Effects of Asymmetric Thickness Mandrel on NC RDB Forming Quality of Ultra-Thin-Walled Tube

Effects of Asymmetric Thickness Mandrel on NC RDB Forming Quality of Ultra-Thin-Walled Tube | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Effects of Asymmetric Thickness Mandrel on NC RDB Forming Quality of Ultra-Thin-Walled Tube lanfang Jang, Wujie Yuan, Heng Li, Xunzhong Guo, Zili Wang, Shuyou Zhang, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5283053/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract To satisfy lightweight design requirements, aerospace ducts frequently employ ultra-thin-walled tubes with a diameter-to-thickness ratio (D/t) exceeding 100.However, ultra-thin-walled tubes present significant forming challenges, and the mandrel plays a critical role in their bending. Therefore, investigating the effect of mandrel structure on the quality of ultra-thin-walled tubes formed through NC bending is of considerable importance. In this study, utilizing the Abaqus nonlinear finite element platform, an asymmetric thickness ball design method is proposed. Based on the positioning of the asymmetric balls within the mandrel, seven distinct designs for asymmetric thickness mandrels are developed. This study conducts a finite element analysis of the NC rotary draw bending (RDB) process for ultra-thin-walled 304 stainless steel tubes and validates the corresponding experiments. The results indicate that as the asymmetric thickness mandrel is positioned further from the mandrel, the stress on the outer side of the tube near the bend initiation first increases and then decreases, while the stress on the inner side of the tube, after the midpoint of the bend, initially decreases and then increases. The use of asymmetric thickness mandrels significantly reduces both the thinning and thickening rates of the tubes, though their impact on improving the ellipticity is less pronounced. The core ball nearest to the mandrel is designated as Ball 1, with subsequent balls further from the mandrel labeled as Ball 2 and Ball 3, respectively. The placement of the asymmetric thickness balls improves the thinning rate in the order: Ball 2 > Ball 1 > Ball 3; enhances the thickening rate in the order: Ball 1 > Ball 2 > Ball 3; and optimizes the ellipticity in the order: Ball 3 > Ball 1 > Ball 2. thin-walled tube NC bending retracting mandrel finite element analysis Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Introduction Due to the material and structural requirements of pipe bending for lightweight and high-strength applications, its use has become increasingly prevalent across high-end industries such as aviation, machinery, automotive, and energy sectors. Consequently, pipe bending occupies a critical position within the field of plastic forming. To meet lightweight requirements, precision machinery often employs ultra-thin-walled pipe fittings with a diameter-to-thickness ratio exceeding 100. These fittings, characterized by large diameters and thin walls, are highly susceptible to defects such as cracking, wrinkling, and cross-sectional distortion during bending. Thus, there is an urgent need to investigate the key theories and technologies related to the bending and forming of ultra-thin-walled pipes, with the goal of improving their forming quality. Usually, for thick-walled pipe fittings (D/t<30), international and domestic scholars have conducted numerous studies. Thin-walled pipe fittings (30 ≤ D/t ≤ 50) is more difficult, so scholars have more research content. Yang et al. [1] surpassed the bending limitations of large-diameter, thin-walled titanium tubes by employing a differential heating method. Cheng et al. [2] elucidated the principle by which the mandrel's support induces deformation in the tubing and systematically examined how various mandrel parameters affect the bending and forming behavior of stainless steel, thin-walled tubes. Guo et al. [3] developed a mechanical model for bend-around forming and analyzed the stress-strain variations in pipe fittings during this process by integrating mechanical simulations with experimental data to investigate the effects of mandrel parameters on wall thickness and cross-sectional morphology of bent pipes. Salem et al. [4] designed and validated the reliability of a novel chain mandrel, using stainless steel tubes as the test subject. Kajikawa et al. [5] investigated how mandrel position, diameter, and tip radius influence the forming quality of fittings, using a D=7 mm, t=0.21 mm copper tube as the study model. Their experiments revealed that when the mandrel's diameter and tip radius were 6.18 mm, defect-free bending of fittings with R=2~3D was achieved. R.J. Gu et al. [6] examined the relationship between the core drawing process and rebound angle in D=38 mm, t=1 mm stainless steel tubes, discovering a linear correlation between the rebound angle and the bending angle as the latter increases. Currently, ultra-thin-walled pipe fittings with a large diameter-to-thickness ratio (D/t ≥ 100) present significant forming challenges due to their large outer diameter and thin-walled structure, making them highly susceptible to defects such as cracking, wrinkling, and cross-sectional distortion. Both domestic and international scholars have conducted numerical simulations and experimental studies on ultra-thin-walled pipe fittings. Hai Liu et al. [7] utilized 10,000 bead grains as fillers in push bending, successfully forming ultra-thin-walled pipe fittings with a diameter of 30 mm and a thickness of 0.3 mm. Xiaosong Wang et al. [8] employed a double-layer tube hydraulic bending process to form an inner tube of stainless steel (D = 180 mm, t = 1 mm) and found that the most severe thinning occurred near a cross-section with a symmetry plane angle of 21°, rather than at the center point. Lanfang Jiang et al. [9] developed core die with various combinations of core ball thicknesses based on 304 stainless steel pipe fittings (D=91.6 mm, t=0.9 mm). They found that the thinning rate can be significantly reduced by utilizing smaller core ball thicknesses at the beginning of the bend and larger ones at the end, and as large as possible in the end bending end, Yiwei Xu et al. [10] proposed and validated the RRS process for 304 austenitic stainless steel thin-walled tubes, demonstrating that the thinning rate of ultra-thin-walled steel tubes can be increased to 90% under ultra-low stress triaxial ring-roll spinning conditions. Sun Kang et al. [11] investigated the hydroforming process for complex cross-section corrugated tubes (D = 50 mm, t = 0.4 mm) and identified optimal process parameters, including a pre-expansion internal pressure of 7.5 MPa. Li Weizhuang et al. [12] studied ultra-thin-walled nickel-copper alloy tubes with submicron structures produced via ball spinning and found that dislocation slip led to the formation of ultrafine laminated (UFL) structures and ultrafine grains (UFGs). The high strain rate and strain gradient promoted dislocation slip, grain refinement, and deformation twinning, ultimately resulting in the successful production of ultra-thin-walled nickel-copper alloy tubes (D = 50 mm, t = 0.4 mm).Yan, J. P. et al. [13] used Inconel 718 to study the effect of the drawing process on the wall thickness of ultra-thin-walled tubes, observing that the average grain size was reduced from 4.9 μm to 2.1 μm. Kunito Nakajima et al. [14] studied ultra-thin-walled rectangular tubes (H0 = 20 mm, W0 = 10 mm, t0 = 0.5 mm) and found that deformation in the height direction decreases when a stacked mandrel is used. L. Cooper et al. [15] employed an automatic orbital gas tungsten carbide arc welding (GTCAW) process to weld ultra-thin-walled CP-2 Ti tubes (D = 2.275 mm, t = 160 ± 10 μm). The aforementioned scholars primarily employed methods such as hydraulic push-bending, spiral welding, and rotational forming to process ultra-thin-walled pipe fittings. While these methods are effective, they have certain limitations, including the requirement for high-pressure environments and the use of internal filler media. Research on CNC bending forming of ultra-thin-walled pipe fittings with large diameter-to-thickness ratios remains limited. Therefore, this study focuses on ultra-thin-walled 304 stainless steel conduits with a diameter of 91.6 mm, a wall thickness of 0.9 mm, and a diameter-to-thickness ratio of 101.8. The research investigates the effects of asymmetric core ball thickness on the forming quality of the pipe fittings. 1 Research programme 1.1 Bend-around forming principle The principle of tube CNC bending forming is illustrated in Figure 1. As shown in the figure, the primary components involved in the bending and forming process are the clamping die, pressure die, wiper die, bending die, bending die inserts, and the mandrel, which consists of two parts: the mandrel shake and the mandrel ball. Prior to bending, one end of the pipe is secured by the clamping die, and the bending die inserts are fixed onto the bending die. The outer side of the bend is compressed by the pressure die, while the inner side is supported by the wiper die, positioned at the tangent point between the pipe and the bending die. During the bending process, the bending die rotates around the Z-axis at an angular speed of ω at point O. The clamping die and the bending die insert clamp the pipe, rotating synchronously with the bending die. Simultaneously, the pressure die moves along the X-axis at a speed of ν, while the wiper die and the mandrel remain stationary. 1.2 The role of the mandrel In ultra-thin-walled tube CNC winding and forming, a mandrel is typically installed inside the tube to provide internal support, preventing the tube from collapsing outward and reducing cross-sectional distortion during bending. Simultaneously, the mandrel works in conjunction with the wiper die to prevent the formation of wrinkles on the inner side of the tube. The mandrel primarily consists of two components: the mandrel itself and the core ball. The mandrel supports the straight section of the pipe, while the core ball supports the bent section. Based on different connection methods, mandrels can be classified into chain mandrels, as shown in Fig. 2(a), and ball-and-socket mandrels, as shown in Fig. 2(b). 