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.     

Gas chroamtographic separation of optically active compounds in glass capillaries

Summary Chiral compounds may be separated by gas chromatography by direct enantiomer separation on optically active stationary phases. More generally the separation can be achieved on conventional stationary phases after formation of diastereoisomeric derivatives. In this work we report on new results in enantiomer separation, indicating that hydrogen bond association is not the only kind of molecular interaction responsible for enantiomer separation. For the separation of a wide variety of chiral compounds with amino or hydroxy groups diastereo-isomeric derivatives may be formed by reaction with L--chlorisovaleryl chloride. The derivatives of amino acids, aliphatic and aromatic amines, amino alcohols and of some alcohols are separated in glass capillaries. Gas chromatography as a separation technique of high selectivity is specifically useful for the separation of mixtures of chemically related components with comparable molecular interactions with the molecules of the stationary phase of a gas chromatographic column. The separation of optically active compounds, particularly, requires highly efficient columns. Glass capillary chromatography is a tool that meets this standard and was applied exclusively in this work.     

Spatial profiling invertebrate ganglia using MALDI MS

The ability of MALDI TOF MS to spatially map peptides and proteins directly from a tissue is an exciting advance to imaging mass spectrometry. Recent advances in instrumentation for MS have resulted in instruments capable of achieving several micron spatial resolution while acquiring high-resolution mass spectra. Currently, the ability to obtain high quality mass spectrometric images depends on sample preparation protocols that often result in limited spatial resolution. A number of sample preparation and matrix deposition protocols are evaluated for spatial profiling of Aplysia californica exocrine gland and neuronal tissues. Such samples are different from mammalian tissues, but make good targets for method optimization because of the wealth of biochemical information available on neuropeptide processing and distribution. Electrospray matrix deposition and a variety of freezing methods have been found to be optimum for these invertebrate tissues, with the exact protocols being tissue dependent.