An upper bound analysis for the reshaping of thick tubes with experimental verification

Reshaping process based on the cold roll forming technology is a process in which noncircular pipes can be produced from circular pipes. A generalized upper bound solution for the deformation of thick tubes between four flat rolls has been formulated. The deforming zone has been divided into two areas namely the “contact region” and the “free region”. Owing to the different physical character of these deforming zones two different kinds of admissible velocity fields have been obtained for each of the deforming zones. To formulate the stream lines and stream surfaces in the deforming zones use has been made of the Bezier curves which have been utilized to define stream lines in such a way that could be manipulated easily to obtain the optimum shape for the upper bound solution to be optimized. This new formulation was used to predict the upper bound on power. Using the theory presented here, the influence of process parameters such as radius of rolls, initial pipe dimensions; amount of roll gap reduction and the roll speed on the final rolled product were investigated. In order to verify the theoretical results, an experimental rolling rig was designed and built which comprised of four flat rolls. Using this rig Al 6101 round pipes were rolled and the variations in the above-mentioned parameters were investigated. Quantities such as energy, wall thickness, and corner radius of the tube were observed and measured. Comparison of theoretical and experimental results showed good agreement and hence demonstrated the capabilities of the new formulation presented here.     

Axisymmetric response of circular plates with piezoelectric layers: an exact solution

Electromechanical responses of symmetric circular laminates consisting of piezoelectric layers are studied, and the influence of surface and interlayer electrodes are involved. The laminates are traction-free on the top and bottom surfaces, but may be subjected to external forces at the lateral edge and to voltages applied across certain layers. Under axisymmetric deformation conditions, an approximate model which employs Kirchhoff hypothesis and incorporates the charge equation of electrostatic is established. Then, a closed-form three-dimensional solution of the laminates is generated in a very straightforward manner by the solution of the approximate model. The three-dimensional solution fulfills all field equations and interface or surface conditions as well as the specified electric edge boundary conditions; the only restriction is that the mechanical edge boundary conditions are satisfied in an average manner, rather than point by point. Thus, according to Saint-Venant's principle the proposed solution is exact in the interior region of the laminates.     

用保角映射法求解多层套扁挤压筒内的应力

生产大型铝型材用扁挤压筒内孔近似椭圆,属多连通域,难以采用常规的单连体平面问题的复变函数保角映射法,应力求解困难。结合单连体复变函数法和留数定理给出了受内压扁挤压筒的应力求解公式和例子,并与有限元模拟结果进行了比较分析。结果表明理论解和数值解在全域范围内基本一致,从而解决了保角映射法应用于扁挤压筒实际工程应力计算的主要障碍,为模具强度设计提供了依据。     

An analysis of hemispherical punch stretching by the energy method

An approach using the energy method in which the total deforming region is divided into several sections of different geometric shapes is suggested for the analysis of axisymmetric sheet metal forming with friction boundary condition. The corresponding solutions are found through optimization of the total energy dissipation with respect to some parameters assumed in the velocity field as well as in the corresponding geometric profile. Computations are carried out for hemispherical punch stretching of normal anisotropic work-hardening materials for several lubrication conditions. The punch load vs stroke relation, geometric configuration and strain components are determined from the computation. The comparison of the computed results with finite element solutions and corresponding experiments shows good agreements of solutions in load vs stroke, deformed profiles and strain distribution for various lubrication conditions. It is thus shown that the present simple approach can be effectively employed for the analysis of axisymmetric sheet metal working processes.     

Effect of Magnetic Field in Lubrication of Synovial Joints

A two region flow model has been developed in this paper in the presence of external magnetic field for the better understanding of synovial joint lubrication mechanism. The model consists of two parallel porous cartilageous surfaces separated by a thin film of non newtonian lubricant representing the synovial fluid which is assumed to behave like a paramagnetic fluid system. In this paper, we have represented the cartilage by a mixture of two interacting continua and synovial fluid by viscoelastic fluid. A transverse magnetic field is applied to the system. We have used the modified form of Darcy’s law given by Zahn and Rosenweig; to describe the penetration dynamics of magnetic fluids through porous media. Because of exact solution not being possible for the governing non-linear partial differential equations, the perturbation method has been used to obtain approximate solutions. The results have been obtained by computational techniques and compared by results available in the literature. In this paper, the possibility of increased efficiency of joint lubrication, particularly in diseased states by the application of applied magnetic fields has been explored. The applied magnetic field increases the load carrying capacity. Which helps in sustaining greater loads. Similarly, the viscoelastic parameter describes the increase in the concentration of the suspended hyaluronic acid molecules which, in turn, increases the overall viscosity of the lubricant, which also helps in sustaining greater loads.     

