Nonlinear Korn Inequalities on a Hypersurface

The authors establish several estimates showing that the distance in W~(1,p),1 p ∞,between two immersions from a domain of R~n into R~(n+1) is bounded by the distance in L~p between two matrix fields defined in terms of the first two fundamental forms associated with each immersion. These estimates generalize previous estimates obtained by the authors and P. G. Ciarlet and weaken the assumptions on the fundamental forms at the expense of replacing them by two different matrix fields.     

A nonlinear shell model of Koiter's type

We define a new two-dimensional nonlinear shell model “of Koiter's type” that can be used for the modeling of any type of shell and boundary conditions and for which we establish an existence theorem. The model uses a specific three-dimensional stored energy function of Ogden's type that satisfies all the assumptions of John Ball's fundamental existence theorem in three-dimensional nonlinear elasticity and that is adapted here to the modeling of thin nonlinearly elastic shells by means of specific deformations that are quadratic with respect to the transverse variable.     

Ramanujan formula for the generalized Stirling approximation

The aim of this paper is to improve Ramanujan’s formula for approximation of the factorial function, starting from Burnside’s formula in contradistinction with the classical formula that starts from Stirling’s formula.     

Estimating the digamma and trigamma functions by completely monotonicity arguments

We establish new sharp bounds for the digamma and trigamma functions, using the complete monotonicity of some class of functions.     

The influence of the substrate nature on the iron repartition in the titania matrix

Fe-doped titania films were deposited by RF sputtering onto different substrates (glass and ITO/glass) in the same deposition run. The rutile nanocrystalline structure of Fe-doped thin films deposited on glass substrates and anatase nanocrystalline structure of Fe-doped thin films deposited on ITO/glass substrates were evidenced by XRD. SEM investigations showed a smooth surface with a dense nanostructure. XPS study evidenced an almost stoichiometric composition with different iron contents. EPR and XPS studies evidenced that iron entered into TiO₂ lattice by substitution, as isolated and dimer species. In Fe-doped thin films deposited on ITO/glass substrates the iron content is ten times higher than in Fe-doped thin films deposited on glass substrates and that a part of them entered as Fe²⁺.     

Continuity in H1-norms of surfaces in terms of the L1-norms of their fundamental forms

The main purpose of this Note is to show how a ‘nonlinear Korn's inequality on a surface’ can be established. This inequality implies in particular the following interesting per se sequential continuity property for a sequence of surfaces. Let ω be a domain in ${\mathbb{R}}^2$, let $\mathit{\theta }:\overline{\omega }\to {\mathbb{R}}^3$ be a smooth immersion, and let ${\mathit{\theta }}^k:\overline{\omega }\to {\mathbb{R}}^3$, $k?1$, be mappings with the following properties: They belong to the space ${\mathit{H}}^1(\omega )$; the vector fields normal to the surfaces ${\mathit{\theta }}^k(\omega )$, $k?1$, are well defined a.e. in ω and they also belong to the space ${\mathit{H}}^1(\omega )$; the principal radii of curvature of the surfaces ${\mathit{\theta }}^k(\omega )$ stay uniformly away from zero; and finally, the three fundamental forms of the surfaces ${\mathit{\theta }}^k(\omega )$ converge in ${\mathit{L}}^1(\omega )$ toward the three fundamental forms of the surface $\mathit{\theta }(\omega )$ as $k\to \infty$. Then, up to proper isometries of ${\mathbb{R}}^3$, the surfaces ${\mathit{\theta }}^k(\omega )$ converge in ${\mathit{H}}^1(\omega )$ toward the surface $\mathit{\theta }(\omega )$ as $k\to \infty$. To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

A New Approach to the Fundamental Theorem of Surface Theory

The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices (a _αβ ) of order two and a field of symmetric matrices (b _αβ ) of order two together satisfy the Gauss and Codazzi-Mainardi equations in a simply connected open subset ω of , then there exists an immersion such that these fields are the first and second fundamental forms of the surface , and this surface is unique up to proper isometries in . The main purpose of this paper is to identify new compatibility conditions, expressed again in terms of the functions a _αβ and b _αβ , that likewise lead to a similar existence and uniqueness theorem. These conditions take the form of the matrix equation

Asymptotic Justification of the Kirchhoff–Love Assumptions for a Linearly Elastic Clamped Shell

The displacement vector of a linearly elastic shell can be computed by using the two-dimensional Koiter's model, based on the a priori Kirchhoff–Love assumptions. These hypotheses imply that the displacement of any point of the shell is an affine function of the transverse variable x ₃. The term independent of x ₃ of this approximation is equal to the displacement vector of the two-dimensional Koiter's model. The term linear in x ₃ depends on the infinitesimal rotation vector of the normal. After an appropriate scaling, we estimate here the difference between the three-dimensional displacement and this affine vector field in the case of shells clamped along their entire lateral face. Besides, in the case of shells with uniformly elliptic middle surface, taking into account the term depending of the rotation of the normal allows to improve the asymptotic estimate between the three-dimensionnal displacement and Koiter's bidimensional displacement. This revised version was published online in August 2006 with corrections to the Cover Date.