Cracks in Tension-Field Elastic Sheets

International audienceWe consider the deformation of a thin elastic sheet which is stiff in traction but very soft in compression, as happens in presence of wrinkling. We use the tension-field material model and explore numerically the response of a thin sheet containing multiple cracks of different geometries, when subjected to applied tension. With a single crack, the stress concentrates along a St-Andrew's cross-shaped pattern, whose branches extend from the crack tips to the corners of the domain; at a (small) distance r from the crack tip, the stress displays the usual $r^{−1/2}$ stress singularity but with an unusual and non-universal angular dependence. A strong interaction between multiple cracks is reported and discussed: in particular, for certain configurations of the cracks, the tensile stiffness of a cracked sheet can be zero even though the sheet is made up of a single component

Combustion dynamics in steady compressible flows

We study the evolution of a reactive field advected by a one-dimensional compressible velocity field and subject to an ignition-type nonlinearity. In the limit of small molecular diffusivity the problem can be described by a spatially discretized system, and this allows for an efficient numerical simulation. If the initial field profile is supported in a region of size l < lc one has quenching, i.e., flame extinction, where lc is a characteristic length-scale depending on the system parameters (reacting time, molecular diffusivity and velocity field). We derive an expression for lc in terms of these parameters and relate our results to those obtained by other authors for different flow settings.Comment: 6 pages, 5 figure

Oscillatory fracture path in thin elastic sheet

We report a novel mode of oscillatory crack propagation when a cutting tip is driven through a thin brittle polymer film. The phenomenon is so robust that it can easily be reproduced at hand (using CD packaging material for example). Careful experiments show that the amplitude and wavelength of the oscillatory crack path scale lineraly with the width of the cutting tip over a wide range of lenghtscales but are independant of the width of thje sheet and the cutting speed. A simple geometric model is presented, which provides a simple but thorough interpretation of the oscillations.Comment: 6 pages, submitted to Comptes Rendus Academie des Sciences. Movies available at http://www.lmm.jussieu.fr/platefractur

From discrete to continuum models of three-dimensional deformations in epithelial sheets

International audienceEpithelial tissue, in which cells adhere tightly to each other and to theunderlying substrate, is one of the four major tissue types in adultorganisms. In embryos, epithelial sheets serve as versatile substratesduring the formation of developing organs. Some aspects of epithelialmorphogenesis can be adequately described using vertex models, in which thetwo-dimensional arrangement of epithelial cells is approximated by apolygonal lattice with an energy that has contributions reflecting theproperties of individual cells and their interactions. Previous studieswith such models have largely focused on dynamics confined to two spatialdimensions and analyzed them numerically. We show how these models can beextended to account for three-dimensional deformations and studiedanalytically. Starting from the extended model, we derive a continuumplate description of cell sheets, in which the effective tissue properties,such as bending rigidity, are related explicitly to the parameters of thevertex model. To derive the continuum plate model, we duly take intoaccount a microscopic shift between the two sublattices of the hexagonalnetwork, which has been ignored in previous work. As an application of thecontinuum model, we analyze tissue buckling by a line tension applied alonga circular contour, a simplified set-up relevant to several situations inthe developmental context. The buckling thresholds predicted by thecontinuum description are in good agreement with the results of directstability calculations based on the vertex model. Our results establish adirect connection between discrete and continuum descriptions of cellsheets and can be used to probe a wide range of morphogenetic processes inepithelial tissues

A nonlinear beam model of photomotile structures

Actuation remains a significant challenge in soft robotics. Actuation by light has important advantages: Objects can be actuated from a distance, distinct frequencies can be used to actuate and control distinct modes with minimal interference, and significant power can be transmitted over long distances through corrosion-free, lightweight fiber optic cables. Photochemical processes that directly convert photons to configurational changes are particularly attractive for actuation. Various works have reported light-induced actuation with liquid crystal elastomers combined with azobenzene photochromes. We present a simple modeling framework and a series of examples that study actuation by light. Of particular interest is the generation of cyclic or periodic motion under steady illumination. We show that this emerges as a result of a coupling between light absorption and deformation. As the structure absorbs light and deforms, the conditions of illumination change, and this, in turn, changes the nature of further deformation. This coupling can be exploited in either closed structures or with structural instabilities to generate cyclic motion

Pulsating Front Speed-up and Quenching of Reaction by Fast Advection

We consider reaction-diffusion equations with combustion-type non-linearities in two dimensions and study speed-up of their pulsating fronts by general periodic incompressible flows with a cellular structure. We show that the occurence of front speed-up in the sense $\lim_{A\to\infty} c_*(A)=\infty$, with $A$ the amplitude of the flow and $c_*(A)$ the (minimal) front speed, only depends on the geometry of the flow and not on the reaction function. In particular, front speed-up happens for KPP reactions if and only if it does for ignition reactions. We also show that the flows which achieve this speed-up are precisely those which, when scaled properly, are able to quench any ignition reaction.Comment: 16p

Pattern of Reaction Diffusion Front in Laminar Flows

Autocatalytic reaction between reacted and unreacted species may propagate as solitary waves, namely at a constant front velocity and with a stationary concentration profile, resulting from a balance between molecular diffusion and chemical reaction. The effect of advective flow on the autocatalytic reaction between iodate and arsenous acid in cylindrical tubes and Hele-Shaw cells is analyzed experimentally and numerically using lattice BGK simulations. We do observe the existence of solitary waves with concentration profiles exhibiting a cusp and we delineate the eikonal and mixing regimes recently predicted.Comment: 4 pages, 3 figures. This paper report on experiments and simulations in different geometries which test the theory of Boyd Edwards on flow advection of chemical reaction front which just appears in PRL (PRL Vol 89,104501, sept2002

Bounding biomass in the Fisher equation

The FKPP equation with a variable growth rate and advection by an incompressible velocity field is considered as a model for plankton dispersed by ocean currents. If the average growth rate is negative then the model has a survival-extinction transition; the location of this transition in the parameter space is constrained using variational arguments and delimited by simulations. The statistical steady state reached when the system is in the survival region of parameter space is characterized by integral constraints and upper and lower bounds on the biomass and productivity that follow from variational arguments and direct inequalities. In the limit of zero-decorrelation time the velocity field is shown to act as Fickian diffusion with an eddy diffusivity much larger than the molecular diffusivity and this allows a one-dimensional model to predict the biomass, productivity and extinction transitions. All results are illustrated with a simple growth and stirring model.Comment: 32 Pages, 13 Figure

A Variational Principle Based Study of KPP Minimal Front Speeds in Random Shears

Variational principle for Kolmogorov-Petrovsky-Piskunov (KPP) minimal front speeds provides an efficient tool for statistical speed analysis, as well as a fast and accurate method for speed computation. A variational principle based analysis is carried out on the ensemble of KPP speeds through spatially stationary random shear flows inside infinite channel domains. In the regime of small root mean square (rms) shear amplitude, the enhancement of the ensemble averaged KPP front speeds is proved to obey the quadratic law under certain shear moment conditions. Similarly, in the large rms amplitude regime, the enhancement follows the linear law. In particular, both laws hold for the Ornstein-Uhlenbeck process in case of two dimensional channels. An asymptotic ensemble averaged speed formula is derived in the small rms regime and is explicit in case of the Ornstein-Uhlenbeck process of the shear. Variational principle based computation agrees with these analytical findings, and allows further study on the speed enhancement distributions as well as the dependence of enhancement on the shear covariance. Direct simulations in the small rms regime suggest quadratic speed enhancement law for non-KPP nonlinearities.Comment: 28 pages, 14 figures update: fixed typos, refined estimates in section