变厚度锥壳方程及其渐近解

本文用薄壳理论导出了壁厚线性变化的锥壳在承受轴对称截荷时的复变量方程，在薄壳理论误差范围内求得了齐次方程的渐近解和工程实用载荷的非齐次解，并给出了算例。     

轴对称圆环壳复变量方程的逼近——渐近解法

求解轴对称圆环壳的复变量方程,一般用级数方法和渐近方法。众所周知,级数解仅对μ=12(1-V~2)~(1/2)(a~2/r_0h)较小的圆环壳才有很好的收敛性;相反,渐近解却对μ值较大的圆环壳才得到很好的近似。本文是在过去工作的基础上,采用逼近——渐近方法,求出了对μ值一致有效的解。计算表明,对于μ=0.5的细环壳,逼近——渐近方法给出的结果和级数解的结果是一致的,而渐近解的误差较大;对于μ=15,逼近——渐近方法给出的结果和渐近解的结果一致。     

轴对称载荷下正交异性圆环壳的一般解

本文导出了在Kirchhoff假设下受轴对称载荷时、材料正交异性的圆环壳的复变量方程,给出了该方程的一般解,该解可用于α=a/R<1,其中a为环壳截面的半径,R为环壳的总体半径,文中并举例说明其应用,结果表明本文所提出的解是很有效的。     

轴对称薄圆环壳方程级数解收敛性的研究

本文研究轴对称薄圆环壳齐次方程级数解的收敛性,证明了每种环壳都有在全域上收敛的级数解。     

球壳轴对称弯曲问题共轭二阶挠度微分方程

提出球壳轴对称弯曲问题共轭二阶挠度微分方程并给出了初等函数解. 球壳微分方程是薄壳理论三大壳之一旋转壳的典型方程. 共轭二阶挠度微分方程是球 壳中微分方程形式最简单的, 是人们最喜爱的挠度微分方程. 挠度微分方程满足边 界条件非常简单, 使球壳的计算得到很大的简化.     

环壳屈曲的渐近解

本文提出分析圆环壳屈曲的一种渐近解析方法,由Sanders非线性平衡方程和壳中面变形协调方程推导出静水外压下环壳的稳定方程,求出了方程的渐近解,理论计算的临界压力值与Fishlowitz的实验结果符合良好,并研究了屈曲前非线性变形对临界载荷的影响。     

两端固支叠层圆柱厚壳轴对称问题的精确解析解

基于柱坐标系下的三维弹性力学基本方程,采用状态空间法得到两端固支单层与叠层圆柱厚壳轴对称问题的精确解析解。为严格满足固支端的边界条件,将固支端的边界位移函数作为状态变量引入状态方程,采用增维方法把非齐次状态方程变为齐次状态方程,并通过层合渐近技术将变系数状态矩阵转为常系数矩阵进行求解。所得到的解不仅严格满足三维弹性力学基本方程,而且严格满足固支边界条件,是真正意义上的三维精确解。算例表明,本研究解与有限元解吻合,具有很高的精度,且关于级数项数和分层数具有很好的收敛性。另外,通过圆柱厚壳各力学量沿径向和轴向的精确分布规律分析了厚径比和跨径比变化对位移和应力分布的影响。     

水平圆柱薄壳非轴对称载荷作用内力分析

近年来大直径圆柱形薄壳结构烟风煤粉管道的使用呈增长趋势.水平圆柱薄壳在非轴对称载荷作用下的内力常按整体梁理论或壳的无矩理论计算,无法反映载荷作用边界的弯曲应力.本文结合圆柱壳的无矩理论和弯矩理论,得到了简支水平圆柱薄壳在任意高度内部积灰(粉)、风、雪、自重、地震作用下的内力解析解,且理论解与三维有限元分析结果非常接近.研究结果可推广到类似的非轴对称变形圆柱壳,并为水平薄壁圆柱壳结构设计提供依据.     

Orr-Sommerfeld方程的数值解法

研究平行流动或近似平行流动,例如平面Poiseuille流及边界层流动的稳定性问题时,若采用线化小扰动理论,则最后归结为解Orr-Sommerfeld方程的特征值问题。对非线性理论来说,只要是弱非线性理论,一般也要顺序解一串Orr-Sommerfeld方程(齐次的或非齐次的)。因此解Orr-Sommerfeld方程,是研究平行或近似平行流动稳定性问题时必然要遇到的问题。在50年代以前,主要利用渐近法求Orr-Sommerfeld方程的特征值,但一般不能      

圆柱壳大开孔的薄壳理论解

采用修正的Ｍｏｒｌｙ方程，以ρ０，Ｙ／Ｒ为小参数，对圆柱壳大开孔的边界条件进行渐近展开，得到了可适用于ρ０≤０．７，情况下的薄壳理论解。     

A novel solution of toroidal shells under axisymmetric loading

μ＝ｒ０／Ｒ０,κ＝１２（１－ν２）μｒ０／ｈ,λ２＝ｉκ,ν—Ｐｏｉｓｏｎ’ｓｒａｔｉｏ,θ—ｔａｎｇｅｎｔｉａｌａｎｇｌｅｏｆｓｈｅｌ,ｒ０—ｒａｄｉｕｓｏｆｔｈｅｍｅｒｉｄｉａｎｃｉｒｃｌｅ,ｈ—ｗａｌｔｈｉｃｋｎｅｓｏｆｓｈｅｌｓ,ｃｏｎｓｔ...     

