Showing papers in "Applied Mathematics and Mechanics-english Edition in 1993"

Saint-venant’s torsion problem for a composite circular cylinder with aninternal edge crack

Abstract: In this paper the writer uses Muskhelishvili single-layer potential function solution and single crack solution for the torsion problem of a circular cylinder to discuss the torsion problem of a composite cylinder with an internal crack, and the problem is reduced to a set of mixed-type integral equation with generalized Cauchy-kernel. Then, by using the integration formula of Gauss-Jacobi, the numerical method is established and several numerical examples are calculated. The torsional rigidity and the stress intensity factors are obtained. The results of these examples fit the results obtained by the previous papers better.

Consolidation theory of unsaturated soil based on the theory of mixture (ii)

Abstract: The present paper uses the mathematics model for consolidation of unsaturated soil developed in ref. [1] to solve boundary value problems. The analytical solutions for one-dimensional consolidation problem are gained by making use of Laplace transform and finite Fourier transform. The displacement and the pore water pressure as well as the pore gas pressure are found from governing equations simultaneously. The theoretical formulae of coefficient and degree of consolidation are also given in the paper. With the help of the method of Galerkin Weighted Residuals, the finite element equations for two-dimensional consolidation problem are derived. A FORTRAN program named CSU8 using 8-node isoparameter element is designed. A plane strain consolidation problem is solved using the program, and some distinguishing features on consolidation of unsaturated soil and certain peculiarities on numerical analysis are revealed. These achievements make it convenient to apply the theory proposed by the author in engineering practice.

Generalized variational inequalities and generalized quasi-variational inequalities

Abstract: Under much weaker hypotheses and in a more general setting some existence theorems of solutions to generalized variational inequalities, generalized quasi-variational inequalities and minimax inequalities are established. The results presented in this paper generalize the corresponding results of [3–13] to the noncompact case, thus improving these results.

Nonlinear bending of simply supported rectangular sandwich plates

Abstract: In this paper, fundamental equations and boundary conditions of the nonlinear bending theory for a rectangular sandwich plate with a soft core are derived by means of the method of calculus of variations. Then the nonlinear bending for a simply supported rectangular sandwich plate under the uniform lateral load is investigated by use of the perturbation method and a quite accurate analytic solution is obtained.

Correction for housner’s equation of bending vibration of a pipe line containing flowing fluid

Abstract: This paper points out that Housner's equation of bending vibration of a pipe line containing flowing fluid is approximate and makes correction to it. An exact form of the vibration equation is given.

The precise and new analysis for a mode III growing crack in an elastic-perfectly plastic solid

Abstract: The near crack line field analysis method has been used to investigate into Mode III quasistatically propagating crack in an elastic-perfectly plastic material. The significance of this paper is that the usual small scale yielding theory has been broken through. By obtaining the general solutions of the stresses and the displacement rate of the near crack line plastic region, and by matching the general solutions with the precise elastic fields (not the usual elastic K-dominant fields) at the elastic-plastic boundary, the precise and new solutions of the stress and deformation fields, the size of the plastic region and the unit normal vector of the elastic-plastic boundary have been obtained near the crack line. The solutions of this paper are sufficiently precise near the crack line region because the roughly qualitative assumptions of the small scale yielding theory have not been used and no other roughly qualitative assumptions have been taken, either. The analysis of this paper shows that the assumingly “steady-state case” for stable crack growth, which has been discussed attentively in previous works, do not exist, and the plastic strains near the crack line do not have singularities. Two most important cases for stable crack growth have been discussed.

Existence of closed orbits for a differential equation model concerning multi-molecule reactions

Abstract: In this paper, the existence of closed orbits for the biochemical reaction model $$i\dot a = \hat H(t)a$$ is discussed, where n is a positive integer and x≥0, y≥0, a>0. We also point out that the equation has no closed orbits or has stable limit cycles arising from Hopf bifurcations under a certain condition of a.

The solution of integral equations with strongly singular kernels applied to the torsion of cracked circular cylinder

Abstract: In this paper, the functions of warping displacement interruption defined on the crack lines are taken for the fundamental unknown functions. The torsion problem of cracked circular cylinder is reduced to solving a system of integral equations with strongly singular kernels. Using the numerical method of these equations, the torsional rigidities and the stress intensity factors are calculated to solve the torsion problem of circular cylinder with star-type and other different types of cracks. The numerical results are satisfactory.

Numerical solutions for singularly perturbed semi-linear parabolic equation

Abstract: In this paper, we discuss singularly perturbed semi-linear parabolic equations for one dimension and two dimension, we find numerical solutions by using both the line-method and the exact difference scheme on a special non-uniform discretization mesh. The uniform convergence in e of the first order accuracy is obtained.

