FEM simulation and optimization on rotating extrusion of bi-metal rod with constant shear friction

This study assuming constant shear friction adopts FEM simulation and the optimization to investigate into extrusion of bi-metal rod which consisted of Aluminum alloy (Core) and Copper alloy (Sleeve) with a rotating die taking tonto account the interface at bi-metal rod under the bonding situation. The forming characteristics, e.g. the effective stress, the effective strain, the velocity field, the extrusion force, the extrusion torque, and die stress can be gained by using FEM commercial software. The FEM optimization for extrusion force combines Taguchi method to establish the L934 orthogonal table that includes four control factors which are rotating angular velocity (ω), frictional factor at the interface of sleeve and die (m1), half die angle (α°), and inner die filet (R). The ranking of influence factors and the optimization combination can be obtained from FEM simulation to provide the reference to the industries.


Introduction
Alcaraz et al. [1] derived an analytical approach to the extrusion of bimetal tubes through a conical die and considered different extrusion conditions and material combinations to investigate the stress levels at critical zones.Chou et al. [2] explored the extrusion of powder/metal composite rods using finite element model and experiments, and the results showed that a smaller core radius can result in a better consolidation state for the powder.Hwang et al. [3] proposed a mathematical model using stream functions to simulate the plastic deformation behaviour of composite rods during traditional extrusion in a conical die (without the rotation of die), which satisfies the incompressibility and velocity boundary conditions inside the die.They conducted experiments to investigate the effects of the aspect ratio on extrusion load and half-diameter ratio at exit, and verified the validity of the model.Kazanowski et al. [4] synthesized experimental and numerical results with the effect of the initial geometry of the billet on the final geometry of the bi-metal rod extrusion without rotation.The results showed that shortening the length of the core material before extrusion improved the geometrical tolerances of the final product and reduced the tendency for material separation within the bi-metal rod.Ma et al. [5,6] used a stable rotating conical die to extrude lead rods and carried out the numerical analysis with the upper bound method.A continuous spherical velocity field and a torsional velocity field constrained within the conical die were conducted, and investigated the hoop sliding between the material and the die.By increasing the rotating speed of the die and reducing the half die angle, more torsional shear strain can be applied to the material.In addition, increasing the rotating angular velocity of the die can also increase the frictional force between the material and the die, resulting in a greater torsional shear strain in the material.Ebrahimi et al. [7] developed a new upper bound approach to derive a mathematical model for power consumption evaluation in tube extrusion process without rotation.The power which changing half die angle and calculated the optimal and critical half die angle could be minimized.It was noted that as reduction ratio increases, half die angle also increases, and change in optimal half die angle is more significant in cases with higher friction.Comparing the load-displacement curves obtained from the numerical and analytical solutions based on upper bound method with experimental results showed a close agreement in trend, it indicates the consistency of the results.Kargin et al. [8] explored the sink drawing process of thin-wall pipes through rotating dies in the plane perpendicular to the drawing axis.A mathematical model was developed to determine the deformation mode, and an application program was created to calculate parameters.Computer simulation was carried out, and it was established that rotation of the drawing die complicates the deformation mode compared to stationary drawing dies.This method could be recommended for the plastic twisting of pipes and ensuring uniform wear of the drawing channel.Haghighat et al. [9] developed a generalized spherical velocity field based on upper-bound analysis for bi-metal tube extrusion process through dies with a moving cylindrical mandrel.The study showed a good agreement between the analytical results, finite element simulation, and experiment.The upper-bound model developed can be used for a fast estimation of the extrusion force of bi-metal tubes and to find the optimal die length that minimizes the extrusion force for a given die shape.Haghighat et al. [10] proposed a kinematically admissible velocity field for bi-metal tube extrusion using a rotating conical die.They evaluated the internal power, friction, and power consumption due to velocity discontinuity at the surface and demonstrated good agreement between the results obtained using the upper bound method and FEM with the rotating die.The bi-metal rod combines the advantages of different metals.Using the transmission shaft as an example, the maximum torsional shear stress occurs on the surface of the transmission shaft.The outer layer can use high-strength metal to resist torsional shear stress.The torsional shear stress occurred near the centre's part of the transmission shaft is relatively lower, so the inner layer can use lightweight metal to reduce weight.That is the objective why the authors investigate into the bi-metal rod using the rotating die.

