Growth and Shape of a Chain Fountain

If a long chain is held in a pot elevated a distance h_1 above the floor, and the end of the chain is then dragged over the rim of the pot and released, the chain flows under gravity down into a pile on the floor. Not only does the chain flow out of the pot, it also leaps above the pot in a "chain-fountain". I predict and observe that if the pot is held at an angle to the vertical the steady state shape of the fountain is an inverted catenary, and discuss how to apply boundary conditions to this solution. In the case of a level pot, the fountain shape is completely vertical. In this case I predict and observe both how fast the fountain grows to its steady state hight, and how it grows quadratically in time if there is no floor. The fountain is driven by an anomalous push force from the pot that acts on the link of chain about to come into motion. I confirm this by designing two new chains, one consisting of hollow cylinders threaded on a string and one consisting of heavy beads separated by long flexible threads. The former is predicted to produce a pot-push and hence a fountain, while the latter will not. I confirm these predictions experimentally. Finally I directly observe the anomalous push in a horizontal chain-pick up experiment.Comment: 6 pages 7 figures and one movi

Pattern selection when a layer buckles on a soft substrate.

If a neo-Hookean elastic layer adhered to a neo-Hookean substrate grows equibiaxially, it will buckle into a topographic pattern. Here, we combine higher order perturbation theory and finite element numerics to predict the pattern formed just beyond the buckling threshold. More precisely, we construct a series of solutions corresponding to hexagonal, square and stripe patterns, and expand the elastic energy for each pattern as a Landau-like energy series in the topography amplitude. We see that, for square and stripe patterns, the elastic energy is invariant under topography inversion, making the instabilities supercritical. However, since patterns of hexagonal dents are physically different to patterns of hexagonal bumps, the hexagonal energy lacks this invariance. This lack introduces a cubic term which causes hexagonal patterns to be formed subcritically and are hence energetically favoured. Our analytic calculation of the cubic term allows us to determine that dents are favoured in incompressible systems, but bumps are favored in sufficiently compressible systems. Finally, we consider a stiff layer sandwiched between an identical substrate and superstrate pair. This system has topography inversion symmetry, so hexagons form supercritically, and square patterns are favoured. We use finite element calculations to verify our theoretical predictions for each pattern, and confirm which pattern is selected. Previous work has used a simplified elastic model (a plate & a linear elastic substrate) that possesses invariance under topography inversion, and hence incorrectly predicted square patterns. Our work demonstrates that large strain geometry is sufficient to break this symmetry and explain the hexagonal dent patterns observed in buckling experiments

Mechanics of invagination and folding: Hybridized instabilities when one soft tissue grows on another.

We address the folding induced by differential growth in soft layered solids via an elementary model that consists of a soft growing neo-Hookean elastic layer adhered to a deep elastic substrate. As the layer-to-substrate modulus ratio is varied from above unity toward zero, we find a first transition from supercritical smooth folding followed by cusping of the valleys to direct subcritical cusped folding, then another to supercritical cusped folding. Beyond threshold, the high-amplitude fold spacing converges to about four layer thicknesses for many modulus ratios. In three dimensions, the instability gives rise to a wide variety of morphologies, including almost degenerate zigzag and triple-junction patterns that can coexist when the layer and substrate are of comparable softness. Our study unifies these results providing understanding for the complex and diverse fold morphologies found in biology, including the zigzag precursors to intestinal villi, and disordered zigzags and triple junctions in mammalian cortex.T. T. acknowledges the Academy of Finland for funding. The computational resources were provided by CSC – IT Center for Science. J.B. acknowledges the 1851 Royal Commission and Trinity Hall Cambridge for funding.This is the author accepted manuscript. The final version is available from APS via http://dx.doi.org/10.1103/PhysRevE.92.02272

Ballooning, bulging, and necking: An exact solution for longitudinal phase separation in elastic systems near a critical point.

Prominent examples of longitudinal phase separation in elastic systems include elastic necking, the propagation of a bulge in a cylindrical party balloon, and the beading of a gel fiber subject to surface tension. Here we demonstrate that if the parameters of such a system are tuned near a critical point (where the difference between the two phases vanishes), then the behavior of all systems is given by the minimization of a simple and universal elastic energy familiar from Ginzburg-Landau theory in an external field. We minimize this energy analytically, which yields not only the well known interfacial tanh solution, but also the complete set of stable and unstable solutions in both finite and infinite length systems, unveiling the elastic system's full shape evolution and hysteresis. Correspondingly, we also find analytic results for the the delay of onset, changes in criticality, and ultimate suppression of instability with diminishing system length, demonstrating that our simple near-critical theory captures much of the complexity and choreography of far-from-critical systems. Finally, we find critical points for the three prominent examples of phase separation given above, and demonstrate how each system then follows the universal set of solutions.UKRI, EPSR

Meniscus instabilities in thin elastic layers.

We consider meniscus instabilities in thin elastic layers perfectly adhered to, and confined between, much stiffer bodies. When the free boundary associated with the meniscus of the elastic layer recedes into the layer, for example by pulling the stiffer bodies apart or injecting air between them, then the meniscus will eventually undergo a purely elastic instability in which fingers of air invade the layer. Here we show that the form of this instability is identical in a range of different loading conditions, provided only that the thickness of the meniscus, a, is small compared to the in-plane dimensions and to two emergent in-plane length scales that arise if the substrate is soft or if the layer is compressible. In all such situations, we predict that the instability will occur when the meniscus has receded by approximately 1.27a, and that the instability will have wavelength λ ≈ 2.75a. We illustrate this by also calculating the threshold for fingering in a thin wedge of elastic material bonded to two rigid plates that are pried apart, and the threshold for fingering when a flexible plate is peeled from an elastic layer that glues the plate to a rigid substrate

Evolving, complex topography from combining centers of Gaussian curvature.

Liquid crystal elastomers and glasses can have significant shape change determined by their director patterns. Cones deformed from circular director patterns have nontrivial Gaussian curvature localized at tips, curved interfaces, and intersections of interfaces. We employ a generalized metric compatibility condition to characterize two families of interfaces between circular director patterns, hyperbolic and elliptical interfaces, and find that the deformed interfaces are geometrically compatible. We focus on hyperbolic interfaces to design complex topographies and nonisometric origami, including n-fold intersections, symmetric and irregular tilings. The large design space of threefold and fourfold tiling is utilized to quantitatively inverse design an array of pixels to display target images. Taken together, our findings provide comprehensive design principles for the design of actuators, displays, and soft robotics in liquid crystal elastomers and glasses

Surface instability in a nematic elastomer

Liquid crystal elastomers (LCEs) are soft phase-changing solids that exhibit large reversible contractions upon heating, Goldstone-like soft modes and resultant microstructural instabilities. We heat a planar LCE slab to isotropic, clamp the lower surface then cool back to nematic. Clamping prevents macroscopic elongation, producing compression and microstructure. We see that the free surface destabilizes, adopting topography with amplitude and wavelength similar to thickness. To understand the instability, we numerically compute the microstructural relaxation of a "non-ideal" LCE energy. Linear stability reveals a Biot-like scale-free instability, but with oblique wavevector. However, simulation and experiment show that, unlike classic elastic creasing, instability culminates in a cross-hatch without cusps or hysteresis, and is constructed entirely from low-stress soft modes.Comment: 6 pages (+22 pages in Supplemental Material), 5 figures (+15 figures in Supplemental Material), 5 supporting videos at https://drive.google.com/drive/folders/18DCJz0DnNKFKmrJ1Ggy_Q40_UhXbj8XH?usp=sharin