Non-uniform curvature and anisotropic deformation control wrinkling patterns on tori

Xiaoxiao Zhang, Patrick T. Mather, Mark John Bowick, Teng Zhang

Research output: Contribution to journalArticle

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 biology, 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 zigzags and segmented labyrinths, 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.

Original languageEnglish (US)
Pages (from-to)5204-5210
Number of pages7
JournalSoft Matter
Volume15
Issue number26
DOIs
StatePublished - Jan 1 2019

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wrinkling
curvature
Stiffness
hexagons
Elasticity
stiffness
Elastic moduli
Geometry
labyrinth
chickens
embryos
biology
Direction compound
elastic properties
shear
radii

ASJC Scopus subject areas

  • Chemistry(all)
  • Condensed Matter Physics

Cite this

Non-uniform curvature and anisotropic deformation control wrinkling patterns on tori. / Zhang, Xiaoxiao; Mather, Patrick T.; Bowick, Mark John; Zhang, Teng.

In: Soft Matter, Vol. 15, No. 26, 01.01.2019, p. 5204-5210.

Research output: Contribution to journalArticle

Zhang, Xiaoxiao ; Mather, Patrick T. ; Bowick, Mark John ; Zhang, Teng. / Non-uniform curvature and anisotropic deformation control wrinkling patterns on tori. In: Soft Matter. 2019 ; Vol. 15, No. 26. pp. 5204-5210.
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