The use of generalised deformation boundaries for the analysis of axisymmetric extrusion through curved dies

A generalised kinematically admissible velocity field is derived for axisymmetric extrusion through curved dies by employing rigid-plastic boundaries expressed in terms of arbitrarily chosen continuous functions. The corresponding upper-bound extrusion pressure is related directly to boundary functions for the plastically deforming region when the die shape, lubrication condition and material characteristics of the billet are given. The proposed method of analysis makes it possible to predict the deformation pattern as well as extrusion pressure. In computation a third-order polynomial is chosen for the die boundary and the bounding function for the plastic region is chosen to be a fourth-order polynomial. The workhardening effect is considered in the formulation. The plastic boundaries as well as stream lines are affected by various process parameters. The theory predicts the relatively faster axial flow at the center than near the die boundary for greater friction factor even with the same die shape. The effects of area reduction and die length are also discussed in relation to extrusion pressure and deformation. Experiments are carried out for steel billets at room temperature. Deformation patterns are measured for several area reductions by the photoetching technique and the extrusion pressure is measured using a load-cell. The predicted extrusion pressure is in excellent agreement with the value computed by the finite element method. The deformation patterns agree well with the experimental observation.     

Inertia effects due to lubricant compressibility in a sliding externally pressurized gas bearing

The sliding externally pressurized gas bearing with parallel surfaces was theoretically investigated taking inertia effects due to lubricant compressibility into account. The ratio of film thickness to bearing width was assumed to be small. An approximate solution of the Navier-Stokes equation for a compressible viscous fluid obeying the barotropic relation was analytically sought to first order in the ratio of film thickness to bearing width. It was found that fluid motion in the directions both perpendicular and parallel to the bearing surface are important in the inertia terms of the equation of motion. The characteristics of the bearing are elucidated by determinations of the load capacity, the mass flow rate, the pressure profile and the stream line. In most cases inertia effects due to compressibility enhance the load capacity irrespective of the direction and the magnitude of the relative velocity.     

通道锥度对透平导向叶片绕流性能的影响及这种大锥度叶片设计方法的改进

本文对透平常用的锥形通道中的三元流动进行了初步探索,文中第一部分研究大锥度扩张（收缩）型透平导叶通道中、在上下游都没有动叶的条件下,锥度对叶栅性能的影响。结果表明：在设计中需要考虑由于实际流面偏离圆柱面所引起的出气角的变化,为此,本文给出了近似计算出气角变化的方法。文中第二部分对上述透平导叶提出了考虑三元流动的近似设计方法。     

精密金属异型挤压塑性成形及模腔优化共形解析

针对异型材挤压塑性成形及模腔建模理论的焦点课题,借助近代复变共形映射理论和金属塑性成形理论成果,将三维金属异型材挤压塑性流动问题转化为二维轴对称成形问题,求解了金属异型塑性流动的流函数、速度场及应变速度场等数学解析模型,建立了金属异型挤压塑性流动通用的三维解析方法。应用能量极值原理,求解了异型材挤压成形及优化模腔工程建模的理论课题。     

金属塑性成形的晶体塑性学有限元模拟研究进展

综述了金属塑性成形过程中晶体塑性有限元模拟的理论背景和应用方面的研究进展,同时总结了国内研究者该领域的研究现状,指出了晶体塑性有限元模拟所要解决的问题及研究重点。晶体塑性理论起源于20世纪20 年代,包括单晶塑性本构理论和多晶塑性本构理论,能够深刻揭示材料变形的规律。与此同时,开始于30多年前的有限元法也已经日益成为求解材料成形理论公式的有效工具。晶体塑性有限元法作为一个强大的模拟工具将二者有机地结合在一起,已经广泛地用于模拟材料的微观结构和各种力学响应,越来越被材料界和力学界的研究者所重视;然而,无论是在理论方面还是应用方面晶体塑性有限元法都还不尽完善。未来晶体塑性有限元模拟的理论研究重点是建立系统的理论架构用于预测由滑移和孪晶引起塑性变形材料的各种力学响应,应用研究重点是运用各种模型模拟其他与织构相关的性能或参数。晶体塑性有限元模拟不仅能够深化人们对材料成形规律的理解, 而且可以不断推进晶体塑性理论的发展。     

