Effect of Forming Parameters on Proﬁle Thinning of Flexible 3D Multi-Point Stretch Bending

: The ABAQUS ﬁnite element simulation software is used to simulate the ﬂexible multi-point three-dimensional stretch bending process of aluminum proﬁles. The effect of process parameters on the web thickness of rectangular proﬁle in ﬂexible multi-point three-dimensional stretch bending is studied by orthogonal experiment and range analysis. The process parameters used in the experiments include pre-stretching value, post-stretching value, the number of multi-point dies and friction coefﬁcient. The optimal combination of process parameters is obtained by numerical simulation and experimental veriﬁcation. When the aluminum proﬁle is completed ﬂexible multipoint stretch bending according to the best parameters, the thickness thinning of outer web and inner web is the smallest. The experimental result is closed to the numerical simulated results. The effectiveness of the numerical simulation is veriﬁed by the corresponding experimental methods.


Introduction
Due to the rapid development of the automobile and high-speed rail industry in recent years, people pay more and more attention to the lightweight design of the structure [1]. In this situation, reducing the weight of cars and high-speed trains cannot only decrease energy consumption and alleviate the growing energy crisis, but also lessen waste emissions. The aluminum is widely used in aerospace, rail vehicles and automobile manufacturing because of its low density, high strength and easy recycling.
The rapid development of the manufacturing industry has made the demand for parts more diversified. Therefore, the stretch bending process of profiles has been widely used, and it has always been a hot issue in the field of material processing. Researchers from various countries have done a lot of research on this process. In the manufacturing process, the distribution of plate thickness and the maximum thinning rate have a great impact on product quality. Nguyen and Trung found that the change of thickness depends on input parameters such as process parameters, die geometry and workpiece material, and determined the relationship between thickness distribution and input parameters [2]. Hussain et al. proposed a novel method to test the thinning limit of the sheet in incremental forming, and verified the thickness distribution by using the law of cosine. Based on the law of cosine, a mathematical expression for predicting the thickness distribution along the depth of the part and the thinning limit of the thin plate is derived. The proposed method can test the thinning limit of sheet metal while reducing processing time and cost [3]. Yang et al. analyzed the principle of thickness deformation in the process of single-point incremental forming, fitted a high-precision deformation zone thickness prediction formula based on the simulation results, and proposed a method to control the thinning rate by changing the forming trajectory [4]. The local thinning below the die entry radius largely depends on the plate die friction during sliding contact. Taleff et al. found that increasing

Model of FE Simulation
The numerical simulation of FMD stretch bending is performed using ABAQUS/ Explicit dynamic. The geometry of the profile and the forming process of profile are symmetrical, the 1/2 model is used for analysis in order to shorten the calculation time and improve the efficiency. The model consists of profiles, clamps, multi-point dies and limit screws. The clamps are used to apply tension and bending moments. The limit screw is designed to limit the displacement of the dies during the vertical bending process. The FMD stretch bending process is a three-dimensional simulation problem, which can cause grid distortion. According to the principle of solid element selection, the solid element C3D8R can be selected for the aluminum profile. C3D8R is a hexahedral linearly reduced integral solid element with 8 nodes for large strain and deformation. In the model, C3D8R solid element is enabled with hourglass control. Multi-point dies, limit screws and clamps are modeled using the rigid element R3D4, since the rigid element is mainly used to simulate very strong components that can be either fixed or moving. Additionally, the size of the rigid body element adopts the default setting, thus the calculation time can be shorten significantly. The classical Coulomb model is chosen to represent the interface friction condition σ f = µσ n . Tangential friction stress is represented by σ f . Friction coefficient is represented by µ. Positive pressure on the contact surface is represented by σ n . The assembled FMD stretch bending model is shown in Figure 1. The horizontal bending angle is 25 • and the vertical bending angle is 15 • . The trajectory of the clamp is shown in Figure 2. where, , is the elastic limit strain, K is the strength coefficient, n is the exponential strain-hardening parameter.

