Bending integral expressions of a cylinder with cracks

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NONLINEAR BENDINGS OF RECTANGULAR SYMMETRICALLY LAMINATED CROSS-PLY PLATES UNDER VARIOUS SUPPORTS

This paper studies the nonlinear bendings of rectangular symmetrically laminated cross-ply plates subjected to uniform pressure under various supports on the basis of[3] by the singular perturbation method offered in[1]. The uniformly valid N-order asymptotic solutions of the deflection and stress function are derived. Analyses and numerical solutions are given for simply supported rectangular laminates and edge displacement zero.     

The first-order approximation of non-Kirchhoff-Love theory for elastic circular plate with fixed boundary under uniform surface loading (III)—Numerical results

Based upon the differential equations and their related boundary conditions given in the previous papers^{[1, 2]}, using a global interpolation method, this paper presents a numerical solution to the axisymmetric bending problem of non-Kirchhoff-Love theory for circular plate with fixed boundary under uniform surface loading. All the numerical results obtained in this paper are compared with that of Kirchhoff-Love classical theory^[3] and E. Reissner's modified theory^[4].     

The solution of integral equations with strongly singular kernels applied to the torsion of cracked circular cylinder

In this paper,the functions of warping displacement interruption defined on the crack lines are taken for the fundamental unknown functions.The torsion problem of cracked circular cylinder is reduced to solving a system of integral equations with strongly singular kernels.Using the numerical method of these equations,the torsional rigidities and the stress intensity factors are calculated to solve the torsion problem of circular cylinder with star-type and other different types of cracks.The numerical results are satisfactory.     

The uniqueness and existence of solution of the characteristic problem on the generalized KdV equation

The generalized KdV equationu ₁+auu_a+μu_a3+eu_a5=0^[1] is a typical integrable equation. It is derived studying the dissemination of magnet sound wave in cold plasma^[2], the isolated wave in transmission line^[3], and the isolated wave in the boundary surface of the divided layer fluid^[4]. For the characteristic problem of the generalized KdV equation, this paper, based on the Riemann function, designs a suitable structure, then changes the characteristic problem to an equivalent integral and differential equation whose corresponding fixed point, the above integral differential equation has a unique regular solution, so the characteristic problem of the generalized KdV equation has a unique solution. The iteration solution derived from the integral differential equation sequence is uniformly convegent in .     

The asymptotic analyeical solution of the strain-hardening elasto-plastic plate containing A circular hole under simple tension

In this paper the general asymptotic analytical solution of plane problem of elasto-plasticity with strain-hardening^[2] is used in solving the problem of an infinitely large plate containing a circular hole under simple tension, and the analytical expressions of stress components of the first two approximations are given. These results are compared with the numerical and the experimental results given by other authors^{[4, 5]}, and a good agreement is obtained. At the end of this paper the authors inspect the correctness of Neuber's formula^[9] for this problem.     

用样条边界元与奇异积分方程法联合求解多裂纹柱体的扭转

通过间解的分离，本文将径向多裂纹柱体的导曲函两个调和函数表示，使问题归为解一组混混合型积分方程。针对方程的特点，本文联合使用三次样条边界法与奇异积分方程的数值方法对所得方程建立了数值法，并对裂纹相交情形作了特殊处理。最后对工程中感兴趣的一些典型的多裂纹柱体的扭转作了例题计算，结果表明，本文方法具有收敛快，精度高的特点。     

Stress intensity factors of the eccentric and edge cracked cylinders

Using the single crack solution and the regular solution of harmonic function,thetorsion problem of a cracked cylinder is reduced to solving a set of mixed-type integralequations which can be solved by combining the numerical method of singular integralequation with the boundary element method.Several numerical examples arecalculated and the stress intensity factors are obtained.     

