分析表面裂纹的一种新方法

本文作者综合了线弹簧模型及边界元法的优点,开发了一种新的线弹簧边界元法.该方法把表面裂纹这一三维问题简化为拟一维问题,可用于分析受到多种载荷作用的含表面裂纹的板.本文对该方法进行了理论分析和数值验证,报告了计算结果.结果表明,该方法经济有效.利用该方法仅使用个人计算机就可以分析表面裂纹问题.     

集中力作用下多铁性板状复合材料的断裂分析

针对多铁性板状复合材料在外表面任一点处存在集中力的界面裂纹问题,建立断裂力学模型.利用Fourier(傅里叶)积分变换和Green(格林)函数推导出该裂纹模型的Cauchy(柯西)奇异积分方程组;通过Chebyshev(切比雪夫)配点法将该方程组离散为对应的代数方程组,进而数值求解裂纹尖端应力强度因子.通过对数值结果的分析可以得到:在外表面集中力作用下,压电层厚度、裂纹长度以及集中力作用位置是影响裂纹尖端应力强度因子的3个主要因素.分析讨论了在该模型下各项参数对应力强度因子的影响规律,可以在工程应用中为此类复合材料的防断裂优化设计提供一定的理论参考.     

谈数学求真精神及教学熏陶

1从学生的一桩误判谈起开学第一天,对高一新生提了这样一个问题:假定有一根绳子,长度恰好为地球赤道长,那么该绳子是紧贴着地球赤道表面而无缝隙的.若将该绳子的长度增加10 m,那么此时绳子与赤道表面所成的缝隙会有多大?你能钻得过去吗?话音刚落,班内同学几乎异口同声,不假思索     

复合超疏水表面制备及其上液滴运动特性研究

采用光刻法制备出微米尺度的硅微方柱结构,并利用化学气相沉积的方法在其上可控地生长定向碳纳米管,由此制备出具有可控比例微纳二级结构的超疏水表面.通过扫描电子显微镜(SEM)测量和接触角、滚动角测试的方法对具有不同微纳结构比例及尺寸的表面形貌和性能进行探查和测试,对制备过程中参数的选择做出了优化处理.得到了表面性质均匀、稳定的二级结构超疏水表面.通过粒子图像测速(PIV)的方法测量在结构表面滚动液滴的内部流动形态,并与一级结构表面进行对比.分析并总结了该文中表面的疏水机理及表面微纳结构比例、形态、尺寸等因素对疏水性能的影响.     

初封载荷下封隔器胶筒表面自由变形特性分析

研究了封隔器胶筒在自由变形阶段受初封载荷作用下内、外表面发生位移变形的特性.依据连续介质力学理论，建立自由变形阶段的有限变形数学模型，给出了胶筒在初封轴向载荷下内、外表面径向变形的过程，得到胶筒非线性变形解析解.通过数值计算，在求解出胶筒外表面自由变形解析式的基础上，进一步分析了容易被忽略的胶筒内表面非线性变形规律和相关参数变化对其密封性能的影响.该变形特性分析可适用于不同型号的封隔器胶筒，为胶筒的密封和可靠性设计提供重要理论依据.     

界面滑移流体动压膜承载能力的形成

运用界面滑移可在两平行平板表面间形成具有承载能力的流体动压膜．在流体入口区,静止平板表面上流体-接触表面的界面剪切强度具有较低值,以在该界面处产生界面滑移,而在流体出口区,静止平板表面上流体-接触表面的界面剪切强度具有足够高的值,以避免在该界面处出现界面滑移．整个运动平板表面上流体-接触表面的界面剪切强度具有足够高的值,以避免在运动平板表面上出现界面滑移．分析表明,这种流体动压接触区具有显著承载能力．使整个接触区具有最大承载能力的流体出口区宽度与入口区宽度的比值为0.5．     

重建极性连续统理论的基本定律和原理(Ⅳ)——表面守恒定律

从普遍均衡定律的平移和转动的不变性出发来重新建立较为完整的微极热力连续统的表面守恒定律,提出了广义的能量动量和能量动量矩张量.给出了Piola型、Cauchy型和Kirchhoff型微极热力连续统的表面守恒定律的具体形式.现有的结果都可以当做是特殊情形从该结果自然地推导出来,并可从归结过程中清楚地看出现有理论的不完整性程度.非局部微极热力连续统的表面守恒定律可通过局部化得到.     