1.3 Asymmetric thickness core ball As illustrated in Fig. 3, the conventional mandrel features equal thicknesses on both its inner and outer sides. In contrast, the asymmetric thickness mandrel is specifically designed with varying thicknesses, with a smaller thickness near the outer side of the tube and a larger thickness near the inner side. Ultimately, the asymmetric thickness core ball is integrated with the conventional mandrel to create an asymmetric thickness mandrel. In Fig. 3, the outer thickness of the mandrel ball is denoted as W1, while the inner thickness is W2. The upper and lower thicknesses of the conventional mandrel ball are equal, both represented as w. 1.4 Asymmetric thickness mandrel programme design For ease of description, the core balls are numbered as 1, 2, and 3. The core ball closest to the core axis is labeled as core ball 1, with the serial numbers increasing as the distance from the core axis increases, resulting in core balls 2 and 3, as illustrated in Figure 4. The asymmetric thickness mandrels are connected via pins between the core balls. Examples of asymmetric mandrels with an outer thickness of W1 = 20 mm and an inner thickness of W2 = 22 mm, as well as Conventional mandrels with a uniform thickness of w = 24 mm, are illustrated in Fig. 5. In the following study, seven different arrangements of asymmetric thickness mandrels were designed to investigate the impact of both the number and arrangement of these mandrels on the forming quality of pipe fittings. A schematic diagram of the asymmetric thickness mandrel is provided in Fig. 6. The results were compared using two control groups: one with conventional mandrel pin connections and the other with conventional mandrel ball connections. The Conventional mandrel uses mandrel balls 1, 2, and 3, with the key difference between pin and ball connections being the method of connecting the mandrel balls. The specific combinations are provided in Table 1. Mandrels with Conventional core ball pin connections were compared to mandrels with symmetric thickness core ball pin connections to verify the feasibility of using asymmetric thickness mandrels. While mandrels with conventional ball joints outperform the pin-connected mandrels in terms of tubing thinning rate and ovality after bending, they result in a higher thickening rate. Therefore, the superior conventional mandrel-ball connection was compared with the experimental group to further validate the feasibility of using asymmetric thickness mandrels. Table 1 Asymmetric thickness mandrel combination methods Core ball combination number Number of asymmetric thickness core balls Core ball usage Core Ball 1 Core Ball 2 Core Ball 3 1 1 asymmetric legacy legacy 2 1 legacy asymmetric legacy 3 1 legacy legacy asymmetric 4 2 asymmetric asymmetric legacy 5 2 asymmetric legacy asymmetric 6 2 legacy asymmetric asymmetric 7 3 asymmetric asymmetric asymmetric Conventional Mandrel Pin Hitch 0 legacy legacy legacy Conventional Mandrel Baseball Hitch 0 legacy legacy legacy 2 Finite element modelling 2.1 Geometric modelling Using dynamic explicit finite element software Abaqus, a three-dimensional elastic-plastic finite element analysis model of the CNC bending process for ultra-thin-walled pipes was developed. Table 2 presents the structural parameters of each forming die used in the CNC tube bending process. Based on pipe bending requirements, empirical formulas, and actual production specifications, the mandrel outreach was determined to be 6 mm, with a spacing of 7 mm between the core ball and the mandrel. Additionally, the gaps between the clamping die, bending die inserts, bending die, pressure die, wiper die, and the pipe were all set to 0 mm. Table 2 Size parameters of forming dies parametric numerical value Clamping die length/mm 215 Length of bending die insert/mm 220 Pressure die length/mm 500 Wiper die length/mm 255 Mandrel length/mm 250 Mandrel diameter/mm 89.2 Ball Diameter/mm 88.9 Thickness of balls/mm 24 Number of balls 3 2.2 Unit type Ultra-thin-walled pipe fittings are modeled as variable shells using a four-node curved shell element (S4R), which allows for the analysis of changes in wall thickness and the distribution of stress and strain during the bending process. The remaining models are established as rigid bodies using a four-node rigid element (R3D4), which remains undeformed by default throughout the bending process. The model applies von Mises' yield criterion, the associated flow rule, and kinematic hardening. The mechanical properties of the 304 ultra-thin-walled stainless steel pipe are presented in Table 3. Each forming die is defined as a rigid body, with 45-gauge steel selected as the material. The mechanical properties of 45-gauge steel are provided in Table 4. Table 3 Mechanical properties of 304 stainless steel parametric numerical value Density ρ / Kg/ mm 3 7.93e-6 Modulus of elasticity E / MPa 2.08e5 Yield strength σ s / MPa 383 Tensile strength σ b / MPa 812 Poisson's ratio υ 0.3 Table 4 Mechanical properties of forming dies parametric numerical value Density ρ / Kg/ mm 3 7.85e-6 Modulus of elasticity E / MPa 2.1e5 Poisson's ratio υ 0.269 2.4 Boundary condition The process parameters for the simulation and analysis of the pipe bending process are provided in Table 5.Based on existing research, the bending die is set to rotate around the Z-axis at 0.1 rad/s at point O. The bending die insert and clamping die rotate synchronously with the bending die. The bending die insert and clamping die rotate in unison with the bending die, while the pressure die moves along the X-axis at a tangential linear velocity of 19 mm/s, matching the bending die's motion. Both the creasing die and mandrel are constrained in all degrees of freedom and remain stationary, while the core ball is unconstrained and retains full degrees of freedom. Table 5 Process parameters for simulation of tube bending parametric numerical value Bending angle θ/ ° 57 Bending radius R/ mm 190 Die speed v/ (mm/ s) 19 Bending angular speed ω/ (rad/s) 0.1 Bending time T/ s 10 The friction between the pipe and the forming dies plays a crucial role in determining the forming quality of the pipe. In this model, Isotropic Coulomb friction is applied by default to define the friction properties, and the friction coefficients are selected based on actual production conditions. The Finite Sliding method is selected for contact tracking to fully capture surface interactions, resulting in higher calculation accuracy. The friction coefficients between the pipe and each forming die are presented in Table 6. Table 6 Friction coefficient between pipe and each forming die contact pair numerical value Clamping dies and fittings 0.9 Bending die inserts and fittings 0.9 Bending dies and fittings 0.45 Compression die and fittings 0.45 Wiper dies and fittings 0.3 Mandrels & Tubing 0.2 Using the dynamic explicit finite element software Abaqus, a three-dimensional elastic-plastic finite element analysis model for the CNC winding and bending forming process of thin-walled pipes was developed, as illustrated in Figure 7. 3 Results and discussion 3.1 Bending and forming quality evaluation index (1) Wall Thinning Rate: During bending, tubing is subjected to tangential tensile stresses, causing the material on the outer side to stretch, which leads to a reduction in wall thicknrsaess. The wall thinning rate is calculated as: In the formula, ξ1 represents the wall thickness reduction rate, t is the initial wall thickness of the tubing, and is the minimum wall thickness after the bending and forming process. (2) Wall Thickness Gain Rate: During bending, the material on the inner side of the fitting is subjected to tangential compressive stresses, causing thickening in the direction of the wall thickness. The wall thickening rate is calculated as: In the formula, ξ2 represents the wall thinning rate, t is the initial wall thickness of the tubing, andis the maximum wall thickness after the bending and forming process. (3) Ellipticity is: During the bending process, the material on the outer side of the bend is subjected to both tangential tensile stress and radial compressive stress, resulting in a reduction in wall thickness and radial displacement. This causes the pipe diameter along the bending radius to decrease and the circumferential diameter to increase. As a result, the originally circular cross-section gradually becomes elliptical due to cross-sectional flattening distortion. Ellipticity is defined as: In the formula, φ represents the ellipticity of the bend, is the maximum outer diameter of the cross-section,is the minimum outer diameter of the cross-section, and D is the original outer diameter of the cross-section. 3.2 Simulation analysis model validation To verify the accuracy and validity of the above finite element analysis model, a bending test was conducted on an ultra-thin-walled 304 stainless steel pipe with dimensions of D60mm × t1mm × R120mm, using the KM-A100-CNC-E120 CNC pipe bender. The bending test was performed at a bending speed of 0.1 rad/s. The structural parameters of each forming die and the bending process were consistent with those used in the simulation analysis, as shown in Figure 8. As shown in Figure 9, a comparison between the wall thickness distribution cloud map from the finite element simulation and the bending test samples of the ultra-thin-walled 304 stainless steel pipe was conducted. It was found that no wrinkling occurred on the inner side of the bent section, and the test samples were largely consistent with the simulation analysis results. The wall thickness values from the simulation calculations of the ultra-thin-walled pipe fittings were extracted and compared with the actual wall thickness values measured after the bending test. The wall thickness variation curves for the inner and outer centerlines of the ultra-thin-walled pipe bends were plotted for each measurement angle. As shown in Figure 10, the simulation results were found to be largely consistent with the wall thickness values obtained from the test measurements. In the outer thinning curve, the largest error occurred at a bending angle of 0°, where the test value was 0.843 mm and the simulation value was 0.807 mm, resulting in a relative error of 4.27%. For the inner thickening curve, the largest error was at a bending angle of 54°, with a test value of 0.968 mm and a simulation value of 0.988 mm, yielding a relative error of 2.08%. The error between the simulation values and the test values remained within a reasonable range, with all discrepancies being less than 10% Figure 11 presents a comparison of the ellipticity curves of ultra-thin-walled pipe fittings at different measurement angles. The ellipticity trends obtained from both the simulation and the test are largely consistent. Both curves show an increasing trend between bending angles of 18° and 48°, reaching the maximum ellipticity at 48°. At this point, the simulation-calculated ellipticity is 2.01%, while the test-measured ellipticity is 1.98%. In summary, by comparing the simulation and test results in terms of bend shape, wall thickness changes, and ellipticity, the results were found to be largely consistent. This indicates that the finite element model for the CNC winding and bending process of thin-walled pipes developed in this paper is reliable. 3.3 Analysis of results after rebound 3.3.1 stress analysis The distribution of equivalent stresses on the outer and inner sides of the bent pipe, derived from the simulation of the numbered 1–7 conventional mandrels, is presented in Table 1 and shown in Figures 12 and 13. As observed in Figure 12, the maximum stress (377.1–502.5 MPa) on the outside of the various fittings occurs consistently at the outer bending angle of 9°–18°. For bending angles between 18° and 57°, except for fitting No. 4, the area of maximum stress (189–314.4 MPa) on the inner side of the fittings is comparable to that of the conventional mandrel-baseball hinge joint. In contrast, the maximum stress area on the inner side of the other fittings is significantly larger than that of the conventional mandrel-pin hinge joint. A comparison of the stress diagrams for the outer side of the fittings at positions Nos. 1 to 3 with those of the conventional mandrels reveals that, before the middle section of the inner bend, the stresses at all three positions are largely the same. After the middle section of the inner bend, the stress increases significantly with the use of an asymmetric thickness mandrel at position No. 3. However, since the maximum stress is relatively low, altering the position of the asymmetric thickness core ball has minimal effect on the maximum stress area on the outer side of the fitting. As shown in Figure 13, at bending angles from 0° to 24°, the maximum stress area (627.8 to 753.2 MPa) on the inside of the fittings with conventional mandrel pin articulation is much smaller than that of the conventional mandrel pin reaming. The maximum stress area on the inside of fittings Nos. 3, 5, 6, and 7 is larger than that of conventional mandrel pin reaming, while the maximum stress area on the other numbered fittings is smaller. The maximum stress area (627.8 to 753.2 MPa) on the inside of fittings with conventional mandrel pin articulation and asymmetric thickness mandrels is smaller than that of conventional mandrel-baseball articulation at bending angles between 30° and 57°. A comparison of the stress diagrams on the inner side of the pipe fittings with conventional mandrel pins at positions Nos. 1 to 3 reveals the following: After the middle section of the inner bend, using an asymmetric thickness mandrel at position No. 2 significantly reduces the stress. Using an asymmetric thickness mandrel at position No. 1 also reduces the stress, though not as much as at position No. 2. Conversely, using an asymmetric thickness mandrel at position No. 3 significantly increases the stress. 