Bulletin of the American Physical Society
APS March Meeting 2021
Volume 66, Number 1
Monday–Friday, March 15–19, 2021; Virtual; Time Zone: Central Daylight Time, USA
Session E16: GeometricallyFrustrated Instabilities in Solid Mechanics IFocus Live

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Sponsoring Units: GSNP DSOFT Chair: Dominic Vella, University of Oxford 
Tuesday, March 16, 2021 8:00AM  8:36AM Live 
E16.00001: Smectic Description of Wrinkle Patterns Invited Speaker: Hillel Aharoni A thin elastic sheet attached to a soft substrate often develops wrinkle patterns when subject to an external forcing or as a result of geometric incompatibility. Such patterns appear spontaneously in a variety of natural systems, ranging from plant tissues to drying paint and from milk skin to human skin. The interplay between elasticity and substrate response favors, at short length scales, a pattern of straight equally spaced wrinkles. This locally preferred order, akin to a twodimensional smectic liquid crystal, is often in conflict with a largescale geometric mismatch or a global constraint. The competition gives rise to a wide variety of complex wrinkle patterns, where different intermediatescale motifs may dominate at different parameter regimes. Among these motifs one finds smooth distortions of the smectic ground state, proliferation of point defects accompanied by amplitude variations, and sharply defined domains separated by thin domain walls. 
Tuesday, March 16, 2021 8:36AM  8:48AM Live 
E16.00002: Exact solutions for wrinkle patterns from geometrically incompatible confinement Ian Tobasco, Yousra Timounay, Desislava V Todorova, Graham C Leggat, Joseph Paulsen, Eleni Katifori Thin elastic shells readily wrinkle in complex and potentially controllable ways. A reliable way of making wrinkles appear is to impose an overall shape on the shell that is different from its natural one. For instance, a shell cut out of a sphere and put onto a planar water bath tends to wrinkle in a mixed “ordereddisordered” fashion, wherein one part exhibits a robust response and a second part behaves statistically instead. In contrast, a shell cut out from a saddle tends to exhibit a totally ordered response with a welldefined wrinkle pattern throughout. We present a simple yet complete set of geometric rules for determining the direction of wrinkling that emerges in a general spherical or saddleshaped shell. We show how the patterned response is set in any case by the medial axis, also known as the skeleton, of the shell. This distinguished, onedimensional set can be the target of control. Underlying this result is a heretofore unnoticed reciprocity between positively and negatively curved wrinkle patterns, as well as a general method based on Lagrange multipliers for finding what we call the shell's "locking stress". 
Tuesday, March 16, 2021 8:48AM  9:00AM Live 
E16.00003: Modeling the behavior of inclusions in circular plates undergoing 2Dto3D shape changes Oz Oshri, Santidan Biswas, Anna Balazs Growth of biological tissues and shape changes in thin synthetic sheets are commonly driven by stimulation of isolated regions (inclusions) in the growing body. These inclusions apply internal forces on their surroundings that, in turn, promote 2D layers to acquire complex 3D structures. We focus on a fundamental building block of these systems, and consider a circular plate that contains an inclusion with dilative strains. We derive an analytical model that predicts the 2Dto3D shape transitions in the system. The solution of our model reveals two distinct patterns in the postbuckling region of the plate. One is an extensive pattern that holds close to the threshold of the instability, and the second is a localized pattern, which preempts the extensive solution beyond the buckling threshold. Then, we utilize these findings to analyze the interaction between two circular inclusions that are embedded in an infinite plate and undergo a buckling instability into a localized state. We show that when the two inclusions are separated far apart the interaction energy between the inclusions decays exponentially. 