复合多层介质在可变温度和集中荷载作用下的位移和应力

研究了多层介质中的热弹性位移和应力.多层介质具有不同厚度,各层又具有不同的弹性性质,最上层表面上作用热荷载和集中荷载.假设各层分别是均匀、各向同性弹性材料,各层相关的位移分量是轴对称的,对称轴为各层表面的垂线.因此,各层应力函数满足无体力的单一方程.利用积分变换法求解了该方程,对由任意多个层数构造的多层介质,给出了其相应层数基础热弹性位移和应力的解析表达式.并对3层介质和4层介质时的数值结果进行了比较.     

一道试题的多角度思考与推广

题目1对直线l上的任意一点M(x,y),点N(4x+2y,x+3y)仍在该直线l上,求直线l的方程.该题是高三模拟训练题中客观题的压轴题,是以直线的方程为背景、以直线的位置关系为载体设计的求直线方程类型的题目.题目设计新颖,表面上是考察直线方程的求法,实质上是考察方     

材料各向异性对热弹性接触问题的影响

研究一个绝热刚性冲头和各向异性弹性传热半空间之间的稳态平面接触问题.由于冲头在半空间表面上的滑移,在接触区域内摩擦生热,并且热辐射到接触区域.之外的区域利用Fourier积分变换,将问题简化为两个奇异积分方程构成的方程组.利用Gauss-Jacobi梯形求积公式,数值地求解该方程组给出了各向异性和热效应的图例.     

Mechanical theorem proving in the surfaces using the characteristic set method and Wronskian determinant

In this paper, we generalize the method of mechanical theorem proving in curves to prove theorems about surfaces in differential geometry with a mechanical procedure. We improve the classical result on Wronskian determinant, which can be used to decide whether the elements in a partial differential field are linearly dependent over its constant field. Based on Wronskian determinant, we can describe the geometry statements in the surfaces by an algebraic language and then prove them by the characteristic set method.     

K-Theory of Surfaces at the Prime 2

We prove that for smooth surfaces over real closed fields, and a class of smooth projective surfaces over a real number field, the map between mod 2 algebraic and étale K-theory is an isomorphism in sufficiently large degrees. For a class of smooth projective surfaces over a real closed field, including rational surfaces, complete intersections and K3-surfaces over the real numbers, we explicate the abutment of the mod 2 motivic cohomology to algebraic K-theory spectral sequence.     

Riemann surfaces defined over the reals

The known (explicit) examples of Riemann surfaces not definable over their field of moduli are those with that field being a subfield of the reals but which cannot be defined over the reals. In this paper we provide explicit families of Riemann surfaces which are definable over the reals but cannot be defined over the field of moduli.     

Asymptotic solutions of a magnetohydrodynamic system which describe smoothed discontinuities

Asymptotic solutions of a nonlinear magnetohydrodynamic system rapidly varying near moving surfaces are described. It is shown that the motion of jump surfaces is determined from a free boundary problem, while the main part of the asymptotics satisfies a system of equations on the moving surface. In the “nondegenerate” case, this system turns out to be linear, while, under the additional condition that the normal component of the magnetic field vanishes, it becomes nonlinear. In the latter case, the small magnetic field instantaneously increases to a value of order 1.     

Parallel mean curvature biconservative surfaces in complex space forms

In this paper, we consider PMC surfaces in complex space forms, and study the interaction between the notions of PMC, totally real and biconservative. We first consider PMC surfaces in a non-flat complex space form and prove that they are biconservative if and only if totally real. Then, we find a Simons-type formula for a well-chosen vector field constructed from the mean curvature vector field and use it to prove a rigidity result for CMC biconservative surfaces in two-dimensional complex space forms. We prove then a reduction codimension result for PMC biconservative surfaces in non-flat complex space forms. We conclude by constructing examples of CMC non-PMC biconservative submanifolds from the Segre embedding and discuss when they are proper-biharmonic.     

