Bulletin of the American Physical Society
2009 APS March Meeting
Volume 54, Number 1
Monday–Friday, March 16–20, 2009; Pittsburgh, Pennsylvania
Session T7: Patterns on Thin Elastic Sheets |
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Sponsoring Units: GSNP DPOLY Chair: Benny Davidovitch, University of Massachusetts Room: 407 |
Wednesday, March 18, 2009 2:30PM - 3:06PM |
T7.00001: The undulating shape of growing ribbons Invited Speaker: The undulating morphology of leaves and petals is now accepted as a consequence of differential growth of the underlying tissue. Various qualitative and quantitative aspects of the buckling patterns seen in both vascular and avascular leaves may thus be ascribed to the distribution of non-uniform growth in the lamina, and have been demonstrated in normal and mutant leaves, as well as in physical models thereof. To understand the different modalities that arise quantitatively, we construct a mathematical model for the stability of an initially flat or curved elastic ribbon with gradients in growth directly motivated by observations of kelp that are capable of phenotypic plasticity in different environments. Using a combination of analysis, numerical simulation, and experimental observations, we map out the phase space of possible shapes for these growing ribbons. In general, we find that as the relative growth strain is increased, the ribbon-like structure first switches to a catenoidal shape before developing undulating edges that can develop on the catenoid's edges. Our framework allows us to delineate the few macroscopic parameters that control the morphology of elongated leaves and flower petals and helps to explain the large variety of observed shapes. [Preview Abstract] |
Wednesday, March 18, 2009 3:06PM - 3:42PM |
T7.00002: The Mechanics of Non-Euclidean Plates in Synthetic and Natural Sheets Invited Speaker: Thin elastic flat plates attain non-trivial configurations when they are confined. I will show that plates with intrinsic non-Euclidean geometry attain multi scale three-dimensional configurations even when they are free of external loading. Such bodies do not have any stress-free configuration, thus current plate theories cannot properly describe their physics. I will present our recent experimental results and our theoretical model for the shaping principles of such plates. Finally, I will show how these principles are manifested during the growth of leaves. [Preview Abstract] |
Wednesday, March 18, 2009 3:42PM - 4:18PM |
T7.00003: Wrinkling patterns on floating elastic films Invited Speaker: A polymer sheet floating on the surface of a fluid is an ideal arena for studying elastic instabilities in thin sheets. In our experiments we use polystyrene sheets whose typical lateral size, $L \quad \sim $ 3 cm, and whose thickness, $t$ ranges from 30 to 300 nm, yielding aspect ratios $L/t$ of up to 10$^{6}$. In their unperturbed state, they lie on the surface of a pool of water, stretched flat by surface tension. We can then generate a rich variety of wrinkling patterns by perturbing the surface locally with capillary forces,\footnote{J. Huang et al., Science 317, 650 (2007).} or with controlled displacements at one or more points on the surface. I will review our understanding of the length scales that characterise these localised patterns. A simple experimental setting in which a multiplicity of these length scales come into play is a situation analogous to an Euler buckling experiment performed on the surface of a fluid. We push two sides of a rectangular sheet towards each other, creating a global pattern of parallel wrinkles whose wavelength is given by a balance between gravitational potential energy of the fluid and bending energy of the sheet. These wrinkles develop a cascade of fine structure at higher wavenumbers close to the uncompressed edges of the sheet. The length scale over which this cascade occurs is the capillary length, whereas the wavenumber at the edge of the sheet reflects a balance between bending energy and surface tension. We discuss the evidence that this is a fundamentally new type of elastic cascade, which proceeds to higher wavenumbers by smooth evolution of the wrinkles, rather than by discrete, sharply localised branching. Work done in collaboration with J. Huang, E. Cerda, B. Davidovitch, W.H. de Jeu, T.P. Russell, C. D. Santangelo [Preview Abstract] |
Wednesday, March 18, 2009 4:18PM - 4:54PM |
T7.00004: Wrinkle to fold transitions: Stress relaxation in lipid monolayers and other elastic thin films Invited Speaker: Surfactants at air/water interfaces are often subjected to mechanical stresses as the interfaces they occupy are reduced in area. The most well characterized forms of stress relaxation in these systems are first order phase transitions. However, once chemical phase transitions have been exhausted, the monolayer undergoes global mechanical relaxations termed collapse. We have previously demonstrated that for lung surfactants, a mixture of lipids and proteins that coats the alveoli to reduce the work of breathing, collapse manifests itself as protrusions of folds into the subphase. These folds remain attached to the monolayer and reversibly reincorporated upon expansion. By studying different types of monolayers, we have shown that this folding transition in monolayers is not limited to lung surfactant films, but rather represents a much more general type of stress relaxation mechanism. Our study indicates that collapse modes are found most closely linked to in-plane rigidity. We characterize the rigidity of the monolayer by analyzing in-plane morphology on numerous length scales. More rigid monolayers collapse out-of-plane \textit{via} a hard elastic mode similar to an elastic membrane, with the folded state being the final collapse state, while softer monolayers relax in-plane by shearing. For the hard elastic mode of collapse, we have further demonstrated experimentally and theoretically that the folded state is preceded by a wrinkled state, and similar wrinkle to fold transitions has been observed in elastic thin films ranging from 2 nm to 10 $\mu $m in thickness of completely different chemical nature (lung surfactant lipid monolayers, gold nanoparticle trilayers, and polyester sheets). [Preview Abstract] |
Wednesday, March 18, 2009 4:54PM - 5:30PM |
T7.00005: Nonlinear dynamics of wrinkle growth and pattern formation in stressed elastic thin films on viscoelastic substrates Invited Speaker: A stressed thin film on a soft substrate can develop complex wrinkle patterns. The onset of wrinkling and initial growth is well described by a linear perturbation analysis, and the equilibrium wrinkle patterns can be analyzed based on an energy approach. In between, the wrinkle pattern undergoes a growth and coarsening process with a peculiar dynamics. By using a proper scaling along with numerical simulations, this paper develops a quantitative understanding of the wrinkling dynamics from initial growth through coarsening toward equilibrium. By considering generally biaxial stresses and anisotropic elastic modulus of the film, we show that a rich variety of wrinkle patterns (e.g., labyrinth, orthogonal, parallel, zigzag, and checkerboard patterns) emerge as a result of the competition between the material anisotropy and the stress anisotropy. [Preview Abstract] |
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