Applied Mathematics and Mechanics (English Edition)

ANALYSIS OF THIN PARALLELOGRAM PLATES BENDING BY SPLINE-FINITE-STRIP METHOD

陈铭俊;谭国焕;张佑啟

1984, 5(6): 1727-1735.

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Spline finite strip has been successfully applied in solving right plates and shells by Cheung et al in 1982. In this paper, the method is extended to the analysis of parallelogram plate. This extension still retains the banded nature of the spline finite strip and only small amount of extra computing effort is required. Furthermore, the discretisation error of the above method is established theoretically as a general case for the spline finite strip method.

CLASSIFICATION OF VARIATIONAL PRINCIPLES IN ELASTICITY

钱伟长

1984, 5(6): 1737-1743.

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In this paper, variational principels in elasticity are classified according to the differences in the constraints used in these principles. It is shown in a previous paper[4] that the stress-strain relations are the constraint conditions in all these variational principles, and cannot be removed by the method of linear Lagrange multiplier. The other possible constraints are four of them: (1) equations of equilibrium, (2) strain-displacement relations, (3) boundary conditions of given external forces and (4) boundary conditions of given boundary displacements. In variational principles of elasticity, some of them have only one kind of such constraints, some have two kinds or three kinds of constraints and at the most four kinds of constraints. Thus, we have altogether 15 kinds of possible variational principles. However, for every possible variational principle, either the strain energy density or the complementary energy density may be used. Hence, there are altogether 30 classes of functional of variational principles in elasticity. In this paper, all these functionals are tabulated in detail.

ON CHARACTERIZATIONS OF STRUCTURAL STABILITY

廖山涛

1984, 5(6): 1745-1750.

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This note takes a sketch of a proof of a characterization theorem for diffeomorphisms on a compact 3-dimensional smooth manifold to be structurally stable.

NEW THEORY FOR EQUATIONS OF NON-FUCHSIAN TYPE——REPRESENTATION THEOREM OF TREE SERIES SOLUTION (Ⅱ)

董明德

1984, 5(6): 1751-1767.

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Our main result consists in proving the representation theorem. Irregular integral is a new type of analytic function, represented by a compound Taylor-Fourier tree series, in which each coefficient of the Fourier series is a Taylor series, while the Taylor coefficients are tree series in terms of equations parameters, higher order correction terms to each coefficient having tree structure with inexhaustible proliferation.The solution obtained is proved to be convergent absolutely and uniformly in the region defined by coefficient functions of the original equation, provided the structure parameter is less than unity. Direct substitution shows that our tree series solution satisfies the equation explicity generation by generation.As compared with classical theory our method not only furnishes explicit expression of irregular integral, leading to the solution of Poincare problem, but also provides possibility of extending the scope of investigation for analytic theory to equations with various kinds of singularities in a unifying way.Exact explicit analytic expression for irregular integrals can be obtained by means of correspondence principle.It is not difficult to prove the convergence of the tree series solution obtained. Direct substitution shows it satisfies the equation.The tree series is automorphic, which agrees completely with Poincaré’s conjecture.

THE LINEAR APPROXIMATION OF THE LINE CONTINUOUS DISTRIBUTION METHOD OF SINGULARITIES IN CREEPING MOTION

吴望一;何青

1984, 5(6): 1769-1776.

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The lineal approximation of the line continuous distribution method of singularities is proposed to treat the creeping motion of the arbitrary prolate axisymmetrical body. The analytic expressions in closed form for the flow field are obtained. The numerical results for the prolate spheroid and Cassini oval demonstrate that the convergence and the accuracy of the proposed method are better than the constant density approximation. Furthermore, it can be applied to greater slender ratio. In this paper the example is yielded to show that the linear approximation of the singularities for the density on the partitioned segments can be utilized to consider the creeping motion of the arbitrary pointed prolaue axisymmetrical body.

