A Bending-Gradient theory for thick laminated plates homogenization

This work presents a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Love-Kirchhoff theory, to which six components are added representing the gradient of the bending moment. The Bending-Gradient theory is an extension to arbitrary multilayered plates of the Reissner-Mindlin theory which appears as a special case when the plate is homogeneous. The new theory is applied to multilayered plates and its predictions are compared to full 3D Pagano's exact solutions and other approaches. It gives good predictions of both deflection and shear stress distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/h goes to infinity

Homogenization of a space frame as a thick plate: Application of the Bending-Gradient theory to a beam lattice

International audienceThe Bending-Gradient theory for thick plates is the extension to heterogeneous plates of Reissner-Mindlin theory originally designed for homogeneous plates. In this paper the Bending-Gradient theory is extended to in-plane periodic structures made of connected beams (space frames) which can be considered macroscopically as a plate. Its application to a square beam lattice reveals that classical Reissner-Mindlin theory cannot properly model such microstructures. Comparisons with exact solutions show that only the Bending-Gradient theory captures second order effects in both deflection and local stress fields

A bending-gradient model for thick plates, I : theory

International audienceThis is the first part of a two-part paper dedicated to a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Kirchhoff-Love theory (3 in-plane stresses and 3 bending moments), to which six components are added representing the gradient of the bending moment. The new theory, called the Bending-Gradient plate theory is described in the present paper. It is an extension to arbitrarily layered plates of the Reissner-Mindlin plate theory which appears as a special case of the Bending-Gradient plate theory when the plate is homogeneous. However, we demonstrate also that, in the general case, the Bending-Gradient model cannot be reduced to a Reissner-Mindlin model. In part two (Lebee and Sab, 2010a), the Bending-Gradient theory is applied to multilayered plates and its predictions are compared to those of the Reissner-Mindlin theory and to full 3D Pagano's exact solutions. The main conclusion of the second part is that the Bending-Gradient gives good predictions of both deflection and shear stress distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/h goes to infinity

Quelques exemples d'application aux composites stratitfiés de la théorie Bending-Gradient pour les plaques

International audienceCe travail présente l'application aux composites fibrés d'une nouvelle théorie de plaque. Ce modèle destiné aux plaques épaisses et anisotropes utilise les six inconnues statiques de la theorie de Kirchhoff-Love auxquelles sont ajoutées six nouvelles inconnues représentant le gradient dumoment de flexion. Nommé théorie Bending-Gradient, ce nouveaumodèle peut être considéré comme une extension aux plaques hétérogènes dans l'épaisseurs du modèle de Reissner-Mindlin ; ce dernier étant un cas particulier lorsque la plaque est homogène. La théorie Bending-Gradient est appliquée aux plaques stratifiées et comparée à la solution exacte de Pagano [1] ainsi qu'à d'autres approches. Elle donne de bonnes prédictions pour la flèche, pour la distribution des contraintes de cisaillement transverse ainsi que pour les déplacements plans dans de nombreuses configurations matérielles

Homogenization of thick periodic plates: Application of the Bending-Gradient plate theory to a folded core sandwich panel

International audienceIn a previous paper from the authors, the bounds from Kelsey et al. (1958) were applied to a sandwich panel including a folded core in order to estimate its shear forces stiffness (Lebée and Sab, 2010b). The main outcome was the large discrepancy of the bounds. Recently, Lebée and Sab (2011a) suggested a new plate theory for thick plates - the Bending-Gradient plate theory - which is the extension to heterogeneous plates of the well-known Reissner-Mindlin theory. In the present work, we provide the Bending-Gradient homogenization scheme and apply it to a sandwich panel including the chevron pattern. It turns out that the shear forces stiffness of the sandwich panel is strongly influenced by a skin distortion phenomenon which cannot be neglected in conventional design. Detailed analysis of this effect is provided

New boundary conditions for the computation of the apparent stiffness of statistical volume elements

International audienceWe present a new auxiliary problem for the determination of the apparent stiffness of a Statistical Volume Element (SVE). The SVE is embedded in an infinite, homogeneous reference medium, subjected to a uniform strain at infinity, while tractions are applied to the boundary of the SVE to ensure that the imposed strain at infinity coincides with the average strain over the SVE. The main asset of this new auxiliary problem resides in the fact that the associated Lippmann-Schwinger equation involves without approximation the Green operator for strains of the infinite body, which is translation-invariant and has very simple, closed-form expressions. Besides, an energy principle of the Hashin and Shtrikman type can be derived from this modified Lippmann-Schwinger equation, allowing for the computation of rigorous bounds on the apparent stiffness. The new auxiliary problem requires a cautious mathematical analysis, because it is formulated in an unbounded domain. Observing that the displacement is irrelevant for homogenization purposes, we show that selecting the strain as main unknown greatly eases this analysis. Finally, it is shown that the apparent stiffness defined through these new boundary conditions "interpolates" between the apparent stiffnesses defined through static and kinematic uniform boundary conditions, which casts a new light on these two types of boundary conditions

Contributions à l'homogénéisation des matériaux aléatoires

Non disponible

Analysis of the mechanical behaviour of soft rockfall barriers

International audienceSoft rockfall barriers are complex structures that generally consist of a metallic net supported by steel posts and cables with brake elements. Several experimental and numerical studies have been carried out to evaluate their behaviour and a technical agreement in EU was recently established to certify these barriers based on experimental tests. Actually, manufacturers develop rockfall kits with their own technical specificities. The objective of the present paper is to determine the intrinsic properties of most common nets technologies and to investigate their influence on the overall mechanical behaviour of the structure. To this end, a comprehensive comparison between the local behaviours of the different nets is first presented using equivalent homogeneous membranes. Results derived for square nets under static concentrated loading illustrate the influence of the manufacturing technology on the deflection and stresses distribution. Then, a numerical and analytical model for the so-called "curtain effect" is developed and validated. In the conclusion, it is focused on the capacity of the pro-posed methodology to study and evaluate the response of the whole barrier

A variational form of the equivalent inclusion method for numerical homogenization

International audienceDue to its relatively low computational cost, the equivalent inclusion method is an attractive alternative to traditional full-field computations of heterogeneous materials formed of simple inhomogeneities (spherical, ellipsoidal) embedded in a homogeneous matrix. The method can be seen as the discretization of the Lippmann--Schwinger equation with piecewise polynomials. Contrary to the original approach of Moschovidis and Mura, who discretized the strong form of the Lippmann--Schwinger equation through Taylor expansions, we propose in the present paper a Galerkin discretization of the weak form of this equation. Combined with the new, mixed boundary conditions recently introduced by the authors, the resulting method is particularly well-suited to homogenization. It is shown that this new, variational approach has a number of benefits: (i) the resulting linear system is well-posed, (ii) the numerical solution converges to the exact solution as the maximum degree of the polynomials tends to infinity and (iii) the method can provide rigorous bounds on the apparent properties of the statistical volume element, provided that the matrix is stiffer (or softer) than all inhomogeneities. This paper presents the formulation and implementation of the new, variational form of the equivalent inclusion method. Its efficiency is investigated through numerical applications in 2D and 3D elasticity