Bulletin of the American Physical Society
2008 APS March Meeting
Volume 53, Number 2
Monday–Friday, March 10–14, 2008; New Orleans, Louisiana
Session B5: Geometry and Elasticity |
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Sponsoring Units: DCMP Chair: Mark Bowick, Syracuse University Room: Morial Convention Center RO1 |
Monday, March 10, 2008 11:15AM - 11:51AM |
B5.00001: Generalized Crumpling: induced singularities in gently deformed elastic sheets Invited Speaker: If a thin disk of elastic material is confined in a shrinking sphere, the deformation of the disk is not smooth but nearly singular when its radius becomes larger than that of the sphere: the curvature at one point diverges as the thickness goes to zero. This talk considers {\it induced} singularities that arise from the interaction of these ``vertex" singularities with their environment. For example, if two vertices are present, the curvature on the line joining them also diverges, forming the familar ridge singularity [1]. Other induced singularities are coming to light. Here we consider two such singularities. The first is the induced vertex at the boundary [2] of a disk that has been compressed until it contains two interior vertices. Asymptotically, the triangular region bounded by the three vertices becomes arbitrarily flat as the sheet thickness goes to zero, while the curvature outside approaches a nonzero limit. The second singularity appears when a vertex is formed by forcing a flat sheet into a circular ring so that the sheet buckles. Then the ring force induces a singular radial curvature in the sheet. Remarkably this curvature just sufficient to make the mean curvature vanish where the the ring contacts the sheet [3]. We explore the generality of this curvature cancellation phenomenon. \newline [1] T. A. Witten {\sl Rev. Mod. Phys} {\bf 79} 643 (2007) \newline [2] E. Cerda, S. Chaieb, F. Melo, and L. Mahadevan, {\sl Nature} London {\bf 401} 46 (1999) \newline [3] T. Liang, T. A. Witten, {\sl Phys. Rev. E} {\bf 73} 046604 (2006) [Preview Abstract] |
Monday, March 10, 2008 11:51AM - 12:27PM |
B5.00002: Columnar and crystalline monolayers on curved substrates Invited Speaker: We study thin self-assembled columns constrained to lie on a curved, rigid substrate. The curvature presents no {\sl local} obstruction to equally spaced columns in contrast to curved crystals for which the crystalline bonds are frustrated. Instead, the vanishing compressional strain of the columns implies that their normals lie on geodesics which converge (diverge) in regions of positive (negative) Gaussian curvature, in analogy to the focussing of light rays by a lens. The bending of the layers generates a pre-stress of geometric frustration in the ground state that exists prior to the inclusion of defects. This simple observation is the basis for a versatile analytical approach to calculate the geometrical forces between dislocations and Gaussian curvature in columnar as well as in crystalline monolayers. The resulting forces play an important role in stress relaxation dynamics, elastic instabilities, and melting. [Preview Abstract] |
Monday, March 10, 2008 12:27PM - 1:03PM |
B5.00003: The shape, stability and dynamics of elastic surfaces Invited Speaker: Bending a thin sheet is easier than stretching it, an observation which has its roots in geometry. We will use this fact to explain some unusual problems in biology, physics and geology. At the everyday scale, I will discuss the morphology of avascular algal blades, the dynamics of defects in an elastic ribbon, and the dynamics of prey capture by certain carnivorous plants. At the geological scale, I will try to explain the shape of island arcs on our planet. Finally, time permitting, I will discuss how we might extend these ideas to the macromolecular scale, to derive a mechanical model for the dynamic instability of a growing microtubule. [Preview Abstract] |
Monday, March 10, 2008 1:03PM - 1:39PM |
B5.00004: Elasticity and capillarity: wet hairs and origami Invited Speaker: Capillary forces are responsible for a large range of everyday observations : the shape of rain droplets, the imbibition of a sponge, the clumping of wet hair into bundles. Although they are often negligible on macroscopic structures, surface capillary forces may overcome volume forces at small scales and deform compliant micro-structures. Capillary-induced sticking can indeed prevent the actuation of mobile elements in micro-electro-mechanical systems (MEMS), or even cause their collapse. Capillary forces also have important consequences in biology such as the buckling of the airway lumen induced by surface tension, which can eventually cause the lethal closure of lung airways (known as neonatal respiratory distress syndrome). We will review a few experimental situations where capillary forces are able to deform two types of objects: rods, and thin sheets. For instance, the nanotubes of a ``carbon nanotube carpet'' self-assemble into conical ``teepee'' structures after the evaporation of a solvent and can produce intriguing cellular patterns. Similarly, macroscopic wet hairs tend to assemble into bundles through a cascade of successive pairings. Comparing attracting capillary forces to bending elasticity, leads to a characteristic ``elasto-capillary'' length. The case of thin sheets is challenging because of geometrical constrains, which generally leads to singularities. Can a thin sheet spontaneously wrap around droplet? We will describe in detail this ``capillary origami'' experiment. [Preview Abstract] |
Monday, March 10, 2008 1:39PM - 2:15PM |
B5.00005: Folding and swirling instabilities of viscous fluid threads in microchannels Invited Speaker: We study the behavior of viscous fluid threads formed by hydrodynamic focusing as they are swept along by the flow of a different outer fluid in hard microfluidic channels. By examining pairs of miscible liquids for which interfacial tension is essentially absent, such as silicone oils having different molecular weights, we reveal a rich variety of fluid instabilities that occur at low Reynolds numbers. When a single thread that propagates stably in the center of a straight channel encounters a divergence in the channel's width, the thread simply dilates if its viscosity is similar to that of the outer fluid. However, due to the extensional flow and deceleration in the diverging channel, a thread that is sufficiently viscous becomes unstable and reduces energy dissipation by performing sinuous bending oscillations, or `folding', rather than dilating. By tuning the flow rates, we reveal a novel period-doubling route to chaotic folding. The folding and stretching of a thread in a diverging channel provides a simple means of mixing viscous liquids and creating controlled viscosity gradients. Moreover, using a sequence of two cross-channels, we make a pair of viscous threads that become unstable when swept along near the walls of a straight channel as a result of the viscous torque induced by the velocity gradient. The amplification of lateral undulations ultimately causes the threads to break up and form an array of viscous swirls, the miscible counterparts of droplets. This swirling instability provides a means for producing discrete and uniform ephemeral swirls, the miscible counterpart of droplets. By injecting three different miscible liquids into a dual cross-channel geometry, we examine the complex patterns that form when several fluid instabilities interact and compete. Overall, we anticipate that these measurements will provide important insight into the behavior of flowing threads in which interfacial tension plays a more substantial role. [Preview Abstract] |
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