Session J52: Focus Session: Extreme Mechanics - Plates

Radial stretching of thin sheets: A prototypical model for morphological complexity

The complex morphologies of thin sheets consist of wrinkles, crumples, folds, creases, and blisters. These descriptive words may sound lucid -- but do they carry any quantitatively distinguishable content? Following the classical approach of pattern formation theory, we seek to impart a universal meaning to these modes of deformation as distinct types of symmetry-breaking instabilities of a flat, featureless sheet. This idea motivates us to consider the general problem of \textit{axisymmetric stretching} of a sheet. A familiar realization of this problem is the ``map maker's conflict'': projecting a flat sheet onto a foundation of spherical shape. Another representative realization is the Lame' set-up: exerting a radial tension gradient on a sheet, which may be free-standing or resting on a solid or liquid foundation. I will introduce a set of \textit{generic parameters: bendability, confinement, stiffness, adhesiveness, }that span a phase space for the morphology of radially stretched sheets. In this phase space, wrinkling, crumpling, folding, creasing and blistering could be identified as primary and secondary symmetry-breaking instabilities.   

Bending of a surface with spontaneous curvature

We interest to curvature deformations that can be described by Helfrich's energy: a quadratic mean curvature term, and a gaussian curvature term. When the surface is not strictly incompressible and presents a nonzero spontaneous mean curvature, we focus on simple cases to show that a priori determination of key features (spontaneous curvature, equilibrium area) may be biased according the expression taken for the energy.   

Deformations of 2D Random Elastic Networks

We study the linear and nonlinear behavior of random 2D elastic networks at the desktop scale. We demonstrate how to fabricate random networks and characterize them with the lattice coordination number Z. We investigate experimentally if there is a relation between the mechanical response and the lattice coordination number Z of the network.   

Packing with a twist: from Wrinkles to Scrolls

We discuss an experimental investigation of a thin elastic sheet in the form of a ribbon with clamped boundary conditions at both ends which is then subjected to a twist by rotating the ends through a prescribed angle. We find that a wrinkling instability appears even at a small twist angle which depends on the aspect ratio of the ribbon, its bending modulus and initial tension. Using x-ray tomography, we show that the pattern of this first instability has an impact on the folding at larger twist angles which can result in ordered configurations including Fermat scrolls. Still further twisting results in a highly compressive packing as in wringing a towel without application of direct radial compression. Implications for developing yarns with novel mechanical and transport properties [Lima, et al., Science 331, 51 (2011)] will be discussed.   

Geometry and Mechanics of Chiral Pod Opening

We study the geometry and mechanics that drive the opening of Bauhinia seeds pods. The pod valve wall consists of two fibrous layers oriented at $\pm$ 45$^{\circ}$ with respect to the pod axis. Upon drying, each of the layers shrinks uniaxially, perpendicularly to the fibers orientation. This active deformation turn the valve into an incompatible sheet with reference saddle-like curvature tensor and a flat (Euclidean) reference metric. These two intrinsic properties are incompatible. The shape is, therefore, selected by a stretching-bending competition. Strips cut from the valve tissue and from synthetic model material adopt various helical configurations. We provide analytical expressions for these configurations in the bending and stretching dominated regimes. Surface measurements show the transition from minimal surfaces (narrow limit) to cylindrical ones (wide limit). Finally, we show how plants use these mechanical principles using different tissue architectures.   

Flat-twisted-helical transition in composed gel sheets and self assembled chiral molecules

We recently presented a new chirality creating mechanism in elastic strips. in such frustrated bodies, the chiral configuration is determined in a competition between bending and stretching energies, controlled by a dimensionless parameter $\tilde {w}=w\sqrt {k/t} $, in which $w$ is the strip's width, $t$ -- its thickness and $k$ - the spontaneous curvature. I will show the geometrical and mechanical equivalence between such elastic strips and self assembled molecules made of twisted elements. I will also show experiments in responsive gels, showing how a continuous variation in $\tilde {w}$ yields an ordered shape transition from flat to twisted and helical shapes and to tubes. Similar transitions have been observed in self assembled macromolecules.   

Curvature and defects in soft membranes with orientational order

Previous research has demonstrated that soft membranes have a coupling between curvature and in-plane orientational order. Defects in the orientational order can induce curvature, and conversely, curvature leads to an effective geometrical potential acting on defects [1]. Recently, our group has done simulations which show that the interaction between curvature and defects depends on several important issues, including the baseline curvature of the membrane (flat, cylinder, sphere, torus), the phase of the defects (radial or tangential), and the relative contribution of in-plane (intrinsic) vs. out-of-plane (extrinsic) variations of the director [2]. To understand the simulations, we develop a theoretical approach that can address those issues. Using this approach, we calculate the energy of defect structures in curved geometries, and determine how the energy varies as a function of the defect position and separation and the membrane distortions. The interaction energy depends on the relative magnitude of intrinsic vs. extrinsic couplings, and on the mechanical properties of the membrane. This approach provides opportunities to design membranes that will relax into selected shapes. [1] AM Turner et al, Rev Mod Phys 82, 1301 (2010). [2] RLB Selinger et al, J Phys Chem B, in press.   