3.3.2 Wall Thickness Reduction Rate The distribution of the overall wall thickness reduction rate on the outer side of the bends for mandrels numbered 1 to 7, as well as the conventional mandrel after rebound, is presented in Figure 14. Analysis of Figure 14(a) reveals that using an asymmetric thickness core ball at position No. 2 reduces the overall thinning rate of the fittings. Using the asymmetric thickness core ball at position No. 1 only improves the thinning rate at bending angles between 36° and 57°, while the use of the core ball at position No. 3 has no significant effect on the overall thinning rate improvement. Therefore, the effect of the asymmetric thickness core ball position arrangement on reducing the thinning rate is as follows: the No. 2 core ball position has the greatest effect, followed by the No. 1 position, while the No. 3 position shows minimal improvement. The superior performance of the No. 2 core ball position is attributed to its constant contact with the tubing during the bending process, which provides the largest contact area. Altering the thickness of the core ball at this position reduces the contact area, resulting in the greatest effect on reducing the thinning rate. In summary, using an asymmetric thickness core ball at the No. 2 position yields the most effective reduction in the pipe thinning rate. Analysis of Figure 14(b) shows that using asymmetric thickness core balls at both positions 1 and 2 effectively reduces the overall thinning rate of the fittings. When core balls are used at positions 2 and 3, the thinning rate is reduced slightly. However, using asymmetric thickness core balls at positions 1 and 3 does not yield significant improvement. Therefore, when using two asymmetric thickness core balls, positions 1 and 2 provide the most effective reduction in the tubing thinning rate. Analysis of Figure 14(c) shows that when asymmetric thickness core balls are used at all positions (1, 2, and 3), the overall thinning rate of the fittings is significantly reduced between 18° and 57°. A combined analysis of Figures 14(a), (b), and (c) shows that using an asymmetric core ball reduces the contact area with the tubing compared to a conventional core ball, thereby improving the tubing thinning rate. The No. 2 core ball position showed the greatest reduction in thinning rate due to its constant contact with the tubing. Therefore, the order of effectiveness in reducing the thinning rate is as follows: No. 7 > No. 4 > No. 2. The greatest reduction in thinning rate, 4.15%, was achieved using the No. 7 mandrel compared to conventional mandrels. 3.3.4 Wall thickening rate The distribution of the overall wall thickening rate on the inner side of the bends for mandrels numbered 1 to 7, as well as the conventional mandrels, is presented in Figure 15. Analysis of Figure 15(a) shows that using an asymmetric thickness core ball at the No. 1 position reduces the thickening rate of the fittings at bending angles between 30° and 42°, but slightly increases it at bending angles between 48° and 54°. The use of asymmetric thickness core balls at the No. 2 and No. 3 positions has a less significant impact on the thickening rate. This occurs because, at the No. 1 position, the high-stress area is smaller compared to the other two positions when bending the tubing between 30° and 57°. Therefore, the best reduction in thickening rate is achieved by using asymmetric thickness core balls at the No. 1 position, while positions No. 2 and No. 3 have minimal impact on reducing the thickening rate. Analysis of Figure 15(b) shows that when asymmetric thickness core balls are used at positions 1 and 2, the tubing thickening rate is reduced between 36° and 42°, but the thinning rate increases between 48° and 57°.When core balls are used at positions 1 and 3, the tubing thickening rate is reduced between 36° and 48°, while the thinning rate slightly increases between 48° and 57°. Using asymmetric thickness core balls at positions 2 and 3 is counterproductive to reducing the overall tubing thickening rate. Therefore, using asymmetric thickness core balls at positions 1 and 3 yields the best reduction in the thickening rate. Analysis of Figure 15(c) shows that when asymmetric thickness core balls are used at positions 1, 2, and 3, the thickening rate is reduced between 0° and 42°, but it increases significantly between 42° and 57°. A combined analysis of Figures 15(a), (b), and (c) shows that, compared to conventional mandrels, the use of asymmetric thickness mandrels can reduce the maximum stress area on the fittings at bending angles between 30° and 57°, which in turn reduces the thickening rate. Therefore, the optimal ranking for reducing the thickening rate is as follows: No. 1 > No. 5 > No. 4 > No. 2. The remaining mandrels are less effective in improving the thickening rate. 3.3.3 Ellipticity The distribution of overall ellipticity for the bends corresponding to mandrels numbered 1 to 7, including conventional mandrels, is presented in Figure 16. Analysis of Figure 16(a) shows that using an asymmetric thickness core ball at the No. 1 position leads to a slight increase in ellipticity between 36° and 57°. At the No. 2 position, ellipticity decreases significantly between 6° and 30°, but increases dramatically beyond 30°. At the No. 3 position, the ellipticity remains similar to that of the conventional mandrel. The use of the asymmetric thickness core ball reduces the support for the tubing, leading to increased ellipticity. The No. 2 position maintains constant contact with the tubing, having the most significant impact on the support effect. It is evident that the use of asymmetric thickness core balls does not significantly reduce the overall ellipticity of the tubing. In terms of controlling the increase in ellipticity, the No. 3 position is the most effective, followed by No. 1, with No. 2 being the least effective. Analysis of Figure 16(b) shows that using two asymmetric thickness core balls does not significantly reduce the overall ellipticity. It decreases ellipticity between 0° and 30° but increases it significantly between 30° and 57°. Using asymmetric thickness core balls at positions 1 and 3 provides the best control over the increase in ellipticity. Analysis of Figure 16(c) shows that using asymmetric thickness core balls at positions 1, 2, and 3 reduces ellipticity between 0° and 42°, but significantly increases it between 42° and 57°. A combined analysis of Figures 16(a), (b), and (c) reveals that asymmetric thickness mandrels reduce support for the tubing compared to conventional mandrels, leading to an increase in ellipticity, particularly between 42° and 57°.Therefore, the optimal ranking for controlling the increase in ellipticity is as follows: No. 5 > No. 3 > No. 1 > No. 6. The remaining mandrels are less effective in improving the ellipticity. 4 Conclusion (1) The use of asymmetric thickness mandrels significantly impacts the stresses on both the inner and outer sides of the tubing, with the effect largely depending on the positional arrangement of the asymmetric thickness mandrel balls within the mandrel. (2) When an asymmetric thickness core ball is positioned progressively farther from the core axis, the stress on the outside of the tubing near the start of the bend initially increases and then decreases, while the stress on the inside of the tubing after the middle of the bend first decreases and then increases. (3) The use of an asymmetric thickness core ball significantly reduces the rates of tubing thinning and thickening compared to a conventional core ball, but it also slightly increases the rate of ellipticity. When using an asymmetric thickness core ball, positioning it at the No. 1 position is more effective in reducing the thickening rate of the fittings, although its effect on reducing the thinning rate is less pronounced. Additionally, the increase in ellipticity is minimized. When using two asymmetric thickness core balls, placing them in positions 1 and 3 provides a more noticeable reduction in both the thinning rate and the increase in ellipticity. However, the effect on reducing the thickening rate is less significant. (4) Among the mandrels with asymmetric core balls, the No. 4 mandrel was the most effective in reducing the overall thinning rate of the tubing, the No. 1 mandrel was the most effective in reducing the overall thickening rate, and the No. 5 mandrel was the most effective in controlling the ellipticity of the tubing. (5) The effectiveness of the asymmetric thickness core ball positions in improving the thinning rate follows this order: core 2 > core 1 > core 3. For improving the thickening rate, the order is: core 1 > core 2 > core 3. In terms of reducing ellipticity, the order is: core 3 > core 1 > core 2. References Yang H, Li H, Ma J, et al. Breaking bending limit of difficult-to-form titanium tubes by differential heating-based reconstruction of neutral layer shifting[J]. International Journal of Machine Tools and Manufacture, 2021, 166: 103742. Cheng C, Chen H, Guo J, et al. 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Deformation Property and Suppression of Ultra-Thin-Walled Rectangular Tube in Rotary Draw Bending[J/OL]. Metals, 2020, 10(8): 1074. DOI:10.3390/met10081074. COOPER L, CROUVIZIER M, EDWARDS S, et al. In situ micro gas tungsten constricted arc welding of ultra-thin walled 2.275 mm outer diameter grade 2 commercially pure titanium tubing[J/OL]. Journal of Instrumentation, 2020, 15(06): P06022-P06022. DOI:10.1088/1748-0221/15/06/P06022. