Bulletin of the American Physical Society
2009 APS March Meeting
Volume 54, Number 1
Monday–Friday, March 16–20, 2009; Pittsburgh, Pennsylvania
Session D9: Elasticity and Geometry of Thin Objects I |
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Sponsoring Units: GSNP Chair: Dominic Vella, DAMTP, Cambridge and LPS ENS, Paris Room: 303 |
Monday, March 16, 2009 2:30PM - 2:42PM |
D9.00001: Can one hear a Kolmogorov Spectrum? Sergio Rica I will talk about a work in collaboration with G. During and C. Josserand on the long- time evolution of waves of a thin elastic plate in the limit of small deformation so that modes of oscillations interact weakly. According to the theory of weak turbulence (successfully applied in the past to plasma, optics, and hydrodynamic waves), this nonlinear wave system evolves at long times with a slow transfer of energy from one mode to another. We derived a kinetic equation for the spectral transfer in terms of the second order moment. We show that such a theory describes the approach to an equilibrium wave spectrum and represents also an energy cascade, often called the Kolmogorov-Zakharov spectrum. We perform numerical simulations that confirm this scenario. Finally, I will discuss recent experiments by A. Boudaoud and collaborators and N. Mordant. [Preview Abstract] |
Monday, March 16, 2009 2:42PM - 2:54PM |
D9.00002: Dynamical Origami Christophe Josserand, Arnaud Antkowiak, Basile Audoly, S\'ebastien Neukirch A drop falling on a thin elastic sheet is rapidly trapped after impact by self-folding of the sheet around the drop. This trapping process, due to capillary forces, occurs on the fast timescale of hydrophobic rebound. The resulting packed drop presents a complex three-dimensional shape, characteristic of the interplay between elasticity and capillarity (Py \textit{et al.}, \textit{Phys. Rev. Lett.} \textbf{98}, 2007). We study experimentally the encapsulation dynamics with high-speed video camera. A shape selection exhibited by the system is evidenced. The role played by the different parameters of impact (drop radius, impact velocity...) in the final shape of this ``dynamical origami'' is eventually discussed. [Preview Abstract] |
Monday, March 16, 2009 2:54PM - 3:06PM |
D9.00003: Mass distribution and geometry of a crumpled ball Anne Dominique Cambou, Narayanan Menon We use X-ray CT scanning to resolve in 3-dimensions the conformation of aluminum sheets with diameters L=7cm to 10cm and thickness T=25 microns, crumpled into spheres with diameters D=1.2cm to 1.5cm. The linear resolution of the reconstructed images is less than 6 microns/voxel. Measurements were made on spheres with average volume fractions, $\phi $ ranging from 0.06 to 0.11. The mass is not homogeneously distributed in the volume: when averaged over several samples, the volume fraction $\phi $(L/D) is found to increase with radius so that the sphere is densest at its surface. The radial dependence of volume fraction appears to be independent of average volume fraction and diameter, D. We also report preliminary measurements of the distribution of curvature in the sphere. [Preview Abstract] |
Monday, March 16, 2009 3:06PM - 3:18PM |
D9.00004: Stress relaxation in thin crumpled sheets Ingo Dierking, Paul Archer Compression of thin crumpled sheets subjected to a constant weight shows a wide range of scaling, covering up to five orders of magnitude [1], i.e. time scales from seconds to weeks. We demonstrate that this scaling behaviour is not smooth, but rather interrupted by sudden changes in height of the uniformly compressed crumple, which we attribute to sudden ridge collapses. Interestingly, when plotting the time laps between successive discontinuous ridge collapses as a function of time, the data falls onto a single linear functionality for all polymer film thicknesses, with a slope of d$\Delta $t/dt=1 over a scaling regime of four orders of magnitude.[2] Further, we investigate the scaling behaviour of thin sheets of different metals to elucidate a possible relation between the scaling parameter and the Young's modulus. Preliminary experiments suggest that scaling is a linear function of the elastic modulus. [1] K. Matan, R.B. Williams, T.A. Witten, S.R. Nagel, Phys. Rev. Lett., \textbf{88}, (2002), 076101. [2] I. Dierking, P. Archer, Phys. Rev. E, \textbf{77}, (2008), 051608. [Preview Abstract] |
Monday, March 16, 2009 3:18PM - 3:30PM |
