Effect of mandrel on the forming quality of thin walled tube

Chao WU，Yong YAN，Zhi-i HU，（Hubei Key Laboratory of Advanced Technology for Automotive Components，Wuhan University of Technology，Wuhan 430070，China）（Hubei Collaborative Innovation Center for Automotive Components Technology，Wuhan 430070，China）

Effect of mandrel on the forming quality of thin walled tube

Chao WU1*
，Yong YAN2，Zhi-1i HU1，2
（1Hubei Key Laboratory of Advanced Technology for Automotive Components，Wuhan University of Technology，Wuhan 430070，China）
（2Hubei Collaborative Innovation Center for Automotive Components Technology，Wuhan 430070，China）

Mandrel shows a great influence on the quality of the tube during tube bending process.Especially，the selection of mandrel type directly affects the forming quality of the tube.In the paper，a 3D simulation model of stainless steel tube was established by finite element platform ABAQUS/Explicit.The influence of five kind mandrels on the quality of bended tube was investigated by simulation and experiment.It is found that the cylinder，round and arc mandrels show a great advantage in controlling the outer wall thinning compared with ball and round ball mandrels.Arc mandrel is benefit to reduce wall thinning of tube bending，while the ball mandrel exhibits the best performance among all the five kind mandrels during tube bending.The finite element simulation results show good agreement with the experimental results，and provide a theoretical guidance for the selection of the mandrel type in the actual production.

Thin-walled tube，NC bending，Mandrel，F(xiàn)inite element simulation

Hydromechatronics Engineering

http：//jdy.qks.cqut.edu.cn

E-mail：jdygcyw@126.com

1 Introduction

As the thin-walled bending tube could meet the lightweight and low cost requirements of the products，it has a wide range of applications in aviation，aerospace，automotive and other fields［1］.However，the thin-walled tube is hollow in structure and the radiusthickness ratio（the ratio of diameter and thickness）is large.It is easy to appear forming defects such as excessive outer wall thinning，wall instability wrinkling and cross section distortion in the bending forming process，which largely affects the forming quality and performance characteristics of the bending tube.To meet the needs of engineering，these defects should be avoided or controlled within a narrow range.It has been proved that a reasonable selection of mandrel could control these defects effectively［2］.

The mandrel is placed inside the tube as a role of support and constraints during the tube bending process.There are various forms of the mandrel，such as the cylinder mandrel，round mandrel，arc mandrel，ball mandrel and the combination of several forms. The influence of the mandrel forms on the quality of thin-walled bending tube is great.To choose the proper mandrel，nowadays the domestic and foreign researchers mostly concentrate on the traditional bending process.Moreover，the small diameter tubes are used as the main study object，which lack the relevant reports for larger diameter thin-walled tube NC bending. In the actual production and processing，it still be dominated by empiricism［3-5］.A theoretical analytic model of the mandrel in the bending process was established［6］.The influence of mandrel parameters on the tube NC bending process was analyzed，and the range of theoretical parameters of the mandrel was obtained.But the article was only based on the one form of mandrel，which lacks the comparison with different kinds of mandrels.The effect of three kinds of man-drels（cylinder mandrel，round mandrel and arc mandrel）on the quality of bended tube was discussed［7］.The influence rule of tube bending under the support of these forms of mandrel was achieved.But the discussion to the forms of mandrel was not comprehensive enough，and the common mandrel in the current wasn’t covered.

The tube NC bending technology is a new process which is developed from the traditional tube bending craft combined with CNC technology.It can make the tube bending process intelligent，flexible and automated，and meet the requirements for high precision and high efficiency of the product［8］.Finite element numerical simulation method is the use of finite element analysis and related critical forming metrics in the actual metal forming process，obtaining the forming limit of metal［9］.The finite element simulation can be used to characterize the thin-walled tube and bending die by selecting the appropriate elements.Thus the contact relationship between the tube and die can be described reasonably and the movement of each bending die can be simulated effectively.As a result，the wall thickness distribution and the stress and strain distribution can be obtained.Therefore，the forming defects can be predicted.

The paper was based on the finite element analysis theory.A 3D simulation model of tube NC bending was established by finite element platform ABAQUS. Based on the explicit integration algorithm，the finite element numerical simulation was used to simulate the tube bending supported by different kinds of mandrels.Finally，an experiment was made as a contrast，in order to provide a theoretical guidance for the choice of the mandrel type in the actual production.

