Research Overview on Rectangular Tube Cross-Section Deformation in Rotary-Draw Bending Process

Rectangular tube bending process forming quality is not only affected by the tube material but also geometric dimensions will also be affected by bending mold and related process parameters. Cross-section deformation produced by the tube during the forming process becomes a major constraint to heightening the quality of the tube. The research work and process of cross-section deformation in the rotary-draw bending process (RDBP) of rectangular tubes in theory analysis, experiment research, and finite-element (FE) simulation in recent years is reviewed in this paper. The development trend of research in this field is also proposed.


Introduction
Metal rectangular tube parts accommodate the demands for good performance, high-strength, and low-weight products, both in terms of components and structural features so they are getting increasingly and widely used in modern aircraft.The tube bending process is a crucial part of plastic forming, used to create thinner material products.[1].Any bending process results in intricate uneven tensile and compressive stress distributions in the portion of the tube being bent.These stress distributions not only result in uneven thickness of the inner and outer walls but also lead to various defects including spring-back, wrinkling, cross-section deformation, over-thinning (cracking), etc. [2][3].
The RDBP is a key forming technology for bending tubes, which can prohibit cross-section deformation and wrinkling effectively.The associated theories and technologies need to be resolved right away in consideration of the new requirements of high efficiency, low consumption, and digitization and intelligence for precision plastic forming.Over the years, numerous academics and engineers spent a great deal of time investigating the theories and approaches for the cross-section deformation of rectangular tubes.The advancement in the field of precision bending forming technology for complex cross-section tubes as well as the improvement of the high quality and economic manufacturing technology level and capability of bending parts are all significantly important from a strategic and urgent practical position.

Research status
Mechanical analysis shows that the rectangular tube section, which experiences distortion under normal compression as the normal size decreases and the transverse size increases, has the largest cross-section deformation in the middle of the bent part.The combined force of the tangential stress of inside and outside of the parts under bending is what causes this distortion.However, the complicated plastic deformation of the nonlinearity of geometric, material, and boundary conditions and multi-factor coupling involved in the RDBP of thin-walled rectangular tubes makes it harder to achieve the forming quality standards.Scholars at home and abroad have viewed cross-section deformation from different perspectives and have conducted extensive research on its principles.

Theoretical analysis
Theoretical analysis, which involves creating an analytical model and calculating differential equilibrium equations directly or approximatively using the energy method, is one of the crucial methods to understanding the metal forming process.The laws and principles summarized in the theoretical analysis are of great practical significance for the conduct of sequential work.
To anticipate the cross-section deformation in bending, Paulsen used the deformation theory of plasticity to analyze this progress.The theory is combined with the energy method.Before buckling, during the onset of buckling, and post-buckling, these models provide precise solutions for cross-section deformation [4].Miller presented a 2D simulation of the extruded tube to predict the deformation, in which the simplicity of the simulation assumes that the variables would remain constant along the length.The deformation can be reduced by appropriate internal pressure [5].Corona proposed a method for predicting cross-section deformation of the arbitrary extruded tubes.This method can also be used to predict the spring-back of tubes produced through the bend-stretch forming process.The establishment of the model greatly reduced the run time and was intentional for cross-section deformation control and parameter optimization [6].Zhao proposed a wrinkling wave function and then developed a predictive model to describe the wrinkling wave of aluminum alloy rectangular tubes and verified the reliability [7].

Experimental texts
Experimental texts are a vital part of the research and the most basic way of studying theories and summarizing laws.Besides, experiments can not only verify finite-element simulation results but also validate the correctness and applicability of the various theories and parameters on the bending process of the tubes.
In Paulsen's experiment, single-room and double-room aluminum alloy extruded profiles with various wall thicknesses and heat tempers were bent.Research has revealed that the most effective approach to minimizing the deformation of sections is by reducing the width of the flange.Additionally, the material's capacity to strain hardens results in beneficial effects on both the initiation of buckling and the extent of deformation in the tensile and compressive flanges.;As the material is bent, its stress and strain gradually cause deformation [8].Corona and Vaze conducted a study on square steel tubes with thin-walled cross-sections under pure bending, utilizing a blend of experiment and analysis.The study found that for low levels of curvature, the tubes experienced uniform cross-sectional deformation due to bending.Besides, the higher height-to-thickness ratios (h/t) showed a greater increase in wrinkling wave compared to those with lower h/t, but the latter exhibited greater cross-section deformation [9].
Although experiments are an important part of the research, it is based on a variety of texts and production conditions.Only by making mistakes in the re-examinations can we get regularity, which wastes a lot of time, manpower, material resources, and financial resources.At the same time, the long production cycle makes it difficult to achieve the requirement of transformation.