On the burst strength and necking behaviour of rotating disks

The behaviour of an elastic-plastic rotating disk is analysed both in the context of three dimensional theory and within the framework of the plane stress approximation. For an axisymmetric disk the possibility of bifurcation into a non-axisymmetric mode is investigated. Computations are also made for the behaviour of a disk with initial imperfections either in the form of a thickness variation or in the form of material inhomogeneities. For a ductile, bored disk of uniform thickness it is found that bursting occurs after the critical bifurcation point, which may occur before or after the point of maximum angular velocity in the axisymmetric solution. Thus, necking has started to develop in the disk, when ductile bursting occurs.     

A numerical model of friction between rough surfaces

This paper describes a computational method to calculate the friction force between two rough surfaces. In the model used, friction results from forces developed during elastic deformation and shear resistance of adhesive junctions at the contact areas. Contacts occur between asperities and have arbitrary orientations with respect to the surfaces. The size and slope of each contact area depend on external loads, mechanical properties and topographies of surfaces. Contact force distribution is computed by iterating the relationship between contact parameters, external loads, and surface topographies until the sum of normal components of contact forces equals the normal load. The corresponding sum of tangential components of contact forces constitutes the friction force. To calculate elastic deformation in three dimensions, we use the method of influence coefficients and its adaptation to shear forces to account for sliding friction. Analysis presented in Appendix A gives approximate limits within which influence coefficients developed for flat elastic half-space can apply to rough surfaces. Use of the method of residual correction and a successive grid refinement helped rectify the periodicity error introduced by the FFT technique that was used to solve for asperity pressures. The proposed method, when applied to the classical problem of a sphere on a half-space as a benchmark, showed good agreement with previous results. Calculations show how friction changes with surface roughness and also demonstrate the method's efficiency.     

Abnormal behavior of a hydrodynamic lubrication journal bearing caused by wall slip

Reynolds lubrication theory assumes that there is no wall slip on the interfaces between the solids and lubricant. During recent years, however, it is found that wall slip often happens. The present paper analyzes the wall slip occurring in a hydrodynamic lubrication journal bearing. If the two surfaces have the same adhesion property wall slip always decreases the oil film load support capacity. If there is wall slip over all of the lubricated surfaces, the hydrodynamic effect of the journal bearing vanishes, and no load support exists. If the two lubricated surfaces have different adhesion properties, the wall slip effect is more complex and may cause the journal bearing to operate in an instable manner. In order to avoid the wall slip, the limiting shear stress at the bearing surface should be higher than that at the journal surface.     

锥形凸台成形的一种叠加滑移线场力学模型

分析了锥形凸台在不同体积成形工艺中的成形机理，基于模块化思想将锥形凸台的成形过程划分为挤压成形与角部充满两个特征时段；将上述两个特征时段的滑移线场力学模型有机联结，构建了锥形凸台成形过程的一种叠加滑移线场力学模型；介绍了叠加滑移线场力学模型的应力场与变形力的求解方法。该模型丰富了金属体积成形过程的特征单元库，可套用于局部成形的应力场与变形力求解，实现局部成形的模具优化设计。     

