Study on cross-section distortion behaviors for tube bending at varying geometry parameters

Tube bending is often plagued by the critical forming defect of cross-section distortion, which increases flow resistance, affects the strength and stiffness of bent-tube, and thus directly influences the usage performance of bent-tube components. To achieve accurate tube bending forming, the distortion behaviors for cross-section during tube bending should be clarified. A finite element (FE) model accounting for the change in elastic modulus was employed for addressing the distortion behaviors of the cross-section for high strength 0Cr21Ni6Mn9N (0Cr21Ni6Mn9N-HS) tube in NC rotary draw bending at varying geometry parameters. The findings indicate that accounting for the change in elastic modulus can aggrandize the distortion ratio of the cross-section, but has no significant impact on the distortion behaviors for the cross-section. The cross-section ovality behaviors of the 0Cr21Ni6Mn9N-HS tube at various geometry parameters for NC rotary draw bending have been clarified, and the ranges of sound parameters have been determined.


Introduction
As an important transportation carrier of various liquids and gases, metallic tube components play an important role in many high-technology fields due to their excellent overall performance and unique hollow structure.The 0Cr21Ni6Mn9N-HS tube has been applied widely in hydraulic and fuel systems of aeronautics and astronautics because of their exceptional properties of high strength, excellent heat stability at elevated temperatures, as well as perfect toughness at low temperatures.However, the low plasticity and large deformation resistance of 0Cr21Ni6Mn9N-HS tubes make them more easily to produce all kinds of defects in NC rotary draw bending, such as wrinkling, wall thickness variation, cross-section ovality as well as spring-back.Moreover, the elastic modulus plays a pivotal role in determining the bending quality of 0Cr21Ni6Mn9N-HS tube [1] , and it usually changes with plastic deformation.Among various forming defects, the cross-section ovality is considered one of the most vital defects, which increases flow resistance, affects the strength and stiffness of bent-tube, and thus directly affects the usage performance of bent-tube.To achieve precise forming, the cross-section distortion should be grimly controlled and accurately predicted.Moreover, various bent tubes have been applied in all kinds of fields to meet various needs, and the cross-section ovality changes with different geometry parameters.Therefore, it is urgent to research the cross-section ovality behaviors upon rotary draw bending of the 0Cr21Ni6Mn9N-HS tube considering elastic modulus change under varying geometry parameters.Up to now, numerous scholars have employed analytical, experimental and FE simulation approaches to research the distortion behaviors of cross-section during tube bending.Through the utilization of energy minimization methodology, Brazier [2] investigated the deformation of circle tubes' cross-section during the process of elastic bending.From that time on, the distortion of cross-section during tube bending has been commonly referred to as the Brazier effect.Using the theory of plastic deformation and the energy method, Paulsen and Welo [3] proposed a model for predicting cross-section deformation during rectangular tube bending, and it was verified by experiment.The findings manifested that the proposed model was an effective approach to evaluate the bendability of the rectangular tube.On the basis of the principle of virtual forces, Liu et al. [4] came up with a novel approach to compute the displacement at random points on the cross-section during tube bending, subsequently the long axes value and short axes value for the deformation cross-section in line with the displacement of each point were calculated, and the theoretical calculation results were verified by tube NC bending experiments in the end.Using NC tube bender, Wu [5] investigated the influences of primitive grain size, temperature and bending velocity on the ellipticity of cross-section during AM30 tube bending, and found that the cross-section flattening decreased with augmenting temperature or with decreasing bending speed and primitive grain size, and medium temperature, small bending speed, as well as fine primitive grain size, could obtain good ellipticity of cross-section.Kale and Thorat [6] proposed a pre-compression method to reduce section flattening in tube bending, and experimentally researched the effects of different precompression amounts on section flattening.They found that higher pre-compression indicated less section flattening, while further pre-compression resulted in difficulties in operations with the increase in energy consumption.Safdarian and Kord [7] experimentally investigated the influence of the main parameters such as mandrel position, pressure of pressure die and mandrel diameter on the flattening of cross-section for BS 3059 steel tubes in NC bending.It was found that the mandrel position was the most conspicuous parameter influencing the ellipticity of cross-section, and the optimal parameters such as 15 mm for mandrel position, 7 MPa for pressure of pressure die and 33.5 mm for mandrel diameter were obtained.By FE analysis, Liang et al. [8] established the reliable 3D-FE model of the TA18 tube upon rotary draw bending on account of the ABAQUS code.They researched the impact of clearance on the ellipticity of cross-section and the change of wall thickness, as well as the sound clearance magnitudes such as 0.075 mm for the mandrel-tube, and 0.1 mm for the other dies-tube.Li et al. [9] conducted FE simulation research on deformation behaviors of tube bending at various push assistant loading conditions in regard to the ellipticity of cross-section, wall thickness change and wrinkling.It was discovered that the direct application of force at the end of the tube was the best approach to boost, the influence of push assistant loading conditions on the outside was more significant than that on the inside, and the effect of simultaneous push and boosting on bending behavior is more significant than that of push alone.Taking the 21-6-9 tube as an objective, Fang et al. [1] investigated the change rule between elastic modulus and plastic strain by cyclic tensile unloading test, built the function expression of elastic modulus and plastic strain, and implanted it into the FE model for tube bending.The findings showed that the elastic modulus decreased at a great lick in the beginning, followed by decreased relaxedly, and tended to be constant at last as the plastic strain augmented.For all forming quality indicators such as spring-back angle or radius, ovality, and wall thinning, regardless of whether the change of elastic modulus is taken into account, the changing trend for each forming index is the same.However, the value for each forming index augments when accounting for variational elastic modulus.Moreover, the forecast precision of the FE model for each forming index was enhanced by more than 10% except for the springback radius.

