Showing papers in "Applied Mathematics and Mechanics-english Edition in 1985"

The solution of large deflection problem of thin circular plate by the method of composite expansion

Abstract: In this paper, the method of composite expansion in perturbation theory is used for the solution of large deflection problem of thin circular plate. In this method, the outer field solution and the inner boundary layer solution are combined together to satisfy all the boundary conditions. In this paper, Hencky's membrane solution is used for the first approximation in outer field solution, and then the second approximate solution is obtained. The inner boundary layer solution is found on the bases of boundary layer coordinate. In this paper, the reciprocal ratio of maximum deflection and thickness of the plate is used as the small parameter. The results of this paper improves quite a bit in comparison with the results obtained in 1948 by Chien Wei-zang.

Involutory transformations and variational principles with multi-varoables in thin plate bending problems

Abstract: In this paper, the generalized variational principles of plate bending problems are established from their minimum potential energy principle and minimum complementary energy principle through the elimination of their constraints by means of the method of Lagrange multipliers. The involutory transformations are also introduced in order to reduce the order of differentiations for the variables in the variation. Furthermore, these involutory transformations become in fact the additional constraints in the variation, and additional Lagrange multipliers may be used in order to remove these additional constraints. Thus, various multi-variable variational principles are obtained for the plate bending problems. However, it is observed that, not all the constraints of variation can be removed simply by the ordinary method of linear Lagrange multipliers. In such cases, the method of high-order Lagrange multipliers are used to remove those constraints left over by ordinary linear multiplier method. And consequently, some functionals of more general forms are obtained for the generalized variational principles of plate bending problems.

The relation of von Kármán equation for elastic large deflection problem and Schrödinger equation for quantum eigenalues problem

Abstract: In this paper the solutions of von Karman for elastic large deflection problem are classified as the several solutions of Schrodinger equation for quantum eigenvalues problem, and we present the transform for elastic large deflection problem from non-linear equation into linear equation. Thus, we create favourable conditions of the adoption of converse scattering method and Backlund transformation. We also discuss the large deflection problem of long and narrow plate. We can study the non-linear transition of elastic thin plate by furnished method from this paper.

Nonlinear kelvin-helmholtz instability

Abstract: A non-linear analysis is presented with derivative expansion method for the interfacial stability of a liquid film adjacent to a subsonic gas flow under the influence of body force and surface tension. The non-linear Rayleigh-Taylor instability is included as a special case. The gas and liquid are considered to be inviscid. Though Nayfeh (1971) gave consideration into this case, there is something omitted in his third-order equation (e.g. p. 213 expression (229)) and inconsistent with his solutions (e.g. the first-order solution (2.31) does not satisfy his initial conditions (2.20)). Besides, in this paper, our solution near the cut-off wave number is extended to include the case of travelling waves and a new conclusion is drawn.

Asymptotic analysis of strongly nonlinear oscillators

Abstract: In this paper, an asymptotic method is presented for the analysis of a class of strongly nonlinear autonomous oscillators. The equations governing the amplitude and phase factor are obtained, and the amplitude and stability of the corresponding limit cycles are determined.

On the sensitive dynamical system and the transition from the apparently deterministic process to the completely random process

Abstract: The detailed analysis of the dynamical process of coin tossing is made. Through calculations, it is illustrated how and why the result is extremely sensitive to the initial conditions. It is also shown that, as the initial height of the mass center of the coin increases, the final configuration, i.e. “head” or “tail”, becomes more and more sensitive to the initial parameters (the initial velocity angular velocity, and the initial orientation), the coefficient of the air drag, and the energy absorption factor of the surface on which the coin bounces. If we keep the “head” upward initially but allow a small range for the change of some other initial parameters, the frequency that the final configuration is “head”, would be 1 if the initial height h of the mass center is sufficiently small, and would be clo to 1/2 if h is sufficiently large. An interesting question is how this frequency changes continuously from 1 to 1/2 as h increases. Detailed calculations show that such a “transition” is very similar to the transition from laminar to turbulent flows. A basic difference between the “transition stage” and the “completely random stage” is indicated: In the “completely random stage”, the deterministic process of the individual case is extremely sensitive to the initial conditions and the dynamical parameters, out the statistical properties of the ensemble are insensitive to the small changes of the initial conditions and the dynamical parameters. On the contrary, in the “transition stage”, both the deterministic process of the individual case and the statistical properties of the ensemble are sensitive to the initial conditions and the dynamical parameters. The mechanism for this feature of the “transition stage” is the existence of the “long-train structure” in the parameter space. The illuminations of this analysis on some other random phenomena are discussed.