Large deflection problem of a clamped elliptical plate subjected to uniform pressure

In this paper, the perturbation solution of large deflection problem of clamped elliptical plate subjected to uniform pressure is given on the basis of the perturbation solution of large deflection problem of similar clamped circular plate (1948), (1954). The analytical solution of this problem was obtained in 1957. However, due to social difficulties, these results have never been published. Nash and Cooley (1959) published a brief note of similar nature, in which only the case λ=a/b=2 is given. In this paper, the analytical solution is given in detail up to the 2nd approximation. The numerical solutions are given for various Poisson ratios v=0.25, 0.30, 0.35 and for various eccentricities λ=1, 2, 3, 4, 5, which can be used in the calculation of engineering designs.     

An exact element method for plane problem

In this paper, based on the step reduction method^[1] and exact analytic method^[2] anew method-exact element method for constructing finite element, is presented. Since the new method doesn ’t need the variational principle, it can be applied to solve non-positive and positive definite partial differential equations with arbitrary variable coefficient. By this method, a quadrilateral noncompatible element with 8 degrees of freedom is derived for the solution of plane problem. Since Jacobi ’s transformation is not applied, the present element may degenerate into a triangle element. It is convenient to use the element in engineering. In this paper, the convergence is proved. Numerical examples are given at the endof this paper, which indicate satisfactory results of stress and displacements can be obtained and have higher numerical precision in nodes.     

Nonlinear MHD Kelvin-Helmholtz instability in a pipe

Nonlinear MHD Kelvin-Helmholtz (K-H) instability in a pipe is treated with the derivative expansion method in the present paper. The linear stability problem was discussed in the past by Chandrasekhar (1961)^[1] and Xu et al. (1981).^[6]Nagano (1979)^[3] discussed the nonlinear MHD K-H instability with infinite depth. He used the singular perturbation method and extrapolated the obtained second order modifier of amplitude vs. frequency to seek the nonlinear effect on the instability growth rate γ. However, in our view, such an extrapolation is inappropriate. Because when the instability sets in, the growth rates of higher order terms on the right hand side of equations will exceed the corresponding secular producing terms, so the expansion will still become meaningless even if the secular producing terms are eliminated. Mathematically speaking, it's impossible to derive formula (39) when γ ₀ ² is negative in Nagano's paper.^[3]Moreover, even as early as γ ₀ ² → O⁺, the expansion becomes invalid because the 2nd order modifier γ₂ (in his formula (56)) tends to infinity. This weakness is removed in this paper, and the result is extended to the case of a pipe with finite depth. Theproject is supported by the National Natural Science Foundation of China.     

Calculation of stresses and deformations of bellows by initial parameter method of numerical integration

By reducing the boundary value problem in stress analysis of bellows into initial value problem, this paper presents a numerical solution of stress distribution in semi-circular arc type bellows based upon the toroidal shell equation of V. V. Novozelov^[8]. Throughout the computation, S. Gill’s method^[1O] of extrapolation is used. The stresses and deformations of bellows under axial load and internal pressure are c-alculated, the results of which agree completely with those derived from the general solution of Prof. Chien Wei-zang^[1-4]. The extrapolation formula presented in this paper greatly promotes the accuracy of discrete calculation.The computer program in BASIC language of Wang 2200 VS computer is included in the appendix.     

Singular perturbation of general boundary value problem for higher order quasilinear elliptic equation involving many small parameters

In this paper applying M. I. Visik’s and L. R. Lyuster-nik’s^[1] asymptotic method and principle of fixed point of functional analysis, we study the singular perturbation of general boundary value problem for higher order quasilinear elliptic equation in the case of boundary perturbation combined with equation perturbation. We prove the existence and uniqueness of solution for perturbed problem. We give its asymptotic approximation and estimation of related remainder term.     

Completely exponentially finite difference methods for problems of turning point

In this paper we construct a completely exponentially fitted finite difference scheme for the boundary value problem of differential equation with turning points, extending Miller’s method^[1] and simplifying the method of the proof We prove the first order uniform convergence of the scheme. The numerical results show that it is better than Hin’s^[2] scheme.     

