Effect of die fillet radius on tube thinning in free bending

Abstract

During the operation of a free-bending die, its fillets inevitably wear out, thereby reducing their “sharpness” and decreasing the forming accuracy of tubes. To clarify the influence law of fillet radius on the wall thickness of freely bent pipes, this study employs a combined approach of finite element simulation and experimental testing to systematically analyze the strain characteristics and wall thickness distribution of the pipes. The results indicate that the outer surface of the pipe exhibits the maximum thinning rate when the mold ceases upward movement. During the mold movement stage, increasing the fillet radius can mitigate tube wall thinning by expanding the contact area between the mold and the pipe. The thinnest region of the bent pipe is located at the end of the radius formation section, with a maximum thinning rate of 5.4%. Furthermore, an increase in mold offset distance, a rise in friction coefficient, and the adoption of thin-walled pipe structures all exacerbate the thinning phenomenon in this vulnerable region, which warrants special attention in engineering practice. With the increase of fillet radius, the maximum thinning rate presents a variation pattern of “first decreasing rapidly and then decreasing extremely slowly”. Within the range where the fillet radius-to-tube radius ratio (R_B/r) is less than 0.33, the rate of decrease is significantly faster than that in the range where R_B/r > 0.33. When R_B/r = 0.33, a sudden enhancement of the thinning-inhibiting effect is observed. Subsequently, further increasing the fillet radius exerts no significant inhibitory effect on the maximum thinning rate. The findings of this study provide a theoretical basis and engineering guidance for the optimization of fillet parameters of free bending dies and the control of pipe forming accuracy.

Introduction

Free bending forming of aviation tubes is a technology that enables the continuous formation of complex tubular curves in three-dimensional space. It allows for variable curvature and multi-dimensional bending, breaking through the “curvature limitation” of the traditional winding process. This technology facilitates the direct fabrication of special-shaped, thin-walled, and high-precision tubular structural components required in the aviation field, making it a “core technology” for enhancing the reliability of key aviation systems. The precision of free bending dies directly determines the forming accuracy of tube workpieces. In practical production, bending dies inevitably undergo wear, which impairs the dimensional precision of the die and subsequently reduces the forming accuracy of tubes. Therefore, to improve the manufacturing accuracy and stability of tube products, there is an urgent need to conduct in-depth research on the impact of die wear on forming accuracy.

Scholars at home and abroad have conducted extensive research on the free bending forming process. However, most of these studies have focused on theoretical research, tube geometric dimensions, material properties, and bending process parameters. In terms of theoretical modeling and tube geometric dimensions, Xuan et al.^1,2 investigated the influence of relative thickness (0.033, 0.067, 0.1) on the stress-strain distribution, neutral layer movement, and formability of tubes; Pepelnjak et al.³ proposed a reverse engineering method to determine the material flow profile, addressing the challenge that “tensile specimens cannot be obtained from thin tubes with a diameter-to-thickness ratio less than 10.” Li et al.⁴ compared spatial and planar involute members, finding that under the same bending radius, spatial members exhibit higher equivalent stress and tangential strain, with more pronounced thickening of the inner wall and thinning of the outer wall. In terms of material parameters, Liu et al.^5,6 studied the cross-sectional deformation and wall thickness distribution of 3A21 aluminum alloy tubes; Zhu et al.⁷ improved the finite element method for H96 rectangular tubes by considering the variable elastic modulus effect and the Bauschinger effect. Zhan et al.^8,9,10 explored the springback mechanism of large-diameter thin-walled CT20 titanium alloy tubes and analyzed the influence of material parameters on springback. H et al. investigated H62 brass tubes using a combination of finite element analysis and bending tests, selecting bending radii ranging from 45 to 100 mm at 5 mm intervals to detect forming defects. Their results showed that as the bending radius increased, the range of wall thickness variation gradually decreased, and the locations of severe forming defects gradually shifted toward both ends of the tube. In terms of forming process parameters, Zhang et al.¹¹ addressed the issue of “insufficient forming accuracy due to springback” during the formation of complex spatial tube components via three-dimensional free bending technology. Based on the discretization method, they established a springback compensation model for variable-curvature tube bending components: specifically, the central axis curve of the tube was discretized, and the “cross-sectional offset springback analysis model of the stress-neutral layer under axial thrust” was utilized to incrementally calculate the spatial compensation angles of different cylindrical helical elements. Guo et al.¹² conducted an in-depth analysis of the key process parameters affecting forming quality and identified the technical parameters required for optimal bending performance. Xuan et al.^1,2 further discussed the influence of feed rate on the formability of thin-walled rectangular tubes. Cheng et al.¹³ studied the effect of friction coefficient, finding that as the friction coefficient increased, the tube wall thickness increased more significantly and deformation became more severe. Li Hai et al. analyzed the influence of mandrel size parameters on stress distribution during bending via a mandrel analytical model. Guo et al.¹⁴ designed a spherical connection structure between the bending die and guide rail via geometric derivation, reducing the bending ratio to 2.5 and optimizing the bending limit parameters. He et al.¹⁵ focused on the influence of deformation transition on the offset distance - bending radius (U-R) relationship, introducing a correction factor k and establishing a U-R correction relationship that accounts for “the bending angle of the transition section + the length of the formed region.” They validated the reliability of this relationship through experiments and simulations. Abd El-Aty et al.¹⁶ first fitted U-R curves under specific conditions using forming results obtained under different eccentricities. They then proposed a mathematical expression for the U-R relationship and applied it to finite element simulations and physical experiments of complex spatially bent parts, verifying the feasibility of the expression. Hu et al.¹⁷ analyzed the geometric relationship among “eccentricity, bending die rotation angle, deformation zone length, and bending radius” in free bending, with a focus on resolving the “non-tangential contact” problem: under ideal conditions, the bending die maintains tangential contact with the outer bend of the tube; however, clearances and material properties can lead to adjustability of the rotation angle, disrupting the tangential state and causing overbending or underbending. Liu et al.¹⁸ proposed a parameter acquisition strategy for variable-curvature spatial tubes based on “segmentation + knowledge base + alternative model.” Specifically, they developed a parameter acquisition method using the knowledge base and established a singular value decomposition-radial basis function alternative model for the transition section to quickly predict the axial profile. Zhang et al.¹⁹ proposed a “three-dimensional free bending forming process that coordinates the movement of the transition section of finned irregular tubes and the bending die” for three-dimensional free bending, addressing the challenges of “unclear transition section mechanism” and “complex cross-sectional deformation of finned special-shaped tubes” to resolve the issue of matching transition section movement. Chen et al.²⁰ studied the influence of sliding or rolling friction on tube forming quality during contact between bending dies of different structures and the tube. Lu et al.²¹ derived the offset distance and offset angle of the neutral layer, as well as the bending moment after tube forming. The derived formulas incorporate key process parameters such as relative bending radius and bending angle, providing a quantitative foundation for controlling the neutral layer position and optimizing blank size design by adjusting forming parameters. Ma et al.²² proposed a generalized analytical method to simulate springback. This method is capable of integrating the effects of process parameters on springback, accurately predicting springback magnitude under different parameter combinations, and providing technical support for process parameter optimization.

