Upper bound analysis of extrusion from square billets through circular and square/rectangular dies

Upper bound elemental technique (UBET) for prediction of extrusion pressure in three-dimensional forward extrusion process is presented. Using square/rectangular billets, the study of the effect of die land length has been extended for the evaluations of extrusion pressures to extrude sections such as circular, square and rectangular shaped sections with power of deformation due to ironing effect at the die land taken into account. The extrusion pressure contributions due to the die land evaluated theoretically for these shaped sections considered are found to increase with die land lengths for any given percentage reduction and also increase with increasing percentage die reductions at any given die land length. The effect of die land lengths on the extrusion pressures increases with increasing complexity of die openings geometry with rectangular section giving the highest extrusion pressure followed by circular with square section die opening, giving the least extrusion pressure for any given die reduction at any given die land lengths. The proper choice of die land length is imperative if excessive pressure buildup at the emergent section is to be avoided so as to maintain good quality and metallurgical structure of the extrudates. This paper was recommended for publication in revised form by Associate Editor Youngseog Lee Ajiboye, Joseph S. received his B.Eng, M.Eng, and PhD degrees in Mechanical Engineering from the University of Ilorin, Nigeria, in 1988, 1995 and 2006 res-pectively. Dr. Ajiboye is a lecturer in the Department of Mechanical Engineering, Uni-versity of Lagos, Nigeria. He is currently a Contract Research Scientist at KAIST Valufacture Institute of Mechanical Engineering, School of Mechanical, Aerospace & Systems Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305 - 701, Korea. Dr. Ajiboye’s research interests include ECAE/P, determination of frictional effects in metal forming operations, upper bound and finite element in plasticity.     

A generalized method for analysis of three-dimensional extrusion of arbitrarily-shaped sections

Despite increasing demand for and application of three-dimensional extrusion of various sections through continuous dies, so far very little work has been done by systematic and general analysis to predict the plastic flow properly. For effective die design, efficient design method and the related method of theoretical analysis are required for extrusion of complicated sections. In generalized three-dimensional extrusion of sections through continuous dies, a new method of die surface representation, using blending function and Fourier series expansion, is proposed by which smooth transitions of die contour from the die entrance to the die exit are obtained. The flow patterns as well as the upper-bound extrusion pressures are obtained on the basis of the derived velocity field. The effects of area reduction, product shape complexity, die length and frictional condition are discussed in relation to extrusion pressure, the distorted grid pattern and distribution of the final effective strain on the cross-section of the extruded billet. As computational examples for arbitrarily-shaped products rounded rectangles and ellipses are chosen for the extruded sections. Experiments are carried out for aluminum alloys at room temperature for a rounded square section and an elliptic section. In order to visualize the plastic flow, the grid-marking technique is employed. The theoretical predictions both in extrusion load and deformed pattern are in good agreement with the experimental results.     

An analytical solution for the spread extrusion of shaped sections

Spread extrusion could be used for manufacturing of wide profiles in the extrusion industry. In this paper a new method of design and analysis has been presented for such a problem. Special dies were designed for profiled sections such as square, rectangular, elliptical and cross shapes. These dies force the material to flow sideways and spread so as to extrude sections with wider dimensions than the initial billet or the maximum container diameter. The geometry of the deforming zone in the die was formulated and based upon that, a kinematically admissible velocity field was derived. Using this velocity, we estimated the field upper bound on extrusion power. Profile sections with different aspect ratios were investigated and the influence of other process parameters such as friction and reduction of area on the extrusion pressure were studied. Optimum die lengths for each die were calculated so as to minimize the extrusion pressure. Finite element analysis for the numerical simulation of the process was also carried out. The finite element results were also used as an aid to the design process of the extrusion dies. Dies were manufactured for different sections such as square, rectangle, and ellipse and cross shapes. Experiments were carried out to obtain data to verify the theory. Comparison of the results showed good agreement between the theoretical, numerical and experimental data. It was concluded the present method could be used to design dies for the spread extrusion of different shaped dies.     

Extrusion of non-symmetric T-shaped sections, an analysis and some experiments

In the present study, previously developed analytical approach (Int. J. Mech. Sci. 42 (2000) 273), based on the upper-bound theory, for the design of three-dimensional off-centric extrusion of arbitrarily shaped dies was applied to the extrusion of T-shaped sections from initially round billets with the experimental verifications. Here, non-symmetric T-shaped sections as well as symmetric ones were considered. A computer program was written to obtain optimum die design which yield the lowest upper-bound for a given reduction in area, die length, its off-centric positioning and the frictional conditions. Computations were carried out both for the converging (ruled-surface) and the smooth curved (advanced-surface) dies. Curvature of the extruded product and simulated deforming grid pattern were also predicted. A sophisticated CAD/CAM package was used in conjunction with the CNC and EDM processes to manufacture the streamlined dies for the extrusion of T-shaped sections. The theoretical predictions were observed to be in good agreement with the experimental results.     

