具有功能梯度加强环的有限尺寸开孔板应力集中问题

基于复变函数理论,结合最小二乘边界配点法,对具有功能梯度加强环的有限尺寸开孔板在任意均布载荷作用下的应力集中问题进行了研究.首先,采用分层均匀化方法,给出了材料参数沿径向任意变化的功能梯度加强环内的复势及孔边应力的半解析解;然后,通过几组数值算例,讨论了组分梯度、加强环厚度、板相对尺寸及偏心率的变化对孔边应力集中的影响.结果表明,通过合理选择功能梯度加强环内材料参数的递变规律及加强环的厚度,可以有效缓解有限尺寸开孔板内的应力集中.     

横观各向同性压电材料中正三角形孔口快速传播裂纹的解析解

利用复变函数方法,通过引入合适的数值保角映射研究了横观各向同性压电材料中正三角形孔口快速传播裂纹的反平面剪切问题,并在电非渗透型与电渗透型两种边界条件下,结合柯西积分,导出了力-电耦合作用下以速度v传播时的Ⅲ型裂纹的动态应力强度因子和电位移强度因子的解析解.最后,考虑面内电载荷和面外机械载荷共同作用,分析了三角形孔尺寸、裂纹尺寸、外载变化对裂尖场强度因子的影响.     

共线裂纹对热电材料转换效率的影响

热电材料是一种能将电能和热能相互转换的功能材料,在工程中扮演着不可或缺的角色.然而由于热电材料转换效率的限制,暂时还不能广泛地投入工业生产和满足日常生活的需要,因此提高热电材料的转换效率就显得尤为重要.提供了一种新的理论模型,在含共线裂纹的有限矩形热电板的顶部和底部边界上分别加载恒定的电势和温度场,构建非线性耦合的电热传输方程,为了降低求解难度本文引入了辅助函数,运用傅里叶变换和奇异积分法,得到了电场和温度场分布的解析表达式,然后,通过改变裂纹和电板的尺寸,用来对比上下板面的电流密度和能量强度因子的变化趋势,最终得到相对热电转换效率公式.数值结果表明,添加合适的裂纹可以提高热电转换效率.当傅里叶积分变换参数n=50时,增大热电板面的高度,电场强度因子和能量强度因子也会增加,但当增加热电板的长度时,电场强度因子和能量强度因子会逐渐减少.     

正交各向异性圆柱体在轴压作用下的应力场

基于材料体积不可压假设,对轴向压缩作用下圆柱试件在加载面内的环向和径向应力分布进行理论分析,计算结果表明:当试件材料本构为正交各向异性时,环向和径向应力分布为半径的幂函数形式;试件材料为横观各向同性时,环向和径向应力为半径的二次函数.在圆柱试件轴线上环向和径向应力相等,且均具有最大值;试件圆周边界上径向应力为0,环向应力具有极小值.通过最大拉伸应变破坏理论对试件环向应变进行分析,获得了产生环向拉伸破坏时的临界轴向载荷;并采用Hill-蔡强度理论对试件圆周边界上计算得到的应力参量进行描述,得到了轴压作用下圆柱试件的Hill-蔡强度理论表达式,其不仅取决于轴向应力和试件材料的基本力学性能,还与试件轴向变形的应变率及应变率随时间的变化率相关.     

轴向均布载荷和约束对石油钻采管柱失稳长度的影响

以石油行业中的直井管柱为研究对象,建立了管柱在轴向均布载荷作用下的屈曲力学模型,概括了三种轴向均布载荷的分布特征.采用特征值屈曲有限元法,提出了分段计算管柱失稳长度的迭代流程.在正反三角形分布载荷作用下,其失稳长度小于三角形分布载荷作用下的失稳长度,前者的最大挠度靠近中和点;在梯形分布载荷作用下,给出了失稳长度随该段及其以上受压段的无量纲曲线.不同的位移约束条件也对管柱失稳长度有较大影响,工程应用中应区分不同的约束条件,方可得出实际的失稳长度.     

