APPLICATION OF THE MODIFIED METHOD OF MULTIPLE SCALES TO THE BENDING PROBLEMS FOR CIRCULAR THIN PLATE AT VERY LARGE DEFLECTION AND THE ASYMPTOTICS OF SOLUTIONS(Ⅱ)

This paper is a continuation of part (Ⅰ), on the asymptotics behaviors of the series solutions investigated in (Ⅰ). The remainder terms of the series solutions are estimated by the maximum norm.     

The analyticity of solutions of the Stefan problem

The Stefan problem of a semi-infinite body with arbitrarily prescribed initial and boundary conditions is studied. One of the objectives of the paper is to investigate the analyticity of the solutions. For this purpose, the prescribed initial and boundary conditions are considered to be series of fractional powers of their arguments. It is found that the exact solutions of the problem for various forms of the initial and boundary conditions can be established in series of parabolic cylinder functions and time t. Existence and convergence of the series solutions are studied and proved. The present solutions include the known exact solutions as special cases. On the basis of the present solutions, the question of the analyticity of solutions of the Stefan problem, raised by Rubinstein in his book, can be answered. Conditions for analyticity of the solutions with various initial and boundary conditions are fully discussed.     

On comparison of exact and series solutions for thin film flow of a third‐grade fluid

The present investigation derives the exact and series solutions for steady thin film flow of a third‐grade fluid. The series solution is constructed by a homotopy analysis method. The obtained solutions are compared and an excellent agreement between these is achieved. Copyright © 2009 John Wiley & Sons, Ltd.     

Antiplane interaction of line crack with an arbitrarily located elliptical inclusion

The antiplane problem of the interaction between a main crack and an arbitrarily located elastic elliptical inclusion near its tip is addressed in the current study. The analysis is based on the use of the complex potentials for the antiplane problem, Laurent series expansion method and an appropriate superposition scheme. The stress intensity factor at the main crack is obtained in a general series form. Explicit asymptotic solutions are also derived by using a perturbation technique and retaining the leading order terms in series expansion. The present solutions are shown to coincide with the Taylor expansion of exact solutions for special cases available in the literature. Discussed are changes in the crack tip stress intensity which can be enhanced or suppressed depending on the location of the elliptical inclusion. The explicit solutions provided herein are well suited for the further quantitative analysis of toughening mechanisms in ceramic composite materials.     

On use of the Dirac delta function in the vibration analysis of elastic structures

The Dirac delta function has often been employed to represent the amplitude of concentrated harmonic forces in the analysis of vibration of elastic structures such as beams and plates. It is known that this function, as represented by a truncated Fourier series, does not provide a true representation of a concentrated force, nevertheless, it is frequently employed and good convergence is usually, though not always, encountered in solutions thereby obtained. In this paper, the nature of the function is discussed and for illustrative purposes it is used to obtain series solutions for some selected beam and plate free vibration problems. In some cases problems are chosen for which exact solutions are already obtainable by analytical means. This permits powerful checks to be made on rates of convergence experienced when the series solutions are investigated. Rates of convergence are discussed in detail and it is explained why convergence is to be expected when analyzing certain families of problems when employing this function and a lack of convergence is to be expected in others.     

Series solutions of unsteady MHD flows above a rotating disk

The series solutions of unsteady flows of a viscous incompressible electrically conducting fluid caused by an impulsively rotating infinite disk are given by means of an analytic technique, namely the homotopy analysis method. Using a set of new similarity transformations, we transfer the Navier–Stokes equations into a pair of nonlinear partial differential equations. The convergent series solutions are obtained, which are uniformly valid for all dimensionless time 0 ≤ τ < ∞ in the whole spatial region 0 ≤ η < ∞. To the best of our knowledge, such kind of series solutions have never been reported. The effect of magnetic number on the velocity is investigated.     

Computing singular solutions to partial differential equations by Taylor series

The Taylor Meshless Method (TMM) is a true meshless integration-free numerical method for solving elliptic Partial Differential Equations (PDEs). The basic idea of this method is to use high-order polynomial shape functions that are approximated solutions to the PDE and are computed by the technique of Taylor series. Currently, this new method has proved robust and efficient, and it has the property of exponential convergence with the degree, when solving problems with smooth solutions. This exponential convergence is no longer obtained for problems involving cracks, corners or notches. On the basis of numerical tests, this paper establishes that the presence of a singularity leads to a worsened convergence of the Taylor series, but highly accurate solutions can be recovered by including a few singular solutions in the basis of shape functions.     