Model of FE Simulation
The numerical simulation of FMD stretch bending is performed using ABAQUS/Explicit dynamic. The geometry of the profile and the forming process of profile are symmetrical, the 1/2 model is used for analysis in order to shorten the calculation time and improve the efficiency. The model consists of profiles, clamps, multi-point dies and limit screws. The clamps are used to apply tension and bending moments. The limit screw is designed to limit the displacement of the dies during the vertical bending process. The FMD stretch bending process is a three-dimensional simulation problem, which can cause grid distortion. According to the principle of solid element selection, the solid element C3D8R can be selected for the aluminum profile. C3D8R is a hexahedral linearly reduced integral solid element with 8 nodes for large strain and deformation. In the model, C3D8R solid element is enabled with hourglass control. Multi-point dies, limit screws and clamps are modeled using the rigid element R3D4, since the rigid element is mainly used to simulate very strong components that can be either fixed or moving. Additionally, the size of the rigid body element adopts the default setting., thus the calculation time can be shorten significantly. The classical Coulomb model is chosen to represent the interface friction condition σf = μσn. Tangential friction stress is represented by σf. Friction coefficient is represented by μ. Positive pressure on the contact surface is represented by σn. The assembled FMD stretch bending model is shown in Figure 1. The horizontal bending angle is 25° and the vertical bending angle is 15°. The trajectory of the clamp is shown in Figure 2.
vertical bending: sin  vertical bending: post stretching: where, L is the length of the profile. δ pre is the amount of pre-stretching. R is the horizontal bending radius. α is the horizontal bending angle. r is the vertical bending radius. β is the vertical bending angle. L 2 represents the length of the profile projected in the vertical plane. L 1 is the projection length of the contact part between the profile and the multi-point dies in the vertical plane. δ po is the amount of post-stretching.

Calculation of the Rectangular Profile Thinning
Thinning amount can be characterized by the change of the distance between the outer centerline of outer web and the inner centerline of outer web, and the distance between outer centerline and inner centerline of inner web respectively. The sectional view of the rectangular profile along the profile is as shown in Figure 3. The aluminum profile is meshed in the ABAQUS software. The original coordinates of the inner and outer adjacent points of the outer web are (x i , y i , z i ), (x (i+1) , y (i+1) , z (i+1) ) (i = 0, 1, 2, 3· · · n). The coordinates after deformation are (x i , y i , z i ) (x (i+1) , y (i+1) , z (i+1) ). Taking A (x 0 , y 0 , z 0 ), B (x 1 , y 1 , z 1 ) as an example, the coordinates after deformation are A (x 0 , y 0 , z 0 ), B (x 1 , y 1 , z 1 ). According to the distance between two points in space, the distance between the original → AB is: 21, 11, x FOR PEER REVIEW 5 of 13 example, the coordinates after deformation are A (x0′, y0′, z0′), B (x1′, y1′, z1′). According to the distance between two points in space, the distance between the original ⃗⃗⃗⃗⃗ is: Distance between AB after deformation is: The spatial equation of the inner centerline of the outer web can be determined by the following formula: Point B is on the inner centerline of outer web, and the normal plane of the point is: Its normal vector is The thickness on the inner side of the outer web is The thinning amount between the two points A and B of the rectangular web is △1=d0-d2, and the maximum value of the thinning amount of the outer web of the profile is △max= max {△(i+1)}. For the same reason, maximum value thinning amount of the inner web is △max′ = max {△ (i+1)′}.   Distance between AB after deformation is: The spatial equation of the inner centerline of the outer web can be determined by the following formula: Point B is on the inner centerline of outer web, and the normal plane of the point is: Its normal vector is The thickness on the inner side of the outer web is The thinning amount between the two points A and B of the rectangular web is 1 = d 0 -d 2 , and the maximum value of the thinning amount of the outer web of the profile is max = max { (i+1) }. For the same reason, maximum value thinning amount of the inner web is max = max { (i+1) }.

