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.     

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.     

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 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.     

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.     

Axisymmetric extrusion through adaptable dies—Part 3: Minimum pressure streamlined die shapes

In Part 1 of this series of papers, six kinematically admissible velocity fields, as well as the power terms, were developed for use in upper bound models for arbitrarily shaped dies for axisymmetric extrusion. Part 2 compared the results obtained in upper bound models for the six velocity fields through a spherical die shape and demonstrated that the sine-based velocity field was the best. In this final part, the application of the sine-1 field to extrusion through streamlined dies is developed. By fixing the values of two additional constants in the radial flow flexibility function, the two surfaces of velocity discontinuity, which separate the deformation zone from the incoming and outgoing material, will have no shear. In effect, the analysis for streamlined dies can be modeled without the surface of velocity discontinuity power terms. The results for an arbitrarily curved streamlined die, as proposed by Yang and Han, using the sine-1 velocity field and the cylindrical velocity field from the work by Yang and Han are compared. It is found that the upper bound model using the sine-1 velocity field predicts lower values for the extrusion pressure. A method to determine a streamlined die shape is proposed. The method allows flexibility between the entrance and exit by the use of a Legendre polynomial series for representation of the die surface. The method is termed an adaptable die design. The adaptable die design method is used to determine streamlined die shapes, which will minimize the pressure required for the extrusion process.     

Analysis of hydrofilm extrusion of elliptic shapes using perturbation method

An approximate method of solution is proposed for the hydrofilm extrusion of elliptic shapes from round billets through optimized curved dies. A modified upper-bound theorem and hydrodynamic lubrication theory are used in combination, in order to analyze metal deformation and fluid flow respectively. The fluid analysis in hydrodynamic lubrication theory is simplified by use of elliptic transformation and perturbation technique. Strain-hardening effect of billet material and viscosity variation of fluid due to pressure are taken into consideration.For several reductions of area, experiments are carried out at room temperature by using mild steel specimens and caster oil as the lubricant.The experimental extrusion pressures are in good agreement with the theoretical predictions.     

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.     

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.     

A generalised CAD/CAM solution to the three-dimensional off-centric extrusion of shaped sections: analysis

In analytical method based on the upper-bound theory is proposed to investigate the three-dimensional off-centric extrusion of arbitrarily shaped sections from arbitrarily shaped billets through linearly converging and smooth curved dies. A set of generalised kinematically admissible velocity fields are derived on the assumption of Bezier-type streamlines that provide compatibility of surface representation with most CAD/CAM systems. To obtain a more realistic non-uniform metal flow, a special velocity function was also incorporated into the derived velocity fields and work hardening effect of the material considered. Based on the method presented, for a given reduction in area, material property, friction condition and off-centric positioning of the exit cross-section, predictions of the deforming grid pattern, curvature of the extruded product as well as upper-bound to the extrusion pressures may be obtained. The above procedure is highlighted and commented upon.     

A numerical model for simulating temperature and speed effects in hot extrusion of rod

In this paper, a two-dimensional numerical model with upper-bound coupled thermal analysis has been developed. The model is capable of simulating the hot rod extrusion process with variable ram speeds. The temperature distributions and the speed effects in hot extrusion are predicted in detail by the proposed numerical model. A generalized kinematically-admissible velocity field without velocity discontinuity is adopted. The temperatures are calculated by considering simultaneously the heat generation due to deformation and friction and heat transfer. A finite-difference method with an implicit time integration scheme is utilized to solve the two-dimensional heat conduction problem. Two mathematical models for variable ram speed profiles are proposd. Ram speed profiles satisfying the exit temperature and the load requirements are obtained. The proposed numerical simulation has been demonstrated to be a powerful tool for the design of hot extrusion processes.     

Analysis of hydrofilm extrusion through optimized curved dies

A new analysis of the hydrofilm extrusion process which includes strain-hardening effects and viscosity variation in the fluid due to pressure is developed. The upper-bound method and hydrodynamic lubrication theory are adopted for the analysis of metal forming and fluid flow respectively.Experiments were carried out at room temperature, for several reductions of area, using axisymmetric curved dies. The theoretical prediction of extrusion pressure shows good agreement with experimental measurements for mild steel using castor oil as the lubricant.     

Upper bound analysis for extrusion at various die land lengths and shaped profiles

The effects of die land lengths, a rarely investigated die extrusion parameter on the die-shaped profiles, on the extrusion pressures are investigated and presented. The analyses of the extrusion pressures by the upper bound method have been extended for the evaluations of the extrusion pressures to complex extruded sections such as square, rectangular, I,- and T-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 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 pressure increases with increasing complexity of die openings geometry with I-shaped section giving the highest extrusion pressure followed by T-shaped section, rectangular, circular-shaped die openings with square section die opening, giving the least extrusion pressure for any given die reduction at any given die land lengths.     

An analysis of hydroforming of longitudinally curved boxes with regular polygonal cross-section

A new kinematically admissible velocity field is suggested for the upper-bound solution of hydroforming of longitudinally curved boxes with regular polygonal cross-section. In order to maintain the uniform wall thickness the back-up fluid pressure is controlled to vary with respect to the punch position during the hydroforming process. The pressure vs punch stroke curves for various process conditions are determined from theoretical analysis. The effects of punch shape, work-hardening exponent, drawing ratio and friction are then analysed and discussed. Experiments are carried out in a hydroforming press according to the computed pressure vs punch stroke curves. The hydroformed specimens have shown very little thickness variation within a 5% range. Thus, the validity of the assumption of uniform wall thickness has been confirmed. The flange deformation from the computation is found to be in good agreement with the experimental observation. When the computed reference curve of pressure vs punch stroke relation has been used, no defect, i.e. no necking or no wrinkling has been observed in the experimental specimens. It has been shown that the present method of analysis can be effectively used for the hydroforming process design of longitudinally curved boxes with regular polygonal cross-section.     

