SELF-SIMILAR CRACK EXPANSION METHOD FOR THE STRESS INTENSITY FACTOR ANALYSIS

The Self-Similar Crack Expansion (SSCE) method is proposed to evaluate stress intensi-ty factors at crack tips, whereby stress intensity factors of a crack can be determined by the crackopening displacement over the crack, not just by the local displacement around the crack tip. The crackexpansion rate is estimated by taking advantage of the crack self-similarity. Therefore, the accuracy ofthe calculation is improved. The singular integrals on crack tip elements are also analyzed and are pre-cisely evaluated in terms of a special integral analysis. Combination of these two techniques greatly in-creases the accuracy in estimating the stress distribution around the crack tip. A variety of two-dimen-sional cracks, such as subsurface cracks, edge cracks, and their interactions are calculated in terms ofthe self-similar expansion rate. Solutions are satisfied with errors less than 0.5% as compared with theanalytical solutions. Based on the calculations of the crack interactions, a theory for crack interactionsis proposed such that for a group of aligned cracks the summation of the square of SIFs at the right tipsof cracks is always equal to that at the left tips of cracks. This theory was proved by the mehtod ofSelf-Similar Crack Expansion in this paper.     

When Attempting Chain Extension,Even Without Solvent,It Is Not Possible to Avoid Chojnowski Metathesis Giving D3

A simple, mild and efficient method to prepare HSi- or HOSi-telechelic, high-molecular-weight polydimethylsiloxane polymers (to 41,600 g·mol⁻¹) using the one-shot hydrolysis of M^HM^H is reported; titration of the water allowed for higher molecular weights (to 153,900 g·mol⁻¹). The “living” character of the chain extension processes was demonstrated by adding a small portion of M^HM^H and B(C₆F₅)₃ (BCF) to a first formed polymer, which led to a ~2-fold, second growth in molecular weight. The heterogeneous reaction reached completion in less than 30 min, much less in some cases, regardless of whether it was performed neat or 50 wt% in dry toluene; homogeneous reactions in toluene were much slower. The process does not involve traditional redistribution, as judged by the low quantities (<3%) of D₄ produced. However, it is not possible to avoid Chojnowski metathesis from M^HDDM^H giving D₃, which occurs competitively with chain extension.     

用拉压不同模量理论分析弯曲板

材料的拉压不同弹性模量性对结构计算的影响是一个不容忽视的问题，本文对不同模量弯曲板进行了较系统的分析，构造了相应的有限元算法，通过算例分析，得出了一些较有价值的结论。     

To gel or not to gel: A prior prediction of gelation in solvent mixtures

The gelation behaviours of low molecular weight gelators 1,3:2,5:4,6-tris(3,4-dichlorobenzylidene)-D-mannitol(G1)and 2,4-(3,4-dichlorobenzylidene)-N-(3-aminopropyl)-D-gluconamide(G2)in 34 solvents have been studied.We found that sample dissolved at low concentrations may become a gel or precipitate at higher concentrations.The Hansen solubility parameters(HSPs)and a Teas plot were employed to correlate the gelation behaviours with solvent properties,but with no success if the concentration of the tests was not maintained constant.Instead,on the basis of the gelation results obtained for the G1 and G2 in single solvents,we studied the gelation behaviours of G1 and G2 in 23 solvent mixtures and found that the tendency of a gelator to form a gel in mixed solvents is strongly correlated with its gelation behaviours in good solvents.If the gelation occurs in a good solvent at higher concentrations,it will take place as well in a mixed solvent(the good solvent plus a poor solvent)at a certain volume ratio.In contrast,if the gelator forms a precipitate in a good solvent at higher concentrations,no gelation is to be observed in the mixed solvents.A gelation rule for mixed solvents is thus proposed,which may facilitate decision making with regard to solvent selection for gel formation in the solvent mixtures in practical applications.     

A Friedrichs–Maz'ya inequality for functions of bounded variation

The aim of this short note is to give an alternative proof, which applies to functions of bounded variation in arbitrary domains, of an inequality by Maz'ya that improves Friedrichs inequality. A remarkable feature of such a proof is that it is rather elementary, if the basic background in the theory of functions of bounded variation is assumed. Nevertheless, it allows to extend all the previously known versions of this fundamental inequality to a completely general version. In fact the inequality presented here is optimal in several respects. As already observed in previous proofs, the crucial step is to provide conditions under which a function of bounded variation on a bounded open set, extended to zero outside, has bounded variation on the whole space. We push such conditions to their limits. In fact, we give a sufficient and necessary condition if the open set has a boundary with σ‐finite surface measure and a sufficient condition if the open set is fully arbitrary. Via a counterexample we show that such a general sufficient condition is sharp.     

On Schrödinger-Virasoro type Lie conformal algebras

In this paper, two classes of Schrödinger-Virasoro type Lie conformal algebras TSV(a,b) and TSV(c) which are non-simple are introduced for some a, b, c∈?. Moreover, central extensions, conformal derivations and free conformal modules of rank 1 of TSV(a,b) and TSV(c) are determined.     

Purity in the category of S-posets

In this paper, purity of S-posets over a pomonoid S is investigated. We first study some basic properties of absolutely po-pure S-posets. Among other results, it is proved that every regular injective S-poset is absolutely po-pure, and every absolutely po-pure inequationally compact S-poset is regular injective. Then, using the notion of semi-finitely presented S-poset based on the finitely induced S-poset congruence, we find an equivalent condition for an S-poset to be 1-po-pure in a regular extension. Finally, a characterization of an absolutely 1-po-pure S-poset is presented.     

On a Class of Non-local Operators in Conformal Geometry

In this expository article, the authors discuss the connection between the study of non-local operators on Euclidean space to the study of fractional GJMS operators in conformal geometry. The emphasis is on the study of a class of fourth order operators and their third order boundary operators. These third order operators are generalizations of the Dirichlet-to-Neumann operator.