Bulletin of the American Physical Society
APS March Meeting 2017
Volume 62, Number 4
Monday–Friday, March 13–17, 2017; New Orleans, Louisiana
Session B23: From Isometry to Reality: Geometric principles, Mechanics, and Morphology of Thin Solid StructuresInvited

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Sponsoring Units: GSOFT GSNP Chair: Benny Davidovich, University of Massachusetts, Amherst Room: New Orleans Theater B 
Monday, March 13, 2017 11:15AM  11:51AM 
B23.00001: Patterns through elastic instabilities, from thin sheets to twisted ribbons Invited Speaker: Pascal Damman Sheets embedded in a given shape by external forces store the exerted work in elastic deformations. For pure tensile forces, the work is stored as stretching energy. When the forces are compressive, several ways to store the exerted work, combining stretching and bending deformations can be explored. For large deflections, the ratio of bending, $\sim Eh^3 \zeta^2/L^4$ and stretching, $\sim Eh \zeta^4/L^4$ energies, suggests that strainfree solutions should be favored for thin sheets, provided $\zeta^2 \gg h^2$ (where $E, \zeta, L \text{ and } h$ are the elastic modulus, the deflection, a characteristic sheet size and its thickness). For uniaxially constrained sheets deriving from the Elastica, strainfree solutions are obvious, i.e., buckles, folds or wrinkles grow to absorb the stress of compression. In contrast, crumpled sheets exhibit ``origamilike'' solutions usually described as an assembly of flat polygonal facets delimitated by ridges focusing strains are observed. This type of solutions is particularly interesting since a faceted morphology is isometric to the undeformed sheet, except at those narrow ridges. In some cases however, the geometric constraints imposed by the external forces do not allow solutions with negligible strain in the deformed state.\\ \\For instance, considering a circular sheet on a small drop, so thin that bending becomes negligible, i.e., $Eh^3/\gamma L^2 \ll 1$. The capillary tension, $\gamma$ at the edge forces the sheet to follow the spherical shape of the drop. Depending on the magnitude of the capillary tension with respect to the stretching modulus, such a sheet on a sphere can be in full tension or subjected to azimuthal compression. These spherical solutions could generate a hoop stress of compression within a small strip at the sheet's edge. The mechanical response of the sheet will generate tiny wrinkles decorating the edge to relax the compression stress while keeping its spherical shape. Finally, twisting a paper ribbon under high tension spontaneously produces helicoidal shapes that also reflect stretching and bending deformations. When the tension is progressively relieved, longitudinal and transverse compressive stresses build. To relax the longitudinal stress while keeping the helicoid shape, the ribbons produce wrinkles that ultimately becomes sharp folds similar to the ridge singularities observed in crumpled paper. The relaxation of the transverse compression stress produces cylindrical solutions. All these examples illustrates the natural tendency of an elastic sheet to stay as close as possible to the imposed shape, i.e. flat, spherical, helicoid. The mechanical response of the elastic sheet aims to relieve the compressive stress by growing a given microstructure, i.e. wrinkles, singularities. In this talk, we will explore the general mechanisms at work, based on geometry and a competition between various energy terms, involving stretching and bending modes. [Preview Abstract] 
Monday, March 13, 2017 11:51AM  12:27PM 
B23.00002: Isometric immersions and selfsimilar buckling in elastic sheets. Invited Speaker: John Gemmer The edges of torn elastic sheets and growing leaves often display hierarchical selfsimilar like buckling patterns. On the one hand, such complex, self similar patterns are usually associated with a competition between two distinct energy scales, e.g. elastic sheets with boundary conditions that preclude the possibility of relieving in plane strains, or at alloyalloy interfaces between distinct crystal structures. On the other hand, within the nonEuclidean plate theory this complex morphology can be understood as low bending energy isometric immersions of hyperbolic Riemannian metrics. In particular, many growth patterns generate residual inplane strains which can be entirely relieved by the sheet forming part of a surface of revolution or a helix. In this talk we will show that this complex morphology (i) arises from isometric immersions (ii) is driven by a competition between the two principal curvatures, rather than between bending and stretching. We identify the key role of branchpoint (or monkeysaddle) singularities, in complex wrinkling patterns within the class of finite bending energy isometric immersions. Using these defects we will give an explicit construction of strainfree embeddings of hyperbolic surfaces that are fractal like and have lower elastic energy than their smooth counterparts [Preview Abstract] 
Monday, March 13, 2017 12:27PM  1:03PM 
B23.00003: Geometric charges in theories of elasticity and plasticity Invited Speaker: Michael Moshe The mechanics of many natural systems is governed by localized sources of stresses. Examples include "plastic events" that occur in amorphous solids under external stress, defects formation in crystalline material, and forcedipoles applied by cells adhered to an elastic substrate. Recent developments in a geometric formulation of elasticity theory paved the way for a unifying mathematical description of such singular sources of stress, as "elastic charges". In this talk I will review basic results in this emerging field, focusing on the geometry and mechanics of elastic charges in twodimensional solid bodies. I will demonstrate the applicability of this new approach in three different problems: failure of an amorphous solid under load, mechanics of Kirigami, and wrinkle patterns in geometricallyincompatible elastic sheets. [Preview Abstract] 
Monday, March 13, 2017 1:03PM  1:39PM 
B23.00004: Gaussian curvature and confinement in thin shells Invited Speaker: Eleni Katifori NonEuclidean shells, when confined, can deform to a broad assortment of large scale shapes and smaller scale wrinkling and folding patterns quite unlike what produced by their flat counterparts. The intrinsic, natural curvature of shells is the central element that allows for this rich morphological landscape, but it is also the source of geometric nonlinearities that renders an analytic treatment of nonEuclidean shells, even under small load, virtually intractable. Understanding the shapes of confined nonEuclidean shells frequently requires tools and approaches that might be nonstandard for flat sheets. In this talk we discuss some snapshots of the morphological landscape of confined curved shells. We use theory, simulations and experiments to explore the large scale deformation of a confined thin spherical shell with an opening. We then proceed to investigate the wrinkling patterns produced by shallow doubly curved shells when external load introduces lateral confinement. From these examples, we see Gaussian curvature emerging as a powerful tool that can shed light on phenomena inaccessible by the mechanics of flat sheets. [Preview Abstract] 
Monday, March 13, 2017 1:39PM  2:15PM 
B23.00005: Sheets shaping liquids and liquids shaping sheets Invited Speaker: Joseph Paulsen An ultrathin elastic sheet floating on a liquid surface sits between two extremes: surface tension can easily bend the film, but cannot cause macroscopic inplane stretching. We demonstrate several striking consequences of this separation of energy scales in two settings. First, we study the wrapping of a water droplet by a polystyrene film that is $\sim$100 nm thick and 3 mm in diameter. The sheet becomes patterned with smallscale wrinkles, crumples, and folds, and the resulting threedimensional shape is highly nonaxisymmetric. Remarkably, we can understand this overall shape with a simple geometric principle: a thin sheet spontaneously maximizes the volume of the enclosed liquid, given a fixed area of the initially flat sheet. Thus, in the limit of zero bending resistance, a sheet can still sculpt the shape of a liquid droplet. Second, we show how the same geometric principle leads to a thicknessindependent folding transition for an annular sheet on a flat bath. In both settings, we uncover an asymptotic regime of ultrathin films, where the gross effects are independent of the thickness, modulus, or even the wettability of the film. Such films provide a robust platform for adding further functionality. [Preview Abstract] 
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