Applied Mathematics and Mechanics (English Edition)

VARIATIONAL PRINCIPLES AND GENERALIZED VARIATIONAL PRINCIPLES IN HYDRODYNAMICS OF VISCOUS FLUIDS

钱伟长

1984, 5(3): 1281-1296.

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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

罗子健;唐才荣

1984, 5(3): 1297-1307.

摘要 ( 525 )

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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 SINGU-LARLY PERTURBED SEMILINEAR SYSTEMS

章国华;刘光旭

1984, 5(3): 1309-1316.

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Some authors employed the method and technique of differential inequalities to obtain fairly general results concerning the existence and asymptotic behavior, as ε→n⁺, of the solutions of scalar boundary value problems εu^u=h(t,y),a<t<b,y(a,ε)=A,y(b,ε)=B. In this paper, we extend these results to vector boundary value problems, under analogous stability conditions on the solution u=xu(t) of the reduced equation 0=h(t, u) Two types of asymptotic behavior are studied, depending on whether the reduced solution u(f) has or does not have a con tinuous first derivative in (a, b) leading to the phenomena of boundary and angular layers.

A FREE RECTANGULAR PLATE ON ELASTIC FOUNDATION

张福范;黄晓梅

1984, 5(3): 1317-1324.

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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 OF THE SCREW TENSOR FOR THE FINITE DISPLACEMENT OF A RIGIDBODY

余燊

1984, 5(3): 1325-1332.

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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

林宗池

1984, 5(3): 1333-1344.

摘要 ( 492 )

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In this paper using the method of "The Two-Variable Expansion Procedure" ^[11] 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 APPLICATION

梁国平;傅子智

1984, 5(3): 1345-1357.

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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

许跃生

1984, 5(3): 1359-1364.

摘要 ( 540 )

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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 is also given.

KANTOROVICH SOLUTION FOR THE PROBLEM OF BENDING OF A LADDER PLATE

谢秀松;王磊

1984, 5(3): 1365-1374.

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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 OR ELASTOPLASTIC BODY WITH POLYGONAL LINE STRAIN——HARDENING

徐孝伟;沈珏铭;邬耀宗

1984, 5(3): 1375-1381.

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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 RECTAN-GULAR THIN PLATES UNDER VARIOUS SUPPORTS

周次青

1984, 5(3): 1383-1397.

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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 ARE THE MIXTURE OF SIMPLY SUPPORTED POR-TIONS AND CLAMPED PORTIONS

陈政清

1984, 5(3): 1399-1408.

摘要 ( 571 )

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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, os that it is enabled 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 socalled 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

林鹏程

1984, 5(3): 1409-1417.

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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 ASYMPTOTIC BEHAVIOR OR SOLUTION DIFFERENTIAL SYSTEM OF SECOND-ORDER WITH VARIABLE COEFFICIENT” (App1ied Mathematics and Mechanics,Vo1.3,No.4,1982)

毛士忠

1984, 5(3): 1419-1423.

摘要 ( 599 )

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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.

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