Arthur Lebée

Laboratoire Navier (ENPC-IFSTTAR-CNRS) UMR8205 · Matériaux et Structures Architecturés

Origami tessellations are particular textured morphing shell structures. Their unique folding and unfolding mechanisms on a local scale aggregate and bring on large changes in shape, curvature and elongation on a global scale. The existence of these global deformation modes allows for origami tessellations to fit non-Trivial surfaces thus inspiring...

This book gives new insight on plate models in the linear elasticity framework taking into account heterogeneities and thickness effects. It is targeted to graduate students how want to discover plate models but deals also with latest developments on higher order models. Plates models are both an ancient matter and a still active field of research....

Starting from simple notions of paper folding, a review of current challenges regarding folds and structures is presented. A special focus is dedicated to folded tessellations which are raising interest from the scientific community. Finally, the different mechanical modeling of folded structures are investigated. This reveals efficient application...

The Bending-Gradient plate theory originally presented in Lebée and Sab (International Journal of Solids and Structures 48 (2011) 2878–2888) is applied to cellular sandwich panels. This theory is the extension of the Reissner–Mindlin theory to heterogeneous plates. Its application clarifies common assumptions made in sandwich theory. It also enable...

This 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 p...

Since the success of bone regenerative medicine depends on scaffold morphological and mechanical properties, numerous scaffolds designs have been proposed in the last decade, including graded structures that are suited to enhance tissue ingrowth. Most of these structures are based either on foams with a random pore definition, or on the periodic re...

La construction de coques minces en béton est coûteuse en matériaux et en main d’œuvre à cause de la fabrication du coffrage qui génère une vaste quantité de déchets. Ces éléments non réutilisables ont un impact négatif sur l’ACV de la construction. Ces difficultés expliquent en partie pourquoi la construction de coques minces est devenue rare à la...

Construction of concrete shells is expensive and generates wastes from the fabrication of formworks. Being non-reusable, these elements have a negative impact on the life-cycle assessment of the construction. The purpose of this research is to design and build a new inexpensive formwork system made of inflatable structures for precast and thin conc...

The principles of origami design have proven useful in a number of technological applications. Origami tessellations in particular constitute a class of morphing metamaterials with unusual geometric and elastic properties. Although inextensible in principle, fine creases allow origami metamaterials to effectively deform non-isometrically. Determini...

This article presents an experiment to characterize the rolling shear modulus Gr,mean of CLT panels and its creep. A four-point bending test is achieved on sandwich beams with skins and wooden core. This allows to isolate the CLT cross-layer and to characterize the shear behaviour. The experiment is performed in a constant environment during 6 mont...

The principles of origami design have proven useful in a number of technological applications. Origami tessellations in particular constitute a class of morphing metamaterials with unusual geometric and elastic properties. Although inextensible in principle, fine creases allow origami metamaterials to effectively deform non-isometrically. Determini...

This paper presents an architectured material featuring significant strain-gradient effects and called pantographic material. It is easy to fabricate, being a plate made of a single and continuous linear elastic material containing voids. The pattern consists of triangles connected by thin junctions and arranged in such a way that two floppy strain...

The multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogeni...

The Asymptotic Expansion Load Decomposition higher-order beam model is based on the classical two-scale asymptotic expansion in the linear elasticity framework.It was successively extended to eigenstrains and to plasticity in small deformations in different papers. The present paper offers a comprehensive and consistent presentation of our approach...

An original experimental setup, dedicated to the measurement of the dynamic response of structures, is presented. Called the Robotized Laser Doppler Vibrometer (RLDV), it consists in the assembly of a fixed point Laser Doppler Vibrometer (LDV) on a 6-axis industrial robot arm. This allows to measure the 3D velocity on the surface of objects with a...

Various forms of timber hollow structural profiles have either been proposed in the literature or are already commercially available. Tests performed on Circular Hollow Section (CHS) beams showed failure modes not usually encountered in timber structures. For relatively thin-walled profiles, a sudden failure in the compression zone, with the openin...

The proper sizing of Cross Laminated Timber (CLT) walls for the construction of high rise buildings requires to take into account their low shear stiffness and their viscoelastic properties and to integrate them into the framework of actual building codes which are all based upon Ayrton-Perry approach of imperfect columns. The present paper starts...

This paper is concerned with the prediction of the propagation of flexural waves in anisotropic laminated plates with relatively high slenderness ratios by means of refined plate models. The study is conducted using the Bending-Gradient theory which is considered as an extension of the Reissner–Mindlin theory to multilayered plates. Two projections...

