Elastic Sheets

Roadmap to the morphological instabilities of a stretched twisted ribbon

Roadmap
We address the mechanics of an elastic ribbon subjected to twist and tensile load. Motivated by the classical work of Green and a recent experiment that discovered a plethora of morphological instabilities, we introduce a comprehensive theoretical framework through which we construct a 4D phase diagram of this basic system, spanned by the exerted twist and tension, as well as the thickness and length of the ribbon. Different types of instabilities appear in various corners of this 4D parameter space, and are addressed through distinct types of asymptotic methods. Our theory employs three instruments, whose concerted implementation is necessary to provide an exhaustive study of the various parameter regimes: (i) a covariant form of the Foppl-von Karman (cFvK) equations to the helicoidal state necessary to account for the large deflection of the highly-symmetric helicoidal shape from planarity, and the buckling instability of the ribbon in the transverse direction; (ii) a far from threshold (FT) analysis which describes a state in which a longitudinally-wrinkled zone expands throughout the ribbon and allows it to retain a helicoidal shape with negligible compression; (iii) finally, we introduce an asymptotic isometry equation that characterizes the energetic competition between various types of states through which a twisted ribbon becomes strainless in the singular limit of zero thickness and no tension.

Reference :
Roadmap to the morphological instabilities of a stretched twisted ribbon, J. Chopin, V. Démery and B. Davidovitch, J. Elasticity, 119, 137-189 (2015)

Wet Granular Materials

Building Designed Granular Towers One Drop at a Time

Tower
A dense granular suspension dripping on an imbibing surface is observed to give rise to slender mechanically stable structures that we call granular towers. Successive drops of grain-liquid mixtures are shown to solidify rapidly upon contact with a liquid absorbing substrate. A balance of excess liquid flux and drainage rate is found to capture the typical growth and height of the towers. The tower width is captured by the Weber number, which gives the relative importance of inertia and capillary forces. Various symmetric, smooth, corrugated, zigzag, and chiral structures are observed by varying the impact velocity and the flux rate from droplet to jetting regime.

Reference :
Building Designed Granular Towers One Drop at a Time, J. Chopin and A. Kudrolli, Phys. Rev. Lett. 107, 208304 (2011)

Adhesives

Liquid Blister Test

We consider a thin elastic sheet adhering to a stiff substrate by means of the surface tension of a thin liquid layer. Debonding is initiated by imposing a vertical displacement at the centre of the sheet and leads to the formation of a delaminated region or ‘blister’. This experiment reveals that the perimeter of the blister takes one of three different forms depending on the vertical displacement imposed. As this displacement is increased, we observe first circular, then undulating and finally triangular blisters. We obtain theoretical predictions for the observed features of each of these three families of blisters. The theory is built upon the Foppl–von Karman equations for thin elastic plates and accounts for the surface energy of the liquid. We find good quantitative agreement between our theoretical predictions and experimental results, demonstrating that all three families are governed by different balances between elastic and capillary forces. Our results may bear on micrometric tapered devices and other systems, where elastic and adhesive forces are in competition.

Reference :

The Liquid Blister Test, J. Chopin, D. Vella and A. Boudaoud, Proc. Roy. Soc. London A 464, 2887 (2008)

Adhesives

Crack Front Dynamics across a Single Heterogeneity

Pinchoffcrack
We study the spatiotemporal dynamics of a crack front propagating at the interface between a rigid substrate and an elastomer. We first characterize the kinematics of the front when the substrate is homogeneous and find that the equation of motion is intrinsically nonlinear. We then pattern the substrate with a single defect. Steady profiles of the front are well described by a standard linear theory with nonlocal elasticity, except for large slopes of the front. In contrast, this theory seems to fail in dynamical situations, i.e., when the front relaxes to its steady shape, or when the front pinches off after detachment from a defect. More generally, these results may impact the current understanding of crack fronts in heterogeneous media.

Reference :
Crack Front Dynamics across a Single Heterogeneity, J. Chopin, A. Prevost, A. Boudaoud, and M. Adda-Bedia, Phys. Rev. Lett. 107, 144301 (2011)

Wet Granular Materials

Pearling and arching instabilities of a granular suspension on a superabsorbing surface

Arches

We show that a granular suspension, composed of particles immersed in a liquid, can form pearls, hooks, and arches when deposited from a nozzle onto a translating substrate that acts as a liquid super-absorber. The removal of the liquid induces a rapid pinning of the contact line leading to mechanically stable structures that are held together by capillary adhesion with shapes that depend on the relative solidification rate. Pearls or hooks form depending on whether the suspension snaps off before or after coming into contact with the substrate. A cylindrical thread with a near circular cross-section and various undulatory structures forms if solidification occurs prior to snap-off. In particular, when the jet solidifies before coming into contact with the substrate, it folds periodically, resulting in arches with a span length determined by the deposition flux and the substrate speed. Period doubling and meandering are observed leading to further structures with vertical and horizontal ripples when the deposition flux is increased.

Reference :
Pearling and arching instabilities of a granular suspension on a superabsorbing surface ,
J. Chopin and A. Kudrolli, Soft Matter 11, 659 (2015)

Elastic Sheets

Disclinations, e-cones, and their interactions in extensible sheets

Fig1AB

We investigate the nucleation, growth, and spatial organization of topological defects with a ribbon shaped elastic sheet which is stretched and twisted. Singularities are found to spontaneously arrange in a triangular lattice in the form of vertices connected by stretched ridges that result in a self-rigidified structure. The vertices are shown to be negative disclinations or e-cones which occur in sheets with negative Gaussian curvature, in contrast with d-cones in sheets with zero-Gaussian curvature. We find the growth of the wrinkled width of the ribbon to be consistent with a far-from-threshold approach assuming a compression-free base state. The system is found to show a transition from a regime where the wavelength is given by the ribbon geometry, to where it is given by its elasticity as a function of the ratio of the applied tension to the elastic modulus and cross-sectional area of the ribbon.

Disclinations, e-cones, and their interactions in extensible sheets ,
J. Chopin and A. Kudrolli, Soft Matter 12, 4457 (2016)