A CLOSED SYSTEM OF EQUATIONS FOR DENSE TWO-PHASE FLOW AND EXPRESSIONS OF SHEARING STRESS OF DISPERSED PHA’E AT A WALL

Inthis paper, each of the two phases in dense two-phase flow is considered as continuous medium and the fundamental equations for two-phase flow arc described in Eulerian form. The generalized constitutive relation of the Bingham fluid is applied to the dispersed phase with the analysis oj physical mechanism of dense two-phase flow. The shearing stress of dispersed phase at a wall is used to give a boundary condition. Then a mathematical model for dense two-phase flow is obtained. In addition, the expressions of shearing stress of dispersed phase at a wall is derived according to the fundamental model of the friclional collision between dispersed-plutse particles and the wall.

Flow stress equation for multipass hot-rolling of aluminum alloys

A series of simple axisymmetric compression tests were carried out on the computer servo controlled Gleeble 1 500 machine when strain rates ranged between 0.05 25 s -1 and deformation temperature 300 500 ℃. The results show that flow stress is related to the Zener Hollonom parameter Z and strain, as well as the static recrystallization fraction between passes during multipass hot deformation of 5182 aluminum alloy. Hence, a modified exponential flow stress equation was presented by considering the values of ln A and β as functions of strain, and by using the uniform softening method and incorporating the static recrystallization fraction between passes to consider the effects of residual strain during multipass hot rolling of 5182 aluminum alloy. The validity of the equation was examined by a typical non isothermal multipass deformation test.

DEVELOPMENT OF CONSTITUTE EQUATIONS FOR Ti-6Al-4V ALLOY UNDER HOT-WORKING CONDITION

The deformaton behavior of Ti - 6Al - 4V alloy under hot - working condition has been studied by compression testing in the temperature range 750 - 950℃ and strain rate range 0.05 - 15s -1. The flow stress decreases with the increase of temperature and with the decrease of strain rate. After a steep initial strain hardening, a flow softening occurs. This softening is mainly ascribed to the temperature rise and dynamic recmptallisation.By a simple extension, a classical sinushyperbolic constitutive equation can be used to describe the flow behavior of Ti - 6Al - 4V alloy. flow stress is described as a function of strain, strain rate and temperature. The parameters Q, n andaare the same at differ- ent deformation conditions, and A is a funciton of strain.

Derivation of a second-order model for Reynolds stress using renormalization group analysis and the two-scale expansion technique

With the two-scale expansion technique proposed by Yoshizawa,the turbulent fluctuating field is expanded around the isotropic field.At a low-order two-scale expansion,applying the mode coupling approximation in the Yakhot-Orszag renormalization group method to analyze the fluctuating field,the Reynolds-average terms in the Reynolds stress transport equation,such as the convective term,the pressure-gradient-velocity correlation term and the dissipation term,are modeled.Two numerical examples：turbulent flow past a backward-facing step and the fully developed flow in a rotating channel,are presented for testing the efficiency of the proposed second-order model.For these two numerical examples,the proposed model performs as well as the Gibson-Launder （GL） model,giving better prediction than the standard k-ε model,especially in the abilities to calculate the secondary flow in the backward-facing step flow and to capture the asymmetric turbulent structure caused by frame rotation.

Evaluation of strongly singular domain integrals for internal stresses in functionally graded materials analyses using RIBEM

An accurate evaluation of strongly singular domain integral appearing in the stress representation formula is a crucial problem in the stress analysis of functionally graded materials using boundary element method.To solve this problem,a singularity separation technique is presented in the paper to split the singular integral into regular and singular parts by subtracting and adding a singular term.The singular domain integral is transformed into a boundary integral using the radial integration method.Analytical expressions of the radial integrals are obtained for two commonly used shear moduli varying with spatial coordinates.The regular domain integral,after expressing the displacements in terms of the radial basis functions,is also transformed to the boundary using the radial integration method.Finally,a boundary element method without internal cells is established for computing the stresses at internal nodes of the functionally graded materials with varying shear modulus.

