A generalized velocity field for plane strain backward extrusion through punches of any shape

In this paper, the process of plane strain backward extrusion process through arbitrarily curved punches, by means of the upper bound method and the finite element method is investigated. A generalized velocity field is developed and by calculating of the internal, shear and frictional powers, the extrusion force is estimated. Then, by using the developed analytical model, optimum punch lengths which minimize the required extrusion forces, are determined for a wedge shaped punch and a streamlined punch shape. The corresponding results for those two punch shapes are also determined by using a finite element code and compared with the upper bound results. This comparison shows that the upper bound predictions are in good agreement with the FE results.     

An analytical approach for the design optimization of bearing land in the metal extrusion of shaped sections

Bearing region plays an important role in controlling material flow and its optimal design could lead to high quality extruded products. On the other hand, too much of bearing causes the process load to increase. Thus, there must be an optimum point where the bearing lands and the extrusion pressure are just the right values. Determining the proper bearing length is often performed using trial and error methods in the extrusion industry and numerical analysis. The aim of this study is to optimize the bearing length in forward extrusion dies using upper bound method for non-axisymmetric sections. A generalized kinematically admissible velocity field is employed to obtain uniform velocity at the exit surface of the die. Dead metal zone and bearing region define the geometry of the deformation zone. The multi-objective optimization using response surface methodology was applied to optimize the relative extrusion pressure and the deviation of the mean value for the velocity at die exit. Using this method, the proper bearing length is determined. Optimization of bearing land is performed for extrusion of rectangular and L-shaped profiles. The proposed analytical method was verified by physical modelling experiments and numerical simulations. A unique answer for the bearing design could be obtained using the suggested method in a few seconds opposing to numerical method which required many timely and costly trials. This method would be useful for die designers to get the appropriate bearing land and at the same time not to increase the process load excessively.

     

A new analytical definition of the dead material zone for forward extrusion of shaped sections

This article aims to provide a new formulation for the analysis of the extrusion process for non-axisymmetric sections. The upper bound theorem has been used to obtain a generalized kinematically admissible velocity field. The geometry of the deforming region has been formulated considering variation of the dead zone size at different angular positions and three-dimensional curved surfaces have been employed to define the entry and exit surfaces of the deformation zone. Using this analytical method, extrusion of square, rectangular and L-shaped sections were analyzed and the effect of shape complexity on material flow and dead material zone (DMZ) formation under different conditions has been investigated. Physical modelling experiments and finite element analysis were carried out to reveal the capability of the proposed theoretical method.     

Cavity formation and enfolding defects in plane strain extrusions using a shaped punch

Summary Expressions are obtained for a best upper bound to the load necessary to cause extrusion through square dies, under conditions of plane strain, using a punch face which is inclined to the axis of the extrusion container. The results are interpreted to suggest how for a given reduction that angle of punch face may be chosen which delays the onset of cavity deformation. The modes of deformation considered suggest the mechanism, whereby the oxide layers on the rear of a billet may become entrained down the centre of an extruded product.     

Lower bound on deformation for dynamically loaded rigid-plastic structures

A lower bound on maximum deformation is determined for rigid-plastic structures subjected to time dependent loads. This bound on deformation amalgamates and slightly extends two previous bounds. It is easily calculated based on an assumed velocity field that is kinematically admissible. Comparisons are made between this bound, the complementary upper bound by Robinson[5], and the analytical solution for maximum deformation of five different structural elements. Thus, characteristics of the structure and applied tractions that affect accuracy of the bound are examined. In two cases, the stress field transition from bending at small deformations to membrane stresses at large deformation is demonstrated.     

A new die profile with high process efficiency

A new technique is used to design die profiles which yield high process efficiency during axi-symmetric extrusion and wire drawing. These profiles have a convex shape. The upper bound of the average ram pressure is calculated for the practical range of reductions and optimal die lengths. For m=0,1 where m is the constant friction factor, and up to 55 per cent reduction the reduced extrusion pressure is the same for this convex die and the optimal length cosine die. Above 55 per cent reduction the pressure is slightly higher than that for the cosine die. Up to a reduction of 55 per cent the optimal lengths for this one are slightly shorter than that of the cosine die, while those values for higher reductions are a little higher for this die. The efficiency of the proposed die exceeds those of conventional conical dies.     

