Journal
PHYSICA D-NONLINEAR PHENOMENA
Volume 240, Issue 19, Pages 1536-1552Publisher
ELSEVIER
DOI: 10.1016/j.physd.2011.07.002
Keywords
Nonlinear elasticity of thin objects; Geometry of hyperbolic surfaces; Pattern formation; Morphogenesis in soft tissue
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Funding
- US-Israel BSF [2008432]
- NSF [DMS-0807501]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [0807501] Funding Source: National Science Foundation
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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-infinity 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. (C) 2011 Elsevier B.V. All rights reserved.
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