In the first part of this article, we extend the formal upscaling of a diffusion–precipitation model through a two-scale asymptotic expansion in a level set framework to three dimensions. We obtain upscaled partial differential equations, more precisely, a non-linear diffusion equation with effective coefficients coupled to a level set equation. As a first step, we consider a parametrization of the underlying pore geometry by a single parameter, e.g. by a generalized “radius” or the porosity. Then, the level set equation transforms to an ordinary differential equation for the parameter. For such an idealized setting, the degeneration of the diffusion tensor with respect to porosity is illustrated with numerical simulations. The second part and main objective of this article is the analytical investigation of the resulting coupled partial differential equation–ordinary differential equation model. In the case of non-degenerating coefficients, local-in-time existence of at least one strong solution is shown by applying Schauder's fixed point theorem. Additionally, non-negativity, uniqueness, and global existence or existence up to possible closure of some pores, i.e. up to the limit of degenerating coefficients, is guaranteed.

In this paper we develop and study numerically a model to describe some aspects of soundpropagation in the human lung, considered as a deformable and viscoelastic porous medium(the parenchyma) with millions of alveoli filled with air. Transmission of sound throughthe lung above 1 kHz is known to be highly frequency-dependent. We pursue the key ideathat the viscoelastic parenchyma structure is highly heterogeneous on the small scaleε and use two-scale homogenization techniques to derive effectiveacoustic equations for asymptotically small ε. This process turns out tointroduce new memory effects. The effective material parameters are determined from thesolution of frequency-dependent micro-structure cell problems. We propose a numericalapproach to investigate the sound propagation in the homogenized parenchyma using aDiscontinuous Galerkin formulation. Numerical examples are presented.

The paper deals with a Dirichlet spectral problem for an elliptic operator with ε-periodic coefficients in a 3D bounded domain of small thickness δ. We study the asymptotic behavior of the spectrum as ε and δ tend to zero. This asymptotic behavior depends crucially on whether ε and δ are of the same order (δ ≈ ε), or ε is much less than δ(δ = ετ, τ < 1), or ε is much greater than δ(δ = ετ, τ > 1). We consider all three cases.

Using the tool of two-scale convergence, we provide a rigorous mathematical setting for the homogenization result obtained by Fleck and Willis [J. Mech. Phys. Solids 52 (2004) 1855–1888] concerning the effective plastic behaviour of a strain gradient composite material. Moreover, moving from deformation theory to flow theory, we prove a convergence result for the homogenization of quasistatic evolutions in the presence of isotropic linear hardening.

The aim of this paper is to provide the correctors associated to the homogenization of a parabolic problem describing the heat transfer. The results here complete the earlier study in [Jose, Rev. Roumaine Math. Pures Appl.54 (2009) 189–222] on the asymptotic behaviour of a problem in a domain with two components separated by an ε-periodic interface. The physical model established in [Carslaw and Jaeger, The Clarendon Press, Oxford (1947)] prescribes on the interface the condition that the flux of the temperature is proportional to the jump of the temperature field, by a factor of orderεγ . We suppose that -1 < γ ≤ 1. As far as the energies of the homogenized problems are concerned, we consider the cases -1 < γ < 1 and γ = 1 separately. To obtain the convergence of the energies, it is necessary to impose stronger assumptions on the data. As seen in [Jose, Rev. Roumaine Math. Pures Appl.54 (2009) 189–222] and [Faella and Monsurrò, Topics on Mathematics for Smart Systems, World Sci. Publ., Hackensack, USA (2007) 107–121] (also in [Donato et al., J. Math. Pures Appl.87 (2007) 119–143]), the case γ = 1 is more interesting because of the presence of a memory effect in the homogenized problem.

Le présent travail concerne la détermination des propriétés mécaniques des âmes nids d'abeilles de forme hexagonale et de leurs versions sur-expansées rectangulaires. Ces âmes en papiers Nomex® sont présentes sur les voiliers multicoques de courses océaniques entièrement réalisés à partir de sandwichs à peaux de tissus carbones. Les caractéristiques mécaniques élastiques de ces âmes sont déterminées par modélisation tridimensionnelle à partir de la théorie de l'homogénéisation des milieux périodiques mise en œuvre dans le cadre de la méthode des éléments finis. L'étude confirme que les symétries des Volumes Élémentaires Représentatifs (VER) conduisent à des propriétés mécaniques homogénéisées orthotropes. Enfin, l'étude des modes de flambement des VERs permet de déterminer les contraintes ultimes de rupture du matériau homogène équivalent. Les techniques d'homogénéisation périodique développées dans cette étude ont été implantées dans le code de calcul éléments finis Cast3M-CEA. Les propriétés mécaniques et les limites à rupture des âmes en nids d'abeilles sont utilisées pour l'étude du renforcement des structures sandwichs proposées sur les trimarans océaniques de 60 pieds.

In this paper we give a general presentation ofthe homogenization of Neumann type problems in periodically perforateddomains, including the case where the shape of the reference hole varies with the sizeof the period (in the spirit of the construction of self-similar fractals).We shows that H 0-convergence holds under the extra assumption thatthere exists a bounded sequence of extension operators forthe reference holes. The general classof Jones-domains gives an example where this result applies. When this assumption fails, another approach, usingthe Poincaré–Wirtingerinequality is presented. A corresponding class where it appliesis that of John-domains, for which the Poincaré–Wirtinger constantis controlled.The relationship between these two kinds of assumptions is also clarified.