叶轮机械三元流动流函数方程组——S1和S2流面的统一数学方程组

随着计算机和计算技术的发展,直接求解叶轮机械三元流动成为新发展的一个问题。本文提出了直接求解叶轮机械三元流动的一个方法。在本文中引入了三元流动流函数。对吴仲华叶轮机械三元流动通用理论中的S1和S2相对流面理论作了统一的数学表达,进一步阐明了流面的意义,对流面理论中连续方程所引入的流片厚度τ作了一个严格的数学推导。所得的结果同流面理论中的结果,和其物理意义都是一致的。这样所得的流动图案是最一般的情况。而我们以往常常使用的任意回转面流动、圆柱面流动、轴对称流动等都是其中某些简化的特例。由所引入的三元流动流函数,从叶轮机械基本方程组,推导得出了包括S1和S2两族流面上流动的统一流函数方程组。由此可以明确看到S1和S2两族流面相互迭代计算的意义和途径。进一步讨论了方程组正问题求解的边界条件和求解方法,从而提出了直接求解三元流动的一种新的方法。在这方法是同时直接求解三元空间内计算节点上ψ1和ψ2两个流函数值。这两个ψ1和ψ2是通过两个联立的流函数方程组迭代求解而得的。解得两个流函数值后,从而再解出所有气动参数。这方法不同于其它直接解速度分量形式的三元流动方程。后者是求解三元空间内计算节点上三个或三个以上未知数。因此看来这个新方法有可能比其他直接解法更容易在计算机上求得完整的三元流动解。     

A coupled stream function-finite element analysis for wire drawing processes

A numerical approach has been developed based on the stream function technique and the finite element analysis to predict required power and temperature rise in wire drawing processes. An admissible velocity field is first proposed using a stream function and then power consumption in the wire drawing is optimized to achieve sensible and unique deformation geometry. In addition, the finite element method together with axi-symmetric Petrov?CGalerkin algorithm is coupled with the deformation model to assess the temperature distribution in both the deforming wire and the die during the process. The work hardening effects are also considered in the model both in the deformation zone and on the velocity discontinuity surfaces. The model can estimate the effects of various process parameters such as drawing velocity and die geometry. In order to evaluate the results of the model, the predictions are compared with the established models including force equilibrium as a lower-bound approach and an upper-bound solution based on the spherical velocity field.     

弹流润滑在摩擦无级变速器疲劳分析中的应用

本文在摩擦传动弹流润滑理论分析与牵引机理试验研究的基础上，进行零件表面牵引力和表层内应力场的研究，而且利用这一方法来分析零件的抗疲劳强度。分析多种设计参数对零件疲劳寿命的影响，研究了壁面附近的滑动对摩擦传动的影响。     

Theoretical aspects of technological mechanics. 7. Axisymmetric problems in plasticity theory. Part 1

As is shown for the example of an axisymmetric problem, there is a fundamental difference between the solution of plastic-deformation problems when using the applied theory of plasticity and when utilizing the engineering theory of the pressure treatment of metals. The axisymmetric problem in plasticity theory is complex and has not been sufficiently studied.     

A Refined Friction Modeling for Lubricated Metal Forming Process

In spite of the existence of a well-developed realistic friction model, it has only been applied to the simple geometric problems, such as the axisymmetric and plane strain conditions, where the formulations of lubricant transport can be decomposed into two characteristic equations. Accordingly, a unified procedure of combining the current lubrication/friction model and finite element code of metal forming has been developed in this article for either steady or unsteady three-dimensional process including both axisymmetric and plane strain cases. In the part of the lubrication analysis, a finite element method is derived for the average Reynolds equation that is appropriated for 3-D metal forming process, and regardless of whether the tooling/workpiece surfaces are in contact or not. With regard to the theory of asperity contact, in addition to the well-known smoothing and roughening effects, significant deformation of asperities could be caused by the elastic microwedges on the tool surface. The availability of the new friction model was proved by a published problem and an axisymmetric stretch forming process was therefore adopted as a benchmark. Numerical results showed that the present analysis provides a good agreement with the measured strain distributions.     

A new generalized upper-bound solution for the ECAE process

A new generalized upper-bound solution for the equal-channel angular extrusion (ECAE) process is presented in this paper. Using mathematical definition of Bezier curves, a streamline was formulated to define a generalized deforming region. Based on this deforming region, a kinematically admissible velocity field was obtained from which upper-bound solutions were computed. By changing the parameters defining the Bezier-shaped streamline which in turn defines the deforming region, the optimization of the upper-bound solution was carried out. Equal-channel angular extrusion through a 90° bend was considered. Using the formulation presented in here, it was possible to predict the shape of the dead metal zone and its variation with frictional conditions. Unlike previous work in which a fixed circular shape had been assumed for the dead metal zone, in this paper, a generalized shaped Bezier curve was used. The optimum value of the extrusion pressure for ECAE was obtained and compared with both experimental and theoretical data from previous works. It was concluded that the present solution gave an improvement over all previous works and the authors’ results were closer to experimental data.