Georges Griso

THE PERIODIC UNFOLDING METHOD IN HOMOGENIZATION

The periodic unfolding method was introduced in 2002 in [Cioranescu, Damlamian, and Griso, C.R. Acad. Sci. Paris, Ser. 1, 335 (2002), pp. 99–104] (with the basic proofs in [Proceedings of the Narvi...

Junction of a periodic family of elastic rods with a 3d plate. Part I

In this second paper, we consider again a set of elastic rods periodically distributed over an elastic plate whose thickness tends here to 0. This work is then devoted to describe the homogenization process for the junction of the rods and a thin plate. We use a technique based on two decompositions of the displacement field in each rod and in the plate. We obtain a priori estimates on each term of the two decompositions which permit to exhibit a few critical cases that distinguish the different possible limit behaviors. Then, we completely investigate one of these critical case which leads to a coupled bending-bending model for the rods and the 2d plate. Résumé Dans ce deuxième article, nous reprenons un ensemble de poutres élastiques périodiquement distribuées sur une plaque élastique dont l’épaisseur tend maintenant vers 0. Il s’agit donc de décrire des modèles d’homogénéisation pour la jonction de poutres et d’une plaque mince. Nous utilisons une technique de décomposition du champ de déplacement à la fois dans chaque poutre et dans la plaque. On obtient des estimations a priori sur chacun des termes de ces décompositions qui mettent en particulier en évidence les cas critiques qui séparent les différents modèles limites possibles. Ensuite, nous analysons en détail un de ces cas critiques pour lequel on obtient un modèle de couplage flexion-flexion entre les poutres et la plaque 2d.

Multiscale Modeling of Elastic Waves: Theoretical Justification and Numerical Simulation of Band Gaps

We consider a three-dimensional composite material made of small inclusions periodically embedded into an elastic matrix; the whole structure presents strong heterogeneities between its different components. In the general framework of linearized elasticity we show that, when the size of the microstructures tends to zero, the limit homogeneous structure presents, for some wavelengths, a negative “mass density” tensor. Hence we are able to rigorously justify the existence of forbidden bands, i.e., intervals of frequencies in which there is no propagation of elastic waves. In particular, we show how to compute these band gaps and illustrate the theoretical results with some numerical simulations.

Homogenization of Periodically Varying Coefficients in Electromagnetic Materials

In this paper, we employ the periodic unfolding method for simulating the electromagnetic field in a composite material exhibiting heterogeneous microstructures which are described by spatially periodic parameters. We consider cell problems to calculate the effective parameters for a Debye dielectric medium in the case of a circular microstructure in two dimensions. We assume that the composite materials are quasi-static in nature, i.e., the wavelength of the electromagnetic field is much larger than the relevant dimensions of the microstructure.

DECOMPOSITION OF DEFORMATIONS OF THIN RODS: APPLICATION TO NONLINEAR ELASTICITY

This paper deals with the introduction of a decomposition of the deformations of curved thin beams, with section of order δ, which takes into account the specific geometry of such beams. A deformation v is split into an elementary deformation and a warping. The elementary deformation is the analog of a Bernoulli–Naviers displacement for linearized deformations replacing the infinitesimal rotation by a rotation in SO(3) in each cross section of the rod. Each part of the decomposition is estimated with respect to the L2 norm of the distance from gradient v to SO(3). This result relies on revisiting the rigidity theorem of Friesecke–James–Muller in which we estimate the constant for a bounded open set star-shaped with respect to a ball. Then we use the decomposition of the deformations to derive a few types of asymptotic geometrical behavior: large deformations of extensional type, inextensional deformations and linearized deformations. To illustrate the use of our decomposition in nonlinear elasticity, we consider a St Venant–Kirchhoff material and upon various scalings on the applied forces we obtain the Γ-limit of the rescaled elastic energy. We first analyze the case of bending forces of order δ2 which leads to a nonlinear extensible model. Smaller pure bending forces give the classical linearized model. A coupled extentional-bending model is obtained for a class of forces of order δ2 in traction and of order δ3 in bending.

Effective constitutive parameters of periodic composites

We present a novel method for analyzing 2D and 3D lossy periodic composites materials, combining an asymptotic multiscale method combined with the unfolding method. The computed effective conductivity for square cylinders (2D), and cubes (3D) suspended in a host isotropic medium are compared with the Maxwell-Garnett mixing formula predictions. The electromagnetic field in a finite lattice of square cylinders using this novel technique is compared with the exact electromagnetic field calculated directly in the heterogeneous lattice.

Application of Multi-Scale Modelling to Some Elastic, Piezoelectric and Electromagnetic Composites

One way to obtain “new materials” is to mix classical ones with periodic heterogeneous microstructures. Using the homogenization method, we rigorously justify the limit models obtained (when the size of the microstructures goes to zero) for composites with very promising properties. We present here three different examples: a new bio-compatible piezoelectric/living cells material designed to improve bone regeneration, a bianisotropic electromagnetic material in a first attempt of photonic crystals modelling and a strongly heterogeneous elastic material exhibiting acoustic band-gaps. We also provide numerical simulations in order to illustrate the influence of the change of intrinsic parameters such as shape and physical characteristics of the micro structures.

Mechanical modeling of the skin

The skin is made of three main layers which are, from the top to the bottom: the epidermis, the dermis and the hypodermis. We consider the dermis as made of a Stokes fluid interacting with a periodic network of elastic fibers, assumed to obey the linearized elasticity law of behaviour. Above and below, the epidermis and the hypodermis are elastic solids. As the dimension of the thickness is very small compared to the two others, we assume periodic boundary conditions in those two planar directions. We study the 3d fluid-structure interaction system in a first part, and in a second part, we make the characteric size of the periodic element of the network go to zero in order to find an homogenized law for the whole skin. Starting from linear elastic materials, we find a viscoelastic law at the limit.

A simplified model for elastic thin shells

We introduce a simplified model for the minimization of the elastic energy in thin shells. This model is not obtained by an asymptotic analysis. The nonlinear simplified model admits always minimizers by contrast with the original one. We show the relevance of our approach by proving that the rescaled minimum of the simplified model and the rescaled infimum of the full model have the same limit as the thickness tends to 0. The simplified energy can be expressed as a functional acting over fields defined on the mid-surface of the shell and where the thickness remains as a parameter.

Homogenization via unfolding in periodic layer with contact

In this work we consider the elasticity problem for two domains separated by a heterogeneous layer. The layer has an e−periodic structure, e 1, including a multiple micro-contact between the structural components. The components are surrounded by cracks and can have rigid displacements. The contacts are described by the Signorini and Tresca-friction conditions. In order to obtain preliminary estimates modification of the Korn inequality for the e−dependent periodic layer is performed. An asymptotic analysis with respect to e → 0 is provided and the limit problem is obtained, which consists of the elasticity problem together with the transmission condition across the interface. The periodic unfolding method is used to study the limit behavior.