Perfectly plastic stress field at a stationary crack tip

Abstract: Under the hypothesis that all the perfectly plastic stress components at a crack tip are the functions of θ only, making use of yield conditions and equilibrium equations, we derive the generally analytical expressions of the perfectly plastic stress field at a crack tip. Applying these generally analytical expressions to the concrete cracks, the analytical expressions of perfectly plastic stress fields at the tips of Mode I. Mode II, Mode III and Mixed Mode I–II cracks are obtained.

Criteria for the existence of solutions to random integral and differential equations

Abstract: In[1], we proved a general random fixed point theorem and gave some applications In this paper, we shall give further applications of the theorem We first obtain a random Darbo's fixed point theorem, using the theorem, we give the criteria for the existence of solutions under compactness hypotheses to nonlinear random volterra integral equations and the Cauchy problem of nonlinear random differential equations Our theorems improve and generalize some main results of Lakshmikanthem[2], Vaughn[3,4] as well as De Blasi and Myjak[5]

On the problems of buckling of an annular thin plate

Abstract: In this paper, problems of buckling of an annular thin plate under the action of in-plane pressure and transverse load are studied by using the method of multiple scales. We obtain N-order uniformly valid asymptotic expansion of the solution. In the latter part of this paper we discuss a particular example, and calculate the critical value of in-plane pressure. We see that the asymptotic expansion obtained by the multiple scales is completely consistent with that of the exact solution.

A semi-analytical method for structure-internal liquid interaction problems

Abstract: A composite structure-internal liquid element is proposed in this paper Through the use of analytical functions in two orthogonal directions, the three-dimensional problem of coupled structure-internal liquid vibration is reduced to a one-dimensional one, resulting in a drastic reduction in computing efforts Cylindrical and conical frastum composite elements are proposed to suit different problems, and examples are presented to demonstrate the accuracy of the method

The stokes flow from half-space into semi-infinite circular cylinder

Abstract: The infinite-series solutions for the creeping motion of a viscous incompressible fluid from half-space into semi-infinite circular cylinder are presented. The results show that inside the cylinder beyond a distance equal to 0.5 times the radius of the tube from the pore opening, the deviation of the velocity profile from Poiseuille flow is less than 1%. The inlet length in this case is comparable to that computed for a finite circular cylinder pore by Dagan et al.[1]. In the half-space outside the cylinder pore region, the flow is strongly affected by the wall. Beyond one radius of the tube from the orifice, the solutions match almost exactly the flow through an orifice of zero thickness given by Sampson[2]. The relationship between the pressure drop and the volumetric flow rate is also computed in the present paper for the semi-infinite tube.

The plane piston problem in a weak gravitational field

Abstract: We analyze a gasdynamical process in the stellar atmosphere that is driven by a “piston” moving with constant velocity in a weak gravitational field. Ahead of the piston, the gas is compressed, and this compressed gas uses part of its internal energy and somewhere its kinetic energy to overcome the applied gravity.

Finite element approximations with multiple sets of functions and quasi-conforming elements for plate bending problems

Abstract: Continuing refs. [1.2]. we try to establish here the mathematical foundation of quasiconforming elements suggested by Prof. Tang Limin and his colleagues for plate bending problems ([3.4]). The main theme used in this paper is the finite element approximations with multiple sets of functions.

