圆柱正交异性体三维弹性问题的一个解法

本文从三维弹性力学基本方程出发,导出了圆柱正交异性体三维弹性静、动态问题的状态方程,并进而求得以位移表达的控制方程.还对横观各向同性体的轴对称问题给出了一个完备闭合解.文中算例计算了一个横观各向同性组合圆柱体的轴对称问题.     

Interaction of a die with a layered elastic foundation

Plane and axisymmetric contact problems for a three-layer elastic half-space are considered. The plane problem is reduced to a singular integral equation of the first kind whose approximate solution is obtained by a modified Multhopp-Kalandiya method of collocation. The axisymmetric problem is reduced to an integral Fredholm equation of the second kind whose approximate solution is obtained by a specially developed method of collocation over the nodes of the Legendre polynomial. An axisymmetric contact problem for an transversely isotropic layer completely adherent to an elastic isotropic half-space is also considered. Examples of calculating the characteristic integral quantities are given. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 165–175, May–June, 2006.     

轴对称横观各向同性热弹性圆柱的分解

为了得到精确的应力场、位移场、温度场,将扭转圆轴的精化理论研究方法推广到轴对称横观各向同性热弹性圆柱。利用Bessel函数以及轴对称横观各向同性热弹性圆柱的通解,给出了轴对称横观各向同性热弹性圆柱的分解定理。根据柱面齐次边界条件获得了精确的精化方程,精化方程可以分解为一阶方程、超越方程、温度方程,从而将横观各向同性热弹性圆柱的轴对称问题分解为轴向拉压问题、超越问题、热-应力耦合问题。超越部分对应端部自平衡情况,可以清晰地了解到端部应力分布对内部应力场的影响,热-应力耦合部分对应无外加应力场时圆柱内部因温度变化引起的热应力。     

FINITE LAYER ANALYSIS FOR SEMI-INFINITE SOILS (Ⅱ)——SPECIAL CASES AND NUMERICAL EXAMPLES

In the present paper reductions of the finite layer mathod once studied in detail by the authors for the elastodynamies of transverse isotropic bodies are given to several special cases. Two-dimensional problems, axisymmetric problems and static problems are discussed, respectively, and this Jinite layer method is also generalized to the problems in which materials possess viscous properties. Two numerical examples have been presented for the axisymmetric case. From these two examples it can be concluded that the finite layer method can be used to analyse semi-infinite layered soils and to deal with the problem of the interaction between soils and structvres.     

Elastodynamic solution of a non-homogeneous orthotropic hollow cylinder

A theoretical method for analyzing the axisymmetric plane strain elastodynamic problem of a non-homogeneous orthotropic hollow cylinder is developed. Firstly, a new dependent variable is introduced to rewrite the governing equation, the boundary conditions and the initial conditions. Secondly, a special function is introduced to transform the inhomogeneous boundary conditions to homogeneous ones. By virtue of the orthogonal expansion technique, the equation with respect to the time variable is derived, of which the solution can be obtained. The displacement solution is finally obtained, which can be degenerated in a rather straightforward way into the solution for a homogeneous orthotropic hollow cylinder and isotropic solid cylinder as well as that for a non-homogeneous isotropic hollow cylinder. Using the present method, integral transform can be avoided and it can be used for hollow cylinders with arbitrary thickness and subjected to arbitrary dynamic loads. Numerical results are presented for a non-homogeneous orthotropic hollow cylinder subjected to dynamic internal pressure. The project supported by the National Natural Science Foundation of China (10172075 and 10002016)     

A receding contact axisymmetric problem between a functionally graded layer and a homogeneous substrate

In this paper, we consider the axisymmetric problem of a frictionless receding contact between an elastic functionally graded layer and a homogeneous half-space, when the two bodies are pressed together. The graded layer is modeled as a nonhomogeneous medium with an isotropic stress–strain law and is subjected over a part of its top surface to normal tractions while the rest of it is free of tractions. Since the contact between the two bodies is assumed to be frictionless, then only compressive normal tractions can be transmitted in the contact area. Using Hankel transform, the axisymmetric elasticity equations are converted analytically into a singular integral equation in which the unknowns are the contact pressure and the receding contact radius. The global equilibrium condition of the layer is supplemented to solve the problem. The singular integral equation is solved numerically using orthogonal Chebychev polynomials and an iterative scheme is employed to obtain the correct receding contact length that satisfies the global equilibrium condition. The main objective of the paper is to study the effect of the material nonhomogeneity parameter and the thickness of the graded layer on the contact pressure and on the length of the receding contact.     

