FEM simulation optimization and slab method on rotating extrusion of thin-walled round tube under constant shear friction

This research considering constant shear friction is to explore the rotating extrusion of thin-walled round tube based on FEM simulation and slab method, and compares the results of both models to realize the variations and acceptance. The effective stress, the effective strain, the velocity field, the average thickness, the extrusion force, and the extrusion torque can be obtained from this study. It reveals the average thickness obtained from rotating extrusion is more uniform than that obtained from no rotating extrusion. The FEM optimization for extrusion force can be combined with Taguchi method, the L934 orthogonal table considers the four control factors which are outer diameter to thickness ratio, frictional factor, rotating angular velocity, half die angle, and three levels. The ranking of influence factors and the optimization combination can be obtained from FEM simulation optimization. Eventually the extrusion force between FEM simulation and slab method is compared under rotating angular velocity, 0.2 rad/s, the maximum error is 10.49 % and the minimum error is -0.29%; the average error is 4.16%, so the trend is in good agreement each other. Therefore, the both models can be verified.


Introduction
Greenwood et al. [1] compared with the traditional wire drawing process without rotating die, and the uniformity of deformation could be improved when the die was rotated with certain degrees.Kong et al. [2] constructed an experimental device for extrusion forming with cyclic torsion.Compared with the monotonous counterpart, the extrusion load was significantly reduced, and the cyclic frequency of the die had considerable impact on the reduction of the extrusion load.The experimental results confirmed that the die or punch as it rotated, its extrusion force dropped, and as the material was twisted by the rotation of the die, the shear strain was created in the process.Kong et al. [3] experiments of extrusion and drawing processes were carried out with or without swing dies.It was especially observed that the extrusion load tended to stabilize after the punch stroke traveled a certain distance during the swing die.Kim et al. [4] adopting the upsetting with the die rotation could transform the harmful effects of friction into beneficial effects, and the material was driven to rotate through reverse friction to reduce the load.Ma et al. [5][6] axisymmetric extrusion was performed through a rotating conical die.It was found that the extrusion load reduced with an increase of the rotating die speed.The greater the friction, the greater the reduction.The extrusion time took a long time to make the extrusion load reach a stable state.A larger frictional angle could be obtained by increasing the rotational angular velocity, but rotational angular velocity being too high may cause slippage at the interface between the billet and the die.Kargin et al. [7] described that as the thin-wall pipe material was drawn through the rotating die to the exit, the tensile axial stress would reach the maximum at the exit.Compared with the stationary die, the rotation of the die increased the axial stress and the die surface pressure, and the wall thickness was increased slightly during the drawing process, thereby reducing the tube length.Li et al. [8] said rotation was mainly suitable for the extrusion of circular cross sections.If it was not circular, it might cause fracture.Observe that the die does not rotate, the support container and the die rotated, and you could observe the effective stress of the material caused by the die rotation compared with the container rotation.Increased and more concentrated in the contact position between the billet and the die.When the die or the support container rotated, it could drive the material in the dead zone of the billet flow to extrude.Haghighat et al. [9] described the extrusion pressure was determined by optimizing the sliding parameters between the die and the material, and the process was analyzed by using the upper bound method.The higher the friction, the greater the impact of the velocity field on the extrusion pressure, and the best half die angle inverse point was compared with it occurred later when the friction was low, and has a significant impact on the research on the rotary extrusion process.Li et al. [10] studied that the same trend was obtained by finite element analysis simulation compared with the upper bound method analysis.As the rotational angular velocity was limited to 0-1 rad/sec, an increase of the rotational angular velocity was shown the extrusion pressure with a downward trend.When the frictional factor was high, the reduction ratio was relatively large, the degree of decline slowed down under the rotational angular velocity exceeded a certain value.