FEM simulation
Figure 1 depicts assembly diagram of rotating extrusion of bi-metal rod.From this figure, the bi-metal rod comprised of Aluminum alloy (Core) and Copper alloy (Sleeve).The flow stress was be done from the experiment [3] shown in Figure 2. The geometry of bi-metal rod is illustrated in Figure 3.The outer layer is Copper alloy (Sleeve); the diameter of outer layer is 30 mm.The inner layer is Aluminum alloy (Core); the diameter of inner layer is 18 mm.Therefore, the ratio of inner diameter to outer diameter (Aspect ratio,  ) is  =0.6.The interface between Aluminum alloy and Copper alloy is sticking friction, indicate the frictional factor (m 2 ) is 1.0 and the frictional factor (m 1 ) at the contact surface between Copper alloy and die is 0.1, 0.2 and 0.3 respectively, and the half die angle () is 15 currently, but changing with half die angle.

FEM analysis
In this paper, the Deform-3D commercial software is used to analyse the forming characteristics of bi-metal rod extrusion using the rotating die.The FEM research results, such as the effective stress, the effective strain, the velocity field, the extrusion force, the extrusion torque, and the die stress, are able to be obtained.The FEM simulation conditions of bi-metal rod with a rotating die are demonstrated in Table 1.Through FEM simulation the velocity fields can be given in Figure 4.In the light of the outer layer harder than the inner layer, and the hard layer (Copper alloy) contacts with the die.Therefore, the velocity field increases with increasing the rotating angular velocity.Table 2 shows the frictional angle () for each rotating angular velocity, when  = 0 rad/s, the  is 0; when  = 2 rad/s, the  is 22.11; when  = 4 rad/s, the  is 50.79.5.As frictional factor (m 1 ) increases to 0.2, the velocity field increases with the increase of the rotating angular velocity.The frictional angle () at each rotating angular velocity is shown in Table 3, as  = 0 rad/s, the  is 0; as  = 2 rad/s, the  is 29.55; as  = 4 rad/s, the  is 50.17.Figure 7 shows effects of rotating angular velocity on the extrusion force for different frictional factors.
It is noted that the extrusion force decreases with increasing rotating angular velocity.And as the frictional factor increases, the extrusion force increases as well.It displays that the extrusion torque increases with increasing the rotating angular velocity.As the rotating angular velocity increases largely, the extrusion torque will be stable, indicates there is some slips, and it more likely to at lower frictional factor.Figure 11 depicts the extrusion forces under various aspect ratios.At reduction of area R A = 25% and 35%, the extrusion force increases with the decrease of aspect ratio.It indicates that smaller aspect ratio, there are more copper alloy fraction of bi-metal rod, it stands for the harder bi-metal rod.
Figure 12 shows the extrusion torques under various aspect ratios (λ).The extrusion torque is increased with the decrease of aspect ratio.The extrusion torque under reduction of area R A = 25% is lower than R A = 35%.

Orthogonal table
Table 4 shows four control factors and three levels.6 depicts the best combination.The extrusion force is 74.04kN.Figure 14 shows the best simulation results, the maximum effective stress occurs on the oblique contact length of hard layer, it is 397 MPa; the maximum effective strain is 2.7 mm/mm, and the maximum velocity happens in the soft layer, it is 44.5mm/sec.

The worst simulation results in orthogonal table
The worst combination is in Table 7, as frictional factor increases to 0.3, half die angle increases to 18, and inner die filet increases to 7mm, it shows that the extrusion force is increased to 103.4kN.In Figure 15, the maximum effective stress occurs on the oblique contact length of hard layer, it is 389MPa, the maximum effective strain is 2.49mm/mm, and maximum velocity occurs in the soft layer, it is 45mm/sec.

The optimal simulation results
Table 8 shows the simulation results of optimization.The extrusion force is 66.45kN, and the S/N ratio is -36.47.In Figure 16, the maximum effective stress occurs on the oblique contact length of hard layer, it is 340MPa, the maximum effective strain is 1.3mm/mm, and the maximum velocity is 62.3mm/sec.

