Int J Adv Manuf Technol DOI 10.1007/s00170-017-0670-x

ORIGINAL ARTICLE

Effect of stress distribution on springback in hydroforming process Zhiying Sun 1 & Lihui Lang 1

Received: 31 December 2016 / Accepted: 13 June 2017 # Springer-Verlag London Ltd. 2017

Abstract Stress of sheet metal forming was analyzed by using the Hill theory, and the moment formula about bending shallow part was derived in sheet hydroforming. The offset distance in the stamping direction is as the size of the springback. The formulas about springback solution were obtained under the action of hydraulic pressure during the stretch bending. According to the formula, with the liquid chamber pressure increased, the force T increased, the moment M decreased, and the springback decreased. The neutral layer of stress moved to the inside of the plate under fluid pressure, and the sections of sheet are mostly in a tensile stress state and less springback after the end of hydroforming. Combined with the results of theoretical analysis and numerical simulation, the experimental method of getting the springback about bending sheet under liquid pressure is proposed, and the 3D model was got by using reverse engineering. It got the springback by comparing the punch surfaces. The test results show that the smaller the springback, the greater the fluid pressure in the forming pressure range about stainless steel metal, which accords with the theory analysis. Keywords Sheet . Hydroforming . Cavity pressure . Springback . Stress

* Zhiying Sun [email protected]

1

School of Mechanical Engineering and Automation, Beijing University of Aeronautics and Astronautics, No. 37, Xueyuan Road, Beijing 100191, China

1 Introduction The research of sheet hydroforming technology is limited to the optimization of process parameters, and the accuracy of springback prediction depends on the accuracy of analyzing the process. Li tao et al. [1, 2] researched simulation analysis of complex parts on multi-stage liquid forming. Hamal researched overflow in the process of liquid forming [3]. Liquid filled forming process had been fully studied, but the influence mechanism of liquid filling on springback is very little. Djavanroodi plotted the limit diagram about the Ti6Al4V and Al6061—T6 in the liquid deep drawing forming by numerical simulation and experimental method [4]. Labergere did the cylindrical experiment with a liquid deep drawing based on the analysis of the liquid chamber for fluid motion velocity and pressure change [5].Wang studied cellular aluminum liquid bulging through the experiment and finite element analysis (FEA) [6].The analysis of hydroforming process also was verified in the experiments. Based on the previous experience of hydroforming process and forming limit, it can help to complete the research process. A. El-Megharbel studied aluminum alloy springback in bending, based on the Hill accurate bending theory, under the condition of uniaxial stress, and the theory model was obtained [7]. Therefore, the results were obtained by combining the hydroforming characteristics. Due to the rapid development of finite element technology, it has made great progress in predicting sheet metal springback. The springback of sheet hydroforming is a complex deformation mechanics problem about nonlinear about geometric, material, and boundary conditions [8]. S.K. Panthi predicted springback of bending by using the finite element method [9, 10] and analyzed the influence of material and geometry parameters on springback. Finn analyzed the U bending, the box forming, and automobile front fender by using the LS-DYNA software [11], and an effective way of

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Fig. 1 Schematic diagram of hydroforming

solving the springback of the thin-walled part was developed. Wagoner et al. [12, 13] studied the simulation accuracy influence on hardening exponent and the bending radius, the tension size for springback. S.F Lajarin et al. [14] analyzed the springback in the two-dimensional state. R. Srinivasan conducted influence of friction parameters and bend force on springback [15]. P. A. Eggertsen had accurate prediction of double curvature model [16, 17]. Most people studied the influence of some parameters in processing on the springback. However, the stress distribution was affected by the process parameters, and the analysis of stress distribution is the most effective and the most direct way to study springback. At present, by summarizing the characteristics of hydroforming and the research results of springback, it is intuitively considered that springback is small due to the pressure in the fluid chamber. However, the action mechanism of the pressure has not been clearly studied. Therefore, when the two-dimensional bending is taken as a research object under the action of liquid pressure, it will further reveal springback mechanism by analyzing internal stress of the sheet.

