THE IMPROVEMENT OF THE GENERAL EXPRESSION FOR THE STRESS FUNCTION φ OF THE TWO-DIMENSIONAL PROBLEM

In this paper,it is pointed that the general expression for the stress functionφ_0 of theplane problem in polar coordinates is incomplete.The problems of the curved bar with anarbitrary distributive load at the boundries can’t be solved by this stress function.For thisreason,we suggest two new stress functions and put them into the general expression.Then,the problems of the curved bar applied with an arbitrary distributive load at r=a,bboundaries can be solved.This is a new stress function including geometric boundaryconstants.     

ANALYSIS OF SYMMETRIC LAMINATED RECTANGULAR PLATES IN PLANE STRESS

Symmetric laminated plates used usually are anisotropic plates. Based on the fundamental equation for anisotropic rectangular plates in plane stress problem, a general analytical solution is established accurately by method of stress function. Therefore the general formula of stress and displacement in plane is given. The integral constants in general formula can be determined by boundary conditions. This general solution is composed of solutions made by trigonometric function and hyperbolic function, which can satisfy the problem of arbitrary boundary conditions along four edges, and the algebraic polynomial solutions which can satisfy the problem of boundary conditions at four corners. Consequently this general solution can be used to solve the plane stress problem with arbitrary boundary conditions. For example, a symmetric laminated square plate acted with uniform normal load, tangential load and nonuniform normal load on four edges is calculated and analyzed.     

A SOLUTION OF COSINE PRESSURES ON A HOLLOW CYLINDER AND THE LIMIT WHEN k→0

A new stress function is found in this paper and then the problems of cosine pressures on a hollow cylinder are solved with the new stress function,which provides the basis for the solution of the problems of space symmetrical deformation of a hollow cylinder.When the pressures do not vary in the axial direction,that is,when k→0,the lame formulae can be deduced.     

A SOLUTION OF COSINE PRESSURES ON A HOLLOW CYLINDER AND THE LIMIT WHEN k→0

A new stress function is found in this paper and then the problems of cosine pressures on a hollow cylinder are solved with the new stress function, which provides the basis for the solution of the problems of space symmetrical deformation of a hollow cylinder. When the pressures do not vary in the axial direction, that is, when k→0 the lame formulae can be deduced.     

The general solution for axial symmetrical bending of nonhomogeneous circular plates resting on an elastic foundation

In this paper,a new method,the exact analytic method,is presented on the basis of stepreduction method.By this method,the general solution for the bending of nonhomogenouscircular plates and circular plates with a circular hole at the center resting,on an elastfcfoundation is obtained under arbitrary axial symmetrical loads and boundary conditions.The uniform convergence of the solution is proved.This general solution can also be applieddirectly to the bending of circular plates without elastic foundation.Finally,it is onlynecessary to solve a set of binary linear algebraic equation.Numerical examples are givenat the end of this paper which indicate satisfactory results of stress resultants anddisplacements can be obtained by the present method.     

Circular shallow spherical shells with central circular hole under simultaneous actions of arbitrary unsteady temperature field and arbitrary dynamic normal load

In this paper, under assumption that tempeature is linearly distributed along the thickness of the shell, we deal with problems as indicated in the title and obtain general solutions of them which are expressed in analytic form.In the first part, we investigate free vibration of circular shallow spherical shells with circular holes at the center under usual arbitrary boundary conditions. As an example, we calculate fundamental natural frequency of a circular shallow spherical shell whose edge is fixed (m=0). Results we get are expressed in analytic form and check well with E. Reissner’s [1]. Method for calculating frequency equation is recently suggested by Chien Wei-zang and is to be introduced in appendix 3.In the second part, we investigate forced vibration of shells as indicated in the title under arbitrary harmonic temperature field and arbitrary harmonic dynamic normal load.In the third part, we investigate forced vibration of the above mentioned shells with initial conditions under arbitrary unsteady temperature field and arbitrary normal load.In appendix 1 and 2, we discuss how to express displacement boundary conditions with stress function and boundary conditions in the case m=1.     

FREE VIBRATION OF ANISOTROPIC RECTANGULAR PLATES BY GENERAL ANALYTICAL METHOD

According to the differential equation for transverse displacement function of anisotropic rectangular thin plates in free vibration, a general analytical solution is established. This general solution, composed of the composite solutions of trigonometric function and hyperbolic function, can satisfy the problem of arbitrary boundary conditions along four edges. The algebraic polynomial with double sine series solutions can also satisfy the problem of boundary conditions at four corners. Consequently, this general solution can be used to solve the vibration problem of anisotropic rectangular plates with arbitrary boundaries accurately. The integral constants can be determined by boundary conditions of four edges and four corners. Each natural frequency and vibration mode can be solved by the determinate of coefficient matrix from the homogeneous linear algebraic equations equal to zero. For example, a composite symmetric angle ply laminated plate with four edges clamped has been calculated and discussed.     

