Three-Dimensional Analytical Solution for an Axisymmetric Biharmonic Problem

In this paper, we propose a method for the solution of the axisymmetric boundary value problem for a finite elastic cylinder with assigned stress and/or displacements acting on the ends and side. The technique utilizes the Love representation, which allows for reduction of the solution of the elastic problem to the search for a biharmonic function on a cylindrical domain. In the solution method suggested here, we write the Love function with a Bessel expansion and analyze in detail the conditions under which it is possible to differentiate the expansion term by term. We show that this is possible only for a restricted class of elastic solutions. In the general case, we introduce two new auxiliary functions of the z-coordinate. In this way, we obtain the general form of the axisymmetric biharmonic function, which is discussed in relation to certain specific boundary conditions applied on the side and ends of the cylinder. We obtain an exact explicit solution of practical interest for a cylinder with free ends and assigned displacements applied to the side.     

Boundary conditions for axisymmetric circular cylinder in one-dimensional hexagonal quasicrystals

Through generalizing the method of a decay analysis technique determining the interior solution developed by Gregory and Wan, a set of necessary conditions on the end-data of axisymmetric circular cylinder in one-dimensional (1D) hexagonal quasicrystals (QCs) for the existence of a rapidly decaying solution is established. By accurate solutions for auxiliary regular state, and using the reciprocal theorem and general solution, these necessary conditions for the end-data to induce only a decaying elastostatic state (boundary layer solution) will be translated into appropriate boundary conditions for the circular cylinder with axisymmetric deformations in 1D hexagonal QCs. The results of the present paper enable us to establish a set of correct boundary conditions, most of which are obtained for the first time. Furthermore, corresponding necessary conditions for the case of axisymmetric circular cylinder with transversely isotropic are derived subsequently, and their isotropic elastic counterparts are also obtained. The accuracy of mixed boundary conditions of isotropic axisymmertic circular cylinder is proved at last.     

On the interactions of surface waves with immersed structures

The fluid forces resulting from wave interaction with large submerged structures may be calculated using numerical procedures based on the solution of the associated boundary-value problem. In this paper, the analysis of wave interaction with a fixed submerged object of arbitrary cross-section and infinite length using a two-dimensional boundary value formation based on linear diffraction theory is summarized. Subsequently, the application of the boundary element method to obtain a solution is presented. The numerical considerations are emphasized with particular reference to computational efficiency. Numerical results are presented in the form of dimensionless wave force plots for various structural shapes. In the case of a bottom-seated half cylinder, for which there exists a closed-form solution, comparisons are made between results generated using both boundary element and equivalent finite element approaches. In the case of a submerged cylinder, comparisons are made between boundary element derived values and experimental results. The boundary element results compare well with both the closed-form solution and the experimental values.     

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。     

Poromechanical response of a finite elastic cylinder under axisymmetric loading

Drained or undrained cylindrical specimens under axisymmetric loading are commonly used in laboratory testing of soils and rocks. Poroelastic cylindrical elements are also encountered in applications related to bioengineering and advanced materials. This paper presents an analytical solution for an axisymmetrically-loaded solid poroelastic cylinder of finite length with permeable (drained) or impermeable (undrained) hydraulic boundary conditions. The general solutions are derived by first applying Laplace transforms with respect to the time and then solving the resulting governing equations in terms of Fourier–Bessel series, which involve trigonometric and hyperbolic functions with respect to the z-coordinate and Bessel functions with respect to the r-coordinate. Several time-dependent boundary-value problems are solved to demonstrate the application of the general solution to practical situations. Accuracy of the numerical solution is confirmed by comparing with the existing solutions for the limiting cases of a finite elastic cylinder and a poroelastic cylinder under plane strain conditions. Selected numerical results are presented for different cylinder aspect ratios, loading and hydraulic boundary conditions to demonstrate the key features of the coupled poroelastic response.     

轴对称一维六方准晶圆柱的精化理论

将轴对称圆柱的精化分析推广到一维六方准晶中轴对称圆柱的研究当中。利用准调和函数的Bessel算子函数表示以及一维六方准晶中的通解,在不做任何预先假设的情况下,给出了一维六方准晶中轴对称圆柱的精化理论。首先,根据准调和函数的Bessel算子函数表示,利用三个一维待定函数,表示出声子场和相位子场的位移场和应力分量。再根据非齐次边界条件,推导出柱面受径向外载时的精化方程。通过舍弃高阶项,推导出了在径向方向受到柱面载荷的近似解。     

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)     

The final approach to steady state in a nonsteady axisymmetric stagnation point heat transfer

An analysis is presented for the transient thermal response of a laminar boundary layer in the vicinity of an axisymmetric stagnation flow on an infinite circular cylinder. The final approach to steady state temperature field is shown to have exponential decay with time. The characteristic factors appearing in the exponents result in the solution of an eigenvalue problem in ordinary linear differential equations. Numerical results are presented for a range of values of the Reynolds number and Prandtl number.     

