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.     

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.     

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.     

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.     

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.     

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.     

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 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.     

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.     

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.     

Upper bound analysis of extrusion of axisymmetric, piecewise homogeneous tubes

A kinematically admissible velocity field for tube extrusion is proposed, which reduces to a kinematically admissible field for solid rods in the limit as the mandrel diameter goes to zero. Although these upper bound calculations of extrusion pressures for solid rods are greater than those obtained by a similar analysis using a different kinematically admissible field, the results are qualitatively similar as regards dead zone formation and the form of the relation between extrusion pressure and extrusion ratio for fixed die cone angle and friction conditions.Analysis of tube extrusion shows that the linear relation between extrusion pressure and the logarithm of the extrusion ratio is preserved to a good approximation for friction-less extrusion in the case where the extrusion ratio is varied by fixing the mandrel size and die cone angle and varying the product diameter. However, the analytical results indicate that the linear relation is not exact. Nonlinearity is more evident as mandrel friction increases. The effective die cone angle for dead zone formation depends on friction conditions on the die and mandrel surfaces. The introduction of a stronger core symmetrically placed in the tube wall results in an increase in the extrusion pressure in approximately direct proportion to the relative strengths of the core and sheath and to the volume fraction of core.     

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.     

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.     

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.     

Analysis and design of flat-die hot extrusion process 1. Three-dimensional finite element analysis

Non-steady-state-coupled three-dimensional analysis is required for investigating complex material flow in practical flat-die hot extrusion processes of aluminum alloys in various channel sections. It is important since the material flow behavior actually determines the amount of distortion of the extruded product. Thus, a non-steady thermo-rigid-viscoplastic finite element program was developed for numerical simulations of the process. Since severely deforming elements of the workpiece can easily interfere with the sharp edges of the flat-die, an automatic remeshing module based on a simple section-sweeping scheme and new contact algorithm were incorporated to allow continuous simulation without manual intervention with less volume loss and computation time. With developed finite element program, non-steady finite element analyses of extrusion processes were carried out for two types of channel-section with constant bearing length of 5 mm. From simulation results, it was found that the exit velocity of the workpiece varied depending on the cross-sectional thickness of the exit and the amount of deflection of the workpiece was not greatly affected by variations of either the workpiece temperature or punch velocity under the present simulation conditions.     

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 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.     

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.