2 Theoretical analysis of sheet metal bending 2.1 Stress of sheet metal drawing under the action of liquid pressure The process of sheet bending under the action of the liquid pressure is shown in Fig. 1. The key problem about the

Fig. 2 The moments and forces acting on the sheet element

Fig. 3 The forces balance acting on the element

springback of the bottom surface is the precision of the bottom surface for a deep drawing part with the small wall [18]. It is mainly manifested by increasing the radius of the bottom, whereby decreases curvature. The whole model of the plate is shown in Fig. 2, σθ is the shear stress of deformation zone, σN is the compressive stress between the plate and the convex die, m is the external bending moment, M is the external bending moments, T is the external force on plate, p is the liquid pressure in die cavity, r is the inside radius on sheet (the side contact with punch), and R is the lateral radius on plate. The following assumptions are used in this article: (1) Materials are in ideal plastic state. (2) Material is subject to the Mises yield criterion. (3) The friction between the plate and mold is ignored. The stress distribution in the deformation zone is shown in Fig. 3, the equilibrium equations for infinitesimal body: dσρ σρ −σθ þ ¼0 dρ ρ

ð1Þ

where ρ is the radius of any position in deformation zone (mm) and ρ0 is the radius of the neutral layer (mm).

Fig. 4 The schematic diagram of springback

Int J Adv Manuf Technol Fig. 5 Double-action equipment of hydroforming

(a) Main body

(b) Control system

Because the lateral side of the plate was pulled, the inside was compressed, according to the Mises yield criterion [19]:

σρ = − σN (the reaction force on the mold), the solution Eq. (4) was obtained.







σθ −σρ ¼ βσs σρ −σθ ¼ βσs

ρ0 ≤ ρ ≤ R r ≤ ρ≤ ρ0

ð2Þ

8 ρ < σρ ¼ βσs ln −p R r : σp ¼ βσs ln −σN ρ

where β is the influence coefficient of intermediate principal stress and σs is the yield stress of plate (MPa). Let Eq. (2) into Eq. (1) 8 dσρ βσs > > ¼ < dρ ρ dσρ βσs > > ¼− : dρ ρ

ð5Þ

ρ0 ≤ ρ≤ R

ð6Þ

r ≤ ρ ≤ ρ0

Equation (6) is obtained from Eq. (2). ρ0 ≤ ρ≤ R

 ρ  8 > < σθ ¼ βσs ln þ 1 −p  R  r > : σθ ¼ βσs ln −1 −σN ρ

ð3Þ

r ≤ ρ ≤ ρ0

Solution of the equation


C 1 ¼ −βσs lnR−p ρ0 ≤ ρ ≤ R C 2 ¼ βσs lnr−σN r ≤ ρ≤ ρ0

σρ ¼ βσs lnρ þ C 1 σρ ¼ −βσs lnρ þ C 2

ρ0 ≤ ρ ≤R r ≤ ρ≤ ρ0

ρ0 ≤ ρ≤ R

ð7Þ

r ≤ ρ≤ ρ0

Radial stress was continuous on the neutral layer; the radius of the neutral layer is Eq. (8): qffiffiffiffiffiffiffiffiffiffiffiffiffiffi p−σN ρ0 ¼ e βσs Rr ð8Þ

ð4Þ

Using the boundary condition, the outer surface of the plate, ρ = R, ρρ = − p; the inner surface of the plate, ρ = r,

According to the balance condition of force in the stamping direction, it can be obtained.

AA

Fig. 6 The shape and dimensions of the punch B

A

A B B

B

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20 Liquid chamber pressure /MPa

AA

X

15

10

5

0

Y

A

0

A

10

20

30

40

50

Punch displacement / mm

Fig. 9 The loading curve of pressure

The bending moment was generated by the tension T: 1 ðR þ r Þ 2

MT ¼ T

ð12Þ

Fig. 7 The blank in experiment

The bending moment on the center of the blank: 2T sin

dα þ pbRdα ¼ σN brdα 2

ð9Þ

dα B is the width of the plate. dα is very small, so sin dα 2 ≈ 2.

σN ¼

T þ pbR br

ð10Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pbðR−rÞþT

Equation (8) can be expressed as ρ0 ¼ e− brβσs Rr. The radius of the neutral layer is decreased when the pressure is increased. The neutral layer moves to the inside of the plate, and the area of tensile stress is increased and the area of pressure stress is decreased on the part section. According to the condition that the internal bending moment and external bending moment of the plate are equal, the following can be obtained: M þ MT ¼ Mσ

ð11Þ

Mσ ¼ Mn þ Mw

where Mn is the bending moment generated by the inner stress of the neutral layer (N.M) and Mw is the bending moment generated by the lateral stress of the neutral layer (N.M) ρ

