应用数学和力学

1987 Vol. 8, No. 7

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Variational Principles in Elasticity with Nonlinear Stress-Strain Relation

Chien Wei-zang

1987, 8(7): 567-577.

Abstract(1778) PDF(1550)

Abstract:
Since 1979, a series of papers have been published concerning the variational principles and generalized variational principles in elasticity such as [1](1979), [6](1980), [2,3](1983) and[4,5](1984). All these papers deal with the elastic body with linear stress-strain relations. In 1985, a book was published on generalized variational principles dealing with some nonlinear elastic body, but never going into detailed discussion. This paper discusses particularly variational principles and generalized variational principles for elastic body with nonlinear stress-strain relations. In these discussions, we find many interesting problems worth while to pay some attention. At the same time, these discussions are also instructive for linear elastic problems. When the strain is small, the high order terms may be neglected, the results of this paper may be simplified to the well-known principles in ordinary elasticity problems.

A Note on Divergence, Rotation and Gradient and Their Associated Theorems

Guo You-zhong, Tai Chen-to

1987, 8(7): 579-590.

Abstract(2496) PDF(860)

Abstract:
In this note, the essence and some supplements for the unified definition of divergence, rotation and gradient advanced by Tai have been presented based on the method of exterior differential form with an expression of vectors of tensors. The main purpose of this note is to introduce the useful expressions and their applications, and to simplify the proofs of many theorems in various field theories, and they are also important because of their utlity for establishing a wide class of principles.

Generalized Variational Principles with Several Arbitrary Parameters and the Variable Substitution and Multiplier Method

Long Yu-qiu

1987, 8(7): 591-602.

Abstract(1966) PDF(652)

Abstract:
The functional transformations of variational principles in elasticity are classified as three patterns: Ⅰ relaxation pattern, Ⅱ augmented pattern and III equivalent pattern.On the basis of pattern Ⅲ, the generalized variational principles with several arbitrary parameters are formulated and their functionals are defined. They are: the generalized principle of single variable u with several parameters, the generalized principle of two variables u, σ with several parameters, the generalized principle of two variables u, ε with several parameters, and the generalized principle of three veriables u, ε, σ with several parameters. From these principles, a series of new forms of equivalent functionals can be obtained. When the values of these parameters are properly chosen, a series of finite element models can be formulated.In this paper, the question of losing effectiveness for Lagrange multiplier method is also discussed. In order to "recover" effectiveness for multiplier method, a modified method, namely, the variable substitution and multiplier method, is proposed.

Asymptotic Analyses of Dynamic Response of Hyperbolic Cooling Tower Shells with Ring-Stiffeners——Perturbation Finite Element Solution

Li Long-yuan, Loo Wen-da

1987, 8(7): 603-610.

Abstract(1701) PDF(462)

Abstract:
In this paper, a new analysing method called perturbational finite element is suggested, by means of which we calculate the dynamic behaviour and response to turbulent wind of hyperbolic cooling tower shell with ring-stiffeners. The results are compared with those of finite element numerical method and show that the method has the advantages of clear physical idea and convenience of calculation and the accuracy of results is assured.

The Singularity Analysis of the Stress and Strain Fields for Mode I Fracture in Elastic-Plastic State

Chen Xiao-ming, Yang Nan-sheng, Guan Zhong-xin

1987, 8(7): 611-616.

Abstract(1622) PDF(459)

Abstract:
The stress and strain singularities of power hardening material for Mode I fracrure are analysed according to the fundamental equations of elastic-plastic mechanics. It is found that the singularities of all stress and strain components do not change in the thick direction, and neither the six stress components nor the six strain components have the same singularity.

Chaotic Behavior in the Helleman Mapping

Cheng Bao-long

1987, 8(7): 617-621.

Abstract(1939) PDF(439)

Abstract:
In this paper, we establish the analytical conditions for the Helleman mapping in which the Smale horseshoe appears. Then we use it to educe the chaotic criterion of Henon maps.

Lubrication Theory for Micropolar Fluids and Its Application to A Journal Bearing with Finite Length

Qiu Zu-gan, Lu Zhang-ji

1987, 8(7): 623-632.

Abstract(1905) PDF(531)

Abstract:
In this paper, the field equation of micropolar fluid with general lubrication theory assumptions is simplified into two systems of coupled ordinary differential equation. The analytical solutions of velocity and microrotat ion velocity are obtained. Micropolar fluid lubrication Reynolds equation is deduced. By means of numerical method, the characteristics of a finitely long journal bearing under various dynamic parameters, geometrical parameters and micropolar parameters are shown in curve form. These characteristics are pressure distribution, load capacity, coefficient of flow flux and coefficient of friction. Practical value of micropolar effects is shown, so micropolar fluid theory further closes to engineering application.

A Method to Calculate Period Doubling Bifurcation

Liu Zheng-rong, Zhou Shi-gang, Liu Er-ning

1987, 8(7): 633-637.

Abstract(1923) PDF(443)

Abstract:
Based on physical meaning of Melnikov function, we establish a method to calculate period doubling bifurcation and discuss this kind of bifurcation of soft spring Duffing system and find that the result is analogous to subharmonic bifurcation, that is, period doubling bifurcation will appear if damping is small and amplitude of excitation is big. This coincides with facts of physics.

The Bending, Stability and Vibrations of Cantilever Rectangular Plates

Cheng Xiang-sheng

1987, 8(7): 639-648.

Abstract(1761) PDF(620)

Abstract:
This paper discusses the problems of the bending, stability and vibrations of cantilever rectangular plates by means of the variational method. In the text a good many calculating examples are illustrated.

Solution of the Connection Problems between a Finite Holed Plate and a Stiffener by Using the Partitioning Concept of the Generalized Variational Method

Chen Yi-heng, Zhou De-jiao, Wang Ping

1987, 8(7): 649-660.

Abstract(1851) PDF(784)

Abstract:
In this paper a new finite element method is presented, in which complex functions are chosen to be the finite element model and the partitioning concept of the generalized variational method is utilized. The stress concentration factors for a finite holed plate welded by a stiffener are calculated and the analytical solutions in series form are obtained. From some computer trials it is demonstrated that the problem of displacement compatibility and continuity of tractions between the holed plate and the stiffener is successfully analysed by using this method. Since only three elements need to be formulated, relatively less storage is required than the usual finite element methods. Furthermore, the accuracy of solutions is improved and the computer time requirements are considerably reduced. Numerical results of stress concentration factors and stresses along the welded-line which may be referential to engineers are shown in tables.