应用数学和力学

1983 Vol. 4, No. 2

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Method of High-Order Lagrange Multiplier and Generalized Variational Principles of Elasticity with More General Forms of Functionals

Chien Wei-zhang

1983, 4(2): 137-150.

Abstract(2705) PDF(920)

Abstract:
It is known^[1] that the minimum principles of potential energy and complementary energy are the conditional variation principles under respective conditions of constraints. By means of the method of La-grange multipliers, we are able to reduce the functionals of conditional variation principles to new functionals of non-conditional variation principles. This method can be described as follows:Multiply undetermined Lagrange multipliers by various constraints, and add these products to the original functionals.Considering these undetermined Lagrange multipliers and the original variables in these new functionals as independent variables of variation,we can see that the stationary conditions of these functionals give these unceter -mined Lagrange multipliers in terms of original variables. The substitutions of Ihese results for Lagrange multipliers into the above functionals lead to the functionals of these non-conditional variation principles.However, in certain cases, some of the undetermined Lagrange multipliers may turn out to be zero during variation.This is a critical state of variation. In this critical state,the corresponding variational constraint cannot be eliminated by means of the simple Lagrange multiplier method. This is indeed the case when one tries to eliminate the constraint condition of stress-strain relation in the variational principle of minimum complementary energy by the me-thod of Lagrange multiplier. By means of Lagrange multiplier method one can only derive, from minimum complementary energy principle,the Hellinger-Reissner principle^[2,3], in which only two types of independent variables, stresses and displacements, exist in the new functional. The strain-stress relation remains to be a constraint,from which one derives the strain from the given stress. Thus the Hellinger-Reissner principle remains to be a conditional variation with one constraint uneliminated.In ordinary Lagrange multiplier method, only the linear terms of constraint conditions are taken into consideration. It is impossible to incorporate this condition of constraint into functional whenever the corresponding Lagrange multiplier turns out to be zero. Hence, we extend the Lagrange multiplier method by considering not only the linear term, but also the high-order terms,such as thequa-dratic terms of constraint in the Taylor's series expansion.We call this method the high order Lagrange multiplier method.With this method we find the more general form of functional of the generalized variational principle ever known to us from the Hellinger-Reissner principle. In particular, this more general form of functional can be all known functionals of existing generalized variational principles in elasticity. Similarly, we can also find the more general form of functional from He-Washizu principle^[4,5].It is also shown that there are equivalent theorem and related equivalent relation between these two general forms of functionals in elasticity.

The Finite Rotation Parameter of a Deformable Body and Gyromagnetic Effect

Chen Zhi-da

1983, 4(2): 151-157.

Abstract(2139) PDF(486)

Abstract:
The derivation of Synge for the finite rotation formula in terms of Euler parameter is put into tensor form.Hence a further clarification of the geometric meaning of the orthogonal transformation obtained in the author's paper[7] is made. The tensor properties of rotation axis vector are also discussed. By means of the method of co-moving coordinate system established in[8],[9], we explain the gyromagnetic effect,and derive a simple formula for calculating body couple which is induced by magnetization.

Homogenized Equations for Steady Heat Conduction in Composite Materials with Dilutely Distributed Impurities

Dai Shi-qiang, Li Jia-chun

1983, 4(2): 159-164.

Abstract(1808) PDF(446)

Abstract:
In this paper, by using the two-space method, homogenized equations for steady heat conduction in the composite material cylinders with dilutely-distributed elliptic cylinders of impurities are derived, and the explicit expressions for the corresponding effective heat conductivity of those which are concerned are obtained. It is also shown that the macroscopic heat conduction is anisotropic when the cross-sections of the impurity cylinders are unidliec-tionally oriented and isotropic when the angular distribution of the cross-sections is uniform.

Subregion Generalized Variational Principles for Elastic Thick Plates

Long Yu-qiu

1983, 4(2): 165-172.

