Differential flow velocities control method for push-bending of the thin-walled tube with a 0.9D bending radius by differential lubrication

The smaller the relative bending radius of the bent tube, the more difficult the tube blank is to be bent. In this paper, the push-bending process for the aluminum tube with a relative bending radius of 0.9D was analyzed through simulation. There was tangential tensile stress concentration occurring at the front endpoint of tube intrados, which increased the risk of cracking. That is because the bending tube’s extrados moves less angular distance than its intrados by the uniform lubrication method, causing excessive deformation of the cross-section. Therefore, a differential lubrication method using the lubricant with a larger friction coefficient at intrados and the lubricant with a smaller friction coefficient at extrados was proposed to increase the angular velocity of the tube’s extrados. It was found that the differential lubrication method helped avoid excessive thinning at the front endpoint of tube intrados through simulations. Furthermore, push-bending experiments of 5A02 aluminum alloy with a relative bending radius of 0.9D were carried out to investigate this effect. The experimental results showed that the differential lubrication method effectively avoided cracking, which agreed with the simulation results.


Introduction
The metallic bent tubular part is an important component in the delivery system and plays a key role in aviation and aerospace [1]. Thin-wall tubes (relative wall thickness of tube t/D ≤ 0.05, t-original tube wall thickness, D-outer diameter of the tube) with a small relative bending radius (R ≤ 1.5D, R-bending radius) have been needed for vital products. Most of these products are fabricated by sheet stamping and then welding in engineering. However, the requirements of lightweight, high strength, and high performance are not fulfilled in this situation. And the integral forming of the thin-walled bent tubes with a small relative bending radius is difficult. The smaller the relative bending radius of the tube, the more difficult the tube to form [2].
Kuboki et al. [3] presented a new shear bending method of tubes for controlling bending radius. The method consists of two steps using a pair of mandrels. The first step makes space inside the tube, and the second step inserts a new mandrel into the space for transferring the mandrel shape to the bent part. The bending radius becomes a target by this method, which is never been achieved by conventional shear bending. Moreover, the experiments revealed that the method applies to copper alloy. Mizumura and Kuriyama [4] studied the effect of the bending method on the hydroforming of a tube in a subsequent process by using experiments and the finite element method. Rotary draw bending and intrusion bending were investigated. A tube was bent into an S-shape by each bending method, and a hydroforming test was carried out with the same hydroforming die. The burst occurred on the intermediate side during axial feeding in both bending cases. They also found that intrusion bending left a larger hydroforming allowance than rotary draw bending, because the wall thickness after intrusion bending was greater than that after rotary draw bending. Ruan et al. [5] investigated the hydroforming method to form the ultrasmall bending radius elbow with a relative bending radius of 0.56D. The preliminary processes were investigated using numerical and experimental ways. Then they presented a high-performance optimized process design to achieve an ultrasmall radius elbow. Zhang et al. [6] analyzed the heatassisted rotary draw bending to form high-strength titanium tubes with a small bending radius of 1.5D. Robust design optimization was investigated for the thermal-mechanical coupled forming process. The optimized solutions reduced the maximum section distortion rate and the maximum wall thickness thinning. Li et al. [7] studied larger diameter thinwalled (D-50 mm, t-1 mm) Al-alloy tube rotary draw 1.5D bending by the multi-objective robust design method. The significant noise factors were variations in tube properties, fluctuations of tube geometries, and friction. And the robust design of mandrel extension length and boosting ratio was realized through simulations and experiments.
The tube with a small relative bending radius has been studied by many researchers using a push-bending method because it helps enhance the formability due to the stress state. Jiang et al. [8] developed a modified push-bending process with a polyurethane rod as a mandrel. The stainless steel bent tube with a relative bending radius of 1.94D was formed. Song et al. [9] demonstrated the granular mediabased thin wall elbow (D-70 mm, t-0.7 mm) 1D pushbending process involves filling a tube with spherical particles in simulations and experiments. The role of granular media in improving wrinkle resistance during bending was validated for large-diameter thin-walled tubes. Liu et al. [10] found that if the wall thickness of the tube is smaller than the particle size, wrinkling significantly depends on micro contact force distribution in granular media. What is more, a simplified formulation was proposed for predicting wrinkling of thin-walled elbow tubes in this situation, validated by experiments of a bent tube with 0.7 mm wall thickness and 1.5D relative bending radius. Kami and Dariani [11] found that the push-bending process by filling a softer rubber rod in the middle of the tube and harder rubbers at both ends reduced the risk of wrinkling, and formed tubes with a relative bending radius of 1.5D successfully.
Recently investigators have examined the effects of lubrication on deformation resistance. Moreover, the friction coefficient is controlled by choosing various lubrication methods to change the material flow. Hama et al. [12] indicated the lubrication condition depended on the position at the die surface and investigated lots of lubrication conditions on the die surface during the square cup sheet hydroforming process. Kaya [13] studied the effect of lubrication conditions on the hydroforming of aluminum alloy tubes and indicated that dry lubricants performed well in the expansion zone, while wet lubricants performed well in the guiding zone. Zhao et al. [14] analyzed rotary draw bending to explore the effect of friction on wrinkling in simulations.
The results showed that the material flow parameters such as the height of the wrinkling wave decreased obviously with the increase of the friction coefficient between die and tube by changing the type of lubricant. Oliveira et al. [15] studied the lubricant performance in the bending of steel and aluminum tubes through experiments. It was found that the surface quality and thinning degree of the bending tube were significantly affected by the lubricant type.
Considering the difficulty for tubes with a relative bending radius of 0.9D to form, in this work, the differential lubrication method is used to reduce the difference in angular velocities of the bending tube's intrados and extrados to reduce the cross-section deformation at the front endpoint of tube intrados. The bend zone of the die cavity is divided into two zones: the inside zone and the outside zone. Differential lubrication is applied to differential zones to explore the influence of forming qualities of 0.9D tubes through simulations and experiments in the push-bending process. The different lubrication methods on the forming properties are analyzed and compared.