In terms of die design, Chen et al.²³ investigated the influence of bending die structure on forming force and forming quality during free bending. Their research revealed that incorporating rollers and steel balls into the bending die enhances the uniformity of circumferential wall thickness distribution in bent pipes. Guo et al.²⁴ developed a novel spherical joint between the bending die and guide, and analyzed the formation mechanism of the elbow arc profile. Maier et al.²⁵ examined the effect of process parameters on bending residual stress in six-axis freely moving dies; for a welded steel pipe with an outer diameter of 42.4 mm, the distance between the bending die center and guide rail end was adjusted to 56 mm along the Z-direction. Hashemi²⁶ studied the influence of key parameters on bending radius during free bending of rectangular copper profiles, with the fixed die fillet set to 4 mm and the distance between fixed and moving die centers approximately 32 mm. Several researchers have noted the significant effect of die fillet radius on free bending performance. Cheng²⁷ investigated the forming behavior with bending die fillets of 1.5, 3.5, and 4.5 mm, proposing that a fillet radius of 4.5 mm should be adopted to reduce internal stress concentration in bent pipes. Guo²⁸ calculated pipe wall thickness under five fillet radii (0, 0.5, 1.5, 2.5, and 3.5 mm), observing notable differences in wall thickness distribution across different fillet sizes. These studies provide valuable guidance for high-precision pipe free bending from the perspectives of pipe dimensions, material properties, process parameters, and die design. However, the influence of die fillet radius on free bending accuracy remains insufficiently explored. In fact, the bending die fillet radius plays a pivotal role in “stress field regulation and defect prevention” for forming quality control: the cavity transition fillet optimizes stress distribution at the pipe bending section, thereby suppressing cracking induced by local stress concentration. During continuous die operation, substantial contact pressure and sliding friction are generated between the fillet contact area and pipe workpiece. Inevitable wear occurs over time, resulting in a gradual increase in fillet radius and subsequent degradation of pipe machining accuracy. Therefore, this study investigates the influence of fillet radius on wall thickness of freely bent pipes. By integrating finite element simulation and experimental investigation, the strain characteristics of pipe free bending were analyzed, and the wall thickness distribution law was examined. This work provides a reference for free-bending die fillet design and service life assessment.

Finite element analysis and experimental analysis verification

Free bend process principle

As shown in Fig. 1, this is a schematic diagram of free bending. In free bending forming, the plastic deformation of the tube is controlled through the coordinated operation of key components. The core components include the bending die, spherical bearing, guide die, clamping die, and pushing die. The pushing die acts as the “power source,” applying an axial force to drive the tube material to move. Spherical bearing can move up and down on the installation plane, driving the bending die to perform synchronous movement. Driven by the spherical bearing, the bending die will form an offset distance (U), and different offset distances will correspond to different bending radii. The inner surface of the bending die is in direct contact with the tube to be processed. Usually, a fillet is made at the contact part of the bend die (R_B as shown in Fig. 1). Obviously, different fillet radii will result in different contact areas and forces between the tube and the die, and the precision of the manufactured tube will also vary to some extent. Moreover, during long-term operation, some degree of wear is inevitable; this wear results in a gradual increase in the fillet radius, which in turn affects the machining accuracy of the tube fitting. To study the influence of fillets on the forming accuracy of tubes, a series of fillet radii (0.5 to 15 mm) were adopted in the subsequent simulation research.

Fig. 1

Free-bending forming principle diagram.

Finite element modeling

Considering the actual working conditions of the die and the requirements for improving calculation speed and efficiency, this study simplified the model during finite element simulation, which mainly consists of six components: spherical bearing, bending die, guiding mechanism, tube blank, clamping mechanism, and pushing die. Among these, the spherical bearing adopted a simplified model and was defined as a rigid body in ABAQUS to enhance calculation efficiency. The bending die maintains a following motion state during the forming process; to enable this following motion, it is defined as a deformable body. The guiding mechanism and tube blank clamping mechanism primarily function to position and clamp the tube blank during pipe forming, thereby preventing axial instability and buckling of the tube blank. These two components were defined as discrete rigid bodies in ABAQUS to further simplify the modeling process. For mesh generation, the four-node double-curvature shell element S4R was selected to characterize the deformation behavior of the tube blank, as it possesses hourglass control and reduced integration capabilities. In addition, tetrahedral elements were employed for mesh division of other components. The finite element model of the CAD model after meshing is shown in Fig. 2.