Orthogonal Experiment and Range Analysis for Best Parameters of Aluminum Profiles in the Process of Three-Dimensional Pulling
In order to find the proper process parameters that minimize the degree of profile thinning in the FMD stretch bending process, the orthogonal experiment was used to predict the pre-stretching amount, post-stretching amount, the number of multi-point dies, and the friction coefficient by the finite element simulation software. The comprehensive discussion and analysis of the values of these parameters is essential. Additionally, the range method is used to discuss the primary and secondary levels of effect and find the best combination of processing parameters. The experimental factors and levels of this orthogonal experiment are shown in Table 1, and the orthogonal experiments are shown in Table 2. The number of flexible units of the multi-point dies is represented by N. The friction coefficient is represented by µ s . The profile thinning degree under horizontal bending stage, the vertical bending stage, and post-stretching stage are calculated respectively. The effect of each processing parameter on profile thinning during the three-dimensional bending process is studied by the range method. The range method can be used to measure the effect level of each processing parameter on the profile thinning during the stretch bending process, and the effect level is greater with expanded range.
The specific method is as follows (the following is an example of the calculation of the factor pre-stretching amount δ pre ): (1) Calculate the comprehensive average of each level of each factor. For example, first, the sum of the data corresponding to the ith level of δ pre is recorded as K iδpre (i = 1, 2, 3), and the arithmetic mean value is calculated as k iδpre . (2) Calculate the range R of the composite mean of each factor and distinguish the primary and secondary factors. Calculate as follows: For example: thinning of the outer web during horizontal bending are: The range analysis method can be used to see the effect degree of different processing stages of the rectangular section processing parameters. The greater difference has determined the greater effect of the processing parameters, and vice versa. The data obtained by the range analysis method is shown in Table 3.
It can be seen from the calculation data of Table 3 that the effect degree of each process parameter is different for different processing processes. And the effect on outer web thinning is: r δpre = 0.028, r δp0 = 0.013 , r N = 0.066 , r µs = 0.034. Effect on inner web thinning: r δpre = 0.026, r δp0 = 0.022 , r N = 0.094 , r µs = 0.028. According to the data analysis, for the thinning of outer and inner webs, the order of the processing parameters is: N > µ s > δ pre > δ po . The number of multi-point dies has the greatest effect on the thinning, and the effect of post-stretching on thinning is minimal.