M n ¼ ∫r0 σθ ρbdρ ρ0 2 r ρ0 2 −r2 b 2 2  ln − − σN ρ0 −r ¼ bβσs 2 4 ρ0 2

ð14Þ

Mw was calculated by the following formula. R

M w ¼ ∫ρ0 σθ ρbdρ  2   R −ρ0 2 ρ0 2 ρ0 b

¼ bβσs − p R2 −ρ0 2 − ln 4 2 R 2

where M is the bending moment (N.M), MT is moment with T bending (N.M), and Mσ is the inner stress produced by bending moment (N.M)

Fig. 8 Numerical simulation model

ð13Þ

Fig. 10 The numerical simulation result

ð15Þ

Int J Adv Manuf Technol

(a) The distribution of internal stress

(b) The distribution of outer stress

Fig. 11 The equivalent stress distribution after forming

The external bending moment can be known by Eq. (11)

Equations (14) and (15) into Eq. (13), the total moment can be generated by internal stress.

 1 M σ ¼ M n þ M w ¼ bβσs R2 þ r2 −2ρ0 2 þ 4 b b σN r2 − pR2 2 2

200

M  ¼ M σ −M T PbðR−rÞþT  1 M ¼ bβσs R2 þ r2 −2Rre− brβσs − 4 1 R½pbðR−rÞ þ T  2

ð16Þ

300

inner outer

150

inner outer

250 200

100

ð17Þ

150 σ 22 /MPa

σ 22 /MPa

50 0 -50

100 50 0 -50

-100

-100

-150

-150

-20

0

20

40

60 80 100 120 140 160 Y /mm

-20

(a) The displacement of punch is12.5mm

20

40

60 80 100 120 140 160 Y /mm

(b) The displacement of punch is25mm 300

300

inner outer

inner outer

200

σ 22 /MPa

200 σ 22 /MPa

0

100

100 0

0

-100 -100 -20

0

20

40

60

80 100 120 140 160

-200 -20

0

20

40

60

80

100 120 140 160

Y /mm

Y /mm

(c) The displacement of punch is 37.5mm

(d) The displacement of punchis50mm

Fig. 12 Tangential stress distribution after forming with pressure 20 MPa

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100

inner outer

100

50

σ 22 /MPa

σ 22 /MPa

150

inner outer

150

0

50 0

-50

-50 -100

-100 -150 -20

0

20

40

60

80

100

120

140

-20

160

0

20

40

(a) The displacement of punch is12.5mm

80

100 120 140 160

(b) The displacement of punch is 25mm

150

150

100

100

50

50

σ 22 /MPa

σ 22 /MPa

60

Y /mm

Y /mm

0 -50

0 -50

inner outer

-100 -20

0

20

40

60

80

inner outer

-100

100 120 140 160

-20

0

20

40

60

80

100 120 140 160

Y /mm

Y /mm

(c) The displacement of punch is 37.5mm

(d) The displacement of punch is 50mm

Fig. 13 The distribution of tangential stress after forming without pressure

Geometric relations are shown in Fig. 4,

2.2 Calculation of springback It is generally considered that making the plate slide along the surface of the mold does not affect the shape of the parts with unloading of the longitudinal tension T [20], so that the part is not attached to the mold, mainly due to the unloading of the external bending moment M. Unloading is equivalent with the elastic moment M in the opposite direction .When elastic deformation is generated in the plate, then curvature change is that   1 1 1 12M e ð1−v2 Þ ¼ − ¼ Δ ρ ρp ρs Et 3

ð18Þ

ρp was the curvature radius before unloading and ρs is the curvature radius after unloading, which are shown in Fig. 4. Springback can also be expressed as the distance Δh between the punch surface and the surface of forming part in the stamping direction.

  1 2 ¼ 2 Δh Δ ρ l

ð19Þ

So,   l2 1 6l 2 M e ð1−v2 Þ Δh ¼ Δ ¼ 2 ρ Et 2

ð20Þ

Through the above analysis, the springback can be decreased as long as the decrease of the moment. Bending moment can be reduced when the liquid chamber pressure P was increased, so the springback of sheet metal forming was reduced because of loading pressure. Also, the springback was reduced when the tension T was increased, and the friction between the plate and the binder could be increased by increasing liquid chamber pressure, so the springback was decreased with increasing the tension T.

