Further investigation into extrusion of trocoidal gear sections considering three-dimensional plastic flow

In the present study, theoretical development proposed in previous work carried out in extrusion of clover sections in relation to a generalized die design method is presented in the extended scope for three-dimensional extrusion of trocoidal gear sections from round billets with experimental verification. Computations are carried out for clover and trocoidal gear sections. The CAD/CAM of the suggested dies is introduced for the experiments. Experiments are carried out for a clover section and a trocoidal gear section with eight teeth. Al 2024 aluminum billets were used as the working material. Half-cut specimens are used for flow visualization of the extrusion process. The theoretical predictions are in good agreement with the experimental results in extrusion load and metal flow.     

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.     

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.     

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.     

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.     

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。     

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.     

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.     

Extrusion of non-symmetric U- and I-shaped sections through ruled-surface dies: numerical simulations and some experiments

In this paper, previously developed analytical approach (Chitkara NR, Celik KF. Int J Mech Sci 2000; 42:273) based on the upper-bound theory was applied to analyse the mechanics of the extrusion of non-symmetric U- and I-shaped sections and the symmetric ones from initially round billets through the ruled-surface dies. To investigate the optimum shape of the designed extrusion dies, that yield the lowest upper-bound for a given reduction in area, die length, off-centric positioning and frictional conditions a computer program was developed. Computations were carried out for various cases and some of the results compared with the experimental verifications. Curvature of the extruded product and simulated deforming grid patterns were also predicted and compared. A sophisticated CAD/CAM package was used in conjunction with the CNC and EDM processes to design and manufacture the streamlined dies for the extrusion of some non-symmetric U- and I-shaped sections. The theoretical predictions were observed to be 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.     

Analysis of hydrofilm extrusion of three-dimensional shapes from round billets

A new method of die construction is proposed, which enables the exact geometrical control of die shape and ensures the initial sealing between billet and die. Using the modified method of die construction the internal metal deformation is analyzed with the aid of a special transformation combined with the upper-bound method. Based on a concept of equivalent friction factor the energy dissipation in fluid film is calculated approximately for three-dimensional hydrofilm extrusion. Computations are carried out for hydrofilm extrusion of various sections such as squares, rectangles, ellipses and clover shapes and various factors affecting the process are discussed. Experiments are performed for clover and square sections using the dies from NC manufacture based on the suggested theoretical design. A reliable sealing system is developed between punch and container. The experimental results are compared with the theoretical prediction and it is shown that the theory is in reasonable agreement with the experimental observation. Various aspects are discussed from the experiment.     

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.     

A die design model for thin section extrusions

In an extrusion plant, the die design procedure is very often a trial-and-error process which requires a number of extrusion trials before reaching a satisfactory design. If the die is not corrected or tuned appropriately, the product may be twisted and warped. In this work, a model is proposed which should guide the preliminary design of extrusion dies. The model is focused on the distortions of the extruded product due to exit velocity variations. The die design model is limited to the extrusion of thin sections through solid dies. Since the die opening is assumed to be long and slender, asymptotic approximations have been used in the analysis. Some die design examples are discussed.     

弹塑性有限元法分析不同型线凹模静液挤压时的应力和应变状态

静液挤压时,如凹模的型线不同,变形区的应力和流动状态是不同的。目前,静液挤压时,广泛采用圆锥模,但在挤压脆性金属时,产品经常会出现缺陷。为了研究产生缺陷的原因,应用弹塑性有限元法计算和分析了等应变型线模、余弦型线模、最短流线模、椭圆型线模及圆锥模等五种不同型线凹模在静液挤压时的应力和应变状态。结果表明,等应变型线凹模挤压时的应力和应变分布最为均匀,挤压力和塑变区内的拉应力最小,适用于挤压脆性金属。计算结果与实验比较,符合很好。计算结果对设计挤压凹模有很大的参考价值。     

基于流函数法的铝型材挤压导流模合理设计

采用模拟试验法和理论分析法研究了导流模内型材挤压的流动机理;提出了确定导流腔最小深度的理论依据;用基于流函数的上限分析法确定了实心薄壁铝型材挤压变形的流线方程、动可容速度场、应变速率场及上限功率;讨论了变形程度等参数对导流模最小理论深度的影响,得出实心薄壁铝型材导流腔深度的理论最小值为导流腔半宽的0.7～0.8倍。理论结果得到了试验证实,为导流模合理设计提供了可靠的依据。     

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.     

Extrusion die design for flow balance using FE and optimization methods

An approximate model has been developed to improve our understanding of flow control at the exit of an extrusion die. This paper deals with the design of traditional die flow correctors used in flat-faced aluminum extrusion dies. The numerical model uses the finite-element method combined with techniques of mathematical programming. Analytical sensitivities have been developed from the discretized finite-element equations in order to compute the necessary derivatives during optimization. The model includes thermal effects, and the extruded material is assumed to be strain-rate and temperature dependent.     

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.     

Modeling and simulation of polymer melts flow in the extrusion process of plastic profile with metal insert

The extrusion technology of plastic profile with metal insert is recently an advanced plastic processing method whose products keeps rising today for their excellent performance. However, the related fundamental research on polymer forming mechanism in the extrusion process of plastic profile with metal insert is lagging behind. With the development of computational fluid dynamics (CFD) theory, numerical method becomes an effective way to investigate such complex material forming problems as in the polymer extrusion process. In the present study, the mathematical model for three-dimensional non-isothermal viscous flow of the polymer melts obeying a Carreau model is developed based on the CFD theory. The Williams–Landel–Ferry equation is employed to involve the temperature dependence of material parameters. A decoupled numerical algorithm based on the penalty finite element method is conducted to predict the rheological behaviors of polymer melts within the complex flow channel. The streamline upwind/Petrov–Galerkin scheme is employed to improve the computational stability for the calculation of temperature field. Based on the theoretical model, the essential flow characteristics of polymer melts in the extrusion process of plastic profile with metal insert is investigated. The distributions of principal field variables like flow velocity, melt temperature, flow stress and pressure drop are predicted. The effects of die structure parameters including the intake angle and the distribution section length upon the melts flow patterns are further discussed. The variations of melt rheological properties versus different processing conditions like the volume flow rate and the metal insert moving velocity are also investigated. Some advice on practical processing operations of the extrusion process of plastic profile with metal insert is accordingly put forward based on the numerical results.