In the three-dimensional finite element modeling of free bending forming, the spherical bearing, guiding mechanism, and tube blank clamping mechanism were all defined as rigid bodies, thus eliminating the need to define their material properties. For the bending die, although it is a rigid entity, its material properties (density, elastic modulus, and Poisson’s ratio) still need to be specified. For the tube blank that undergoes bending deformation, corresponding material properties were assigned, and the specific material parameters are listed in Tables 1 and 2^29,30.

A contact model was developed in the interaction module, and this contact condition was applied globally to all contact interfaces. Based on the aforementioned modeling, appropriate friction properties were selected to characterize the transmission behavior of normal pressure and tangential shear stress between the tube blank and the die surface during the loading process. For cold forming processes, the classical Coulomb friction model is commonly employed to describe the frictional contact state between the workpiece and the die. The friction coefficient is typically determined through experimental tests. In the forming process investigated in this study, the friction coefficient for each frictional contact state was set as a constant value of 0.06¹³, independent of field variables such as relative sliding speed, contact surface pressure, and ambient temperature. Additionally, the friction coefficient was assumed to exhibit isotropy in different directions. Regarding load setup, the guide die was fully fixed. The push die was permitted to move solely along the axial direction of the pipe (with its other five degrees of freedom constrained), and its axial speed was set to 30 mm/s. Spherical bearings were retained with only one degree of freedom in the vertical direction. The bending die and the tube blank were maintained in a free state to ensure unconstrained bending deformation, thereby confining all movements entirely within the preset plane.

In the finite element simulation of free bending forming, to obtain the most accurate simulation results, the analysis step was set to explicit dynamic analysis, with the mass scaling factor set to 100 in each analysis step. The control type was designated as “Target Time Increment”. As a key parameter in the analysis step settings, the target time increment specifies the desired time step size, which directly influences computational efficiency, accuracy, and stability. The time increment for explicit analysis is strictly constrained by the Courant stability condition, expressed by the formula: Δt_max ≈ L_min/c. Herein, L_min denotes the minimum element characteristic length of the model, and c represents the wave velocity in the material. The target time increment must be less than or equal to Δt_max; otherwise, the program will automatically adjust it to Δt_max (or a smaller value), resulting in unstable calculations. A target value that is too small will significantly increase the number of calculation steps and reduce efficiency, while an excessively large target value will be compulsorily reduced by the program, rendering such a setting meaningless. In this simulation, the target time increment was set to 10^− 5.

Fig. 2

Finite element model.

Table 1 Tube material parameters.

Table 2 Other mold material parameters.

Free bending experiment and verification

To verify the reliability of the established finite element model, this study employed the engineering prototype developed by Zhejiang Jingmasong Intelligent Manufacturing Co., Ltd. to conduct free bending forming tests. The test device and its core working principle are presented in Fig. 3. During the free bending forming process, all components function in synergy to facilitate the forming of the tube: under the joint constraints of the spherical bearing and the guide die, the bending die is capable of rotating around the axis of the guide die. This enables flexible adjustment of its spatial posture, thereby laying the foundation for adjusting the tube’s bending direction. The device employs a push-type drive mode to drive the tube to move stably along the axial direction, ensuring that the tube continuously enters the bending area as required for the forming process. The central controller precisely coordinates the coordinated operation of the X, Y, and Z axes at a specific matching speed ratio based on the preset forming path, ultimately enabling the continuous bending forming of tubes with multiple curvature characteristics. The materials of the tubes used in the experiments are presented in Table 1, and the dimensions of the mold are presented in Table 2. Upon completion of the tests, the thickness variations of the tubes were measured, with the Olympus 38DL PLUS ultrasonic thickness gauge employed as the measurement instrument. This instrument has a measurement range spanning from 0.080 to 635.00 mm, and its thickness measurement accuracy is ± 0.001 mm. The measurement setup is presented in Fig. 4.

In this study, the wall thicknesses of the inner and outer arcs of the pipe were primarily measured. After the pipe was bent into shape, the positions of the initial bending points on the inner and outer arcs were accurately located, as indicated by Point 1 in Fig. 5. Different measurement points were selected along the inner and outer arcs, respectively, as presented in Fig. 5. An ultrasonic thickness gauge was employed to measure the thickness at each measurement point. The outer diameter at each corresponding point was measured using a vernier caliper, and the cross-sectional distortion rate of the pipe was calculated based on these outer diameter data.

Fig. 3

Free bending engineering prototype.

Fig. 4

Measurement of pipe wall thickness and cross-sectional distortion: (a) measurement of pipe wall thickness (b) measurement of cross-sectional distortion.

Fig. 5

The measurement positions of the inner and outer wall thickness of the pipe: (a) inner wall measurement position, (b) outer wall measurement position.

To ensure consistency with the finite element simulation parameters, multiple measurements of the wall thickness thinning rate and distortion rate were performed on the formed pipe specimens. The experimental results are presented in Table 3.

Table 3 Experimental measurement results.

The finite element simulation results and experimental results for different mold offset distances are presented in Fig. 6. Meanwhile, the distortion rate and wall thickness thinning rate of the experimental section were measured, and the experimental results were compared with the finite element simulation results, as illustrated in Fig. 7.

Fig. 6

Simulation and experimental results of free bending forming of pipes: (a) Simulation result, (b) experimental result.

