各向同性弹性力学求解新体系正交关系的研究

在弹性力学求解新体系中，将文献[3]对偶向量进行重新排序后，提出了一种新的对偶微分矩阵L，对于各向同性3维弹性力学问题发现了一种新的正交关系，文中证明了这种正交关系的成立，对于各向同性问题，新的正交关系包含文献[3]的正交关系。     

弹性力学的一种正交关系

在弹性力学求解新体系中，将对偶向量进行重新排序后，提出了一种新的对偶微分矩阵，对于有一个方向正交的各向异性材料的三维弹性力学问题发现了一种新的正交关系．将材料的正交方向取为z轴，证明了这种正交关系的成立．对于z方向材料正交的各向异性弹性力学问题，新的正交关系包含弹性力学求解新体系提出的正交关系。     

薄板理论的正交关系及其变分原理

利用平面弹性与板弯曲的相似性理论，将弹性力学新正交关系中构造对偶向量的思路推广到 各向同性薄板弹性弯曲问题，由混合变量求解法直接得到对偶微分方程并推导了对应的变分 原理. 所导出的对偶微分矩阵具有主对角子矩阵为零矩阵的特点. 发现了两个独立的、对称 的正交关系，利用薄板弹性弯曲理论的积分形式证明了这种正交关系的成立. 在恰当选择对 偶向量后，弹性力学的新正交关系可以推广到各向同性薄板弹性弯曲理论.     

平面扇形域问题的新正交关系和变分原理

通过构造新的对偶向量, 用空间的环向坐标数学上比拟Hamilton体系的时间变量, 在平面弹性扇形域问题中导出了一个斜对角Hamilton算子. 该算子具有主对角元为零, 斜对角元是非零对称算子的结构特性. 得到两个独立的、对称的子正交关系. 恰当选择对偶向量后, 直角坐标系下各向同性平面弹性问题的新正交关系被推广到 极坐标情形. 根据控制微分方程的弱(积分)形式及相应的边界条件, 建立了对应边值问题的变分原理, 并提出了相应的泛函表达式.     

非均匀多跨半无限长板条弯曲问题的一般解

1.引言B. S. Ramachandra Rao等人1971年提出了半无限长板条弯曲问题的一种解法,但只限于分析等厚度和一种材料构成的板条。本文导出了非均匀多跨半无限长板条对应的特征函数的一组新正交关系,它和文献[1]中的正交关系有着完全不同的形式。最后利用这组正交关系求得了问题的一般解。不难看出,文献[1]中研究的问题是本文的一个特例。     

特征函数展开解法的研究

以常微分方程的理论为基础，利用新的对偶变量、对偶微分矩阵和正交关系，以单连续坐标弹性体系为例，建立了与弹性力学求解新体系平行的特征函数展开解法．并将正交关系应用于可对角化边界条件的处理，实现了求解待定系数方程组的解耦，求得问题的显式封闭解．     

辛体系下平面热黏弹性圣维南问题的解析解

借助积分变换，将辛体系引入平面热黏弹性问题，建立了基本问题的对偶方程，并将全 部圣维南问题归结为满足共轭辛正交关系的零本征值本征解问题. 同时，利用变量代换和本 征解展开等技术给出了一套求解边界条件问题的具体方法. 算例讨论了几种典型边界条件问 题，描述了热黏弹性材料的蠕变和松弛特征，体现了这种辛方法的有效性.     

等参数正交异性板平面问题的解法

关于文中指出了现有文献中具有等值复参数正交异性板的平面问题的一些错误，给出了相应正确的应力场和位移场表达式；并提出一种以各向同性板平面问题比拟正交异性板平面问题的方法。     

不可压缩流体平面流动分离的充分判据

本文研究了不可压缩流体平面流动的分离判据问题。得到了比文献[1]更为普遍的分离的充分判据,即定理一。文中指出,MRS判据,零涡判据,奇点判据以及文献[2]的判据在非定常流动中均不是充分的。     

各向异性和各向同性介质中反平面剪切波的相似性

近年来,刘殿魁等发表了一系列文章讨论各向异性介质中反平面剪切波的多种问题。与反平面剪切波对应的平衡问题是柱体的扭转问题。在五十年代以前,各向异性柱体的扭转问题被看作回异于各向同性柱体的扭转同题而曾受到众多的研究。在专著[8]§10.3,§10.4中,作者通过适当的坐标变换,首次将各向异性柱体的扭转问题转化为一个相当的各向同性的问题,从而建立了两者之间的相似性。有了这个相似性后,各向异性柱体的扭转问题就无需作专门的研究。而可把它归并入各向同性的问题。对反平面剪切     

RESEARCH ON AN ORTHOGONAL RELATIONSHIP FOR ORTHOTROPIC ELASTICITY

IntroductionAsymplecticsystematicmethodology[1- 3]forelasticitywasestablishedbyZhongWan_xie .Hepresentedcreativelythedualvectorsandthesymplecticorthogonalrelationshipandopenedaworkplatformparalleledtothetraditionalelasticity[4 - 9].AnewdualvectorandanewdualdifferentialmatrixLwerepresentedforasymplecticsystematicmethodologyfortwo_dimensionalelasticityandaneworthogonalrelationshipwasdiscoveredforisotropicplaneproblems[4 ]byLuoJian_hui.Theneworthogonalrelationshipisgeneralizedfororthotropicelas…     