基于数值分析的轮胎胶料共挤出口模设计

在轮胎胶料挤出生产中,口模的结构对共挤出产品的质量起着关键的作用,但在实际生产中,共挤出口模的设计主要依靠经验,口模设计质量的控制需要经过多次试模和修模。采用流体计算软件Polyflow进行轮胎胶料(TWD40/FS10)的共挤成型过程的三维数值模拟,分析共挤出过程中的速度、压力分布以及胶料熔体的流动情况,并通过试验验证模拟的可行性。针对初始口模设计的不足,提出增加窄缝区域高度和扩充口模入口的方法,有效解决初始模具设计中速度分布不均的问题,改善了口模中熔体的流动,提高了挤出质量,同时还降低了挤出过程中的能耗。利用数值方法可对共挤出口模结构进行设计,提出的口模设计方法可以对同类轮胎胶料共挤出口模结构的设计提供指     

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.     

Axisymmetric extrusion through adaptable dies—Part 1: Flexible velocity fields and power terms

An adaptable die is one that not only produces the correct geometrical shape, but also is designed through an adaptable method to impart other desirable properties to the product or process. In this first part of a series of papers, six kinematically admissible velocity fields are developed for use in upper bound models for axisymmetric extrusion through various dies, including extrusion through adaptable dies. Three base velocity fields are presented:

(1) assuming proportional angles in the deformation zone,

(2) assuming proportional areas in the deformation zone, or

(3) assuming proportional distances from the centerline in the deformation zone.

The base velocity is modified by an additional term comprised of two functions. One function allows extra flexibility in the radial direction, and the second function allows extra flexibility in the angular direction. There are two forms of the second function, which meet the required boundary conditions. The flexibility function in the radial direction is represented by a series of Legendre polynomials, which are orthogonal over the deformation region. The power terms derived for these velocity fields for use in upper bound models are also presented.Part 2 of this series compares the results obtained in upper bound models for the six velocity fields for a spherical extrusion die. In Part 3, the use of the best velocity field for extrusion through streamlined dies is developed to determine the adaptable die shape, which minimizes the required extrusion pressure. Additionally, the adaptable die shape is compared with results from Yang and Han for arbitrarily curved and streamlined dies.     

An analysis of axisymmetric hydrostatic bulging by the upper-bound method

An upper-bound solution for the analysis of hydrostatic bulging is found by minimizing the plastic energy consumption rate divided by the fluid volume change rate. The corresponding upper-bound pressure is found through optimization with respect to some parameters given in the assumed velocity field. Computations are carried out for axisymmetric hydrostatic bulging of isotropic work-hardening sheet metal. The computed results are then compared with those by the finite-element method and by the finite-difference method as well as the reported experimental results. The comparison shows that the present method renders excellent predictions in pressure and strain distributions with much reduced computational time, which are still in close agreement with those obtained by the numerical methods.     

Axisymmetric extrusion through adaptable dies—Part 2: Comparison of velocity fields

In Part 1 of this series of papers, six kinematically admissible velocity fields, along with the power terms, were developed for use in upper bound models for arbitrarily shaped dies for axisymmetric extrusion. The three base velocity fields in the deformation zone were derived:

(1) assuming proportional angles in the deformation zone,

(2) assuming proportional areas in the deformation zone, or

(3) assuming proportional distances from the centerline in the deformation zone.

In each case the base velocity was modified by an additional term comprised of two functions, each function containing pseudo-independent parameters. One function allows extra flexibility in the radial direction, and the second function allows extra flexibility in the angular direction. In Part 2, the results obtained in upper bound models for the six velocity fields for extrusion through a spherical die are compared to one another. The velocity fields are compared based upon: (a) the base velocity field, (b) the number and distribution of pseudo-independent parameters in the flexible functions, and (c) the form of the angular flexible function. A spherical extrusion die shape is used to evaluate and compare the three velocity fields. The results demonstrate that the sine-based velocity field is the best. Furthermore, a natural boundary condition exists which allows the shear surface associated with the streamlined portion of a die to energetically disappear. Part 3 uses the best velocity field to determine an adaptable die shape, which minimizes the extrusion pressure and compares the shape to the arbitrarily curved and streamlined die shape of Yang and Han.     

凹模型线对挤压过程的数值模拟和试验研究

本文用数值拟和试验研究相结合的方法分析凹模型线对挤压过程变形状态的影响。模拟计算采用刚塑性有限元法,试验研究用坐标网格法和密栅云纹技术。文章对文献中提出的圆锥凹模、余弦曲线凹模、等应变率曲线凹模、双曲线凹模、椭圆曲线凹模和最短流线凹模,以及另外新设计的正弦曲线凹模,进行数值模拟,得到了各种型线模挤压时的速度场、应力场和应变场,以及由于变形热效应引起的试件和模具温升。并用光刻网格法和密栅云纹法进行试验校核。计算结果与实验结果相互吻合良好,研究结果表明七种曲线凹模中,以新设计的正弦曲线凹模最佳,余弦曲线凹模次之,这两种型线凹模的挤压力低,而且内部应变分布均匀。