Nonlinear Shell Models of Kirchhoff-Love Type: Existence Theorem and Comparison with Koiter’s Model

Acta Mathematicae Applicatae Sinica, English Series - We define two nonlinear shell models whereby the deformation of an elastic shell with small thickness minimizes ad hoc functionals over sets of...     

Organocuprates as initiators for the polymerization of methyl methacrylate in THF

Several types of lithium organocuprates (I) e.g. R₂CuLi, RR'CuLi, R₃CuLi₂, R₂CuLi₂X (X- CN, SCN), were prepared and used as polymerization initiators for methyl methacrylate in THF at temperatures ranging from −30 to +20 ^*C. Polymerization initiated by a mixed homocuprate, e.g. lithium n-butyl (thien-2-yl)cuprate was studied in more detail. Kinetic measurements indicated an overall large (−4 to −14 kcal) apparent negative activation energy of polymerization in the −30 + +20 ^*C temperature range. The reaction was shown to have a kinetic order of about 1.5 with respect to initiator concentration. Conductance measurements indicate that the equivalent conductance increases with initiator concentration in the 10⁻² + 10⁻¹ M range indicating the formation of multiple ions formed by intermolecular association. The data are consistent with a polymerization involving aggregated ion pairs of as yet undetermined composition. The negative activation energy is tentatively interpreted as due to the exothermic formation of aggregates existing as highly solvated ion pairs.     

New nonlinear estimates for surfaces in terms of their fundamental forms

We establish several estimates of the distance between two surfaces immersed in the three-dimensional Euclidean space in terms of the distance between their fundamental forms, measured in various Sobolev norms. These estimates, which can be seen as nonlinear versions of linear Korn inequalities on a surface appearing in the theory of linearly elastic shells, generalize in particular the nonlinear Korn inequality established in 2005 by P. G. Ciarlet, L. Gratie, and C. Mardare.     

Nonlinear estimates for hypersurfaces in terms of their fundamental forms

A sufficiently regular hypersurface immersed in the $(n+1)$-dimensional Euclidean space is determined up to a proper isometry of ${\mathbb{R}}^{n+1}$ by its first and second fundamental forms. As a consequence, a sufficiently regular hypersurface with boundary, whose position and positively-oriented unit normal vectors are given on a non-empty portion of its boundary, is uniquely determined by its first and second fundamental forms. We establish here stronger versions of these uniqueness results by means of inequalities showing that an appropriate distance between two immersions from a domain ω of ${\mathbb{R}}^n$ into ${\mathbb{R}}^{n+1}$ is bounded by the $L^p$-norm of the difference between matrix fields defined in terms of the first and second fundamental forms associated with each immersion.     

Asymptotic Considerations Shedding Light on Incompressible Shell Models

The incompressible singularity found in 3D elasticity when Poissons ratio approaches 1/2 is not present in classical shell models, nor in the limit models obtained from 3D elasticity when performing an asymptotic analysis with respect to the thickness parameter. However, some specific shell models – such as the 3D-shell model – do retain the incompressible singularity. These observations raise the issue of how adequately shell models can represent incompressible conditions, which this paper aims at investigating. We first perform a combined asymptotic analysis of 3D elasticity with respect to both the thickness parameter and Poissons ratio and we obtain a commuting property, which is very valuable as a justification of the concept of an incompressible shell, and substantiates the use of classical shell models with incompressible materials. We then show that the 3D-shell model does not enjoy a similar commuting property; nevertheless we propose a simple modification of this model for which commuting is obtained, hence consistency with incompressibility is recovered. We also illustrate our discussions with some numerical results. Mathematics Subject Classifications (2000) 74K25, 74G10, 74S05.A. Münch: Current address: Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, 16 route de Gray, 25030 Besancon Cedex, France.     

On Pfaff systems with coefficients and their applications in differential geometry

We prove that a Pfaff system with coefficients in , p>2, in a simply-connected open subset Ω of has at least a nontrivial solution of class provided that its coefficients satisfies a compatibility condition in the distributional sense. If in addition the set Ω is connected, the Cauchy problem associated with the Pfaff system has a unique solution. An application of this result is that the fundamental theorem of surface theory holds under the assumption that the first and second fundamental forms are respectively of class and , with p>2, and satisfy together the Gauss and Codazzi–Mainardi equations in the distributional sense.     

The distance between fixed points of some pairs of maps in Banach spaces and applications to differential systems

Let T be a λ-contraction on a Banach space Y and let S be an almost λ-contraction, i.e. sum of an (ε, λ)-contraction with a continuous, bounded function which is less than ε in norm. According to the contraction principle, there is a unique element u in Y for which u = Tu: If moreover there exists v in Y with v = Sv, then we will give estimates for ‖u−v‖. Finally, we establish some inequalities related to the Cauchy problem.     

On the fundamental theorem of surface theory under weak regularity assumptions

We consider a symmetric, positive definite matrix field of order two and a symmetric matrix field of order two that together satisfy the Gauss and Codazzi–Mainardi equations in a connected and simply connected open subset of $\text{R}{}^2$. If these fields are of class C² and C¹ respectively, the fundamental theorem of surface theory asserts that there exists a surface immersed in the three-dimensional Euclidean space with the given matrix fields as its first and second fundamental forms. The purpose of this Note is to prove that this theorem still holds true under the weaker regularity assumptions that these fields are of class W^1,∞_loc and L^∞_loc respectively, the Gauss and Codazzi–Mainardi equations being then understood in a distributional sense. To cite this article: S. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004).