Matched asymptotic expansions in nonlinear fracture mechanics—I. longitudinal shear of an elastic perfectly-plastic specimen

A method of analysis based upon matched asymptotic expansions is proposed for a cracked specimen which is subjected to longitudinal shear (mode III) loading. This gives the small-scale yielding estimate of linear fracture mechanics as a first approximation, and provides systematic refinements which take account of the nonlinear interaction between the elastic and the plastic regions. Explicit solutions can be generated for any specimen which is amenable to a linear elastic analysis. Fracture parameters, such as crack opening displacement and the Jintegral, are expressed as power series in the ratio of applied stress to yield stress, and three terms are given explicitly. These are defined from linear elastic solutions alone. The edge-cracked strip and cracking from a semi-circular notch are studied as examples. Comparison with an exact solution for the former geometry suggests that the three-term expansions give useful results up to 75 % of limit load. The latter example is new and shows the effect of a notch on a crack at loads beyond the normal range of validity of linear elastic fracture mechanics.     

An elastic-plastic analysis of an asymmetric cracked specimen subjected to longitudinal shear

Some recent elastic-plastic analyses of cracked specimens subjected to symmetric mode III loading are extended to include asymmetric loading and geometry. Solutions are given for arbitrary work hardening behaviour in any specimen that is amenable to a linear elastic analysis. It is shown that asymmetry has a major influence on the shape of the plastic zone, but does not affect the J-integral unil the loading is well into the large scale yielding range. In particular the “plastic zone corrected” estimate of J, obtained by elastically solving a problem for a crack longer than the actual one, is shown to remain a valid two-term asymptotic expansion in the presence of asymmetry. The general results are applied to a crack at an angle to a uniform stress field in a power law hardening material. The growth of the plastic zone is displayed graphically for various hardening exponents and crack orientations. No other asymmetric solution is available, but values of J are compared with those obtained from a fully plastic analysis in the symmetric case.     

Asymmetric alkylation of aromatic aldehydes with dialkylzinc catalyzed by novel amino derivatives of hydroxytetrahydropyran

Novel, specially prepared, tetrahydropyran‐based γ‐amino alcohols (S)‐2‐(aminomethyl)‐3‐hydroxy‐6‐ethoxy(phenoxy)‐tetrahydropyrans ( I ) (amino = n‐Bu₂N, piperidinyl, pyrrolidinyl, azetidinyl) were tested as catalysts in the asymmetric addition of Et₂Zn and n‐Bu₂Zn to (hetero)aromatic aldehydes. In most cases the phenoxy derivatives of I acted more enantioselectively than the ethoxy ones. The dibutylamino derivaties showed the least enantioselectivity; the pyrrolidinyl derivatives were more active as catalysts than piperidinyl and azetidinyl compounds. The highest enantioselectivity was observed in the addition of Et₂Zn to benzaldehyde in the presence of (S)‐2‐(N‐pyrrolidinylmethyl)‐3‐hydroxy‐6‐phenoxytetrahydropyran. The corresponding alcohol was prepared with 72% ee (R‐configuration). The addition of dibutylzinc proceeded slowly and less selectively. The alkylation of (hetero)aromatic aldehydes with Et₂Zn and n‐Bu₂Zn was also studied in the presence of the known optical inductor (1S,2R)‐N,N‐dibutylnorephedrine. Some chiral aromatic secondary alcohols were synthesized in high chemical yields and up to 93% ee enantioselectivity. Copyright © 1999 John Wiley & Sons, Ltd.     

Inequalities of Rellich Type

Functional Analysis and Its Applications - Hardy inequalities have been important topics of research for a century, and in the past twenty years or so, there has been a deluge of important papers...     

Sobolev‐type inequalities for potentials in grand variable exponent Lebesgue spaces

We introduce a new scale of grand variable exponent Lebesgue spaces denoted by . These spaces unify two non‐standard classes of function spaces, namely, grand Lebesgue and variable exponent Lebesgue spaces. The boundedness of integral operators of Harmonic Analysis such as maximal, potential, Calderón–Zygmund operators and their commutators are established in these spaces. Among others, we prove Sobolev‐type theorems for fractional integrals in . The spaces and operators are defined, generally speaking, on quasi‐metric measure spaces with doubling measure. The results are new even for Euclidean spaces.     

Synthesis,psychotropic and anticancer activity of 2,2‐dimethyl‐5‐[5′‐trialkylgermyl(silyl)‐2′‐hetarylidene]‐1,3‐dioxane‐4,6‐diones and their analogues

A series of 2,2‐dimethyl‐5‐(5′‐R‐hetarylidene)‐1,3‐dioxane‐4,6‐diones has been synthesized for examing a structure–activity relationship. Furyl and thienyl derivatives of Meldrum's acid possess neurotropic activity comprising both depriming and activating components. Comparison of acute toxicity of carbon, silicon and germanium analogues in the furan series of the compounds has demonstrated that the germanium derivative is 11.5 times less toxic than the carbon analogue and four times less toxic than the silicon derivative. 2,2‐Dimethyl‐5‐(5′‐triethylsilyl‐2′‐thenylidene)‐1,3‐dioxane‐4,6‐dione has moderate toxicity with the highest neurotropic and cytotoxic activity Copyright © 2003 John Wiley & Sons, Ltd.     

Fast,simple, and sensitive high‐performance liquid chromatography method for measuring vitamins A and E in human blood plasma

Vitamins A and E are fat‐soluble vitamins that play important roles in several physiological processes. Monitoring their concentrations is needed to detect deficiency and guide therapy. In this study, we developed a high‐performance liquid chromatography method to measure the major forms of vitamin A (retinol) and vitamin E (α‐tocopherol and γ‐tocopherol) in human blood plasma. Vitamins A and E were extracted with hexane and separated on a reversed‐phase column using methanol as the mobile phase. Retinol was detected by ultraviolet absorption, whereas tocopherols were detected by fluorescence emission. The chromatographic cycle time was 4.0 min per sample. The analytical measurement range was 0.03–5.14, 0.32–36.02, and 0.10–9.99 mg/L for retinol, α‐tocopherol, and γ‐tocopherol, respectively. Intr‐aassay and total coefficient of variation were <6.0% for all compounds. This method was traceable to standard reference materials offered by the National Institute of Standards and Technology. Reference intervals were established using plasma samples collected from 51 healthy adult donors and were found to be 0.30–1.20, 6.0–23.0, and 0.3–3.2 mg/L for retinol, α‐tocopherol, and γ‐tocopherol, respectively. In conclusion, we developed and validated a fast, simple, and sensitive high‐performance liquid chromatography method for measuring the major forms of vitamins A and E in human plasma.