Int J Adv Manuf Technol 10 10 0MPa( numerical simulation) 20MPa( numerical simulation) 0MPa( experiment) 20MPa( experiment)

9 8

6

Springback /mm

h /mm

8

4 2

6 5

0 -20

7

0

20

40

60

80

100

120

140

4 80

160

Y /mm

100 120 140 160 180 200 220 240 260 Elastic modulus(GPa)

Fig. 14 Springback in simulation analysis

Fig. 16 Effect of elasticity modulus on the springback

3 Springback of sheet hydroforming

3.1 Springback test equipment The sheet-forming machine used in this experiment was a double-action press, as shown in Fig. 5. Cylinder nominal force of drawing was 7500 kN, cylinder side force was 3000 kN, and maximum pressure of working was 100 MPa. The automatic operation of the sheet hydroforming could be achieved by using PLC (programmable controller) control and the experimental parameters were set and modified through the interaction interface of Win CC, and the experimental data could be monitored and recorded immediately. The liquid medium is emulsion. 3.2 Numerical simulation analysis of sheet drawing The radius of die is 8 mm, and the gap between die and binder is 2.2 mm in experiment. The double-layer plates are used in experiment, and one plate which was in contact with the punch was cut into three pieces; the other one remained complete in order to seal the liquid chamber. The upper part of the metal is bent. The shape and size of die is shown in Fig. 6 in experiment . The hydroforming technology and springback are very complex. In order to reduce the computation time, the 1/4

9.5 9.0 Springback /mm

In this paper, a set of experimental device was designed to study the springback of sheet forming, the distribution characteristic and law of springback were obtained, and the theoretical results were verified by FEA.

model was used to calculate because the model was symmetrical (Fig. 7). The model is shown in Fig. 8, the models of molds and sheet are built according to the actual size. The punch, binder, and die were defined as rigid bodies with the R3D4 unit. Thickness stress was produced in the sheet by liquid pressure, so in order to study the liquid pressure, the C3D8R was used in sheet. The middle sheet in thickness direction was divided into three layers; the other was divided into one layer. The friction between models is the general contact. In the forming process, forming and springback after unloading were continuous. They would be divided into two processes during the FEA. The dynamic explicit was used in the forming process, and the static implicit was used in the springback process. Two methods to solve the springback were generally analyzed: one method was without die; this springback was the process of elastic deformation using the incremental method. To separate the mold and part, reverse forming was used as the mechanical boundary condition before the calculation, and then iterative calculation was finished until the contact stress was 0. The other method was with mold, and the process of springback could be simulated by the method. It is similar to the forming, but the mold moved to

8.5 8.0 7.5 7.0 6.5 6.0 100

150

200

250

Yield strength /MPa

Fig. 15 The distribution of equivalent strain after hydroforming

Fig. 17 Effect of yield strength on the springback

300

350

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Springback /mm

8

7

6

5

4 0.0

0.5

1.0

1.5

2.0

2.5

Fig. 19 The specimens in experiment

Plate thickness /mm

Fig. 18 Effect of blank thickness on the springback

the opposite direction. When all nodes on the plates were divorced from the die, it is over. The calculation time was very long when contacting with the mold, so the efficiency of the method without mold is better. Plate is at least fixed six freedoms which are the X, Y, Z, XY, YZ, and ZX when the springback was done. The quarter model was used, so symmetric is applied in boundary condition. At that time, the center of the plate only needed to beam the translational freedom. 3.3 The analysis of stress and strain in the forming process With the maximum liquid chamber pressure of 20 MPa and the drawing depth of 50 mm in the forming process, the loading curve is shown in Fig. 9, the FEA result is shown in Fig. 10. Equivalent stress of outer surface was shown in Fig. 11.The inner surface and the outer surface of the stress was different in forming process. The equivalent stress of the outer surface is greater than the inner surface. It explains that the outer is larger than the inside in plastic deformation degree. In the further analysis, stress σ22 which is in the long axis direction of the blank was studied. The changes about stress along the long axis direction should be shown in Fig. 12. The inner surface of plate bears compressive stress in the initial stage of forming, with the drawing depth increased; the compressive stress gradually becomes tensile stress. All cross sections of the plate are mostly in a tensile stress state after forming, and the stress in outer surface is larger than the inner surface. There was disparity stress between internal surface and external surface, thus the moment was formed. When Table 1 Mechanical properties of SUS304

Element

Value

Yield strength, (MPa) Ductility,% Anisotropy coefficient, r Strain hardening, n Hardening coefficient, k(MPa)