Slab method 2.1. Assumptions
Before derivation of slab model with constant shear friction, the main basic hypothesis are as below: 1.The die is assumed to be a rigid body, and the tube extruded is a perfect plastic material.2. Regardless of the shear stress distribution on the vertical section, the stress is uniformly distributed, i.e. the axial stress ( z  ), circumferential stress (   ) and radial stress ( r  ) are regarded as the principal stresses.3. Due to axisymmetric extrusion, the radial stress ( r  ) equals to the circumferential stress (   ). 4. The plastic deformation keeps uniform, and the plane still remains to be a plane.5.The von Mises yield criterion is adopted.6. Constant shear friction is assumed, namely τ=mk.
In the light that the rotating angular velocity ( ) is applied to the die, the frictional shear stress ( ) no longer applies on the axial direction, but acts on the angle  (Frictional angle) direction away from the z-axis, so that the frictional shear stress ( ) is able to be decomposed into the frictional shear stress along the half die direction ( cos   ) and the circumferential frictional shear stress ( sin   ), as shown in Figure 1 and Figure 2.    First the force equilibrium equation along the z direction is taking


, is very small, it can be ignored, and the following formula can be obtained: Furthermore, the force equilibrium equation in the radial direction (r direction) is taking, that is Arranging Eq. ( 1) then gets von Mises yield criterion is expressed as below: Owing to the axis-symmetry, then r     , Eq. ( 2) is able to be rearranged as follows (5) Eq. ( 1) is solved as B value can be replaced by Eq. ( 5), then B becomes Assuming constant shear friction, mk    value keeps constant.Integrating Eq. ( 6) on both sides Since c is an integral constant, it is obtained from an boundary condition.
At the exit, the axial stress   Because the cross-section area where 2 o r is outer radius of the tube at exit; 2 i r is inner radius of the tube at exit; t is the wall thickness.Therefore Eq. ( 10) is reformulated as where or D is outer diameter at a certain position in the forming zone.
Since the radial stress   r (12) In order to find the axial stress ( z1  ) at entry, substituting area ( 1 A ) at entry into Eq.( 11), the axial stress ( z1  ) at entry can be obtained as  Multiplying the positive axial stress at entry by the entry area, the extrusion force ( F ) can be obtained: Considering the influence of shear stress during forming, extra extrusion force should be included ( s F ).
Therefore, the total extrusion force ( t F ) becomes Once the half die angle is determined, the projection contact length ( L ) is given by using the following geometry.
The geometry of extrusion process is demonstrated in Figure 4. Using the geometry in Figure 4, the radius    at a certain position can be obtained as follows: The extrusion torque can be derived as below

Results of slab method
Figure 5 illustrates variations of the extrusion force with frictional factor ( m ) for distinct half die angles ( ).It notes that as the frictional factor ( m ) is increased under the smaller half die angle ( ), the extrusion force is increased greatly.

FEM Simulation
With a view conducting DEFORM-3D simulation analysis in this study, the various datum used are arranged in Table 2. Step distance Distance per step (mm/step) 0.1 Figure 8 is the flow stress-strain diagram of A286 super alloy steel.The solid line is a true stress-strain curve measured from the experiment, and the dash line is an estimated flow stress obtained after Curve Fitting.

Result of FEM simulation
Figure 9 is the analysis and simulation diagram without rotating die (  = 0 rad/sec), frictional factor ( m ) = 0.05, half die angle (  ) = 10.It shows the maximum velocity is 1.07 mm/sec, and the thickness distribution can be observed from the partial enlarged view from the tube formed.The maximum thickness is 0.449mm, the minimum thickness is 0.417mm, the average thickness is 0.432mm, and the extrusion force is 875.071N.Figure 10 is the analysis and simulation diagram of rotating angular velocity (  ) = 0.4 rad/sec, frictional factor ( m ) = 0.05, half die angle ( ) = 10.It demonstrates the maximum velocity is 1.06 mm/sec, and the thickness distribution can be observed from the partial enlarged view from the tube formed.The maximum thickness is 0.44mm, the minimum thickness is 0.412mm, the average thickness is 0.431mm, and the extrusion force is 823.736N.Compared with the state without rotation, the extrusion force is reduced by 5.87%.Figure 12 depicts comparison of extrusion force between slab method and FEM simulation under the frictional factor ( m ) = 0.05, the rotating angular velocity (  ) = 0.2 rad/sec.When changing half die angle ( ), it can be seen from the figure that at half die angle ( ) = 8, the inverse points of the extrusion forces for both models are happened.It notes that the maximum error is 10.49%, the minimum error is -0.29%, and the average error is 4.16%.The maximum extrusion force diagram in the L 9 3 4 orthogonal table is shown in Figure 14, the parameters combinations are A3(  =18), B1(  =0 rad/sec), C3( m =0.15), D2(D o1 /t=12.8),and the thickness variation is 13.54%, the extrusion force is 1986.613N.16 is the S/N ratio factor response table.The optimal parameters' combinations are A1( =10), B2(  =0.2 rad/sec), C1( m =0.05), D3(D o1 /t=16).
Figure 17 illustrates the optimal extrusion force diagram, the thickness variation value is 5.06%, and the extrusion force is 682.335N.