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-5283053","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":373862277,"identity":"fff3e236-83c7-431b-bf92-5a7506a2517f","order_by":0,"name":"lanfang 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Engineering","correspondingAuthor":false,"prefix":"","firstName":"Wujie","middleName":"","lastName":"Yuan","suffix":""},{"id":373862279,"identity":"57a50ac3-fa7b-4fc4-b4c8-3bb0e235b312","order_by":2,"name":"Heng Li","email":"","orcid":"","institution":"Northwestern Polytechnical University School of Materials Science and Engineering","correspondingAuthor":false,"prefix":"","firstName":"Heng","middleName":"","lastName":"Li","suffix":""},{"id":373862280,"identity":"f4694475-675e-4a72-ae82-f6518599ba95","order_by":3,"name":"Xunzhong Guo","email":"","orcid":"","institution":"Nanjing University of Aeronautics and Astronautics College of Material Science \u0026 Technology","correspondingAuthor":false,"prefix":"","firstName":"Xunzhong","middleName":"","lastName":"Guo","suffix":""},{"id":373862281,"identity":"3c56d8fb-10ec-4119-92ef-dfcacbdb49ef","order_by":4,"name":"Zili Wang","email":"","orcid":"","institution":"Zhejiang University School of Mechanical 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Engineering","correspondingAuthor":false,"prefix":"","firstName":"Hao","middleName":"","lastName":"Pan","suffix":""}],"badges":[],"createdAt":"2024-10-17 13:09:28","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5283053/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5283053/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":69252937,"identity":"4d32ac58-1623-4cf5-abfb-fc85fe5b7020","added_by":"auto","created_at":"2024-11-18 11:57:11","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":128681,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of rotary draw bending of tube\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-5283053/v1/0f6e1a56f9a7c4e8cc6160c9.png"},{"id":69253225,"identity":"cba47604-1666-46aa-9032-4b3cd5233d47","added_by":"auto","created_at":"2024-11-18 12:05:10","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":404379,"visible":true,"origin":"","legend":"\u003cp\u003eCommon mandrel connections\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-5283053/v1/17b2f75aa400d4fc4cd2ace0.png"},{"id":69253227,"identity":"3e84cf02-675c-41d8-87c2-6f6d50ea6c3d","added_by":"auto","created_at":"2024-11-18 12:05:10","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":178968,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of the asymmetric thickness ball\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-5283053/v1/1bc9f8cabbfacfc71e8f0618.png"},{"id":69252926,"identity":"abaa4920-9f0f-4ca3-b8ca-a74479e58cf5","added_by":"auto","created_at":"2024-11-18 11:57:10","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":199540,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of the ball arrangement\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-5283053/v1/636504ec55113c43c228379e.png"},{"id":69252925,"identity":"ff29fd5f-4b92-4e56-9a3d-41453eae9d1e","added_by":"auto","created_at":"2024-11-18 11:57:10","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":118459,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of the asymmetric thickness ball and conventional ball\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-5283053/v1/6b76d7b727adcee667336dd9.png"},{"id":69254377,"identity":"57f64563-adec-4acb-bfee-7e3300d09c88","added_by":"auto","created_at":"2024-11-18 12:13:10","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":52261,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of asymmetric thickness mandrel\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-5283053/v1/7a7664d1f1c60d8ab0f5c9e2.png"},{"id":69254378,"identity":"9a8671cf-1cd8-4a0a-a50b-942bf022607f","added_by":"auto","created_at":"2024-11-18 12:13:10","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":179294,"visible":true,"origin":"","legend":"\u003cp\u003eFinite element analysis model of pipe NC bending forming process\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-5283053/v1/b1bf998cdcb86c1644cacc43.png"},{"id":69254722,"identity":"e4f74422-223b-4c9b-9468-ab9c30e02374","added_by":"auto","created_at":"2024-11-18 12:21:10","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":433856,"visible":true,"origin":"","legend":"\u003cp\u003eKM-A100-CNC-E120 tube bending machine\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-5283053/v1/ed0822ec3254be554a9a8150.png"},{"id":69252928,"identity":"d673a029-5e2e-4125-9c6f-c3fd7ee043df","added_by":"auto","created_at":"2024-11-18 11:57:10","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":109129,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of bent tube obtained by experiment and simulation\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-5283053/v1/d1071b5135ad487bbe31a425.png"},{"id":69252941,"identity":"596bf1f1-9324-4e90-8cc1-7d7abca7b4c4","added_by":"auto","created_at":"2024-11-18 11:57:11","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":33565,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of wall thickness variation between simulation and experiment for ultra-thin wall tubes\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-5283053/v1/6abd76a41fc17d9d7b096cab.png"},{"id":69253228,"identity":"6e015d64-546a-442b-9fe8-12abe77d3505","added_by":"auto","created_at":"2024-11-18 12:05:10","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":21613,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of wall thickness ellipticity between simulation and experiment for ultra-thin wall tubes\u003c/p\u003e","description":"","filename":"11.png","url":"https://assets-eu.researchsquare.com/files/rs-5283053/v1/291cf708134bc8dfd97ad7b2.png"},{"id":69252933,"identity":"dc97d165-63a4-4a28-bec7-d9dc592976de","added_by":"auto","created_at":"2024-11-18 11:57:10","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":182743,"visible":true,"origin":"","legend":"\u003cp\u003eThe equivalent force diagram of the outer ultra-thin-walled bends\u003c/p\u003e","description":"","filename":"12.png","url":"https://assets-eu.researchsquare.com/files/rs-5283053/v1/6949024baea0e0179d4bba57.png"},{"id":69252936,"identity":"dc79057e-198f-4188-ad81-50045020e04b","added_by":"auto","created_at":"2024-11-18 11:57:11","extension":"png","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":194786,"visible":true,"origin":"","legend":"\u003cp\u003eThe equivalent force diagram of the inner ultra-thin-walled bends\u003c/p\u003e","description":"","filename":"13.png","url":"https://assets-eu.researchsquare.com/files/rs-5283053/v1/aff48cd78a99d51d5291f791.png"},{"id":69254381,"identity":"949c229d-26bc-4174-b30e-b9b9d4bd5013","added_by":"auto","created_at":"2024-11-18 12:13:11","extension":"png","order_by":14,"title":"Figure 14","display":"","copyAsset":false,"role":"figure","size":74484,"visible":true,"origin":"","legend":"\u003cp\u003eVariation curve of wall thickness thinning rate of bent tube with No.1~7 ATMs vs. conventional mandrels\u003c/p\u003e","description":"","filename":"14.png","url":"https://assets-eu.researchsquare.com/files/rs-5283053/v1/463f1e3215138d34fbb355e5.png"},{"id":69254380,"identity":"088a1695-3172-433e-8c44-669d88e7222a","added_by":"auto","created_at":"2024-11-18 12:13:10","extension":"png","order_by":15,"title":"Figure 15","display":"","copyAsset":false,"role":"figure","size":73256,"visible":true,"origin":"","legend":"\u003cp\u003eVariation curve of wall thickness thickening rate of bent tube with No.1~7 ATMs vs. conventional mandrels\u003c/p\u003e","description":"","filename":"15.png","url":"https://assets-eu.researchsquare.com/files/rs-5283053/v1/f7294b5215bed2402f60fbe2.png"},{"id":69253231,"identity":"816e8d0c-4162-465d-a33f-b42c2b3df904","added_by":"auto","created_at":"2024-11-18 12:05:11","extension":"png","order_by":16,"title":"Figure 16","display":"","copyAsset":false,"role":"figure","size":64490,"visible":true,"origin":"","legend":"\u003cp\u003eVariation curve of ellipticity rate of bent tube with No.1~7 ATMs vs. conventional mandrels\u003c/p\u003e","description":"","filename":"16.png","url":"https://assets-eu.researchsquare.com/files/rs-5283053/v1/59f1b4ed9105650d5640e52d.png"},{"id":70882192,"identity":"bba3662a-634e-4698-8504-2b3aacfd9c1b","added_by":"auto","created_at":"2024-12-08 23:57:10","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3231969,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5283053/v1/f106d434-2e09-4fc7-aa52-a0f44870c10f.pdf"}],"financialInterests":"","formattedTitle":"Effects of Asymmetric Thickness Mandrel on NC RDB Forming Quality of Ultra-Thin-Walled Tube","fulltext":[{"header":"Introduction","content":"\u003cp\u003eDue to the material and structural requirements of pipe bending for lightweight and high-strength applications, its use has become increasingly prevalent across high-end industries such as aviation, machinery, automotive, and energy sectors. Consequently, pipe bending occupies a critical position within the field of plastic forming. To meet lightweight requirements, precision machinery often employs ultra-thin-walled pipe fittings with a diameter-to-thickness ratio exceeding 100. These fittings, characterized by large diameters and thin walls, are highly susceptible to defects such as cracking, wrinkling, and cross-sectional distortion during bending. Thus, there is an urgent need to investigate the key theories and technologies related to the bending and forming of ultra-thin-walled pipes, with the goal of improving their forming quality.\u003c/p\u003e\n\u003cp\u003eUsually, for thick-walled pipe fittings (D/t\u0026lt;30), international and domestic scholars have conducted numerous studies. Thin-walled pipe fittings (30\u0026nbsp;\u0026le;\u0026nbsp;D/t\u0026nbsp;\u0026le;\u0026nbsp;50) is more difficult, so scholars have more research content.\u003c/p\u003e\n\u003cp\u003eYang et al. [1] surpassed the bending limitations of large-diameter, thin-walled titanium tubes by employing a differential heating method. Cheng et al. [2] elucidated the principle by which the mandrel\u0026apos;s support induces deformation in the tubing and systematically examined how various mandrel parameters affect the bending and forming behavior of stainless steel, thin-walled tubes. Guo et al. [3] developed a mechanical model for bend-around forming and analyzed the stress-strain variations in pipe fittings during this process by integrating mechanical simulations with experimental data to investigate the effects of mandrel parameters on wall thickness and cross-sectional morphology of bent pipes. Salem et al. [4] designed and validated the reliability of a novel chain mandrel, using stainless steel tubes as the test subject. Kajikawa et al. [5] investigated how mandrel position, diameter, and tip radius influence the forming quality of fittings, using a D=7 mm, t=0.21 mm copper tube as the study model. Their experiments revealed that when the mandrel\u0026apos;s diameter and tip radius were 6.18 mm, defect-free bending of fittings with R=2~3D was achieved. R.J. Gu et al. [6] examined the relationship between the core drawing process and rebound angle in D=38 mm, t=1 mm stainless steel tubes, discovering a linear correlation between the rebound angle and the bending angle as the latter increases.\u003c/p\u003e\n\u003cp\u003eCurrently, ultra-thin-walled pipe fittings with a large diameter-to-thickness ratio (D/t \u0026ge; 100) present significant forming challenges due to their large outer diameter and thin-walled structure, making them highly susceptible to defects such as cracking, wrinkling, and cross-sectional distortion. Both domestic and international scholars have conducted numerical simulations and experimental studies on ultra-thin-walled pipe fittings. Hai Liu et al. [7] utilized 10,000 bead grains as fillers in push bending, successfully forming ultra-thin-walled pipe fittings with a diameter of 30 mm and a thickness of 0.3 mm. Xiaosong Wang et al. [8] employed a double-layer tube hydraulic bending process to form an inner tube of stainless steel (D = 180 mm, t = 1 mm) and found that the most severe thinning occurred near a cross-section with a symmetry plane angle of 21\u0026deg;, rather than at the center point. Lanfang Jiang et al. [9] developed core die with various combinations of core ball thicknesses based on 304 stainless steel pipe fittings (D=91.6 mm, t=0.9 mm). They found that the thinning rate can be significantly reduced by utilizing smaller core ball thicknesses at the beginning of the bend and larger ones at the end, and as large as possible in the end bending end, Yiwei Xu et al. [10] proposed and validated the RRS process for 304 austenitic stainless steel thin-walled tubes, demonstrating that the thinning rate of ultra-thin-walled steel tubes can be increased to 90% under ultra-low stress triaxial ring-roll spinning conditions. Sun Kang et al. [11] investigated the hydroforming process for complex cross-section corrugated tubes (D = 50 mm, t = 0.4 mm) and identified optimal process parameters, including a pre-expansion internal pressure of 7.5 MPa. Li Weizhuang et al. [12] studied ultra-thin-walled nickel-copper alloy tubes with submicron structures produced via ball spinning and found that dislocation slip led to the formation of ultrafine laminated (UFL) structures and ultrafine grains (UFGs). The high strain rate and strain gradient promoted dislocation slip, grain refinement, and deformation twinning, ultimately resulting in the successful production of ultra-thin-walled nickel-copper alloy tubes (D = 50 mm, t = 0.4 mm).Yan, J. P. et al. [13] used Inconel 718 to study the effect of the drawing process on the wall thickness of ultra-thin-walled tubes, observing that the average grain size was reduced from 4.9 \u0026mu;m to 2.1 \u0026mu;m. Kunito Nakajima et al. [14] studied ultra-thin-walled rectangular tubes (H0 = 20 mm, W0 = 10 mm, t0 = 0.5 mm) and found that deformation in the height direction decreases when a stacked mandrel is used. L. Cooper et al. [15] employed an automatic orbital gas tungsten carbide arc welding (GTCAW) process to weld ultra-thin-walled CP-2 Ti tubes (D = 2.275 mm, t = 160 \u0026plusmn; 10 \u0026mu;m).