This article discusses design strategies to improve the mechanical behavior of elastic gridshells with singularities. The advantage of meshing with one or more singularity is to allow a wider range of surfaces to be meshed with equal-length, quadrilateral meshes, known as Chebyshev nets. However, the application of this meshing process will influen...

The present paper introduces a new elastoplastic beam model for reinforced concrete based on a higher-order beam model previously developed (Int J Numer Methods Eng. https://doi.org/10.1002/nme.5926, 2018). Steel and concrete are both defined as elastoplastic materials. The beam model represents the concrete body whereas rebars are given a specific...

The High Resolution Wavevector Analysis (HRWA) is presented and its application illustrated. Extending the High Resolution Wavenumber Analysis method [1] to 2D signals, it allows the wide-band and local characterization of the linear elastic behavior of anisotropic plates. The method belongs to the family of experimental wavenumber-based characteri...

In this paper, a closed-form approach is presented to estimate rapidly the equivalent stiffness of boards used in cross laminated timber (CLT) panels from local orthotropic behavior at ring scale for varying sawing patterns. It is first assumed that narrow edges are glued. In this case, closed-form Reuss and Voigt bounds are derived for the equival...

The High-Resolution Wavenumber Analysis (HRWA) is presented. It identifies complex wavenumbers and amplitudes of waves composing the harmonic response of a beam. Based on the frequency dependence of these wavenumbers, experimental dispersion equations of various beam mechanisms (e.g bending, torsion) can be retrieved. The HRWA method is compared to...

In this paper, a new methodology for the experimental determination of the CLT equivalent cross-layer shear elastic modulus is suggested using a wooden core sandwich beam with Carbon Fiber Reinforced Polymer skins under four-point bending. The stiffness contrast between the wooden layer and the CFRP skins ensures that the bending stiffness of the s...

This paper is devoted to the mathematical justification of the Bending-Gradient theory which is considered as the extension of the Reissner-Mindlin theory (or the First Order Shear Deformation Theory) to heterogeneous plates. In order to rigorously assess the well-posedness of the Bending-Gradient problems, we first assume that the compliance tenso...

The basic structural units of adsorbing microporous materials such as clays and cementitious materials are flexible nanolayers. The flexibility of these layers is reported to play a crucial role in the structuration of these materials, potentially affecting therefore the thermo-mechanical behavior of such materials. Adsorbing fluids are structured...

A new higher‐order elasto‐plastic beam model is derived and implemented in this paper. The elastic kinematics of the element reduced kinematic approximation is based on a higher‐order elastic beam model using the asymptotic expansion method to extend its kinematics. This model introduces new degrees of freedom associated to arbitrary loads as well...

In a former paper by the authors (Franzoni et al., 2017), the elastic behavior of Cross Laminated Timber (CLT) and timber panels having periodic gaps between lateral lamellae has been analyzed. A thick plate homogenization scheme based on Finite Elements computations has been applied. The predicted behavior was in agreement with experimental result...

A higher-order beam model based on the asymptotic expansion method was suggested by Ferradi et al.[11] Introducing new degrees of freedom specific to the applied loads into the kinematics of the beam, this model yields fast and accurate results. The present paper focuses on the extension of this model to the case of arbitrary eigenstrains expressed...

Origami tessellations are particular textured morphing shell structures. Their unique folding and unfolding mechanisms at a local scale aggregate and bring on large changes in shape, curvature and elongation at a global scale. The existence of these global deformation modes allows for origami tessellations to fit non-trivial surfaces. This paper ch...

In the pre­sent pa­per, the in­flu­ence of pe­ri­odic gaps be­tween lamel­las of Cross Lam­i­nated Tim­ber (CLT) on the panel’s elas­tic be­hav­ior is an­a­lyzed by means of a pe­ri­odic ho­mog­e­niza­tion scheme for thick plates hav­ing pe­ri­odic geom­e­try. Both small gaps, due to the fab­ri­ca­tion process of not-glu­ing lat­eral lamel­las, and...

This is the first part of a two-part paper presenting the generalization of Reissner thick plate theory (Reissner in J. Math. Phys. 23:184–191, 1944) to laminated plates and its relation with the Bending-Gradient theory (Lebée and Sab in Int. J. Solids Struct. 48(20):2878–2888, 2011 and in Int. J. Solids Struct. 48(20):2889–2901, 2011). The origina...