THE BÄCKLUND TRANSFORMATION AND NONLINEAR SUPERPOSITION FORMULA OF SOLUTIONS FOR THE LIOUVILLE’S EQUATION IN HIGHER DIMENSIONS

黄迅成

1984, 5(6): 1777-1782.

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In this paper, we show that Backlund transformation derived by Leibbrandt et al. for the Liouville’s equation in three spatial dimensions, ▽²a=expa, ▽²=-∂_x²+∂_y²+∂_z² can be decomposed into several Backlund transformations for the same equation in two spatial dimensions, moreover, the superposition formula which is derived from this transformation is actually invalid, thus the discussions based on that formula is incorrect as well. We also considered some results about the Liouville’s equation in N spatial dimensions.

INTERPOLATION WITH BOUNDARY CONDITION USING BIVARIATE QUADRIC SPLINES CONNECTED WITH TRIANGULAR PARTITION

李稚娴

1984, 5(6): 1783-1789.

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In this paper, we study the relations between the exis-tance uniqueness of interpolation with boundary condition using bivariate quadric splines connected with triangular partition and positions of interpolating points. After proving the existence uniqueness of interpolation problems at center, vertex and partial center, we give the construction methods for these three interpolating functions.

THE SOLUTION OF DEFLECTION OF ELASTIC THIN PLATE BY THE JOINT ACTION OF DYNAMICAL LATERAL PRESSURE, FORCE IN CENTRAL SURFACE AND EXTERNAL FIELD ON THE ELASTIC BASE

沈惠川

1984, 5(6): 1791-1801.

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In this paper the Euler equation of the deflection of elastic thin plate is reduced to the equation with Schröd-inger form by the principle of quantum electro-dynamics. Then we can obtain the general solution of deflection of elastic thin bending plate by the joint action of dynamical lateral pressure, force in central surface and external field on the elastic base.

ONE-DIMENSIONAL PISTON PROCESS IN STRONG GRAVITATIONAL FIELD

唐泽眉

1984, 5(6): 1803-1811.

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In this paper, the dynamic process driven by one-dimensional piston in strong gravitational field was studied on the Cartesian, cylinderical and spherical coordinates. The gasdynamic equations were numerically solved by the characteristic method. The solution which satisfies the velocity condition at piston and the boundary conditions connect the flow region and the quiet region is obtained. The present paper analyses especially the influence of coordinate systems on the field of compressible flow, uniform flow and rarefaction flow region, the shock velocity and the temperature distribution at the piston.

DECOMPOSITION OF SYMMETRIC TENSOR AND ITS APPLICATION

王敏中

1984, 5(6): 1813-1816.

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In this paper any symmetric tensor is decomposed into the sum of two tensors. One of them is a "type of stress" tensor, and another is a "type of strain" tensor. The inner product space of symmetric tensor is decomposed into the sum of two orthogonal subspaces. The geometric meaning of several principles in the theory of elasticity, is given.

VARIATIONAL PRINCIPLES OF ELASTIC-VISCOUS DYNAMICS IN LAPLACE TRANSFORMATION FORM, F. E. M. FORMULA-TION AND NUMERICAL METHOD

金问鲁

1984, 5(6): 1817-1823.

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The author gives variational principles of elastic-viscous dynamics in spectral resolving form^[1], it will be extended to Laplace transformation form in this paper, mixed variational principle of shell dynamics and variational principle of dynamics of elastic-viscous-porous media are concerned, for the latter, F. E. M. formulation has been worked out.Variational principles in Laplace transformation form nave concise forms, for the sake of utilizing F. E. M. conveniently it is necessary to find values of preliminary time function at some instants, when values of Laplace transformation at some paints are known, but there are no efficient methods till now. In this paper, a numerical method for finding discrete values of preliminary function is presented, from numerical example we see such a method is efficient.By combining both methods stated above, variational principles in Laplace transformation form and numerical method, a quite wide district of solid dynamic problems can be solved by ths aid of digital computers.