\u003c/p\u003e\n\u003cp\u003eThe aforementioned scholars primarily employed methods such as hydraulic push-bending, spiral welding, and rotational forming to process ultra-thin-walled pipe fittings. While these methods are effective, they have certain limitations, including the requirement for high-pressure environments and the use of internal filler media. Research on CNC bending forming of ultra-thin-walled pipe fittings with large diameter-to-thickness ratios remains limited. Therefore, this study focuses on ultra-thin-walled 304 stainless steel conduits with a diameter of 91.6 mm, a wall thickness of 0.9 mm, and a diameter-to-thickness ratio of 101.8. The research investigates the effects of asymmetric core ball thickness on the forming quality of the pipe fittings.\u003c/p\u003e"},{"header":"1 Research programme","content":"\u003cp\u003e\u003cstrong\u003e1.1 Bend-around forming principle\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe principle of tube CNC bending forming is illustrated in Figure 1. As shown in the figure, the primary components involved in the bending and forming process are the clamping die, pressure die, wiper die, bending die, bending die inserts, and the mandrel, which consists of two parts: the mandrel shake and the mandrel ball. Prior to bending, one end of the pipe is secured by the clamping die, and the bending die inserts are fixed onto the bending die. The outer side of the bend is compressed by the pressure die, while the inner side is supported by the wiper die, positioned at the tangent point between the pipe and the bending die. During the bending process, the bending die rotates around the Z-axis at an angular speed of \u0026omega; at point O. The clamping die and the bending die insert clamp the pipe, rotating synchronously with the bending die. Simultaneously, the pressure die moves along the X-axis at a speed of \u0026nu;, while the wiper die and the mandrel remain stationary. \u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e1.2 The role of the mandrel\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eIn ultra-thin-walled tube CNC winding and forming, a mandrel is typically installed inside the tube to provide internal support, preventing the tube from collapsing outward and reducing cross-sectional distortion during bending. Simultaneously, the mandrel works in conjunction with the wiper die to prevent the formation of wrinkles on the inner side of the tube. The mandrel primarily consists of two components: the mandrel itself and the core ball. The mandrel supports the straight section of the pipe, while the core ball supports the bent section. Based on different connection methods, mandrels can be classified into chain mandrels, as shown in Fig. 2(a), and ball-and-socket mandrels, as shown in Fig. 2(b).\u003c/p\u003e\n\u003cp\u003e1.3 Asymmetric thickness core ball\u003c/p\u003e\n\u003cp\u003eAs illustrated in Fig. 3, the conventional mandrel features equal thicknesses on both its inner and outer sides. In contrast, the asymmetric thickness mandrel is specifically designed with varying thicknesses, with a smaller thickness near the outer side of the tube and a larger thickness near the inner side. Ultimately, the asymmetric thickness core ball is integrated with the conventional mandrel to create an asymmetric thickness mandrel. In Fig. 3, the outer thickness of the mandrel ball is denoted as W1, while the inner thickness is W2. The upper and lower thicknesses of the conventional mandrel ball are equal, both represented as w.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e1.4 Asymmetric thickness mandrel programme design\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eFor ease of description, the core balls are numbered as 1, 2, and 3. The core ball closest to the core axis is labeled as core ball 1, with the serial numbers increasing as the distance from the core axis increases, resulting in core balls 2 and 3, as illustrated in Figure 4.\u003c/p\u003e\n\u003cp\u003eThe asymmetric thickness mandrels are connected via pins between the core balls. Examples of asymmetric mandrels with an outer thickness of W1 = 20 mm and an inner thickness of W2 = 22 mm, as well as Conventional mandrels with a uniform thickness of w = 24 mm, are illustrated in Fig. 5.\u003c/p\u003e\n\u003cp\u003eIn the following study, seven different arrangements of asymmetric thickness mandrels were designed to investigate the impact of both the number and arrangement of these mandrels on the forming quality of pipe fittings. A schematic diagram of the asymmetric thickness mandrel is provided in Fig. 6. The results were compared using two control groups: one with conventional mandrel pin connections and the other with conventional mandrel ball connections. The Conventional mandrel uses mandrel balls 1, 2, and 3, with the key difference between pin and ball connections being the method of connecting the mandrel balls. The specific combinations are provided in Table 1.\u003c/p\u003e\n\u003cp\u003eMandrels with Conventional core ball pin connections were compared to mandrels with symmetric thickness core ball pin connections to verify the feasibility of using asymmetric thickness mandrels. While mandrels with conventional ball joints outperform the pin-connected mandrels in terms of tubing thinning rate and ovality after bending, they result in a higher thickening rate. Therefore, the superior conventional mandrel-ball connection was compared with the experimental group to further validate the feasibility of using asymmetric thickness mandrels.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 1\u003c/strong\u003e\u0026nbsp; Asymmetric thickness mandrel combination methods\u0026nbsp;\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"336\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd rowspan=\"2\" style=\"width: 72px;\"\u003e\n \u003cp\u003eCore ball combination number\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd rowspan=\"2\" style=\"width: 72px;\"\u003e\n \u003cp\u003eNumber of asymmetric thickness core balls\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd colspan=\"3\" style=\"width: 193px;\"\u003e\n \u003cp\u003eCore ball usage\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003eCore Ball 1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003eCore Ball 2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003eCore Ball 3\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 72px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 72px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003easymmetric\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003elegacy\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003elegacy\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 72px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 72px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003elegacy\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003easymmetric\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003elegacy\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 72px;\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 72px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003elegacy\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003elegacy\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003easymmetric\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 72px;\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 72px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003easymmetric\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003easymmetric\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003elegacy\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 72px;\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 72px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003easymmetric\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003elegacy\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003easymmetric\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 72px;\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 72px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003elegacy\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003easymmetric\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003easymmetric\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 72px;\"\u003e\n \u003cp\u003e7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 72px;\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003easymmetric\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003easymmetric\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003easymmetric\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 72px;\"\u003e\n \u003cp\u003eConventional Mandrel Pin Hitch\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 72px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003elegacy\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003elegacy\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003elegacy\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 72px;\"\u003e\n \u003cp\u003eConventional Mandrel Baseball Hitch\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 72px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003elegacy\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003elegacy\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 64px;\"\u003e\n \u003cp\u003elegacy\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e"},{"header":"2 Finite element modelling","content":"\u003cp\u003e\u003cstrong\u003e2.1 Geometric modelling\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eUsing dynamic explicit finite element software Abaqus, a three-dimensional elastic-plastic finite element analysis model of the CNC bending process for ultra-thin-walled pipes was developed. Table 2 presents the structural parameters of each forming die used in the CNC tube bending process. Based on pipe bending requirements, empirical formulas, and actual production specifications, the mandrel outreach was determined to be 6 mm, with a spacing of 7 mm between the core ball and the mandrel. Additionally, the gaps between the clamping die, bending die inserts, bending die, pressure die, wiper die, and the pipe were all set to 0 mm.