An Extension of the Complex Variable Method in Plane Elasticity to Domains with Corners: A Notch Problem

A severe limitation on the complex variable method in plane elasticity is the requirement that the region in which the stresses and strains are sought admits a rational conformal mapping into the half-plane or the unit circle. Of course, this limitation has not prevented the solution of many fundamental and interesting problems. However, the extension of this method to more general regions remains a source of debate. We here present an attempt in this direction. If the conformal mapping of the region is an algebraic function, it is still possible to find mathematical conditions defining the complex-valued stress function, but its explicit representation now requires the solution of a singular, linear, complex-valued, integro-differential equation.     

Periodic elliptic holes in anisotropic elasticity

The problem of collinear periodic elliptic holes in an anisotropic medium is examined in this paper. By means of Stroh formalism and the conformal mapping method, explicit full domain solutions for the periodic hole problems are presented. The solutions are valid not only for plane problems but also for antiplane problems and the problems whose implane and antiplane deformations are coupled. The stress concentration around the holes is analysed.     

AN ANALYSIS FOR DIFFRACTION OF FLEXURAL WAVES AND DYNAMIC STRESS CONCENTRATIONS IN THICK PLATES WITH A CAVITY

By using the complex variable method and conformal mapping,the diffraction of flexu-ral waves and dynamic stress concentrations in thick plates with a cavity have been studied.A generalsolution of the stress problem of the thick plate satisfying the boundary conditions on the contour of anarbitrary cavity is obtained.By employing the orthogonal function expansion technique,the dynamicstress problem can be reduced to the solution of an infinite algebraic equation series.As an example,the numerical results for the dynamic stress concentration factor in thick plates with a circular,ellipticcavity are graphically presented.The numerical results are discussed.     

各向异性平板开孔动应力集中问题的研究

采用各向异性平板弯曲波动理论及摄动方法，对正交各向异性平板开孔弯曲波的散射及动应力集中问题进行了分析研究，得到了此种平板稳态弯曲波动问题的渐近形式的分析解。同时采用保角映射技术，为求解正交各向异性平板开孔弹性波的散射及动应力集中问题提供了一种统一规范的方法。     

General solutions of plane problem for power function curved cracks

A new exact and universal conformal mapping is proposed. Using Muskhelishvili's complex potential method, the plane elasticity problem of power function curved cracks is investigated with an arbitrary power of a natural number, and the general solutions of the stress intensity factors (SIFs) for mode I and mode II at the crack tip are obtained under the remotely uniform tensile loads. The present results can be reduced to the well-known solutions when the power of the function takes different natural numbers. Numerical examples are conducted to reveal the effects of the coefficient, the power, and the projected length along the x-axis of the power function curved crack on the SIFs for mode I and mode II.     

EVALUATION OF DEGENERATE SCALE FOR N-GON CONFIGURATION IN ANTIPLANE ELASTICITY

This paper provides a solution for the degenerate scale for N-gon configuration in antiplane elasticity using the conformal mapping function, and the lower and upper bounds for the degenerate scale for N = 3, 4, 5, 6, 7, 8, 10, 20, 30, 40, 50, 100, 150 and 200, are evaluated.     

采用精确化理论求解厚板任意形开孔动应力集中

论文基于复变函数与保角映射法,采用平板弯曲振动精确化方程[1],对含任意形开孔平板中弹性波散射与动应力集中问题进行了研究.利用正交函数展开的方法将待解的问题归结为对一组无穷代数方程组的求解.作为算例,计算了自由边界条件下圆孔和椭圆孔的动弯矩集中系数的数值结果,并对板厚与孔径比对动弯矩分布的影响做了分析研究.结果表明:入射波数、平板厚度和椭圆偏心率等参数对动弯矩的分布都有很大的影响.在较低频率和平板较薄的情况下,基于文献[1]的方程与基于Mindlin板的动弯矩结果在数值分布上是基本一致的;在较高频率和平板较厚的情况下,基于文献[1]的方程与基于Mindlin板的动弯矩结果在数值分布计算结果相差较大.由于文献[1]给出的平板振动精确化方程是在没有任何工程假设条件下得到的,因此本文的分析计算结果更精确一些.     

含有非圆形双孔的无限平板中应力的解析解研究

无限平板中含有任意形状单个孔的问题可以使用复变函数方法获得其应力解析解.对于无限平板中含有两个圆孔或两个椭圆孔的双连通域问题,也可以利用多种方法进行求解,比如双极坐标法、应力函数法、复变函数法以及施瓦茨交替法等.其中复变函数中的保角变换方法是获得应力解析解的一个重要方法.但目前尚未见到用此方法求解无限板中含有一个正方形孔和一个椭圆孔的问题.当板在无穷远处受有均布载荷和孔边作用垂直均布压力时,利用保角变换方法可以求解板中含有两个特定形状孔的问题.该方法将所讨论的区域映射成象平面里的一个圆环,其中最关键的一步是找出相应的映射函数.基于黎曼映射定理,提出了该映射函数一般形式,并利用最优化方法,找到了该问题的具体映射函数,然后通过孔边应力边界条件建立了求解两个解析函数的基本方程,获得了该问题的应力解析解.运用ANSYS有限单元法与结果进行了对比.研究了孔距、椭圆形孔大小和两孔布置方位对边界切向应力的影响,以及不同载荷下两孔中心线上应力分布规律.      

