具有孔洞的周期热弹性全平面问题的复势

基于对已知实部的周期解析函数及其原函数的表示和多值性分析,导出了周期调和变温下具有周期分布孔洞的无限体平面弹性问题的两个复势函数的一般表达式,并讨论了几种特殊情形下的结果.     

双周期平面裂纹问题的特征展开-变分方法

根据弹性力学的变分原理,利用双周期问题位移场的双准周期性质和应力应变场的双周期性质,构造了双周期平面问题的单胞泛函变分表达式. 然后结合针对裂纹问题的复应力函数特征展开式,发展了基于单胞模型的双周期裂纹平面问题的特征展开-变分方法. 由于该方法考虑了最一般的双周期边界条件,因而能够分析一般非对称排列的双周期裂纹问题. 通过结果的收敛性分析说明了该方法具有计算效率和精度都高的优点. 最后,对于裂纹呈平行四边形排列的情况,分析了不同的裂纹排列对应力强度因子的影响.      

周期桶状垫圈全平面应变问题

本文运用Ｍｕｓｋｈｅｌｉｓｈｖｉｌｉ复变方法研究周期桶状垫圈全平面应变问题。将此特殊的三维弹性问题归结为求解三个复应力函数所满足的解析函数边值问题。对相同材料和不同材料（但μ￣＋＝μ￣－）两类问题分别进行了讨论，并获得了其封闭解。     

弹性椭圆夹杂纵向剪切问题

获得纵向剪切下弹性椭圆夹杂问题的精确解。将复变函数的分区全纯函数理论，Cauchy型积分和Riemann边值问题相结合，求得各复势函数之间的解析关系，从而得到问题的封闭形式解，并给出了界面应力的解析表达式。本文解答与已有文献结果一致。本文发展的分析方法，为求解复杂多连通域的平面弹性问题提供了一条有效途径。     

广义平面应变问题

一般情况下平面应变问题归结为一个双调和方程、一个调和方程及轴向应变应满足平面坐标x、y 的线性函数条件下的求解.进一步分析证明,轴向载荷分布必须是x、y 的线性函数类问题都能按广义平面应变公式进行分析.     

均布荷载作用下悬臂磁电弹性梁的解析解

对磁电弹性平面问题进行了研究，给出了用拟调和位移函数表达的通解，进而以试凑法按平面应力问题推导出了均布荷载作用下悬臂磁电弹性粱的解析解，所得解有易于理解、便于校对、形式统一简洁的特点。本文还将计算结果与压电材料和弹性材料相应结果进行了分析、比较，为验证各种数值计算方法提供了参考依据。     

弹性椭圆夹杂的线性温变问题

研究了线性温变作用下椭圆夹杂的热弹性问题。通过构造辅助函数，将复变函数的分区全纯函数理论，Riemann边值问题和Cauchy型积分相结合，求得各分区之间的解析关系，从而获得了无穷远均匀加载和线性温变共同作用下椭圆夹杂平面热弹性场的封闭形式解。从本文解答的特殊情况可直接得到已有的若干结果，并可得到一些具有实际意义的新结果。本文发展的分析方法，为求解复杂多连通域的平面热弹性问题提供了一条有效途径。     

多孔有限大弹性薄板弯曲应力集中问题

应用弹性力学的复变函数理论,采用多保角变换的方法,推出了含有任意多孔有限大弹性薄板弯曲的多复变量应力函数的表达式.在内边界上进行复Fourier级数展开,在外边界采用配点法来确定应力函数的未知系数,从而计算有限大弹性薄板的应力场.本文以外边界为矩形,内边界为任意多椭圆孔的有限薄板为例,编制了相应的计算程序,进行了算例分析.结果表明本方法对处理多孔有限大弹性平面问题是简单且行之有效的.     

压电材料椭圆夹杂界面开裂问题的电弹性耦合解

本文研究了在反平面剪切和面内电场的共同作用下，压电材料椭圆夹杂的界面开裂问题，假定夹杂是刚性的导体，采用复变函数保角变换和级数展开方法，可确定压电材料基体的复势表达式，进而求得夹杂界面开裂的电弹性耦合的能量释放率。     

含双周期裂纹的一维六方准晶电弹性全平面应变基本问题

针对电极化方向沿准周期方向的情况,讨论了周期平面内含双周期裂纹的一维六方准晶的电弹性全平面应变第一基本问题和第二基本问题。根据力的叠加原理,将三维的应力系统分解为线性独立的两组二维应力状态;运用复应力函数方法将弹性平衡状态的求解归结为求解正则型积分方程,并证明了其唯一可解性。结果表明:一维六方准晶双周期断裂问题在广义双周期边界条件下的力学解是存在并且唯一的。此结果为运用各种方法研究该基本问题的正确性和一致性提供了理论保证。     

Dynamic stress concentration around elliptic cavities in saturated poroelastic soil under harmonic plane waves

Potential function and complex function in the elliptic coordinate system are employed to solve the problem of scattering harmonic plane waves by multiple elliptic cavities in water saturated soil medium. The steady state Biot’s dynamic equations of poroelasticity are uncoupled into Helmholtz equations via given potentials. The stresses and pore water pressures are obtained by using complex functions in elliptic coordinates with certain boundary conditions. Finally, the dynamic stresses for the case of two interacting elliptic cavities are obtained and discussed in details via a numerical example.     