Conclusions and future works 5.1. Conclusions
Through a series of research results, some major contributions could be summarized as below: 1.The rotating velocity can effectively reduce the extrusion force, but if the rotating velocity () is higher than 3 rad/s in such conditions, which will cause a slight slip.2. The friction angle () increases with the increase of the rotating velocity (). 3. The rotating velocity is to make the increase of extrusion torque and the decrease of extrusion force.
For extrusion force, the better the smaller friction factor and half die angle.The inner die filet is the least influence on the extrusion force, so it can be selected for convenient processing.4. For example, when the reduction of area is 25%, the aspect ratio is 0.6 at the entry, under the die without rotation the aspect ratio is around 0.596 at the exit; under the die with rotation whereas the aspect ratio at exit can be improved to 0.598.It indicates that the bonding at exit under the die with rotation is better than that without rotation.

Future works
In the near future, simulation can be performed for various friction types.Moreover establishing the slab method is to compare the FEM 3D simulation, and simulation of die stress is able to be conducted as well.

Figure 3 .
Figure 3.The geometrical dimensions of bi-metal rod.

2 .
The analysis resultsFigure6illustrates variabilities of the frictional angle () with rotating angular velocity for different frictional factors.Under rotating angular velocity  = 2 rad/s, the frictional angle at frictional factor m 1 = 0.1 is lower than one at frictional factor m 1 = 0.2.As rotating angular velocity increases to  = 4 rad/s, the frictional angles at friction factors 0.1 and 0.2 are almost same, it indicates that the sliding situation is might occur.

Figure 6 .
Figure 6.Variabilities of the frictional angle with rotating angular velocity for different frictional factors.

Figure 8
Figure 8 depicts effects of rotating angular velocity on extrusion torque for different frictional factors.It displays that the extrusion torque increases with increasing the rotating angular velocity.As the rotating angular velocity increases largely, the extrusion torque will be stable, indicates there is some slips, and it more likely to at lower frictional factor.

Figure 9
Figure 9 depicts effects of half die angle on the extrusion force for different rotating angular velocities.The extrusion force increases with increasing half die angle.When the rotating angular velocity of die increases to 4 rad/s, the extrusion force decreases compared to 0 rad/s (without rotation).Figure 10 demonstrates variabilities of extrusion torque with half die angle for different frictional factors.The extrusion torque decreases with the increase of half die angle.The extrusion torque increases with increase of frictional factor.

Figure 10
Figure 9 depicts effects of half die angle on the extrusion force for different rotating angular velocities.The extrusion force increases with increasing half die angle.When the rotating angular velocity of die increases to 4 rad/s, the extrusion force decreases compared to 0 rad/s (without rotation).Figure 10 demonstrates variabilities of extrusion torque with half die angle for different frictional factors.The extrusion torque decreases with the increase of half die angle.The extrusion torque increases with increase of frictional factor.

Figure 7 .
Figure 7. Effects of rotating angular velocity on the extrusion force for different frictional factors.

Figure 8 .
Figure 8. Effects of rotating angular velocity on extrusion torque for different frictional factors.

Figure 9 .
Figure 9. Effects of half die angle on the extrusion force for different rotating angular velocities.

Figure 10 .
Figure 10.Variabilities of extrusion torque with half die angle for different frictional factors.

Figure 13 .
Figure 13.The fishbone diagram established from Taguchi method.

Figure 15 .
Figure 15.The worst simulation results.4.5.Factors response diagramFigure16is the factors response diagram for optimization of extrusion force.It can be seen that the ranks influence of B (m 1 ) is the first, the second is C (), the third is A (), and the forth is D (R).The optimal parameters experiment for extrusion force is conducted by selecting the levels with the maximum S/N ratio for each factor, which are A3 ( = 4 rad/s), B1 (m 1 = 0.1), C1 ( = 12), D1 (R = 3 mm).

Figure 16 .
Figure 16.The factor response diagram for optimization of extrusion force.

Table 1 .
The FEM simulation conditions of bi-metal rod with a rotating die.

Table 2 .
The frictional angle for each rotating angular velocity.

Table 3 .
The frictional angle for each rotating angular velocity.

Table 5
is L 9 3 4 orthogonal table form Taguchi method.

Table 4 .
The table of control factors and levels.Variations of extrusion torque with aspect ratio for various area reductions.Figure11.Effects of aspect ratio on the extrusion force for various area reductions.The best simulation results in orthogonal tableTable

Table 6 .
The best combination in orthogonal table.The best simulation results in orthogonal table.

Table 7 .
The worst combination in orthogonal table.

Table 8 .
The optimization simulation results.