A group-theoretic approach to the bifurcation analysis of elastic frameworks with symmetry

We present a general approach to the bifurcation analysis of elastic frameworks with symmetry. While group-theoretic methods for bifurcation problems with symmetry are well known, their actual implementation in the context of elastic frameworks is not straightforward. We consider frames comprising assemblages of Cosserat rods, and the main difficulty arises from the nonlinear configuration space, due to the presence of (cross-sectional) rotation fields. We avoid this via a single-rod formulation, developed earlier by one of the authors, whereby the governing equations are embedded in a linear space. The field equations comprise the assembly of all rod equations, supplemented by compatibility and equilibrium conditions at the joints. We demonstrate their equivariance under the symmetry-group action, and the implementation of group-theoretic methods is now natural within the linear-space context. All potential generic, symmetry-breaking bifurcations are predicted apriori. We then employ an open-source path-following code, which can detect and compute simple, onedimensional bifurcations; multiple bifurcation points are beyond its capabilities. For the latter, we construct symmetry-reduced problems implemented by appropriate substructures. Multiple bifurcations are rendered simple, and the path-following code is again applicable. We first analyze a simple tripod framework, providing all details of our methodology. We then treat a hexagonal space frame via the same approach. The tripod and the hexagonal dome both exhibit simple and double bifurcation points.Comment: preprin

Bifurcation et stabilité des matériaux cellulaires à l'aide de la théorie des groupes

Les méthodes issues de la théorie des groupes permettent de résoudre un certain nombre des problèmes posés par le haut degré de symétrie interne de certains matériaux cellulaires et ainsi d'étudier, de manière exhaustive, le champ des états possibles coexistants dans un tel système. On propose ici d'étudier le comportement post-bifurqué de matériaux cellulaires à murs courbes, sollicités en grandes transformations, dans leur plan. Plusieurs types de comportements non-linéaires sont considérés ainsi que plusieurs symétries d'empilement

Un modèle d'endommagement pour des plaques en béton armé sous séisme

Afin d'effectuer des calculs sismiques sur des bâtiments industriels, nous proposons un modèle macroscopique d'endommagement pour le béton armé dédié à des éléments plaques, formulé dans le cadre standard généralisé. Les armatures assurant une pente post-élastique toujours positive, on évite les problèmes de localisation dus au comportement adoucissant du béton. L'utilisation d'éléments de structure permet d'autre part de gagner en temps de calcul. On présente des comparaisons avec des expérimentations

Selecting generalized continuum theories for nonlinear periodic solids based on the instabilities of the underlying microstructure

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Generalized continuum media confronted to long and short wavelength instabilities in architected materials

In the context of architected materials, it has been observed that both long-wavelength instabilities leading possibly to localization and short-wavelength commensurate to cell-size instabilities leading to the apparition of a deformation pattern could occur. This work compares the ability of two families of higher order equivalent media, namely strain-gradient and micromorphic media, to capture both mesoscale cell-commensurate and long-wavelength macroscopic instabilities in those materials. The studied architected material consists in a very simple one-dimensional arrangement of non-linear springs, thus allowing for analytical or nearly analytical treatment of the problem, dismissing any uncertainties or imprecisions coming from a numerical method. A numerical solving of the problem is then used to compare the post-buckling predictiion of both models. The study concludes that, even on a very simple case, it is impossible for a strain-gradient Taylor-series expansion type of homogenization method to capture the cell-commensurate instabilities while the micromorphic medium can capture both instabilities but fails to converge properly in the post-buckling regime when localization appears. Micromorphic media are thus the family of equivalent continuum model that are to prefer when dealing with the possibility of patterning inside a structured medium, but if localization is to consider, it would be interesting to combine both strategies into a micromorphic, gradient enhanced equivalent medium

Symétries et Instabilités dans les matériaux

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Selecting generalized continuum theories for nonlinear periodic solids based on the instabilities of the underlying microstructure

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Formulation d'un modèle homogénéisé de plaque en béton armé pour des applications sismiques