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 2\u003c/strong\u003e\u0026nbsp; Size parameters of forming dies\u0026nbsp;\u003c/p\u003e\n\u003cdiv align=\"\"\u003e\n \u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"340\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 52.0588%;\"\u003e\n \u003cp\u003eparametric\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47.9412%;\"\u003e\n \u003cp\u003enumerical value\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 52.0588%;\"\u003e\n \u003cp\u003eClamping die length/mm\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47.9412%;\"\u003e\n \u003cp\u003e215\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 52.0588%;\"\u003e\n \u003cp\u003eLength of bending die insert/mm\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47.9412%;\"\u003e\n \u003cp\u003e220\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 52.0588%;\"\u003e\n \u003cp\u003ePressure die length/mm\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47.9412%;\"\u003e\n \u003cp\u003e500\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 52.0588%;\"\u003e\n \u003cp\u003eWiper die length/mm\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47.9412%;\"\u003e\n \u003cp\u003e255\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 52.0588%;\"\u003e\n \u003cp\u003eMandrel length/mm\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47.9412%;\"\u003e\n \u003cp\u003e250\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 52.0588%;\"\u003e\n \u003cp\u003eMandrel diameter/mm\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47.9412%;\"\u003e\n \u003cp\u003e89.2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 52.0588%;\"\u003e\n \u003cp\u003eBall Diameter/mm\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47.9412%;\"\u003e\n \u003cp\u003e88.9\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 52.0588%;\"\u003e\n \u003cp\u003eThickness of balls/mm\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47.9412%;\"\u003e\n \u003cp\u003e24\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 52.0588%;\"\u003e\n \u003cp\u003eNumber of balls\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 47.9412%;\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cstrong\u003e2.2 Unit type\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eUltra-thin-walled pipe fittings are modeled as variable shells using a four-node curved shell element (S4R), which allows for the analysis of changes in wall thickness and the distribution of stress and strain during the bending process. The remaining models are established as rigid bodies using a four-node rigid element (R3D4), which remains undeformed by default throughout the bending process.\u003c/p\u003e\n\u003cp\u003eThe model applies von Mises\u0026apos; yield criterion, the associated flow rule, and kinematic hardening. The mechanical properties of the 304 ultra-thin-walled stainless steel pipe are presented in Table 3. Each forming die is defined as a rigid body, with 45-gauge steel selected as the material. The mechanical properties of 45-gauge steel are provided in Table 4.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 3\u003c/strong\u003e\u0026nbsp; Mechanical properties of 304 stainless steel\u0026nbsp;\u003c/p\u003e\n\u003cdiv align=\"\"\u003e\n \u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"340\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 51.4706%;\"\u003e\n \u003cp\u003eparametric\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003enumerical value\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 51.4706%;\"\u003e\n \u003cp\u003eDensity \u003cem\u003e\u0026rho;\u003c/em\u003e/ Kg/ mm\u003csup\u003e3\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003e7.93e-6\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 51.4706%;\"\u003e\n \u003cp\u003eModulus of elasticity \u003cem\u003eE\u003c/em\u003e/ MPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003e2.08e5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 51.4706%;\"\u003e\n \u003cp\u003eYield strength \u003cem\u003e\u0026sigma;\u003csub\u003es\u003c/sub\u003e\u003c/em\u003e/ MPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003e383\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 51.4706%;\"\u003e\n \u003cp\u003eTensile strength \u003cem\u003e\u0026sigma;\u003csub\u003eb\u003c/sub\u003e/\u003c/em\u003e MPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003e812\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 51.4706%;\"\u003e\n \u003cp\u003ePoisson\u0026apos;s ratio \u0026upsilon;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003e0.3\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 4\u003c/strong\u003e\u0026nbsp; Mechanical properties of forming dies\u0026nbsp;\u003c/p\u003e\n\u003cdiv align=\"\"\u003e\n \u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"340\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 51.4706%;\"\u003e\n \u003cp\u003eparametric\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003enumerical value\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 51.4706%;\"\u003e\n \u003cp\u003eDensity \u003cem\u003e\u0026rho;\u003c/em\u003e/ Kg/ mm\u003csup\u003e3\u003c/sup\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003e7.85e-6\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 51.4706%;\"\u003e\n \u003cp\u003eModulus of elasticity \u003cem\u003eE\u003c/em\u003e/ MPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003e2.1e5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 51.4706%;\"\u003e\n \u003cp\u003ePoisson\u0026apos;s ratio \u0026upsilon;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003e0.269\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cstrong\u003e2.4 Boundary condition\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe process parameters for the simulation and analysis of the pipe bending process are provided in Table 5.Based on existing research, the bending die is set to rotate around the Z-axis at 0.1 rad/s at point O. The bending die insert and clamping die rotate synchronously with the bending die. The bending die insert and clamping die rotate in unison with the bending die, while the pressure die moves along the X-axis at a tangential linear velocity of 19 mm/s, matching the bending die\u0026apos;s motion. Both the creasing die and mandrel are constrained in all degrees of freedom and remain stationary, while the core ball is unconstrained and retains full degrees of freedom.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 5\u003c/strong\u003e\u0026nbsp; Process parameters for simulation of tube bending\u0026nbsp;\u003c/p\u003e\n\u003cdiv align=\"\"\u003e\n \u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"340\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 51.4706%;\"\u003e\n \u003cp\u003eparametric\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003enumerical value\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 51.4706%;\"\u003e\n \u003cp\u003eBending angle \u0026theta;/ \u0026deg;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003e57\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 51.4706%;\"\u003e\n \u003cp\u003eBending radius R/ mm\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003e190\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 51.4706%;\"\u003e\n \u003cp\u003eDie speed v/ (mm/ s)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003e19\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 51.4706%;\"\u003e\n \u003cp\u003eBending angular speed \u0026omega;/ (rad/s)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003e0.1\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 51.4706%;\"\u003e\n \u003cp\u003eBending time T/ s\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eThe friction between the pipe and the forming dies plays a crucial role in determining the forming quality of the pipe. In this model, Isotropic Coulomb friction is applied by default to define the friction properties, and the friction coefficients are selected based on actual production conditions. The Finite Sliding method is selected for contact tracking to fully capture surface interactions, resulting in higher calculation accuracy. The friction coefficients between the pipe and each forming die are presented in Table 6.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 6\u003c/strong\u003e\u0026nbsp; Friction coefficient between pipe and each forming die\u003c/p\u003e\n\u003cdiv align=\"\"\u003e\n \u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"340\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 51.4706%;\"\u003e\n \u003cp\u003econtact pair\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003enumerical value\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 51.4706%;\"\u003e\n \u003cp\u003eClamping dies and fittings\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003e0.9\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 51.4706%;\"\u003e\n \u003cp\u003eBending die inserts and fittings\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003e0.9\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 51.4706%;\"\u003e\n \u003cp\u003eBending dies and fittings\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003e0.45\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 51.4706%;\"\u003e\n \u003cp\u003eCompression die and fittings\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003e0.45\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 51.4706%;\"\u003e\n \u003cp\u003eWiper dies and fittings\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003e0.3\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 51.4706%;\"\u003e\n \u003cp\u003eMandrels \u0026amp; Tubing\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48.5294%;\"\u003e\n \u003cp\u003e0.2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eUsing the dynamic explicit finite element software Abaqus, a three-dimensional elastic-plastic finite element analysis model for the CNC winding and bending forming process of thin-walled pipes was developed, as illustrated in Figure 7.\u0026nbsp;\u003c/p\u003e"},{"header":"3 Results and discussion","content":"\u003cp\u003e\u003cstrong\u003e3.1 Bending and forming quality evaluation index\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e(1) Wall Thinning Rate: During bending, tubing is subjected to tangential tensile stresses, causing the material on the outer side to stretch, which leads to a reduction in wall thicknrsaess. The wall thinning rate is calculated as:\u003c/p\u003e\n\u003cp\u003e\u003cimg src=\"data:image/png;base64,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\" height=\"85\" width=\"371\"\u003e\u003c/p\u003e\n\u003cp\u003eIn the formula, \u0026xi;1 represents the wall thickness reduction rate, t is the initial wall thickness of the tubing, and\u0026nbsp;is the minimum wall thickness after the bending and forming process.\u003c/p\u003e\n\u003cp\u003e(2) Wall Thickness Gain Rate: During bending, the material on the inner side of the fitting is subjected to tangential compressive stresses, causing thickening in the direction of the wall thickness. The wall thickening rate is calculated as:\u003c/p\u003e\n\u003cp\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"373\" height=\"95\"\u003e\u003c/p\u003e\n\u003cp\u003eIn the formula, \u0026xi;2 represents the wall thinning rate,\u0026nbsp;\u003c/p\u003e\n\u003cp\u003et is the initial wall thickness of the tubing, andis the maximum wall thickness after the bending and forming process.\u003c/p\u003e\n\u003cp\u003e(3) Ellipticity is: During the bending process, the material on the outer side of the bend is subjected to both tangential tensile stress and radial compressive stress, resulting in a reduction in wall thickness and radial displacement. This causes the pipe diameter along the bending radius to decrease and the circumferential diameter to increase. As a result, the originally circular cross-section gradually becomes elliptical due to cross-sectional flattening distortion. Ellipticity is defined as:\u003c/p\u003e\n\u003cp\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"425\" height=\"85\"\u003e\u003c/p\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003eIn the formula, \u0026phi; represents the ellipticity of the bend,\u0026nbsp;is the maximum outer diameter of the cross-section,is the minimum outer diameter of the cross-section, and D is the original outer diameter of the cross-section.