AMCE-2023
However, in the FE models used in the aforementioned research, a constant value was assumed for the elastic modulus in most studies, which did not change with plastic deformation increasing.In this way, particularly for the materials with great deformation resistance and small ductility for instance the 0Cr21Ni6Mn9N-HS tube, the simulative findings will deviate from the experiment ones larger to a certain degree.In addition, most of the studies are ref the impact of process paraments on the ellipticity of cross-section during tube bending, while the impact of the geometrical parameters on that is less studied.Therefore, in this study, taking into account the change in elastic modulus, the elastic-plastic 3D-FE model of 0Cr21Ni6Mn9N-HS tubes upon NC rotary draw bending is developed on account of ABAQUS software.Using the model, the deformation behaviors for cross-section upon tube NC rotary draw bending under various geometrical parameters are addressed, and the sound geometrical parameter ranges are obtained, which satisfy the aviation standard.The results can provide the basis for choosing the sound geometry parameters and optimizing mold design to realize accurate bending forming of the 0Cr21Ni6Mn9N-HS tube.

FE model accounting for variation in elastic modulus
Taking 0Cr21Ni6Mn9N-HS tube with dimensions of 9.53 × 0.51 mm (D ×t, D refers to tube outside diameter, and t refers to tube initial wall thickness) as an objective, the entire process FE model in NC rotary draw bending for tube with a bending radius R=3D was built on account of the ABAQUS platform, as illustrated in Figure 1.The process of tube bending and retraction mandrel involved the use of an explicit algorithm, whereas the die removal process involved the use of an implicit algorithm.On account of the ABAQUS/Explicit platform, a user material subroutine VUMAT considering elastic modulus change was developed based on the Mises yield criterion and inserted into the bending FE model.For the die removal process, the user material subroutine UMAT considering elastic modulus change was also designed and implanted into the spring-back FE model by using ABAQUS/Standard code.Further information regarding the FE modeling process including materials parameters, friction model, mesh size, boundary conditions as well as loading conditions, etc., can be obtained from Fang et al.'s work [1] .To validate the accuracy and reliability of the FE model, the bending tests were conducted on a tube NC bender machine.The experimental conditions match the simulated ones.Figure 2 shows the distortion ratio of cross-section ΔD (ΔD = (D -D′)/D ×100%, D′ represents the tube diameter in the direction of bending radius after experiencing spring-back) obtained from simulation and experimentation.It is found that the variation tendency of simulative results with varied elastic modulus is similar to that with constant elastic modulus, but only the values considering elastic modulus change increase, and these are closer to the experimental values.The primary cause lies in the fact that the tangential tension/compression stress is of greater magnitude before and after spring-back when elastic modulus change is considered [1] , and the corresponding resultant force directed towards the crosssectional center of the tube is greater.Thus, the variation in elastic modulus leads to a greater distortion in the cross-section compared to an invariable elastic modulus.The relative errors pertaining to the maximum distortion ratio of the cross-section between the simulative results with varied elastic modulus or constant elastic modulus and experimental results are 5.23% and 40.76%, respectively.The prediction precision is improved by 35.53%, which indicates that the FE model incorporating change in elastic modulus is characterized by enhanced accuracy and reliability, and can ideally depict the authentic process of bending deformation.Therefore, the FE model can be employed for further exploring distortion behaviors of the cross-section of 0Cr21Ni6Mn9N-HS tubes during NC rotary draw bending.