The overall shape and size of the elbow workpiece numerically obtained via the finite element method (FEM) are in good agreement with the experimental results. Compared with the experimental data, the error in the thickness change rate derived from the simulations is less than 9.6%, as shown in Fig. 6. Compared with the experimental data, the finite element thinning rate error is 9.6%, and the distortion rate error is 5.1%. These results demonstrate that the FEM for free bending can effectively and accurately predict the plastic deformation process during the free bending of tubes. This method thus provides a valuable tool for investigating the effects of the bending die radius on the forming quality of metal tubes.

Fig. 7

Comparison of simulation results with experimental data: (a) section distortion rate, (b) outer wall thinning rate.

Results and discussions

The tube is further segmented based on its different bending radii. As shown in Fig. 8, the tube fitting during the bending process can be divided into three primary sections: S₁ (bending radius establishment section), S₂ (stable forming section) and S₃ (pre-formed section). Specifically, for S₁, the bending die undergoes displacement, and the curvature of the tube undergoes a continuously changing process. In contrast, S₂ refers to the stable forming section; during this stage, the offset distance of the bending die remains constant, and the tube curvature stays unchanged. To refine the analysis, S₂ (stable forming section) is further subdivided into two sub-sections based on stress variations: S₂₁ (initial stable section) and S₂₂ (stable arc section). Each section exhibits distinct boundaries in terms of mechanical properties and thickness variation. Notably, each section corresponds to a specific length of the elbow tube to be bent and is associated with unique bending parameters. Except for the bending direction parameter, the bending parameters of a subsequent bending section can be independent of those of the preceding section. The segmented analysis proposed herein provides an important theoretical foundation for future digitalization efforts in tube bending.

Fig. 8

The diagram of the stress nephogram.

The validated finite element model is used to investigate how variations in fillet radius affect the quality of pipe forming. This study analyzes the changes in contact area, contact force and other bending parameters resulting from variations in the fillet radius of the bending die during the pipe bending process.

Free bending differs fundamentally from winding bending. When the pipe is pressed into the bending die, the contact position between the pipe blank and the die evolves dynamically over time. The contact surfaces corresponding to different fillet radii are extracted, as illustrated in Fig. 9. Taking the fillet radius R_B=15 mm as an example: during the upward movement of the die, the contact area rapidly increases from 0 to 222 mm ² and fluctuates within this range over the 0–3.3 s period, with a fluctuation amplitude of 20 mm². This fluctuation indicates that the contact between the pipe blank and the die is not perfectly smooth.

When the die displacement U reaches 10 mm (i.e., within the 3.3–23 s period), the die stops moving, and the pipe enters a constant-radius bending stage. In the initial stable phase (S₂₁), the contact area decreases sharply due to the cessation of die movement: by 7.5 s, it drops to 167.8 mm², representing a 26% reduction compared to the average contact area recorded in the 0–3.3 s period. Notably, after 7.5 s, slight fluctuations (amplitude: 20 mm²) persist in the contact area as the pipe advances, which is attributed to the inherently non-smooth nature of the pipe-die contact.

As shown in Fig. 9, the contact area between the pipe and the die varies significantly with the fillet radius. To quantify this relationship, the maximum and average contact areas under different fillet radii were extracted, and their variation curves are presented in Fig. 9. Both the maximum and average contact areas increase gradually with an increase in R_B. In particular: When R_B/r < 0.5 (where r denotes the pipe radius), the contact area increases linearly with R B; When R_B/r > 0.5, the growth rate of the contact area slows down noticeably. This observation demonstrates that the effect of the fillet radius on the contact area follows a complex nonlinear behavior, rather than a simple linear correlation.

Fig. 9

Change in contact area during bending.

Fig. 10

Change of contact area of the stable.

Contact forces between the die and the pipe were extracted, and the time-dependent variation curve of contact force is presented in Fig. 11. Taking R_B=15 mm as an example: during the forming process, the contact force rapidly increases from 0 to 3431 N and exhibits slight oscillations within the 1.2–3.3 s period. This phenomenon indicates that during the upward movement of the die, the contact force rises sharply and oscillates as a result of relative extrusion and frictional interactions between the die and the pipe. As the die ceases movement, the contact force decreases to 2256 N and remains in a state of mild fluctuation. Combined with the contact area variation curve (Fig. 9), it is observed that for fillet radii R_B>2.5 mm, both the contact force and contact area in the bending radius establishment stage (S₁) are larger than those in the stable forming stage (S₂). This is attributed to the combined effects of the die’s upward displacement and the pipe’s feeding motion. In contrast, for the small fillet radius R_B=2.5 mm, there is no significant difference in the average contact area or contact force between stage S₁ and stage S₂. This suggests that with a relatively sharp die fillet, the pipe can be bent under a lower die extrusion force, leading to negligible variations in contact area and contact force between the bending radius establishment and stable forming stages. Notably, when R_B=2.5 mm (small fillet radius), the fluctuations in both contact force and contact area are significantly amplified. At t = 1.55s, both the contact area and contact force drop to zero. This indicates that during the bending radius establishment stage with a small fillet die, the upward movement and pressure application of the die result in discontinuous extrusion or even transient loss of contact between the die and the pipe. Furthermore, this discontinuous contact behavior exerts a notable influence on the wrinkling susceptibility of the pipe during forming.

Fig. 11

Change in contact force during Bending.

Contact forces between the pipe and the die under different fillet radii were extracted, as presented in Fig. 12. As the die fillet radius increases, the contact force gradually rises initially and then plateaus—exhibiting a trend consistent with the contact area curve (Fig. 10). This indicates that employing a smaller fillet radius reduces the contact area and lowers the contact force. With increasing fillet radius, both the contact force and contact area increase; however, when the fillet radius ratio R_B/r exceeds 0.83, further increases in the fillet radius lead to stabilization of both the contact force and contact area.