Principle of push-bending
The principle of push-bending is shown in Fig. 1. The punch provides thrust to make the straight tube bend along the die cavity. The sectional cylindrical polyurethane rubbers are thrust between the punch and the flexible rod to support the tube blank to avoid wrinkling.
As Fig. 2 shows, the tube blank is divided into the front zone, middle zone, and rear zone. And front endpoint of tube intrados (FEI) is the endpoint where the front zone and the middle zone meet at the intrados. The die cavity is divided into the input zone, bend zone, and output zone, as shown in Fig. 3. During the forming process, the tube Fig. 1 Schematic diagram of tube push-bending goes through the input zone, the bend zone, and the output zone in turn. The material of the tube in the input zone is considered as having no deformation. The material in the bend zone is bending. Similarly, the material in the output zone is straightening.
The wall thickening and wall thinning are interactive defects that restrict product quality in the push-bending tubes. Material accumulation leads to wall thickening at the intrados where compressive stress occurs. Conversely, material distribution leads to wall thinning at the extrados where tensile stress occurs. As Eq. 1 shows, the maximum wall thickening degree and the maximum wall thinning degree quantitatively represent two quality indexes.
where t 0 is the initial wall thickness of the tube, t max is the maximum wall thickness at the intrados of the bent tube, and t min is the minimum wall thickness at the extrados of the bent tube, as shown in Fig. 4.   Tables 1 and 2. The contact between the punch and tube is defined as nodeto-surface. And other contacting surfaces are defined as surface-to-surface. The tube was predefined as a variable body and its mesh type was S4R. The number of tube elements was 7884. The rubber was predefined as a variable body and its mesh type was C3D8R. The number of rubber elements was 4070. The die and the punch were predefined as discrete rigid and their mush type was R3D4. The push displacement of the punch was 85 mm. The Mooney-Rivlin strain energy was chosen as the constitutive relation of rubbers. The hardness of the rubber was A80, its diameter was 29 mm, and its height was 10 mm. The friction coefficient between the tube and the die was 0.05. The Coulomb friction model is selected to set the friction coefficient.
Step time is 10 s.
Simulation contours of the maximum principal stress of the tube are shown in Fig. 6, where a positive value is expressed as tensile stress and a negative value is expressed as compressive stress. The push-bending process can be divided into three stages according to the stress value of FEI.
Stage I, from 0 to 4.5 s, is called the pre-bending stage when the stress value of FEI is zero. The front zone of the   Fig. 6 Simulation contours of maximum principal stress of the tube tube has been fed into the bend zone of the die cavity, while the FEI has been staying in the input zone. Stage II, from 4.5 to 6.5 s, is called as bending stage when the stress value of FEI is beyond zero. It is apparent from Fig. 7 that a tangential tensile stress concentration has been occurring when the FEI is fed into the bend zone of the die cavity. Stress concentration generally causes tube cracking at FEI.
Stage III, from 6.5 to 10 s, is called as straightening stage when the stress value of FEI has been above zero. The FEI has been fed into the rear zone of the die cavity.