Transient thermal stresses in an orthotropic hollow cylinder for axisymmetric problems

For the thermoelastic dynamic axisymmetric problem of a finite orthotropic hollow cylinder, one comes closer to reality to involve the effect of axial strain than to consider the plane strain case only. However, additional mathematical difficulties should be encountered and a different solution procedure should be developed. By the separation of variables, the thermoelastic axisymmetric dynamic problem of an orthotropic hollow cylinder taking account of the axial strain is transformed to a Volterra integral equation of the second kind for a function of time, which can be solved efficiently and quickly by the interpolation method. The solutions of displacements and stresses are obtained. It is noted that the present method is suitable for an orthotropic hollow cylinder with an arbitrary thickness subjected to arbitrary axisymmetric thermal loads. Numerical comparison is made to show the effect of the axial strain on the displacements and stresses. The project supported by the National Natural Science Foundation of China (10172075) and China Postdoctoral Science Foundation (20040350712)     

A theoretical solution of cylindrical shells for axisymmetric plain strain elastodynamic problems

A method is developed for the transient responses of axisymmetric plain strain problems of cylindrical shells subjected to dynamic loads. Firstly, a special function was introduced to transform the inhomogeneous boundary conditions into the homogeneous ones. Secondly, using the method of separation of variables, the quantity that the displacement subtracts the special function was expanded as the multiplication series of Bassel functions and time functions. Then by virtue of the orthogonal properties of Bessel functions, the equation with respect to the time variable was derived, of which the solution is easily obtained. The displacement solution was finally obtained by adding the two parts mentioned above. The present method can avoid the integral transform and is fit for arbitrary loads. Numerical results are presented for internally shocked isotropic and cylindrically isotropic cylindrical shells and externally shocked cylinders, as well as for an externally shocked, cylindrically isotropic cylindrical shell that is fixed at the internal surface. Contributed by Ding Hao-jiang Foundation item: the National Natural Science Foundation of China (10172075) Biography: Ding Hao-jiang (1934-)     

On the stress-strain state of a transversally isotropic body with a paraboloidal inclusion

A closed solution is presented for the three-dimensional problem of the stress-strain state of an unbounded elastic body with a soldered-in transversally isotropic inclusion in the form of a paraboloid of revolution. Here, it is assumed that the body is under axisymmetric tension (compression). A solution of the corresponding problem for a paraboloidal recess is obtained as a special case. Podil’chuk [2, 3] has investigated similar problems for isotropic bodies with an inclusion assuming the form of a paraboloid of revolution or an elliptical paraboloid. Translated from Prikladnaya Mekhanika, Vol. 34, No. 11, pp. 16–22, November, 1998.     

Indentation of a smooth axisymmetric punch into a transversely isotropic layer

We consider an axisymmetric contact problem for a transversely isotropic layer on a rigid base. The problem is reduced to a Fredholm integro-differential equation of the first kind for the contact pressure under the punch. The integral equation is solved by the collocation method [1]. Some numerical results are presented.     

Angular vibrations of an ellipsoid of revolution in a confined viscous fluid

Vibrations of bodies in confined viscous fluids have been studied on a number of occasions, transverse vibrations of rods being the main subject of investigation [1–3]. The present authors [4] have considered the general problem of translational vibrations of an axisymmetric body in an axisymmetric region containing a low-viscosity fluid. The present paper follows the same approach and considers the problem of small angular vibrations of an ellipsoid of revolution in a circular cylinder with flat ends. In the general case, the hydrodynamic coefficients in the equation of motion of the ellipsoid are determined numerically for different values of the dimensionless geometrical parameters using Ritz's method. In the case of an unconfined fluid, analytic dependences in terms of elementary functions are obtained for the hydrodynamic coefficients. The theoretical results agree well with experimental investigations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 34–39, March–April, 1984.     

Axisymmetric buckling of transversely isotropic circular and annular plates

This paper presents a general solution of the three-dimensional governing equations for the axisymmetric buckling problem of transversely isotropic media. The solution is expressed by a displacement function that satisfies a homogeneous fourth-order partial differential equation. Using this general solution, the axisymmetric buckling of circular and annular plates is investigated and exact solutions are obtained for appropriate boundary conditions. Numerical results are considered for clamped and simply supported circular and annular plates in comparison with existent results.     

FINITE LAYER ANALYSIS FOR SEMI-INFINITE SOILS(Ⅱ)——SPECIAL CASES AND NUMERICAL EXAMPLES

In the present paper reductions of the finite layer mathod once studied in detail by the authors for the elastodynamics of transverse isotropic bodies are given to several special cases. Two-dimensional problems, axisymmetric problems and static problems are discussed, respectively, and this finite layer method is also generalized to the problems in which materials possess viscous properties. Two numerical examples have been presented for the axisymmetric case. From these two examples it can be concluded that the finite layer method can be used to analyse semi-infinite layered soils and to deal with the problem of the interaction between soils and structures. This paper is based on a portion of the author’s dissertation submitted in partial fulfillment of the requirements for the degree of Ph. D at Shanghai Institute of Appl. Math & Mech., Shanghai.     

The solution of weak formulation for axisymmetric problem of orthotropic cantilever cylindrical shell

Weak formulation of equilibrium equations including boundary conditions of laminated cylindrical shell is presented, and thermal stresses mixed state equation for axisymmetric problem of closed cantilever cylindrical shell is established. A unified approach and weak solutions are obtained for closed laminated cantilever cylindrical shell of arbitrary thickness under thermal and mechanical loadings. The equation and boundary conditions proposed in this paper are weakened, the method of this paper would be easy to popularize in dynamics analysis of elasticity. Foundation item: the National Natural Science Foundation of China (19392305)     

Free axisymmetric vibration of transversely isotropic laminated circular plates

Based on the three dimensional elastic theory, the state equation of the axisymmetric free vibration of transversely isotropic circular plates is established. Taking the advantage of finite Hankel transform, two exact solutions are derived for two boundary conditions, i. e. the rigid-slipping boundary and elastic simply supported boundary. Finally, numerical results are presented and compared with those of FEM. The project is supported by the National Natural Science Foundation of China and the Zhejiang Provincial Natural Science Foundation.     