Fig. 12

The relationship between rounded corners and contact force.

Fig. 13

Stress cloud map of the pipe during free bending Process (U = 8 mm): (a) Inner stress nephogram, (b) Outer stress nephogram.

Stress data for the inner and outer arcs were extracted from the finite element stress contour diagram (in Fig. 13), respectively, as illustrated in Fig. 14. For the inner arc, significant stress fluctuations in the pipe material are observed in the radius establishment section (S₁). Specifically, when the fillet radius (R_B) is 2.5 mm, the stress fluctuation amplitude of the inner arc within the 0–20 mm range reaches 45 MPa; minor fluctuations (amplitude: 25 MPa) also occur in the initial stable section (S₂₁). During the initial stable stage, the cessation of mold movement leads to reductions in both contact force and contact area, which induces stress fluctuations on the inner side of the pipe. Such stress waves are highly likely to trigger wrinkling and instability of the inner pipe wall. As reported by Cheng et al.³¹ (Fig. 15), wrinkling was observed in both the radius establishment section and S₂₁ of freely bent pipes through combined finite element simulation and experimental tests. In contrast, the inner arc stress remains relatively stable in the stable arc Sects. (60–140 mm, Fig. 14), suggesting that this section is less susceptible to wrinkling—a conclusion further validated by Cheng et al.’s research³¹.

Comparison of the inner arc stress variation under different mold fillet radii reveals that in the stable arc section, the pipe stress decreases with an increase in the mold fillet radius. This appears to contradict the finding in Fig. 10 that pipe contact stress increases with larger mold fillet radii. However, this apparent contradiction can be resolved by considering the mold geometry: a larger fillet radius results in a less abrupt mold profile and a larger contact area with the pipe. Although a substantial external force is required during processing, the stress generated within the pipe material is relatively low.

Fig. 14

Stress on the inner side of the pipe.

Fig. 15

Image of wrinkles on the inner side of the pipe²⁹: (a) wrinkled pipes with eccentricity of 14 mm, (b) wrinkled pipes with eccentricity of 14.5 mm.

The outer arc of the pipe was analyzed as shown in Fig. 16. It can also be found that the pipe fluctuates in both the radius establishment section S₁ and the initial stability section S₂₁. However, since the stress acting on the outer pipe wall is tensile, wrinkling deformation does not occur here. Instead, this tensile effect induces a significant reduction in pipe wall thickness. Notably, the mold fillet radius (R_B) exerts a pronounced influence on the stress peak in the radius establishment section: at the 9 mm position, the stress peaks corresponding to R_B values of 0.5 mm, 1.5 mm, 2.5 mm, and 4.0 mm are 175 MPa, 190 MPa, 204 MPa, and 270 MPa, respectively—indicating that the stress peak increases with an increasing mold fillet radius. In contrast, in the stable arc section, the stress peaks for the same R_B values are 131 MPa, 75 MPa, 57 MPa, and 33 MPa, respectively, showing an inverse trend where a larger mold fillet radius leads to lower stress on the pipe’s outer side.

A comparison of stress variation characteristics between the radius establishment section and the stable arc section reveals that the effect of mold fillet radius on pipe stress differs significantly between these two regions. This discrepancy suggests that outer arc stress in the radius establishment section is primarily governed by contact force, whereas stress in the stable arc section is dominated by contact area. During the radius establishment phase, the mold moves upward to actively compress the pipe; a larger mold fillet radius requires a higher forced lifting force, thereby increasing the pipe’s stress peak. In the stable arc section, the mold remains stationary: driven by the clamping mechanism, the pipe is passively pressed against the mold. As the mold fillet radius increases, the contact area between the mold and pipe expands, consequently reducing the stress acting on the pipe.

Fig. 16

Stress on the inner side of the pipe.

Furthermore, the strain contour plot from the finite element analysis (FEA) was utilized, as illustrated in Fig. 17. The axial strain is positive on the outer surface of the pipe and negative on the inner surface, indicating that the outer surface is subjected to axial tension while the inner surface experiences axial compression. Circumferential and radial strains exhibit a variation trend opposite to that of the axial strain—i.e., the outer surface undergoes circumferential and radial compression, whereas the inner surface is subjected to circumferential and radial tension. Inner and outer arc strains were extracted from the strain contour plot, and the axial variation curve of the pipe’s equivalent strain was plotted, as presented in Fig. 18.

Fig. 17

Strain cloud diagram of the bent pipe (U = 8 mm, R_B=10 mm): (a) radial strain cloud image, (b) axial strain cloud picture, (c) circumferential strain cloud picture, (d) equivalent strain cloud image.

As illustrated in Fig. 18, it denotes the strain variation curve of the inner arc. As observed in Fig. 18a, within the bending radius formation section S₁, the equivalent strain of the pipe’s inner arc increases from 0 to 0.07 as the die moves upward. Upon entering the initial stable stage S₂₁, small fluctuations in the equivalent strain are observed due to the reduction in contact force. Subsequently, during the subsequent stable stage (S₂₂), the equivalent strain remains essentially constant. Notably, for the small fillet radius (R_B = 2.5 mm), an abrupt change in the pipe’s equivalent strain occurs in the stable bending section: at the 390 mm position, the strain decreases from 0.06 to 0.056, which is distinctly different from the trends of other curves. This suggests that the fillet radius exerts a more significant influence on the pipe’s stable section than on the bending radius formation section and the initial stable section.