THE CIRCUMFERENTIAL STRESS-STRAIN PRODUCT CRITERION OF MIXED MODE BRITTLE FRACTURE

林拜松

1984, 5(6): 1825-1829.

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This paper presents a new criterion of mixed mode brittle fracture, i.e. the circumferential stress-strain product criterion. This criterion is shown to be in good agreement with known experimental data.

ANALYSIS OF POSTBUCKLING STRENGTH OF STRINGER STIFFENED CYLINDRICAL SHELLS UNDER AXIAL COMPRESSION

邵文蛟

1984, 5(6): 1831-1851.

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An analysis of the postbuckling strength of stringer stiffened cylindrical shells, subject to axial compression is described. The method used in this paper is based on plastic analysis extending Murray’s method which was used to analyse postbuckling behaviour of stiffened plates loaded axially and in bending. The mechanism tripping of stringer and crumpling of shell plate is described based on how the test specimens are deformed after buckling.Finally the theoretical analyses are compared with the experimental results of steel specimens. The theoretical results coincide quite well with the experimental data. It should therefore be possible to use the method described here to analyse postbuckling strength of stringer stiffened cylindrical shells and to estimate energy absorption capabilities in relation to collision studies.

THE TEMPERATURE FIELDS AROUND THE TIP OF A FAST RUNNING CRACK

汪懋骅

1984, 5(6): 1853-1858.

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When a crack is running, the temperature rise is a quite important actual problem, which not only depends on some material constants, but also the propagation velocity and the distribution of the heat resource density. In this paper, the shape of plastic zone around the crack tip and the density of heat resource have been discussed and the model of the temperature fields has been proposed. The numerical results with PMMA are given and comparedwith other theories and experimental results.

SPLINE SECTORIAL ELEMENTS

袁驷

1984, 5(6): 1859-1866.

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In this paper, the quadratic and cubic splines local interpolation on a sectorial element in polar coordinates is discussed and a class of spline sectorial elements for analyses of plane and thin problems are presented. A reasonable treatment of the assumed displacement fields for elements with nodes at the origin (r=0) has been made so that the elements can not only characterize the geometrical properties at the origin but also remove the singularity of strains and stresses there. Some numerical examples are given to show the efficiency of the proposed elements.

THE DESCRIPTION OF THE CHARACTERISTICS OF THE STRUCTURE AND THE QUANTITY IN FIXED PANSYSTEMS THEOREMS

李贵华

1984, 5(6): 1867-1874.

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Pansymmetry is the abstract of symmetry, stability and other concepts in physics and so on. Fixed pansystems theorems portray a typical pansymmetry of systemic structure. The present paper complements and extends the work in [1]-[3] concerned in fixed pansystems theorems. It gives in finite case the structural character of fixed subsets; the criterion of existence of the least fixed subset, and the numbering formula of fixed subsets and minimal fixed subsets.

NOTE ON THE CRITICAL VARIATIONAL STATE IN ELASTICITY THEORY

刘成群

1984, 5(6): 1875-1881.

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Recently Prof. Chien Wei-zang^[1] pointed out that in certain cases, by means of ordinary Lagrange multiplier method, some of 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 can not be eliminated by means of simple Lagrange multiplier method. This is indeed the case when one tries to eliminate the constraint condition of strain-stress relation in variational principle of minimum complementary energy by the method 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 type of independent variables, stresses and displacements, exist in the new functional. Hence Prof. Chien introduced the high-order Lagrang multiplier method bu adding the quadratic terms.

to original functions.The purpose of this paper is to show that by adding

to original functionals one can also eliminate the constraint condition of strain-stress by the high-order Lagrange multiplier method. With this methods, we find more general form of functional of generalized variational principle ever known to us from Hellinger-Reissner principle. In particular, this more general form of functional can be reduced into all known functions of existing generalized variational principles in elasticity. Similarly, we can also find more general form of functional from H. C. Hu-Washizu prin-ciple^[4,5].

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