Expression of Dirichlet boundary conditions in terms of the strain tensor in linearized elasticity

In a previous work, it was shown how the linearized strain tensor field $\mathbf{e}:=\frac{1}{2}(\mathbf{?}{\mathit{u}}^T+\mathbf{?}\mathit{u})\in {\mathbb{L}}^2(\Omega )$ can be considered as the sole unknown in the Neumann problem of linearized elasticity posed over a domain $\Omega ?{\mathbb{R}}^3$, instead of the displacement vector field $\mathit{u}\in {\mathit{H}}^1(\Omega )$ in the usual approach. The purpose of this Note is to show that the same approach applies as well to the Dirichlet–Neumann problem. To this end, we show how the boundary condition $\mathit{u}=\mathbf{0}$ on a portion ${\Gamma }_0$ of the boundary of Ω can be recast, again as boundary conditions on ${\Gamma }_0$, but this time expressed only in terms of the new unknown $\mathbf{e}\in {\mathbb{L}}^2(\Omega )$.     

Mixed oxide growth on combinatorial aluminium–gadolinium alloys — a thermodynamic and first-principles approach

Metal surfaces covered with oxides have attracted considerable scientific attention in various applications. In particular, anodic films fabricated by cost-effective anodizing have been widely used in nano-structured engineering to provide various surface functionalities. However, understanding of alloy film stability, having individual elements with widely varying structures and morphologies, is very limited due to lack of thermodynamic information and effects of electrolyte chemistry. This requires many tedious efforts on a trial and error basis in selecting suitable electrolytes that can produce the protective film at high efficiency on alloys having mixed chemistries. It is, therefore, crucial to develop a combination of high throughput theoretical analysis and automated rapid localized electrochemical probing that provides a fast and simple solution for electrolyte choice and paves the way to the remarkable expansion of industrial applications of oxides. Herein, we demonstrate that combinatorial Al–Gd alloys covering 1.0 to 10.0 at.% Gd can be oxidized into ultra-thin anodic films of controlled thickness through a selection of electrolyte based on thermodynamics (phosphate buffer with a pH of 8.20). We propose that growth of anodic films on alloys at high efficiency is possible if Gibbs free energy minimization criteria would be systematically contemplate.

Graphical abstract

     

Completely monotone functions and the Wallis ratio

The aim of the paper is to improve known estimates of the Wallis ratio. Moreover, we show that these improvements are valid, because certain functions involving the continuous version of the Wallis ratio are completely monotone.     

Synthesis and structure of tridentate bis(phosphinic amide)-phosphine oxide complexes of yttrium nitrate. Applications of 31P,89Y NMR methods in structural elucidation in solution

The synthesis and characterisation of a tridentate ligand containing two diphenylphosphinic amide side-arms connected through the ortho position to a phenylphosphine oxide moiety and the 1:1 and 2:1 complexes formed with yttrium nitrate are reported for the first time. The free ligand (R(P1)*,S(P3)*)-11 is obtained diastereoselectively by reaction of ortho-lithiated N,N-diisopropyl-P,P-diphenylphosphinic amide with phenylphosphonic dichloride. Complexes [Y((R(P1)*,S(P3)*)-11)(NO(3))(3)] and [Y((R(P1)*,S(P3)*)-11)(2)(NO(3))](NO(3))(2) were isolated by mixing ligand 11 with Y(NO(3))(3)·6H(2)O in acetonitrile at room temperature in a ligand to metal molar ratio of 1:1 and 2:1, respectively. The 1:1 derivative is the product of thermodynamic control when a molar ratio of ligand to yttrium salt of 1:1 is used. The new compounds have been characterised both as the solid (X-ray diffraction) and in solution (multinuclear magnetic resonance). In both yttrium complexes the ligand acts as a tridentate chelate. The arrangement of the two ligands in the 2:1 complex affords a pseudo-meso structure. Tridentate chelation of yttrium(III) in both complexes is retained in solution as evidenced by (89)Y NMR data obtained via(31)P,(89)Y-HMQC, and (89)Y,(31)P-DEPT experiments. The investigation of the solution behaviour of the Y(III) complexes through PGSE NMR diffusion measurements showed that average structures in agreement with the 1:1 and 1:2 stoichiometries are retained in acetonitrile.     

Orientation-preserving condition and polyconvexity on a surface: Application to nonlinear shell theory

We propose in this paper a definition of a “polyconvex function on a surface”, inspired by the definitions set forth in other contexts by J. Ball (1977) [3] and by J. Ball, J.C. Currie, and P.J. Olver (1981) [5]. When the surface is thought of as the middle surface of a nonlinearly elastic shell and the function as its stored energy function, we show that it is possible to assume in addition that this function is coercive for appropriate Sobolev norms and that it satisfies specific growth conditions that prevent the vectors of the covariant bases along the deformed middle surface to become linearly dependent, a condition that is the “surface analogue” of the orientation-preserving condition of J. Ball. We then show that a functional with such a polyconvex integrand is weakly lower semi-continuous, a property which eventually allows to establish the existence of minimizers. We also indicate how this new approach compares with the classical nonlinear shell theories, such as those of W.T. Koiter and P.M. Naghdi.     

Best estimates of the generalized Stirling formula

The aim of this paper is to introduce an approximations family of the factorial function that contains Stirling’s formula, Burnside’s formula and Gosper’s formula. The parameters which provide the best approximations are indicated. Finally, numerical computations are made to show the superiority of our formulas over other known formulas.     

Sharp inequalities related to Gosper's formula

The purpose of this Note is to construct a new type of Stirling series, which extends the Gosper's formula for big factorials. New sharp inequalities for the gamma and digamma functions are established. Finally, numerical computations which demonstrate the superiority of our new series over the classical Stirling's series are given.     

Boundary conditions in intrinsic nonlinear elasticity

The intrinsic formulation of the displacement-traction problem of nonlinear elasticity is a system of partial differential equations and boundary conditions whose unknown is the Cauchy–Green strain tensor field instead of the deformation as is customary. We explicitly identify here the boundary conditions satisfied by the Cauchy–Green strain tensor field appearing in such intrinsic formulations.