Figure 18b presents the variation curve of axial strain, whose characteristics are analogous to those of the equivalent strain curve. Similar to the equivalent strain, the axial strain exhibits slight fluctuations in the initial stable stage (S₂₁) but remains relatively stable during the subsequent stable stage (S₂₂). Depicted in Fig. 18c is the radial strain variation curve: unlike the linear growth observed in Fig. 18a, b for other strains, the radial strain in the bending radius formation section (S₁) does not increase linearly. Instead, as the die ascends, fluctuations in radial strain intensify—further indicating unstable radial material flow of the pipe. This unstable radial flow renders the inner side of the pipe susceptible to wrinkling in the bending radius formation section. Notably, for the small fillet radius (R_B = 2.5 mm), an abrupt change in radial strain occurs in the stable bending section (S₂₂), which corresponds to a sharp thickening of the pipe’s inner surface.

Figure 18d illustrates the circumferential strain variation curve. It is observed that for R_B = 2.5 mm, a downward abrupt change in circumferential strain occurs in S₂₂, indicating a significant reduction in circumferential material flow for the die with a small fillet radius. By comparing with the radial strain variation curve, it can be inferred that under the action of the small fillet radius die, during the axial compression of the pipe’s inner side, the material primarily accumulates in the radial direction with minimal circumferential flow. Consequently, dies with small fillet radii are highly likely to induce an increase in the radial wall thickness of the pipe’s inner arc in the stable bending section (S₂₂).

Fig. 18

Influence of fillet radius on strain on the inner tube wall (U = 25 mm): (a) inner equivalent strain, (b) inner axial strain, (c) inner radial strain, (d) inner circumferential strain.

The average strain of the pipe’s inner arc stable section (S₂₂) was extracted, and the strain variation characteristics under different fillet radii were analyzed, as presented in Fig. 19a. The equivalent strain curve exhibits an initial rapid increase followed by stabilization: it rises sharply when the fillet radius (R_B) ranges from 0 to 0.33r, and stabilizes once R_B reaches the critical value of 0.33r. This intuitively illustrates the trend of equivalent strain initially increasing and subsequently plateauing with increasing fillet radius. As observed in Fig. 19b, the axial strain is negative, indicating that the pipe’s inner surface is in an axial compression state. With increasing fillet radius, the axial strain gradually increases, and the compression degree intensifies. Figure 19c depicts the radial strain variation curve with fillet radius: for R_B ranging from 0 to 0.33r, radial strain decreases drastically as R_B increases; in the interval of 0.33r to r, radial strain remains essentially stable with further increases in R_B. The circumferential strain variation with fillet radius is shown in Fig. 19d: within the range of 0–0.33r, circumferential strain increases linearly with R_B, while in the 0.33r–r interval, it increases at a slower rate. By comparing Fig. 19b–d, it can be inferred that during the axial compression of the pipe’s inner surface, material flows in both radial and circumferential directions. The fillet radius exerts a considerable influence on the material flow characteristics. Specifically, for R_B between 0 and 0.33r, increasing R_B leads to a decrease in radial strain and an increase in circumferential strain. This indicates that increasing the fillet radius to a certain extent can promote circumferential material flow, thereby mitigating the wall thickness growth rate of the pipe.

Fig. 19

Influence of fillet radius on strain of stable circular arc section on the inner tube wall: (a) mean equivalent strain, (b) mean axial strain, (c) mean radial strain, (d) Mean circumferential strain.

Figure 20 presents the strain variation curve of the pipe’s outer arc. As observed in Fig. 20a, a significant abrupt change in equivalent strain occurs within the 0–200 mm axial range. This is attributed to the transition of contact friction between the die and the pipe from static to kinetic friction at the onset of bending—accompanied by a distinct discontinuity in frictional force. Furthermore, certain fluctuations in contact force arise as the die ascends, persisting until the pipe material enters the initial stable stage (S₂₁). In the pipe’s stable bending section (S₂₂), the equivalent strain remains relatively stable along the axial direction, with variations observed under different fillet radii. Notably, for R_B = 2.5 mm in Fig. 20a, a downward abrupt change in equivalent strain also occurs in the stable section, indicating weaker plastic flow of the pipe under small fillet radii.

Figure 20b depicts the axial strain variation curve of the pipe’s outer arc. Within the 0–200 mm range, slight axial compressive strain is observed, resulting from the axial compression effect induced by the clamping force applied by the clamping mechanism and the friction between the pipe and the die at the initial bending moment. As the die moves upward, the axial strain increases rapidly, decreases slightly in the initial stable stage (S₂₁), and subsequently stabilizes. As illustrated in Fig. 20b, the axial strains in the stable bending section for R_B = 2.5 mm, 7.5 mm, and 15 mm are 0.043, 0.046, and 0.049, respectively. These results indicate that increasing the fillet radius can mitigate the axial strain on the pipe’s outer surface to a certain extent.

Figure 20c depicts the radial strain variation curve. The radial strain is compressive, indicating that the pipe exhibits a radial thinning tendency. During the bending radius formation section (S₁), the radial strain of the pipe material fluctuates significantly, with the minimum radial strain achieved in the critical transition zone between the bending radius formation section (S₁) and the initial stable stage (S₂₁) (at the 293 mm position in Fig. 20c). In the stable bending section (S₂₂), the radial strain remains relatively stable. Furthermore, for a fillet radius of R_B = 2.5 mm, an upward abrupt change in radial strain is observed.

Figure 20d illustrates the circumferential strain variation curve. Circumferential strain exhibits slight fluctuations at the onset of bending. As the die ascends, the circumferential strain decreases sharply from 0 to a minimum value of -0.03 with minor fluctuations, and stabilizes in the stable bending section (S₂₂).

Fig. 20

Influence of fillet radius on strain on the outer tube wall: (a) outer equivalent strain, (b) outer axial strain, (c) outer radial strain, (d) Outer circumferential strain.

