Isabelle Gruais

The influence of lateral boundary conditions on the asymptotics in thin elastic plates

Here we investigate the limits and the boundary layers of the three-dimensional displacement in thin elastic plates as the thickness tends to zero in each of the eight main types of lateral boundary conditions on their edges: hard and soft clamped, hard and soft simple support, friction conditions, sliding edge, and free plates. Relying on construction algorithms [M. Dauge and I. Gruais, Asymptotic Anal., 13 (1996), pp. 167--197], we establish an asymptotics of the displacement combining inner and outer expansions. We describe the two first terms in the outer expansion: these are Kirchhoff--Love displacements satisfying prescribed boundary conditions that we exhibit. We also study the first boundary layer term: when the transverse component is clamped, it has generically nonzero transverse and normal components, whereas when the transverse component is free, the first boundary layer term is of bending type and has only its nonzero in-plane tangential component.

Homogenization of a conductive suspension in a Stokes–Boussinesq flow

Radiant spherical suspensions have an ϵ-periodic distribution in a tridimensional incompressible viscous fluid governed by the Stokes–Boussinesq system. We perform the homogenization procedure when the radius of the solid spheres is of order ϵ3 (the critical size of perforations for the Navier-Stokes system) and when the ratio of the fluid/solid conductivities is of order ϵ6, the order of the total volume of suspensions. Adapting the methods used in the study of small inclusions, we prove that the macroscopic behavior is described by a Brinkman–Boussinesq type law and two coupled heat equations, where certain capacities of the suspensions and of the radiant sources appear.

Edge layers in thin elastic plates

Abstract This paper deals with the asymptotics of the displacement of a thin elastic 3D plate when it is submitted to various boundary conditions on its lateral face: namely, hard and soft clamped conditions, and hard support. Of particular interest is the influence of the edges of the plate where boundary conditions of different types meet. Relying on general results of [M. Dauge and I. Gruais, Asymptotics of arbitrary order for a thin elastic clamped plate. I: Optimal error estimates. Asymptotic Anal. 13 (1996) 167–197] and [M. Dauge and I. Gruais, Asymptotics of arbitrary order for a thin elastic clamped plate. II: Analysis of the boundary layer terms, Asymptotic Anal. (1996) to appear] for the hard clamped case, we see that the clamped plate (hard and soft) admit strong boundary layers, in which are concentrated the edge layers, while the hard supported plate has no edge layer and even no boundary layer at all in certain situations. We conclude with hints about corner layers, in the case when the mean surface of the plate itself is polygonal.

Diffusion in a highly rarefied binary structure of general periodic shape

We study the homogenization of a diffusion process that takes place in a binary structure formed by an ambiental connected phase surrounding a suspension of very small particles of general form distributed in an ε-periodic network. The asymptotic distribution of the concentration is determined for both phases, as ε → 0, assuming that the suspension has mass of unity order and vanishing volume. Three cases are distinguished according to the values of a certain rarefaction number. When it is positive and finite, the macroscopic system involves a two-concentration system, coupled through a term accounting for the non-local effects. In the other two cases, where the rarefaction number is either infinite or going to zero, although the form of the system is much simpler, some peculiar effects still account for the presence of the suspension.

Heat transfer models for two-component media with interfacial jump

The paper deals with the asymptotic behaviour of the heat transfer in a bounded domain having an -periodic structure formed by two interwoven components separated by an interface on which the heat flux is continuous and the temperature subjects to a first-order jump condition. We study the cases when the orders of magnitude with respect to of the ratio between the two conductivities and of the jump transmission coefficient are, respectively, and , with and . We derive the macroscopic laws and the effective coefficients obtained by the two-scale convergence technique of the homogenization theory.

Homogenizing media containing a highly conductive honeycomb substructure

The present paper deals with the homogenization of the heat conduction which takes place in a binary three-dimensional medium consisting of an ambiental phase having conductivity of unity order and a rectangular honeycomb structure formed by a set of thin layers crossing orthogonally and periodically. We consider the case when the conductivity of the thin layers is in inverse proportion to the vanishing volume of the rectangular honeycomb structure. We find the system that governs the asymptotic behaviour of the temperature distribution of this binary medium. The dependence with respect to the thicknesses of the layers is also emphasized. We use an energetic method associated to a natural control-zone of the vanishing domain. Mathematical Subject Classification (2000). 35B27, 35K57, 76R50. Keywords. homogenization, conduction, fine-scale, honeycomb structure.

Homogenization of a highly rarefied binary structure of finite diffusivity

The homogenization of a binary structure made of very small particles of general shape is performed when the diffusivity is finite and when the size of the particles has a critical value with respect to a rarefaction number. The limit problem involves an auxiliary function which can be interpreted as the solution of a problem of cellular type, thus filling the gap with classical methods of multiple scales.

Homogenization of fluid–porous interface coupling in a biconnected fractured media

The modelization of mass transfer through biconnected fractured porous media is studied by homogenizing the coupling between the Darcyan percolation and the viscous Stokes flow on their interface governed by the Beavers–Joseph law. The case of high transmission coefficients is considered. The asymptotic behaviour is completely described with the help of the solutions of some specific local problems and of a nonhomogeneous Neumann problem defined by the effective permeability tensor.

The effective permeability of fractured porous media subject to the Beavers–Joseph contact law

We are interested in the asymptotic behaviour of a fluid flow contained in a microscopic periodic distribution of fissures perturbating a porous medium where the Darcy law is valid, when the coupling between both systems is modeled by the Beavers–Joseph interface condition. As the small period of the distribution tends to zero, the interface condition is preserved on a microscopic scale under the additional assumption that the permeability coefficients behave like the squared period of the distribution which is also the squared size of the fissures. Moreover, the resulting pressure is purely macroscopic unlike the velocity field which also depends on the microscopic variable.

Asymptotic heat equation for crossing superconductive thin walls

This work deals with the homogenization of the nonstationary heat conduction which takes place in a binary three-dimensional medium consisting of an ambiental phase having conductivity of unity order and a set of highly conductive thin walls crossing orthogonally and periodically. This situation covers in fact three types of microstructures, usually called: box-type (or honeycomb), gridwork and layered. The study is based on the energetic procedure of homogenization associated to a control-zone method, specific to the geometry of the microstructure and to the singularity of the conductivity coefficients. In the present case the main result is the system that governs the asymptotic behaviour of the temperature distribution in this binary medium. It displays a significant increase of the conductivity due to the superconductive thin walls, revealing their seemingly paradoxal behaviour of having an everlasting action on the environment, in spite of an obvious vanishing volume. Moreover, the dependence of this behaviour with respect to the relative thicknesses of the walls can be detailed..