Axisymmetric contact problem of cubic quasicrystalline materials

The axisymmetric elasticity theory of cubic quasicrystal was developed in Ref. [1]. The axisymmetric elasticity problem of cubic quasicrystal is reduced to a single higher-order partial differential equation by introducing a displacement function, based on which, the exact analytic solutions for the elastic field of an axisymmetric contact problem of cubic quasicrystalline materials are obtained for universal contact stress or contact displacement. The result shows that if the contact stress has order −1/2 singularity on the edge of the contact domain, the contact displacement is a constant in the contact domain. Conversely, if the contact displacement is a constant, the contact stress must have order −1/2 singularity on the edge of the contact domain. Project supported by the National Natural Science Foundation of China (No. 19972011).     

Axisymmetric smooth contact for an elastic isotropic infinite hollow cylinder compressed by an outer rigid ring with circular profile

A contact problem for an infinitely long hollow cylinder is considered. The cylinder is compressed by an outer rigid ring with a circular profile. The material of the cylinder is linearly elastic and isotropic. The extent of the contact region and the pressure distribution are sought. Governing equations of the elasticity theory for the axisymmetric problem in cylindrical coordinates are solved by Fourier transforms and general expressions for the displacements are obtained. Using the boundary conditions, the formulation is reduced to a singular integral equation. This equation is solved by using the Gaussian quadrature. Then the pressure distribution on the contact region is determined. Numerical results for the contact pressure and the distance characterizing the contact area are given in graphical form. The English text was polished by Yunming Chen     

多层横观各向同性圆柱壳的三维弹性理论解

本文从三维弹性理论出发,用特征函数法研究多层横观各向同性圆柱壳的轴对称问题.把位移和应力分量的齐次解表达成特征函数展开式,并把特解部分用Fourier级数表示.以多层圆柱壳的内、外柱面作为齐次边界,同时考虑层间的连续条件,推导出问题的特征方程并用Muller法求解.文中运用传递矩阵技术处理多层问题,并用边界型最小二乘配点法处理端部边界条件.作为实例,对双层圆柱壳作了数值计算.     

On Compressible Materials Capable of Sustaining Axisymmetric Shear Deformations. Part 4: Helical Shear of Anisotropic Hyperelastic Materials

Conditions on the form of the strain energy function in order that homogeneous, compressible and isotropic hyperelastic materials may sustain controllable static, axisymmetric anti-plane shear, azimuthal shear, and helical shear deformations of a hollow, circular cylinder have been explored in several recent papers. Here we study conditions on the strain energy function for homogeneous and compressible, anisotropic hyperelastic materials necessary and sufficient to sustain controllable, axisymmetric helical shear deformations of the tube. Similar results for separate axisymmetric anti-plane shear deformations and rotational shear deformations are then obtained from the principal theorem for helical shear deformations. The three theorems are illustrated for general compressible transversely isotropic materials for which the isotropy axis coincides with the cylinder axis. Previously known necessary and sufficient conditions on the strain energy for compressible and isotropic hyperelastic materials in order that the three classes of axisymmetric shear deformations may be possible follow by specialization of the anisotropic case. It is shown that the required monotonicity condition for the isotropic case is much simpler and less restrictive. Restrictions necessary and sufficient for anti-plane and rotational shear deformations to be possible in compressible hyperelastic materials having a helical axis of transverse isotropy that winds at a constant angle around the tube axis are derived. Results for the previous case and for a circular axis of transverse isotropy are included as degenerate helices. All of the conditions derived here have essentially algebraic structure and are easy to apply. The general rules are applied in several examples for specific strain energy functions of compressible and homogeneous transversely isotropic materials having straight, circular, and helical axes of material symmetry.     

The radially nonhomogeneous elastic axisymmentric problem

The plane axisymmetric problem with axisymmetric geometry and loading is analyzed for a radially nonhomogeneous circular cylinder, in linear elasticity. Considering the radial dependence of the stress, the displacements fields and of the stiffness matrix, after a series of admissible functional manipulations, the general differential system solving the problem is developed. The isotopic radially inhomogeneous elastic axisymmetric problem is also analyzed. The exact elasticity solution is developed for a radially nonhomogeneous hollow circular cylinder of exponential Young’s modulus and constant Poisson’s ratio and of power law Young’s modulus and constant Poisson’s ratio. For the isotropic elastic axisymmetric problem, a general expression of the stress function is derived. After the satisfaction of the biharmonic equation and making compatible the stress field’s expressions, the stress function and the stress and displacements fields of the axisymmetric problem are also deduced. Applications have been made for a radially nonhomogeneous hollow cylinder where the stress and displacements fields are determined.