# Non-uniform Curvature and Anisotropic Deformation Control Wrinkling Patterns on Tori

## TL;DR

This study explores how the stiffness of an inner core influences wrinkling patterns on a torus, revealing transitions from hexagonal to stripe patterns driven by anisotropic deformation and stress states.

## Contribution

It introduces a finite element simulation framework to analyze how core stiffness controls pattern transitions and anisotropic deformation in toroidal structures.

## Key findings

- Hexagonal patterns form with stiff cores.
- Stripe patterns develop with soft cores.
- Hybrid patterns emerge at intermediate stiffness.

## Abstract

We investigate wrinkling patterns in a tri-layer torus consisting of an expanding thin outer layer, an intermediate soft layer and an inner core with a tunable shear modulus, inspired by pattern formation in developmental biologies, such as follicle pattern formation during the development of chicken embryos. We show from large-scale finite element simulations that hexagonal wrinkling patterns form for stiff cores whereas stripe wrinkling patterns develop for soft cores. Hexagons and stripes co-exist to form hybrid patterns for cores with intermediate stiffness. The governing mechanism for the pattern transition is that the stiffness of the inner core controls the degree to which the major radius of the torus expands this has a greater effect on deformation in the long direction as compared to the short direction of the torus. This anisotropic deformation alters stress states in the outer layer which change from biaxial (preferred hexagons) to uniaxial (preferred stripes) compression as the core stiffness is reduced. As the outer layer continues to expand, stripe and hexagon patterns will evolve into Zigzag and segmented labyrinth, respectively. Stripe wrinkles are observed to initiate at the inner surface of the torus while hexagon wrinkles start from the outer surface as a result of curvature-dependent stresses in the torus. We further discuss the effects of elasticities and geometries of the torus on the wrinkling patterns.

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Source: https://tomesphere.com/paper/1901.01129