Session Y55: Shell Buckling II

Mechanics of turning a page

A mundane activity of turning a page arises not only in reading, but also in production of paper and textile. Here we analyze the shape and stability of a thin sheet of paper, bound to a book in one side and compressed from the other side, to gain mechanical understanding of the page turning. We start with describing the shape of a sheet that is buckled between the bound end and gripping fingers, using the Euler elastica. Then we predict the fate of the bent page when the external load is removed, to determine the condition for the page to turn to the other side. Upon corroborating our theory with experiments of various geometry and materials of thin sheets, we suggest the shortest path of fingers to turn pages for both lazy readers and efficient manufacturing robots.   

Swimming through shell buckling

Under pressure, a hollow elastic sphere becomes unstable and
collapses. It reinflates back when the pressure is decreased. The
hysteresis in shape sequence and in deformation velocity associated
to this cycle makes this simple object a good candidate for becoming a
swimmer able to move at low or high Reynolds number.

We explore this possibility through a macroscopic experiment in fluids
of varying viscosities so as to explore different flow regimes. We
show that not only the shape hysteresis leads to swimming but the fast
buckling phase is an efficient mechanism for propulsion that implies
inertial effects and subtle coupling between shape post-buckling
oscillations and fluid flow patterns. Our modeling shows that such an
inertial regime could even be reached at microscopic scale.

We anticipate that a conveyor made of a few of microbubbles with
different shell thicknesses, would constitute a microrobot whose 3D
displacement can be remotely controlled by ultasounds.

Ref : A. Djellouli, P. Marmottant, H. Djeridi, C. Quilliet and G.
Coupier, PRL 119, 224501 (2017).
  

Buckled Bundles and Beyond

Defects can elastically buckle 2D crystalline membranes (like graphene) into 3D shapes. Using a combination of numerical simulations and continuum elasticity theory we show how the same is true for columnar structures (like nanotubes or protein fibers). This discovery builds upon the recently uncovered mapping between the inter-filament spacing in bundles and the metric properties of a curved surface. We show that shape instabilities are controlled by a single material-dependent parameter that characterizes the ratio of bending to cohesion energies. Along with a host of previously unknown shape equilibria—the filamentous analogs to the conical and saddlelike shapes of defective membranes—we find a profoundly asymmetric response to positive and negative disclinations in the infinite length limit that is without parallel to the membrane analog.   

Dynamic buckling of rings and annuli

When an elastic ring is subject to an external pressure, it is known to buckle as a 'figure of 8'; similarly an annulus subject to a strong internal tension is known to form two folds. Here we explore how these examples of mode-2 buckling change under dynamic loading. In particular, we show that the presence of inertia means that higher modes are observed at the onset of instability, and present results for the observed mode number as a function of the driving pressure. Our results are in accord with experiments in which an elastic ring is rapidly pulled inwards by surface tension. Surprisingly, we find that while inertia is required to observe higher modes, the actual selected mode does not depend on inertia. We explain this, and reconcile our observation of higher modes with earlier experiments that exhibited mode 2, by understanding the growth rate of instability.   

Membrane morphology beyond polyhedra

Conventional homogeneous or heterogeneous elastic crystalline membrane exhibits buckling transition from sphere to polyhedra. However, their morphologies do not go beyond polyhedra and cannot be externally controlled. We study the morphology of a closed crystalline membrane co-assembled with super-paramagnetic particles which is called magnetoelastic membrane. With additional competition from magnetic dipole-dipole interaction on top of elasticity, magnetoelastic membrane shows novel morphology beyond polyhedra. By systematically changing two dimensionless control parameters, we find a rich class of membrane morphologies. The morphology of magnetoelastic membrane can be manipulated by external magnetic field, providing promising applications for membrane shape control, design of micro-containers and targeted drug delivery systems.   