Capillary induced buckling of floating sheets

When a water droplet is deposited over a thin floating sheet, radial wrinkles appear in the vicinity of the droplet as a result of capillary forces exerted at the contact line [1]. However, determining the stress state at the contact line is still challenging and limits the full description of the wrinkling pattern. In order to avoid this contact line ambiguities, we propose the experimental study of the buckling of a macroscopic annulus floating on the surface of water and submitted to a difference in surface tension between its inner and outer edges. This particular configuration allows to generate radial wrinkles on the membrane with well defined border conditions. The topography of the wrinkled patterns are precisely measured using a synthetic Schlieren technique. Based on the standard buckling theory, we develop scaling laws for the buckling threshold of the annulus as well as for the wave length and radial extension of the wrinkles, which are compared to our experimental results and numerical simulations. \\[4pt] [1] J. Huang, M. Juszkiewicz, W.H. de Jeu, E. Cerda, T. Emrick, N. Menon, and T.P. Russell. Capillary wrinkling of floating thin polymer films. Science, 317(5838):650-653, 2007.   

The wrinkle transition of a sheet on a drop

A thin sheet subject to confinement will wrinkle in order to relieve compressive stress. We discuss the case of a circular sheet living on the surface of a liquid drop. The pressure of the drop forces the sheet to be non-planar, which may induce confinement along the outer edge of the sheet. We show that, in the limit of very thin, highly bendable sheets, the system is governed by a single confinement parameter. This parameter determines if and where wrinkles appear on the sheet. Comparison to experimental results provides the first detailed confirmation of a new far-from-threshold theory to describe such ultra-thin sheets. According to this model, the transition to the wrinkled state represents the loss of axisymmetry in the height field, while the stress field maintains its symmetry.   

Transition from Wrinkling to Crumpling in a Sheet Floating on a Drop

An ultrathin* circular polystyrene sheet floating on the surface of a water drop stretches radially and compresses along its circumference as the curvature of the drop increases. The compression is at first fully relaxed by a wrinkle pattern extending inward from the edge. When the wrinkles occupy too large a fraction of the area of the sheet, sharp, localized, crumpled features continuously emerge. We show that the onset of crumpling is a primary symmetry breaking transition of the stress field. We experimentally characterize this transition from wrinkling to crumpling by studying the distribution of gaussian curvature in the film, measured by optical profilometry. *Typical dimensions are tens of nanometers in thickness and millimeters in lateral size.   

Supported conical defects

In this work we study the elastic equilibrium configurations of a hyperbolic conical defect (a flat disc with a single negative Gaussian curvature condensation), supported on a rigid plane. Originating from the study of strictureplasty, this problem which seems to be a natural extension to the D-cone problem displays a distinct behavior.   

Stamping and wrinkling of elastic plates

In classical Euler buckling a beam is found to buckle with the lowest mode as a compressive strain is applied. Higher modes are however observed if the amplitude of the out-of-plane displacement is bounded by geometrical constraints. What is the limit when the maximum amplitude prescribed is decreased to zero? We show that the wavelength tends towards a finite value dictated by the thickness of the beam. This one-dimensional model is used to describe the compression of a circular elastic plate into an hemispherical mold.   

Deterministic Wrinkling Patterns of Thin Polymeric Coatings on Soft Substrates

Wrinkling surface patterns in soft materials have become increasingly important for a broad range of applications including stretchable electronics, microfluidics, thin-film property measurement, wetting and adhesion, and other surface area and topology controlled phenomena. Thermal and swelling mismatch between the thin surface layer and the soft substrate lead to spontaneous formation of buckling-induced disordered labyrinth patterns, which exhibit a mechanistically-determined short wavelength, but an undetermined and highly varied long wavelength. In this paper, analytical and computational models are presented to create deterministic wrinkling patterns through directed buckling methods, which capture the physics of the instabilities governing the formation of multiple wavelength wrinkling patterns, providing a predictive tool for design of deterministic wrinkling patterns. The fabrication of the deterministic patterns is accomplished using novel chemical vapor deposition processes. The role of these patterns in providing multifunctional performance is illustrated and discussed.   

Wrinkles in reinforced membranes

We study, through model experiments, the buckling under tension of an elastic membrane reinforced with a more rigid strip or a fiber. In these systems, the compression of the rigid layer is induced through Poisson contraction as the membrane is stretched perpendicularly to the strip. Although strips always lead to out-of-plane wrinkles, we observe a transition from out-of-plane to in plane wrinkles beyond a critical strain in the case of fibers embedded into the elastic membranes. The same transition is also found when the membrane is reinforced with a wall of the same material depending on the aspect ratio of the wall. We describe through scaling laws the evolution of the morphology of the wrinkles and the different transitions as a function of material properties and stretching strain.   

Mechanics of Graphene Electronics

Graphene, a monolayer of tightly-packed carbon atoms, has demonstrated great academic and industrial promises for integrating superior properties of nanomaterials and nanostructures into novel macroscale devices. Here, we demonstrate a simple method to enable over 200{\%} reversible deformation of continuous large-area graphene sheet (over 1cm x 1cm) on polymer substrates. By patternning large-area graphene on a pre-stretched polymer layer by 200{\%}, the graphene film develops hierachical patterns including wrinkles with wavelengthes on the order 10$\sim $100 nm and delaminated buckles with wavelengths on the the order of 1$\mu$m. If the polymer is stretched again ($<$100{\%}), the wrinkled region relaxes and the graphene on this region becomes flat. As the stretch further increases (over 100{\%}), the graphene on delaminated buckles slides toward the flat regions, decreasing the amplitude of the buckles. The relaxation of the wrinkles and buckles enables the large deformation of graphene electrode without fracture. We further demonstrate potential applications of the graphene electrodes capable of large deformation. For example, a polymer film can be sandwiched between two graphene electrodes. As a voltage is applied between the two graphene electrodes, the polymer can achieve an actuation strain over 200{\%}.