General solution for large deflection problems of nonhomogeneous circular plates on elastic foundation

In this paper, a new method is presented based on [1]. It can be used to solve the arbitrary nonlinear system of differential equations with variable coefficients. By this method, the general solution for large deformation of nonhomogeneous circular plates resting on an elastic foundation is derived. The convergence of the solution is proved. Finally, it is only necessary to solve a set of nonlinear algebraic equations with three unknowns. The solution obtained by the present method has large convergence range and the computation is simpler and more rapid than other numerical methods.Numerical examples given at the end of this paper indicate that satisfactory results of stress resullants and displacements can be obtained by the present method. The correctness of the theory in this paper is, confirmed.     

Solution of the plane stress problems of strain-hardening materials described by power-law using the complex pseudo-stress function

In the present paper,the compatibility equation for the plane stress problems of power-law materials is transformed into a biharmonic equation by introducing the so-calledcomplex pseudo-stress function,which makes it possible to solve the elastic-plastic planestress problems of strain hardening materials described by power-law using the complexvariable function method like that in the linear elasticity theory.By using this generalmethod,the close-formed analytical solutions for the stress,strain and displacementcomponents of the plane stress problems’of power-law materials is deduced in the paper,which can also be used to solve the elasto-plastic plane stress problems of strain-hardeningmaterials other than that described by power-law.As an example,the problem of a power-law material infinite plate containing a circular hole under uniaxial tension is solved byusing this method,the results of which are compared with those of a known asymptoticanalytical solution obtained by the perturbation method.     

Applying the generalized step-function to solve the bending problems of nonuniform beams with variable cross section

The bending problems of nonuniform beams with variable cross-section can be approximated by that of a step beam under sectionally uniform load (including both concentrated forces and couples). In this paper, the concept of Heaviside function {x-a}⁰ will be generalized, and a new function {x-a}⁰, n=0,1,2…,will be defined, which may be named as a generalized step function. The rules of operation will also be given to {x-a}ⁿ{x-b}⁰. The reciprocal of the flexural of rigidity 1/EJ and the bending moment M(x) can all be expressed in terms of {x-a}ⁿ,and substituted into the differential equation of the elastic curve of the beam respectively. Thus we may establish a set of unified method to solve various types of bending problems of straight beams. The general solution of the deflection equation will be given.     

Some anomalies of the curved bar problem in classical elasticity

The classical formulation of the homogeneous problem of a curved bar loaded only by and end force involves the assumption of an appropriate stress function with four arbitrary constants and the determination of these constants from the boundary conditions. Since there are five boundary conditions, four on the curved edge and one at the end, the solution is only possible because the coefficient matrix of the resulting algebraic equations is singular. This in turn means that certain inhomogeneous problems in which the curved edges are loaded by sinusoidally varying tractions cannot be solved using apparently appropriate stress functions.Additional stress functions which resolve this difficulty are introduced and an example problem is solved, which exhibits qualitatively different behavior from that in more general cases of loading. The problem is then reconsidered as a limiting case of sinusoidal loading of arbitrary wavelength. It is shown that the solution of the latter problem appears to become unbounded as the special case is approached, but that when the end conditions have been correctly satisfied by superposing an appropriate multiple of the end-loaded solution, the limit can be approached regularly and the correct special solution is recovered. The limiting process reveals a general procedure for determining the additional stress functions required for the special case.Similar relationships between homogeneous and inhomogeneous solutions for other common geometries are discussed.     

The curved bar subjected to an arbitrarily distributed loading: Another paradox in the two-dimensional theory of elasticity

The problem of the curved bar subjected to an arbitrarily distributed loading on the surfacesr=a andr=b is solved by using the method of complex functions and expanding the boundary conditions atr=a andr=b into Fourier series. Then another paradox in the two-dimensional theory of elasticity is discovered, i. e., the classical solution becomes infinite when the curved bar is subjected to a uniform loading or when the angle included between the two ends of the curved bar 2 is equal to 2 and the curved bar is subjected to a sine or cosine loading. In this paper the paradox is resolved successfully and the solutions for the paradox are obtained. Moreover, the modified classical solution which remains bounded as 2 approaches 2 is provided.     