FE simulation
Optimizing the process and tooling of bending is a complicated task since there are multiple factors and defects.From a mathematical perspective, it is a difficulty coupled with multiple conditions that need to be analyzed from multiple perspectives.However, in practice, the most common approach to obtaining feasible bending parameters is through empirical data or the traditional "trial and error" method based on the operator's expertise [5].However, for precise tube bending with complex cross-sections with small diameters and thick walls, the above method is insufficient.A computer-based tool can perform the process simulation and numerical calculation and summarize the rules and theories.

Figure2. Illustration of FE model for the RDBP
Liu and Yang presented a reliable model of thin-walled rectangular tubes (TWRT) under RDBP.Then they studied the influence of related parameters between the wiper die and tube and deformation behavior based on ABAQUS/Explicit [10].Zhang and Liu investigated how material parameters and element size affect the deformity of rectangular tubes during the RDBP.According to the study, a higher hardenability value can decrease the tangential stress on the sinuous part.This leads to a uniform distribution of stress, and the maximum original yield stress, hardenability value, and strength factor can be found at the location of the greatest degree of deformation of the cross-section.[11].A regression prediction model to determine the wrinkling wave height in TWRT made of aluminum alloy during the RDBP was developed by Liu and Li.The model combined finite-element (FE) simulation with orthogonal regression analysis to study the forming limit without the occurrence of wrinkling [12].The 3D-FE model was utilized to determine the clearance between the different dies and tubes.The results indicate that reducing the clearance between the mandrel, bending die, and tube significantly weakens the tube's deformation.[13].
Zhao and Yang et al. found that increasing the initial lead of the mandrel can make the maximum tangential compress stress decrease [14].A study to examine the impact of various forming parameters on the RDBP of TWRT made of aluminum alloy was conducted.The research explored the distance change values of each node in the cross-section and found that the bending deformation of the pipe and the cross-section had the maximum numerical change near their respective symmetry axes [15].Liu and Liu compared the accuracy of FE models based on three different yield criteria in describing the anisotropic characteristics of heterogeneous rectangular pipes through experiments and research.This search showed that the Hill'48 yield criteria are more accurate in describing the deformation of the pipes [16].An experimental study on the influence of core on the deformation of cross-section and it was determined that the number of balls had the greatest impact on the tubes [17].
Hashemi and Niknam provided an innovative flexible bending machine to overcome the chance of cross-section deformation and investigated the key parameters in flexible bending with the ABAQUA finite-element method [18].Cheng et al. studied the deformation behavior of TWRT with continuously varying radii through free bending.Additionally, they estimated the equivalent stress and distribution of strain.These findings indicate that the deformation ratio of the cross-section increases as the feeding speed decreases [19].After comparing the results of the deformation of the double-ridged rectangular tubes using different yield criteria with experimental data, it was concluded that the Barlat/Lian yield criterion is applicable [20].

Advances in Tooling Research
In recent years, there has been a growing focus on developing new tube bending techniques and innovative tooling designs.This is particularly important in aviation, aerospace, and automotive industries where lightweight and high-performance tubes are in high demand, along with specific bending specifications.Engineers and scholars have been working hard to meet these demands.Aiming at the limitation of traditional bending molds that can only bend pipes of a single diameter, an innovative idea is proposed to bend pipes of different diameters on a single bending mold by adjusting the groove shape of the bending mold reasonably.Although the allowable outer diameter range of the pipes is limited, it can fully meet the quality requirements of the pipes [21].The Liu Yuli research group at Northwestern Polytechnical University has invented a thin-walled rectangular tube composite mold that can simultaneously achieve H-bending and E-bending.The new rectangular tube bending mold invented by the Harbin Institute of Technology has effectively solved the problem of poor-quality tube bending during the process of forming rectangular cross-section tubes using existing bending molds.Research on rotary-draw bending molds is limited to only one or two process problems encountered during production experiments.In short, bending tools should move away from traditional production methods towards more flexible and efficient methods such as intelligent CNC machining.

Trends and Challenges in Rectangular Tube
There is a new trend in the increased demand for the specificity of rectangular tubes in specific application scenarios that will lead to more complex cross-sections, thinner wall thicknesses, and lighter weights.To further improve the performance of thin-walled rectangular tubes, new advanced materials should gradually focus on the application of tubes including high-strength lightweight alloy material with good comprehensive performance like aluminum alloy tubes, magnesium alloy tubes, and titanium alloy tubes.It is worth noticing that CNC pipe bending machines have become the main technology for achieving precise and efficient pipe bending machines, thus bending machines should develop towards high automation and efficiency.
Though extensive research has been conducted on the deformations that occur in the RDBP, accurate prediction, and effective control of thin-walled bends under multimode constraints remains a challenging task [22].

Conclusions
Accurately predicting and effectively controlling failures and instability during bending is crucial for high-precision production.This requires understanding the mechanisms and impact patterns of defects and instability.The application of FE numerical simulation methods to capture cross-sectional distortion during the bending and forming process of TWRT considers both computational accuracy and efficiency requirements and is a mainstream method in this research field with promising development prospects.