Symmetry groups in nonlinear elasticity: An exercise in vintage mathematics

submitted to Comm. Pure Appl. Anal.International audienceThis manuscript aims at characterizing energy densities and constitutive laws of transversely isotropic materials, orthotropic elastic materials and materials with non orthogonal families of fibers. It makes explicit references to results that are scattered over the literature and, although said to be well-known, are not always easy to locate. Direct proofs that are thought to be new and simplified expressions of constitutive laws for materials with two preferred directions are given

A constitutive law for cross-linked actin networks by homogenization techniques

Inspired by experiments on the actin driven propulsion of micrometer sized beads we develop and study a minimal mechanical model of a two-dimensional network of stiff elastic filaments grown from the surface of a cylinder. Starting out from a discrete model of the network structure and of its microscopic mechanical behavior we derive a macroscopic constitutive law by homogenization techniques. We calculate the axisymmetric equilibrium state and study its linear stability depending on the microscopic mechanical properties. We find that thin networks are linearly stable, whereas thick networks are unstable. The critical thickness for the change in stability depends on the ratio of the microscopic elastic constants. The instability is induced by the increase in the compressive load on the inner network layers as the thickness of the network increases. The here employed homogenization approach combined with more elaborate microscopic models can serve as a basis to study the evolution of polymerizing actin networks and the mechanism of actin driven motion.Comment: 19 pages, 7 figure

A Lagrangian formulation for the Oldroyd B fluid and the second principle of thermodynamics

We show that the Oldroyd B fluid model is the Eulerian form of a Lagrangian model with an internal variable that satisfies the second principle of thermodynamics under some conditions on the initial value of the internal variable. We similarly derive a compressible version of the Oldroyd B model and several nonlinear versions thereof. We also derive Lagrangian formulations of the Zaremba-Jaumann and Oldroyd A fluid models. We discuss whether or not these new models satisfy the second principle

Simulation et contrôle gestuel en temps réel d'une large variété d'objets physiques, flexibilité, plasticité, rupture, comportement de multitude

International audienceL'équipe de recherche de I'ACROE travaille sur le thème de I'animation par ordinateur, basée sur I'utilisation de modèles physiques, depuis 1981. Cette recherche est guidée par la réalisation d'un système complet pour I'animation d'images et la création musicale : le "modeleur-simulateur Cordis-Anima", qui doit permettre la simulation en temps réel d'objets manipulables, audibles et visibles . Cordis est dédié à la création musicale, Anima à I'animation, mais I'un et l'autre ont en commun une forte base conceptuelle et matérielle. Ce papier, bien que présentant l'analyse menant à cette base commune, est plus particulièrement dédié à I'animation par ordinateur

Elastic limit of square lattices with three point interactions

26 pagesInternational audienceWe derive the equivalent energy of a square lattice that either deforms into the three-dimensional Euclidean space or remains planar. Interactions are not restricted to pairs of points and take into account changes of angles. Under some relationships between the local energies associated with the four vertices of an elementary square, we show that the limit energy can be obtained by mere quasiconvexification of the elementary cell energy and that the limit process does not involve any relaxation at the atomic scale. In this case, it can be said that the Cauchy-Born rule holds true. Our results apply to classical models of mechanical trusses that include torques between adjacent bars and to atomic models

Comparison between advected-field and level-set methods in the study of vesicle dynamics

International audiencePhospholipidic membranes and vesicles constitute a basic element in real biological functions. Vesicles are viewed as a model system to mimic basic viscoelastic behaviors of some cells, like red blood cells. Phase field and level-set models are powerful tools to tackle dynamics of membranes and their coupling to the flow. These two methods are somewhat similar, but to date no bridge between them has been made. This is a first focus of this paper. Furthermore, a constitutive viscoelastic law is derived for the composite fluid: the ambient fluid and the membranes. We present two different approaches to deal with the membrane local incompressibility, and point out differences. Some numerical results following from the level-set approach are presented

Mathematical modelling of the atherosclerotic plaque formation

International audienceThis article is devoted to the construction of a mathematical model describing the early formation of atherosclerotic lesions. Following the work of El Khatib, Genieys and Volpert, we model atherosclerosis as an inflammatory disease. We consider that the inflammatory process starts with the penetration of Low Density Lipoproteins cholesterol in the intima. This phenomenon is related to the local blood flow dynamics. Using a system of reaction-diffusion equations, we first provide a one-dimensional model of lesion growth. Then we perform numerical simulations on a two-dimensional geometry mimicking the carotid artery. We couple the previous mathematical model with blood flow and we provide a model in which the lesion appears in the area of lower shear stress

Homogenization of hexagonal lattices

International audienceWe characterize the macroscopic e ffective behavior of a graphene sheet modeled by a hexagonal lattice of elastic bars, using Gamma-convergence

MODELES COMPORTEMENTAUX. Vers une approche instrumentale de la synthèse d'images

International audienceLes travaux présentés ici ont un triple objectif : 1) proposer une extension de la notion d'objet, par l'élaboration de la notion d'objet physique ; 2) proposer des modèles d'objets physiques qui présentent un certain degré de constructivisme, grâce à une modularité conceptuelle et algorithmique ; 3) permettre une certaine instrumentalité de la simulation, afin que l'opérateur humain actionne réellement les objets simulés et dispose pour ce contrôle des informations visuelles et tactiles représentatives du comportement dynamique des objets