含双圆孔平板弹性波散射与动应力分析

利用复变量及局部坐标系方法,对平板中含双圆孔弹性波的衍射与动应力集中问题进行了研究,给出了不同方向入射弯曲波条件下该问题一般解的函数逼近序列和边界条件的表达式用正交函数展开的方法将待解的问题归结为对一组无穷代数方程组的求解给出了双圆孔附近的动应力集中系数的数值结果,分析了孔间距对动应力分布的影响     

基于解析解和新准则的深埋隧道断面形状优化

针对深埋隧道断面形状优化问题,考虑复杂受力下洞周可能存在拉应力情况,提出基于摩尔-库伦屈服准则的新的优化准则,基于解析解用模拟退火法获得洞周最小破坏范围的最优断面形状．优化模型中,以断面映射函数系数为优化的设计变量,利用混合罚函数使优化孔形满足一定的约束条件,设立孔边最大有效应力（与屈服准则相关）最小的优化准则．针对存在内压的深埋隧道断面形状优化问题,对不同内压、不同侧压系数、不同宽高比限制等情况进行优化计算,并对比两种优化准则在该情况下优化结果的异同．结果表明,对压应力主导优化的情况,新准则与原准则没有差别;对可能出现拉应力的复杂应力情况,新准则有更好的适应性,可以获得更优断面形状．     

ELECTRO-MAGNETO-THERMOELASTIC PLANE WAVES IN MICROPOLAR SOLID INVOLVING TWO TEMPERATURES

<正>The model of equations of micropolar generalized magneto-thermoelasticity is introduced within the context of the theory of two temperatures generalized thermoelasticity and we consider a problem of an isotropic homogeneous micropolar medium taking into account the heat effects and allowing the magnetic field effects.A plane wave analysis is employed to obtain the exact formulas of the two temperatures(conductive and mechanical),displacement components, micro-rotation components,stresses,couple stresses,induced electric current,electric field and magnetic field.Arbitrary application is chosen to enable us to get the complete solution.The considered variables are presented graphically and discussions are made for the results.     

Stress analysis of notched anisotropic finite plates under mechanical and hygrothermal loads

Summary Notch-induced stress concentrations in anisotropic composite materials depend on their directional material properties, especially for uniaxially reinforced composites with high-modulus fibres. The design of notched high-performance composites requires therefore a special proof of their notched strength, which includes the structural parameters of the fibre/matrix combination, fibre orientation and layer arrangement. The assessment of the effects of the finite outer boundary is of practical importance when dimensioning critical notched regions. An anisotropic plate with finite dimensions and a hole in its center will be used here to model stress concentrations. The calculation is based on conformal mappings combined with complex-valued stress functions. The outer boundary is described using point-matching and the least-squares method. The solutions are conducive to the assessment of the essential influencing factors of material properties, geometry and loads. Notched finite plates made of fibre/matrix composites, mainly carbon-fibre reinforced polymers, will be presented as illustrations. Received 29 June 1998; accepted for publication 22 October 1998     

On Symmetries and Anisotropies of Classical and Micropolar Linear Elasticities: A New Method Based upon a Complex Vector Basis and Some Systematic Results

A complete and unified study of symmetries and anisotropies of classical and micropolar elasticity tensors is presented by virtue of a novel method based on a well-chosen complex vector basis and algebra of complex tensors. It is proved that every elasticity tensor has nothing but 1-fold, 2-fold, 3-fold, 4-fold and ∞-fold symmetry axes. From this fact it follows that the crystallographic symmetries plus the isotropic symmetry are complete in describing the symmetries of any kind of classical elasticity tensors and micropolar elasticity tensors. Further, it is proved that for each given integer m>>2 every classical Green elasticity tensor with an m-fold symmetry axis must have at least m elastic symmetry planes intersecting each other at this symmetry axis. From this fact and the aforementioned fact it follows that for all possible material symmetry groups, there exist only eight distinct symmetry classes for classical Green elasticity tensors, which correspond to the isotropy group and the seven crystal classes S ₂, C _2h , D _2h , D _3d , D _4h , D _6h and O _h , while it is shown that there exist twelve distinct symmetry classes for any other kind of elasticity tensors, including the classical Cauchy elasticity tensor and the micropolar elasticity tensors, which correspond to the eight subgroup classes just mentioned and the four crystal classes S ₆, C _4h , C_6h and T _h . From these results, it turns out that all possible elasticity symmetry groups are nothing but the full orthogonal group, the transverse isotropy groups C _{∞ h} and D _{∞ h} , and the nine centrosymmetric crystallographic point groups except C _6h and D _6h . This revised version was published online in August 2006 with corrections to the Cover Date.     

带裂纹的椭圆孔口问题的应力分析

断裂现象与材料和结构中的孔洞、缺口或裂纹等缺陷密切相关,这是因为缺陷附近的应力集中明显.该文利用复变方法,通过保角映射研究了带裂纹的椭圆孔洞的平面弹性问题,给出了应力强度因子的解析解.并由此计算了两互相垂直的裂纹问题.     

集中力作用下含尖孔无限域的反平面剪切问题

论文采用了一种新的方法求解含孔元发域的反平面剪切问题，以闭合形式导出了在集中力作用下，孔口尖点的应力强度因子表达式。同时，给出了对称翼状尖孔、内摆线状尖孔、震状尖孔及圆孔径向边裂纹的应力强度因子，其结果在理论上和应用中都具有重要的意义。