The Effect of Various Process Parameters on Thinning
In order to explore the effects of various process parameters on profile thinning respectively, numerical simulation of the parameters is shown in Table 4. And the number of multi-point dies effect on rectangular profiles thinning is shown in Figure 4.
As can be seen from Figure 4a, with the increase of the number of multi-point dies, the thinning degree of the outer web of the rectangular profile decreases sharply. At the end of the horizontal bending, vertical bending and post-stretching process, the same law is observed. This is because the more the number of dies, the closer the die surface is to the overall die, and the better the forming effect of the profile. Similarly, the thinning degree of the inner web of the rectangular profile decreases with the increase of the number of dies. However, from the comparison of Figure 4a,b, it can be concluded that the thinning amount of the inner web is less than that of the outer web. This is due to the fact that the outer surface of the profile is stretched and the inner surface is compressed during the stretch-bending forming process.
The effect of friction coefficient on thinning is shown in Figure 5. It can be seen from Figure 5a,b that after the horizontal bending, the friction coefficient has little effect on the thinning of the outer and inner webs of the profile, which can be ignored. The amount of thinning of the outer web and inner web of the profile increases with the increase of the friction coefficient during the vertical bending forming and the post-stretching forming process, respectively. This is because the greater the friction coefficient, the greater the friction force in the forming process, which hinders the forming process. Therefore, it is necessary to increase the stretching amount to meet the forming effect of the profile, resulting in more serious profile thinning.
Post-stretching is the last process of FMD stretch-bending forming. Therefore, in the horizontal bending forming and vertical bending forming process, the amount of posstretching will not affect the thickness changes of the outer web and the inner web. Figure 6 shows the effect of post-stretching amount on the thickness change of rectangular profile after post-stretching forming. It can be seen from Figure 6a,b that when the amount of post-stretching increases from 0.8% to 1.2% of the profile length, the thickness reduction of the outer web increases from 0.122 mm to 0.152 mm, and the thickness reduction of the inner web increases from 0.098 mm to 0.12 mm. Therefore, the greater the amount of post-stretching, the more serious the thinning of the profile.  As can be seen from Figure 4a, with the increase of the number of multi-point dies, the thinning degree of the outer web of the rectangular profile decreases sharply. At the end of the horizontal bending, vertical bending and post-stretching process, the same law is observed. This is because the more the number of dies, the closer the die surface is to the overall die, and the better the forming effect of the profile. Similarly, the thinning degree of the inner web of the rectangular profile decreases with the increase of the number of dies. However, from the comparison of Figure 4a,b, it can be concluded that the thinning amount of the inner web is less than that of the outer web. This is due to the fact that the outer surface of the profile is stretched and the inner surface is compressed during the stretch-bending forming process.
The effect of friction coefficient on thinning is shown in Figure 5. It can be seen from Figure 5a,b that after the horizontal bending, the friction coefficient has little effect on the thinning of the outer and inner webs of the profile, which can be ignored. The amount of thinning of the outer web and inner web of the profile increases with the increase of the friction coefficient during the vertical bending forming and the post-stretching forming process, respectively. This is because the greater the friction coefficient, the greater the friction force in the forming process, which hinders the forming process. Therefore, it is necessary to increase the stretching amount to meet the forming effect of the profile, resulting in more serious profile thinning.  As can be seen from Figure 4a, with the increase of the number of multi-point dies, the thinning degree of the outer web of the rectangular profile decreases sharply. At the end of the horizontal bending, vertical bending and post-stretching process, the same law is observed. This is because the more the number of dies, the closer the die surface is to the overall die, and the better the forming effect of the profile. Similarly, the thinning degree of the inner web of the rectangular profile decreases with the increase of the number of dies. However, from the comparison of Figure 4a,b, it can be concluded that the thinning amount of the inner web is less than that of the outer web. This is due to the fact that the outer surface of the profile is stretched and the inner surface is compressed during the stretch-bending forming process.
The effect of friction coefficient on thinning is shown in Figure 5. It can be seen from Figure 5a,b that after the horizontal bending, the friction coefficient has little effect on the thinning of the outer and inner webs of the profile, which can be ignored. The amount of thinning of the outer web and inner web of the profile increases with the increase of the friction coefficient during the vertical bending forming and the post-stretching forming process, respectively. This is because the greater the friction coefficient, the greater the friction force in the forming process, which hinders the forming process. Therefore, it is necessary to increase the stretching amount to meet the forming effect of the profile, resulting in more serious profile thinning. Post-stretching is the last process of FMD stretch-bending forming. Therefore, in the horizontal bending forming and vertical bending forming process, the amount of posstretching will not affect the thickness changes of the outer web and the inner web. Figure  6 shows the effect of post-stretching amount on the thickness change of rectangular profile after post-stretching forming. It can be seen from Figure 6a,b that when the amount of post-stretching increases from 0.8% to 1.2% of the profile length, the thickness reduction of the outer web increases from 0.122 mm to 0.152 mm, and the thickness reduction of the inner web increases from 0.098 mm to 0.12 mm. Therefore, the greater the amount of poststretching, the more serious the thinning of the profile. The effect of the pre-stretching on thinning is as shown in Figure 7. The thinning degree of the outer web and the inner web increases with the increasing of the pre-stretching amount. When the pre-stretching amount is 0.8% after completion of post-stretching, the outer web is thinned by 0.12 mm and the inner web is thinned by 0.094 mm. In general, in the FMD stretch-bending process, the thickness of the profile after horizontal bending is much smaller than that after vertical bending. The thinning of the profile after the post-stretching forming is the largest. And no matter which forming stage the profile is in, the thinning degree of the outer web is always greater than that of the inner web.

Process Parameter Optimization and Experimental Verification
The result in 3.2 is based on the analysis of thinning degree between the outer and inner webs. The smaller value has exhibited the better forming effect of profile. Therefore, it is determined that the number of multi-point dies is N = 12 and the friction coefficient is 0.1. This combination can be represented by N3f1. When the pre-stretching amount is 0.8% and the post-stretching amount is 1%, the effect of thinning is not obviously. The conclusions which are obtained by orthogonal test and range analysis are not necessarily in full compliance with the actual situation of numerical simulation. Therefore, in order to obtain more accurate conclusions, it is necessary to narrow the experimental interval. The effect of the pre-stretching on thinning is as shown in Figure 7. The thinning degree of the outer web and the inner web increases with the increasing of the pre-stretching amount. When the pre-stretching amount is 0.8% after completion of post-stretching, the outer web is thinned by 0.12 mm and the inner web is thinned by 0.094 mm.
Metals 2021, 11, x FOR PEER REVIEW 9 of 13 post-stretching increases from 0.8% to 1.2% of the profile length, the thickness reduction of the outer web increases from 0.122 mm to 0.152 mm, and the thickness reduction of the inner web increases from 0.098 mm to 0.12 mm. Therefore, the greater the amount of poststretching, the more serious the thinning of the profile. The effect of the pre-stretching on thinning is as shown in Figure 7. The thinning degree of the outer web and the inner web increases with the increasing of the pre-stretching amount. When the pre-stretching amount is 0.8% after completion of post-stretching, the outer web is thinned by 0.12 mm and the inner web is thinned by 0.094 mm. In general, in the FMD stretch-bending process, the thickness of the profile after horizontal bending is much smaller than that after vertical bending. The thinning of the profile after the post-stretching forming is the largest. And no matter which forming stage the profile is in, the thinning degree of the outer web is always greater than that of the inner web.

