Shape selection in non-Euclidean plates

John A. Gemmer, Shankar C. Venkataramani

Research output: Contribution to journalArticle

20 Scopus citations

Abstract

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 W2 ,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 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 W2,2, 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.

Original languageEnglish (US)
Pages (from-to)1536-1552
Number of pages17
JournalPhysica D: Nonlinear Phenomena
Volume240
Issue number19
DOIs
StatePublished - Sep 15 2011

Keywords

  • Geometry of hyperbolic surfaces
  • Morphogenesis in soft tissue
  • Nonlinear elasticity of thin objects
  • Pattern formation

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics

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