应用数学和力学

1982 Vol. 3, No. 6

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On the Equivalence of Non-Conforming Element and Hybrid Stress Element

Theodore Hsueh-huang Pianl

1982, 3(6): 715-719.

Abstract(1950) PDF(814)

Abstract:
This paper is intended to show that the non-conforming element by Wilson^[3] et al. and the hybrid stress element by Pian^[2] are equivalent.

Solitary Waves at the interface of a Two-Layer Fluid

Dai Shi-qiang

1982, 3(6): 721-731.

Abstract(2111) PDF(950)

Abstract:
In this paper, we discuss the solitary waves at the interface of a two-layer incompressible inviscid fluid confined by two horizontal rigid walls, taking the effect of surface tension into account.First of all, we establish the basic equations suitable for the model considered, and hence derive the Korteweg-de Vries(KdV) equation satisfied by the first-order elevation of the interface with the aid of the reductive perturbation method under the approximation of weak dispersion.lt is found that the KdV solitary waves may be convex upward or downward. It depends on whether the signs of the coefficients a and u of the KdV equation are the same or not. Then we examine in detail two critical cases, in which the nonlinear effect and the dispersion effect cannot balance under the original approximation. Applying other appropriate approximations, we obtain the modified KdV equation for the critical case of first kind(a=0 and conclude that solitary waves cannot exist in the case considered as μ>0, but may still occur as μ<0, being in the form other than that of the KdV solitary wave. As for the critical case of second kind (μ=0), we deduce the generalized KdV equation, for which a kind of oscillatory solitary waves may occur. In addition, we discuss briefly the near-critical cases. The conclusions in this paper are in good agreement with some classical results which are extended considerably.

The Finite Element Scheme for Ordinary Differential Equation with Small Parameter

Wu Qi-guang

1982, 3(6): 733-736.

Abstract(1920) PDF(553)

Abstract:
In this short paper the author has constructed a speical finite element scheme and investigated the convergence of this scheme.

Recent Advances in the Method of Weighted Residuals on Solid Mechanics in China

Xu Ci-da

1982, 3(6): 737-742.

Abstract(2232) PDF(773)

Abstract:
This paper presents a review of research work in recent years on the method of weighted residuals (MWR) on solid mechanics in the People's Republic of China. MWR, as a kind of mathematical method by which approximate solutions of differential equations may be obtained, is being extensively used in the fields of fluid mechanics, heat transfer, etc. In China,prompted by needs, this method has been developed to be used on solid mechanics in recent years and has also been recognized as having merits over other methods.

The Analytical Solution of G. I. Taylor’s Theory of Plastic Deformation in impact of Cylindrical Projectiles and its Improvement

Chien Wei-zhang

1982, 3(6): 743-756.

Abstract(2081) PDF(634)

Abstract:
The theory of plastic deformation in the impact of cylindrical projectiles on rigid targets was first introduced by G. I. Taylor(1948)^[1]. The importance of this theory lies in the fact that the dynamic yield strength of the materials can be determined from the measurement of the plastic deformation of flat-ended cylindrical projectiles. From the experimental results^[2] we find that the dynamic yield strength is independent of impact velocity, and that it is higher than the static yield strength in general, and several times higher than the static yield strength in certain cases. This gives an important foundation for the study of elastoplastic impact problems in general. However, it is well known that the complexity of differential equations in Taylor's theory compelled us to use the troublesome numerical solution. In this paper, the analytical solution of all the equations in Taylor's theory is given in parametrical form and the results are discussed in detail.In the latter part of this paper, the method of calculation of impulse of impact is improved by considering the processes of radial' movement of materials. The analytical solution of the improved theory shows that it gives better agreement with the experimental results than that of original Taylor's theory.

Nonstationary Random Vibration Analysis of Linear Elastic Structures with Finite Element Method

Jin Wen-lu

1982, 3(6): 757-766.