Boundary element method to solve torsion problem of cracked cylinder

Separating the discontinuous solution by use of the single crack solution, together with the regular solution of harmonic function, the torsion problem of a cracked cylinder is reduced to solving a set of mixed-type integral equations and its numerical technique is then proposed by combining the numerical method of singular integral equation with the boundary element method. Several numerical examples are calculated which will be useful to engineering practice. The method proposed is characterized by its fine accuracy and convenience for using, which can be extended to the cases of multiple crack.The project supported by National Natural Science Foundation of China.     

Solution of the laminar duct flow problem by a boundary integral equation method

Summary A boundary integral equation method is proposed for approximate numerical and exact analytical solutions to fully developed incompressible laminar flow in straight ducts of multiply or simply connected cross-section. It is based on a direct reduction of the problem to the solution of a singular integral equation for the vorticity field in the cross section of the duct. For the numerical solution of the singular integral equation, a simple discretization of it along the cross-section boundary is used. It leads to satisfactory rapid convergency and to accurate results. The concept of hydrodynamic moment of inertia is introduced in order to easily calculate the flow rate, the main velocity, and the fRe-factor. As an example, the exact analytical and, comparatively, the approximate numerical solution of the problem of a circular pipe with two circular rods are presented. In the literature, this is the first non-trivial exact analytical solution of the problem for triply connected cross section domains. The solution to the Saint-Venant torsion problem, as a special case of the laminar duct-flow problem, is herein entirely incorporated.     

Existence and comparison results for solutions of random integral and differential equations

This paper is the continuation of [1]. In this paper, we give another criteria of the existence of solutions for nonlinear random Volterra integral. A comparison theorem and the existence of random extremal solutions are also obtained by using the notion of ordering with respect to a cone. Our theorems generalize the corresponding results of Vaughan^[2,3] and Lakshmikantham^[4,5].     

域外奇点法解杆的弹塑性扭转问题

本文提出一种借助于沙丘比拟的求解杆弹塑性扭转问题的域外奇点法,这种方法可降低所求问题的维数,有效地避免解的奇异性。它具有方法简单,不需要数值积分,计算时间短和精度高等优点。     

Variation of the stress intensity factor along the front of a 3-D rectangular crack subjected to mixed-mode load

Summary  The singular integral equation method is applied to the calculation of the stress intensity factor at the front of a rectangular crack subjected to mixed-mode load. The stress field induced by a body force doublet is used as a fundamental solution. The problem is formulated as a system of integral equations with r ⁻³-singularities. In solving the integral equations, unknown functions of body-force densities are approximated by the product of polynomial and fundamental densities. The fundamental densities are chosen to express two-dimensional cracks in an infinite body for the limiting cases of the aspect ratio of the rectangle. The present method yields rapidly converging numerical results and satisfies boundary conditions all over the crack boundary. A smooth distribution of the stress intensity factor along the crack front is presented for various crack shapes and different Poisson's ratio. Received 5 March 2002; accepted for publication 2 July 2002     

Asymmetric problem of a row of revolutional ellipsoidal cavities using singular integral equations

This paper deals with numerical solution of singular integral equations of the body force method in an interaction problem of revolutional ellipsoidal cavities under asymmetric uniaxial tension. The problem is solved on the superposition of two auxiliary loads; (i) biaxial tension and (ii) plane state of pure shear. These problems are formulated as a system of singular integral equations with Cauchy-type singularities, where the unknowns are densities of body forces distributed in the r, θ, z directions. In order to satisfy the boundary conditions along the ellipsoidal boundaries, eight kinds of fundamental density functions proposed in our previous papers are applied. In the analysis, the number, shape, and spacing of cavities are varied systematically; then the magnitude and position of the maximum stress are examined. For any fixed shape and size of cavities, the maximum stress is shown to be linear with the reciprocal of squared number of cavities. The present method is found to yield rapidly converging numerical results for various geometrical conditions of cavities.