The Half-plane with an Edge Crack in Plane Elastostatics

The Wiener-Hopf technique is applied to the problem of a half-planewith a crack perpendicular to its edge. A closed solution isobtained giving the stress and displacement fields in the formof eigenfunction expansions, valid over the whole region, exceptthe neighbourhood of the tip of the crack. The behaviour inthis region is obtained by an asymptotic technique. The coefficientsof the expansions involve a generalized factorial function,and a method for the computation of this function is given,thus enabling numerical values of the stresses and displacementsto be obtained for the given cases of stretching and bendingthe half-plane.     

相应于一类对称自对偶李代数的顶点代数结构

设gsp是给定的对称自对偶李代数,给出了相应于gsp的一个顶点代数结构.     

Vertex Operators and the Class Algebras of Symmetric Groups

We exhibit a vertex operator that implements the multiplication by power sums of Jucys–Murphy elements in the centers of the group algebras of all symmetric groups simultaneously. The coefficients of this operator generate a representation of , to which operators multiplying by normalized conjugacy classes are also shown to belong. A new derivation of such operators based on matrix integrals is proposed, and our vertex operator is used to give an alternative approach to the polynomial functions on Young diagrams introduced by Kerov and Olshanski. Bibliography: 18 titles.     

Vertex Operators for Standard Bases of the Symmetric Functions

We present formulas for operators which add a row or a column to the partition indexing the power, monomial, forgotten, Schur, homogeneous and elementary symmetric functions. As an application of these operators we show that the operator that adds a column to the Schur unctions can be used to calculate a formula for the number of pairs of standard tableaux the same shape and height less than or equal to a fixed k.     

Shock Waves in Plane Symmetric Spacetimes

We consider Einstein's equations coupled to the Euler equations in plane symmetry, with compact spatial slices and constant mean curvature time. We show that for a wide variety of equations of state and a large class of initial data, classical solutions break down in finite time. The key mathematical result is a new theorem on the breakdown of solutions of systems of balance laws. We also show that an extension of the solution is possible if the spatial derivatives of the energy density and the velocity are bounded, indicating that the breakdown is really due to the formation of shock waves.     

含椭圆孔或裂纹压电介质平面问题的基本解

应用复变函数的方法，并基于精确的电边界条件，导出了含一椭圆孔或裂纹的横观各向同性压电体在任意集中力和集中电荷作用下的复变函数解，即Ｃｒｅｎ函数解·叠加该解，得到了裂纹表面作用任意集中载荷或分布载荷时的一般解·这些解不但澄清了从前文献中一些不合理的结果，同时也为应用边界元法求解更复杂的压电介质断裂力学问题提供了基本解·     

平面应变和反平面应变复合型裂纹尖端的各向异性塑性应力场

在裂纹尖端的理想塑性应力分量都只是θ的函数的条件下,利用平衡方程,各向异性塑性应力应变率关系、相容方程和Hill各向异性屈服条件,本文导出了平面应变和反平面应变复合型裂纹尖端的各向异性塑性应力场的一般解析表达式.将这些一般解析表达式用于复合型裂纹,我们就可以得到Ⅰ-Ⅲ、Ⅱ-Ⅲ及Ⅰ-Ⅱ-Ⅲ复合型裂纹尖端的各向异性塑性应力场的解析表达式.     

平面应变和反平面应变复合型裂纹尖端的理想塑性应力场

在裂纹尖端的理想塑性应力分量都只是θ的函数的条件下.利用平衡方程.应力应变率关系、相容方程和屈服条件,本文导出了平面应变和反平面应变复合型裂纹尖端的理想塑性应力场的一般解析表达式.将这些一般解析表达式用于复合型裂纹.我们就可以得到Ⅰ-Ⅲ、Ⅱ-Ⅲ及Ⅰ-Ⅱ-Ⅲ复合型裂纹尖端的理想塑性应力场的解析表达式.     

关于平面应变和反平面应变复合型裂纹尖端的理想塑性应力场

文献[1]在裂纹尖端的理想塑性应力分量都只是θ的函数的条件下,利用平衡方程、应力应变率关系、相容方程和屈服条件导出了平面应变和反平面应变复合型裂纹尖端的理想塑性应力场的一般解析表达式。但文献[1]对应力应变率关系式中的比例因子λ(r,θ)作了很多限制,即假定λ与θ无关,并假定λ=c或cr-1。本文取消了对λ的这些限制。而文献[1]所研究的λ=crn(n=0或-1)的情形,只是本文的一个特殊情况。     

Plane Partitions and Characters of the Symmetric Group

In this paper we show that the existence of plane partitions, which are minimal in a sense to be defined, yields minimal irreducible summands in the Kronecker product of two irreducible characters of the symmetric group S(n). The minimality of the summands refers to the dominance order of partitions of n. The multiplicity of a minimal summand equals the number of pairs of Littlewood-Richardson multitableaux of shape (, ), conjugate content and type . We also give lower and upper bounds for these numbers.     