Abstract(1916) PDF(661)

Abstract:
At present, the finite element method is an efficient method for analyzing structural dynamic problems. When the physical quantities such as displacements and stresses are resolved in the spectra and the dynamic matrices are obtained in spectral resolving form, the relative equations cannot be solved by the vibration mode resolving method as usual. For solving such problems, a general method is put forward in this paper. The excitations considered with respect to nonstationary processes are as follows:P(t)={P_i(t)},P_i(t)=a_i(t)P_i⁰(t), a_i(t) is a time function already known. We make Fourier transformation for the discretized equations obtained by finite element method, and by utilizing the behaviour of orthogonal increment of spectral quantities in random process^[1], some formulas of relations about the spectra of excitation and response are derived. The cross power spectral denisty matrices of responses can be found by these formulas, then the structrual safety analysis can be made. When a_i(t)=l (i=1,2,…n), the. method stated in this paper will be reduced to that which is used in the special case of stationary process.

W. Shockley’s Equation and Its Limitation

Bai Zhe

1982, 3(6): 767-770.

Abstract(3732) PDF(1131)

Abstract:
The problem of P-n junction in Ref.[1] is discussed in this paper. We consider that Tsai Shu-tang's view that W.Shock-ley's equation cannot be applied to all circumstances is correct, but we cannot say, because of this, that W. Shockley's method of treatment and its conclusion for P-n junction are wrong. In this paper we demonstrated that W. Shockley's equation merely describes ideal P-n junction model.

Application of the Method of Split Rigidities to Anisotropic Laminated Shallow Shells

Wang Zhen-ming, Liu Guo-xi, Lü Ming-shen

1982, 3(6): 771-780.

Abstract(2048) PDF(475)

Abstract:
In this paper, according to the method stated by Hu Hai-chang in [3], on the basis of [1], the method of split rigidities is generalized for the purpose of solving the problems of lateral deflection, stability and lateral vibration for anisotropic laminated shallow shells,and a simple and practical approximate method is obtained,in which the errors and computing work are comparatively small.

The Relation between Markov Process Theory and Kolmogoroff’s Theory of Turbulence and the Extension of Kolmogoroff’s Laws——I. The Analysis of the Relation between the Two Theories

Yue Zeng-yuan, Zhang Bin

1982, 3(6): 781-789.

Abstract(1950) PDF(646)

Abstract:
The whole paper consists of two parts (Part Ⅰ and Part Ⅱ). In part X/we shall analyze the relation between the two theories of turbulence involving large Reynolds number:the Markov process theory from the La-grangian point of view and Kolmogoroff's theory from the Eulerian point of view. It will be pointed out that the Reynolds number needed for the Markovian description of turbulence should be as large as.that..needed for Kolmogoroff's second hypothesis, that the eddies of the period of order T,(the self-correlation time scale of the random velocity u) and the eddies of the period of order T_u (the self-correlation time scale of the random force f) correspond to the energy-containing eddies and the eddies in the dissipation range respectively, and that T_*≤t≤β^-1 the time interval for the applicability of Richardson's law during two-particle's dispersion, corresponds to the inertial subrange in Kolmogoroff's theory. Thus, these two theories reflect the property of the turbulence involving very large Reynolds number arguing from different-aspects.In Part Ⅱ, by using physical analysis in Part I, we shall establish in a certain way the quantitative relation between these tvo'theories. In terms of this relation and the results of the study of two-particle's dispersion motion, we shall obtain the structure functions, the correlation functions and the energy spectrum, which are applici-able not only to the inertial subrange,but also to the whole range with the wave number less than that in the inertial subrange.Kolmogoroff's "2/3 law" and "-5/3 law" are the asymptotic solutions with respect to the present result in the inertial subrange. Thus, the present result is an extension of Kolmogoroff's Jaws.

A Note on the Minimum Complementary Energy Theorem

Fu Bao-lian

1982, 3(6): 791-792.

Abstract(1713) PDF(897)

Abstract:
In this paper we propose a very simple and clear method which can prove that the minimum complementary energy theorem is equivalent to the condition of single-valued displacements and displacement boundary condition of the deformable body.