Integral theory for the dynamics of nonlinear nonholonomic system in noninertial reference frames

Abstract: This paper establishes the integral theory for the dynamics of nonlinear nonholonomic system in noninertial reference frame. Firstly, based on the Routh equation of the relative motion of nonlinear nonholonomic system gives the first integral of the system. Secondly, by using cyclic integral or energy integral reduces the order of the equation and obtains generalized Routh equation and Whittaker equation respectively. Thirdly, derives canonical equation and variation equation and by using the first integral constructs integral invariant. And then, establishes the basic integral variants and the integral invariant of Poincare-Cartan type. Finally, we give a series of deductions.

Generalized dynamic model for multibodies manipulator

Abstract: In this paper the general dynamical equations were given for multibodies manipulator. The system is a topologic tree structure consisting of arbitrary number of rigid bodies. The hinges allow the rotational and/or translational motion. In consideration of influence of friction the dynamic equations are established by means of Newton-Euler’s method. Further, the equations are separated by way of constructing the distribution matrices and a group of force and motion equations are obtained.

Some multiplicity results for an elastic beam equation at resonance

Abstract: This paper deals with multiplicity results for nonlinear elastic equations of the type $$\begin{gathered} - d^4 u/dx^4 + \pi ^4 u + g(x,u) = e(x) (0< x< 1) \hfill \\ u(0) = u(1) = u''(0) = u''(1) = 0 \hfill \\ \end{gathered}$$ where g:[0, 1]×R→R satisfies Caratheodory condition e∈L2[0,1]. The solvability of this problem has been studied by several authors, but there isn't any multiplicity result unitl now to the author's knowledge. By combining the Lyapunov-Schmidt procedure with the technique of connected set, we establish several multiplicity results under suitable condition.

Similarity solutions of round jets and plumes

Abstract: The k-e turbulence model, considering the effect of buoyancy on turbulent kinetic energy and its dissipation rate, is adopted to present a mathematical model for round plumes and jets. There are similarity solutions in the uniform environment. Taking into account the conservation of momentum and heat flux. Finite Analytic Method is applied to obtain the similarity functions of velocity, temperature and turbulent kinetic energy. The agreement between the calculated and experimental data is good.

Hamiltonian system and simpletic geometry in mechanics of materials (iii)-flexure and free vibration of plates

Abstract: The methodology presented in Part 1 is employed to deal with flexure and free vibration of anisotropic plates.

Control of transpiration cooling system and its characteristics

Abstract: This paper gives the mathematical model of three-dimensional transpiration cooling control system in heat shield. Generally, it's a non-linear control system on variable-domain mixed up with both distributed and concentrated parameters. This paper points out that the thermal and ablative problems of the shield can be solved solely with the coolant flow only under one-dimensional incompressible or steady condition. In regard to the surface ablating problem of the thermal shield, the control schemes of the system, including its simplified condition and the characteristics of one-dimensional point control is suggested here. Solutions of the equilibrium state are given with or without phase change as far as the coolant is concerned.

A theoretical model of the coherent structure in the wall region of a turbulent boundary layer

Abstract: In this paper, the formation of the coherent structures in the wall region of a turbulent boundary layer was studied, using the nonlinear theory of the hydrodynamic stability. The spanwise and streamwise wavelengths of the most amplified unstable wave obtained by this study were found in good agreement with the experiments, which makes the distinct feature of this study in the present paper, as the basis of the stability analysis, a more rational velocity profile has been used, which is different from that of the turbulent mean flow. And also, the new nonlinear theory was used. The result is useful in understanding of the quasi-periodicity of the coherent structure in the turbulent boundary layer.

Singularity under a concentrated force in elasticity

Abstract: We first discuss singularity problem of a sort of partial differential equation involving δ function. Using this result we, then have the answer to various singularity problems in elasticity due to the presentation of a concentrated force. Lastly corresponding conclusions in vibration problem are drawn.

Nonlinear bending of the shallow spherical shells with variable thickness under axisymmetrical loads

Abstract: Based on the differential equation of the nonlinear bending of shallow spherical shells with variable thickness under axisymmetrical loads, this paper studies the numerical solution of the nonlinear differential equation by means of interpolating matrix method. The analysis of the results indicates that the suggested method is easy to implement and obtains the same high accuracy for both the displacements and the internal forces.

Boundary layer development of pulsatile blood flow in a tapered vessel

Abstract: Assuming that the tapered angle is small, the problems of developing flow under unsteady oscillatory condition are studied in this paper. The formula of velocity distribution is obtained. The analyses for the results show that the blood flow in a converging tapered vessel remains a developing flow throughout the length, and the effects of tapered angle on the developing flow are increased with the increment of the tapered angle.

Vibration problems of flexible circular plates with initial deflection

Abstract: In this paper, the differential equations of flexible circular plates with initial deflection are derived. The stability of motion is investigated in phase plane. The periodical solutions of nonlinear vibration for circular plates with initial deflection are obtained by use of Galerkin method and Lindstedt-Poincare perturbation method. The effect of initial deflection on the dynamic behavior of the flexible plates are also discussed.