Process Parameter Optimization and Experimental Verification
The result in 3.2 is based on the analysis of thinning degree between the outer and inner webs. The smaller value has exhibited the better forming effect of profile. Therefore, it is determined that the number of multi-point dies is N = 12 and the friction coefficient is 0.1. This combination can be represented by N3f1. When the pre-stretching amount is 0.8% and the post-stretching amount is 1%, the effect of thinning is not obviously. The conclusions which are obtained by orthogonal test and range analysis are not necessarily in full compliance with the actual situation of numerical simulation. Therefore, in order to obtain more accurate conclusions, it is necessary to narrow the experimental interval. In general, in the FMD stretch-bending process, the thickness of the profile after horizontal bending is much smaller than that after vertical bending. The thinning of the profile after the post-stretching forming is the largest. And no matter which forming stage the profile is in, the thinning degree of the outer web is always greater than that of the inner web.

Process Parameter Optimization and Experimental Verification
The result in 3.2 is based on the analysis of thinning degree between the outer and inner webs. The smaller value has exhibited the better forming effect of profile. Therefore, it is determined that the number of multi-point dies is N = 12 and the friction coefficient is 0.1. This combination can be represented by N3f1. When the pre-stretching amount is 0.8% and the post-stretching amount is 1%, the effect of thinning is not obviously. The conclusions which are obtained by orthogonal test and range analysis are not necessarily in full compliance with the actual situation of numerical simulation. Therefore, in order to obtain more accurate conclusions, it is necessary to narrow the experimental interval. In order to verify the optimal combination, it is necessary to compare the k i between pre-stretching amount and the post-stretching amount. The k i value of the pre-stretch and post stretching (the smaller k i value represents the lower effect degree of the level), the two process parameters of the pre-stretching amount and the post-stretching amount are respectively numerically simulated and verified by experiments. A total of four combina-tion experiments, the combination of pre-stretching amount and post-stretching amount is shown in Table 5. The numerical simulation results of four groups of experiments with pre-stretching amount and post-stretching amount are shown in Table 6. The results show that the outer and inner webs of experiment δ pre 2δ po 2N3µ s 1 have the lowest thinning degree. The number of multi-point dies 12, the friction coefficient 0.1, and the pre-stretching amount 1%, post-stretching 1% are the best combination of process parameters. Table 5. Optimal combination design of process parameters of pre-stretching amount and poststretching amount.

