Buckling of thin plates in the weakly and strongly nonlinear regimes

Curvature-induced stiffness and the spatial variation of wavelength in wrinkled sheets

Wrinkle patterns in compressed thin sheets are ubiquitous in nature and technology, from the furrows on our foreheads to crinkly plant leaves, from ripples on plastic-wrapped objects to the protein film on milk. The current understanding of an elementary descriptor of wrinkles--their wavelength--is restricted to deformations that are parallel, spatially uniform, and nearly planar. However, most naturally occurring wrinkles do not satisfy these stipulations. Here we present a scheme that quantitatively explains the wrinkle wavelength beyond such idealized situations. We propose a local law that incorporates both mechanical and geometrical effects on the spatial variation of wrinkle wavelength. Our experiments on thin polymer films provide strong evidence for its validity. Understanding how wavelength depends on the properties of the sheet and the underlying liquid or elastic subphase is crucial for applications where wrinkles are used to sculpt surface topography, to measure properties of the sheet, or to infer forces applied to a film.

Anisotropic blistering instability of highly ellipsoidal shells

The formation of localized periodic structures in the deformation of elastic shells is well documented and is a familiar first stage in the crushing of a spherical shell such as a ping-pong ball. While spherical shells manifest such periodic structures as polygons, we present a new instability that is observed in the indentation of a highly ellipsoidal shell by a horizontal plate. Above a critical indentation depth, the plate loses contact with the shell in a series of well-defined "blisters" along the long axis of the ellipsoid. We characterize the onset of this instability and explain it using scaling arguments, numerical simulations, and experiments. We also characterize the properties of the blistering pattern by showing how the number of blisters and their size depend on both the geometrical properties of the shell and the indentation but not on the shell's elastic modulus. This blistering instability may be used to determine the thickness of highly ellipsoidal shells simply by squashing them between two plates.

Lipid domain depletion at small localized bends imposed by a step geometry

Natural processes in biological cells rely on molecules to be in the right place at the right time to maintain the dynamics of living processes. When lipids in bilayer membranes move and mix, they experience kinetic and thermodynamic barriers that affect the time scales of their locations and associations with each other. One of these barriers is that of the membrane shape. Using spin coating as a deposition technique, we formed multilamellar supported lipid bilayers on topologically patterned substrates with defined step rise heights of 13 and 27 nm measured by atomic force microscopy. Each step rise imposed two ridges on the lipid bilayers, and the ridge angles were measured by atomic force microscopy. The lipid composition of this system was 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC), 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC), and cholesterol (4:4:2), doped with a fluorescent lipid, which displays liquid-ordered-liquid-disordered (Lo-Ld) phase coexistence upon cooling to 25 °C. The DPPC-rich Lo domains in the upper bilayers were established to have boundaries and positions that responded to local forces. We found that these Lo domains were depleted at the location of each step rise. We employed an equation for local bending at a ridge and demonstrate that Lo domain densities at each rise correspond to these energies. Remarkably, an energy barrier greater than 1k(B)T is erected at a small deflection (1.3°) from planar geometry at the ridge, resulting in depletion of the majority of the optically visible Lo domains from the step rise. This work provides a means to design substrates that, in conjunction with supported lipid bilayers, provide defined localized topological energy barriers that can be used in biomembrane engineering. It also provides a method for easily analyzing the energetics of cusp-like shapes in cellular membrane structures.

Diffusion and reaction-diffusion in fast cellular flows

Cellular structures, as the rolls generated by Rayleigh-Bénard instability, have always been an important topic in nonlinear science. The diffusion of a passive scalar in a given steady cellular flow becomes an interesting question in the limit of a large Péclet number, often realistic. The main result there is that the effective diffusion is somewhere in between the molecular diffusion and the "turbulent" diffusion. A new added twist to this is the reaction-diffusion case, where the front speed is the laminar propagation velocity (without flow) times the Péclet number to the power 1/4. I refine this last result and give the behavior of the prefactor in the Zel'dovich limit of a narrow reaction zone.

Scaling of the buckling transition of ridges in thin sheets

When a thin elastic sheet crumples, the elastic energy condenses into a network of folding lines and point vertices. These folds and vertices have elastic energy densities much greater than the surrounding areas, and most of the work required to crumple the sheet is consumed in breaking the folding lines or "ridges." To understand crumpling it is then necessary to understand the strength of the ridges. In this work, we consider the buckling of a single ridge under the action of inward forcing applied at its ends. We demonstrate a simple scaling relation for the response of the ridge to the force prior to buckling. We also show that the buckling instability depends only on the ratio of strain along the ridge to the curvature across it. Numerically, we find for a wide range of boundary conditions that ridges buckle when our forcing increases their elastic energy by 20% over their resting state value. We also observe a correlation between neighbor interactions and the location of initial buckling. Analytic arguments and numerical simulations are employed to prove these results. Implications for the strength of ridges as structural elements are discussed.

Swelling-induced morphology in ultrathin supported films of poly(d,l-lactide)

In this study we describe a surface morphology that arises when ultrathin supported films of poly(d,l-lactide) are immersed in water. The films are initially flat with a rms roughness of approximately 2 nm. After immersion the surfaces of the films are covered with craters. The craters have a narrow distribution of sizes and are typically micrometers in diameter. They have depths in the 10-100 nm range. In situ atomic force microscopy shows that the craters occur as a result of a blistering process, which occurs when the films delaminate from the silicon substrate. The films buckle away from the substrate to give a nonzero initial diameter and then the blisters proceed to grow until they reach a maximum size. At any point during the growth process, the blisters can be made to collapse by removing the films from water. This phenomenon is explained in terms of a laterally confined swelling film, which has a buckling instability and releases excess strain energy by wrinkling. An expression for the initial buckling wavelength is extracted using the expressions for a buckling plate. Information about the mechanical properties of the films and the surface interaction between the film and substrate can also be obtained by considering the kinetics of blister growth.

Singularities, structures, and scaling in deformed m-dimensional elastic manifolds

The crumpling of a thin sheet can be understood as the condensation of elastic energy into a network of ridges that meet in vertices. Elastic energy condensation should occur in response to compressive strain in elastic objects of any dimension greater than 1. We study elastic energy condensation numerically in two-dimensional elastic sheets embedded in spatial dimensions three or four and three-dimensional elastic sheets embedded in spatial dimensions four and higher. We represent a sheet as a lattice of nodes with an appropriate energy functional to impart stretching and bending rigidity. Minimum energy configurations are found for several different sets of boundary conditions. We observe two distinct behaviors of local energy density falloff away from singular points, which we identify as cone scaling or ridge scaling. Using this analysis, we demonstrate that there are marked differences in the forms of energy condensation depending on the embedding dimension.