Recent work has shown yet another fascinating example. When the shell is a catenoid-helicoid or a Möbius band, their geometric properties induce topological solitons in their elastic deformations [1- 4], even when these surfaces have vastly different geometries and topologies. We argue these solitons may appear for the same reason. We provide a systematic strategy to study the connection between solitons, geometry, and mechanics when the undeformed surface is a minimal surface. By localizing the isometry of the minimal-surface associate family and interpreting it as a goldstone mode, we explore the appearance of these solitons and how elastic deformations reflect the topology and geometry of the undeformed surface.

[1] Bartolo, D., & Carpentier, D. (2019). Topological elasticity of nonorientable ribbons. Physical Review X, 9(4), 041058.

[2] Sun, K., & Mao, X. (2021). Fractional excitations in non-Euclidean elastic plates. Physical Review Letters, 127(9), 098001.

[3] Núñez, C. N. V., Poli, A., Stanifer, E., Mao, X., & Arruda, E. M. (2023). Fractional topological solitons in nonlinear viscoelastic ribbons with tunable speed. Extreme Mechanics Letters, 61, 102027.

[4] Guo, X., Guzmán, M., Carpentier, D., Bartolo, D., & Coulais, C. (2023). Non-orientable order and non-commutative response in frustrated metamaterials. Nature, 618(7965), 506-512.