Figure 21a presents the variation curve of the pipe’s outer arc strain. With increasing fillet radius (R_B), the equivalent strain of the outer arc in the stable bending section (S₂₂) gradually increases: the growth rate is relatively rapid within the R_B range of 0–0.33r, while it slows down significantly in the 0.33r–r interval. This trend indicates that increasing the fillet radius enhances the contact pressure between the die and the pipe, thereby promoting material flow on the pipe’s outer surface. However, this enhancement effect on material flow gradually diminishes when the fillet radius is sufficiently large.

Figure 21b illustrates the axial strain variation with R_B. The axial strain is positive, confirming that the pipe’s outer surface is in an axial tension state. As R_B increases, the axial strain gradually rises and tends to stabilize when R_B reaches 0.83r. Depicted in Fig. 21c is the radial strain variation curve with R_B: the radial strain is negative, indicating that the pipe’s outer surface is under radial compression. Within the 0–0.33r range, the radial strain increases significantly with increasing R_B, reflecting a pronounced thinning tendency of the pipe’s outer wall. In the 0.33r–r interval, the radial strain decreases slowly as R_B increases and stabilizes at R_B = 0.83r.

Figure 21d depicts the circumferential strain variation with R_B. Within the 0–0.33r range, the circumferential strain decreases sharply with increasing R_B; in the 0.33r–r interval, the decreasing rate slows down noticeably.

Fig. 21

Influence of fillet radius on strain of stable circular arc on the outer tube wallz; (a) mean equivalent strain in the outer stable arc section, (b) mean axial strain in the outer stable arc section, (c) mean radial strain in the outer stable arc section (d) mean circumferential strain in the outer stable arc section.

The variation in inner wall thickness can be observed in Fig. 22a. In the bending radius establishment section (S₁), the wall thickness increases sharply to a peak. This is attributed to the rapid radial movement of the bending die in S₁, which establishes a stable offset distance. Under the combined action of the radial force exerted by the die and the axial feed thrust, the tube fitting undergoes rapid bending. A significant portion of the tube material on the inner side is rapidly compressed, leading to a sharp increase in inner wall thickness and thus forming a distinct peak. Once the bending die completes its radial offset movement, it stops quickly and remains stationary. Subsequently, under the effect of the static die, the tube undergoes constant-radius bending forming. Consequently, the compressive strain of the material decreases, and the tube’s inner wall thickness starts to decrease. However, during the transition of the die from a dynamic to a static state, the abrupt change in force induces certain strain fluctuations at the S₂₁ stage. These fluctuations, in turn, cause slight thickness variations within the stable arc segment. After entering the S₂₂ section, the thickness remains essentially stable and gradually decreases in the S₂₃ section.

The variation in the outer thickness is presented in Fig. 22b, with its variation trend being opposite to that of the inner side. During the radial movement of the mold, the outer material of the tube thins rapidly under the combined effect of mold movement and axial feed motion. The thinnest region appears at the instant the mold ceases movement—i.e., at the critical position corresponding to the constant bending radius—where the tube wall thickness reaches its minimum. Once the mold ceases movement, the tube is formed with a constant bending radius inside the mold. However, due to the abrupt change in the forces acting on the tube, the strain fluctuates during the subsequent descending process. Consequently, thickness fluctuations are also observed in the initial segment during the constant-radius forming process. In the S₂₂ and S₂₃ regions, the outer tube thickness remains stable. In contrast to the thickening observed on the inner side, the outer side exhibits the minimum degree of thinning. This phenomenon can be attributed to the applied tensile stress, which induces tangential deformation on the outer side of the tube. This tangential deformation, in turn, leads to radial compression within the cross-section, ultimately resulting in the thinning of the outer thickness.

Fig. 22

Thickness of different R_B: (a) inner thickness variation, (b) outer thickness variation.

The thinnest region of the pipe during bending is located at the junction of S₁ and S₂. Specifically, when the mold ceases upward movement, the outer wall thickness at the pipe-mold contact point reaches its minimum value. This phenomenon is attributed to the rapid thinning of the pipe material induced by the extrusion of the mold’s fillet during the mold’s movement. In other words, the mold’s upward displacement accelerates the thinning of the pipe’s outer wall.

Furthermore, as illustrated in Fig. 23a, the maximum thinning rate of the pipe material first decreases sharply and then declines extremely slowly with an increase in the fillet radius (R_B). Notably, for small fillet radii (R_B < 0.33r, where r denotes the pipe radius), the maximum thinning rate decreases at a significantly higher rate with increasing R_B compared to that observed for large fillet radii (R_B > 0.33r). This indicates that during the mold’s upward movement, increasing the fillet radius alleviates the stress on the pipe’s outer side by expanding the contact area, thereby mitigating the outer wall thinning. However, this inhibitory effect on the thinning rate undergoes a sudden transition when R_B = 0.33r. The primary mechanism is that while an excessively large fillet radius enhances the contact area, the corresponding contact force increases sharply. Consequently, when R_B/r > 0.33, further increasing the fillet radius exerts a negligible effect on suppressing the maximum thinning rate of the pipe’s outer side.

Figure 23b presents the variation curve of the thinning rate in the stable section of the pipe’s outer wall with the fillet radius. As R_B increases, the thinning rate of the stable radius section rises rapidly from 0.17% to 0.36% and subsequently increases at an extremely slow rate. For the small fillet radius range (R_B/r < 0.33), the contact force between the pipe and the mold increases with R_B, leading to enhanced radial strain in the stable section of the pipe’s outer side and a corresponding rise in the thickness reduction rate. In contrast, for the large fillet radius range (R_B/r > 0.33), further increasing R_B results in an extremely slow growth rate of the thinning rate.

