Shape Selection in the Non-Euclidean Model of Elasticity

Abstract

In this dissertation we investigate the behavior of radially symmetric non-Euclidean plates of thickness t with constant negative Gaussian curvature. We present a complete study of these plates using the Föppl-von Kármán and Kirchhoff reduced theories of elasticity. Motivated by experimental results, we focus on deformations with a periodic profile. For the Föppl-von Kármán model, we prove rigorously that minimizers of the elastic energy converge to saddle shaped isometric immersions. In studying this convergence, we prove rigorous upper and lower bounds for the energy that scale like the thickness t squared. Furthermore, for deformation with n-waves we prove that the lower bound scales like nt² while the upper bound scales like n²t². We also investigate the scaling with thickness of boundary layers where the stretching energy is concentrated with decreasing thickness. For the Kichhoff model, we investigate isometric immersions of disks with constant negative curvature into R³, and the minimizers for the bending energy, i.e. the L² 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 H² into R³. In elucidating the connection between these immersions and the nonexistence/ singularity results of Hilbert and Amsler, we obtain a lower bound for the L^∞ 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 numerically have 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 grow approximately exponentially with the radius of the disc.