Distributed branch points and the shape of elastic surfaces with constant negative curvature

Abstract

We develop a theory for distributed branch points and investigate their role in determining the shape and influencing the mechanics of thin hyperbolic objects. We show that branch points are the natural topological defects in hyperbolic sheets, they carry a topological index which gives them a degree of robustness, and they can influence the overall morphology of a hyperbolic surface without concentrating energy. We develop a discrete differential geometric (DDG) approach to study the deformations of hyperbolic objects with distributed branch points. We present evidence that the maximum curvature of surfaces with geodesic radius $R$ containing branch points grow sub-exponentially, $O(e^{c\sqrt{R}})$ in contrast to the exponential growth $O(e^{c^{\prime }R})$ for surfaces without branch points. We argue that, to optimize norms of the curvature, i.e. the bending energy, distributed branch points are energetically preferred in sufficiently large pseudospherical surfaces. Further, they are distributed so that they lead to fractal-like recursive buckling patterns.

arXiv comment59 pages, 20 figures. Major revisions including new proofs with weakened hypotheses, expanded discussion and additional references. Some images are not at their original resolution to keep them at a reasonable size. Comments are very welcome and much appreciated

References in corpus (15)

• Elastic theory of unconstrained non-Euclidean plates
• Geometrically incompatible confinement of solids
• Lower bounds for the energy in a crumpled elastic sheet - A minimal ridge
• A Riemannian Approach to Reduced Plate, Shell, and Rod Theories
• On the Plants Leaves Boundary, "Jupe à Godets" and Conformal Embeddings
• Shape selection in non-Euclidean plates
• Subdivision Shell Elements with Anisotropic Growth
• The shape of the edge of a leaf
• Discretization of asymptotic line parametrizations using hyperboloid patches
• Isometric immersions, energy minimization and self-similar buckling in non-Euclidean elastic sheets
• Scaling laws for non-Euclidean plates and the $W^{2,2}$ isometric immersions of Riemannian metrics
• Defects and boundary layers in non-Euclidean plates
• Isometric bending requires local constraints on free edges
• Continuum mechanics of moving defects in growing bodies
• Buckling and wrinkling from geometric and energetic viewpoints

Cited by in corpus (1)

• Nature's forms are frilly, flexible, and functional