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3.2 Simulation analysis model validation\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eTo verify the accuracy and validity of the above finite element analysis model, a bending test was conducted on an ultra-thin-walled 304 stainless steel pipe with dimensions of D60mm\u0026nbsp;\u0026times;\u0026nbsp;t1mm\u0026nbsp;\u0026times;\u0026nbsp;R120mm, using the KM-A100-CNC-E120 CNC pipe bender. The bending test was performed at a bending speed of 0.1 rad/s. The structural parameters of each forming die and the bending process were consistent with those used in the simulation analysis, as shown in Figure 8.\u003c/p\u003e\n\u003cp\u003eAs shown in Figure 9, a comparison between the wall thickness distribution cloud map from the finite element simulation and the bending test samples of the ultra-thin-walled 304 stainless steel pipe was conducted. It was found that no wrinkling occurred on the inner side of the bent section, and the test samples were largely consistent with the simulation analysis results.\u003c/p\u003e\n\u003cp\u003eThe wall thickness values from the simulation calculations of the ultra-thin-walled pipe fittings were extracted and compared with the actual wall thickness values measured after the bending test. The wall thickness variation curves for the inner and outer centerlines of the ultra-thin-walled pipe bends were plotted for each measurement angle. As shown in Figure 10, the simulation results were found to be largely consistent with the wall thickness values obtained from the test measurements. In the outer thinning curve, the largest error occurred at a bending angle of 0\u0026deg;, where the test value was 0.843 mm and the simulation value was 0.807 mm, resulting in a relative error of 4.27%. For the inner thickening curve, the largest error was at a bending angle of 54\u0026deg;, with a test value of 0.968 mm and a simulation value of 0.988 mm, yielding a relative error of 2.08%. The error between the simulation values and the test values remained within a reasonable range, with all discrepancies being less than 10%\u003c/p\u003e\n\u003cp\u003eFigure 11 presents a comparison of the ellipticity curves of ultra-thin-walled pipe fittings at different measurement angles. The ellipticity trends obtained from both the simulation and the test are largely consistent. Both curves show an increasing trend between bending angles of 18\u0026deg; and 48\u0026deg;, reaching the maximum ellipticity at 48\u0026deg;. At this point, the simulation-calculated ellipticity is 2.01%, while the test-measured ellipticity is 1.98%.\u003c/p\u003e\n\u003cp\u003eIn summary, by comparing the simulation and test results in terms of bend shape, wall thickness changes, and ellipticity, the results were found to be largely consistent. This indicates that the finite element model for the CNC winding and bending process of thin-walled pipes developed in this paper is reliable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3.3 Analysis of results after rebound\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3.3.1 stress analysis\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe distribution of equivalent stresses on the outer and inner sides of the bent pipe, derived from the simulation of the numbered 1\u0026ndash;7 conventional mandrels, is presented in Table 1 and shown in Figures 12 and 13.\u003c/p\u003e\n\u003cp\u003eAs observed in Figure 12, the maximum stress (377.1\u0026ndash;502.5 MPa) on the outside of the various fittings occurs consistently at the outer bending angle of 9\u0026deg;\u0026ndash;18\u0026deg;. For bending angles between 18\u0026deg;\u0026nbsp;and 57\u0026deg;, except for fitting No. 4, the area of maximum stress (189\u0026ndash;314.4 MPa) on the inner side of the fittings is comparable to that of the conventional mandrel-baseball hinge joint. In contrast, the maximum stress area on the inner side of the other fittings is significantly larger than that of the conventional mandrel-pin hinge joint.\u003c/p\u003e\n\u003cp\u003eA comparison of the stress diagrams for the outer side of the fittings at positions Nos. 1 to 3 with those of the conventional mandrels reveals that, before the middle section of the inner bend, the stresses at all three positions are largely the same. After the middle section of the inner bend, the stress increases significantly with the use of an asymmetric thickness mandrel at position No. 3. However, since the maximum stress is relatively low, altering the position of the asymmetric thickness core ball has minimal effect on the maximum stress area on the outer side of the fitting.\u003c/p\u003e\n\u003cp\u003eAs shown in Figure 13, at bending angles from 0\u0026deg;\u0026nbsp;to 24\u0026deg;, the maximum stress area (627.8 to 753.2 MPa) on the inside of the fittings with conventional mandrel pin articulation is much smaller than that of the conventional mandrel pin reaming. The maximum stress area on the inside of fittings Nos. 3, 5, 6, and 7 is larger than that of conventional mandrel pin reaming, while the maximum stress area on the other numbered fittings is smaller. The maximum stress area (627.8 to 753.2 MPa) on the inside of fittings with conventional mandrel pin articulation and asymmetric thickness mandrels is smaller than that of conventional mandrel-baseball articulation at bending angles between 30\u0026deg;\u0026nbsp;and 57\u0026deg;.\u003c/p\u003e\n\u003cp\u003eA comparison of the stress diagrams on the inner side of the pipe fittings with conventional mandrel pins at positions Nos. 1 to 3 reveals the following: After the middle section of the inner bend, using an asymmetric thickness mandrel at position No. 2 significantly reduces the stress. Using an asymmetric thickness mandrel at position No. 1 also reduces the stress, though not as much as at position No. 2. Conversely, using an asymmetric thickness mandrel at position No. 3 significantly increases the stress.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3.3.2 Wall Thickness Reduction Rate\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe distribution of the overall wall thickness reduction rate on the outer side of the bends for mandrels numbered 1 to 7, as well as the conventional mandrel after rebound, is presented in Figure 14.\u003c/p\u003e\n\u003cp\u003eAnalysis of Figure 14(a) reveals that using an asymmetric thickness core ball at position No. 2 reduces the overall thinning rate of the fittings. Using the asymmetric thickness core ball at position No. 1 only improves the thinning rate at bending angles between 36\u0026deg;\u0026nbsp;and 57\u0026deg;, while the use of the core ball at position No. 3 has no significant effect on the overall thinning rate improvement.\u003c/p\u003e\n\u003cp\u003eTherefore, the effect of the asymmetric thickness core ball position arrangement on reducing the thinning rate is as follows: the No. 2 core ball position has the greatest effect, followed by the No. 1 position, while the No. 3 position shows minimal improvement. The superior performance of the No. 2 core ball position is attributed to its constant contact with the tubing during the bending process, which provides the largest contact area. Altering the thickness of the core ball at this position reduces the contact area, resulting in the greatest effect on reducing the thinning rate. In summary, using an asymmetric thickness core ball at the No. 2 position yields the most effective reduction in the pipe thinning rate.\u003c/p\u003e\n\u003cp\u003eAnalysis of Figure 14(b) shows that using asymmetric thickness core balls at both positions 1 and 2 effectively reduces the overall thinning rate of the fittings. When core balls are used at positions 2 and 3, the thinning rate is reduced slightly. However, using asymmetric thickness core balls at positions 1 and 3 does not yield significant improvement. Therefore, when using two asymmetric thickness core balls, positions 1 and 2 provide the most effective reduction in the tubing thinning rate.\u003c/p\u003e\n\u003cp\u003eAnalysis of Figure 14(c) shows that when asymmetric thickness core balls are used at all positions (1, 2, and 3), the overall thinning rate of the fittings is significantly reduced between 18\u0026deg;\u0026nbsp;and 57\u0026deg;.\u003c/p\u003e\n\u003cp\u003eA combined analysis of Figures 14(a), (b), and (c) shows that using an asymmetric core ball reduces the contact area with the tubing compared to a conventional core ball, thereby improving the tubing thinning rate. The No. 2 core ball position showed the greatest reduction in thinning rate due to its constant contact with the tubing. Therefore, the order of effectiveness in reducing the thinning rate is as follows: No. 7 \u0026gt; No. 4 \u0026gt; No. 2. The greatest reduction in thinning rate, 4.15%, was achieved using the No. 7 mandrel compared to conventional mandrels.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3.3.4 Wall thickening rate\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe distribution of the overall wall thickening rate on the inner side of the bends for mandrels numbered 1 to 7, as well as the conventional mandrels, is presented in Figure 15.\u003c/p\u003e\n\u003cp\u003eAnalysis of Figure 15(a) shows that using an asymmetric thickness core ball at the No. 1 position reduces the thickening rate of the fittings at bending angles between 30\u0026deg; and 42\u0026deg;, but slightly increases it at bending angles between 48\u0026deg; and 54\u0026deg;. The use of asymmetric thickness core balls at the No. 2 and No. 3 positions has a less significant impact on the thickening rate. This occurs because, at the No. 1 position, the high-stress area is smaller compared to the other two positions when bending the tubing between 30\u0026deg; and 57\u0026deg;.\u003c/p\u003e\n\u003cp\u003eTherefore, the best reduction in thickening rate is achieved by using asymmetric thickness core balls at the No. 1 position, while positions No. 2 and No. 3 have minimal impact on reducing the thickening rate.\u003c/p\u003e\n\u003cp\u003eAnalysis of Figure 15(b) shows that when asymmetric thickness core balls are used at positions 1 and 2, the tubing thickening rate is reduced between 36\u0026deg;\u0026nbsp;and 42\u0026deg;, but the thinning rate increases between 48\u0026deg;\u0026nbsp;and 57\u0026deg;.When core balls are used at positions 1 and 3, the tubing thickening rate is reduced between 36\u0026deg;\u0026nbsp;and 48\u0026deg;, while the thinning rate slightly increases between 48\u0026deg;\u0026nbsp;and 57\u0026deg;. Using asymmetric thickness core balls at positions 2 and 3 is counterproductive to reducing the overall tubing thickening rate. Therefore, using asymmetric thickness core balls at positions 1 and 3 yields the best reduction in the thickening rate.\u003c/p\u003e\n\u003cp\u003eAnalysis of Figure 15(c) shows that when asymmetric thickness core balls are used at positions 1, 2, and 3, the thickening rate is reduced between 0\u0026deg;\u0026nbsp;and 42\u0026deg;, but it increases significantly between 42\u0026deg;\u0026nbsp;and 57\u0026deg;.\u003c/p\u003e\n\u003cp\u003eA combined analysis of Figures 15(a), (b), and (c) shows that, compared to conventional mandrels, the use of asymmetric thickness mandrels can reduce the maximum stress area on the fittings at bending angles between 30\u0026deg; and 57\u0026deg;, which in turn reduces the thickening rate. Therefore, the optimal ranking for reducing the thickening rate is as follows: No. 1 \u0026gt; No. 5 \u0026gt; No. 4 \u0026gt; No. 2. The remaining mandrels are less effective in improving the thickening rate.