D9.00005: The compensation of Gaussian curvature in developable cones is local Jin Wang, Thomas Witten We use the angular deficit scheme[1] to determine numerically the distribution of Gaussian curvature in developable cones(d-cones)[2] formed by forcing a flat elastic sheet into a circular container so that the sheet buckles. This provides a new way to confirm the vanishing of mean-curvature[3] at the rim where the sheet touches the container. This angular deficit scheme also allows us to explore the potential role of the Gauss-Bonnet theorem in explaining the mean-curvature vanishing phenomenon. The theorem's global constraint on curvature resembles the global conditions observed to be relevant for vanishing mean curvature. However, our result suggests that the Gauss-Bonnet theorem does not explain the vanishing of mean-curvature. \newline [1] V. Borrelli, F. Cazals, and J.-M. Morvan, {\sl Computer Aided Geometric Design} {\bf 20}, 319 (2003). \newline [2] E. Cerda, S. Chaieb, F. Melo, and L. Mahadevan, {\sl Nature} {\bf 401}, 46 (1999). \newline [3] T. Liang and T. A. Witten, {\sl Phys. Rev. E} {\bf 73}, 046604 (2006). [Preview Abstract] |
Monday, March 16, 2009 3:30PM - 3:42PM |
D9.00006: Buckling Thin Disks and Ribbons with Non-Euclidean Metrics Christian Santangelo I consider the problem of a thin membrane on which a metric has been prescribed, for example by lithographically controlling the local swelling properties of a polymer thin film. While any amount of swelling can be accommodated locally, geometry prohibits the existence of a global strain-free configuration. To study this geometrical frustration, I introduce a perturbative approach. I compute the optimal shape of an annular, thin ribbon as a function of its width. The topological constraint of closing the ribbon determines a relationship between the mean curvature and number of wrinkles that prevents a complete relaxation of the compression strain induced by swelling and buckles the ribbon out of the plane. These results are then applied to thin, buckled disks, where the expansion works surprisingly well. I identify a critical radius above which the disk in-plane strain cannot be relaxed completely. [Preview Abstract] |
Monday, March 16, 2009 3:42PM - 3:54PM |
D9.00007: Fragmentation of an elastica Nicolas Vandenberghe, Emmanuel Villermaux When a thin rod is submitted to an axial force greater than its critical buckling load it takes the shape of an {\it elastica}. As the load further increases, a rod made of a brittle material breaks suddenly. More than two fragments may be formed during this fragmentation. In this work we discuss the sequence of events that lead to the final broken state with two or more fragments. We show that the criterion for breaking is not trivial. In particular, we investigate the effect of the duration of the loading and we show that at a given load the waiting time before breaking is broadly distributed. We discuss the consequences of the time delayed breaking on the distributions of fragment sizes and fragment numbers. [Preview Abstract] |
Monday, March 16, 2009 3:54PM - 4:06PM |
D9.00008: Relaxation of a plastic fold Morgan Cervo, Narayanan Menon Crumpled objects have been observed to show stress relaxation when confined to a constant volume, and to show creep when subjected to a constant load. These relaxation processes are described by logarithmic (or other similarly slow) functional dependences on waiting time. In an effort to understand the microscopic elements responsible for this slow collective relaxation, we study the mechanics of a single fold in a thin strip of polycarbonate sheet (typical dimensions: thickness t=0.127 mm, length L=14 cm, and width w=2cm). We create folds of different initial opening angles by placing the strip under varying loads. We then measure the opening angle as a function of time. We find that even one isolated fold is sufficient to mimic the relaxation behavior of the composite crumpled sheet: the unfolding process is logarithmic in time. The unfolding rate depends on sheet thickness, but surprisingly is independent of initial opening angle. We have observed qualitatively similar behavior in metal and paper sheets. [Preview Abstract] |
Monday, March 16, 2009 4:06PM - 4:18PM |