On multiple circular inclusions in plane magnetoelasticity

A general series solution to the magnetoelastic problem of interacting circular inclusions in plane magnetoelasticity is provided in this paper. By the use of complex variable theory and Laurent series expansion method, the general expression of the magnetic and the magnetoelastic complex potentials for the circular inclusion problem is derived. Expanding the definition of the Airy’s stress function of pure elastic field into the magnetoelastic field and applying the superposition method, the general expression then can be reduced to a set of linear algebraic equations and solved in a series form. An approximate closed form solution for the case of two arbitrarily located inclusions is also provided. For illustrating the effect of the pertinent parameters, the numerical results of the interfacial magnetoelastic stresses are displayed in graphic form.     

Plane problem of thermoelectromagnetoelasticity for multiply connected bodies

Generalized complex potentials of the plane problem of thermoelectromagnetoelasticity are introduced. Expressions for the basic characteristics of the thermoelectromagnetoelastic state, boundary conditions for the complex potentials, and the general form of these functions for a multiply connected plate are obtained. The potentials are used to solve the problem for an elliptic disk with constant temperature at the edge and a concentrated heat source at the center     

The effect of initial stresses on harmonic stress fields within the stratified half plane

The effect of initial stresses on dynamic (harmonic) stress fields within an elastic stratified half plane is investigated. It is assumed that the point-located harmonic force acting on the free plane of the layer by which the half plane is stratified causes this stress field. By employing displacement potentials and the exponential Fourier transform the governing system of partial differential equations of motion is solved. The necessary inverse transformations including rigorous mathematical complexity is performed numerically. The analysis of the numerical results, which shows the influence of the homogeneous initial stresses on the distribution of the stresses on the inter-medium plane, is made. These analyses are examined for various problem parameters and it is assumed that the material of both the layer and the half plane is homogeneous, isotropic, compressible and linearly elastic. It has been observed that the initial stresses may change significantly the values of the superimposed harmonic stresses.     

Three-dimensional elasticity based on quaternion-valued potentials

One of the most fruitful and elegant approach (known as Kolosov–Muskhelishvili formulas) for plane isotropic elastic problems is to use two complex-valued holomorphic potentials. In this paper, the algebra of real quaternions is used in order to propose in three dimensions, an extension of the classical Muskhelishvili formulas. The starting point is the classical harmonic potential representation due to Papkovich and Neuber. Alike the classical complex formulation, two monogenic functions very similar to holomorphic functions in 2D and conserving many of interesting properties, are used in this contribution. The completeness of the potential formulation is demonstrated rigorously. Moreover, body forces, residual stress and thermal strain are taken into account as a left side term. The obtained monogenic representation is compact and a straightforward calculation shows that classical complex representation for plane problems is embedded in the presented extended formulas. Finally the classical uniqueness problem of the Papkovich–Neuber solutions is overcome for polynomial solutions by fixing explicitly linear dependencies.     

On complex potentials for finite plane deformation of a transversely isotropic material

We consider the finite deformation of plane equilibrium problem for a transversely isotropic layer, using the complex variable approach. We give the general expression for the pertinent complex potentials and state the corresponding fundamental problems. We discuss in detail the boundary value problem for fundamental problem-one. As an application of the espoused method, an analytical solution of “Lame's problem” for an infinite layer is obtained. The nonlinear effect of this is highlighted in the obtained figure.     

PLANE ELASTIC PROBLEM ON RIGID LINES ALONG CIRCULAR INCLUSION

The plane elastic problem of circular-arc rigid line inclusions is considered. The model is subjected to remote general loads and concentrated force which is applied at an arbitrary point inside either the matrix or the circular inclusion. Based on complex variable method, the general solutions of the problem were derived. The closed form expressions of the sectionally holomorphic complex potentials and the stress fields were derived for the case of the interface with a single rigid line. The exact expressions of the singular stress fields at the rigid line tips were calculated which show that they possess a pronounced oscillatory character similar to that for the corresponding crack problem under plane loads. The influence of the rigid line geometry, loading conditions and material mismatch on the stress singularity coefficients is evaluated and discussed for the case of remote uniform load.     

The stability of the interface between two bodies compressed along interface cracks. 2. Exact solutions for the case of equal roots

The problem of buckling of the interface between two bodies is considered for the case where several plane cracks are located in the interface and the bodies are compressed along the cracks (along the interface of two different materials). The studies were carried out for a plane problem using the three-dimensional linearized theory of stability of deformable bodies. The complex variables and potentials of the above-mentioned linearized theory are used. This problem is reduced to the problem of linear conjugation of two analytical functions of a complex variable. The exact solution of the above-mentioned buckling problem is obtained for the case where the roots of the basic equation are equal. Some mechanical effects are analyzed under general conditions (elastic, elastoplastic, compressible, incompresible, isotropic, and orthotropic bodies). S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 5, pp. 66–73, May, 2000.