Abstract(1914) PDF(582)

Abstract:
In this paper, the subregion generalized variational principle for elastic thick plates is proposed. Its main points may be stated as follows:1. Each subregion may be assigned arbitrarily as a potential region or complementary region. The subregion variational principles of potential energy, complementary energy and mixed energy represent three special forms of this principle.2. The number of independent variational variables in each sub-region may be assigned arbitrarily. Any one of the subregions may be assigned as a one-variable-region, two-variable-region or three-variable-region.3. The conjunction conditions of displacements and stresses on each interline of neighbouring subregions may be relaxed. On the basis of this principle the finite element analysis of non-conforming elements for thick plates can be formulated.Different finite element models for thick plates can be obtained by different applications of this principle. In particular,the subregion mixed variational principle for thick plates may be applied to formulating the subregion mixed finite element method for thick plates.

Vector Analysis of Spatial Mechanisms——(Ⅰ) Configuration Analysis of Spatial Mechanisms by Means of the Method of Vector Decomposition

Yu Xin

1983, 4(2): 173-178.

Abstract(1888) PDF(442)

Abstract:
In the present series of four papers we offer various vectorial methods for analysing the configuration,kinematics and dynamics of spatial mechanisms, In the process,we obtain vectorial solutions by means of pure vectorial procedures. In part(Ⅰ)we treat the configurations of the R-G-G-R and the H-R-G-R mechanisms by the method of vector decomposition.

The Relation between the Markov Process Theory and Kolmogoroff’s Theory of Turbulence and the Extension of Kolmogoroff’s Laws—(Ⅱ) The Extension of Koimogoroff’s“2/3 Law” and “-5/3 Law”

Yue Zeng-yuan, Zhang Bin

1983, 4(2): 179-191.

Abstract(3232) PDF(546)

Abstract:
Based on the physical analysis in part 1 of this paper, we quantitative relation between the Markov process theory of two-particle's in a dispersion turbulence of very large Reynolds number and the Kolmogoroff's theory. In terms of this relation and the results of two-particle s dispersion,we shall obtain the structure functions, the correlation functions and the energy spectrum, which are applicable not only to the inertial subrange,but also to the whole range of the wave number Iess than that in the inertial subrange. The Kolmogoroff's "2/3 law" and "-5/3 Law" are the aspmptotic case of the present result for small r(or large k), Thus, the present result is an extension of Kolmogoroff's laws.

Propagation of a Long Wave in a Curved Duct——(Ⅱ) Applications of Matched Expansion Method to Long Wave Propagation through a Hole with Variable Cross Section

Fang Guang-xiong, Gu Sheng-shi

1983, 4(2): 193-208.

Abstract(1990) PDF(596)

Abstract:
Only the case in which the parameter ε=ka≤1 is considered in this paper, where k is the wave number and a is the characteristic radius of the cross-section of the hole. The general asymptotic expansion of the complex velocity potential of a long wave propagating in the hole with variable cross-section is obtained by regular perturbation: The methods of matched asymptotic expansion are employed to calculate the reflection coefficients, scattering coefficients and radiation coefficients at the open ends of the hole when a long wave propagates through it, which may be open at both ends or only at one end. Three examples of different kinds of holes are given to show the way to solve such two-dimensional or three-dimensional problems.

The Spatial Static and Dynamic Analyses of Tall Building Structures

Wang Xing-xiang, Che Wei-yi, Yu Yong-sheng

1983, 4(2): 209-219.

Abstract(2195) PDF(571)

Abstract:
In this paper,the static and dynamic analyses of tall buildings are studied as the space system by the application of "JIGFEX-Ⅱ" programme.Suggestions have been made for the calculation of earthquake forces and torques.Besides that,a practical approach is proposed for realizing the analysis of complicated structures on minicomputers or on medium sized computers.Reference data and charts from practical engineering applications are also presented.

Studies of Nonlinear Theories for Thin Shells

Wu Shen-rong

1983, 4(2): 221-231.

Abstract(2434) PDF(945)

Abstract:
In order to formulate the equations for the study here, the Fourier expansions upon the system of orthonormal polynomials areused.It may be considerably convenient to obtain the expressions of displacements as well as stresses directly from the solutions.Based on the principle of virtual work the equilibrium equations of various orders are formulated. In particular, the system of third-order is given in detail, thus providing the reference for accuracy analysis of lower-order equations. A theorem about the differentiation of Legendre series term by term is proved as the basis of mathematical analysis. Therefore the functions used are specified and the analysis rendered is no longer a formal one.The analysis will show that the Kirchhoff-Love's theory is merely of the first-order and the theory which includes the transverse deformation but keeps the normal straight is essentially of the first-order, too.