Fundamental solution for bonded materials with a free surface parallel to the interface. Part I: Solution of concentrated forces acting at the inside of the material with a free surface

The problem studied in this paper is that of a coated semi-infinite plane subjected to a concentrated force in the upper thin layer (or film). The elastic properties of the coating material are different from those of the substrate, and a perfect bond is assumed between the two materials. The exact solutions of stress functions in a series form are obtained by the method of image. The terms in series form of the stress functions correspond to the image points from the lower order to the higher. The recurrence relations of the stress functions are given, i.e., the stress functions corresponding to the higher order image points are determined by the lower ones. Hence, from the original stress functions for an infinite plane subjected to a concentrated force, the explicit formulas of all terms of the stress function series can be derived. Also, through comparisons between the theoretical results and the numerical results by FEM, it is verified that the convergence rate of the solutions is very rapid. In most practical cases only the first several image points are sufficient to ensure the accuracy of the solutions.     

Heat transfer on a cylinder in accelerated slip flow

The axisymmetric laminar boundary layer flow along the entire length of a semi-infinite stationary cylinder under an accelerated free-stream is investigated. Considering flow at reduced dimensions, the boundary layer equations are developed with the conventional no-slip boundary condition for tangential velocity and temperature replaced by a linear slip-jump boundary condition. Asymptotic series solutions are obtained for the heat transfer coefficient in terms of the Nusselt number. These solutions correspond to prescribed values of the momentum and temperature slip coefficients and the index of acceleration. Heat transfer at both small and large axial distances is determined in the form of series solutions; whereas at intermediate distances, exact and interpolated numerical solutions are obtained. Using these results, the heat transfer along the entire cylinder wall is evaluated in terms of the parameters of acceleration and slip.     

A new method for Fourier series expansions: Applications in rotor-seal systems

Complex nonlinearities of rotor-seal systems make it difficult to implement some widely used techniques for nonlinear vibrations. This is partly due to the fact that it is rather cumbersome to expand the nonlinear terms into Fourier series. In this paper, a novel Fourier series expansion method is proposed to circumvent this difficulty. The incremental harmonic balance method is utilized to obtain the solutions of a rotor-seal system, where the complicated nonlinearities are handled by the expansion method. Periodic, double-periodic and triple-periodic solutions are obtained in excellent agreement with numerical results, which shows the validity and efficacy of the proposed solution procedures.     

Solution of bending of cantilever rectangular plates under uniform surface-load by the method of two-direction trigonometric series

The bending of a cantilever rectangular plate is a very complicated problem in thetheory of plates.For a long time,there have been only approximate solutions for thisproblem by energy methods and numerical methods.since 1979,Prof.F.V.Chang of Tsing Hua University obtained,by the method ofsuperposition,a series of analytic solutions for cantilever rectangular plates under uniformload and concentrated load.In this paper,the two-direction trigonometric series is used to obtain the solution forthe bending of cantilever rectangular plates under uniform load.The obtained results arecompared with the results by the method of superposition.The comparison shows that theresults of these two methods are in good agreement,hence they are mutually confirmed to becorrect.     

New solutions of Novozhilov's equation of toroidal shells

New solutions are obtained for Novozhilov’s equation of toreidal shells having general slenderness ratio 0<a<1 (a=a/R). In contrast to the results by continued fractiontechnique, the exponents and expansion coefficients of our series solutions are all closed and explicit. The series satisfies shell equation identically. Convergence proof is also demonstrated.Explicit expressions for boundary effect and monodromy indices are also given. Finally, we discuss the possibility of applying the present method to solve the fundamental system of equations for elastic shells with rotational symmetry.     

Recursive-operator method in vibration problems for rod systems

Using linear differential equations with constant coefficients describing one-dimensional dynamical processes as an example, we show that the solutions of these equations and systems are related to the solution of the corresponding numerical recursion relations and one does not have to compute the roots of the corresponding characteristic equations. The arbitrary functions occurring in the general solution of the homogeneous equations are determined by the initial and boundary conditions or are chosen from various classes of analytic functions. The solutions of the inhomogeneous equations are constructed in the form of integro-differential series acting on the right-hand side of the equation, and the coefficients of the series are determined from the same recursion relations. The convergence of formal solutions as series of a more general recursive-operator construction was proved in [1]. In the special case where the solutions of the equation can be represented in separated variables, the power series can be effectively summed, i.e., expressed in terms of elementary functions, and coincide with the known solutions. In this case, to determine the natural vibration frequencies, one obtains algebraic rather than transcendental equations, which permits exactly determining the imaginary and complex roots of these equations without using the graphic method [2, pp. 448–449]. The correctness of the obtained formulas (differentiation formulas, explicit expressions for the series coefficients, etc.) can be verified directly by appropriate substitutions; therefore, we do not prove them here.     

