Date of Award

December 2020

Degree Type


Degree Name

Doctor of Philosophy (PhD)


Mechanical and Aerospace Engineering


Teng Zhang


compression, curved surface, gradient curvature, Neo-Hookean, wrinkles

Subject Categories



This work focuses on understanding the wrinkling patterns on a curved surface, i.e., tori and cones, with varying curvatures and global deformation. By tuning the geometry confine-ment and material properties, we can control the surface morphologies in these systems which can potentially be applied to wide applications in bio-inspired devices, microfluidic, and optical.

First, we investigate the wrinkles on a tri-layer torus consisting of an expanding thin outer layer, an intermediate soft layer, and an inner core with a tunable shear modulus. This study is inspired by pattern formations and evolutions 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 direc-tion 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 the 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 be-cause of curvature-dependent stresses in the torus. We further discuss the effects of elasticities and geometries of the torus on the wrinkling patterns.

The second study focuses on wrinkles on a cone with a variable mean curvature but zero Gaussian curvature. Compared to a torus whose Gaussian curvature and mean curvature vary with surface location simultaneously, the zero Gaussian curvature of cones can enable us to ex-plore the mechanisms of governing the pattern formations and evolutions on curved structures in a more controlled way. Besides, the wrinkles on cone-like structures share similar patterns to the micro-channels found in plants. The cones we adopt in this study are truncated () for the sake of computation convergence and are composed of two layers, a stiff thin film bonded to a compliant substrate and the thin film is under uniform expansion to drive the wrinkling instabil-ities. To simplify the work, we maintain the same material properties of the system while first changing the top radius which affects the wrinkling patterns on the axial surface. Wrinkling pat-tern transits from uniformly distributed (i.e., constant wrinkle number) to bifurcated (i.e., varied wrinkle number) during the film expanding and more bifurcations emerge on the cone with smaller in both vertical and circumferential directions. To obtain the wrinkling patterns on a larger area, we double the bottom radius that provides a lower cone angle (the angle between the inclined surface to the horizontal plane) and obtain a distinctive phenomenon, defect-free zone (same wrinkles numbers) separated by defect-rich zone (varied wrinkle numbers) uniform-ly. We establish a phase diagram of the wrinkle patterns by cone angle, which shows a larger compressive strain is needed for a higher angle to trigger the wrinkling instability both uniform wrinkle and bifurcated wrinkle. Further, we discuss the effect of the boundary condition on the wrinkling patterns by comparing the results of whether the bottom surface is all-direction fixed or vertical direction fixed.

The reveal of the wrinkling patterns on the curved surfaces will shed light on the funda-mental understanding of the morphogenesis in animals and plants. The knowledge generated from our simulations can further provide guidelines for designing multifunctional engineering devices with novel surface morphologies, which can be potentially generated by the wrinkling mechanism.


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Available for download on Friday, January 27, 2023

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