Wrinkles

Wrinkles: what are they and why are they interesting?

A thin object like a beam or a sheet, subjected to a sufficiently large compressive force, will buckle out of plane. Usually, this results in a sinusoidal shape whose wavelength is equal to the beam length (or 2x or 4x its value, depending on the boundary conditions). However, sometimes environmental factors lead to the creation of a very different sinusoidal shape, with many wrinkles covering a surface. This phenomenon occurs in nature, especially in biology, but is also of value for engineering. The mechanics of how and why wrinkles form is particularly intereting to me as an applied mathematician!

Wrinkle branching: patterns that vary in space

In this project we studied how wrinkles merge or branch in systems where material properties (and hence wrinkle wavelengths) vary in space.

We found that even very complicated branching profiles can be described as a superposition of sinusoidal modes, whose amplitudes evolve in space according to rules that can be derived from the beam equation.

Wrinkling of a bilayer with spatially varying stiffness: from wrinkle branching to cascades, D. O’Kiely, M. Xin, P. Lecointre & D. Vella, Proceedings of the Royal Society A (2025)

The fast and flexible: dynamic buckling

In this project our team studied dynamic buckling in rings of elastic material, and in doing so developed a new, dynamic take on a classic surface tension demonstration.

There is a demonstration of surface tension commonly used in outreach, where a loop of string is placed on a soap film, where it is held and floats freely. The part of the soap film inside the string circle is then broken, and the soap film remaining outside of the string circle pulls it taut. In doing so, it demonstrates that the soap film acts to minimise its surface area.

If the soap film outside the loop is broken instead, and the inner soap film remains intact, it will pull the loop radially inward. Depending on the geometry of the loop, it can buckle in one of the two ways shown to the left. Our ultimate goal in the project was to learn about the number of oscillations in the loop during this buckling process, and it turned out to depend on the inertia in an interesting an non-trivial way. You can learn more about this system in

Dynamic buckling of an elastic ring in a soap film, F. Box, O. Kodio, D. O'Kiely, V. Cantelli, A. Goriely & D. Vella, Phys. Rev. Letters (2020)

or in the video to the left, created by Finn Box. It has also been featured by Fuck Yeah Fluid Dynamics.

Dynamic wrinkling of floating elastic sheets

Thin elastic sheets cannot sustain compressive loads, and buckle under confinement. For a thin elastic sheet floating at a liquid interface, indenting the sheet vertically creates an azimuthal confinement by drawing material radially inward. This gives rise to a distintive wrinkling pattern, with wrinkles aligned radially like the spokes on a bicycle wheel. The equilibrium wavelength of these wrinkles is determined by a balance between the bending stiffness of the sheet, which favours long-wavelength deformations, and factors such as liquid buoyancy, which favour short-wavelength deformations.

If the indentation is fast, dynamic effects come into play, and the behaviour is significantly different. This research project formed a major component of my postdoc with Dominic Vella in OCIAM at the University of Oxford. The primary experimentalist on this project was Finn Box, now at the University of Manchester.

Read about our research in more detail in:

Dynamics of wrinkling in ultrathin elastic sheets, F. Box, D. O'Kiely, O. Kodio, A. A. Castrejón-Pita & D. Vella, Proc. Nat Acad. Sci (2019)

Impact on floating thin elastic sheets, D. O'Kiely, F. Box, O. Kodio, J. Whiteley & D. Vella, Phys. Rev. Fluids (2020)

Out-of-plane buckling in glass manufacture

In addition to the thickness variations described above, glass sheets undergoing redraw may also buckle out of plane, forming ripples that render the final product unusable. These ripples form due to localized regions of lateral compression that occur when the sheet is stretched along its length. We investigated the growth of ripples due to compression in a glass sheet, and the feedback between the out-of-plane deformation and the consequent change in stresses.

Read about glass buckling in more detail in:

Out-of-plane buckling in two-dimensional glass drawing, D. O'Kiely, C. J. W. Breward, I. M. Griffiths, P. D. Howell & U. Lange, Journal of Fluid Mechanics (2019)

or in my PhD thesis

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