基于有限元的金属基短纤维复合材料MMC的一种统计蠕变模型

在有限元分析的基础上建立了一个单向应力状态下金属基短纤维复合材料(MMC)的统计蠕变模型.首先建立细胞模型并进行有限元分析,得到了单向应力状态下材料细观尺寸及载荷方向对宏观蠕变响应的影响规律.通过在细胞模型中增加一界面层(考虑材料特性和厚度)来研究基体和纤维的界面对MMC宏观蠕变响应的影响.基于细胞模型的数值结果,提出了一适用于纤维平面随机分布的随机统计模型,该模型考虑了纤维的断裂.通过试验获得纤维的统计分布规律.分析结果表明随机统计模型可以满意地描述试验结果.进一步讨论了材料细观尺寸,纤维的断裂特性以及界面层的材料特性和厚度对MMC宏观蠕变响应的影响.     

一维六方准晶中带光滑顶点的正三角形孔边裂纹的反平面问题

本文通过引入合适的保角映射,利用Stroh公式和复变函数方法研究一维六方准晶材料中含光滑顶点的正三角形孔边裂纹的反平面问题,得到正三角形孔边裂纹尖端的场强度因子和能量释放率的表达式.通过数值算例,讨论了裂纹长度和正三角形孔口边长比值对等效场强度因子和能量释放率的影响,以及耦合系数和机械载荷对能量释放率的影响规律.结果表明:裂尖等效场强因子只与裂纹长度有关,而孔洞大小对其影响可忽略;裂纹长度、耦合系数和机械载荷总是促进裂纹扩展.     

热电材料中含椭圆夹杂问题的精确解

主要研究了热电材料中含椭圆夹杂问题.假定受到无穷远处的热流和电流荷载条件下,采用保角变换和复变函数方法研究了热电材料中的椭圆夹杂问题,得到了基体和夹杂中的温度场和电场的复势表达式,还通过数值算例分析了椭圆夹杂物对热流和电流的影响.     

基于修正偶应力理论的Timoshenko微梁模型和尺寸效应研究

基于修正偶应力理论,将Timoshenko微梁的应力、偶应力、应变、曲率等基本变量,描述为位移分量偏导数的表达式.根据最小势能原理,推导了决定Timoshenko微梁位移场的位移场控微分方程.利用级数法求解了任意载荷作用下Timoshenko简支微梁的位移场控微分方程,得到了反映尺寸效应的挠度、转角及应力的偶应力理论解.通过对承受余弦分布载荷Timoshenko简支微梁的数值计算,研究了Timoshenko微梁的挠度、转角和应力的尺寸效应,分析了Poisson比对Timoshenko微梁力学行为及其尺寸效应的影响.结果表明:当截面高度与材料特征长度的比值小于5时,Timoshenko微梁的刚度和强度均随着截面高度的减小而显著提高,表现出明显的尺寸效应;当截面高度与材料特征长度的比值大于10时,Timoshenko微梁的刚度与强度均趋于稳定,尺寸效应可以忽略;材料Poisson比是影响Timoshenko微梁力学行为及尺寸效应的重要因素,Poisson比越大Timoshenko微梁刚度和强度的尺寸效应越显著.该文建立的Timoshenko微梁模型,能有效描述Timoshenko微梁的力学行为及尺寸效应,可为微电子机械系统（MEMS）中的微结构设计与分析提供理论基础和技术参考.     

压电介质内裂纹问题的精确解

在压电介质断裂力学分析中,人们常假定裂纹面上的电位移法向分量为零,可是实验表明,这一假设将导致错误的结果。本文基于精确的电边界条件,并应用Stroh公式的方法,导出了含裂纹压电介质在无限远处均匀外载作用下二维问题的精确解。结果表明:(ⅰ)应力强度因子与各向同性材料相同,而电位移强度因子取决于材料常数和机械载荷,但与电载荷无关;(ⅱ)能量释放率大于纯弹性各向异性材料内的值,即总是正的,且与电载荷无关;(ⅲ)裂纹内所含空气的介电常数对介质内的场强无影响。     

Linear analysis and energy budget of viscous liquid jets in both axial and radial electric fields