The Application of the Finite Element Method to Solving Biot’s Consolidation Equations

Zheng Jia-dong, Hu Hui-zhi, Xu Hong-jiang, Zhu Ze-min, Yin Zong-ze

1982, 3(6): 793-805.

Abstract(2289) PDF(809)

Abstract:
Biot's theory of consolidation of saturated soil.regards the consolidation process as a coupling problem between stress of elastic body and flow of fluid existing in pofes^[1]. It can more precisely reflect the mechanism of consolidation than Terzhigi's theory^[2]. In this article, we obtain the general Biot's finite element equations of consolidation with classical variational principles. The equations have clear physical meaning and have been applied to analysing the consolidation of Bajiazui earth dan. The computational results are in accord with engineering practice.

An Upper-Bound Solution for the Force of Combined Backward-Forward Extrusion

Wu Shi-chun, Tang Cai-rong, An Jiang-shui, Bin Feng

1982, 3(6): 807-816.

Abstract(1772) PDF(442)

Abstract:
A rigid-triangle velocity field for combined backward-forward extrusion based on the experiments and the slip-line field is proposed in this paper. The flow separation point in the rigid-triangle velocity field is defined in accordance with the slip-line theory. A formula of minimum upper bound solution for the punch pressure of the combined extrusion is derived. The values from this formula are compared with those from the slip-line solution.

Solution for Free Vibration Problem of the Membrane with Unequal Tension in Two Directions

Qian Guo-zhen

1982, 3(6): 817-824.

Abstract(1955) PDF(769)

Abstract:
In this paper, we obtain the analytic solution of free vibration frequency and mode shapes of rectangle, circle and elliptic membranes. The approximate solution of membrane with arbitrary boundary is also obtained. All of these membranes are acted on by unequal tension in two directions.For the rectangle membrane, in this paper we transform its vibration equation into one of usual membranes by trnasforming the coordinate, thus it is easy to get the solution. For the circle membrane, first we transform the coordinate in the same way we deal with the rectangle membrane. Next we transform the vibration equation into the Mathieu equation, then we get a formula of frequency of that membrane with some Mathieu function's property. In the solution the elliptic membrane is similar to that of the circle membrane.Finally, some examples are given.

Decompositions of Bäcklund Transformations for the Korteweg-de Vries Equation

Huang Xun-cheng

1982, 3(6): 825-828.

Abstract(1814) PDF(520)

Abstract:
In this paper, decomposition relations such B_a=S_a^-1B₁S_a, connecting scale transformations and Bäcklund transformations for the Korteweg-de Vries K-dV, modified K-dV, higher-order K-dV and cylindrical equations, are obtained.

Dynamical Stress Function Tensor

Shen Hui-chuan

1982, 3(6): 829-834.

Abstract(1623) PDF(768)

Abstract:
This paper generalizes the concept of stress function tensor in static elasticity into the more general case of continuum dynamics, and finds out the expressions for dynamical stress function tensor.

A Calculation Method of Blood Flow Velocity Distribution in Smaller Blood Vessels

Lin Xie-xian

1982, 3(6): 835-841.

Abstract(1926) PDF(834)

Abstract:
The calculative method presented in this paper is based on an improvement of boundary conditions for a micro-continuum fluid model with blood flow assuming that the blood cell velocity at blood vessel wall is unequal to zero.As for steady state flood flow equation (flow in vitre-a rigid circular tube) presented by Eringen, the magnitude of the blood cell gyroscopic velocity at blood vessel wall and the slope of the blood cell gyroscopic velocity distribution curve at the axis of the blood vessel are assumed. From the above-mentioned assumptions the calculating method of velocity distribution curve in blood vessel is derived. The curve calculated by this method is compared with the test curve measured by Bugliarello and Hayden.Tho results obtained by Turk, Sylvester and Ariman as well as with this method are compared with each other, too.