构造极坐标中Airy应力函数的观察法

通过引入Airy应力函数，平面问题可以归结为在给定的边界条件下求解一个双调和方程．因此对双调和函数性质的研究将有利于平面问题的求解．首先给出一个有关双调和函数的引理，并分别从复变和微分两种角度提供该引理的证明．借助这个引理，提出了一种构造极坐标中Airy应力函数的观察法．最后，举例说明了该观察法在几个经典平面问题中的应用．这些例子说明，利用本的观察法可以将某些平面问题应力函数构造的过程简单化。     

SOLUTION OF THE PLANE STRESS PROBLEMS OF STRAIN-HARDENING MATERIALS DESCRIBED BY POWER-LAW USING THE COMPLEX PSEUDO-STRESS FUNCTION

In the present paper,the compatibility equation for the plane stress problems of power-law materials is transformed into a biharmonic equation by introducing the so-calledcomplex pseudo-stress function,which makes it possible to solve the elastic-plastic planestress problems of strain hardening materials described by power-law using the complexvariable function method like that in the linear elasticity theory.By using this generalmethod,the close-formed analytical solutions for the stress,strain and displacementcomponents of the plane stress problems’of power-law materials is deduced in the paper,which can also be used to solve the elasto-plastic plane stress problems of strain-hardeningmaterials other than that described by power-law.As an example,the problem of a power-law material infinite plate containing a circular hole under uniaxial tension is solved byusing this method,the results of which are compared with those of a known asymptoticanalytical solution obtained by the perturbation method.     

板弯曲求解新体系及其应用

建立平面弹性与板弯曲的相似性理论,给出了板弯曲经典理论的另一套基本方程与求解方法,然后进入哈密顿体系用直接法研究板弯曲问题．新方法论应用分离变量、本征函数展开方法给出了条形板问题的分析解,突破了传统半逆解法的限制．结果表明新方法论有广阔的应用前景．     

基于平面偶应力-Reissner/Mindlin板比拟的偶应力有限元

偶应力理论的有限元列式面临本质性的C1连续性困难. 平面偶应力理论和Reissner/Mindlin板弯曲理论之间的比拟关系表明这两个理论系统的有 限元的同一性，而R/M板有限元并不存在C1连续性困难. 因此，研究将R/M板单元转化为具有一般位移自由度的平面偶应力单元的一般方法. 根据这一方法，将典型的8节点Serendipity型R/M板单元Q8S转化为一个4节点12 自由度的四边形平面偶应力单元，数值结果表明该单元具有良好的精度和收敛性     

On the representation of three-dimensional elasticity solutions with the aid of complex valued functions

In this paper the representation of three-dimensional displacement fields in linear elasticity in terms of six complex valued functions is considered. The representation includes the complex Muskhelishvili formulation for plane strain as a special case. The completeness of the complex representation for regular solutions is shown and a relationship to the Neuber/Papkovich solutions is given.     

On displacement methods in planar anisotropic elasticity

Two displacement formulation methods are presented for problems of planar anisotropic elasticity. The first displacement method is based on solving the two governing partial differential equations simultaneously/ This method is a recapitulation of the orignal work of Eshelby, Read and Shockley [7] on generalized plane deformations of anisotropic elastic materials in the context of planar anisotropic elasticity.The second displacement method is based on solving the two governing equations separately. This formulation introduces a displacement function, which satisfies a fourth-order partial differential equation that is identical in the form to the one given by Lekhnitskii [6] for monoclinic materials using a stress function. Moreover, this method parallels the traditional Airy stress function method and thus the Lekhnitskii method for pure plane problems. Both the new approach and the Airy stress function method start with the equilibrium equations and use the same extended version of Green's theorem (Chou and Pagano [13], p. 114; Gao [11]) to derive the expressions for stress or displacement components in terms of a potential (stress or displacement) function (see also Gao [10, 11]). It is therefore anticipated that the displacement function involved in this new method could also be evaluated from measured data, as was done by Lin and Rowlands [17] to determine the Airy stress function experimentally.The two different displacement methods lead to two general solutions for problems of planar anisotropic elasticity. Although the two solutions differ in expressions, both of the depend on the complex roots of the same characteristic equation. Furthermore, this characteristic equation is identical to that obtained by Lekhnitskii [6] using a stress formulation. It is therefore concluded that the two displacement methods and Lekhnitskii's stress method are all equivalent for problems of planar anisotropic elasticity (see Gao and Rowlands [8] for detailed discussions).     