In their countless industrial applications, axisymmetric and non-axisymmetric instabilities are respectively responsible for electrospraying and electrospinning. A linear method and energy budget have been applied in this study to investigate the instability of viscous jets under both the axial and radial electric fields; the liquid was taken to be a leaky dielectric and the gas a perfect dielectric; the effect of a parabolic velocity profile was considered and compared to that of a uniform velocity, and the energy analysis explained the physical mechanisms to an extent. The liquid viscosity and parabolic velocity profile had a combined effect on jet instability. Work induced by the parabolic velocity profile consisted of two parts: the energy transferred from the basic flow to the disturbances, and the influence of the corresponding shear stresses. At low viscosities, these influences were positive enough to prevail over the viscous dissipation, enhancing axisymmetric instability. However, the parabolic velocity profile functioned differently in small and large wavenumber regions, and the response to the axial electric fields varied in different regions, accounting for the dual effects of axial electric fields on axisymmetric instability. Also, under the interplay between the strong axial electric fields and the parabolic velocity profile, two distinct unstable regions emerged for the non-axisymmetric mode. The effects of the radial electric fields were less sensitive, whether or not the parabolic velocity profile was considered. In summary, the parabolic velocity profile was significant–especially for charged jets with weak viscosity and strong axial electric intensity. The effects of axial electric fields in the atomization zone, and the effects of fluid permittivity coupled with the axial electric fields, were also investigated here.     

The hodograph method in electrodynamics of continuous nonlinearly conducting media PMM vol. 31, no. 6, 1967, pp. 1077–1080

In a wide class of conductors, the presence of strong electric fields disrupts the linearity of the relationship between the current density and the electric field strength. Plasma and semiconductors exhibit the most observable deviations from the Ohm's law. This lack of linearity results from the fact that the current carriers assume nonequilibrium states. During the collisions, the electrons transfer their energy to the ions, molecules e.t.c. relatively slow because of their small mass. At the same time, they possess arbitrarily large free paths and can, therefore absorb large amount of energy from the electric field between successive collisions. Thus their temperature may differ appreciably from the equilibrium temperature due to the heating effect of the field.

Since initial equations are nonlinear, we may experience a difficulty in determining the current distribution in media with these properties. However, using the hodograph transformation we can linearise the electrodynamic equations. This, and use of approximate methods, give us an effective procedure tor computing the electric fields in nonlinearly conducting media.     

Perturbation analysis of radiative effect on free convection flows in porous medium in the presence of pressure work and viscous dissipation

The effect of thermal radiation with a regular three-parameter perturbation analysis has been studied for the effects in some free convection flows of Newtonian fluid-saturated porous medium. The effects of the thermal radiation, permeability of the porous medium, pressure stress work and viscous dissipation on the flows and temperature fields have been included in the analysis. Four different vertical flows have been analyzed, those adjacent to an isothermal surface, uniform heat flux surface, a plane plume and flow generated from a horizontal line energy source, and, a vertical adiabatic surface. Rosseland approximation is used to describe the radiative heat flux in the energy equation. The numerical results of the perturbation analysis for four conditions are solved numerically by the fourth-order Runge–Kutta integration scheme. Numerical values of the main physical quantities are the skin friction and a heat transfer and total heat and mass convected downstream are presented in a tabular form with the parameters characterizing the radiation, permeability of the porous medium, pressure stress work and viscous dissipation. The obtained results are compared and a representative set is displayed graphically to illustrate the influences of the radiation, permeability of the porous medium, pressure stress work and viscous dissipation on the velocity and the temperature profiles.     

Homogenization of the stationary periodic Maxwell system in the case of constant permeability

The homogenization problem in the small period limit for the stationary periodic Maxwell system in ℝ³ is considered. It is assumed that the permittivity η^ε(x)=η(ε⁻x), ε > 0, is a rapidly oscillating positive matrix function and the permeability μ₀ is a constant positive matrix. For all four physical fields (the electric and magnetic field intensities, the electric displacement field, and the magnetic flux density), we obtain uniform approximations in the L ₂(ℝ³)-norm with order-sharp remainder estimates. __________ Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 41, No. 2, pp. 3–23, 2007 Original Russian Text Copyright ? by M. Sh. Birman and T. A. Suslina Dedicated to the memory of the great mathematician Mark Grigor’evich Krein Supported by RFBR grants no. 05-01-01076-a, 05-01-02944-YaF-a.     