杯外径不相等的杯杯型轴对称复合挤压凸模单位压力

本文建立了杯外径不相等的杯杯型轴对称复合挤压的运动学容许速度场.在此基础上,用上限法分析了此类复合挤压的变形过程,确定了此类复合挤压变形不同阶段所需的上凸模平均单位压力.研究结果表明:作用于上凸模上的平均单位压力随着反挤变形程度的变化呈近似抛物线型;随着正挤变形程度的变化呈单调增函数.在挤压过程中,随着上凸模的逐渐压入,变形体的变形模式发生改变,即由早期阶段的稳态流动进入晚期阶段的非稳态流动.而正挤、反挤部分的变形程度组合是除剩余坯料相对厚度以外影响变形模式转换的主要因素.本文的理论分析结果可较为方便地运用于实际生产中的工艺设计.实验表明:理论分析结果与实际结果颇为一致.     

The extrusion of a model yield stress fluid imaged by MRI velocimetry

Extrusion tests were performed by forcing a well-characterized model yield stress fluid from a cylindrical cartridge through various cylindrical extrusion dies using a variety of different piston velocities. In this study the Bingham number within the die ranged from 0.1 to 10. MRI techniques allowed for the non-invasive determination of the local velocity within the extruded material in the range [0.015; 20 mm s^?1]. The velocity profile within a very long die was determined by MRI and agreed very well with the analytical results for the flow of a Herschel–Bulkley fluid within a conduit using parameters determined from independent rheometrical tests, validating both the rheological approach and the accuracy of the MRI techniques. Although the velocity was determined by MRI in the upper and lower zones separately, the intersection of these zones showed great agreement, providing an entire view of the extrusion process. In the range of Bingham number studied, the velocity field for a given contraction ratio appeared similar when scaled by the piston velocity, with a dimpled acceleration zone above the die and lateral dead zones varying negligibly with the piston velocity. For a further analysis the experimental results were compared with the results of numerical simulations. Finite element simulations using an elastic solids model were performed to provide this comparison. It was found that this model did well in representing the characteristics of extrusion flow seen in the experiments; an aspect that was not present in the biviscous simulations. The MRI results show that for the range of values studied, both the piston velocity and the contraction ratio have little effect on the characteristics of the flow, including the size and location of the apparent dead zones. It was found that with an appropriate scaling the central, longitudinal velocity follows a master curve. A decreasing contraction ratio, on the other hand, appears to increase the size of the weak velocity region, in contrast with the simulation results.     

竖直振动管中颗粒毛细效应的离散元模拟

将一根细管插入填充有颗粒的静止容器中并对管施加竖直振动,颗粒将在管内发生上升运动,并最终稳定在一定高度,这一现象与液体毛细效应类似,被称为颗粒毛细效应.为探究颗粒毛细效应过程中伴随的颗粒尺度动力学行为及机理,基于离散元方法建立颗粒运动模型,对颗粒毛细效应动力学过程和特性开展数值模拟研究.模拟再现了文献中实验得到的颗粒毛细效应全过程,给出了管内颗粒柱高度随时间的演变规律,结果表明,受到颗粒系统参数的影响,本模拟条件下颗粒毛细效应过程呈现单周期上升、倍周期上升和倍周期稳定三个阶段,在倍周期上升阶段颗粒柱上升速度逐渐减小,平缓过渡到稳定阶段.在此基础上,分析了管内颗粒速度场和填充率分布随时间的演变特性,揭示了颗粒毛细效应过程中由容器传输到管内的颗粒的占比分布.研究发现,管内不同高度位置颗粒的运动并不同步,随着管的振动,管内出现速度波,速度波的传播引起管内颗粒出现膨胀和压缩交替的情况,从而管内颗粒填充率随时间发生周期性波动;在上升阶段,越接近管壁由容器传输到管内的颗粒占比越大,在稳定阶段,管内上层颗粒的对流引起容器传输到管内的颗粒占比发生反转.      

An upper bound solution for the force of combined backward-torward extrusion

A rigid-triangle velocity field for combined backward-forward extrusion based on the experiments and the slip-line field is proposed in this paper. The flow separation point in the rigid-triangle velocity field is defined in accordance with the slip-line theory. A formula of minimum upper bound solution for the punch pressure of the combined extrusion is derived. The values from this formula are compared with those from the slip-line solution and with experimental results. The formula of upper bound solution can be used in practice.     