Fig. 23

Thinning rate on the outer side of the pipe: (a) maximum thinning rate, (b) average thinning rate of the pipe stable radius section.

By comparing Fig. 23a, b, it is evident that the fillet radius (R_B) exhibits opposite trends with the maximum thinning rate of the radius-forming section and the average thinning rate of the radius-stable section. This phenomenon arises from the combined effect of the upward movement of the radius-forming section of the die and the horizontal extrusion of the pipe material, which results in a larger contact area between the pipe and the die compared to the contact extrusion area under a stationary die. Specifically, in the radius-forming section, the die’s influence on the pipe is primarily dependent on expanding the contact area: increasing R_B rapidly enhances the contact area, thereby reducing the stress on the pipe’s outer wall and lowering the maximum thinning rate. In contrast, for the radius-stable section (S₂₂) under the action of the stationary die, increasing R_B mainly intensifies the contact force between the pipe and the die, consequently exacerbating the thinning of the pipe’s outer wall in the stable section.

Furthermore, a comparison between the maximum thinning rate of the radius-forming section and the average thinning rate of the radius-stable section (S₂₂) reveals that the former is significantly higher than the latter. This is because the pipe is subjected to a greater force from the die during its upward movement. Notably, under small fillet radii (R_B < 0.33r), the contact stability between the die and the pipe is compromised, which tends to induce stress concentration in the pipe and further increase the outer wall thinning rate.

A key concern in engineering practice is the maximum thinning rate of the pipe. To address this, the variation curves of the maximum thinning rate with R_B under varying friction coefficients, mold offset displacements, and pipe wall thicknesses were analyzed. Figure 24 presents the variation of the maximum thinning rate with R_B for different friction coefficients. As observed, the maximum thinning rate of the pipe gradually increases with an increase in the friction coefficient. This indicates that poor lubrication leads to greater frictional resistance at the pipe-die interface, which enhances the radial strain of the pipe in the radius-forming section and consequently further exacerbates pipe thinning.

Figure 25 illustrates the variation of the maximum thinning rate with R_B for different mold offset displacements. In this Fig. 25, the mold offset displacements are 10 mm, 15 mm, 20 mm, and 25 mm, corresponding to bending radii of 58 mm, 42.8 mm, 37 mm, and 34.1 mm, respectively. It can be seen that a larger mold offset displacement results in a smaller pipe bending radius, and the maximum thinning rate of the pipe gradually increases accordingly. This confirms that pipes bent with small radii are more susceptible to severe thinning.

Figure 26 shows the variation of the maximum thinning rate with R_B for pipes of different wall thicknesses. The results indicate that the maximum thinning rate decreases with an increase in pipe wall thickness. This is because a greater wall thickness reduces the radial strain experienced by the pipe, thereby lowering its maximum thinning rate—highlighting that thin-walled pipes are more prone to thinning.

Collectively, the analysis of Figs. 24, 25 and 26 demonstrates that the die exerts the most significant influence on the pipe material when the fillet radius is small (R_B < 0.33r). As R_B continues to increase beyond this threshold, the influence of the fillet radius on the maximum thinning rate is substantially diminished.

Fig. 24

The variation curve of the maximum thinning rate with the radius of the fillet under different friction coefficients.

Fig. 25

The variation curve of the maximum thinning rate with the fillet radius under different mold offset distances.

Fig. 26

The variation curve of the maximum thinning rate with the radius of the fillet under different pipe wall thicknesses.

Conclusions

The forming accuracy of tubes is directly determined by the accuracy of free bending dies: During long-term production, die wear inevitably degrades forming accuracy, yet this issue remains insufficiently addressed in existing research both domestically and internationally. In this study, the effect of die fillet radius on the wall thickness of aviation tubes formed via free bending was investigated using finite element simulation and experimental validation. The key research findings are as follows:

• 1. Experimental results reveal that the thinnest region of the outer wall in free bending forming is located at the end of the radius-forming section—specifically, the outer wall contact interface between the pipe and the die at the moment the die ceases upward movement—with a maximum thinning rate of up to 5.4%. Notably, a larger die offset distance, higher friction coefficient, and the use of thin-walled pipes further exacerbate the thickness reduction of this vulnerable region. This critical position warrants special attention in engineering applications.

• 2. As the die fillet radius (R_B) increases, the maximum thinning rate of the pipe first decreases sharply and then declines at an extremely slow rate. Specifically, the rate of decrease is significantly higher in the range of R_B/r < 0.33 (where r denotes the pipe radius) than in the range of R_B/r > 0.33. A sudden transition in the inhibitory effect on thinning occurs when R_B/r = 0.33; beyond this threshold, further increasing the fillet radius exerts no significant suppression on the maximum thinning rate.

• 3. The die fillet radius (R_B) exhibits completely opposite trends with respect to the maximum thinning rate of the radius-forming section and the average thinning rate of the radius-stable section. In the radius-forming section, the combined effect of the die’s upward movement and the pipe’s horizontal extrusion results in a substantially larger pipe-die contact area compared to the contact extrusion area in the radius-stable section (where the die remains stationary). Increasing R_B rapidly expands this contact area, which effectively disperses the stress acting on the pipe’s outer wall and thus reduces the maximum thinning rate. In contrast, the die is stationary in the radius-stable section; here, the primary effect of increasing R_B is to enhance the pipe-die contact force. This elevated contact force intensifies the plastic deformation of the pipe’s outer wall, consequently leading to an increase in the average thinning rate of the radius-stable section.

This study has clarified the intrinsic relationship between die fillet radius and wall thickness evolution during the free bending forming of aviation tubes, thereby providing theoretical references and technical support for the optimal design of free bending dies, the extension of die service life, and the enhancement of forming accuracy and stability for high-performance aviation tubular components.