Solution of boundary layer flow and heat transfer of an electrically conducting micropolar fluid in a non-Darcian porous medium

In this paper, we have studied the effects of radiation on the boundary layer flow and heat transfer of an electrically conducting micropolar fluid over a continuously moving stretching surface embedded in a non-Darcian porous medium with a uniform magnetic field has been analyzed analytically. The governing fundamental equations are approximated by a system of nonlinear locally similar ordinary differential equations which are solved analytically by applying homotopy analysis method (HAM). The effects of Darcy number, heat generation parameter and inertia coefficient parameter are determined on the flow. Convergence of the obtained series solution is discussed. The homotopy analysis method provides us with a new way to obtain series solutions of such problems. This method contains the auxiliary parameter which provides us with a simple way to adjust and control the convergence region of series solution. By suitable choice of the auxiliary parameter, we can obtain reasonable solutions for large modulus.     

Mechanical behavior of functionally graded material plates under transverse load—Part I: Analysis

An elastic, rectangular, and simply supported, functionally graded material (FGM) plate of medium thickness subjected to transverse loading has been investigated. The Poisson’s ratios of the FGM plates are assumed to be constant, but their Young’s moduli vary continuously throughout the thickness direction according to the volume fraction of constituents defined by power-law, sigmoid, or exponential function. Based on the classical plate theory and Fourier series expansion, the series solutions of power-law FGM (simply called P-FGM), sigmoid FGM (S-FGM), and exponential FGM (E-FGM) plates are obtained. The analytical solutions of P-, S- and E-FGM plates are proved by the numerical results of finite element method. The closed-form solutions illustrated by Fourier series expression are given in Part I of this paper. The closed-form and finite element solutions are compared and discussed in Part II of this paper. Results reveal that the formulations of the solutions of FGM plates and homogeneous plates are similar, except the bending stiffness of plates. The bending stiffness of a homogeneous plate is Eh³/12(1 − ν²), while the expressions of the bending stiffness of FGM plates are more complicated combination of material properties.     

安全壳中钢衬壳热屈曲问题理论与实验研究(Ⅱ)──实验部分

钢衬壳热屈曲问题是核工程安全壳设计中的主要问题,但实验研究方面的文章发表得不多文中以200兆瓦核电站安全壳中钢衬壳为研究对象,采用局部1：1模型,测得了钢衬壳热屈曲温度和应变载荷,给出了钢衬壳屈曲和初始后屈曲过程中挠度和温度关系、以及膜应变和温度关系,实验测得钢衬壳具有局域屈曲的现象,实验屈曲载荷与理论结果符合较好     

Effects of thermophoresis and thermal radiation in mixed convection three-dimensional flow of Jeffrey fluid

Mixed convection three-dimensional flow of Jeffrey fluid is studied in the presence of thermal radiation and thermophoresis. The relevant problems are formulated, and series solutions are presented for velocities, temperature, and concentration. Convergence of series solutions is obtained graphically and numerically. Effects of different emerging parameters on the velocities, temperature, and concentration fields are plotted and discussed.     

Symmetry reductions and exact solutions to the ill-posed Boussinesq equation

In this paper, using the Lie symmetry analysis method, we study the ill-posed Boussinesq equation which arises in shallow water waves and non-linear lattices. The similarity reductions and exact solutions for the equation are obtained. Then the exact analytic solutions are considered by the power series method, and the physical significance of the solutions is considered from the transformation group point of view.     

Auto-Darboux Transformation and Exact Solutions of the Brusselator Reaction Diffusion Model

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On the Stokes entry flow into a semi-infinite circular cylindrical tube

The problem of Stokes entry flow into a semi-infinite circular cylindrical tube was studied in this paper. A new kind of series solutions was derived. Their evident difference from the solutions in References [1,2] is that the present solutions don ’t involve infinite integral. So they are favourable for calculation. We calculated an example by allocated method and obtained satisfied results.