应用数学和力学

1984 Vol. 5, No. 3

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Variational Principles and Generalized Variational Principles in Hydrodynamics of Viscous Fluids

Chien Wei-zhang

1984, 5(3): 305-322.

Abstract(1927) PDF(1051)

Abstract:
In this paper, the variational principles of hydrodynamic problems for the incompressible and compressible viscous fluids are established. These principles are principles of maximum power losses. Their generalized variational principles are also discussed on the basis of Lagrangian multiplier methods.

A Model for Simulation of Friction Phenomenon between Dies and Workpiece

Z.J. Luo, C. R. Tang, B. Avitzur, C. J. Van Tyne

1984, 5(3): 323-335.

Abstract(1748) PDF(431)

Abstract:
Based on the interaction of asperities and upperbound approach a mathematical model for simulation of friction phenomenon between dies and workpiece is proposed. Optimizing the mathematical model with respect to several variables it is found that in addition to adhering, tearing, ploughing, etc., asperities workpiece can move wave-like along the surface layer and under certain circumstances they may disappear. If the asperities wavily move along the surface layer the friction coefficient depends on the geometry of asperities. However, the bonding strength of asperities, has no significant influence on friction coefficient. The depth of the plastic deformation layer is related to the geometry of asperities, too. The soundness of the prerequisite of the proposed model and some analytical results were verified by experiments.

Boundary and Angular Layer Behavior in Singularly Perturbed Semilinear Systems

K. W. Chang, G. X. Liu

1984, 5(3): 337-344.

Abstract(1812) PDF(473)

Abstract:
Some authors employed the method and technique of differential inequalities to obtain fairly general results concerning the existence and asymptotic behavior, as ε→0⁺, of the solutions of scalar boundary value problems εu^u=h(t,y),a

A Free Rectangular Plate on the Elastic Foundation

Chang Fo-van, Hwang Xiao-mei

1984, 5(3): 345-353.

Abstract(1757) PDF(675)

Abstract:
In the theory of elastic thin plates, the bending of a rectangular plate on the elastic foundation is also a difficult problem. This paper provides a rigorous solution by the method of superposition. It satisfies the differential eguation, the boundary conditions of the edges and the free corners. Thus we are led to a system of infinite simultaneous eguations. The problem solved is for a plate with a concentrated load at its center. The reactive forces from the foundation should be made to be in equilibrium with the concentrated force to see whether our calculation is correct or not.

The Existence and Uniqueness Theorem of the Screw Tensor for the Finite Displacement of a Rigidbody

Yu xin

1984, 5(3): 355-362.

Abstract(1616) PDF(512)

Abstract:
The existence and uniqueness theorem of the screw tensor for the finite displacement of a rigidbody is proposed and then proved using the screw calculus. As a conseguence, formulae are obtained for determining the screw tensor in terms of the finite displacement data of the rigidbody.

Again Discussing about Singular Perturbation of General Boundary Value Problem for Higher Order Elliptic Equation Containing Two-Parameter

Lin Zong-chi

1984, 5(3): 363-375.

Abstract(1879) PDF(550)

Abstract:
In this paper using the method of "The Two-Variable Expansion Procedure" we again discuss the construction of asymptotic expression of solution of general boundary value problem for higher order ellitptic equation containing two-parameter whose boundary condition is more general than [1], We give asymptotic expression of solution as well as the estimation corresponding to the remainder term.

Mixed Hybrid Penalty Finite Element Method and Its Applications

Liang Guo-ping, Fu Zi-zhi

1984, 5(3): 377-390.

Abstract(1982) PDF(627)

Abstract:
The penalty and hybrid methods are being much used in dealing with the general incompatible element, With the penalty method convergence can always be assured, but comparatively speaking its accuracy is lower, and the condition number and sparsity are not so good. With the hybrid method, convergence can be assured only when the rank condition is satisfied. So the construction of the element is extremely limited. This paper presents the mixed hybrid penalty element method, which combines the two methods together. And it is proved theoretically that this new method is convergent, and it has the same accuracy, condition number and sparsity as the compatible element. That is to say, they are optimal to each other.Finally, a new triangle element for plate bending with nine freedom degrees is constructed with this method (three degreesof freedom are given on each corner——one displacement and tworotations), the calculating formula of the element stiffness matrix is almost the same as that of the old triangle element for plate bending with nine degrees of freedom. But it is converged to true solution with arbitrary irregrlar triangle subdivision. If the true solution u∈H³ with this method the linear and quadratic rates of convergence are obtianed for three bending moments and for the displacement and two rotations respectively.

