Isometric immersions, energy minimization and self-similar buckling in non-Euclidean elastic sheets

The edges of torn plastic sheets and growing leaves often display hierarchical buckling patterns. We show that this complex morphology (i) emerges even in zero strain configurations, and (ii) is driven by a competition between the two principal curvatures, rather than between bending and stretching. We identify the key role of branch-point (or "monkey-saddle") singularities in generating complex wrinkling patterns in isometric immersions, and show how they arise naturally from minimizing the elastic energy.Comment: 6 pages, 6 figures. This article supersedes arXiv:1504.0073

Most probable transition paths in piecewise-smooth stochastic differential equations

We develop a path integral framework for determining most probable paths in a class of systems of stochastic differential equations with piecewise-smooth drift and additive noise. This approach extends the Freidlin-Wentzell theory of large deviations to cases where the system is piecewise-smooth and may be non-autonomous. In particular, we consider an $n-$dimensional system with a switching manifold in the drift that forms an $(n-1)-$dimensional hyperplane and investigate noise-induced transitions between metastable states on either side of the switching manifold. To do this, we mollify the drift and use $\Gamma-$convergence to derive an appropriate rate functional for the system in the piecewise-smooth limit. The resulting functional consists of the standard Freidlin-Wentzell rate functional, with an additional contribution due to times when the most probable path slides in a crossing region of the switching manifold. We explore implications of the derived functional through two case studies, which exhibit notable phenomena such as non-unique most probable paths and noise-induced sliding in a crossing region.Comment: 38 pages, 9 figure

Nature's forms are frilly, flexible, and functional

A ubiquitous motif in nature is the self-similar hierarchical buckling of a thin lamina near its margins. This is seen in leaves, flowers, fungi, corals and marine invertebrates. We investigate this morphology from the perspective of non-Euclidean plate theory. We identify a novel type of defect, a branch-point of the normal map, that allows for the generation of such complex wrinkling patterns in thin elastic hyperbolic surfaces, even in the absence of stretching. We argue that branch points are the natural defects in hyperbolic sheets, they carry a topological charge which gives them a degree of robustness, and they can influence the overall morphology of a hyperbolic surface without concentrating elastic energy. We develop a theory for branch points and investigate their role in determining the mechanical response of hyperbolic sheets to weak external forces. We also develop a discrete differential geometric (DDG) framework for applications to the continuum mechanics of hyperbolic elastic sheets.Comment: 35 pages, 26 figure

Tipping in a Low-Dimensional Model of a Tropical Cyclone

A presumed impact of global climate change is the increase in frequency and intensity of tropical cyclones. Due to the possible destruction that occurs when tropical cyclones make landfall, understanding their formation should be of mass interest. In 2017, Kerry Emanuel modeled tropical cyclone formation by developing a low-dimensional dynamical system which couples tangential wind speed of the eye-wall with the inner-core moisture. For physically relevant parameters, this dynamical system always contains three fixed points: a stable fixed point at the origin corresponding to a non-storm state, an additional asymptotically stable fixed point corresponding to a stable storm state, and a saddle corresponding to an unstable storm state. The goal of this work is to provide insight into the underlying mechanisms that govern the formation and suppression of tropical cyclones through both analytical arguments and numerical experiments. We present a case study of both rate and noise-induced tipping between the stable states, relating to the destabilization or formation of a tropical cyclone. While the stochastic system exhibits transitions both to and from the non-storm state, noise-induced tipping is more likely to form a storm, whereas rate-induced tipping is more likely to be the way a storm is destabilized, and in fact, rate-induced tipping can never lead to the formation of a storm when acting alone. For rate-induced tipping acting as a destabilizer of the storm, a striking result is that both wind shear and maximal potential velocity have to increase, at a substantial rate, in order to effect tipping away from the active hurricane state. For storm formation through noise-induced tipping, we identify a specific direction along which the non-storm state is most likely to get activated

Shape selection in non-Euclidean plates

We investigate isometric immersions of disks with constant negative curvature into $\mathbb{R}^3$, and the minimizers for the bending energy, i.e. the $L^2$ norm of the principal curvatures over the class of $W^{2,2}$ isometric immersions. We show the existence of smooth immersions of arbitrarily large geodesic balls in $\mathbb{H}^2$ into $\mathbb{R}^3$. In elucidating the connection between these immersions and the non-existence/singularity results of Hilbert and Amsler, we obtain a lower bound for the $L^\infty$ norm of the principal curvatures for such smooth isometric immersions. We also construct piecewise smooth isometric immersions that have a periodic profile, are globally $W^{2,2}$, and have a lower bending energy than their smooth counterparts. The number of periods in these configurations is set by the condition that the principal curvatures of the surface remain finite and grows approximately exponentially with the radius of the disc. We discuss the implications of our results on recent experiments on the mechanics of non-Euclidean plates