Curvature-Driven Morphing of Non-Euclidean Shells

Abstract

We investigate how thin structures change their shape in response to non-mechanical stimuli that can be interpreted as variations in the structure's natural curvature. Starting from the theory of non-Euclidean plates and shells, we derive an effective model that reduces a three-dimensional stimulus to the natural fundamental forms of the mid-surface of the structure, incorporating expansion, or growth, in the thickness. Then, we apply the model to a variety of thin bodies, from flat plates to spherical shells, obtaining excellent agreement between theory and numerics. We show how cylinders and cones can either bend more or unroll, and eventually snap and rotate. We also study the nearly-isometric deformations of a spherical shell and describe how this shape change is ruled by the geometry of a spindle. As the derived results stem from a purely geometrical model, they are general and scalable.

arXiv comment13 pages, 5 figures

References in corpus (9)

• Capillary origami: spontaneous wrapping of a droplet with an elastic sheet
• Elastic theory of unconstrained non-Euclidean plates
• A Geometric Theory of Growth Mechanics
• Morphing of Geometric Composites via Residual Swelling
• Conical defects in growing sheets
• Dynamics of a $Volvox$ Embryo Turning Itself Inside Out
• A Geometric Theory of Thermal Stresses
• Elasticity and Glocality: Initiation of Embryonic Inversion in $Volvox$
• Pattern selection and multiscale behavior in metrically discontinuous non-Euclidean plates

Cited by in corpus (9)

• Static bistability of spherical caps
• Curvature-Induced Instabilities of Shells
• Geometry of wave propagation on active deformable surfaces
• Snapping of Bistable, Prestressed Cylindrical Shells
• Morphoelasticity of Large Bending Deformations of Cell Sheets during Development
• The nonlinear buckling behavior of a complete spherical shell under uniform external pressure and homogenous natural curvature
• A tug-of-war between stretching and bending in living cell sheets
• Some observations on variational elasticity and its application to plates and membranes
• Energies for elastic plates and shells from quadratic-stretch elasticity