2 Stress and strain analysis

In the action of bending moment M，the stressstrain condition of the tube is shown in Fig.1.In the process of tube bending，the outer wall of the tube suffers a two-way tensile stress，and the inner wall is subjected to the action of three-way compressive stress.The outer wall of tube is thinning under the action of the tangential tensile stress σθ；the instability wrinkling occurs on the inner wall of tube in the role of the tangential compressive stress σo；the cross section tends to distort under the action of tensile and compressive stress at the same time.

Fig.1 Stress-strain condition of tube bending

The wall is easily to crack if the outer wall thinning of tube is too large during the bending process，as shown in Fig.2（a）.When the inner wall thickening of tube is serious，the wall instability wrinkling may appear due to the material cannot maintain a stable deformation，as shown in Fig.2（b）.And the combining action of large tensile and compressive stress leads to the lateral diameter of the tube increasing，the normal diameter reducing，and makes it elliptical，as shown in Fig.2（c）.

Fig.2 The faiIure modes of the tube bending process

3 Finite element model

The process of tube forming is complex，mainly involves three nonlinear problems of mechanics：geometric nonlinearity，namely large displacement，and large rotation or large deformation occurred during the tube bending process；physical nonlinearity，that is material nonlinearity；boundary nonlinearity，that is non-linear relationship caused by the contact friction between the die and workpiece［10-11］.Therefore，based on the finite element platform ABAQUS/Explicit，a finite element modeling was built to describe the process in this paper，which can solve the nonlinear problems of tube bending simulation effectively.

3.1 Geometric model

According to the theory of NC tube bending，combined with the actual bending process，a three-dimensional model was used to describe the tube，clamp die，die insert，pressure die，wiper die，bend die，mandrel（flexible balls）and other components.The established geometric model is shown in Fig.3. Among them，the tube which will be out of shape is deformable body，and the rest are rigid bodies.

Fig.3 Finite eIement modeI for NC tube bending

3.2 Material parameters

The 304 stainless steel was used to simulate the tube NC bending.The outside diameter of the tube（D）is 60 mm；the wall thickness of the tube（t）is 0.73 mm；the bending radius（R）is 120 mm.The density of tube（ρ）is 7 800 kg/m3；the elastic modulus（E）is 207 GPa；the Poisson＇s ratio（λ）is 0.28. The uniaxial tensile test was conducted according to GB/T228-2002“tensile test methods for metallic materials at room temperature”.The basic parameters in the mechanical properties of the tube were obtained. The material model is shown in formula（1）and the tensile specimen is shown in Fig.4.

The material hardening occurs in the process of tube NC bending.Thus the exponent function is used to characterize the hardening behavior of material［12］

where K is intensity factor，1 426 MPa；ε0is prestrain，0.028；n is hardening exponent，0.5.

Fig.4 TensiIe specimen materiaI

The 304 stainless steel stress-strain curve was obtained through data processing，as shown in Fig.5.

3.3 Element type

Since the four-node double curvature shell elements S4R（4-node doubly curved thin shell，reduced integration，hourglasscontrol，andfinitemembrane strains）has a feature of hourglass control and reduced integration，the element type is used to describe the tube.Five integration points was chosen in the thickness direction.The discrete rigid shell elements R3D4（4-node 3D bilinear rigid quadrilateral）was used to describe the dies，such as bend die，pressure die，wiper die，clamp die，die insert，and mandrel（flexible balls）［13］.

Fig.5 The stress-strain reIation of tube

3.4 Contact and friction conditions

The appropriate contact algorithm can make the motion of dies in the simulation closer to the actual bending process.The contact interfaces contain tube/ clamp die，tube/die insert，tube/pressure die，tube/ wiper die，tube/bend die and tube/mandrel（balls）. Surface to surface contact mode was selected to describe the contact algorithm between dies and tube. According to the actual process，the“small sliding mode”was chosen to characterize the movement type between tube and clamp die and the“finite sliding mode”was used to describe the rest of the contact interface［14］.The Coulomb friction was selected to describe the contact between the tube and dies.The coefficients of friction between the contact interfaces are shown in Table 1.

