Asymptotic analysis of contact problems between an elastic material and thin-rigid plates

We consider an elastic material in contact with a three-dimensional rigid plate of varying thickness. We suppose that a perfect adhesion occurs along thin zones disposed in a self-similar way on the interface between the two materials. We suppose that the elasticity coefficients in the plate depend on its thickness and tend to infinity as this thickness tends to zero. We derive the effective material properties using Γ-convergence methods.     

Asymptotic analysis of a transport model of organic material through a stratified porous medium: Concentration effect

We consider the transport through capillarity of an organic material inside a porous medium, using Leverett’s model. We first prove an existence result for a weak solution of this nonlinear evolution problem, using a regularization process. We then describe the asymptotic behavior of the solution, when the permeability _kε$k_{\epsilon }$ of the porous medium is associated to a scalar function which only depends on the third variable, assuming that _kε$k_{\epsilon }$ (resp. the inverse of _kε$k_{\epsilon }$) converges to some measure _λ∗${\lambda }_{\ast }$ (resp. λ^∗${\lambda }^{\ast }$). We use Γ-convergence arguments in order to describe this asymptotic behavior. We finally characterize the asymptotic behavior of the problem, considering special choices of the permeability _kε$k_{\epsilon }$, which correspond to stratified porous media, and give a numerical test for a 1D model.     

Homogenization of an incompressible viscous flow in a porous medium with double porosity

We study the homogenization of an incompressible viscous flow in a porous medium with double porosity. We derive a macroscopic model with local Navier–Stokes system in the large cavities, Darcy law in the thinner porous rock, and a contact law between the two. We use Γ-convergence methods associated with multi-scale convergence notions in order to get this limit law. We exhibit a critical ratio between the two scales of the pores.     

Homogenization of rectangular cross-section fibre-reinforced materials: bending–torsion effects

We study the homogenization of an elastic material in contact with periodic parallel elastic rectangular cross-section fibres of higher rigidity. The interactions between the matrix and the fibres are described by a local adhesion contact law with interfacial adhesive stiffness parameter depending on the period. Assuming that the Lamé constants in the fibres and the stiffness parameter have appropriate orders of magnitude, we derive a class of energy functionals involving extension, flexure and torsion terms.     

On the interface boundary conditions between two interacting incompressible viscous fluid flows

We consider two incompressible viscous fluid flows interacting through thin non-Newtonian boundary layers of higher Reynolds? number. We study the asymptotic behaviour of the problem, with respect to the vanishing thickness of the layers, using Γ-convergence methods. We derive general interfacial boundary conditions between the two fluid flows. These boundary conditions are specified for some particular cases including periodic or fractal structures of layers.     

Interfacial Contact Model in a Dense Network of Elastic Materials

Functional Analysis and Its Applications - We consider a dense network of elastic materials modeled by a dense network of elastic disks. More specifically, we consider a dense network of elastic...     

Derivation of a contact law between a free fluid and thin porous layers via asymptotic analysis methods

We describe the asymptotic behaviour of an incompressible viscous free fluid in contact with a porous layer flow through the porous layer surface. This porous layer has a small thickness and consists of thin channels periodically distributed. Two scales are present in this porous medium, one associated to the periodicity of the distribution of the channels and the other to the size of these channels. Proving estimates on the solution of this Stokes problem, we establish a critical link between these two scales. We prove that the limit problem is a Stokes flow in the free domain with further boundary conditions on the basis of the domain which involve an extra velocity, an extra pressure and two second-order tensors. This limit problem is obtained using Γ-convergence methods. We finally consider the case of a Navier–Stokes flow within this context.