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

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Abstract

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.

Introduction

Extrusion is one of the most important metal forming processes due to its high productivity, lower cost and increased physical properties of the material. In metal forming industry, many kinds of non-symmetric rods are produced by the extrusion process. For example, extrusion from a round billet into a bar with various cross-sectional shapes such as ‘T’, ‘L’, ‘U’ and ‘I’, etc., by using aluminium alloys, steels, super alloys and titanium alloys are widely used. However, the process itself is difficult to analyse due to complex die shape and the rotational component of velocity that can cause unpredictable behaviour of the metal flow during extrusion. Metal flow does not remain in a radial plane passing along the longitudinal axis of the die and this behaviour results in unusual stress and strain distribution in the deforming material. The complexity grows even further in determining the optimum location of the exit section of the extrusion die particularly in the three-dimensional (3-D) extrusion of shaped sections. Therefore, attempts to obtain the optimum die configuration is still done mostly by applying the intuitive and empirical methods. Some analytical methods for predicting the metal flow in order to obtain the optimum die configurations for various cross-sectional shapes have been proposed by some workers. The literature survey of the related work to the problem was earlier reviewed in Ref. [1].

Most of the research work that has been done thus far, very specific to certain cross-sections considers that the exit section is not only symmetric but is also situated symmetrically along one of the planar axis of the die [2], [3], [4], [5], [6]. There is to the authors’ knowledge, no research work so far that has either been presented or done on the non-symmetric shaped sections or has considered the effect of their positionings except those of Kiuchi et al. [5] who applied his general theoretical analysis to only symmetric T-sections and some on the L sections. Chitkara and Celik [7] investigated recently the extrusion of non-symmetric T-shaped sections in their early report. Investigating the extrusion of non-symmetric shaped sections and effect of their positioning however, is a useful area because non-symmetric sections are widely used and positioning of the exit section not only affects the required extrusion pressure but varies considerably both the curvature of and the soundness of the extruded product. In previous paper by the Author's [8] it was suggested that off-centric positioning may be advantageous under certain conditions to obtain smaller extrusion pressures. Besides these, in some cases a curved extruded product may be required that would thus necessitate correct positioning of the exit section with respect to the entry section. Again, where horizontal extrusion machines are used even for the centric extrusion the extruded product may also come out bent due to the increased frictional resistance at the bottom part of the billet. In practice, most of the times a straight product is achieved by adding die lands to the die, which increases the extrusion load and deformation of the metal significantly. However, this may also be achieved by using correct positioning to obtain a straight extruded product.

In the present work, as an other application of the previous work of the Author's [1] to the generalised CAD/CAM solution to 3-D off-centric extrusion to both the symmetric and non-symmetric U- and I-shaped sections through the ruled-surface dies from an initially round billet in each case was carried out. In each of these cases, the process was analysed and the effect of positioning on the extrusion pressure was investigated with some supporting experiments in a few cases. Using the given theoretical approach, the extrusion pressure was computed in each case in terms of the cross-section of the product, reduction in area, die length, frictional conditions and off-centric position of the exit cross-section. The CAD/CAM of the extrusion dies is also discussed along with details of the experimental investigation. To show the validity of the computational method followed and the results obtained, some of the streamlined dies (ruled surface dies) were designed and manufactured using CAD/CAM, CNC machining and the EDM process and the experimental investigation conducted.

Section snippets

Construction of a general extrusion die for non-symmetric shaped sections

Detailed mathematical formulations of the theoretical approach followed for the extrusion of a non-symmetric shaped section from an initially round billet through a ruled surface die was given earlier in Ref. [1]. Here, to analyse the extrusion process for both the symmetric and non-symmetric U- and I-shaped sections only the necessary formulations applied are given briefly. Typical Fig. 1, Fig. 1) show schematically the perspective views of the off-centred extrusion of the shaped symmetric U-

Procedure for derivation of functions F1, F2, for I- and U-shaped sections in order to determine kinematically admissible velocity fields: sections situated at off-centric positions

A general unequal flanged non-symmetric I-shaped section within a circle of the type shown in Fig. 3(a)(i) is bounded with vertical and horizontal lines, that includes most of the differing cases of the unsymmetrical deforming regions (both the internal and external). Here, firstly the deforming regions for the I-shaped section are analysed and then a similar procedure is adopted to analyse the non-symmetric U-shaped section.

To analyse the problem, the number of segments into which the exit

The upper bound solution

The total power consumption, J, during extrusion through the die is the sum of the power losses due to the plastic deformation inside the die, (Wi), due to velocity discontinuities at entry, (We), and at exit, (Wf), and that due to frictional resistance at the interface between the material and the die, (Ws). Then the total power consumption can be expressed asJ=Wi+We+Wf+Ws.

The individual power loss components are evaluated as per procedure followed for the case of extrusion of a

Improved velocity field for non-uniform flow distribution

The present method given earlier in the text, assumes once again that the axial velocity, Vz, is constant at a given cross-section in the deformation zone. However, this conflicts with the experimental evidence. To adjust the present analysis to account for the change in axial velocity along the radial directions at any axial distance, Z an appropriate function, G(u,q,t), was introduced into the above set of equations, which obeys the boundary conditions. Then the kinematically admissible

Computer aided design of the extrusion dies

Extrusion of a non-symmetric section from an initially round billet was mainly carried out by coinciding the central axis of the billet either with the centre of gravity, i.e., centroid of the exit cross-section or with the centre of area of the exit section. The centre of area is that due to the intersection of two perpendicular lines parallel to the X- and Y-axis which divides the exit cross-section either vertically or horizontally into two equal areas. This is shown in typical Fig. 5(a) for

Variation of relative extrusion pressure with relative die length

Using the types of U- and I-shaped exit sections illustrated in Figs. 5(a), (d) and denoted as the theoretical model 1 and model 4, respectively, plots of relative extrusion pressure, p/σ1 (p/Y) versus relative die length, L/R values were computed theoretically for different off-centric positions, e1,e2 at friction factors, m=0.2 and 0.4. These are shown in typical Figs. 7(a) and (b), respectively. Reduction in area

Comparison of the author's results with other's research work

As there was no work so far to the authors’ knowledge that has been carried out on the off-centric extrusion of the U- and I-shaped sections, respectively, through the ruled-surface dies, some comparison between the author's analysis and that of the others could only be made by considering only the centric positionings for the symmetrically extruded shapes of the I-shaped sections that were investigated by Kiuchi et al. [5].

Numerically estimated results on this basis following the current

Conclusions

Previously developed analytical methods based on the upper-bound technique was applied to the investigation of a few cases of 3-D non-symmetrical extrusion of U- and I-shaped sections from initially round billets through the ruled-surface dies and the results compared with the somewhat limited experimental investigation. These suggest in general the following:

(i)That on the whole, the comparison of the estimated extrusion pressures and those obtained from the experimental investigation for the

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