Airy, Beltrami, Maxwell, Einstein and Lanczos Potentials Revisited

The purpose of this paper is to revisit the well known potentials, also called stress functions, needed in order to study the parametrizations of the stress equations, respectively provided by G.B. Airy (1863) for 2-dimensional elasticity, then by E. Beltrami (1892), J.C. Maxwell (1870) for 3-dimensional elasticity, finally by A. Einstein (1915) for 4-dimensional elasticity, both with a variational procedure introduced by C. Lanczos (1949, 1962) in order to relate potentials to Lagrange multipliers. Using the methods of Algebraic Analysis, namely mixing differential geometry with homological algebra and combining the double duality test involved with the Spencer cohomology, we shall be able to extend these results to an arbitrary situation with an arbitrary dimension n. We shall also explain why double duality is perfectly adapted to variational calculus with differential constraints as a way to eliminate the corresponding Lagrange multipliers. For example, the canonical parametrization of the stress equations is just described by the formal adjoint of the components of the linearized Riemann tensor considered as a linear second order differential operator but the minimum number of potentials needed is equal to for any minimal parametrization, the Einstein parametrization being “in between” with potentials. We provide all the above results without even using indices for writing down explicit formulas in the way it is done in any textbook today, but it could be strictly impossible to obtain them without using the above methods. We also revisit the possibility (Maxwell equations of electromagnetism) or the impossibility (Einstein equations of gravitation) to obtain canonical or minimal parametrizations for various equations of physics. It is nevertheless important to notice that, when n and the algorithms presented are known, most of the calculations can be achieved by using computers for the corresponding symbolic computations. Finally, though the paper is mathematically oriented as it aims providing new insights towards the mathematical foundations of general relativity, it is written in a rather self-contained way.

MATHEMATIC MODEL AND ANALYTIC SOLUTION FOR CYLINDER SUBJECT TO UNEVEN PRESSURES

According to the inverse solution of elasticity mechanics, a stress function is constructed which meets the space biharmonic equation, this stress functions is about cubic function pressure on the inner and outer surfaces of cylinder. When borderline condition that is predigested according to the Saint-Venant＇s theory is joined, an equation suit is constructed which meets both the biharmonic equations and the boundary conditions. Furthermore, its analytic solution is deduced with Matlab. When this theory is applied to hydraulic bulging rollers, the experimental results inosculate with the theoretic calculation. Simultaneously, the limit along the axis invariable direction is given and the famous Lame solution can be induced from this limit. The above work paves the way for mathematic model building of hollow cylinder and for the analytic solution of hollow cvlinder with randomly uneven pressure.

Steady rotation of a composite sphere in a concentric spherical cavity

The problem of steady rotation of a composite sphere located at the centre of a spherical container has been investigated. A composite particle referred to in this paper is a spherical solid core covered with a permeable spherical shell. The Brinkman＇s model for the flow inside the compos- ite sphere and the Stokes equation for the flow in the spheri- cal container were used to study the motion. The torque ex- perienced by the porous spherical particle in the presence of cavity is obtained. The wall correction factor is calculated. In the limiting cases, the analytical solution describing the torque for a porous sphere and for a solid sphere in an un- bounded medium are obtained from the present analysis.

Analysis of mode Ⅲ crack perpendicular to the interface between two dissimilar strips

The present work is concerned with the problem of mode Ⅲ crack perpendicular to the interface of a bi-strip composite. One of these strips is made of a functionally graded material and the other of an isotropic material, which contains an edge crack perpendicular to and terminating at the interface. Fourier transforms and asymptotic analysis are employed to reduce the problem to a singular integral equation which is numerically solved using Gauss-Chebyshev quadrature formulae. Furthermore, a parametric study is carried out to investigate the effects of elastic and geometric characteristics of the composite on the values of stress intensity factor.

On the Relationship of the Discrete Model of the Nuclei of Linear and Planar Defects and the Continuum Models of Defects in Crystalline Materials

A physical and mathematical model of the transition from a discrete model of linear and flat defects nuclei to continuum models of defects such as dislocations and disclinations and their combinations is presented, where the tensors of energy-momentum and angular momentum of an alternating field are considered, for which the type and structure of the Maxwell stress tensor σif αβ are given and the corresponding angular momentum tensor, using the dynamic equation for the evolution of internal stresses and the correlation between the stresses σif αβ in the defect core and the elastic stresses σelik in its environment, obtains elastic displacement and deformation fields identical to these fields from Burgers and Frank vectors of continuous models. The spectral density of the autocorrelation functions of the velocity of photoelectrons Ψβ⊥(β) and cations , which transforms into linear spectra as T → 0, is considered reflecting the existence of threshold values of oscillation and rotations currents of photoelectrons and cations at all stages of plastic deformation and fracture. The features of the process of sliding linear defects in metals are disclosed.