Cracking at FEI of the tube
During the push-bending process, the FEI is prone to cracking. This is because the tangential tensile stress is concentrated at the FEI in stage 2, as shown in Fig. 8.
To reduce the tangential tensile stress concentration at the FEI is to reduce the excessive deformation there. As Fig. 9 shows, the diameter of the tube blank is D, the bend zone radius is R, and the wall thickness is t. A micro-element of the cross-section at the FEI is taken, whose length is dl.
The extrados length of the bend zone is greater than the intrados length. When the micro-element enters the bend zone, the extrados moves more distance than the intrados. It indicates that the material of the extrados must be fed at a higher speed than it of the intrados. The difference Δ of the material flow angular velocity between intrados and extrados of the micro-element is given by: where in is the angular velocity of the micro-element intrados. out is the angular velocity of the micro-element extrados. in is the angle of the micro-element intrados. out is the angle of the micro-element extrados.
As Fig. 10 shows, after the micro-element feeds dl, the material at the extrados of the tube needs to move an extra distance dp . So that the extrados and the intrados of the tube are consistent. The deformed cross section C is close to the original cross-section C 0 , as a result, to reduce the tensile stress concentration at the FEI. The equation for dp is:  According to Eq. 4, reducing dp help reduce Δ . The extrados elongation of the bending micro-element is δl out , and the intrados compression of it is δl in . So the extrados length of the bending micro-element is dl + δl out , and the intrados length of it is (dl − δl in ) . The equation for d in and d out is: Simultaneous Eqs. 3 and 5: As Eq. 6 shows, dp can be reduced by increasing δl in and δl out . It indicates that the Δ can be reduced by increasing the compression at the intrados and the elongation at the extrados. However, increasing the compression at the intrados is prone to wrinkling. As a result, increasing the elongation at the extrados is the best way to reduce Δ .
It is known that lubrication affects the fluidity of deforming metal material. It can reduce Δ by using the differential lubrication method that has the lubricant with a larger friction coefficient at intrados and the lubricant with a smaller friction coefficient at extrados (LISE). So that it helps reduce the tensile stress concentration at FEI and eliminate the cracking at it.

Differential lubrication methods for push-bending simulation
Because the cracking at FEI is caused by a big difference Δ of the material flow angular velocity between intrados and extrados, differential lubrication methods for push-bending of 0.9D radius are proposed. The difference in material flow angular velocity between intrados and extrados is controlled by the difference friction coefficient. The bend zone is divided into two zones: the inside zone and the outside zone, as shown in Fig. 11. The boundary between inside zone and outside zone is in the middle of the tube. Table 3 shows the differential lubrication scheme. 1# uses a uniform lubrication method with a larger friction coefficient (LU), which is the control group. The friction coefficients are both 0.05. 2# uses a uniform lubrication method   with a smaller friction coefficient (SU). The friction coefficients are both 0.02, which raises the material flow angular velocities of both intrados and extrados in contrast with 1#. 3# uses LISE differential lubrication method where in is 0.05 and out is 0.02, which raises the material flow angular velocity of extrados in contrast with 1#. 4# uses the differential lubrication method using the lubricant with a smaller friction coefficient at intrados and the lubricant with a larger friction coefficient at extrados (SILE) where in is 0.02 and out is 0.05, which raises the material flow angular velocity of intrados, in contrast with 1#.

Influence of differential lubrication method on cracking
The dp represents the arc length that the extrados of the micro-element lags behind the intrados, as shown in Fig. 12. Moreover, the greater the value of dp, the greater the compressive stress concentration at FEI, and the easier it is to crack at it. It is apparent from Fig. 13 that differential lubrication methods affect dp. At stage II, dp values are 3#, 1#, 2#, and 4# from smallest to largest. From this figure, it can be seen that the LISE differential lubrication method results in the lowest value of dp in numerical simulation. SU lubrication method does little to help reduce dp, while the SILE differential lubrication method results in the highest value of dp.
It can be seen from Fig. 14 that the differential lubrication methods affect the thickness at FEI. At stage II, thickness values at FEI are 3#, 1#, 2#, and 4# from smallest to largest. It indicates that the LISE differential lubrication method results in the highest value of wall thickness at FEI in numerical simulation. SILE differential lubrication method and SU lubrication method result in thinner FEI wall thickness.
The above indicates that the LISE differential lubrication method is the best lubrication method to reduce Δ and the thinning at FEI, compared with SILE differential lubrication method and SU lubrication method. Figure 15 shows that the wall thickness values of intrados of tubes increase first and then decrease. The results of maximum wall thickness degree, as shown in Table 4, show that the LISE lubrication method increases the maximum wall thickness at the intrados most and the SILE lubrication method decreases it. Moreover, there are no significant differences between the SU lubrication method and the LU lubrication method.  Table 5, show the simulation values of maximum wall thinning degree. It indicates that the SILE differential lubrication method decreases the minimum wall thickness at the intrados least, while the SU lubrication method decreases it most. And the SILE differential lubrication method increases the maximum wall thinning rate less than the SU lubrication method.