Internally Pressurized Radially Polarized Piezoelectric Cylinders

The piezoelectric phenomenon has been exploited in science and engineering for decades. Recent advances in smart structures technology have lead to a resurgence of interest in piezoelectricity, and in particular, in the solution of fundamental boundary-value problems. In this paper, we develop an analytic solution to the axisymmetric problem of an infinitely long, radially polarized, radially orthotropic piezoelectric hollow circular cylinder. The cylinder is subjected to uniform internal pressure, or a constant potential difference between its inner and outer surfaces, or both. An analytic solution to the governing equilibrium equations (a coupled system of second-order ordinary differential equations) is obtained. On application of the boundary conditions, the problem is reduced to solving a system of linear algebraic equations. The stress distributions in the cylinder are obtained numerically for two typical piezoceramics of technological interest, namely PZT-4 and BaTiO₃. It is shown that the hoop stresses in a cylinder composed of these materials can be made virtually uniform throughout the cross-section by applying an appropriate set of boundary conditions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Global stability analysis of axisymmetric boundary layer over a circular cylinder

This paper presents a linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cylinder. The base flow is parallel to the axis of the cylinder at inflow boundary. The pressure gradient is zero in the streamwise direction. The base flow velocity profile is fully non-parallel and non-similar in nature. The boundary layer grows continuously in the spatial directions. Linearized Navier–Stokes (LNS) equations are derived for the disturbance flow quantities in the cylindrical polar coordinates. The LNS equations along with homogeneous boundary conditions forms a generalized eigenvalues problem. Since the base flow is axisymmetric, the disturbances are periodic in azimuthal direction. Chebyshev spectral collocation method and Arnoldi’s iterative algorithm is used for the solution of the general eigenvalues problem. The global temporal modes are computed for the range of Reynolds numbers and different azimuthal wave numbers. The largest imaginary part of the computed eigenmodes is negative, and hence, the flow is temporally stable. The spatial structure of the eigenmodes shows that the disturbance amplitudes grow in size and magnitude while they are moving towards downstream. The global modes of axisymmetric boundary layer are more stable than that of 2D flat-plate boundary layer at low Reynolds number. However, at higher Reynolds number they approach 2D flat-plate boundary layer. Thus, the damping effect of transverse curvature is significant at low Reynolds number. The wave-like nature of the disturbance amplitudes is found in the streamwise direction for the least stable eigenmodes.     

Transient heat conduction in two-dimensional functionally graded hollow cylinder with finite length

In this paper a thick hollow cylinder with finite length made of two dimensional functionally graded material (2D-FGM) subjected to transient thermal boundary conditions is considered. The volume fraction distribution of materials, geometry and thermal boundary conditions are assumed to be axisymmetric but not uniform along the axial direction. The finite element method with graded material properties within each element is used to model the structure and the Crank–Nicolson finite difference method is implemented to solve time dependent equations of the heat transfer problem. Two-dimensional heat conduction in the cylinder is considered and variation of temperature with time as well as temperature distribution through the cylinder are investigated. Effects of variation of material distribution in two radial and axial directions on the temperature distribution and time response are studied. The achieved results show that using two-dimensional FGM leads to a more flexible design so that transient temperature, maximum amplitude and uniformity of temperature distributions can be modified to achieve required specifications by selecting a suitable material distribution profile in two directions.     

The elastic sphere under nonsymmetric loading

Existing solutions to boundary value problems arising from an elastic sphere subjected to a body force have been primarily restricted to axisymmetric, conservative loading. In this paper, a method for solving the displacement equations governing the static equilibrium of an elastic sphere subjected to an arbitrary body force and surface displacement is presented. The solutions are obtained in terms of three vector spherical harmonics and expressions for the displacement and stress fields are presented. Additionally, a short discussion indicating extension of these solutions to dynamic problems is included.This research was supported in part by an Organized Research Grant, Southwest Texas State University, 1979.     

Natural frequency analysis of 2D-FGM thick hollow cylinder based on three-dimensional elasticity equations

In this paper, natural frequencies characteristics of a thick hollow cylinder with finite length made of two-dimensional functionally graded material (2D-FGM) based on three-dimensional equations of elasticity is considered. The axisymmetric conditions are assumed for the 2D-FGM cylinder. The material properties of the cylinder are varied in the radial and axial directions with power law functions. Effects of volume fraction distribution and FGM configuration on the natural frequencies of a simply supported cylinder are analyzed. Also, the effects of length and thickness of the cylinder are considered for different material distribution profiles. Three-dimensional equations of motion are used and the eigen value problem is developed based on direct variational method. Finite element method with graded material characteristics within each element of the structure is used for solution. The study shows that the 2D-FGM cylinder exhibit interesting frequency characteristics when the constituent volume fractions and its configuration are varied.     

