Shape selection in non-Euclidean plates

• Published in: Physica. D 2011-09, Vol.240 (19), p.1536-1552
• Main Authors: Gemmer, John A., Venkataramani, Shankar C.
• Format: Article
• Language: English
• Subjects: Bending; Curvature; Disks; Exact sciences and technology; Geometry of hyperbolic surfaces; Immersion; Joints; Mathematical analysis; Morphogenesis in soft tissue; Nonlinear elasticity of thin objects; Norms; Pattern formation; Physics; Plates
• ISSN: 0167-2789; 1872-8022
• DOI: 10.1016/j.physd.2011.07.002

We investigate isometric immersions of disks with constant negative curvature into 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 H 2 into 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 ∞ 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 disk. We discuss the implications of our results on recent experiments on the mechanics of non-Euclidean plates. ► Low bending energy isometric immersions of the hyperbolic plane. ► Non-Euclidean plates and isometric immersions. ► Small-slope approximation and non-Euclidean plates. ► Periodic rippling in non-Euclidean plates.