D9.00009: Shape and trajectory of a tumbling elastic sheet of paper Mike Robitaille, Arshad Kudrolli Inspired by wind dispersal of winged seeds and gliders, we study the flight of a tumbling piece of paper to explore the competing effect of inertia, lift, drag, and elasticity on its aerodynamics. Above a critical aspect ratio, a rigid rectangular sheet is well known to exhibit autorotation, leading to a lift force which causes it to drift away from the vertical as it falls through air. Less known is that the fact that the sheet buckles and bends along the axis of rotation when the rigidity of the sheet is reduced. We measure the deflection of the paper as a function of aspect ratio, and find its speed and angle of descent with high speed imaging. We find that the rotation speed is lower when the sheet is bent, than when it is unbent. The sheet deflection increases above a critical aspect ratio reaching a maximum before decreasing. The angle of descent is well described by a simple model balancing the gravitational, lift and drag forces acting on the sheet. [Preview Abstract] |
Monday, March 16, 2009 4:18PM - 4:30PM |
D9.00010: Shape and trajectory of a tumbling elastic sheet of paper II Daniel Tam, Michael Robitaille, Arshad Kudrolli, John Bush We investigate the dynamical coupling between the tumbling motion and elastic deformation of a paper strip freely falling in air. Recent experiments suggest the existence of a critical length above which the strip bends as it tumbles. We demonstrate that this bending is caused by the centripetal force associated with its tumbling motion. A simple theory predicts that bending occurs above a critical length in much the same way that buckling occurs in a compressed beam. We further discuss the influence of bending on the trajectory of paper strips, as well as biological implications for the dispersal of seed pods. [Preview Abstract] |
Monday, March 16, 2009 4:30PM - 4:42PM |
D9.00011: Nonlinear response of tensed membranes Peker Milas, Benny Davidovitch We study the response of elastic membranes under tension$ T$, to localized normal forces $F$. Focusing on simple geometries, characterized by translational or radial symmetries, we calculate the membrane shape for a range of values of F and T by numerically solving the appropriate FvK equations. We find that the linear regime, where membrane displacement is proportional to F, vanishes in the asymptotic limit$F/T<<1$, and characterize scaling properties of the resulting nonlinear response. We discuss the relevance of our results to the puzzling scaling behavior of the length of radial wrinkles, recently found in ``drop on membrane'' experiments (Huang\textit{ et al. }Science 2007). [Preview Abstract] |
Monday, March 16, 2009 4:42PM - 4:54PM |
D9.00012: Period Fissioning and Other instabilities of stressed elastic membranes Benny Davidovitch We study the shapes of elastic membranes under the simultaneous exertion of tensile and compressive forces when the translational symmetry along the tension direction is broken. We predict a multitude of novel morphological phases in various regimes of a 2-dimensional parameter space$(\varepsilon ,\nu )$, defined by the relevant mechanical and geometrical conditions. The parameters $\varepsilon ,\nu $are, respectively, the ratio between compression and tension, and the wavelength contrast along the tension direction. In particular, our theory associates the repetitive period fissioining pattern, recently observed on wrinkled membranes floating on liquid and subject to capillary forces (J. Huang \textit{et al.}) to the morphology in the asymptotic regime $(\varepsilon <<1,\nu >>1)$where tension is dominant and the wavelength contrast is large. [Preview Abstract] |
Monday, March 16, 2009 4:54PM - 5:06PM |
D9.00013: Experimental study of the dynamics of crumpling Hillel Aharoni, Eran Sharon We experimentally measure the temporal evolution of crumpled configurations of thin elastic sheets. In our experiment, elastic hydrogel sheets swell inside a hard spherical shell, free of gravitational and plastic effects. We observe the dynamic evolution of structures in the sheet as confinement ratio increases, and analyze the statistical nature of the elastic energy localization around singularities. [Preview Abstract] |
Monday, March 16, 2009 5:06PM - 5:18PM |
D9.00014: Geometry Induced Charge Separation on a Helicoidal Ribbon Avadh Saxena, Victor Atanasov, Rossen Dandoloff Helical ribbons are ubiquitous in nature including in the carbon based nanostructures such as graphene. We derive an effective geometry-induced quantum potential for a particle confined on a helicoidal ribbon. This potential leads to the appearance of localized states at the rim of the helicoid. In this geometry the twist of the ribbon plays the role of an effective transverse electric field on the surface and thus this is reminiscent of the quantum Hall effect. We also calculate the effective polarization and discuss the consequences of these findings. [Preview Abstract] |
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