加强板的弯矩函数列式

本文首先谇薄板弯曲问题矩函数的物理意义,据此,将弯矩函数列式推广到具有加强条的薄板弯曲问题,给出了与平面弹性问题完全对应的余能原理。     

A necessary and sufficient set of boundary integral equations with indirect unknowns for plate bending problem

A boundary integral representation of plane biharmonic function is established rigorously by the method of unanalytical continuation in the present paper. In this representation there are two boundary functions and four constants which bear a one to one correspondence to biharmonic functions. Therefore the set of boundary integral equations with indirect unknowns based on this representation is equivalent to the original differential equation formulation.     

On plane Λ-fractional linear elasticity theory

Non-local plane elasticity problems are discussed in the context of Λ-fractional linear elasticity theory. Adapting the Λ-fractional derivative along with the Λ-fractional space, where geometry and mechanics are valid in the conventional way, non-local plane elasticity problems are solved with the help of biharmonic functions. Then, the results are transferred into the initial plane.Applications are presented to homogeneous and the fractional beam bending problem.     

On the Solution of an Elliptical Inhomogeneity in Plane Elasticity by the Equivalent Inclusion Method

Stress analysis of an elliptical inhomogeneity in an infinite isotropic elastic plane is a classical elasticity problem, which is usually solved by means of the complex variable formulation. In this work, we demonstrate that an alternative method of solution for such a problem, via the equivalent inclusion method, may be more convenient and straightforward without recourse to complex potentials or curvilinear coordinates. The explicit analytical solution can be derived through simple algebraic manipulation, although the longitudinal eigenstrain component should be handled with care in the case of plane strain. Since the exterior Eshelby tensor for an elliptical inclusion is available in closed-form, the present study provides a full field stress solution expressed in Cartesian coordinates. Furthermore, the in-plane stress components are represented in terms of Dundurs’ parameters. The solution methodology and the convenient formulae of the stress concentration may be of practical use to the engineers in developing benchmarks for design evaluation.     

The solution of elasticity problems for the half-space by the method of Green and Collins

The paper reviews the method of complex potential functions developed by Green and Collins as applied to axisymmetric mixed boundary value problems in elasticity for the half-space. It is shown how the method can be applied to problems in several coupled potential functions such as adhesive and frictional contact problems, to problems involving annular regions and to problems in thermoelasticity. Attention is given to the question of choosing a formulation which leads to a well-behaved numerical solution. Tables are given of the most commonly needed inversion formulae and of expressions for total load and stress intensity factor.     

New solution system for circular sector plate bending and its application

Instead of the biharmonic type equation, a set of new governing equations and solving method for circular sector plate bending is presented based on the analogy between plate bending and plane elasticity problems. So the Hamiltonian system can also be applied to plate bending problems by introducing bending moment functions. The new method presents the analytical solutions for the circular sector plate. The results show that the new method is effective. Project supported by National Natural Science Foundation (No. 19732020) and the Doctoral Research Foundation of China.     

应力函数及其对偶理论在有限元中的应用

借助于Cosserat连续介质模型，探讨了应力函数和位移对避免有限元C$^{1}$ 连续性困难的互补性作用. 通过对应力函数对偶理论的深入分析，为将应力函数列式得到的 余能单元转化为具有一般位移自由度的势能单元提供了严格的理论基础，在此基础上， 给出应用应力函数构造有限元的一般方法.     