A theoretical solution of cylindrically isotropic cylindrical tube for axisymmetric plane strain dynamic thermoelastic problem

A dynamic thermoelastic solution of a cylindrically isotropic cylindrical tube or solid cylinder with axisymmetric plane deformations is developed. Since there exist thermal boundary conditions and tractions on the two surfaces of a cylindrical tube, the problem under consideration is with inhomogeneous boundary conditions. Therefore we introduce a special function to transform the inhomogeneous boundary conditions to homogeneous ones for an unknown function. Then by using the method of separation of variables, the unknown function can be expressed as the multiplication series of Bessel functions and unknown time functions. Thirdly, by virtue of the orthogonal properties of Bessel functions, the equations about these unknown time functions are derived and the solutions are obtained. Finally, the displacement is obtained by adding the two parts mentioned above. By means of the present method, integral transform can be avoided. It is suitable for arbitrary thermal loads and mechanical loads. Numerical results are also presented for thermal shocked, cylindrically isotropic cylindrical tube and solid cylinder. Project supported by the National Natural Science Foundation of China (No. 10172075 and No. 10002016).     

Cracked infinite cylinder with two rigid inclusions under axisymmetric tension

This paper considers the problem of an axisymmetric infinite cylinder with a ring shaped crack at z = 0 and two ring-shaped rigid inclusions with negligible thickness at z = ±L. The cylinder is under the action of uniformly distributed axial tension applied at infinity and its lateral surface is free of traction. It is assumed that the material of the cylinder is linearly elastic and isotropic. Crack surfaces are free and the constant displacements are continuous along the rigid inclusions while the stresses have jumps. Formulation of the mixed boundary value problem under consideration is reduced to three singular integral equations in terms of the derivative of the crack surface displacement and the stress jumps on the rigid inclusions. These equations, together with the single-valuedness condition for the displacements around the crack and the equilibrium equations along the inclusions, are converted to a system of linear algebraic equations, which is solved numerically. Stress intensity factors are calculated and presented in graphical form.     

求解几何非线性桩-土耦合系统的微分求积单元法

将桩-土系统看成在土层中嵌入了一根等圆截面桩的空间轴对称弹性体,在几何非线性的条件下建立了具有间断性条件的桩-土系统的非线性控制方程,并运用微分求积方法(DQEM)来求解了该问题.提出了利用DQEM求解非线性空间轴对称问题中处理单元之间连接条件(包括间断性条件)及边界条件的离散化方法,最终得到了一组离散化的非线性DQEM代数方程,运用Newton-Raphson迭代方法求解非线性代数方程组可以得到每个节点处的位移,进一步可以得到系统的应力和应变.给出了两个数值算例,并与有限元解进行了比较,它们是非常吻合的.将看到,由于在采用DQEM求解时只布置了较少的节点,因此,该文方法具有较小的计算工作量、较高的精度、良好的收敛性以及应用广泛等优点.该文提出的处理连接条件的方法是一个一般的方法,由于它在数学上遵循了求解边值问题的思路,因此,数学上也是严谨的.     

轴对称压电-压磁圆柱夹层结构的圣维南端部效应

研究了半无限长轴对称压电-压磁夹层结构的圆柱体圣维南端部效应的衰减问题。圆柱的端部承受自平衡磁电弹载荷;圆柱的内外表面为机械自由表面,但承受不同的电磁边界条件,即电学短路或电学开路及磁学短路或磁学开路边界条件。基于横贯各向同性压电或压磁材料在轴对称圆柱坐标系下的本构方程,推导了关于衰减率的特征方程并求得问题的数值解。结果表明,边界条件、内外径之比、材料厚度比对结构的衰减率都有显著的影响。     

Axisymmetric viscoplastic deformation by the boundary element method

This paper presents a boundary element formulation and numerical implementation of the problem of small axisymmetric deformation of viscoplastic bodies. While the extension from planar to axisymmetric problems can be carried out fairly simply for the finite element method (FEM), this is far from true for the boundary element method (BEM). The primary reason for this fact is that the axisymmetric kernels in the integral equations of the BEM contain elliptic functions which cannot be integrated analytically even over boundary elements and internal cells of simple shape. Thus, special methods have to be developed for the efficient and accurate numerical integration of these singular and sensitive kernels over discrete elements. The accurate determination of stress rates by differentiation of the displacement rates presents another formidable challenge.A successful numerical implementation of the boundary element method with elementwise (called the Mixed approach) or pointwise (called the pure BEM or BEM approach) determination of stress rates has been carried out. A computer program has been developed for the solution of general axisymmetric viscoplasticity problems. Comparisons of numerical results from the BEM and FEM, for several illustrative problems, are presented and discussed in the paper. It is possible to get direct solutions for the simpler class of problems for cylinders of uniform cross-section, and these solutions are also compared with the BEM and FEM results for such cases.     

Solving the stress problem for solid cylinders with different end conditions

The paper proposes an approach to solving a spatial stress problem for solid circular cylinders under axisymmetric surface loading. Two types of boundary conditions at the ends are examined: simply supported or clamped. The circumferential variable is separated using Fourier series for the former type of boundary conditions, and spline-approximation in longitudinal coordinate is used for the latter type of boundary conditions. The resulting one-dimensional problems are solved by the stable discrete-orthogonalization method, evaluating indeterminate forms on the cylinder axis in the governing equations. Radial displacements and circumferential and longitudinal stresses are plotted __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 6, pp. 24–31, June 2006.