Leibniz’ Formula of Generalized Difference with Respect to a Class of Differential Operators and Recurrence Formula of Their Green’s Function

Xu Yue-sheng

1984, 5(3): 391-398.

Abstract(2016) PDF(449)

Abstract:
In this paper, Leibniz' formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green function about the operators has also been given.

Kantorovich Solution for the Problem of Bending of a Ladder Plate

Xie Xiu-song, Wang Lei

1984, 5(3): 399-409.

Abstract(1955) PDF(594)

Abstract:
Based on the Kantorovich approximation solution for a rectangular plate in bending, this paper deals with the solutions for the ladder plate with various boundary conditions. The deflection of the plate is expressed in a first-order displacement function w(x,y)=(x,y)v(y) where the u(x,y) in x direction is the generalized beam function. By making use of the principle of least potential energy, the variable coefficients differential equations for v(y) may he established. By solving is, these differential euqations and making use of the boundary conditions, the accurate solutions of v(y) in y direction may be obtained. Then the displacement function w(x,y) is the solution for the problem of the bending of the ladder plate with a better degree of approximation.

A Finite Element Method for Stress Analysis of Elastoplastic Body with Polygonal Line Strain-Hardening

Xu Xiao-wei, Shen Jue-ming, Wu Yao-zong

1984, 5(3): 411-417.

Abstract(1785) PDF(391)

Abstract:
In this paper, the stress-strain curve of material is fitted by polygonal line composed of three lines. According to the theory of proportional loading in elastoplasticity, we simplify the complete stress-strain relations, which are given by the increment theory of elastoplasticity. Thus, the finite element equation with the solution of displacement is derived. The assemblage elastoplastic stiffness matrix can be obtained by adding something to the elastic matrix, hence it will shorten the computing time. The determination of every loading increment follows the von Mises yield criteria. The iterative method is used in computation. It omits the redecomposition of the assemblage stiffness matrix and it will step further to shorten the computing time. Illustrations are given to the high-order element application departure from proportional loading, the computation of unloading fitting to the curve and the problem of load estimation.

Nonlinear Bendings for the Orthotropic Rectangular Thin Plates under Various Supports

Zhou Ci-qing

1984, 5(3): 419-436.

Abstract(1669) PDF(653)

Abstract:
In this paper, the nonlinear bandings for the orthotropic rectangular thin plates under various supports are studied.The uniformly valid asymptotic solutions of the displacement ω and stress function Φ are derived by the perturbation offered in [1].

The Application of Weinstein-Chien’s Method——The Upper and Lower Limits of Fundamental Frequency of Rectangular Plates with Edges Which Are a Mixture of Simply Supported Portions and Clamped Portions

Cheng Zheng-qing

1984, 5(3): 437-446.

Abstract(1907) PDF(403)

Abstract:
In this paper, the method of relaxed boundary conditions is applied to rectangular plates with edges which are a sort of the mixture of simply supported portions and clamped portions, so that the lower limit of fundamental frequency of such plates is evaluated. A kind of polynomial satisfying the displacement boundary conditions is designed, wich makes it possoble to evaluate the upper limit of fundamental frequency by Ritz method. The practical calculation examples solved by these methods have given satisfactory results. At the end of this paper, it is pointed out that the so-called exact solution of such plates usually evaluated by the force superposition method is essentially a kind of lower limit of solution, if the truncated error of series which occurs in actual calculation is considered.

Bending of Corner-Supported Rectangular Plate under Concentrated Load

Lin Peng-cheng

1984, 5(3): 447-456.

Abstract(1885) PDF(609)

Abstract:
In this paper the solution for the bending of corner-supported rectangular plate under concentrated load at any point (α/2, η) of the middle line of the plate is given by means of a conception called modified simply supported edges and the method of superposition. Some numerical example is presented. The solution obtained by this method checks very nicely with what was obtained by G.T. Shih^[3] by means of spline finite element method when η=d/2. This shows that this method of solution is satisfactory.

Discussion on “The Boundedness and Symptotic Behavior of Solutions Differential System of Second-Order with Variable Coefficient”

Mao Shi-zhong, Li Li

1984, 5(3): 457-460.

Abstract(1781) PDF(483)

Abstract:
After reading the article "The Boundedness and Asymptotic Behavior of Solution of Differential System of Second-Order with Variable Coefficient" in "Applied Mathematics and Mechanics", Vol.3, No.4, 1982, we would like to put forward a few points to discuss with the author and the readers.