239 55 1.02 0.502 1426

the external force is unloaded after the forming, the springback appeared. Numerical simulation is carried out without liquid chamber pressure, and the changes of σ22, which is the stress along the long axis direction after the end of the forming, are shown in Fig. 13. The outer surface is under the tensile stress, and the inner surface is in the compressive stress state. And by contrasting the results, it shows that the pressure makes neutral layer to move towards the inside of the plate. The chamber pressure is between 2 and 20 MPa, and the velocity of punch is 6000 mm/s through finite element analysis (FEA). Because the chamber pressure is the main factor in the hydroforming process, when the pressure is too high, the neutral layer will be removed from the plate profile. In the analysis of springback, the section position is the y-axis in Fig. 8; Δh is the springback value. Comparison was performed with fluid chamber pressures at 2 and 20 MPa, and it was found that the bigger the fluid chamber pressure, the smaller the springback. So, the springback of parts can be reduced by the pressure, and the results between the numerical simulation and experiment are consistent, which is shown in Fig. 14. The equivalent strain is shown in Fig. 15. When the hydroforming ends, the strain is greater than zero, which indicated that plastic deformation occurred. But the equivalent strain is smaller at the bottom of parts, and the plastic deformation is smaller, so the springback value after forming is larger.

Fig. 20 Contrast between the punch and formed part

Int J Adv Manuf Technol 10

0MPa 10MPa 20MPa

h /mm

8 6 4 2 0 -20

0

20

40

60

80

100 120 140 160

Y/mm

When the pressures are respectively 0, 10, and 20 MPa, the parts by hydroforming are shown in Fig. 19, and the comparison between part and punch is shown in Fig. 20. The measurement results along the long axis direction are shown in Fig. 21. The springback decreased with fluid chamber pressure increased, which is consistent with theoretical analysis result, so it verified the accuracy of theoretical analysis. The maximum springback decreases by 8.9% when the liquid chamber pressure is 10 MPa than without pressure, and springback decreases by 26% when the pressure is 20 MPa than without pressure.

Fig. 21 The springback with different pressure

4 Conclusion 3.4 Material property parameters influence on springback 3.4.1 Effect of elastic modulus on springback When other parameters remain unchanged, the elastic modulus parameters were set to 100, 150, 210, and 250 Gpa, which were respectively simulated. Springback was compared at the Y 150 mm, as shown in Fig. 16. In short, the springback was decreased with the increase of elastic modulus. The smaller the elastic part of plastic deformation, the greater the elastic modulus and the lesser the springback caused after unloading, which can also be launched by Eq. (20), so the analysis result is consistent with previous theories. 3.4.2 Effect of yield strength on Springback When other parameters remain unchanged, the yield strength parameters were set to 126, 236, 284, and 323 MPa, which were respectively simulated. The springback was compared at y150mm, as shown in Fig. 17. Springback decreased when the yield strength increased. Equation (17) shows that the larger the yield strength, the greater the moment caused by deformation, so springback is greater after unloading. 3.4.3 Effect of sheet thickness on Springback Remaining other conditions unchanged, thickness of plate is respectively provided with 0.5, 1, 1.5, and 2 mm, which were simulated. The springback at Y 150 mm was shown in Fig. 18. The conclusion shows springback decreased with the thickness of sheet increased. Equation (20) shows that the larger the thickness of the sheet, the smaller the springback. So, the results were consistent with the theoretical analysis.

Due to the two-way pressure in the thickness direction of sheet during hydroforming process, the friction continues to exist, and stress difference is small. The springback is small after unloading. The following conclusions are obtained through research: (1) With increasing the liquid chamber pressure, and reducing the external moment, the neutral layer moved to inside. The region of tensile stress was increased and springback decreased in the section of parts. At the same time liquid chamber pressure could make the friction between the sheet and the binder increased, the tensile stress T was increased, thus the springback would be decreased. (2) The state of stress about the outer and inner surface of sheet was affected by the liquid chamber pressure. Springback was smaller when the liquid chamber pressure was higher. (3) The experimental results showed that the larger the hydraulic pressure, the smaller the springback, and it was consistent with the theoretical analysis. When the liquid chamber pressure was 10 MPa, the springback was reduced by 8.9%. When the pressure was 20 MPa, the springback was reduced by 26%.

Acknowledgments The authors gratefully acknowledge the financial support from National Science and Technology Major Project with Grant No.2014ZX04002041.

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3.5 The springback test The material SUS304 is used in the experiment, and the mechanical properties of SUS304 are shown in Table 1.

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