ELASTOSTATIC SOLUTOINS FOR SEMI-INFINITE ORTHOTROPIC CANTILEVERED STRIPS

The elastostatic solutions of semi-infinite orthotropic cantilevered strips with traction free edges and loading at infinity are governed by the differential equation with Based on the work of [10] for case,.this paper completes the case for isotropic materials and the case for orthotropic materials. The solutions of the above problems have important application in the properly formulated boundary conditions of plate theories for prescribed displacement edge data.

An Analytical Study on Distribution of Staticaly Determinate Streses in Particulate Half Space

General solution of stresses solved from the two dimensiona l system of equilibrium equations in Cartesian coordinates is characterized by the presence of two families of characteristic lines along which initial stresses and discontinuities in them are transmitted intact far down to infinity.This is against our intuition and not verifiable by experimental findings. For the fundamental case of infinite uniform pressure on the upper surface,a comparison between solutions from equilibrium equations in Cartesian coordinates and from those in polar coordinates is carried out in details.The semi infinite characteristic lines in the former are bent up to exponential spirals with both ends on the upper surface in the latter.Thus,the transmission pattern from solution in polar coordinates comes closer to actual situation.However,in polar reference frame,the solution for distribution of stresses in particulate half space under surface strip pressure or so can then only be obtained from boundary value problem of second order partial differential equation.

Hot Deformation Behavior and Flow Stress Prediction of Ultra Purified 17% Cr Ferritic Stainless Steel Stabilized with Nb and Ti

The hot deformation behavior of ultra purified 17% Cr ferritic stainless steel stabilized with Nb and Ti was investigated using axisymmetric hot compression tests on a thermomechanical simulator.The deformation was carried out at the temperatures ranging from 700 to 1 100℃ and strain rates from 1to 10s-1.The microstructure was investigated using electron backscattering diffraction.The effects of temperature and strain rate on deformation behavior were represented by Zener-Hollomon parameter in an exponent type equation.The effect of strain was incorporated in the constitutive equation by establishing polynomial relationship between the material constants and strain.A sixth order polynomial was suitable to represent the effect of strain.The modified constitutive equation considering the effect of strain was developed and could predict the flow stress throughout the deformation conditions except at800℃in 1s-1 and at 700℃in 5and 10s-1.Losing the reliability of the modified constitutive equation was possibly ascribed to the increase in average Taylor factor at 800℃in 1s-1 and the increase in temperature at 700℃in 5and10s-1 during hot deformation.The optimum window for improving product quality of the ferritic stainless steels was identified as hot rolling at a low finisher entry temperature of 700℃,which can be achieved in practical production.

Effect of squeeze casting process on microstructures and flow stress behavior of Al-17.5Si-4Cu-0.5Mg alloy

The effects of squeeze casting process on microstructure and flow stress behavior of Al-17.5Si-4Cu-0.5Mg alloy were investigated and the hot-compression tests of gravity casting and squeeze casting alloy were carried out at 350-500°C and 0.001-5s-1.The results show that microstructures of Al-17.5Si-4Cu-0.5Mg alloys were obviously improved by squeeze casting.Due to the decrease of coarse primary Si particles,softα-Al dendrite as well as the fine microstructures appeared,and the mechanical properties of squeeze casting alloys were improved.However,when the strain rate rises or the deformation temperature decreases,the flow stress increases and it was proved that the alloy is a positive strain rate sensitive material.It was deduced that compared with the gravity casting alloy,squeeze casting alloy（solidified at 632 MPa）is more difficult to deform since the flow stress of squeeze casting alloy is higher than that of gravity casting alloy when the deformation temperature exceeds 400°C.Flow stress behavior of Al-17.5Si-4Cu-0.5Mg alloy can be described by a hyperbolic sine form with Zener-Hollomon parameter,and the average hot deformation activation energy Q of gravity casting alloy and squeeze casting alloy is 278.97 and 308.77kJ/mol,respectively.

NUMERICAL SOLUTIONS FOR A NEARLY CIRCULAR CRACK WITH DEVELOPING CUSPS UNDER SHEAR LOADING

In this paper, we study the behavior of the solution at the crack edges for a nearly circular crack with developing cusps subject to shear loading. The problem of finding the resulting force can be written in the form of a hypersingular integral equation. The equation is then trans-formed into a similar equation over a circular region using conformal mapping. The equation is solved numerically for the unknown coefficients, which will later be used in finding the stress intensity factors. The sliding and tearing mode stress intensity factors are evaluated for cracks and displayed graphically. Our results seem to agree with the existing asymptotic solution.