Analysis of three-dimensional extrusion of sections through curved dies by conformal transformation

A new method of analysis is proposed for the extrusion of arbitrarily shaped sections through curved die profiles. A kinematically admissible velocity field is found by deriving the equation of a stream line. Conformal transformation of a unit circle onto a section is utilized in the derivation. The upper-bound method is then applied to determine the extrusion pressure for the rigid-perfectly plastic material. The redundant work relating to the velocity discontinuities at the entrance and the exit is included in the formulation. The general formulation for an arbitrary cross section is obtained by use of conformal transformation. The upper-bound pressure for extrusion through curved die profiles is computed for a complex section with a curved boundary. Two curved die profiles widely used are chosen to compare the effects of die profiles. From the derived velocity field, the upper-bound extrusion pressures are also computed for the extrusion of regular polygons and rectangles of various aspect ratios. The effects of sectional shape, die profile and interfacial friction at the die surface are discussed.     

Effects of extrusion variables on temperature distribution in axisymmetric extrusion process

A numerical method was developed to simulate the non-steady-state temperature distributions during forward extrusion process. The velocity, strain rates, and strain fields within the deformation zones during extrusion were obtained, using upper bound method of analysis to obtain internal heat generations coupled to the necessary heat transfer conduction equations. The computer program written in C++ language essentially simulates the extrusion process and takes into account extrusion variables such as material properties, friction conditions, extrusion velocity, extrusion ratio, die preheat temperature, billet height, percentage reduction in area, and die land length. The effects of billet height and percentage reduction in area on the temperature distributions within the dead metal zone give good agreements with experimental results. It is found that the higher the billet's heights and higher the percentages reduction in areas, the higher the temperature rises during the extrusion process. The die land zone shows increasing temperature rise with increasing friction coefficient, while increasing friction coefficient has no effect on the dead zone temperature. Also, increasing speed of deformation shows an increasing dead zone temperature rise than a more gradual die land temperature rise. It can be stated that the extrusion temperature increases proportionally to the increase of the container temperature.     

Application of an upper bound method to off-centric extrusion of square sections, analysis and experiments

The previously given analytical method [1], which was based on the upper-bound theory, was used to design the streamlined dies and to investigate the three dimensional off-centric extrusion of circular sections from initially circular billets through linearly converging (ruled-surface) and smooth curved (advancedsurface) dies. For a reasonably correct upper-bound to the load, a set of generalised kinematically admissible velocity fields were derived on the assumption of Bezier-type streamlines by incorporating a special velocity function that takes care of the non-uniform material flow. Based on the present method, for a given reduction in area, material property, friction condition and off-centric positioning of the exit cross-section, computations were carried out to predict the upper-bound to the extrusion pressures, the deforming grid patterns and curvature of the extruded product. Experiments were carried out for some off-centric circular sections with varying die lengths and reduction in areas. A sophisticated CAD/CAM package was used in conjunction with the CNC and EDM processes to manufacture the streamlined dies for the off-centric extrusion of circular rods. Both the solid and split-type specimens made of tellurium lead under both the lubricated and the dry conditions were used for experimental investigations in order to assess pressures, deformation modes and to visualise the deforming grid patterns. The theoretical predictions were observed to be in good agreement with the experimental.     

Numerical and experimental study on the three-dimensional extrusion of square section from square billet through a polynomial shaped curved die

The geometry of die profile plays a major role in reducing the extrusion pressure and ensuring the smooth flow of material. In general, the extrusion process is mostly affected by billet geometry, die geometry, and interface frictional force at the die billet geometry. In the present investigation, an analysis using three-dimensional upper bound method using fifth-order die profile function has been carried out for extrusion of square sections from square billet. The extrusion pressure and optimum die length have been computed by multivariable optimization technique. The present die shape profile is found to be superior to many other profiles. The results obtained will help in design of optimum die profile and investigation of its performance.     