Bending Solutions for Inhomogeneous and Laminated Elastic Plates

A procedure has been developed in previous papers for constructing exact solutions of the equations of linear elasticity in a thick plate of inhomogeneous isotropic linearly elastic material in which the elastic moduli depend in any specified manner on a coordinate normal to the plane of the plate. The essential idea is that any solution of the classical thin plate or classical laminate theory equations (which are two-dimensional theories) generates, by straightforward substitutions, a solution of the three-dimensional elasticity equations for the homogeneous material. Recently this theory has been formulated in terms of functions of a complex variable. It was shown that the displacement and stress fields in the inhomogeneous material could be expressed in terms of four complex potentials that are analytic functions of the complex variable ζ = x + iy in the mid-plane of the plate. However, the analysis performed so far applies only to the case of a plate with traction-free upper and lower faces. The present paper extends these solutions to the case where the plate is bent by a pressure distribution applied to a face.     

Magneto-thermo-elastic stresses in an infinitely long cylindrical conductor carrying a uniformly distributed axial current

A formulation of two-dimensional magneto-thermo-elastostatic boundary value problems is presented in this paper. It is found that since the Lorentz pondermotive body force is conservative in this case, a nonhomogeneous biharmonic equation for the stress function can be derived. The effects of the Lorentz body force and Joule heating on the elastic deformation can thus be easily examined. As an illustrative example, stresses induced in an infinitely long cylindrical conductor carrying a uniformly distributed axial current is analysed.     

Green’s functions for plane problem under various boundary conditions and applications

Green’s functions of a point dislocation as well as a concentrated force for the plane problem of an infinite plane containing an arbitrarily shaped hole under stress, displacement, and mixed boundary conditions are stated. The Green’s functions are obtained in closed forms by using the complex stress function method along with the rational mapping function technique, which makes it possible to deal with relatively arbitrary configurations. The stress functions for these problems consist of two parts: a principal part containing singular and multi-valued terms, and a complementary part containing only holomorphic terms. These Green’s functions can be derived without carrying out any integration. The applications of the Green’s functions are demonstrated in studying the interaction of debonding and cracking from an inclusion with a line crack in an infinite plane subjected to remote uniform tension. The Green’s functions should have many other potential applications such as in boundary element method analysis. The boundary integral equations can be simplified by using the Green’s functions as the kernels.     

Influence of couple-stresses on stress concentrations around the cavity

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Experimental study of plane elastic contact problems by the pseudocaustics method

The optical method of pseudocaustics can be used for the experimental solution of plane elasticity, smooth contact problems for finite or infinite media in contact of arbitrary shape. This technique constitutes an alternative to the various numerical and experimental techniques for the approximate solution of plane elasticity contact problems. The success in the application of the method of pseudocaustics to plane elasticity contact problems is due to the possibility inherent in this method of the direct determination of the derivative Φ'(z) of the complex potential Φ(z) of N.I. Muskhelishvili along the boundaries of the media in contact. Then, the Muskhelishvili complex potentials, Φ(z) and Ψ(z), completely characterizing the state of stress and strain in a plane elastic medium, can easily be determined at any point of the media in contact after simple algebraic calculations. Two applications of the proposed method to contact problems of practical interest are also made.     

Applications of the method of complex functions to dynamic stress concentrations

The complex function method used in the solution of static stress concentration around an irregularly shaped cavity in an infinite elastic plane is generalized to the case of dynamic loading. This paper presents the solutions of two dimensional elastic wave equations in terms of complex wave functions, and general expressions for boundary conditions for steady state incident waves. Dynamic stresses around a cavity of arbitrary shape are then expressed in series of complex ‘domain functions’, the coefficient of the series can be determined by truncating a set of infinite algebraic equations. Results of dynamic stress concentration factors for circular and elliptical cavities are given in this paper.     

The simple layer potential method of fundamental solutions for certain biharmonic problems

A novel formulation of the method of fundamental solutions for the numerical solution of plane biharmonic problems, based on the simple layer potential representation of Fichera, is presented. The applicability and accuracy of the method are demonstrated by examining its performance on a set of practical problems arising in Stokes fluid flow.