Forming Process δ pre δ po Combination
Horizontal bending max

The FMD Stretch Bending Experiments
In the experiments, the FMD stretch bending machine is shown in Figure 8a. Figure 8b shows the experimental parts of rectangular profile. The material used in the experiment is 6005A aluminum alloy. The processing parameters of the FMD stretch bending machine are set according to δ pre 2δ po 2N3µ s 1, δ pre 3δ po 2N3µ s 1, δ pre 2δ po 3N3µ s 1, and δ pre 3δ po 2N3µ s 1 respectively. As shown in Figure 9a,b, the thickness of the formed rectangular profile is measured with a thickness gauge. Due to the large length of the profile, the thickness gauge cannot measure each position of the profile. Therefore, as shown in Figure 9c, we divide the profile, cut the profile with a wire cutting machine, measure the thickness of the outer and As shown in Figure 9a,b, the thickness of the formed rectangular profile is measured with a thickness gauge. Due to the large length of the profile, the thickness gauge cannot measure each position of the profile. Therefore, as shown in Figure 9c, we divide the profile, cut the profile with a wire cutting machine, measure the thickness of the outer and inner webs of the middle section of each part of the profile, and get the thickness change of the profile after each process. As shown in Figure 9a,b, the thickness of the formed rectangular profile is measured with a thickness gauge. Due to the large length of the profile, the thickness gauge cannot measure each position of the profile. Therefore, as shown in Figure 9c, we divide the profile, cut the profile with a wire cutting machine, measure the thickness of the outer and inner webs of the middle section of each part of the profile, and get the thickness change of the profile after each process. Taking experiment N3μs1δpre2δpo2 as an example, the thickness of profiles after horizontal bending, vertical bending and post-stretching forming are measured respectively, and the maximum thinning value is calculated. As shown in Tables 7 and 8, it is necessary to measure three times and take the average value of the measurement results. The remaining three groups of experiments also need to be measured in sequence.  Taking experiment N3µ s 1δ pre 2δ po 2 as an example, the thickness of profiles after horizontal bending, vertical bending and post-stretching forming are measured respectively, and the maximum thinning value is calculated. As shown in Tables 7 and 8, it is necessary to measure three times and take the average value of the measurement results. The remaining three groups of experiments also need to be measured in sequence. The experimental measurement results are shown in Figure 10. In each process stage, the thinning degree of the profile processed according to the N3µ s 1δ pre 2δ po 2 parameter is smallest. Outer web is thinned by 0.101 mm. The inner web is thinned by 0.087 mm, and the results are close to the finite element simulation results in Table 6. Therefore, the finite element model can predict the thinning degree of the profile.
Post-stretching 0.088 0.086 0.087 0.087 The experimental measurement results are shown in Figure 10. In each process stage, the thinning degree of the profile processed according to the N3μs1δpre2δpo2 parameter is smallest. Outer web is thinned by 0.101 mm. The inner web is thinned by 0.087 mm, and the results are close to the finite element simulation results in Table 6. Therefore, the finite element model can predict the thinning degree of the profile.

Conclusions
Taking the rectangular profile as the research object, the thickness change of the profile in the FMD stretch-bending process is studied. The effect of process parameters on the thickness of the profile is obtained through numerical simulation and experiments. The work of this paper is summarized as follows: 1. The degree of thinning of the outer web and inner web of the rectangular profile decreases significantly with the increase of the number of dies, and gradually increases with the increase of the friction coefficient and the pre-stretching amount. The amount of post-stretching only affects the final forming result. As the amount of poststretching increases, the thickness of the outer web and the inner web also shows an increasing trend. And the degree of thinning of the outer web of the profile is greater than that of the inner web. 2. Using orthogonal test and range analysis, it is determined that the number of multipoint dies has the greatest effect on the thinning of the profile during FMD stretch bending, and the amount of post-stretching is the smallest. The combination of process parameters that minimizes the degree of thinning of the profile during the bending process is N = 12, μs = 0.1, δpre = 1%, and δpo = 1%. 3. Based on the abstraction and reasonable simplification of the forming process principle, the finite element simulation software ABAQUS is used to simulate the FMD Figure 10. Experimental results.

Conclusions
Taking the rectangular profile as the research object, the thickness change of the profile in the FMD stretch-bending process is studied. The effect of process parameters on the thickness of the profile is obtained through numerical simulation and experiments. The work of this paper is summarized as follows: 1.
The degree of thinning of the outer web and inner web of the rectangular profile decreases significantly with the increase of the number of dies, and gradually increases with the increase of the friction coefficient and the pre-stretching amount. The amount of post-stretching only affects the final forming result. As the amount of post-stretching increases, the thickness of the outer web and the inner web also shows an increasing trend. And the degree of thinning of the outer web of the profile is greater than that of the inner web.

2.
Using orthogonal test and range analysis, it is determined that the number of multipoint dies has the greatest effect on the thinning of the profile during FMD stretch bending, and the amount of post-stretching is the smallest. The combination of process parameters that minimizes the degree of thinning of the profile during the bending process is N = 12, µ s = 0.1, δ pre = 1%, and δ po = 1%. 3.
Based on the abstraction and reasonable simplification of the forming process principle, the finite element simulation software ABAQUS is used to simulate the FMD stretch bending process. The simulation results are basically consistent with the experimental results, and the trend is also consistent. Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.

Data Availability Statement:
The datasets used or analysed during the current study are available from the corresponding author on reasonable request.