楔型向错偶极子和裂纹的干涉效应

研究了晶体材料中一个楔型向错偶极子与裂纹的弹性干涉效应．运用复变函数方法获得了复势函数和应力场的封闭形式解答,导出了裂纹尖端应力强度因子和作用在向错偶极子中心点像力的解析表达式．获得了向错偶极子的位置、方向和偶臂长度对裂纹尖端应力强度因子的影响规律,并讨论了裂纹附近向错偶极子的平衡位置．结果表明向错偶极子靠近裂纹尖端时,对应力强度因子有明显的屏蔽或反屏蔽作用．     

Symmetric Sextics in the Real Projective Plane and Auxiliary Conics

In the present paper, the already-known list of the rigid isotopy classes of nonsingular curves of degree 6 in containing a symmetric curve is obtained in an elementary way. To prove that a curve in a given rigid isotopy class is not symmetric, we study the position of the curve with respect to auxiliary conics. Bibliography: 21 titles.     

半无限平面裂纹构型横向应力的Green函数

针对各向同性弹性无限大板中半无限裂纹,用解析函数方法求解了裂尖处横向应力的Green函数.加载情况为一任意集中力作用于任意一内点处.用叠加法求解了复势,它给出该平面问题的弹性解.通过渐近分析抽取复势的非奇异部分.基于该非奇异部分,用一种直接方法求解了横向应力的Green函数.进一步,用叠加法得到了一对对称和反对称集中力加载时的Green函数.然后,用得到的Green函数来预测铁电材料双悬臂梁试验中畴变引起的横向应力.用力电联合加载引起的横向应力来判断试验中所观察到的稳定和不稳定裂纹扩展行为.预测结果和试验数据基本吻合.     

Simplicity of a Vertex Operator Algebra Whose Griess Algebra is the Jordan Algebra of Symmetric Matrices

Let r ∈ ? be a complex number, and d ∈ ?_≥2 a positive integer greater than or equal to 2. Ashihara and Miyamoto [4 Ashihara , T. , Miyamoto , M. ( 2009 ). Deformation of central charges, vertex operator algebras whose Griess algebras are Jordan algebras . Journal of Algebra 32 : 1593 – 1599 . [Google Scholar]] introduced a vertex operator algebra V _𝒥 of central charge dr, whose Griess algebra is isomorphic to the simple Jordan algebra of symmetric matrices of size d. In this article, we prove that the vertex operator algebra V _𝒥 is simple if and only if r is not an integer. Further, in the case that r is an integer (i.e., V _𝒥 is not simple), we give a generator system of the maximal proper ideal I _r of the VOA V _𝒥 explicitly.     

对称迭层矩形板的平面应力分析

常用的对称迭层板为各向异性板．根据平面应力问题的基本方程精确地用应力函数解法求得了各向异性板的一般解析解．推导出平面内应力和位移的一般公式,其中积分常数由边界条件来决定．一般解包括三角函数和双曲函数组成的解,它能满足4个边为任意边界条件的问题．还有代数多项式解,它能满足4个角的边界条件．因此一般解可用以求解任意边界条件下的平面应力问题．以4边承受均匀法向和切向载荷以及非均匀法向载荷的对称迭层方板为例,进行了计算和分析．     

Elastostatics in the presence of a temperature distribution or inhomogeneity

The equilibrium of an inhomogeneous elastic body is analyzed theoretically and numerically, with special emphasis on the case when the inhomogeneity arises from a given temperature distribution. The case when the inhomogeneity (or variation in temperature) is small is treated via linearization, the corresponding expansion of the solution in terms of an appropriate small parameter being justified by means of the implicit function theorem. For certain stored-energy functions suggested by the problem of cooling basalt rock, an exact solution to the linearized problem is found. A direct minimization of the energy using a finite-element algorithm is found to agree with the linearized solution as 0.

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Résumé L'équilibre d'un corps élastique inhomogène est étudié théoriquement et numériquement, en particulier dans le cas où l'homogenéité résulte d'une distribution de température donnée. Le cas où l'inhomogenéité (ou variation de la température) est petite est traité par linéarisation, le développement correspondant de la solution en termes d'un paramètre convenable étant justifié par le théorème des fonctions implicites. Pour diverses fonctions d'énergie interne suggérées par le problème du refroidissement du basalte, on donne une solution exacte du problème linéarisé. Une minimisation directe de l'énergie utilisant une méthode d'éléments finis est en accord avec les solutions linéarisées quand 0.

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Dedicated to Klaus Kirchgässner on the occasion of his 60th birthday     

十次对称二维准晶中的楔形裂纹问题

研究了集中力作用下十次对称二维准晶中的楔形裂纹问题.采用推广的Stroh公式给出了应力和位移的一般解,在此基础上,讨论了声子场环向应力和相位子场环向应力的变化规律.作为特例,给出了十次对称二维准晶半无限裂纹问题应力和位移的解析表达式.     

含内裂纹的有限圆盘的平面问题

本文用弹性理论复变函数方法讨论了在内部任意位置含直线裂纹的有限圆盘在一般载荷作用下的平面问题,得到了复应力函数和应力强度因子用级数表示的一般表达式,并对此问题讨论了三种特殊情形,即裂纹受均布压应力,均布剪应力和圆盘匀速旋转的情形,其中还给出了计算应力强度因子的近似式.计算结果表明,对位于圆盘内部且不靠近外边界的中、小裂纹,这些近似式给出好的或较好的近似.     

集中力作用下的半平面边缘裂纹问题

讨论了半平面中的倾斜裂纹问题。集中力作用于裂纹表面上,或作用于开裂半平面的边界上。利用有理保角映像函数方法求解这个问题,同时得到了闭合形式的解。最后,给出了二个数值例子和计算结果。