TabIe1 The friction condition of contact interface

3.5 Boundary conditions

Based on the ABAQUS platform，the two kinds of constraints，those are Rotation and Velocity，were used to apply load among dies.The dies，such as bend die，clamp die and die insert，could only rotate around the bending center，and all the remaining five degrees of freedom were constrained.The pressure die could only move along the bending direction，the moving speed of pressure die was equal to the linear velocity of bend die in the bending process.All the mobile and rotational degrees of freedom of wiper die were constrained in the bending process.The mandrel was fixed in the bending process.If it contains balls，the corresponding connecting unit is used to describe the relationship between the mandrel and balls.

4 Mandrel type

Since the mandrel plays an important role in the bending process，the mandrel is usually used in the tube NC bending.There are many types of mandrel shapes.For the tube with different relative bending radius R/D or radius-thickness ratio D/t，the proper mandrel should be chosen［15］.According to the structure characteristic of the mandrel，the mandrel can be roughly classified into two categories：one is the integral mandrel，such as cylinder mandrel（a），round mandrel（b）and arc mandrel（c）；the other is articulated mandrel（there are several parts in front of mandrel），such as ball mandrel（d）and round ball mandrel（e）.

The structure of cylinder mandrel is simple，the cost of that is low，and it is easy to manufacture.The end of the round mandrel is hemispherical and the end of arc mandrel is near parabolic.The front of ball mandrel is formed by a plurality of balls.The relationship between the mandrel and the ball or the each ball is similar to the hinge connection.Thus the ball can be adaptive rotated with the rotation of dies in a certain angle.Round ball mandrel is a deformed structure of ball mandrel，but the hemispherical ball is used at the end.

These common forms of the mandrel all can play a supporting role in tube bending.But to the different types of mandrel，there are certainly differences in the supporting effect.It lacks of persuasiveness and credibility only by theoretical analysis.In order to achieve the optimal supporting effect，it also needs to obtain the effects of different forms of mandrel to the quality of the tube by the finite element simulation results.

Fig.6 The common types of mandreI

5 Numerical analysis of mandrels

The five kinds of mandrels were used in the finite element simulation models respectively.The bending angles of the tube were 90°.In order to make the tube inside better supported by mandrel，a reasonable overhang of the mandrel should be selected.For the cylinder mandrel，round mandrel and arc mandrel，the overhang of the mandrel was 10 mm.And for the ball mandrel，the Mandrel parameters show great influence on the forming of the tube，as shown in Fig.7. Therefore，the reasonable parameters were chosen. The outer wall thinning and inner wall thickening were used to measure the change of the wall thickness，and the computation formulas were as follows.

where t is original wall thickness of tube（mm）；tminis the minimum wall thickness（mm）；tmaxis the maximum wall thickness（mm）.

Fig.7 The forming tube under different parameters of the baII mandreI

5.1 Effect of mandrels on wall thinning

After the simulation finished，the tube was cut along the centerline of the tube.Then the thickness values of the outer wall from the pressure die to clamp die were read by the Postprocessor of ABAQUS.In according to formula（2），the wall thinning of the tube was calculated，and got the data processed.Then the data was imported to the Graphics software（Origin Pro），thus the wall thinning curves were obtained. The contrast curves of outer wall thinning under the support of five kinds of mandrels were obtained，as shown in Fig.8.

As shown in Fig.8，under the support of articulated mandrel，such as ball mandrel and round ball mandrel，the outer wall thinning of tube is significantly higher than which of the integral mandrel，such as cylinder mandrel，round mandrel and arc mandrel. The effect on wall thinning is little difference between ball mandrel and round ball mandrel，but the rest three have some differences.Compared with other mandrels，arc mandrel is benefit to reduce wall thinning of tube in the bending process.

Fig.8 Effects of different mandreIs on waII thinning of the tube

Fig.9 is the diagram of tube supported by round mandrel（articulated mandrel）and ball mandrel（integral mandrel）respectively.As shown in the figure，the support angle is bigger when the tube is supported by ball mandrel（support angle is the angle between the line which is from bending center O to the tangent point of the front of mandrel and original bending plane that go across bending center O），which makes the contact area between tube and the internal wall of tube increase.Thus the frictional resistance increases when the tube rotates around the bend die in the bending process，so the tensile stress of the outer tube will increase.That results in the material fiber of tube outside increased，so the wall thinning is larger.

