# Reception and Posters

Monday, May 16, 2011 - 5:00pm - 6:00pm

Lind 400

**Differential Growth and Ripples in Thin Elastic Sheets**

John Gemmer (University of Arizona)

In this poster we illustrate that isometric immersions of the hyperbolic

plane into three dimensional Euclidean space with a periodic profile

exist. These surfaces are piecewise smooth but have vastly lower bending

energy then their smooth counterparts and could explain why periodic

hyperbolic surfaces are proffered in nature.**Stable Phase-tip Splitting via Global Bifurcation**

Andras Sipos (Cornell University)

We consider multi-phase equilibria of elastic solids under anti-plane shear.We use global bifurcation methods to determine paths of equilibria in the presence of small interfacial energy. In an earlier paper the rigorous existence of global bifurcating branches was established. The stability of the solutions along these branches are difficult to determine. By an appropriate numerical representation of the second variation we show, that phase-tip splitting at the boundary (which is typically observed in experiments with shape memory alloys) appears for stable solutions of our model.**Remote Control Jell-O**

David Hamby (GELgothic)

I propose Jell-O as a building material. The concept stems from a question in blobby architecture on transformable walls. Phase transition gels are known to expand and contract up to a thousand fold. Tiny wireless stimulators mixed inside these gels could direct local shape changes. A sum of small volume changes would, in theory, yield the overall shape desired. The immediate goal of creating a set of prototype gel models was to provide visual aids as a basis for discussion with other disciplines. Starting by experimenting with rigid and flexible molds, a series of 10-centimeter jiggly gel objects was formed and photographed. Next, as a proof of concept, a 1-meter pneumatic robot was designed and constructed to demonstrate motion via selective volume displacement. Following the successes of the gel mold objects and robot control experiments, the two components will now be mixed for preliminary tests of a “slosh-bot.”**Shaping via active deformation of thin elastic sheets**

Eran Sharon (Hebrew University)

I will present our theoretical framework and experimental techniques, developed for constructing thin elastic sheets that undergo a known, nonuniform active deformation (or growth) and calculating their equilibrium configurations. The poster includes two limit examples: 1) Non-Euclidean plates, in which the lateral growth is uniform along the thickness of the sheet, but varies across its surface. 2) An incompatible shell, in which the lateral growth is uniform across the surface, but varies along the sheet thickness, leading to double spontaneous curvature. interesting configurations and transitions, relevant to biological and chemical systems will be presented**Buckling of swelled ribbons: conical normals and minimal resonances**

Abstract

Bryan Chen (University of Pennsylvania)

Differential growth processes play a prominent role in shaping leaves and biological tissues. Using both analytical and numerical calculations, we consider

the shapes of closed, elastic strips which have been subjected to an inhomogeneous pattern of swelling. The stretching and bending energies of a closed strip are frustrated by compatibility constraints between the curvatures and metric of the strip. To analyze this frustration, we study the class of “conical” closed strips with a prescribed metric tensor on their center line. The resulting strip shapes can be classified according to their number of wrinkles and the prescribed pattern of swelling. We use this class of strips as a variational ansatz to obtain the minimal energy shapes of closed strips and find excellent agreement with the results of a numerical bead-spring model.**Prestrained Kirchhoff shell theory: the derivation and the analysis**

Applying methods of Calculus of Variations, we introduce and justify a variant of Kirchhoff theory for thin 3d shells, valid in presence of residual stresses. The effective 2d energy is the relative bending, appropriately modified while in the varying thickness and/or incompressible materials situation**Metric-induced wrinkling of an elastic thin film**

Peter Bella (New York University)

We study the wrinkling of a thin elastic sheet caused by a prescribed

non-Euclidean metric. This is a model problem for the folding patterns

seen, e.g., in torn plastic membranes and the leaves of plants.

Following the lead of other authors we adopt a variational viewpoint,

according to which the wrinkling is driven by minimization of an elastic

energy subject to appropriate constraints and boundary conditions. Our

main goal is to identify the scaling law of the minimum energy as the

thickness of the sheet tends to zero. This requires proving an upper bound

and a lower bound that scale the same way. The upper bound is relatively

easy, since nature gives us a hint. The lower bound is more subtle, since

it must be ansatz-free.