Discussion
These simulation results suggest that the risks of cracking at FEI from high to low are 2# > 4# > 1# > 3#. There is no significant difference in the thickening at intrados. And the thinning of extrados thickness from large to small is 2# > 3# > 4# > 1#.
The LISE differential lubrication method helps to reduce the risk of cracking at FEI. It is because the LISE differential lubrication method reduces dp most, which means the difference in angular velocity between intrados and extrados is the least. And it results in the highest value of maximum wall thickening at intrados. The maximum thinning rate at extrados is smaller than the BU lubrication method but greater than the LU lubrication method and SILE differential lubrication method.
The SILE differential lubrication method's angular velocity difference between intrados and extrados is the largest, for increasing the dp most. And its wall thickness at FEI thins most. As a result, the SILE differential lubrication method has the highest risk of cracking at FEI.
The SU lubrication method increases the dp and angular velocity difference between intrados and extrados. So it still has a high risk of cracking in contrast with the LU lubrication method. And its wall thickness at extrados thins most.

Experimental design
The equipment of the 0.9D push-bending experiment is shown in Fig. 17. The tube blank was fully covered with polytetrafluoroethylene (PTFE) film. Table 6 shows the differential lubrication scheme of the bend zone, which corresponds to the simulation scheme. To apply differential lubrication, MOS 2 lubricant was sprayed on the specific zone of die surface. And the MOS 2 lubricant beyond the boundary was wiped out.

Fig. 16
Simulated thickness distribution at the extrados  Figure 18 shows the 0.9D bent tubes after forming. There was no cracking at the 3# tube. And 1#, 2#, and 3# cracked. The experimental results were consistent with the simulations. Figure 19 shows that the intrados and extrados wall thickness of 3# between the experiment and simulation were in good agreement. The wall thickness of 3# met the 0.9D elbow standard.

Experimental results and discussion
Therefore, spraying MOS 2 on the outside zone of the die cavity and fully covering the PTFE film on the tube blank is the best way to push-bend the 0.9D aluminum alloy tube in this case. The use of the LISE differential lubrication method is the optimal lubrication method to eliminate cracking at FEI.
The thickness value at tube intrados declines. It is because the material accumulates at the 0-45° intrados, meeting the simulated results. Though the maximum wall meets the 0.9D elbow standard, it is meaningful to make further prospects to uniform the wall thickness distribution. The inside zone can be considered to be divided into zones and use differential lubrication method to let 0-45° intrados materials flow better.

Conclusion
(1) A tangential tensile stress concentration at FEI occurs in stage II of the simulated push-bending process. It is affected by the difference in the angular velocities between the intrados and extrados of the tube. And it can cause the tube to crack at FEI.  (2) A 0.9D push-bending finite element model was established to control the difference in the angular velocities between intrados and extrados by the differential lubrication methods. The LISE differential lubrication method is beneficial to reduce the difference in the angular velocities. And it reduces the thinning at FEI in stage II, which means it can reduce the risk of cracking. But it slightly increases the wall maximum thickness at intrados and decreases the wall thickness at extrados, in contrast with the LU lubrication method. The SILE differential lubrication method and the SU lubrication method both increase the difference and the thinning at FEI in contrast with the LU lubrication method, which means they increase the risk of cracking. (3) The 0.9D tube push-bending experiments were carried out to bend the 5A02 aluminum 0.9D tube. The LISE differential lubrication method by fully covering the tube with PTFE film and spraying MOS 2 on the outside zone of the die cavity formed the no-cracking 0.9D tube successfully. And the SILE differential lubrication method and uniform lubrication method formed cracking 0.9D tubes. The experimental results were consistent with the simulation results.