The effect of selected parameters on temperature distribution in axisymmetric extrusion process

A numerical method was developed to simulate the transient temperature distributions during forward extrusion process. The computer program simulates the extrusion process and takes into account some extrusion variables such as extrusion velocity, extrusion ratio, die preheat temperature, and percentage reduction in area. It can be seen that the higher the percentages reduction in areas, the higher the temperature rises during the extrusion process. Also, increasing speed of deformation shows an increasing dead zone temperature rise than a more gradual die land temperature rise. It is further seen that extrusion temperature increase is a function of the container temperature.     

An analysis for extrusion of helical shapes from round billets

A method of analysis is proposed for three-dimensional extrusion of a helical shape from a round billet. It is reported that a helical shape can be made by hot extrusion through a square die. In this paper, it is suggested that a helical shape be effectively cold extruded through a continuous die with appropriate lubrication. The extrusion of helical shapes can find practical application in some useful products. However, the analytical method regarding this kind of extrusion has not been attempted so far.A kinematically admissible velocity field is derived for the extrusion model where a round billet is extruded into a twisted helical section with a long elliptic cross section. The axis of the cross section is rotating during extrusion. By assuming proper stream surfaces, the velocity field is obtained by deriving the equation of a stream line. Then, an upperbound solution is formulated for the rigid-perfectly plastic material. Computation for the upperbound pressure is carried out for various process variables such as reduction of area, friction, rotation of axis, aspect ratio of a product, die length and overall die profiles.     

An analytical formulation as an alternative to FEM software giving strain and stress distributions for the three-dimensional solution of extrusion problems

A new improved analytical method based on the upper bound theorem is presented for the solution of extrusion problems. This method has been formulated in such a way as to eliminate the deficiencies in the previous works in order to enable it as a powerful analytical tool as an alternative to finite element software. The proposed formulation is a general method of solution, which could be used for the analysis of many bulk forming processes. However, in this paper, the forward extrusion of a square section from a round billet has been presented as an example. Kinematically admissible velocity fields were computed which gave a more physically realistic material flow patterns as compared to previous works. The distribution of strain and stress as well as the results for the effect of process parameters on the extrusion load and die geometry was also given. These results were compared with FEM data to observe the accuracy and effectiveness of the present method. For the extrusion of square sections from round billets, detailed analysis of strain distribution on the exit section was carried out using the components of the power due to internal deformation, interface friction, and velocity discontinuities at the entry and exit surfaces. Comparison with experimental data was made, and verification of the theoretical results was carried out. The improvement of the results computed using the present method was shown by comparison with the previous works.     

A new formulation for three-dimensional extrusion and its application to extrusion of clover sections

The metal flow in the extrusion process is an important factor in controlling the mechanical property of the extruded products. It is, however, difficult to predict the metal flow in three-dimensional extrusion of complicated sections due to the difficulty in representing the geometry of the die surface and in expressing the corresponding velocity field. In this study a new kinematically admissible velocity field for a generalized three-dimensional flow is derived, in which the flow is bounded by the die surface expressed by an analytic function. Then, by applying the upper-bound method to the derived velocity field, the flow pattern as well as the upper-bound extrusion pressure are obtained. As a computational example, extrusion of clover sections from round billets is chosen. A new method of die surface representation is proposed by which there is a smooth transition of die contour from the die entrance to the die exit. Computation is carried out for work-hardening materials such as aluminium and steel. The analysis takes into account the effect of product shape complexity, lubrication condition and reduction of area on extrusion pressure, average effective strain and distribution of effective strains on the cross-section of the extruded product.     

Round-to-channel section extrusion through linearly converging die: a three-dimensional analysis

Metal flow in the extrusion process is an important factor in controlling the mechanical properties of the extruded products. It is, however, difficult to predict the metal flow in three-dimensional extrusion of sections due to the involvement of re-entrant corners. The present work is an attempt to find an upper bound solution for the extrusion of channel section from round billet through the taper die. The rigid-perfectly plastic model of the material is assumed, and the spatial elementary rigid region (SERR) technique is presented for which the kinematically admissible velocity field is found out by minimizing the plastic dissipation of power. The presented analysis allows for specification of process control parameters and their relation to extrusion load, equivalent die angle, reduction ratios and friction factor.     