5.2 Effect of mandrels on wall thickening

Due to its structural limitations of the mandrel，the severe wrinkling defects will appear in the inner wall of bending tube when it is supported by integral mandrel，such as cylinder mandrel，round mandrel and arc mandrel，as shown in Fig.10（Avg 75%is the default threshold criteria.If the relative nodal variation for each node is less than averaging threshold，the contributing values from elements are averaged at that node）.

Fig.9 The supporting diagram of the tube by different mandreIs

Fig.10 The forming diagram under different mandreIs

By the analysis，it can be seen that when the tube is supported by integral mandrel，the plastic deformation zone of tube get small rigid bound area by mandrel.Thus the tangential pressure stress of the tube inside is large and the gap between the mandrel and inner wall of tube is wide，which provides sufficient space for the accumulation of material in the bending process.Thus leads to the serious wrinkling in the inner wall after tube bending.

Due to the severe wrinkling of the forming tube，it cannot play a normal function in engineering applications.Thus it has no practical significance to discuss the effects of these mandrels on wall thickening of the tube.To obtain the impact of mandrel structure on the inner wall thickening，just two mandrels（ball mandrel and round ball mandrel）are discussed here.

The simulation results were calculated in according to formula（3），and got the data processed.The inner wall thickening contrast curve under two kinds of mandrels supported are obtained，as shown in Fig.11.

As shown in Fig.11，the effect on wall thickening is approximate between ball mandrel and round ball mandrel，all of which are small.The inner wall does not appear obvious wrinkling，and it shows that the inner wall of the tube can be better constrained by the articulated mandrel.However，there are some differences in wall thickening at different positions.When the measurement position is greater than 45°，ball mandrel shows advantage in control wall thickening.

Fig.11 Effects of different mandreIs on waII thickening of the tube

5.3 Comparison with different mandrels

The effect of five kinds of mandrels on outer wall thinning，inner wall thickening and instability wrinkling of the tube has been discussed.It is found that integral mandrel，such as cylinder mandrel，round mandrel and arc mandrel，is benefit to reduce wall thinning of tube bending.But the severe wrinkling will occur at the inner wall of tube.While the instability wrinkling could get better controlled under the support of articulated mandrel（ball mandrel and round ball mandrel）.Though the thinning of the outer wall is slightly larger than that supported by the integral mandrel，the wall thinning can be controlled within the range allowed by the industry standard as long as the mandrel parameters are reasonable selected. Thus the performance of tube can be satisfied.

In addition，the bending section of tube can be fully supported and restricted under the support of articulated mandrel.Thus the cross section distortion can be significantly improved.The supporting diagram is shown in Fig.12.

Fig.12 The supporting diagram of the tube by baIIs

In the process of tube bending，the lateral of the tube suffers tangential tensile stress，and the inner is subjected to the tangential compressive stress.Under the role of tensile and compressive stress at the same time，the cross section has a tendency to distortion. However，the ball of ball mandrel can reach the sensitive area where the section of tube is easy to be out of shape.The support forces N′1and N′2，which are the forces that the balls give to the inner wall，are in the opposite direction of tangential tensile stress N1and tangential pressure stress N2respectively.Therefore，the appropriate choice of the ball can offset part of the normal compressive stress，and the cross section distortion gets better controlled.

Integrating the indicators，it is indicated that the tube exhibits the better performance under the support of articulated mandrel（ball mandrel and round ball mandrel），as shown in Fig.13.Especially the wrinkling and cross section distortion get a great degree of improved.These two forms of defects will seriously restrict the actual use of performance of tube.To ensure the well comprehensive quality of tube，now the articulated mandrel，such as ball mandrel，is widely used as an internal support structure in CNC bending machine.

Fig.13 The forming diagram under different mandreIs

6 Experimental verification

It needs to be tested by experiment to determine whether the numerical results are of practical significance.Compared to several other mandrels，the ball mandrel exhibits the best performance.Therefore，the ball mandrel is chosen as the research object of the experiment in the paper.

Experimental conditions：with a stainless steel tube as the research object，the outer diameter of the tube is 60 mm，the wall thickness is 0.73 mm，the bending radius is 120 mm，and the diameter of the mandrel is 57.6 mm.With three balls at the front of mandrel，the diameter of the ball is 57.5 mm，the thickness of the ball is 19 mm，and the interval between balls is 3 mm.The special lubricant is used to lubricate mandrel，balls，wiper die and wall of tube in the experiment.Besides，it must ensure that there is no relative sliding between the clamping die and tube.