齐次扩容精细算法

钟万勰院士创立的线性定常系统的精细算法HPD具有非常重要的工程实用价值。对于非齐次线性定常系统，钟构造了在一个积分步长内将激励项线性化的处理方法LHPD，Lin^[3]等通过Fourier级数展开和寻找有解析形式的特解的方法，构造了HPD－F算法，这两种算法有一个共同点，即算法的实现需要求解系统矩阵及相关长阵的逆矩阵，数学上，也即隐含要求系统的矩阵及其相关矩阵非奇异，这样，就产生以下两个问题：1．当系统矩阵及其相关矩阵奇异时，如何设计这类动力响应问题的精细格式？2．算法的实现，需要设计高精度的矩阵求逆算法，而矩阵求逆的工作量是奶大的.本文借助齐次扩容技巧，设计了求解非齐次线性定常系统的一类新的精细算法－齐次扩容精细算法HHPD。该算法不涉及矩阵求逆运算，有效地解决 上述两个问题，并且具有设计合理，易于实现等特点，本文最后就几个典型算例，应用齐次扩容精细算法求解，与文献相比，数值结果更为理想。     

Revisiting displacement functions in three-dimensional elasticity of inhomogeneous media

The paper presents a three-dimensional solution to the equilibrium equations for linear elastic transversely isotropic inhomogeneous media. We assume that the material has constant Poisson’s ratios, and its Young’s and shear moduli have the same functional form of dependence on the co-ordinate normal to the plane of isotropy. We show, apparently for the first time, that stresses and displacements in such an inhomogeneous transversely isotropic elastic solid can be represented in terms of two displacement functions which satisfy the second- and fourth-order partial differential equations. We examine and discuss key aspects of the new representation; they include the relationship between the new displacement functions and Plevako’s solution for isotropic inhomogeneous material with constant Poisson’s ratio as well as the application of the new representation to some important classes of three-dimensional elasticity problems. As an example, the displacement function is derived that can be used to determine stresses and displacements in an inhomogeneous transversely isotropic half-space which is subjected to a concentrated force normal to a free surface and applied at the origin (Boussinesq’s problem).     

THEORY OF ELASTICITY OF AN ANISOTROPIC BODY FOR THE BENDING OF BEAMS

A theory of elasticity for the bending of orthogonal anisotropic beams has been developed by analogy with the special case, which can be obtained by applying the theory of elasticity for bending of transversely isotropic plates to the problems of two deminsions. In this paper, we present a method to solve the problems of bending of orthogonal anisotropic beams and a new theory of the deep-beam whose ratio of depth to length is larger. It is pointed out that Reissner's theory to account for the effect of transverse shear deformation is not very approximate in the components of stress,     

SOME IDENTITY RELATIONS BETWEEN PLANE PROBLEMS FOR VISCO-AND ELASTICITY

I.IntroductionThereareimportantapplicationsfortheor}:ofplanea'iscoelasticit}'Inthefieldsofgeology,miningandconstructingetc.,butformostproblemsofviscoelastici[}'.theirsolutionsareobtainedfromthecorrespondingelasticsolutionsb}'"leansofthecorrespondenceprinc…     

On the reduction of the normality conditions in equality-constrained optimization problems in mechanics

A large class of problems in mechanics leads to the minimization of an objective function under equality constraints. In fact, inequality constraints can always be transformed into equality constraints by means of slack variables. The classical approach to solve equality-constrained problems relies on Lagrange multipliers, whose first-order normality conditions (FONC) lead to a system of nonlinear algebraic equations. This system of equations involves as many equations as unknowns, composed of the design variables and Lagrange multipliers, and hence, is amenable to a host of solution methods. In this paper, two methods to eliminate the Lagrange multipliers are reported, by which a reduced system of normality conditions is obtained. Reduction is conducted here either symbolically or numerically using an isotropic orthogonal complement L of the Jacobian matrix of the equality constraints. The relations thus resulting are cast into what is termed the dual form of the FONC. When the problem allows for symbolic calculations, a semi-graphical approach is applied, which leads to the global optimum of the problem at hand. However, the main novelty of the paper lies in an algorithm that returns the stationary points of a constrained optimization problem without requiring the closed-form expressions of the dual form of the FONC. Moreover, numerically efficient and stable procedures are given for the intermediate computational steps. The application of this algorithm is demonstrated with three examples from mechanics.     

On the relationship between the solutions of static contact problems of elasticity and electroelasticity for a half-space

The paper establishes the relationship between the static contact problems of elasticity and electroelasticity (in the absence of friction) for a transversely isotropic half-space whose surface is the isotropy plane. This makes it possible to avoid solving the electroelastic problem by finding all the characteristics of electroelastic contact from known cases of purely elastic interaction. Moreover, the electroelastic state of the half-space can be fully described using a known harmonic function, which is a solution of the purely elastic problem. The approach is exemplified by solving contact problems of electroelasticity for flat, elliptic, two circular, conical, and paraboloidal (circular and elliptic in plan) punches __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 11, pp. 69–84, November 2006.