Drag of a rotating sphere

We consider the problem of steady incompressible viscous fluid flow about a rotating sphere, with the flow specified on a sphere of finite radius, which reduces to the solution of the complete Navier-Stokes equations.The dimensionless stream functions and circulai velocity are sought in the form of series in powers of the Reynolds numbers, which converge for small values of this number. Recurrence formulas are derived for determining the coefficients of these series. The pressure, rotational resistance torque, and drag are determined. It is established that the rotating sphere has higher drag than a stationary sphere. The leading term of the series in powers of the Reynolds number for the drag and resistive torque is calculated.     

Upper Bound Analysis for Extrusion of T-section Bar from Square Billet Through Square Dies

The present study is devoted to the upper bound analysis of extrusion of T-section bars from square billets through square dies using the modified SERR (Spatial Elementary Rigid Region) technique. Optimized values of the non-dimensional average extrusion pressure at various area reductions have been computed and compared with experimental results available in literature. The optimization process in this study consisted of three stages: (i) optimization of the extrusion pressure with respect to appropriate system variables, (ii) determination of the optimum scheme of discretization of the deformation zone for each of the three formulations (single-point, double-point and triple-point), and (iii) identifying the formulation that gives the lowest upper bound.     

Development of upper bound technique for analysis of fracture in metal forming

Summary An attempt is made to modify the plasticity theory for compressible materials and then to derive the upper bound theorem or the second extremum principle which incorporate normal velocity discontinuity; this discontinuity is considered to be a measure of fracture. The theorem is applied to predict the occurrence of fracture, central bursting in extrusion or drawing. The occurrence of fracture appears to depend on the value of the parameters in the theorem; these parameters may be related to ductility or workability of the material in question. The upper bound technique is then applied to multi-stage extrusion of a carbon steel for which the values of the parameters are given with some assumptions. It is thus shown that the stage at which central bursting occurs agrees with experimental results and that it apparently depends on the extrusion condition.

---

Ein obere-schranken-konzept zur bestimmung von bruchvorgängen bei der umformung von metallen

Übersicht Es wird der Versuch unternommen, mit einer modifizierten Plastizitätstheorie für kompressible Werkstoffe den Obere-Schrankensatz bzw. das zweite Extremum-Prinzip herzuleiten, wenn die Normalgeschwindigkeit auf Sprungflächen unstetig ist. Diese Unstetigkeit wird als Indikator von Bruchvorgängen angesehen. Angewandt wird diese Hypothese auf die Vorhersage des Bruchs, d. h. der Materialtrennung im Kern beim Durchdrück- bzw. Durchziehverfahren. Das Auftreten von Brüchen erweist sich als abhängig von den Parametern des Ansatzes. Diese Parameter lassen sich mit der Umformbarkeit des Materials in Zusammenhang bringen. Angewandt wird das Obere-Schranken-Konzept auf das mehrstufige Durchpressen eines Kohlenstoff-Stahls, für den die Parameterwerte mit Hilfe einiger Parameterwerte festgelegt werden. Davon ausgehend wird gezeigt, daß die Preßstufe bei Materialtrennung mit experimentellen Ergebnissen übereinstimmt und offensichtlich von den Durchpreßbedingungen abhängt.

     

An upper bound approach on yield surfaces of porous materials

Summary Triaxial deformation of a porous material is analyzed. A material model is proposed where the material consists of spherical cells each of which has a spherical pore in the center of the matrix. The velocity field in the matrix is assumed and the upper bound approach is attempted. The yield surfaces of the porous materials with various volume fractions of pores v _f are obtained as ellipsoids whose axes become smaller with increasing v _f.They coincide with the yield surfaces which have been proposed by the authors earlier.

---

Ein Obere-Schranken-Ansatz für den Fließort von porösem Material

Übersicht Analysiert wird die dreiachsige Deformation eines porösen Werkstoffs, der durch kugelförmige Zellen mit sphärischen Löchern im Zentrum der Matrix modelliert wird. Für die Anwendung des Obere-Schranken-Satzes wird ein Geschwindigkeitsfeld in der Matrix angesetzt. Damit erhält man als Fließort von porösen Werkstoffen mit verschiedenen Porenvolumen-Anteilen v _fEllipsoide, deren Hauptachsen mit zunehmendem v _fkleiner werden. Sie stimmen mit den Fließorten, welche die Autoren schon früher halbexperimentell und halb-analytisch ermittelt haben, überein.

