The limits of hamiltonian structures in three-dimensional elasticity,shells, and rods

Summary This paper uses Hamiltonian structures to study the problem of the limit of three-dimensional (3D) elastic models to shell and rod models. In the case of shells, we show that the Hamiltonian structure for a three-dimensional elastic body converges, in a sense made precise, to that for a shell model described by a one-director Cosserat surface as the thickness goes to zero. We study limiting procedures that give rise to unconstrained as well as constrained Cosserat director models. The case of a rod is also considered and similar convergence results are established, with the limiting model being a geometrically exact director rod model (in the framework developed by Antman, Simo, and coworkers). The resulting model may or may not have constraints, depending on the nature of the constitutive relations and their behavior under the limiting procedure. The closeness of Hamiltonian structures is measured by the closeness of Poisson brackets on certain classes of functions, as well as the Hamiltonians. This provides one way of justifying the dynamic one-director model for shells. Another way of stating the convergence result is that there is an almost-Poisson embedding from the phase space of the shell to the phase space of the 3D elastic body, which implies that, in the sense of Hamiltonian structures, the dynamics of the elastic body is close to that of the shell. The constitutive equations of the 3D model and their behavior as the thickness tends to zero dictates whether the limiting 2D model is a constrained or an unconstrained director model. We apply our theory in the specific case of a 3D Saint Venant-Kirchhoff material andderive the corresponding limiting shell and rod theories. The limiting shell model is an interesting Kirchhoff-like shell model in which the stored energy function is explicitly derived in terms of the shell curvature. For rods, one gets (with an additional inextensibility constraint) a one-director Kirchhoff elastic rod model, which reduces to the well-known Euler elastica if one adds an additional single constraint that the director lines up with the Frenet frame. This paper is dedicated to the memory of Juan C. Simo This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan Simo.     

A simple yet accurate boundary element method for continuum dielectric calculations

A simple yet accurate method for calculating electrostatic potentials using the boundary element continuum dielectric method is presented. It is shown that the limiting factor in accuracy is not the evaluation of integrals involving the interaction between boundary elements but rather a proper estimation of the self-polarization of a patch upon itself. We derive a sum rule that allows us to calculate this important self-polarization term in a self-consistent and simple way. Intricate integration schemes used in previous treatments are consequently rendered unnecessary while concurrently achieving at least comparable accuracy over earlier methods. In some model systems for which analytic solutions are available, the computed surface polarization charge and reaction field energy are correct to better than six significant figures. An application of the method to the calculation of hydration free energies is presented. Good agreement with experimental values is obtained.     

On estimating pressure build-up in severely accelerated,shocked flow

A new approach to the calculation of the high pressures characterizing the flow field in front of a piston undergoing severe acceleration over the short term is presented. In contrast with previous approaches where the computational domain is altered but which stop short of transforming velocities, here the problem is solved in an accelerating non-Euclidean co-ordinate system where the piston is stationary. The method is applied to a study of the problem of premature sabot separation. Through use of Harten's second-order-accurate TVD scheme, flow simulations are performed for both 1D and 3D axisymmetric geometries. The simple 1D model gives pressure profiles surprisingly close to those of the more physical 3D model.     

Thermal stability of poly(mono-n-alkylitaconates) and mono-n-alkylitaconate-co-vinylpyrrolidone copolymers I

Poly(monoitaconates) containing octyl, decyl and dodecyl groups and random monoalkylitaconate-co-vinylpyrrolidone copolymers were studied by thermogravimetric analysis. Copolymers of mono-n-octylitaconate (MOI), mono-n-decylitaconate (MDI), and mono-n-dodecylitaconate (MDoI), respectively, with N-vinyl-2-pyrrolidone (VP) of different compositions were studied by dynamic thermogravimetric analysis. The thermal stability of the copolymers depends on the structure of the monoitaconate comonomer and on the composition of the copolymer The kinetic analysis of the degradation data shows that the thermal decomposition of these copolymers can be described by several kinetic orders depending on the copolymer and on the composition. The relative thermal stability of the copolymers increases as the VP content increases and as the length of the side chain of the itaconate increases, following the same trend as the flexibility of the copolymers in solution.     

The ternary system CsCl−NaCl−LaCl3

The ternary system CsCl?NaCl?LaCl₃ was investigated by means of differential thermal analysis and X-ray powder diffraction analysis. There exists one congruently melting compound, Cs₂NaLaCl₆, crystallizing with the cubic elpasolite structure. No quasi-binary section exists for the whole system, however three binaries range from the ternary compound Cs₂NaLaCl₆ to NaCl, CsLa₂Cl₇ and Cs₃LaCl₆ resp., dividing the system in three areas of composition: one triangle, Cs₃LaCl₆?Cs₂NaLaCl₆?CsLa₂Cl₇, containing additionally a compound Cs₂LaCl₅ below 510°C, and the two areas CsCl?NaCl?Cs₂NaLaCl₆?Cs₃LaCl₆ and Cs₂NaLaCl₆?NaCl?LaCl₃?CsLa₂Cl₇, containing a mixed crystal range between LaCl₃ and Na₃La₅Cl₁₈. These areas could be further divided in five triangles, so that the whole system contains six Alkemade triangles.     

Chromatographic characterization of thermally upgraded products from alternative fuels

An overview is presented of the analytical approaches developed by our research group over the last ten years for analysis of alternative fuel, both biomass and fossil. The alternative fuels are analyzed successively by PLC-8 (preparative liquid chromatography–group-type) fractionation and high resolution gas chromatography. Some of the possibilities for fractionation and characterization of alternative fuels are herein exemplified with sugar cane bagasse pyrolysis products.     

Effects of a controlled N2 and O2 addition to Ar on the analytical parameters of the GD-OES

The influence of the controlled addition of N₂ and of O₂ to Ar on the analytical parameters of a dc glow discharge has been investigated for bulk samples of the pure metals Al, Ti, Fe, Ni, Cu, and Ag. The constant voltage discharge mode at 1000 V was applied with a mean power of about 1 W/mm². The N₂ and O₂ concentrations in the discharge gas Ar were varied in the range 0–3 mass-%, corresponding to a partial pressure of about 4 · 10^?3 hPa. The general effect of the gaseous addition is the decrease of the sputtering rate with an increasing concentration of N₂ or O₂. Atomic processes, which might be responsible for the observed effects, are discussed.