Tuesday, March 16, 2021 9:00AM  9:12AM Live 
E16.00004: Developable zigzag wrinkling in a thin annulus contracted inwards Anshuman Pal, Luka Pocivavsek, Thomas Witten The Lamé system – an annulus subjected to radial tensile loads at both boundaries so that the material circles wrinkle – serves as a prototype for studying twodimensional (2d) wrinkling. While this system has been extensively studied [1], an open question is whether it can attain a fully strainfree (or developable) state postbuckling. In this work, we analyse a novel geometric Lamé system that displays such a strainfree state of azimuthal wrinkling. Here, we pull in the inner boundary of a thin, unsupported elastic annulus by a radial distance Δ, leaving the outer boundary free. Using finiteelement simulations, we show that, as Δ is increased, the annulus transitions from a smooth radially wrinkled configuration to a faceted developable morphology of regularly alternating flat triangles and radial cones, with the cone tips located on the inner boundary, and with wavelength much shorter than the radius. We account theoretically for the observed scaling behaviour of the energy and stress, and offer a novel kinetic mechanism for the wavelength selection based on the structure of wrinklons [2]. 
Tuesday, March 16, 2021 9:12AM  9:24AM Live 
E16.00005: Wrinkling of viscous bubble films Alexandros Oratis, John W M Bush, Howard A Stone, James C Bird Mechanical instabilities have recently seen a resurgence in their application towards the design of new complex materials. Even though these instabilities were traditionally considered a route towards failure, they are now being utilized to design structures with advanced functionality. The wrinkling pattern adopted by a thin elastic sheet when subjected to compression, is such a mechanical instability that has been extensively studied and used to design flexible electronics and surfaces with tunable patterns. The wrinkling instability can also be observed in thin viscous sheets. A bubble resting at the surface of a very viscous liquid develops a wrinkling pattern during its collapse. Here we show that the underlying wrinkling dynamics of viscous films are similar to those in elastic sheets. We combine experiments and analytical modeling to show that the number of wrinkles follows a scaling arising from the simultaneous balance of compression, bending and inertia. 
Tuesday, March 16, 2021 9:24AM  9:36AM Live 
E16.00006: Two instabilities underlying curvatureinduced delamination patterns Benjamin Davidovitch, Finn Box, Vincent Demery, Dominic Vella We adderess the delamination of a thin solid film from an adhesive, sphericallyshaped substrate. Scaling arguments suggest that delamination may occur through two distinct instabilities. The first is local  triggered solely by a compressive, curvatureinduced component of the film stress. This instability is akin to the delamination of a uniaxiallycompressed film from a flat substrate, and is thus reminiscent of Euler buckling and wrinkling instabilities, hence we refer to it as ``Eulerlike". The second instability occurs when the total strain energy, associated with both tensile and compressive components of the stress and integrated over the film, reaches a a separate threshold. We refer to this instability as ``Gausslike". 
Tuesday, March 16, 2021 9:36AM  9:48AM Live 
E16.00007: Measuring the liftoff force for d cones Tianyi Guo, Xiaoyu Zheng, Peter PalffyMuhoray The buckling of slender objects is of considerable current interest [1] [2]. One example is the formation of d cones: a normal force applied to the center of a thin circular elastic disk supported by the rim of a hollow right circular cylinder at first results in a cylindrically symmetric cone. At a critical force, the cone buckles, and a portion of the disk lifts from the rim, forming a d cone. We have measured the force as function of displacement of the cone center for Mylar disks, and determined the liftoff force as function of radius and thickness. The results follow a simple scaling law. 
Tuesday, March 16, 2021 9:48AM  10:00AM Live 
E16.00008: Finite Curved Creases in Infinite Isometric Sheets Aaron Mowitz, Thomas Witten Geometric stress focusing, e.g. in a crumpled sheet, creates pointlike vertices that terminate in a characteristic local crescent shape. The observed scaling of the size of this crescent is an open question in the stress focusing of elastic thin sheets. We address this question by modeling the observed crescent with a more geometric approach: we treat the crescent as a curved crease in an isometric sheet. Although curved creases have already been studied extensively, the crescent in a crumpled sheet has features not previously addressed. These features together with the general constraints of isometry lead to constraints linking the surface profile to the creaseline geometry. We construct several examples obeying these constraints, showing finite curved creases are fully realizable. This approach gives new information about the asymptotic shape of the crease, and provides a new viewpoint for explaining the scaling of the crescent size. 