弹性曲杆的稳定性问题

本文给出空间任一曲杆在弯扭联合作用下的稳定性问题的一般讨论,并且给出了曲杆某一平衡状态的扰动量所满足的方程组(28)—(36),在适当的边界条件下,这些扰动量的非零解对应于临界状态。文末用这组方程具体讨论了五个实际例子,这些例子有些结果是新的,有些是用新的方法去处理老问题。     

On the Generalized Plane Stress Problem,the Gregory Decomposition and the Filon Mean Method

In the present paper a stress general solution is obtained for the generalized plane stress problem with planar body forces, and it is demonstrated that only body force of biharmonic type ensures the compatibility with the generalized plane stress assumptions (σ ₃₃=0). Inspired by the Filon perspective of average values, two more generalized plane stress problems with weak assumptions on the out-of-plane stress averages (\(\bar{\sigma}_{33}=0\) or \(\nabla^{2} \bar{\sigma}_{33}=0\)) are studied, and the averages of the corresponding stress fields are expressed by the Airy stress functions. The authors also provide an alternative proof of the Gregory decomposition theory.     

The load-bearing capacity of a continuous-flow squeeze film of liquid

A new form of squeeze film system is described in which the movement of one plate towards the other is simulated by the continuous volume generation of liquid over the plate area. The liquid exudes from 1580 holes distributed uniformly over the lower plate surface. An advantage of the system is that there are no moving parts, but it is important to evaluate the device using Newtonian liquids in order to compare the load bearing capacity with that predicted by equations developed for orthodox squeeze film systems. Liquid maldistribution is shown to be a problem which may be solved in various ways, one of which is to ensure that the pressure drop through the plate is high relative to that in the squeeze film.Results obtained using Newtonian liquids make satisfactory comparison with theoretical predictions, though liquid inertia probably makes a lower contribution to load bearing than is the case for an orthodox squeeze film. Liquid maldistribution is allowed for on a theoretical basis or corrected by the use of a distributor plate placed below the perforated surface.Preliminary tests using viscoelastic solutions (based on polyacrylamide of high molecular weight) suggest that the load bearing properties of the squeeze film are significantly enhanced. A load 600 per cent greater than the theoretical load is obtained in one case, the suggestion being made that this is due to stress of viscoelastic origin.Nomenclature D Exit diameter of holes in spinnerette - F ₁ to F ₆ Vertical force on top plate due to flow in squeeze film, defined by (1), (8), (11), (12), (13) and (14) respectively - h Plate separation - h _L Distance of distributor plate from lower surface of spinnerette (function of r) - I ₀ Modified Bessel function of first kind, order 0 - I ₁ Modified Bessel function of first kind, order 1 - K ₀ Modified Bessel function of second kind, order 0 - L Length of hole, based on diameter D, giving same pressure drop as actual spinnerette holes - dm/dt Mass flowrate of liquid - N Total number of holes in spinnerette (1580) - p Isotropic pressure in squeeze film - P _RES Isotropic pressure in reservoir behind lower plate of spinnerette - p–P _RES - (dp/dr)_s Pressure gradient in squeeze film - (dp/dr)_L Pressure gradient in lower film below spinnerette when distributor plate is used - Q Total liquid volume flowrate - q _s Volume flowrate through squeeze film at radius r - q _L Volume flowrate through lower film at radius r - r radial coordinate - R radius of upper disc - - v Velocity of upper disc relative to lower one (simulated by Q/R ² in continuous flow system) - V _R Average radial liquid velocity at radius R - V _S Liquid exit velocity from single hole - V _r V V _z Point velocity components in r, and z directions respectively - z Axial coordinate - Parameter in (8) (3ND ⁴/32LR ² h ³) - Viscosity of liquid - Density of liquid - _rz Shear stress     

The determination of a general time creep compliance relation of linear viscoelastic materials under constant load and its extension to nonlinear viscoelastic behavior for the Burger model

Linear viscoelastic materials yield a creep function which only depends on time if creep experiments are performed under constant stress ₀. In practice, this condition is very difficult to realize, and as a consequence, the experiments are performed under constant force. For small strains the difference between the conditions of constant stress and constant force is negligible. Otherwise, the decrease in cross-section has to be taken into account and leads to increasing stress in the course of time for creep experiments under constant load. The Boltzmann superposition principle is solved under the condition of constant load and for strains . The creep complicance C(t; ₀) defined by the ratio becomes, in principle, dependent on the initial stress ₀. As a consequence, a set of creep compliance curves cannot be approximated with a simple parameter fit. Already the application of the solution on the Burger model yields a creep compliance curve with all three creep ranges. Furthermore, the mathematical structure of the time creep compliance relation of the Burger model allows nonlinear viscoelastic extension via the introduction of the yield strength _max and a nonlinearity parameter n _l . The creep behavior of PBT and PC can be described in the range of long times up to initial stresses ₀, being 75% for PBT and 60% for PC of the yield stress _max with only two or one free fit parameter, respectively.     