Conclusions
The main analysis results are able to be arranged as below: 1. Suitable rotating angular velocity and friction can reduce extrusion force.).The rank of influence is A (Half die angle) > D (Ratio of outer diameter to thickness at entry) > C (Frictional factor) > B (Rotating angular velocity), and the optimal extrusion force is 682.335N. 4. The extrusion force between FEM simulation and slab method is compared under rotating angular velocity, 0.2 rad/s, the maximum error is 10.49 % and the minimum error is -0.29%; the average error is 4.16%, so the trend is in good agreement each other.

Figure 1 .
Figure 1.Diagrammatic sketch for rotating extrusion of thin-walled round tube.

Figure 2 .
Figure 2. Diagrammatic sketch of the applied direction of frictional shear stress in rotating extrusion.

Figure 3
Figure 3 is the small stress element diagram of the thin-walled round tube formed in rotating extrusion, and the force equilibrium equations along z direction and r direction are obtained from this diagram.

Figure 3 .
Figure 3. Small stress element diagram of thin-walled round tube in rotating extrusion.

Figure 5 .
Figure 5. Variations of the extrusion force with frictional factor for distinct half die angles.

Figure 6 Figure 6 .
Figure6illustrates variations of the extrusion force with frictional angle (  ) for distinct half die angles ( ).It can be observed that when the frictional angle (  ) is small, the extrusion force becomes greater.As the frictional angle (  ) increases, when the half die angle (  ) is higher, the extrusion force is reduced greater.

Figure 7
Figure7depicts effects of half die angle ( ) on the extrusion force for distinct frictional angles (  ) and frictional factors ( m ), it can be observed that as the frictional factor ( m ) is low, the reverse point the point occurs at the smaller half die angle ( ).If the die is rotated, the extrusion force becomes lower, and the optimal half die angle gets smaller.

Figure 7 .
Figure 7. Effect of half die angle ( ) on the extrusion force for distinct frictional angles (  ) and frictional factors ( m ).

Figure 11
Figure 11 is the extrusion force changing with the rotating angular velocity (  ) under the half die angle ( ) = 10.It reveals that the extrusion force is 860N without rotation, and the extrusion force decreases

Figure 12 .
Figure 12.Comparison of extrusion force between slab method and FEM simulation under frictional factor ( m ) = 0.05, rotating angular velocity (  ) = 0.2 rad/sec.4. Taguchi method Figure 13 depicts the fishbone diagram of extrusion force optimization.It can be observed from the figure that there are 4 control factors and 3 levels, respectively: A. (Half die angle), B. (Rotating angular velocity), C. (Frictional factor), D. (Ratio of outer diameter to thickness at entry, D o1 /t).

Figure 14 .
Figure 14.The maximum extrusion force diagram in the L 9 3 4 orthogonal table.

Figure 15 .
Figure 15.The minimum extrusion force diagram in the L 9 3 4 orthogonal table.

Table 1
sums up the foregoing derivation formulae.

Table 1 .
Analysis formulae table for thin-walled round tube extruded by die rotation under constant shear friction.

Table 3 is
L 9 3 4 orthogonal table, after FEM analysis, the extrusion forces are obtained for each simulation experiment, and then the S/N ratio can be obtained through calculation.

Table 4
table.demonstrates the influence ranking of control factors in the L 9 3 4 orthogonal table, and the ranking of influence is A (Half Die Angle) D (Ratio of Outer Diameter to Thickness) C (Frictional Factor) B (Rotating Angular Velocity).Figure

Table 4 .
The influence ranking of control factors in the L 9 3 4 orthogonal table.