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.     

Partitioning Boolean lattices into chains of subsets

We prove that the Boolean lattice of all subsets of an n-set can be partitioned into chains of size four if and only if n9.Research supported in part by N.S.F. grant DMS-8401281.Research supported in part by N.S.F. grant DMS-8406451.     

Migratory-insertion reactions involving the thiocarbonyl ligand. dihapto-thioacyl complexes from the rearrangement of arylthiocarbonyl complexes of osmium(II). Structure of Os(η2-CSR)(η1-O2CCF3)(CO)(PPh3)2

Reaction of HgR₂ with OsHCl(CS)(PPh₃)₃ yields red, five-coordinate, OsRCl-(CS)(PPh₃)₂ (R = p-tolyl). From this have been derived the compounds OsRX(CS)(PPh₃)₂ with X = Br, I, S₂CNEt₂, O₂CMe, O₂CCF₃. These compounds add an additional ligand, MeCN, CO or CNR to form colourless, six coordinate arylthiocarbonyl complexes, which undergo migratory-insertion reactions to form red, dihapto-thioacyl complexes. The crystal structure of a representative example, Os(η²-CSR)(η¹-O₂CCF₃)(CO)PPh₃)₂ has been determined. The red equant crystals are orthorhombic, space group P2₁2₁2₁, a 11.584(1), b 19.184(2), c 18.90(1) Å, V 4199 ų, Z  4. The structure was solved by conventional heavy-atom methods and refined by full-matrix least-squares employing anisotropic thermal parameters for all non-hydrogen atoms except the carbon atoms of the triphenylphosphines. The final R factor is 0.057 for 2868 observed reflections.The coordination geometry in the monomeric complex is that of an octahedron distorted by the constraints of the ligands. The triphenyl phosphine ligands are mutually trans; the equatorial plane contains carbonyl, monohapto-trifluoroacetate, and dihapto-thioacyl ligands. Bond distances and angles are OsP 2.405, 2.407(4) Å; POsP 173.9(1)°; OsCO 1.83(2) Å; Os-O (trifluoroacetate) 2.206(11) Å; OsC (thioacyl) 1.91(2); OsS 2.513(6); CS 1.72 Å. The CS bond length implies a reduction in bond order from 2.0 to approx. 1.5 upon coordination to the metal.The η²-thioacyl ligand in Os(η²-CSR)Cl(CNR)(PPh³)₂ is methylated with methyl triflate and further reaction with LiCl produces the thiocarbene complex OsCl₂(C[SMe]R)(CNR)(PPh₃)₂.     

A mild and efficient process for detoxifying polychlorinated biphenyls

Polychlorinated biphenyls (PCBs) were effectively dechlorinated to biphenyl under mild conditions using an aqueous solution of sodium hypophosphite in the presence of a palladium catalyst.     

Chemistry and Biology of Cylindrols: Novel Inhibitors of Ras Farnesyl-Protein Transferase from Cylindrocarpon lucidum

Farnesyl-protein transferase (FPTase) is an enzyme responsible for the farnesylation of Ras protein. Farnesylation is required for cell-transforming activity in several tumor-types, and therefore, inhibition of FPTase activity may be a potential target for anticancer drugs. Our continued search for novel inhibitors led to the isolation of a number of bicyclic resorcinaldehyde cyclohexanone derivatives named here cylindrols A(1) to A(4), cylindrols B and B(1), and a number of known compounds, from Cylindrocarpon lucidum. The compounds were isolated by bioassay-guided separation using Sephadex LH-20, silica gel, and reverse phase HPLC. Structures were elucidated by extensive application of 2D NMR and X-ray crystallography. The determination of absolute stereochemistry was accomplished by CD measurements. Chemical transformations of the most abundant compound resulted in a number of key derivatives which were critical for the evaluation of structure activity relationship. These compounds are members of ascochlorin family and showed a wide range of inhibitory activity (0.7 &mgr;M to >140 &mgr;M) against FPTase. The FPTase activity was noncompetitive with respect to both substrates. Isolation, structures, chemical transformations, and FPTase activity are discussed in detail.     

Personal reflections on receiving the Roger Adams Award in organic chemistry

The author reflects on his early experiences as a chemist, and on the subsequent shift in emphasis that his research has undergone from mechanistic and synthetic organic chemistry to natural products chemistry. Finally, the extension of the field of natural products chemistry into the emerging discipline of chemical ecology is noted. This essay concludes with a consideration of the importance of including science in the curricula of all college and university students.     

Infrared spectra of CH3-CrH, CH3-WH, CH2=WH2, and CH[triple bond]WH3 formed by activation of CH4 with Cr and W atoms

Laser-ablated W atoms react with CH4 in excess argon to form the CH3-WH, CH2=WH2, and CH[triple bond]WH3 molecules with increasing yield in this order of product stability. These molecules are identified from matrix infrared spectra by isotopic substitution. Tungsten methylidene and methylidyne hydride molecules are reversibly interconverted by alpha-H transfers upon visible and ultraviolet irradiations. Matrix infrared spectra and DFT/B3LYP calculations show that CH[triple bond]WH3 is a stable molecule with C3v symmetry, but other levels of theory were required to describe agostic distortion for CH2=WH2. Analogous reactions with Cr gave only CH3-CrH, which is calculated to be by far the most stable product.     

Structure of a fluorinated azoxy compound: fluoro(trifluoromethyl)-diazene-2-oxide,CF3N(O)NF

A gas-phase electron diffraction study of the azoxy compound which was synthesized by the reaction of CF3NO with N2F4 in a Pyrex glass vessel results in a trans CF3N(O)NF structure (F trans to CF3), although quantum chemical calculations (MP2 and B3LYP) predict a greater stability of the cis CF3NN(O)F isomer by about 12 kcal/mol. The CF3 group eclipses the N=N double bond. The following skeletal geometric parameters (r(a) values with 3sigma uncertainties) were obtained: N=N 1.287(15) A; N=O 1.231(6) A; N-F 1.380(6) A; N-C 1.498(6) A; N=N=O 131.2(13) degrees; N=N-F 103.5(13) degrees; N=N-C 114.0(12) degrees. The bond lengths in CF3N(O)NF are compared to those in azo, nitryl, and nitrosyl compounds with fluorine and/or CF3 substituents.