Common fixed point theorems for (sub) compatible and set-valued generalized nonexpansive mappings in complete convex metric spaces

Abstract: In this paper, using some conditions of (sub) compatibility between a set-valued mapping and a single-valued mapping, we establish a necessary and sufficient condition and a sufficient condition for set-valued generalized nonexpansive mappings to have a unique common fixed point in complete convex metric spaces. The results improve, extend and develop the main results in [2–7].

Boundary value problems for second order delay differential equations

Abstract: In this paper, using a fixed point principle and existence principle given in [1], we study the boundary value problems for second order differential equations. Some new existence results are obtained.

Analytical solution of flow of second-order non-newtonian fluids through annular pipes

Abstract: This paper presents an analytical solution to the unsteady flow of the second-order non-Newtonian fluids by the use of intergral transformation method. Based on the numerical results, the effect of non-Newtonian coefficient Hc and other parameters on the flow are analysed. It is shown that the annular flow has a shorter characteristic time than the general pipe flow while the correspondent velocity, average velocity have a smaller value for a given Hc. Else, when radii ratio keeps unchanged, the shear stress of inner wall of annular flow will change with the inner radius compared with the general pipe flow and is always smaller than that of the outer wall.

Four-point explicit difference schemes for the dispersive equation

Abstract: A class of three-level explicit difference schemes for the dispersive equation u1=auzzz are established. These schemes have higher stability and involve four mesh points at the middle level. Their local truncation errors are O(τ+h) and stability conditions are from |R|≤0.25 to |R|≤10, where |R|=|a|τ/h3, which, is much better than |R|≤0.25[1].

A modification of Taylor-Galerkin Finite Element Method and its application

Abstract: Two basic hypothesises of Taylor-Galerkin Finite Element Method are studied in this paper. One of them which is unreasonable is redefined. The only hypothesis becomes the standpoint of Generalized Finite Element. We use this idea to analysis stream function-vorticity equations with Modified Taylor-Galerkin Finite Element Method, and give the two-step solving method, which makes the solving process more reasonable than ever before. Several computational examples reveal that the results of this new method are satisfied.

Chaotic behavior of a nonlinear oscillator

Abstract: Behavior of bifurcation and chaos in a forced oscillator $$\bar x_1 + \bar \delta \dot x_1 + \omega _0^2 x_1 - \bar \beta x_1^2 = \bar f\cos \omega \tau$$ containing a square nonlinear term is investigated by using Mel'nikov method and digital computer simulations.

The analyses of the three-dimensional stress structure near the crack tip of mode i ct specimens in elasticplastic state(i)—the analyses of constraint parameters and fracture parameters

Abstract: In the present paper, three dimensional analyses of some general constraint parameters and fracture parameters near the crack tip. of Mode I CT specimens in two different thicknesses are carried out by employing ADINA program. The results reveal that the constraints along the thickness direction are obviously separated into two parts: the keeping similar high constraint field (Z1) and rapid reducing constraints one (Z2). The two fields are experimentally confiremed to correspond to the smooth region and the shear lip on the fracture face respectively. So the three dimensional stress structure of Mode I specimens can be derived through discussing the two fields respectively. The distribution of the Crack Tip Opening Displacement (CTOD) along the thickness direction and the three dimensional distribution of the void growth ratio (Vg) near the crack tip are also obtained. The two fracture parameters are in similar trends along the thickness direction, and both of them can reflect the effect of thickness and that of the loading level to a certain degree.

Problem of hydrodynamic pressure on suddenly starting vessel

Abstract: In this paper, Lagrangian method is applied to discuss the problem of the hydrodynamic pressure on a suddenly starting vessel. The free surface profile and the coefficients of the hydrodynamic pressure on the vessel wall are obtained. And it is verified that the singularity of the pressure near the free surface is only logarithmic.

Incompatible curved quadrilateral plate bending element with 12-degrees of freedom

Abstract: This paper presents a new curved quadrilateral plate element with 12-degree freedom by the exact element method[1]. The method can be used to arbitrary non-positive and positive definite partial differential equations without variation principle. Using this method, the compatibility conditions between element can be treated very easily, if displacements and stress resultants are continuous at nodes between elements. The displacements and stress resultants obtained by the present method can converge to exact solution and have the second order convergence speed. Numerical examples are given at the end of this paper, which show the excellent precision and efficiency of the new element.

An iteration algorithm for solving postbuckling equilibrium path of simply-supported rectangular plates under biaxial compression

Abstract: In this paper, on the basis of von Karman large deflection equations and its double trigonometric series solution, we present a simple, fast and effective iteration algorithm for solving simply-supported rectangular plate subjected to biaxial compression.