The Linear Growth of the Density Wave in Galaxies

Tang Ze-mei

1983, 4(2): 233-237.

Abstract(1989) PDF(464)

Abstract:
The aim of this article is to study the linear growth of the density wave in galaxies by means of numerically resolving unsteady, two-dimensional hydrodynamic equations coupled with Poisson equation under the condition that the local asymptotic solution of linear density wave is given.as an initial value. The results show that the perturbed peak density of linear density wave grows to the same order as the basic state density during merely tens of million years,the spiral pattern emerging which has barred structure in its inner region. The angular velocity of the spiral pattern and the growth rate of perturbed density vary gradually with changes in spatial place and time. The approximate property of quasistationary spiral structure hypothesis is discussed in this paper.

Damping Feature of Dynamic Problem and “Velocity” Finite Element Method

Yang Zhen-rong, Yan Rong-1ing

1983, 4(2): 239-246.

Abstract(2131) PDF(857)

Abstract:
In order to reduce the amount of computation and storage of dynamic problem, this paper based on [16] is intended to analyse damping feature, and study the relations among the damping and the material as well as frequencies and the size of mesh of finite element, besides giving the estimation theorem of maximum norm and a corollary.Examples have been analyzed numerically with limited norm. The influence of damping on the dynamic tense stress is assumed to be limited, in value, but it can be botli positive and negative.This means that to regard damping as always tending to decrease the stress incline is incorrect.The feature of "velocity" finite element method is summarized further in the paper. Some necessary numericsl results are given in the appendix.

The Generalized Galerkin’s Equations of the Finite Element, the Boundary Variational Equations and the Boundary Integral Equations

Niu Xiang-jun

1983, 4(2): 247-254.

Abstract(2000) PDF(485)

Abstract:
Based on [1], we have further applied the variational principle of the variable boundary to investigate the discretization analysis of the solid system and derived the generalized Ga-lerkin's equations of the finite element, the boundary variational equations and the boundary integral equations.These e-quations indicate that the unknown functions of the solid system must satisfy the conditions in the element S_a or on theboundaries Γ_a.These equations are applied to establishing the discretization equations in order to obtain the numerical solution of the unknown functions. At a time these equations can be used as the basis for the simplified calculation in various corresponding cases.In this paper, the results of boundary integral equations show that the calculation Γ_a of integration is not accurate along the surface of interior element S_a by J-integral suggested by Rice [2].

A Two-Dimensional Lagrangian Difference Method of Movable Meshes——A Computing Method for Compressible Radiational Hydrodynamic Equations with Several Materials

Chang Qian-shun

1983, 4(2): 255-267.

Abstract(2151) PDF(551)

Abstract:
In this paper we give the method of movable meshes uhichdonot need to follow surface of cell. This method merely computes the motion of the center of cell. The surface of cell is formed by the center of cell by rule. The Lagrantfian mesh thus formed allows slip between cells and the variety of neighborhood. In order to imporve the stability of the method and avoid unnecessary slips of cell, we give two velocities for the cell.

Mechanical Conditions for Fracture of Anisotropic Bodies and Their Geometry in Stress Space

Han Yu-ying

1983, 4(2): 269-276.

Abstract(1823) PDF(539)

Abstract:
The conditions for fracture of anisotropic bodies and their geometry in stress space are proposed in this paper.The analytical formulae expressing the fracture conditions are established front the viewpoint of energy theory for crack propagation.In stress space the limiting surface corresponding to the fracture conditions derived for anisotropic solids is quadratic. It is an ellipsoid in case the mean stress is greater than zero and it is hyperboloid in case the mean stress is smaller than zero.The conclusions formed by the author in the present paper have certain generality. Some results obtained by predecessors appear to be special cases with respect to the present theory.

The Hamilton’s Principle of Nonholonomic Variable Mass Systems

Ge Zheng-ming, Cheng Yi-he

1983, 4(2): 277-288.

Abstract(2175) PDF(981)

Abstract:
The Hamilton's principle is extended to the most general,non-holonomic variable mass systems. The Hamilton's principle of nonholonomic variable mass systems is obtained and is illustrated with examples.