The closest elastic tensor of arbitrary symmetry to an elasticity tensor of lower symmetry

The closest tensors of higher symmetry classes are derived in explicit form for a given elasticity tensor of arbitrary symmetry. The mathematical problem is to minimize the elastic length or distance between the given tensor and the closest elasticity tensor of the specified symmetry. Solutions are presented for three distance functions, with particular attention to the Riemannian and log-Euclidean distances. These yield solutions that are invariant under inversion, i.e., the same whether elastic stiffness or compliance are considered. The Frobenius distance function, which corresponds to common notions of Euclidean length, is not invariant although it is simple to apply using projection operators. A complete description of the Euclidean projection method is presented. The three metrics are considered at a level of detail far greater than heretofore, as we develop the general framework to best fit a given set of moduli onto higher elastic symmetries. The procedures for finding the closest elasticity tensor are illustrated by application to a set of 21 moduli with no underlying symmetry.Comment: 48 pages, 1 figur

The stereochemistry of SRN1 reactions in Nitroacenaphthenes

The reaction of the 2-chloro-1,l-dialkyl-6-nitroacenaphthenes (1)-(3) with azide and p- toluenethiolate ions takes place by the SRN1 mechanism, to give the substitution products (27)-(32), despite the fact that the nitro group and the chlorine-bearing benzylic carbon are attached to different aromatic rings. The reaction of the stereoisomers (2) and (3) of 2-chloro-1-ethyl-1-methyl-6-nitroacenaphthene takes place through an effectively planar benzylic radical (35), which is preferentially attacked from the face remote from the a-ethyl group. The presence of geminal alkyl groups α to the reaction site in the rigid acenaphthene system restricts the scope of the substitution processes to sterically unhindered nucleophiles, and the reaction fails with reagents such as sodium p- toluenesulfinate (23) and the salts, (24) and (25), derived from 2-ethylmalononitrile and 2-nitropropane

Nucleophilic substitution reactions of thienyl neopentyl substrates

The reaction of the chlorides (4)-(6), which are both neopentylic and thenylic , were studied. The chloride (4), unlike its analogue (13) in the benzene series, undergoes ready solvolysis with alcohols to give the corresponding ethers, e.g. (7)-(9). The chlorides (5) and (6) react more slowly than (4) but undergo methanolysis to give the methyl ethers (11) and (12) respectively. In the dipolar aprotic solvents, dimethyl sulfoxide and dimethylformamide, the reactions of the chlorides (4), (5) and (6) with the thiolate salt (16) appear to proceed by an SN1-like, an SN(AEAE) and an SRNl process respectively

Substitution reactions of nitrothiophenes. 6. Disparate mechanisms for substitution reactions at neopentyl carbons bearing 4- and 5-nitrothienyl groups

The reaction of the (5-nitro-2-thienyl)- and (4-nitro-2-thienyl) neopentyl chlorides (9 and 10) with ptoluenesulfmate, azide, and p-toluenethiolate ions proceeds smoothly and in high yield (70-95%) under mild conditions (20 °C, Me2SO) to give the sulfones 14 and 15, the azides 16 and 17, and the sulfides 18 and 19, respectively. The mechanisms of these substitutions are quite different, however. The substitutions in the 5-nitro series take place by the SRn1 mechanism, whereas those in the 4-nitro series take place by the ionic SN (aeae) process, which involves initial attack of a nucleophile at the 5-position of the thiophene ring