Three-dimensional analysis of round-to-angle section extrusion through straight converging die

The increasing interest in the modeling of metal-forming processes in recent years has brought the development of different analytical and/or numerical technique. However, due to the complexity nature of the problem, most of the attempts are made with plain strain assumptions. Among the different techniques used, the upper bound method is a convenient tool for evaluating the rate of work in processes involving predominantly plastic deformation of rigid/perfectly plastic material. The present study is an endeavor to remodel and apply the spatial elementary rigid region technique for analyzing extrusion of angle-section bars from round billets through the linearly converging die. Optimized values of the nondimensional average extrusion pressure at various area reductions have been computed and compared with experimental results. It is observed that the proposed technique can be used effectively with adequate accuracy to predict the optimal die geometry which requires a minimal forming stress at different reduction of areas and friction conditions.     

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.     

Axisymmetric extrusion and drawing through a die of arbitrary form

Axisymmetric extrusion and drawing through a die of arbitrary shape is analyzed by the Orowan method. The stress state in a representative element is determined by solving a boundary value problem of plasticity theory.     

Sensitivity analysis on ram speed of a direct extrusion process model using a porthole die through CEL method

Prediction of machine tool chatter requires the characterization of dynamic of the machine-tool-workpiece system by means of frequency response functions (FRFs). Uncertainties of the measured FRFs result in uncertainties of the calculated stability diagrams, therefore robustness of stability prediction against parameter perturbations is of high importance. Although there exist methods to determine robust stability in terms of stability radii, these methods either give a conservative estimate of the real uncertainties or are limited to perturbations of a few modal parameters, only. In this paper, a frequency-domain approach is presented to determine robust stability boundaries using the measured FRFs directly without any modal parameter identification. The method is based on an envelope fitting around the measured FRFs combined with some considerations of the single-frequency method. The application of the method is demonstrated in case of a turning operation, where the machine tool structure is characterized by a series of FRF measurements.     

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.     

Axi-symmetric tube extrusion through a flat-faced circular die: Numerical construction of slip-line fields and associated velocity fields

A numerical method of construction of axi-symmetric slip-line fields and their associated velocity fields [13–15] is applied to investigate the problem of axi-symmetric tube extrusion through smooth and rough flat-faced circular dies and the salient results presented in a computographic manner. A reasonably satisfactory agreement was observed between the estimated values of the non-dimensionalised specific extrusion pressures in each case when these were compared with the analytical and experimental work of Kudo [6] and that carried out by the authors. The deficiency in the technique used, the difficulties encountered and also some specific errors noticed during the construction of the fields, are highlighted and the results commented upon.     

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.     

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.     

Forward extrusion through steadily rotating conical dies. Part II: theoretical analysis

Forward extrusion of a cylindrical rod from a round billet was carried out through steadily rotating conical dies. Die rotation was shown to decrease the extrusion load and impose a twist shear strain into the partially extruded billet. The material twisting occurred both inside the container and inside the convergent die. However, not all the rotary work was transferred into shearing the bulk of the material, and this led to circumferential slippage at the rotating tool/material interface. The twisting degree on the outer surface of the material is quantified by a simple measurement technique. The influences of various process parameters on the extrusion load are also studied, including die semi-cone angle, die rotating speed and lubrication condition.     

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.     

Plastic flow through conical converging dies, using a viscoplastic constitutive equation

The influence of the working speed in drawing or extrusion processes is shown when a viscoplastic constitutive equation is used and when the yield condition is replaced by a yield inequality. It is shown that the drawing (or extrusion) pressure is increased when the speed of the process or when the viscosity coefficient of the material is increased, or when the diameter of the die is decreased or when the mean yield stress of the billet is reduced. This increase is more significant for higher reduction ratios and semi-cone angles. It is shown that the optimum semi-cone angle is also dependent on the speed of the process as is the critical semi-cone angle, which corresponds to the possible formation of a dead-zone near to the die surface. The pressure on the die surface is variable and depends on the speed of the process also. From the numerical examples given it is evident that in many practical circumstances the influence of speed may be significant even for very low working speeds.     

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.     

Round-to-hexagon drawing through straightly converging dies: an application of the SERR technique

The modelling of metal forming processes in recent years has brought the development of different analytical and/or numerical techniques. The SERR technique is eminently suitable for analyzing drawing/extrusion of sections having re-entrant corners. However, in its present form, it cannot be used to analyse forming processes where the deformation zone has curved boundaries, since the elementary regions will no longer remain rigid. The present study is an endeavor to remodel this technique so that it can handle round billets. The circular cross section of the round billet is approximated by a regular polygon of equal area and the number of sides of the polygon is progressively increased until convergence of the drawing stress is achieved. As a test, the drawing of hexagon section bars from round billets through the straightly converging dies is analyzed     

Evaluation of a pyramid die extrusion for a hollow aluminum profile using FE simulation

The pyramid die extrusion for a hollow aluminum profile was analyzed to investigate the potential of such innovative dies. For this purpose, the pyramid and conventional porthole dies were respectively designed for a given hollow aluminum profile. And the extrusion process was comprehensively studied by performing different types of finite element simulation, such as the analysis of steady state, transient state and billet skin tracking. The effects of pyramid angle on the evaluation parameters of extrusion, such as extrusion load, material flow, exit temperature, length of transverse weld, quality of longitudinal weld, back end defect and die stress were overall analyzed and compared with the conventional porthole die. Through this study, the advantages and shortcomings of pyramid die were well concluded, which should be important information for die designers and makers.

     

Axi-symmetric tube extrusion through a smooth conical or cosine die and over a conical or ogival mandrel: numerical construction of axi-symmetric slip-line fields and associated velocity fields

The numerical method of construction of axi-symmetric slip-line fields and their associated velocity fields suggested earlier for rod extrusion (Chitkara NR, Butt MA. Axi-symmetric rod extrusion through smooth conical, cosine and lat-faced dies, International Journal of Mechanical Sciences, submitted) and tube extrusion (Chitkara NR, Butt MA. Axi-synnetric tube extrusion through a flat-faced circular die: International Journal of Mechanical Sciences, 1997;39(3);341–366) is employed to construct slip-line fields and their associated velocity fields for a few cases of forward tube extrusion through smooth, rigid conical and cosine dies and over a smooth rigid conical or ogival mandrel. The computographic plots of slip-line fields and associated velocity fields are given in the form of weighted and directed velocity vectors as are the normal pressure distribution on both the die and the mandrel surfaces. The values of the non-dimensionalised mean extrusion pressures, are compared in each case with a similar case of plane strin extrusion and the results commented upon.     

反挤压杯型件凹模受力分析

用二种不同方法分析了冷挤压凹模侧壁的受力情况，比较了它们各自的特点并说明了产生误差的原因，给实际设计凹模提供了参考。     

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.     

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.