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# The Mathematics of Wrinkles and Folds

The following is a short introduction to an invited lecture to be presented at the upcoming 2018 SIAM Annual Meeting (AN18) in Portland, Ore., from July 9-13. Look for feature articles by other AN18 invited speakers introducing the topics of their talks in future issues.

Robert V. Kohn, New York University.
The wrinkling and folding of thin elastic sheets is familiar to most: our skin wrinkles, a crumpled sheet of paper folds, and a flat sheet stretched over a round surface must wrinkle or fold. Similar patterns occur in a wide variety of settings. Their study is important for a number of reasons:

(a) In some situations, wrinkling patterns are highly ordered and reproducible. Such patterns can be useful when measuring the physical properties of sheets or designing templates for self-assembly, for example.

(b) Wrinkled configurations are local minima of a variational problem—the elastic energy of the sheet—with a rather special structure. Understanding their properties is a problem in energy-driven pattern formation, a current frontier in the calculus of variations.

(c) We would like to understand the features of low-energy configurations in specific settings; this will help us separate universal phenomena from those that depend, for example, on the history of loading.

(d) Strong analogies exist between the wrinkling of elastic sheets and pattern formation in other physical systems, such as liquid crystals, ferromagnets, and superconductors. Progress in any of these areas has the potential to yield insight for the others.

What kind of mathematics is this? The elastic energy of a thin sheet consists of a nonconvex membrane energy (which prefers isometry) plus a small coefficient times bending energy (which penalizes curvature). The bending term is a singular perturbation; its small coefficient is the sheet thickness squared. The patterns and defects in thin sheets arise from energy minimization — but not in the same way that minimal surfaces arise from surface area minimization. Rather, analysis of wrinkles and folds involves the asymptotic character of minimizers in the limit as the sheet thickness tends to zero.

When a disk-shaped flat sheet is wrapped around a sphere, it wrinkles to avoid compression in the longitudinal direction. Image courtesy of [1].
What kind of methods are useful? Simulation is of limited utility, since the problems are highly nonconvex and very stiff. Bifurcation theory is also of limited utility, because the configurations of interest lie deep in the bifurcation diagram. Instead, focusing on the energy scaling law—the minimum energy’s dependence upon the thickness of the sheet (and other relevant physical parameters)—has been fruitful. Optimizing the energy within a particular ansatz gives an upper bound on the minimum energy. Obtaining ansatz-free lower bounds is a key mathematical challenge. The lower and upper bound demonstrate the adequacy of the ansatz when they are close to agreement, and the underlying arguments help explain why certain configurations are preferred.

Wrinkling is observed in a diverse array of different situations. When a sheet is under tension, any wrinkles must be parallel to the tensile direction, and understanding the length scale of wrinkling is the main goal. The situation is more complicated when wrinkling serves to avoid biaxial compression. In such situations, even the orientation of the wrinkling is far from clear.

The 2018 SIAM Annual Meeting is co-located with the 2018 SIAM Conference on Mathematical Aspects of Materials Science. My invited talk, which is part of both meetings, will develop the aforementionted topics, focusing on recent examples in which the identification of energy scaling laws has produced some interesting surprises.

References
[1] Bella, P., & Kohn, R.V. (2017). Wrinkling of a thin circular sheet bonded to a spherical substrate. Phil. Trans. Roy. Soc. A, 375(2093), 20160157.

 Robert V. Kohn is the Silver Professor of Mathematics in the Department of Mathematics at New York University's Courant Institute of Mathematical Sciences.