Flow of a simple non-newtonian fluid past a sphere

Summary Creeping flow past a sphere is solved for a limiting case of fluid behaviour: an abrupt change in viscosity.List of Symbols d _ij Component of rate-of-deformation tensor - F _d Drag force exerted on sphere by fluid - G ^(d) Coefficients in expression for _ij in terms of d _ij - G _YOJK ^(d) Coefficients in power series representing G ^(d) - R Radius of sphere - r Spherical coordinate - V Velocity of fluid very far from sphere - v _i Component of the velocity vector - x Dimensionless radial distance, r/R - x _i Rectangular Cartesian coordinate - Dimensionless quantity defined by (26) - ^(d) Potential defined by (7) - Value of x denoting border between Regions 1 and 2 as a function of - ₁, ₂ Lower and upper limiting viscosities defined by (10) - Spherical coordinate - * Value of for which =1 - Value of denoting border between regions 1 and 2 as a function of x - Newtonian viscosity - _ij Component of the stress tensor - Spherical coordinate - ₁, ₂ Stream functions defined by (12) and (14) - Second and third invariants of the stress tensor and of the rate-of-deformation tensor, defined by (3)     

Dirichlet problem of higher order elliptic equation with perturbation both in differential operator and in boundary

This paper is the continuation of article [7]. It gives further results about the asymptotic expression for the solution of higher order elliptic equation in the case of boundary perturbation combined with operator perturbation. When unperturbed problemA ₀ is not on the spectrum, the asymptotic expression for the solution of perturbation problemA may be expanded with respect to the small parameter . WhileA ₀ is on the spectrum, the asymptotic expression of the solution contains negative powers of the small parameter . The approximation of arbitrary order to the solution is considered and the recursive formula for the general term and the estimation of remainder term are given.     

An approach to the estimation of polymer melt elasticity

An effective method has been proposed to estimate the primary normalstress difference versus shear rate curves at temperatures relevant to the processing conditions only from the knowledge of the melt flow index, the molecular-weight distribution and the glass transition temperature of the polymer. The method involves the use of a unified curve obtained by coalescing the elastic response curves of various grades in terms of the modified normal-stress coefficient ₁ (MFI)² and a modified shear rate . Unified curves have been reported for low density polyethylene, high density polyethylene, polypropylene and nylon.Nomenclature C ₁ constant in eq. (4) - J _e steady state compliance (cm²/dyne) - proportionality constant in eq. (6) - L load (kg) - L ₁ load (kg) at ASTM test conditions - L ₂ load (kg) at required conditions - MFI melt flow index (gm/10 min) - number average molecular weight - weight average molecular weight - z-average molecular weight - (z+1)-average molecular weight - n slope of the shear stress vs. shear curve on a log-log scale - N ₁ primary normal-stress difference (dynes/cm²) - Q molecular weight distribution expressed as - T ₁is> temperature (K) at condition 1 - T ₂is> temperature (K) at condition 2 - T _g glass transition temperature (K) - T _s standard reference temperature equal toT _g + 50 K - shear rate (s^–1) - ₀ zero-shear viscosity (poise) - apparent viscosity (poise) - density (g/cm³) - ₁₂ shear stress (dynes/cm²) - ₁₁– ₂₂ primary normal-stress difference (dynes/cm²) - _1,0 zero shear rate primary normal-stress coefficient (dynes/cm² · sec²) - ₁ primary normal-stress coefficient (dynes/cm² · sec²) - ₂ secondary normal-stress coefficient (dynes/cm² · sec²) NCL-Communication No. 3106     

A DEVELOPMENT OF BEM TO TWO DIMENSIONAL ELASTICITY

This paper presents a new boundary integral equatiòn for two-dimensional elasticity with the stress component σ_(ij)t_it_j as one of the boundary values,where t_i is the direction cosine of the tangent on the boundary.This form of BEM hasan advantage in that the stress component σ_(ij)t_it_j on the boundary can be calculateddirectly from the numerical solution.The present formulation for plane problems usestwo kernels,the one is logarithmically singular and the other is 1/r singular.Theeffectiveness of the approach is also discussed through test examples.