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3.3.3 Ellipticity\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe distribution of overall ellipticity for the bends corresponding to mandrels numbered 1 to 7, including conventional mandrels, is presented in Figure 16.\u003c/p\u003e\n\u003cp\u003eAnalysis of Figure 16(a) shows that using an asymmetric thickness core ball at the No. 1 position leads to a slight increase in ellipticity between 36\u0026deg;\u0026nbsp;and 57\u0026deg;. At the No. 2 position, ellipticity decreases significantly between 6\u0026deg;\u0026nbsp;and 30\u0026deg;, but increases dramatically beyond 30\u0026deg;. At the No. 3 position, the ellipticity remains similar to that of the conventional mandrel. The use of the asymmetric thickness core ball reduces the support for the tubing, leading to increased ellipticity. The No. 2 position maintains constant contact with the tubing, having the most significant impact on the support effect.\u003c/p\u003e\n\u003cp\u003eIt is evident that the use of asymmetric thickness core balls does not significantly reduce the overall ellipticity of the tubing. In terms of controlling the increase in ellipticity, the No. 3 position is the most effective, followed by No. 1, with No. 2 being the least effective.\u003c/p\u003e\n\u003cp\u003eAnalysis of Figure 16(b) shows that using two asymmetric thickness core balls does not significantly reduce the overall ellipticity. It decreases ellipticity between 0\u0026deg;\u0026nbsp;and 30\u0026deg;\u0026nbsp;but increases it significantly between 30\u0026deg;\u0026nbsp;and 57\u0026deg;. Using asymmetric thickness core balls at positions 1 and 3 provides the best control over the increase in ellipticity.\u003c/p\u003e\n\u003cp\u003eAnalysis of Figure 16(c) shows that using asymmetric thickness core balls at positions 1, 2, and 3 reduces ellipticity between 0\u0026deg;\u0026nbsp;and 42\u0026deg;, but significantly increases it between 42\u0026deg;\u0026nbsp;and 57\u0026deg;.\u003c/p\u003e\n\u003cp\u003eA combined analysis of Figures 16(a), (b), and (c) reveals that asymmetric thickness mandrels reduce support for the tubing compared to conventional mandrels, leading to an increase in ellipticity, particularly between 42\u0026deg; and 57\u0026deg;.Therefore, the optimal ranking for controlling the increase in ellipticity is as follows: No. 5 \u0026gt; No. 3 \u0026gt; No. 1 \u0026gt; No. 6. The remaining mandrels are less effective in improving the ellipticity.\u003c/p\u003e"},{"header":"4 Conclusion","content":"\u003cp\u003e(1) The use of asymmetric thickness mandrels significantly impacts the stresses on both the inner and outer sides of the tubing, with the effect largely depending on the positional arrangement of the asymmetric thickness mandrel balls within the mandrel.\u003c/p\u003e \u003cp\u003e(2) When an asymmetric thickness core ball is positioned progressively farther from the core axis, the stress on the outside of the tubing near the start of the bend initially increases and then decreases, while the stress on the inside of the tubing after the middle of the bend first decreases and then increases.\u003c/p\u003e \u003cp\u003e(3) The use of an asymmetric thickness core ball significantly reduces the rates of tubing thinning and thickening compared to a conventional core ball, but it also slightly increases the rate of ellipticity. When using an asymmetric thickness core ball, positioning it at the No. 1 position is more effective in reducing the thickening rate of the fittings, although its effect on reducing the thinning rate is less pronounced. Additionally, the increase in ellipticity is minimized. When using two asymmetric thickness core balls, placing them in positions 1 and 3 provides a more noticeable reduction in both the thinning rate and the increase in ellipticity. However, the effect on reducing the thickening rate is less significant.\u003c/p\u003e \u003cp\u003e(4) Among the mandrels with asymmetric core balls, the No. 4 mandrel was the most effective in reducing the overall thinning rate of the tubing, the No. 1 mandrel was the most effective in reducing the overall thickening rate, and the No. 5 mandrel was the most effective in controlling the ellipticity of the tubing.\u003c/p\u003e \u003cp\u003e(5) The effectiveness of the asymmetric thickness core ball positions in improving the thinning rate follows this order: core 2\u0026thinsp;\u0026gt;\u0026thinsp;core 1\u0026thinsp;\u0026gt;\u0026thinsp;core 3. For improving the thickening rate, the order is: core 1\u0026thinsp;\u0026gt;\u0026thinsp;core 2\u0026thinsp;\u0026gt;\u0026thinsp;core 3. In terms of reducing ellipticity, the order is: core 3\u0026thinsp;\u0026gt;\u0026thinsp;core 1\u0026thinsp;\u0026gt;\u0026thinsp;core 2.\u003c/p\u003e"},{"header":"References","content":"\n\u003col\u003e\n\u003cli\u003eYang H, Li H, Ma J, et al. Breaking bending limit of difficult-to-form titanium tubes by differential heating-based reconstruction of neutral layer shifting[J]. International Journal of Machine Tools and Manufacture, 2021, 166: 103742. \u003c/li\u003e\n\u003cli\u003eCheng C, Chen H, Guo J, et al. Investigation on the influence of mandrel on the forming quality of thin-walled tube during free bending process[J]. Journal of Manufacturing Processes, 2021, 72: 215-226.\u003c/li\u003e\n\u003cli\u003eGuo, X.Z, Cheng, C, Guo J.X, et al. Influence of mandrel structure on forming quality in free bending for thin -walled tube[J]. Forging \u0026amp; Stamping Technology, 2021,46: 127-136.\u003c/li\u003e\n\u003cli\u003eSalem M, Farzin M, Kadkhodaei M, et al. A chain link mandrel for rotary draw bending: experimental and finite element study of operation[J]. The International Journal of Advanced Manufacturing Technology, 2015, 79(5-8): 1071-1080.\u003c/li\u003e\n\u003cli\u003eKAJIKAWA S, WANG G, KUBOKI T, et al. Prevention of defects by optimizing mandrel position and shape in rotary draw bending of copper tube with thin wall[J]. Procedia Manufacturing, 2018, 15: 828-835.\u003c/li\u003e\n\u003cli\u003eGU R J, YANG H, ZHAN M, et al. Research on the springback of thin-walled tube NC bending based on the numerical simulation of the whole process[J]. Computational Materials Science, 2008, 42(4): 537-549.\u003c/li\u003e\n\u003cli\u003eLIU H, ZHANG S H, CHENG M, et al. DEM simulation of bead packs as fillers in thin-wall tube push bending process[C/OL]//THE 11TH INTERNATIONAL CONFERENCE ON NUMERICAL METHODS IN INDUSTRIAL FORMING PROCESSES: NUMIFORM 2013. Shenyang, China, 2013: 708-713[2023-11-05]. https://pubs.aip.org/aip/acp/article/1532/1/708-713/877953. DOI:10.1063/1.4806899.\u003c/li\u003e\n\u003cli\u003eWANG X, LI F. Analysis of wall thickness variation n the hydro-bending of a double-layered tube[J/OL]. The International Journal of Advanced Manufacturing Technology, 2015, 81(1-4): 67-72. DOI:10.1007/s00170-015-7188-x.\u003c/li\u003e\n\u003cli\u003eJIANG L, LIN Y, LI H, et al. A new mandrel design with mandrel ball thickness variation for the bending process of aviation ultra-thin-walled tubes[J/OL]. The International Journal of Advanced Manufacturing Technology, 2022, 122(3-4): 1805-1819. DOI:10.1007/s00170-022-09954-y.\u003c/li\u003e\n\u003cli\u003eXU Y, ZHAO C, WANG Q, et al. Ultra-low stress triaxiality ring-roller spinning: microstructure, plastic behavior, and cumulative large deformation mechanism[J/OL]. Journal of Materials Processing Technology, 2024, 324: 118240. DOI:10.1016/j.jmatprotec.2023.118240.\u003c/li\u003e\n\u003cli\u003eResearch on hydroforming process of stainless steel bellows with ultra-thin wall-All Databases[EB/OL]. [2024-03-09]. https://webofscience.clarivate.cn/wos/alldb/full-record/CSCD:7610752.\u003c/li\u003e\n\u003cli\u003eLI W, XU Y, WANG C, et al. Preparation and forming mechanism of ultrathin-walled Ni-Cu alloy tubes with submicrometer structures by ball spinning[J/OL]. The International Journal of Advanced Manufacturing Technology, 2022, 121(7-8): 5427-5437. DOI:10.1007/s00170-022-09738-4.\u003c/li\u003e\n\u003cli\u003eYAN J P, ZHAO R, MENG B, et al. Analysis of the properties and microstructure of ultra-thin tube[J/OL]. IOP Conference Series: Materials Science and Engineering, 2022, 1270(1): 012023. DOI:10.1088/1757-899X/1270/1/012023.\u003c/li\u003e\n\u003cli\u003eNAKAJIMA K, UTSUMI N, SAITO Y, et al. Deformation Property and Suppression of Ultra-Thin-Walled Rectangular Tube in Rotary Draw Bending[J/OL]. Metals, 2020, 10(8): 1074. DOI:10.3390/met10081074.\u003c/li\u003e\n\u003cli\u003eCOOPER L, CROUVIZIER M, EDWARDS S, et al. In situ micro gas tungsten constricted arc welding of ultra-thin walled 2.275 mm outer diameter grade 2 commercially pure titanium tubing[J/OL]. Journal of Instrumentation, 2020, 15(06): P06022-P06022. DOI:10.1088/1748-0221/15/06/P06022.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"thin-walled tube, NC bending, retracting mandrel, finite element analysis","lastPublishedDoi":"10.21203/rs.3.rs-5283053/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5283053/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eTo satisfy lightweight design requirements, aerospace ducts frequently employ ultra-thin-walled tubes with a diameter-to-thickness ratio (D/t) exceeding 100.However, ultra-thin-walled tubes present significant forming challenges, and the mandrel plays a critical role in their bending. Therefore, investigating the effect of mandrel structure on the quality of ultra-thin-walled tubes formed through NC bending is of considerable importance. In this study, utilizing the Abaqus nonlinear finite element platform, an asymmetric thickness ball design method is proposed. Based on the positioning of the asymmetric balls within the mandrel, seven distinct designs for asymmetric thickness mandrels are developed. This study conducts a finite element analysis of the NC rotary draw bending (RDB) process for ultra-thin-walled 304 stainless steel tubes and validates the corresponding experiments. The results indicate that as the asymmetric thickness mandrel is positioned further from the mandrel, the stress on the outer side of the tube near the bend initiation first increases and then decreases, while the stress on the inner side of the tube, after the midpoint of the bend, initially decreases and then increases. The use of asymmetric thickness mandrels significantly reduces both the thinning and thickening rates of the tubes, though their impact on improving the ellipticity is less pronounced. The core ball nearest to the mandrel is designated as Ball 1, with subsequent balls further from the mandrel labeled as Ball 2 and Ball 3, respectively. The placement of the asymmetric thickness balls improves the thinning rate in the order: Ball 2 \u0026gt; Ball 1 \u0026gt; Ball 3; enhances the thickening rate in the order: Ball 1 \u0026gt; Ball 2 \u0026gt; Ball 3; and optimizes the ellipticity in the order: Ball 3 \u0026gt; Ball 1 \u0026gt; Ball 2.\u003c/p\u003e","manuscriptTitle":"Effects of Asymmetric Thickness Mandrel on NC RDB Forming Quality of Ultra-Thin-Walled Tube","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-11-18 11:57:06","doi":"10.21203/rs.3.rs-5283053/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"85a60031-d291-4dfc-8b06-7417be1a571b","owner":[],"postedDate":"November 18th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-12-08T23:49:02+00:00","versionOfRecord":[],"versionCreatedAt":"2024-11-18 11:57:06","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-5283053","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-5283053","identity":"rs-5283053","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}