口模压缩段对塑料挤出流动影响的有限元分析

在塑料挤出成形过程中，压缩段是调节模头流道各部分流量(流速)的主要区段，对熔体的流动具有重要影响。本文采用有限元数值分析方法，计算了熔体在口模内流动的速度场和压力场，定量分析了压缩比和压缩角等压缩段模具结构参数对挤出速度分布、挤出流量和挤出流动均匀性等的影响规律，为优化流道结构参数，提高挤出流动均匀性的研究提供了基础。     

Theoretical analysis of extrusion of rectangular, hexagonal and octagonal composite clad rods

Composite clad rods with non-axisymmetric cross-sectional areas are commercially important owing to their extensive industrial applications such as in electrodes, conductors and chemical devices. For instance, in the processing of superconductor wire, increasing the packing density involves closely packing superconductor rods which have a hexagonal cross-section next to each other inside the high purity copper tube. During extrusion process, non-uniform deformation tends to occur because the core and sleeve of a composite clad rod is usually composed of materials with different mechanical properties. The first recognized paper on round-to-square drawing/extrusion with variable corner radius was presented by Boer et al. in 1979. In this study, we present a model based on upper-bound theorem to analyze the extrusion of composite clad rods with non-axisymmetric cross-section. Velocity fields for both core and sleeve are generated with the assistance of a product's cross-sectional profile functions. Products with rectangular, hexagonal and octagonal sections are chosen as the study objects. Also discussed herein are numerical results for various process variables such as semi-die angle, reduction of area, frictional condition of die, and product shape complexity. According to these results, the extrusion pressure and product dimensional change are closely related to the process variables.     

Formulation of a new generalized kinematically admissible velocity field with a variable axial component for the forward extrusion of shaped sections

For most three-dimensional analytical solutions proposed for the extrusion of shaped sections, the axial component of the velocity vector has been assumed to be constant at each cross section throughout the deforming zone. This shortcoming means that these velocity fields are not in accordance with the reality of the extrusion problem, and hence, the upper bounds based on such fields give high values for the extrusion pressure. To overcome this, a new formulation has been presented in this paper for which a kinematically admissible velocity field has been developed using a variable axial velocity component. For this purpose, curved surfaces of velocity discontinuities at the entry and exit have been proposed and incorporated into the formulation for the extrusion of shaped sections from circular billets. As an example, a square profiled section has been chosen for the extrusion problem. The upper bound on extrusion pressure was computed using the new formulation. It was shown that the initial velocity discontinuity surface at the entry to the deforming region was flat, and as one travels into the deforming region towards the exit section, the velocity discontinuity surface gradually became convex, having the highest convexity at the exit section. This was contrary to what has been suggested in the literature so far. The measure of convexity depends on the extrusion parameters which have been investigated in this work. Experiments were also carried out to verify the theoretical results, and good agreements were observed between the two. Comparison of the present results with similar previous works showed good improvements as well.     

NUMERICAL DESIGN OF DIE LAND FOR SHAPE EXTRUSION

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腔室截面形状对扩张室液压脉动衰减器滤波特性的影响

将气体消声器设计理论中的格林函数法扩展到计算和分析具有矩形、正方形截面的扩张室液压脉动衰减器的滤波特性,在平面波截止频率范围内,这两种截面型式脉动衰减器的插入损失理论曲线与实验测量结果吻合较好,证明了该方法同样可用于计算矩形和正方形截面的扩张室压力脉动衰减器的滤波特性。而针对圆形截面,为避免坐标系变换带来的麻烦,引入消声器声学特性研究中最常使用的一维解析法,其计算结果也与实验测量值吻合良好。通过对这三种不同截面扩张室脉动衰减器插入损失的比较,可以得出：控制扩张室腔体截面周长一定的前提下,在2 kHz测试频带内,圆形截面具有最优的滤波特性,正方形截面次之,而矩形截面脉动衰减性能最差。     

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.     

Analysis and die design of flat-die hot extrusion process 2. Numerical design of bearing lengths

This paper deals with the assignment of bearing lengths for the control of material flow in the flat die hot extrusion. The design process makes the use of the three-dimensional non-steady analysis using the thermo-rigid–viscoplastic finite element method that includes an automatic remeshing module. The exit velocity distribution of the workpiece obtained from the analysis results was used to find appropriate values for the factors used in the proposed bearing length design equation. This equation for designing bearing lengths is a function of the cross-sectional thickness and distance from the die center of die exit section. A geometric factor was included in formulation of the design equation to consider the end region of the die exit. The appropriate values of factors were determined from three-dimensional analyses of flat-die hot extrusion processes with single and double channel-sections. The analysis of a flat-die hot extrusion process with a L-section was used to verify the proposed design equation. It was found that the design equation determined bearing lengths that resulted in a fairly uniform exit velocity distribution throughout the extruded section. From the results of this study, it was found that the proposed design equation can be effectively used to estimate appropriate bearing lengths.