Large deflections of deep orthotropic spherical shells under radial concentrated load: asymptotic solution

Nonlinear behavior of deep orthotropic spherical shells under inward radial concentrated load is studied. The singular perturbation method is developed and applied to Reissner’s equations describing axially symmetric large deflections of thin shells of revolution. A small parameter proportional to the ratio of shell thickness to the sphere radius is used. The simple asymptotic formulas describing load–deflection diagrams, maximum bending and membrane stresses of the structure are derived. The influence of boundary conditions on the behavior of the shell by large deflections is considered. Obtained asymptotic solution is in close agreement with the experimental and numerical results and has the same accuracy (in the asymptotic meaning) as the given equations of nonlinear theory of thin shells.     

Torsional-mode transients in variable-thickness shells of revolution

For thin shells of revolution the existence of torsional-vibration modes, uncoupled from bending and extensional modes, has been established[1]. Here a linear second-order differential equation for the uncoupled torsional stress mode is obtained and its solution for impact loading of shells is sought. The mode-superposition method which utilizes the natural modes of vibration predicted by elementary theory, is, in general, not satisfactory for sharp impact loading as many modes are often required for convergence. Hence we employ two novel techniques for solving the impact problems. Firstly a formal asymptotic procedure, based on extensions to geometrical optics, is employed to generate asymptotic wavefront expansions. Rigorous justifications for this formal technique are provided in an appendix. Secondly a transform technique whereby solutions are sought in terms of Bessel functions is discussed and applied to particular impact loading problems. The Bessel function solutions found here can be used to determine the natural frequencies of the shells. Shells both finite and infinite in extent are discussed and reflections at a stress-free end are examined.     

Mathematical model of micropolar elastic thin plates and their strength and stiffness characteristics

A number of hypotheses were formulated using the properties of an asymptotic solution of boundary-value problems of the three-dimensional micropolar (moment asymmetric) theory of elasticity for areas with one geometrical parameter being substantially smaller than the other two (plates and shells). A general theory of bending deformation of micropolar elastic thin plates with independent fields of displacements and rotations is constructed. In the constructed model of a micropolar elastic plate, transverse shear strains are fully taken into account. A problem of determining the stress-strain state in bending deformation of micropolar elastic thin rectangular plates is considered. The numerical analysis reveals that plates made of a micropolar elastic material have high strength and stiffness characteristics.     

On second order asymptotic solutions of axial symmetrical problems ofr>0 thin uniform circular toroidal shells with a large parametera 2/R0h

According to the classical shell theory based on the Love-Kirchhoff assumptions, the basic differential equations for the axial symmetrical problems of r>0 thin uniform circular toroidal shells in bending are derived, and the second order asymptotic solutions are given for r>0 thin uniform circular toroidal shells with a large parameter a²/R₀h. In the resent paper, the second order asymptotic solutions of the edge problems far from the apex of toroidal shells are given, too. Their errors are within the margins allowed in the classical theory based on the Love-Kirchhoff assumptions.     

The Solution Asymptotics of Classical and Nonclassical, Static and Dynamic Boundary-Value Problems for Thin Bodies

An asymptotic method for solution of classical and nonclassical boundary-value problems of the theory of elasticity for thin bodies (beams, rods, plates, and shells) is expounded. Studies on the asymptotic theory of thin bodies are reviewed. Asymptotic results are compared with those obtained by other applied theories. The asymptotic approach has been found out to be related to Saint Venant's principle. The correctness of this principle is mathematically proved for one class of problems. A fundamentally new asymptotics in the components of the stress tensor and the displacement vector is revealed in considering new classes of problems. On their basis, the applicability domains are outlined for various models of understructures. Solutions are obtained to certain classes of dynamic problems for thin bodies, particularly, those simulating seismic effects. The resonance conditions are established and ways of preventing them are pointed out.     

Effect of Heat Generation on Forced Convection Through a Porous Saturated Duct

The effect of heat generation on the flow characteristics of the fully developed forced convection through a porous duct is investigated analytically on the basis of Brinkman?CForchheimer model. The duct is bounded by two isoflux plates. For solving momentum equation a regular asymptotic expansion method is used for hyper-porous materials and a matched asymptotic expansion method is used for low-porous materials. This solution permits a uniform solution for the energy equation to find the temperature distribution as well as Nusselt number. A numerical solution is found here to check the accuracy of the asymptotic one.     

Exponential dichotomies of nonlinear discrete systems and its application to numerical analysis and computation

In this pepar we consider the upwind difference scheme of a kind of boundary value problems for nonlinear, second order, ordinary differential equations. Singular perturbation method is applied to construct the asymptotic approximation of the solution to the upwind difference equation. Using the theory of exponential dichotomies we show that the solution of an order-reduced equation is a good approximation of the solution to the upwind difference equation except near boundaries. We construct correctors which yield asymptotic approximations by adding them to the solution of the order-reduced equation. Finally, some numerical examples are illustrated.