The experiment of thin-walled tube bending was carried out on CNC bending machine.After the completion of the experiment，the special cutting device was used to cut the forming tube along the centerline of the tube.The cutting sample is shown in Fig.14. The wrinkling on the inner wall is a typical defect in the tube bending.Under this experimental condition，the wrinkling also occurred in the simulation.To some extent，the inner wall has been validated by comparing the wrinkling.On the other hand，the out wall thinning was chosen to verify the reliability of simulation by numerical calculations.Thus the accuracy and effectiveness of the simulation results would be better verified.The calipers was used to measure the thickness of outer wall at five positions（0°，30°，45°，60° and 90°）.The results were calculated in according to formula（2）.The wall thinning obtained by numerical simulation are 7.7%，14.9%，15.3%，14.8%and 10.8%，while the wall thinning achieved by the experiment are 7.8%，15.1%，15.8%，13.7%and 10.2%.The contrast curve between simulation results and experimental measurements is shown in Fig.15.

As shown in Fig.15，under the same conditions，that is，the original parameters stay the same in the numericalsimulationandexperiment，thewall thinning is approximate at the same position.The maximum error of the wall thinning is 7.4%.The finite element simulation results shows good agreement with the experimental results，which indicates the thin-walled tube NC bending can be simulated by the finite element simulation reliably.Besides，the finite element numerical simulation method can reduce the number of tests and cost，which has certain guiding significance for the actual production.

Fig.14 The experiment sampIe

Fig.15 Comparison between the simuIation data and experimentaI resuIts

7 Conclusions

In order to study the effect of different mandrels on the forming quality of thin-walled tube，the finite element numerical simulation methods，combined with experimental validation，are used in the paper.The following conclusions are drawn from the present study.

1）The outer wall thinning of tube is significantly higher than that of the integral mandrel under the support of articulated mandrel，such as ball mandrel and round ball mandrel.

2）Among the five kinds of mandrels，arc mandrel is benefit to reduce wall thinning of tube bending，while the instability wrinkling and cross section distortion can be better controlled under the support of ball mandrel.

3）The distortion of the cross section significantly improved as the bending section of tube is supported by balls during tube bending.It is found that the type of mandrel is the main factor that influences the cross section distortion of the tube.

4）The ball can be adaptive rotated with the rotation of the tube under the support of the ball mandrel. Thus，the forming quality of the bending tube is better.

5）The thin-walled tube NC bending can be simulated by the finite element simulation reliably for the finite element simulation results showing good agreement with the experimental results.

Acknowledgements

This paper is supported by the Fundamental Research Funds for the Central Universities（2014-IV-042）.

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芯棒形式對薄壁管彎曲成形質(zhì)量的影響

吳 超1*，嚴 勇2，胡志力1，2
1.武漢理工大學(xué)現(xiàn)代汽車零部件技術(shù)湖北省重點實驗室，武漢 430070
2.汽車零部件技術(shù)湖北省協(xié)同創(chuàng)新中心，武漢 430070

薄壁管彎曲成形過程中芯棒對彎管的成形質(zhì)量影響較大，芯棒形式的選取是否合理直接關(guān)系著管件成形質(zhì)量的優(yōu)劣。采用有限元平臺ABAQUS/Explicit建立不銹鋼管三維仿真模型，討論了圓柱式、圓頭式、弧式、球窩式和圓頭球窩式5種形式的芯棒對彎管成形質(zhì)量的影響，并對球窩式芯棒進行了實驗驗證。結(jié)果表明：在球窩式和圓頭球窩式等關(guān)節(jié)式芯棒的支撐下彎曲管件的外壁減薄率比圓柱式、圓頭式以及弧式芯棒等整體式芯棒高；弧式芯棒較有利于減輕管件彎曲后外壁減薄情況；球窩式芯棒有利于獲得綜合性能更好的成形管件；有限元仿真模型能夠較可靠地模擬實際數(shù)控彎曲成形，為生產(chǎn)加工芯棒的選取提供了理論指導(dǎo)。

薄壁管；數(shù)控彎曲；芯棒；有限元仿真

10.3969/j.issn.1001-3881.2015.12.009Document code：A

TG386.43

21 November 2014；revised 12 February 2015；accepted 8 March 2015

*Corresponding author：Chao WU，Master degree candidate.

E-mail：wuchaoktzdk@163.com