In the first part of this two-part paper (Lebée and Sab in On the generalization of Reissner plate theory to laminated plates, Part I: theory, doi:10. 1007/ s10659-016-9581-6, 2015), the original thick plate theory derived by Reissner (J. Math. Phys. 23:184–191, 1944) was rigorously extended to the case of laminated plates. This led to a new plate...

In a recent work, a new plate theory for thick plates was suggested where the static unknowns are those of the Kirchhoff-Love theory, to which six components are added representing the gradient of the bending moment (Lebée and Sab, Int J Solids Struct, 48(20):2878-2888, 2011a). This theory, called the Bending-Gradient theory, is the extension to mu...

In the present paper, the experimental deflection of Cross Laminated Timber floors exposed to fire is predicted with advanced and simplified methods. The accurate modelling is based on heat transfer prediction and reduced stiffness, while the simplified methods are based on the Reduced Cross Section Method (RCSM) of EN 1995 1-2 (CEN, 2004). Then, a...

In the present paper, the bending behavior of Cross Laminated Timber panels is investigated by means of the linear elastic exact solution from Pagano (1970, 1969). The resulting stresses are the input for a wood failure criterion, which can point out the first-crack load and the respective dominant failure mode. Heterogeneous layers are modeled as...

Cross Laminated Timber (CLT) floors or roofs have well-established advantages which lead to an increasing application in massive timber construction. With the recent development of high-rise timber buildings, CLT panels having spaced voids can lead to lighter and more acoustically efficient floors. However, such voids can decrease the bending perfo...

In this paper, we apply the asymptotic expansion method to the mechanical problem of beam equilibrium, aiming to derive a new beam model. The asymptotic procedure will lead to a series of mechanical problems at different order, solved successively. For each order, new transverse (in-plane) deformation and warping (out of plane) deformation modes ar...

In this paper, the linear buckling of a heterogeneous thick plate is studied using the Bending-Gradient theory which is an extension of the Reissner-Mindlin’s plate theory to the case of heterogeneous plates. Reference results are taken from a 3D numerical analysis using finite-elements and applied to Cross-Laminated-Timber panels which are thick a...

In this paper, the resolution of the linear buckling problem is presented using the Bending-Gradient theory which is an extension of the Reissner's plate theory to the case of heterogeneous plates. Reference results are taken from a 3D numerical analysis using finite-elements and applied to Cross Laminated Timber walls which are thick and highly an...

This chapter first provides a non-exhaustive bibliography in order to point out difficulties encountered when deriving sandwich panels' shear force stiffness for a Reissner plate theory. Then, the membrane and bending behavior of sandwich panels is derived using the thin plate theory, and the contrast assumption for sandwich panels is stated. Next,...

This chapter focuses on centro-symmetric periodic plates undergoing prescribed external transverse forces along the thickness direction. After explaining the asymptotic expressions of the 3D stress and 3D strain solutions in terms of the plate curvature, and hence its bending moment, corrective terms are added to better represent the effect of the...

This chapter illustrates the substitution of a homogeneous elastic 2D Kirchhoff-Love plate model to the heterogeneous 3D model. First, the 3D elastic problem is stated, which is followed by the description of a Kirchhoff-Love model by introducing the homogenized plate stiffness tensors. Then, the chapter illustrates the determination of the homogen...

This chapter talks about thick symmetric laminated plate subjected to out-of-plane loading. The extension of the Reissner approach suitable for such plates is presented. The idea is to minimize the stress energy over the subset of stress fields having the Kirchhoff-Love distribution inside the plate. In the case of laminated plates, this distributi...

This chapter talks about the homogeneous thick plates subjected to out-of-plane loading. The theorem of the minimum of the complementary energy is used to derive the well-known Reissner-Mindlin plate model, suitable for such plates. The 3D statically admissible trial stress fields are obtained from the Kirchhoff-Love expression of the stress. These...

This chapter examines the Bending-Gradient model for thick symmetric laminate plate subjected to out-of-plane loading. The Bending-Gradient problem is first summarized in a synthetic way. The purpose of the Bending-Gradient model is to substitute a reduced 2D plate model for the full 3D model and to reconstruct the 3D solution fields from the 2D so...

This chapter examines the asymptotic behavior of a homogeneous elastic plate, as its thickness goes to zero. It begins with the presentation of the 3D problem for a homogeneous plate. The Kirchhoff-Love plate model is derived from purely static considerations. Then, the two-energy principle is used to establish that this 2D model accurately capture...

In order to give a more comprehensive illustration of the features of the bending-gradient theory, this chapter discusses the homogenization scheme to space frames. In the chapter, a space frame is a unit-cell made of connected beams periodically reproduced in a plane and which “from far” can be considered as a plate. Initially, a square beam latti...