Solution of sector plate by fourier-bessel series

Abstract: In this paper a solution of deflection in the form of Fourier-Bessel double series with supplementary terms is proposed to analyse bending and vibration problems of thin elastic sector plate with various edge conditions This solution is suitable to a wider range, convenient for calculation and it is in an analytical form As computational examples, the distribution curves of deflection and bending moment of plates with various sector angles, simply supported or clamped along the radial edges under uniform or concentrated load are obtained and the result are compared with the numerical results of related references Thus the range of application of the Fourier series method with supplementary terms is extended Frequencies and nodal lines in free vibration of plates with various sector angles simply supported along the radial edges are also given in this paper

Elastoplastic analysis of cylindrically orthotropic composite thick-walled tube

Abstract: In the present paper, Tsai-Hill Yield Criterion in a state of plane strain is derived for incompressible and compressible materials respectively. the elastoplastic stress and displacement fields of cylindrically orthotropic composite thick-walled tube under uniform radial pressure are studied. The formulas of the yield pressure, limit pressure and region of shakedown are obtained.

Application of the reciprocal theorem for calculating the natural frequencies of rectangular elastic thin plates

Abstract: This paper further extends the applications of the reciprocal theorem to calculating the natural frequencies of rectangular elastic thin plates on the basis of [1]. Applying the presented method, there is no need to solve governing differential equations, it is only necessary to solve a simple integral equation after using the reciprocal theorem between the basic system and the actual system.

Solution of bending of cantilever rectangular plates under uniform surface-load by the method of two-direction trigonometric series

Abstract: The bending of a cantilever rectangular plate is a very complicated problem in the theory of plates. For a long time, there have been only approximate solutions for this problem by energy methods and numerical methods.

A discussion on “The large deflection problem of circular thin plate with variable thickness under uniformly distributed loads”

Abstract: “The large deflection problem of circular thin plate with variable thickness under uniformlydistributed loads”has been solved by using the small parameter method and modified iterationmethod jointly in the ref.[1].The solution of ref.[1]and its special results are correct.But,theprocedure in the...

Forced vibrations of elastic shallow shell due to the moving mass

Abstract: This paper discusses the forced vibrations of the elastic shallow shell due to the moving mass by means of the variational method. In the text mentioned above a series of problems such as the forced vibrations, resonance conditions and critical speed and so forth.

Equi-strength design for statically indeterminate beams

Abstract: In this paper a method for equi-strength design of statically indeterminate beams ispresented,based on the principle of minimum complementary energy.And an analyticalexpression is derived for the stiffness variation of single or multi-span beams under theapplication of arbitrarily distributed lo...

One-dimensional nonuniform temperature flow with heat transfer by convection

Abstract: In ref. [1], V. E. Najenov studied the conditions that when the viscosity of the liquid is an exponential function of temperature, the pipe flow, having steady heat transfer, is one dimensional and with nonuniform temperature. For plane canal and circular pipe he still studied the velocity and the temperature fields.

The mixed mode brittle fracture criteria in siliding mode fracture

Abstract: It is well-known that the present mixed mode brittle fracture criteria are all the opening mode fracture criterion. We consider that mixed mode brittle fracture of sliding mode fracture exists too. Hence we propose three criteria of mixed mode brittle fracture of sliding mode fracture;: the radial shearing stress criterion, the maximum shearing stress criterion and the distortional strain-energy-density criterion. Thus, we can overall explain the phenomena of brittle fracture in the structural elements with cracks.

The tshb technique for material testing at high rates of strain

Abstract: The material testing technique of Torsional Split Hopkinson Bar (TSHB) is investigated in this paper. It can solve nearly all the problems of Split Hopkinson Pressure Bar (SHPB). Furthermore, accurate experimental results can be obtained in large deformation condition. In this paper some dynamic stress-strain curves of some engineering materials are also given which are obtained from a TSHB apparatus made by ourselves.