An Invariant Theory of Spacelike Surfaces in the Four-dimensional Minkowski Space

We consider spacelike surfaces in the four-dimensional Minkowski space and introduce geometrically an invariant linear map of Weingarten-type in the tangent plane at any point of the surface under consideration. This allows us to introduce principal lines and an invariant moving frame field. Writing derivative formulas of Frenet-type for this frame field, we obtain eight invariant functions. We prove a fundamental theorem of Bonnet-type, stating that these eight invariants under some natural conditions determine the surface up to a motion. We show that the basic geometric classes of spacelike surfaces in the four-dimensional Minkowski space, determined by conditions on their invariants, can be interpreted in terms of the properties of the two geometric figures: the tangent indicatrix, and the normal curvature ellipse. We apply our theory to a class of spacelike general rotational surfaces.     

计算机辅助几何设计中极小曲面造型的研究进展

极小曲面在工程领域有着广泛应用,因此将其引入计算机辅助几何设计领域具有重要意义.详细概述了近年来计算机辅助几何设计领域中极小曲面造型的研究工作,按照造型方法的不同,可将现有工作分为精确造型方法和逼近造型方法两类.精确造型方法主要包括两个部分:某些特殊极小曲面的控制网格表示与构造;等温参数多项式极小曲面的挖掘与性质.逼近造型方法主要包括3个部分t基于数值计算的逼近方法;基于线性偏微分方程的逼近方法;基于能量函数最优化的逼近方法.最后对这些方法进行了分析比较,并讨论了极小曲面造型中有待进一步解决的问题.     

On the optimization of surface textures for lubricated contacts

The pressure field that develops inside a lubricated contact obeys an elliptic equation known as Reynolds equation, with coefficients that depend on the shape of the contacting surfaces. The load-carrying capacity of a contact, defined as the integral of the pressure field, is an important performance indicator that should be as high as possible to avoid wear and damage of the surfaces. In this article, the effect of arbitrary uniform periodic textures on the load-carrying capacity of lubricated devices known as thrust bearings is investigated theoretically by means of homogenization techniques and first-order perturbation analysis. It is shown that the untextured shape is a local optimum for the load-carrying capacity of the homogenized pressure field. This is proved for bearings of general shape and considering both incompressible and compressible models for the lubricant. The homogenization technique however implies an error. Suitable bounds for the effect of this error are provided in a simplified case.     

On the Parameterization of Rational Ringed Surfaces and Rational Canal Surfaces

Ringed surfaces and canal surfaces are surfaces that contain a one-parameter family of circles. Ringed surfaces can be described by a radius function, a directrix curve and vector field along the directrix curve, which specifies the normals of the planes that contain the circles. In particular, the class of ringed surfaces includes canal surfaces, which can be obtained as the envelopes of a one-parameter family of spheres. Consequently, canal surfaces can be described by a spine curve and a radius function. We present parameterization algorithms for rational ringed surfaces and rational canal surfaces. It is shown that these algorithms may generate any rational parameterization of a ringed (or canal) surface with the property that one family of parameter lines consists of circles. These algorithms are used to obtain rational parameterizations for Darboux cyclides and to construct blends between pairs of canal surfaces and pairs of ringed surfaces.     

Non-hyperelliptic Riemann surfaces with real field of moduli but not definable over the reals

The known examples of explicit equations for Riemann surfaces whose field of moduli is different from their field of definition, are all hyperelliptic. In this paper we construct a family of equations for non-hyperelliptic Riemann surfaces, each of them is isomorphic to its conjugate Riemann surface, but none of them admit an anticonformal automorphism of order 2; that is, each of them has its field of moduli, but not a field of definition, contained in \mathbb R{{\mathbb R}} . These appear to be the first explicit such examples in the non-hyperelliptic case.