This chapter describes the closed-form solutions for the Bending-Gradient model in the case of cylindrical bending, and compares them to the exact solutions from Pagano. The laminated plate configuration is presented first as well as the corresponding stress localization fields. The distance between the Reissner-Mindlin model and the bending-gradie...

This chapter examines the asymptotic behaviour of a laminated elastic plate, as its thickness goes to zero. First, it presents the 3D problem for a laminated plate. The 3D elasticity problem can be formulated thanks to the complementary energy theorem. The Kirchhoff-Love plate model is derived by using static methodology for homogeneous plates. The...

This chapter provides a recap of the theory of linear elasticity. It talks about deformable solids in quasi-static equilibrium (no inertia forces), and provides the notations and the vocabulary of a model theory. Tensors will be used to represent the physical quantities which describe an elastic solid such as the displacement vector, the strain ten...

There is a close link between the action of folding and the design of structures. Folding is a fabrication process; folding generates new shapes; folding gives structural thickness to a surface. This link was identified for a while by architects, designers and engineers. However, recent advances in folding simulations and digital fabrication proces...

In this paper, the authors endeavour to develop design formulas for reciprocal systems using homogenization techniques. The theoretical background for homogenizing periodic beams systems as Kirchhoff-Love plates is first recalled. Then it is applied to a square reciprocal system. It is found that only biaxial bending (i.e. positive Gaussian curvatu...

Présentation pour la réception du prix Daniel Valentin de l'Association pour les MAtériaux Composites (AMAC)

In a recent work, a new plate theory for thick plates was suggested where the static unknowns are those of the Kirchhoff-Love theory, to which six components are added representing the gradient of the bending moment [1]. This theory, called the Bending-Gradient theory, is the extension to multilayered plates of the Reissner-Mindlin theory which app...

The 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...

Ce 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 nouveaum...

In 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 Bend...

This 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 p...

Ce 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 nouveaum...

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...

Sandwich panels are widespread in everyday life. Their structural efficiency is well-known and is a central criterion in possible applications. This Ph.D. thesis is dedicated to the study of a new sandwich panel core which might replace honeycomb in some applications: the chevron pattern. In order to compare this new core to other ones, an accurate...

Using [S. Kelsey, R. Gellatly and B. Clark, The shear modulus of foil honeycomb cores: a theoretical and experimental investigation on cores used in sandwich construction, Aircraft Engineering and Aerospace Technology 30, No. 10, 294–302 (1958)] unit load method, upper and lower bounds for the effective transverse shear moduli of a chevron folded c...

Since the wide acceptation of Reissner-Mindlin plate theory, the derivation of a constitutive equation for the transverse shear behavior of heterogenous plates has risen many difficulties. A first proposal for laminates was to take the average of transverse shear stiffness as constitutive equation. This is equivalent to assume a uniform shear defor...

In this paper, the Cosserat multiparticle model (CM2) for 3D periodically layered materials is proposed in order to reproduce both size and boundary effects in these materials. This model can handle n- phase periodically layered materials with 4n+1 kinematic variables at each 3D geometric point: two in-plane displacements and two rotations per phas...

Cecchi and Sab homogenization method (Cecchi and Sab in Int. J. Solids Struct. 44(18-19):6055-6079, 2007) for the derivation of the effective Reissner-Mindlin shear moduli of a periodic plate is applied to sandwich panels including chevron pattern. Comparison with existing bounds (Lebee and Sab in Int. J. Solids Struct., 2010) and full 3D finite el...

Les panneaux sandwichs sont des éléments de structure omniprésents au quotidien. Leur efficacité structurelle n'est plus à démontrer. Elle est même un élément déterminant dans le marché qui leur est associé. Ce mémoire de doctorat s'intéresse à un nouveau type d'âme de panneau sandwich qui pourrait être amené à supplanter le nid d'abeilles dans cer...

En se basant sur la méthode proposée par Kelsey et al. [1], les bornes supérieures et inférieures de la raideur en cisaillement transverse d'une âme pliée en module à chevrons sont déterminées analytiquement et comparées au calcul par éléments finis. On observe que ces bornes sont généralement assez larges et qu'il existe des configurations géométr...

En se basant sur la méthode proposée par Kelsey et al. [1], les bornes supérieures et inférieures de la raideur en cisaillement transverse d'une âme pliée en module à chevrons sont déterminées analytiquement et comparées au calcul par éléments finis. On observe que ces bornes sont généralement assez larges et qu'il existe des configurations géométr...