Onétudie le comportement asymptotique de la solution de l'équation de Laplace dans un domaine dont une partie de la frontière est fortement oscillante. La motivation de ce travail est l'étude d'unécoulement longitudinal dans un domaine infini borné inférieurement par une paroi et supérieurement par une paroi rugueuse. Cette dernière est un plan recouvert d'aspérités périodiques dont la taille dépend d'un petit paramètre ε. On fait l'hypothèse de rugosité forte,à savoir que la hauteur des aspérités reste constante. A l'aide d'un correcteur de couche limite, on obtient une approximation non oscillante de la solution qui est d'ordre ε 3/2) en norme H 1 .

In this work, a refined interelement diffuse fracture theoretical model, based on a cohesive finite element approach, is proposed for concrete and other quasibrittle materials. This model takes advantage of a novel micromechanics‐based calibration technique for reducing the artificial compliance associated with the adopted intrinsic formulation. By means of this technique, the required values for the elastic stiffness parameters to obtain nearly invisible cohesive interfaces are provided. Furthermore, the mesh‐induced toughening effect, essentially related to the artificial crack tortuosity caused by the different orientations of the interelement cohesive interfaces, is numerically investigated by performing comparisons with an additional fracture model, newly introduced for the purpose of numerical validation. These comparisons are presented to assess the reliability and the numerical accuracy of the proposed fracture approach.

We consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction Ωε, which is the union of a domain Ω0 and a large number 2N of thin rods with variable thickness of order [Formula: see text]. The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are ε-periodically alternated. The Robin conditions are given on the lateral boundaries of the thin rods. Using the method of matched asymptotic expansions, we construct the asymptotic approximation for the solution as ε → 0 and prove the corresponding estimates in the Sobolev space H1(Ωε).

The main purpose of this paper is to derive a wall law for a flow over a very rough surface. We consider a viscous incompressible fluid filling a 3-dimensional horizontal domain bounded at the bottom by a smooth wall and at the top by a very rough wall. The latter consists in a plane wall covered with periodically distributed asperities which size depends on a small parameter ε > 0 and with a fixed height. We assume that the flow is governed by the stationary Stokes equations. Using asymptotic expansions and boundary layer correctors we construct and analyze an asymptotic approximation of order $\mathcal{O}(\varepsilon^{3/2-\gamma})$ ($\gamma>0$ being arbitrary small) in the $H^1$ norm for the velocity, and in the $L^2$ norm for the pressure. We derive an effective boundary condition of Navier type, then expressing the boundary layer terms in terms of the homogenized solution and the solution of a cell problem we obtain an effective approximation in the whole domain of the flow

Recent progresses in nanotechnology have clearly shown that the incorporation of nanomaterials within concrete elements leads to a sensible increase in strength and toughness, especially if used in combination with randomly distributed short fiber reinforcements, as for ultra high-performance fiber-reinforced concrete (UHPFRC). Current damage models often are not able to accurately predict the development of diffuse micro/macro-crack patterns which are typical for such concrete structures. In this work, a diffuse cohesive interface approach is proposed to predict the structural response of UHPFRC structures enhanced with embedded nanomaterials. According to this approach, all the internal mesh boundaries are regarded as potential crack segments, modeled as cohesive interfaces equipped with a mixed-mode traction-separation law suitably calibrated to account for the toughening effect of nano-reinforcements. The proposed fracture model has been firstly validated by comparing the failure simulation results of UHPFRC specimens containing different fractions of graphite nanoplatelets with the available experimental data. Subsequently, such a model, combined with an embedded truss model to simulate the concrete/steel rebars interaction, has been used for predicting the load-carrying capacity of steel bar-reinforced UHPFRC elements enhanced with nanoplatelets. The numerical outcomes have shown the reliability of the proposed model, also highlighting the role of the nano-reinforcement in the crack width control.