Tuesday, March 16, 2021 10:00AM  10:12AM Live 
E16.00009: Instability of thin sheets under torquefree geometricallyincompatible confinement Meng Xin, Benjamin Davidovitch Geometricallyincompatibe confinement (GIC) of thin sheets and shells, namely, imposing a nonzero Gaussian curvature different from the 
Tuesday, March 16, 2021 10:12AM  10:24AM Live 
E16.00010: Counting wrinkles on cones Xiaoxiao Zhang, Joseph Paulsen, Teng Zhang Wrinkles on curved surfaces have a rich phenomenology and are relevant to various biological and engineering structures. Consider the problem of a soft elastic cone covered by a stiff thin sheet with excess circumferential length. When moving down along the surface of the cone away from the tip, the changing circumference leads to an incompatibility between wrinkles of fixed wavelength (favored by a local balance of the elastic energy of the sheet with the substrate) and a fixed number of wrinkles (which would avoid the energy of inserting new wrinkles into the pattern). This can create a library of very rich wrinkling patterns, which might be connected to hierarchical microchannels found on the Sarracenia trichome. Here, we carry out largescale finite element simulations to systematically examine the effect of the cone angle on the wrinkle splitting along the axial and circumferential directions. For small cone angles, we observe a hierarchical pattern with defectfree zones (constant wrinkle wavelength) separated by defectrich zones where new wrinkles emerge. Our findings closely resemble the mesoscale structure of wrinkle patterns in an annulus film floating on water and highlight the generality of the multiscale wrinkling structures governed by geometry incompatibility. 
Tuesday, March 16, 2021 10:24AM  10:36AM Live 
E16.00011: CurvatureDriven Propulsion of Floating Films Monica Ripp, Joseph Paulsen Small floating objects often clump together, due to the longrange capillary interactions between their boundary menisci. Previous studies of this “Cheerios effect” [1] have focused on rigid solids, but much less is known about analogous behaviors for floating objects that are easily deformed by surface tension. Here the behaviors may be much more subtle, and can be strongly influenced by geometric incompatibilities between the solid and the fluid. We focus on a model system where a thin polymer film (~100 nm) is confined to the curved water interface in an overfilled petri dish. We measure the trajectories of initially planar films and curved shells released from various starting positions. By altering the geometry and thickness of the films and the viscosity of the fluid, we build an empirical description of this system, including regimes where the drag force is dominated by either the liquid viscosity or inertia. In all cases, the spontaneous translation is directed towards a region where the curvature of the meniscus is similar to the rest curvature of the film. This suggests a novel route to fast, parallel sorting of interfacial shells by their rest curvature. 
Tuesday, March 16, 2021 10:36AM  10:48AM Live 
E16.00012: Solid Domains on Giant Lipid Vesicles: Growth of 2D Crystals on Curved Surfaces Hao Wan, Maria Santore The growth of twodimensional crystals on surfaces has aroused much attention in recent decades. Theory predicts, and early experiments on 2D colloidal assembly are starting to show, how evolving crystal patterns on surfaces of nonzero Gaussian curvature produce complex patterns. In this study, we investigate how solid lipid domains grow in the nanometricallythin bilayer membranes of giant unilamellar vesicles and how the growth pattern results from the interplay of membrane tension and curvature. This system is interesting and sophisticated because it differs from colloidal system in the following aspects including the smaller molecular rather than colloidal dimensions of building blocks and tension at the edge of the solid domains. Here, we focus on how solid domains assume a flowerlike shape as they grow on giant vesicles. We find that solid domains appear hexagonal when grown under low tension but grows into 6petal flowers after reaching critical size under high tension, with the petals bending to envelop the vesicle. By varying the membrane tension and curvature, flowers with different petal length and secondary features can be observed. 
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