应用数学和力学

1980 Vol. 1, No. 1

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1980, 1(1): 1-3.

Abstract(1995) PDF(630)

Abstract:

Unified Theory of Variational Principles in Non-linear Theory of Elasticity

Guo Zhong-heng

1980, 1(1): 5-23.

Abstract(2175) PDF(703)

Abstract:
The purpose of this paper is to introduce and to discuss several main variational principles in nonlinear theory of elasticity,namely the classic potential energy principle, complementary energy principle, and other two complementary energy pri nciples(Levinson principle and Fraeijs de Veubeke principle), which are widely discussed in recent literatures.At the same time, the generalized variational principles are given also for all these principles. In this paper, Systematic derivation and rigorous proof are given to these variational principles on the unified bases of principle of virtual work, and the intrinsic relations between these principles are also indicated. It is shown that, these principles have unified bases, and their differences are solely due to the adoption of different variables and Legendre transformation. Thus, various variational principles constitute an organized totality in an unified frame. For those variational principles not discussed in this paper, the same frame can also be used.a diagram is given to illustrate the interrelationships between these principles.

Finite Element Analysis of Axisymmetric Elastic Body Problems

Chien Wei-zang

1980, 1(1): 25-35.

Abstract(2323) PDF(986)

Abstract:
Linear form functions are commonly used in a long time for a toroidal volume element swept by a triangle revolved about the symmetrical axis for general axisymmetrical stress problems. It is difficult to obtain the rigidity matrix by exact integration, and instead, the method of approximate integration is used. As the locations of element close to the symmetrical axis, the accuracy of this approximation deteriorates very rapidly. The exact integration have been suggested by various authors for the calculation of rigidity matrix. However, it is shown in this paper that these exact integrations can only be used for those axisymmetric bodies with central hole. For solid axisymmetric body, it can be proved that the calculation fails due to the divergent property of rigidity matrix integration. In this paper a new form function is suggested. In this new form function, the radial displacement u vanishes as radial coordinates r approach to zero. The calculated rigidity matrix is convergent everywhere, including these triangular toroidal element closed to the symmetrical axis. This kind of element is useful for the calculation of axisymmetric elastic solid body problems.

Some Applications of Perturbation Method in Thin Plate Bending Problems

Kiang Fu-ru

1980, 1(1): 37-53.

Abstract(1762) PDF(691)

Abstract:
In this paper,problems of bending of thin plates under the combined action of lateral loading and in-plane forces are studied by means of perturbation method.

On An Eigenvalue Problem Related to the Buckling of Sandwich Beams

Hu Hai-chang

1980, 1(1): 55-62.

Abstract(1739) PDF(521)

Abstract:
This paper gives various properties of eigenvalue problems related to the buckling of sandwich beams. The applications of eigenfunctions are also indicated.

Variation Transformation Analysis(Ⅰ)

Wu Xue-mou

1980, 1(1): 63-69.

Abstract(1660) PDF(647)

Abstract:
This paper is concerned with operator variation from the transformation point of view,and presents some new concepts and new relations.Related problems and concepts include:convex operator,reciprocity set and reciprocity principles,H-generalized solution and operator-differential equation,etc.

Nonlinear Stability of Thin Elastic Circular Shallow Spherical Shell under the Action of Uniform Edge Moment

Yeh Kai-yuan, Liu Zen-huai, Chang Chuan-dzi, Shue Ih-fan

1980, 1(1): 71-87.

Abstract(2066) PDF(671)

Abstract:
In 1939. the importance of the nonlinear feature in the shell buckling problem was first pointed out in a most spectacular manner by von Karman and Tsien,but the mathematical difficulty is so great that progress has been slow after the first attempts.According to our experience, we should face the difficult barrier of solving nonlinear differential equations and an effective, simple, accurate method is required.On the other hand, modern rapid developements in technics, such as aeronautical, naval, structure, precise instrument manufacturing and automatic control engineerings require keenly these problems to have accurate, reliable theory results for direct design use, With such background, we suggest the modified iteration method and have worked out four cases for the purpose of certaining it, This is the one, in which nonlinear stability of thin elastic circular shallow spherical shell under the action of uniform edge moment are considered.As a special case, w e also investigate large deflection of circular plate under the same load.

Various Reciprocal Theorems and Variational Principles in the Theories of Nonlocal Micropolar Linear Elastic Mediums

Tai Tien-min

1980, 1(1): 89-106.

Abstract(1859) PDF(609)

Abstract:
In the first part of our paper, we have extended the concepts of the classical convolution and the "convolution scalar product" given by I. Hlavacek and presented the concepts of the "convolution vector" and the "convolution vector scalar product", which enable us to extend the initial value as well as the initial-boundary value problems for the equation with the operator coefficients to those for the system of equations with the operator coefficients.In the second part of this paper, based on the concepts of the convolution vector and the convolution vector scalar product, two fundamental types of reciprocal theorems of the non-local micro-polar linear elastodynamics for inhomogeneous and anisotropic solids are derived.In the third part of this paper, based on the concepts and results in the first and second parts as well as the Lagrange multiplies method which is presented by W. Z. Chien, four main types of variational principles are given for the nonlocal micropolar linear elastodynamics for inhomogeneous and anisotropic solids. These are the counterparts of the variational principles of Hu-Washizu type, Hellinger-Reissner type and Gurtin type in classical elasticity as well as Hlavacek type and lesan type in local micropolar and nonlocal elasticity. Finally, we have proved the equivalence of the last two main variational principles which are given in this paper.

An Asymptotic Elastic Plastic Analysis in Plane Strain Deformation near a Crack Tip

Hsueh Dah-wei

1980, 1(1): 107-113.

Abstract(1813) PDF(648)

Abstract:
In this Paper,an asymptotic elastoplastic analysis in plane strain deformation near a crack tip is established. The special case of which is the well-known Irwin's solution in elastic material.

An Iteration Method for Integral Equations Arising from Axisymmetric Loading Problems

Yun Tain-quan

1980, 1(1): 115-123.

Abstract(1890) PDF(618)

Abstract:
Let the concentrated forces and the centers of pressure with unknown density functions x(ξ) and y(ξ) respectively be distributed along the axis z outside the solid, then one can reduce an axismmetric loading problem of solids of revolution to two simultaneous Fredholm integral equations. An iteration method for solving such equations is duscussed. A lemma equivalent to E. Rakotch's contractive mapping theorem and a theorem concerning the convergent proof of the iteration method are presented.

“Velocity” Finite Element Method for Dynamic Problem

Yang Zhen-rong

1980, 1(1): 125-138.

Abstract(1744) PDF(679)

Abstract:
After analysing the essential features of successive integration method taking displacement as variable by N. M. Newmark and E. L. Wilson et al, this paper presents a "Velocity" Element Method, taking velocity as variable for the solution of the initial value problem.A simplified scheme is offered for the non-damping system, and the stability is also discussed. Owing to the fact that this simplified scheme for non-damping and apparent static damping is explicit in form, it is unnecessary to solve the algebraic system of equations at every time interval, consequently the amount of computation is greatly reduced. For non-linear dynamic problems, this scheme may be used to obtain fairly good initial values for iteration.An extended form of "elocity" Element is presented for the arbitrary damping system. For the non-linear cases, the incremental Velocity iteration scheme is adopted and its convergence proved. Some discussions have been given on artificial damping and the effect of the parameter.Finally, the results of numerical calculatio of some typical problem are given in the appendix.