Results and discussion
Taking the specification of the 0Cr21Ni6Mn9N-HS tube with a diameter of 9.53 mm as well as a wall thickness of 0.51 mm as an objective, the impacts of geometry parameters on cross-section deformation behaviors upon tube NC rotary draw bending were investigated.The relative bending radius R/D is set to 3 when the impacts of bending angles β on cross-section deformation are studied during tube bending, and the value of β ranges from 15 to 180 degrees, where the angle interval is 15 degrees or 30 degrees.When the impacts of the R/D on cross-section distortion are investigated in tube bending, the β is set to 180º, and the R/D is set to 2, 2.5, 3, 3.5 and 4, respectively.
The deformation ratio of the cross-section for the 0Cr21Ni6Mn9N-HS tube upon NC rotary draw bending at various β accounting for the variation in elastic modulus or not is illustrated in Figure 3.As evidenced by the data, the trend of the cross-sectional deformation rate remains similar whether or not the change in elastic modulus is taken into account, but the distortion ratio of cross-section increases when considering varied elastic modulus, compared to that without such consideration.The maximum distortion ratio of the cross-section at different β is below 3% without exceeding the aviation standard limit of 5%.; (b) Elastic modulus invariableness.
When the β is below 60°, the cross-sectional distortion ratio curves take on a distinct shape that closely resembles that of a parabola from pressure die end to clamp die end, and the maximum distortion ratio of cross-section aggrandizes bit by bit with an increase in the β.When the β is greater than 60°, the distortion ratio of the cross-section exhibits a sharp increase initially, followed by a gradual decrease, then a tardy augmentation and finally a rapid decline from pressure die end to clamp die end.The results in the current investigation are distinct from those obtained from the NC rotary draw bending of the TA18-HS tube [8] .The reason for this is that the plastic deformation does not reach a steady state when the β is less than 60°, which results in augmenting the ovality of the cross-section with augmenting the β.The plastic deformation attains a steady state when the β is more than 60°.Therefore, the distortion of the cross-section displays a steady deformation state.The ellipticity for the cross-section is smaller in the neighborhood of the bending section or incipient bending section due to support from the mandrel or restraint by the die cavity, while the ellipticity of the cross-section is larger in the middle part because the tube is without mandrel support and lies in dangling state.
The deformation ratio of the cross-section for the 0Cr21Ni6Mn9N-HS tube bending at various R/D taking into account the elastic modulus variation or not is manifested in Figure 4.The findings described in Figure 4 reveal that the tendencies of the cross-sectional distortion ratio remain consistent regardless of whether the alteration in elastic modulus is taken into consideration.However, the magnitude of the cross-sectional ovality rate increases when the variation in elastic modulus is taken into account.
The distortion ratio of the cross-section augments as the R/D decreases in both cases.The primary factor behind the phenomenon is that the extent of plastic deformation increases with a decrease in the R/D.When the R/D is equal to 2, the maximum distortion rate for cross-section (8.68%) taking into account the change in elastic modulus has exceeded the maximum limit of 5% of aviation standard.However, the maximum distortion ratio of the cross-section without considering the change in elastic modulus does not surpass the aviation standard limit of 5% when the R/D=2.The aforementioned findings suggest that the distortion ratio of the cross-section is more dependable and secure when accounting for the variation in elastic modulus.Thus, in the case of acquiring a qualified bent-tube component, the R/D of NC rotary draw bending must exceed 2 for 0Cr21Ni6Mn9N-HS tubes.Moreover, when the R/D is equal to 2, the distortion ratio of the cross-section considering the change in elastic modulus augments suddenly at first, then reduces slowly and diminishes quickly in the end, which does not show the normal change tendency.Namely, the distortion ratio of cross-section augments quickly at the initial stage, then diminishes tardily, after that increases gently and finally declines sharply as the angle θ augments from 0° to 180°.; (b) Elastic modulus invariableness.

Conclusions
(1) The distortion behaviors of the cross-section of the 0Cr21Ni6Mn9N-HS tube upon NC rotary draw bending remain similar regardless of whether changes of elastic modulus are considered, but only the magnitude of the cross-sectional distortion ratio aggrandizes when such changes are taken into account.
(2) The distortion of cross-section augments with decreasing the R/D, while it increases with increasing the β before 60°, then it changes non-significant with increasing the β after 60°.
(3) Taking into account the index of cross-section ovality, the rational parameter ranges of the 0Cr21Ni6Mn9N-HS tube upon NC rotary draw bending are that the β can reach 180 degrees, and the R/D should exceed 2.0.

Figure 1 .
Figure 1.Entire process FE model of the 0Cr21Ni6Mn9N-HS tube upon NC rotary draw bending.
varying elastic modulus Simulation results using a fixed elastic modulus

Figure 2 .
Figure 2. Comparing the simulation findings with experiment ones regarding the ellipticity of crosssection.