     

An elementary consideration of some extrusion defects

Summary It is shown that the best upper bounds to the load in direct extrusion when the fault known as piping is anticipated are less than the best upper bounds obtainable by the use of similar deformation patterns, when piping is assumed not to be going to occur. A similar approach is made to the case in which a short slug is pierced by two equal size punches and which results in faults at the container wall; this instance is important because it is analogous to the new method by which gudgeon pins are formed.     

Vibration analysis of rotating twisted and open conical shells

Based on a non-linear strain–displacement relationship of a non-rotating twisted and open conical shell on thin shell theory, a numerical method for free vibration of a rotating twisted and open conical shell is presented by the energy method, where the effect of rotation is considered as initial deformation and initial stress resultants which are obtained by the principle of virtual work for steady deformation due to rotation, then an energy equilibrium of equation for vibration of a twisted and open conical shell with the initial conditions is also given by the principle of virtual work. In the two numerical processes, the Rayleigh–Ritz procedure is used and the two in-plane and a transverse displacement functions are assumed to be algebraic polynomials in two elements. The effects of characteristic parameters with respect to rotation and geometry such as an angular velocity and a radius of rotating disc, a setting angle, a twist angle, curvature and a tapered ratio of cross-section on vibration performance of rotating twisted and open conical shells are studied by the present method.     

????????????????????????????????

在重轨万能轧制过程中,首先建立了简化的三维理论模型. 然后分别给出了轨腰、轨头及轨 底的运动学许可速度场以及相应的应变速度场和剪应变速度强度,并求出了相应变形区的塑 性变形功率、速度间断面上消耗的功率以及由于摩擦产生的摩擦功率. 最后根据上限原理分 别求解了水平辊和两个立辊的轧制力上限解. 通过比较可知,二辊轧制理论公式误差很大因 此不能用于万能轧制过程,而上限法求得的轧制力近似解大于轧制现场数据但最大误差不超 过13\%, 因此根据上限原理进行轧制工艺参数设定及优化是比较可靠的.     

Unsteady circular Couette flow of a Bingham plastic with the Augmented Lagrangian Method

Numerical simulations are undertaken for unsteady flows of an ideal Bingham fluid in a circular Couette viscometer. The main difficulties in such simulations are caused by the non-differentiability of the constitutive equation and the need to determine the position and shape of the yield surface separating the yielded zones from the unyielded ones. In this work, these difficulties are overcome by using a numerical method based on variational inequalities, i.e. the augmented Lagrangian/Uzawa method. The start-up and cessation of circular Couette flows of a Bingham fluid are solved numerically assuming that only one of the cylinders is rotating. An improved theoretical upper bound for the stopping time in the case of cessation is derived. The numerical estimates for the stopping time compare well with the theoretical bounds. Moreover, with the adopted method the evolution of the velocity profiles and the locations of yielded/unyielded surfaces are accurately calculated. In flow cessation, we observe an interesting effect, namely the appearance of a small unyielded region adjoined to the outer cylinder shortly before cessation.     

花键轴冷挤压成形后的三维光塑性模拟研究

本文利用光塑性方法,以聚碳酸酯（Ｐｏｌｙｃａｒｂｏｎａｔｅ）为材料研究了矩形花键轴冷挤压成形过程,获得了花键挤压成形的三维塑性应变场分布,提出其变形的应变分布特征,并对典型截面的应变进行了计算分析;文中还通过对聚碳酸酯材料的实验,得到了使花键齿形充满的最佳坯料尺寸,为实际工艺生产优选出合理的工艺参数。     

A plasticity model for cellular materials with open-celled structure

Based on a rigid-plastic material model that obeys the von Mises yield criterion, the plastic behavior of foams with an open-celled structure is studied in this paper using a single unit cell. An approximate continuum plasticity model is developed within the framework of the upper bound theorem of plasticity to describe the yield behavior of foams. The microscopic velocity fields are derived for the unit cell, which satisfy the incompressibility and the kinematic boundary conditions, and expressed in macroscopic rate of deformation. From the microscopic velocity fields, a macroscopic yield function is developed for foams under multi-axial stresses and includes the effects of the hydrostatic stress due to the void presence and growth. The dependency of the derived yield surfaces of foams on their relative densities is studied. The plastic behavior of foams is also studied numerically using the finite element method. The newly developed plasticity model is compared with the finite element analysis results and other available foam models and then correlated with the finite element results.