Buckling for an elastoplastic plate with an incremental constitutive relation

We study the buckling of an elastoplastic plate loaded on its edges by a planar thrust(t)F(x ₁ ,x ₂ ) depending on time. We consider an incremental constitutive relation which permits the distinction between loading and unloading. This could be obtained by taking a Prandtl-Reuss relation, but, for technical reasons, we restrict ourselves to an approximation of this relation. We give a necessary condition for the existence of a nontrivial solution (together with the zero one) after a timet ₀ ; an important feature is that this condition takes into account the whole story of the loading and cannot be expressed in terms of its value at a single time. We prove the existence of bifurcated branches, for some sufficient condition involving the slope of at timet ₀ , and describe their local behavior; we point out the fact that buckling can actually occur when the loading is decreasing.Part of this work was done while the authors were visiting the I.M.A. in Minneapolis, Minnesota, USA.     

Structural transition in SF6 clusters

SF₆ clusters are produced during the free jet expansion of a Ne-SF₆ mixture and studied by electron diffraction methods. Present experiments have been performed at constant nozzle diameter, stagnation pressure and stagnation temperature, adjustable parameters being the SF₆ mole fraction X and the nozzle-to-skimmer distancex _s /d. It is possible to warm up SF₆ clusters by increasing either X, and thus the cluster size, orx _s /d, which makes them collide with background molecules downstream of the Mach disk. In both cases, cold clusters made of several hundreds of molecules experience a structural transition, similar to that observed in bulk material, from the triclinic to the bcc plastic structure. A molecular dynamics simulation accounts correctly for intermediate stages of the transition which are visible in experimental patterns. Contrary to bulk results, cluster results show that the structural transition occurs over some temperature range.     

Preparation du complexe cationique [η5-cyclopentadienyl fer-η6-pentamethylaniline]+ et de quelques amides et sulfonamides derives

The [η⁵-cyclopentadienyl-η⁶-pentamethylanilineiron]⁺ cation has been prepared from ligand exchange between ferrocene and pentamethylaniline. Acidic chlorides (CH₃COCl, C₆H₅COCl) and sulfonyl chlorides (p-CH₃C₆H₄SO₂Cl, camphorsulfonyl-d₁₀ chloride) react in situ with deprotonated species after treatment of this cation with t-BuOK in THF. An acid—base reaction between amide or sulfonamide derivatives and deprotonated species takes place and can limit the yield; the mechanism is discussed. Amide derivatives also have been prepared by direct action of acidic chlorides with [η⁵-C₅H₅Fe-η⁶-C₆-(CH₃)₅NH₂]⁺ in acetone solution. An optically active sulfonamide has been prepared.     

Discrete Homogenization in Graphene Sheet Modeling

Graphene sheets can be considered as lattices consisting of atoms and of interatomic bonds. Their bond lengths are smaller than one nanometer. Simple models describe their behavior by an energy that takes into account both the interatomic lengths and the angles between bonds. We make use of their periodic structure and we construct an equivalent macroscopic model by means of a discrete homogenization technique. Large three-dimensional deformations of graphene sheets are thus governed by a membrane model whose constitutive law is implicit. By linearizing around a prestressed configuration, we obtain linear membrane models that are valid for small displacements and whose constitutive laws are explicit. When restricting to two-dimensional deformations, we can linearize around a rest configuration and we provide explicit macroscopical mechanical constants expressed in terms of the interatomic tension and bending stiffnesses.     

Asymptotic theory and analysis for displacements and stress distribution in nonlinear elastic straight slender rods

An asymptotic expansion method is applied to nonlinear three-dimensional elastic straight slender rods. Nonlinear ordinary differential equations for approximate displacements and explicit formulas for approximate stress distributions are obtained. Mathematical properties of these models are studied.

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Résumé On applique la méthode des développements asymptotiques à des poutres tridimensionnelles droites, élancées et non linéairement élastiques. On en déduit des équations différentielles ordinaires non linéaires pour des déplacements approchés, ainsi que des formules explicites pour des approximations des distributions de contraintes. On étudie les propriétés mathématiques de ces modèles.

     

A justification of nonlinear properly invariant plate theories

A single asymptotic derivation of three classical nonlinear plate theories is presented in a setting which preserves the frame-invariance properties of three-dimensional finite elasticity. By a successive scaling of the external loading on the three-dimensional body, the nonlinear membrane theory, the nonlinear inextensional theory and the von Kármán equations are derived as the leading-order terms in the asymptotic expansion of finite elasticity. The governing equations of the nonlinear inextensional theory are of particular interest where 1) plane-strain kinematics and plane-stress constitutive equations are derived simultaneously from the asymptotic analysis, 2) the theory can be phrased as a minimization problem over the space of isometric deformations of a surface, and 3) the local equilibrium equations are identical to those arising in the one-director Cosserat shell model. Furthermore, it can be concluded that with a regular, single-scale asymptotic expansion it is not possible to obtain a system of plate equations in which finite membrane strain and finite bending strain occur simultaneously in the leading-order term of an asymptotic analysis.     

Zur Moleculargewichtsbestimmung

Ohne Zusammenfassung     

Lemme du mouvement rigide infinitésimal en coordonnées lipschitziennes et application aux coques de régularité minimale

We establish a version of the infinitesimal rigid displacement lemma in curvilinear Lipschitz coordinates. We give an application to linearly elastic shells whose midsurface and normal vector are both Lipschitz. To cite this article: S. Anicic et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

The infinitesimal rigid displacement lemma in Lipschitz co‐ordinates and application to shells with minimal regularity

We establish a version of the infinitesimal rigid displacement lemma in curvilinear Lipschitz co‐ordinates. We give an application to linearly elastic shells whose midsurface and normal vector are both Lipschitz. Copyright © 2004 John Wiley Sons, Ltd.