Thermoelectric problem for an axisymmetric ellipsoid particle in the liquid metal: Analytical solution and numerical modeling

A thermo-electric problem is solved analytically for an electrically conducting particle in a form of an ellipsoid of revolution immersed in the liquid metal and subjected to a temperature gradient. It is shown that the density of the thermoelectric current is constant inside the particle and its value depends on the eccentricity of the ellipse in the meridian plane of the ellipsoid, but does not depend on the size of the particle. Another parameter which affects the value of the thermoelectric current is the orientation of the ellipsoid with respect to the imposed temperature gradient. The vector of the thermoelectric current inside the particle and the vector of the imposed thermal gradient are co-planar, but a planar angle between these vectors exist and its value is also a function of the eccentricity of the ellipse and its orientation in a thermal field. Limiting minimal and maximal value of the thermoelectric current inside a very elongated particle are found and compared with values obtained in simulations for a dendrite grain. Numerical simulation performed with FEM software for two orientations of an elongated ellipsoid with respect to the imposed thermal gradient provided results similar to analytical solutions with the relative error less than 0.1%.     

环空管内粘弹性流体不定常旋转流的解及流动特性分析

本文用Hankel积分变换法分别求得二阶流体和Maxwell流体在环管内不定常旋转流运动方程的解析解,据此可以分析环管内旋转速度和切应力的分布与变化特征;流体物性参数、管道环隙大小等参量在解析公式中有明确反映,便于定性分析和讨论,本解可以为钻探工程和高分子加工工艺的设计提供理论依据,另外还可用来分析双筒粘度计的流动状态和应力特征,拟合曲线,确定材料的粘弹性参数,在对这种流体进行特性分析时,我们发现,Maxwell流体的旋转流动在起动初期表现为方波振荡,振动的幅度和周期随Ha(物质常数)的增大而增大,此种现象还是首次发现,可能对实际应用有一定的意义。     

Thermal stress analysis of current-carrying media containing an inclusion with arbitrarily-given shape

The presence of inclusions in metal-based composites subjected to an electric current or a heat flux induces thermal stresses. Inclusion geometry is one of the important parameters in the stress distribution. In this study, the plane problem of an arbitrarily-shaped inclusion embedded in an infinite conductive medium is investigated based on the complex variable method. The shape of the inclusion is defined approximately by a polynomial conformal mapping function. Faber series and Fourier expansion techniques are used to solve the corresponding boundary value problems. The obtained results show that the shape, bluntness and rotation angle of the inclusion have a significant effect on the stress concentration around the inclusion induced by the far-field electric current. In addition, for the considered inclusion-matrix system under given electric loading, a lower amount of the Von Mises stress concentration than that around a circular inclusion could be achieved by appropriate selection of the inclusion shape and orientation.     

Error measurement and FEM benchmark for phase field modeling

Solid-state phase-transformation often deals with coupled field variables like mechanical stress or electric potential on different scales compared to the size of phase transition. Further, highly nonlinear equations rule the phase parameter. Thus, required mesh density to approximate phase field variables is often higher than for coupled fields. Finite element techniques applying h- or p-adaptivity can help to increase the efficiency of such simulations. Therefore, it is important to define a benchmark test to verify numerical approaches. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

含椭圆形夹杂的压电体平面应变问题

利用复变函数理论,对在无限远处均匀应力和电位移载荷作用下的含有椭圆形弹性夹杂的横观各向同性压电材料,作了力电分析．在该文有限元结果和前人相关理论解的基础上,提了出一个可接受的认为弹性夹杂体内的应力场为常应力场的假设．在采用了不导通电边界条件之后,获得了以复势形式表示的压电基体的和弹性夹杂体内部的应力场解．     

Fundamental solutions of uniform loads over triangular elements in a three-dimensional piezoelectric medium

In this paper, fundamental solutions of uniform loads over triangular elements in an infinite transversely isotropic piezoelectric three-dimensional space are derived. The triangle element can be parallel or vertical to the plane of isotropy and the uniform load can be mechanical and electric types, oriented in an arbitrary orientation. The solutions are expressed simply as a linear combination of three kinds of elementary functions – linear, trigonometric and logarithm functions. Three methods of superposition are employed to verify the obtained fundamental solutions. Numerical examples are also presented for the extended displacements and stresses induced by both mechanical and electric loads on the vertical and horizontal triangles.