Swelling and warpage of orthotropic plates

While isotropic in-plane swelling problems for elastic sheets have been studied extensively in
recent years, many shape-programmable materials, including nematic solids and 3d-printed
structures, are anisotropic. Inspired by the warpage of engineered wood panels, we examine
plates in which both the in-plane swelling and the material stiffness are highly orthotropic,
leading to multiple separations in energy scales. For these orthotropic plates, in-plane swelling
can produce configurations that approximate cylindrical shapes and twisted ruled surfaces.   

Transverse edge extension underlies compression, buckling and wrinkling in thin solids

When a rectangular sheet whose short edges are clamped is stretched, elongated wrinkles appear, indicating the presence of transverse compression. So far, the mechanism by which longitudinal tension and edge clamping act jointly to generate transverse compression has not been clarified. Here we employ analytic tools and numerical simulations to compare this problem with a new variant, where instead of stretching and edge clamping one only imposes edge-localized transverse strain, pulling the corners outward. We find that, despite the absence of longitudinal tension in the new variant, both models exhibit similar transverse stress profile in planar state, thus revealing that the generic origin of compression is the transverse extension of edges relative to the bulk. This similarity in planar stress profiles underlies similarity in the near-threshold buckling patterns exhibited by the two models. In contrast, the two models are sharply distinct in their respective far-from-threshold regime: the deflection of the stretched sheet consists of small wavelength wrinkles whereas the edge-extended sheet does not develop wrinkles. This indicates the role of longitudinal tension in providing an effective substrate resistance, which is crucial for the emergence of wrinkle patterns.   

A geometric theory of wrinkling for confined shells: Part 1

The problem of joining a planar sheet to a surface with a different metric is a familiar frustration. Flat bandages don’t stick as well to curved knuckles or elbows, and maps of the earth exaggerate areas near the poles. We study the deformations of ultrathin (∼100 nm) elastic shells, which we manufacture on spherically-curved substrates and then transfer to a flat water bath. The sheets respond by forming distinct domains filled by smooth parallel wrinkles or by disordered buckled patterns. We show that the selection of these domains and the orientation of wrinkles within them depends sensitively on the shape of the boundary of the film. Remarkably, these complex patterns may be predicted by a theoretical model wherein the exposed surface area of the water bath is minimized. The derivation and solution of this model will be presented in the next talk. (This is part 1 in a 3-talk series).
  

A geometric theory of wrinkling for confined shells: Part 2

Thin elastic shells readily take on shapes wildly different from their own. Motivated by the puzzle of determining the wrinkle patterns exhibited by shallow shells floating on a water bath, we obtain a fully rigorous asymptotic expansion of the energy (elastic and otherwise) valid in the high-frequency limit. After renormalizing by the typical energy of wrinkling, we derive a coarse-grained model in which an elastically compatible pattern is assigned energy proportional to the difference between its intrinsic undeformed area and its projected area in the plane. Energetically optimal patterns therefore maximize their projected area. Surprisingly, this limiting model turns out to be explicitly solvable in a large variety of cases, including for shells whose (possibly non-constant) curvature is of a known sign. We demonstrate our methods in many cases of interest, offering an ansatz-free explanation for the geometry of wrinkle patterns seen in confined elastic shells. (This is part 2 in a 3-talk series.)   

A geometric theory of wrinkling for confined shells: Part 3

Materials engineered through surface patterning are used for a broad array of applications, including flexible electronic and microfluidic devices, electronic skin, and many others. Microfabrication techniques based on elastic instabilities have attracted much attention because of their relative simplicity and potential for technological innovation. We employ Gaussian curvature as a mechanism for pattern formation: when a shallow curved shell is placed upon a liquid surface, well-defined domains of unidirectional wrinkles are formed. In the third part of this series of talks, we use finite element simulations and a theoretical approach based on the minimization of the elastic energy, to probe how the global arrangement of the patterns and wrinkling amplitude depends on the shape and curvature of the shell. We finally consider cases of shells with highly non-trivial boundary geometries and we demonstrate how this setup can be employed to harness surface structures with complex, yet predictable and controllable, topography. (This is part 3 in a 3-talk series).