Safety reassessments are periodically performed on the EDF nuclear power plants and the recent seismic reassessments leaded to the necessity of taking into account the non-linear behaviour of materials when modeling and simulating industrial structures of these power plants under seismic solicitations. A large proportion of these infrastructures is composed of reinforced concrete buildings, including reinforced concrete slabs and walls, and literature seems to be poor on plate modeling dedicated to seismic applications for this material. As for the few existing models dedicated to these specific applications, they present either a lack of dissipation energy in the material behaviour, or no micromechanical approach that justifies the parameters needed to properly describe the model. In order to provide a constitutive model which better represents the reinforced concrete plate behaviour under seismic loadings and whose parameters are easier to identify for the civil engineer, a constitutive model dedicated to reinforced concrete plates under seismic solicitations is proposed: the DHRC (Dissipative Homogenised Reinforced Concrete) model. Justified by a periodic homogenisation approach, this model includes two dissipative phenomena: damage of concrete matrix and internal sliding at the interface between steel rebar and surrounding concrete. An original coupling term between damage and sliding, resulting from the homogenisation process, induces a better representation of energy dissipation during the material degradation. The model parameters are identified from the geometric characteristics of the plate and a restricted number of material characteristics, allowing a very simple use of the model. Numerical validations of the DHRC model are presented, showing good agreement with experimental behaviour. A one dimensional simplification of the DHRC model is proposed, allowing the representation of reinforced concrete bars and simplified models of rods and wire mesh.Le parc d'installations nucléaires dont EDF a la charge est soumis régulièrement à des réévaluations de sureté. Concernant le risque sismique, ces réévaluations ont mené à la décision de prendre en compte les comportements non-linéaires des matériaux dans les structures lors des modélisations et des simulations numériques des bâtiments composant ce parc d'installations assez souvent réalisés en béton armé. Dans ce contexte, le besoin de disposer de modélisations représentatives du comportement de plaques en béton armé soumises à des sollicitations sismiques est fort et il s'est avéré que la littérature ne proposait que très peu de ce type de modélisation. Afin de répondre à ce besoin tout en cherchant à maîtriser la phase d'identification des paramètres du modèle, un modèle de plaque en béton armé pour des applications sismiques est proposé dans ce travail. Ce modèle, DHRC (Dissipative Homogenised Reinforced Concrete), est construit par une approche d'homogénéisation périodique. Il couple deux phénomènes dissipatifs : l'endommagement de la matrice de béton et le glissement interne à l'interface entre les barres de renfort en acier et le béton avoisinant. Ce couplage original entre endommagement et glissement permet une meilleure représentativité de la dissipation d'énergie au cours des cycles de chargement induite par la dégradation du matériau. Les paramètres du modèle global sont identifiés par une procédure d'homogénéisation automatisée et la résolution numérique de problèmes cellulaires. La procédure s'appuie sur les caractéristiques géométriques de la microstructure de la plaque et un nombre très restreint de caractéristiques matériaux, permettant ainsi à l'ingénieur de l'utiliser de manière simple et maîtrisée. Le modèle implanté dans le code de calculs par éléments finis ASTER est validé numériquement sur plusieurs structures tests sous des chargements simples et combinés. Sa capacité à simuler le comportement expérimental de structures voiles est également analysée. Des perspectives d'enrichissement du modèle à l'échelle microscopique sont proposées. Enfin, une simplification unidimensionnelle de ce modèle permet de représenter le comportement de barres en béton armé pour des représentations simplifiées de type bielles ou treillis ou pour modéliser le comportement de poteaux en béton armé en traction-compression

Generalized continuum models confronted to cell-commensurate instabilities in structured media

International audienceThis work focuses on the ability for two generalized continuum models to capturemesoscale cell-commensurate instabilities in structured media using a very simple base model.Indeed, [1] have shown that a periodic arrangement of atoms, linked by complex interactomicpotential could lead to short-wave commensurate and incommensurate instabilities and [3] proposesa quasi-continuum model based on strain gradients able to capture both long and shortwavelength instabilities. On the other hands, using non-linear springs,[2] studied the abilityof a strain-gradient model to capture long wavelength instabilities. As a consequence, the exampleof a periodic arrangement of non-linear springs, as presented in Fig.1a) below has beenproposed. This simple example has been shown to display both long and short-wavelength instabilitiesdepending on the values of the model’s non-linear parameters (Fig.1b)).Two generalized media have been compared: micromorphic and second order strain-gradientmedia. It is thus shown that the short-wavelength bifurcation points can only be captured by amicromorphic-type medium while the long-wavelength bifurcation points are captured by bothmodels. This concludes that if short-wavelength instabilities are possible, micromorphic-typemedia are more appropriate to describe, in a contiunous way, the buckling of the mesoscopicstucture

Selecting generalized continuum theories for nonlinear periodic solids based on the instabilities of the underlying microstructure

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