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Coupling effects of material properties and the bending angle on the springback angle of a titanium alloy tube during numerically controlled bending
Materials and Design 31 (2010) 2001–2010
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Coupling effects of material properties and the bending angle on the springback angle of a titanium alloy tube during numerically controlled bending Z.Q. Jiang a, H. Yang a, M. Zhan a,*, X.D. Xu b, G.J. Li b a b
School of Materials Science and Engineering, State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, China Chengdu Aircraft Industry (Group) Corporation Ltd., Chengdu 610092, China
a r t i c l e
i n f o
Article history: Received 15 June 2009 Accepted 16 October 2009 Available online 21 October 2009 Keywords: Titanium alloy tube NC bending Springback Numerical simulation Coupling effects Stepwise regression analysis
a b s t r a c t The significant springback after the numerically controlled (NC) bending of a titanium alloy tube has an important influence on the precision of the shape and size of the bent tube. This springback depends on the material properties of the tube, the bending angle, and especially their coupling effects. The influence of some material properties and the bending angle on the springback angle in the NC bending of a TA18 tube were investigated using a three-dimensional (3D) elastic–plastic finite element model. Using multivariate and stepwise analyses, the coupling effects of the bending angle and the material properties on the springback angle during NC bending were revealed. It was observed that Young’s modulus, yield stress, the strain hardening coefficient and exponent, and the thickness anisotropy exponent, as well as interactions of these parameters with the bending angle, have a significant influence on the springback angle. The bending angle, yield stress, and hardening coefficient have positive effects on the springback angle, and Young’s modulus, the hardening exponent, and the thickness anisotropy exponent have negative effects. The influence of the material properties of the titanium alloy increases with the bending angle. Young’s modulus and the strain hardening coefficient and exponent have the greatest influence on the springback angle. The results will be very useful in predicting, compensating for and controlling the springback of titanium alloy tubes during NC bending. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction The demand for bent tubes with high pressure resistance, a high strength/weight ratio and long life, as well as the demand for bending processes with high precision and manufacturing efficiency for the aviation and aerospace industries, have led to increased interest in and application of numerically controlled (NC) tube bending technology for the rotary bending of titanium alloy tubes [1,2]. Titanium alloys exhibit poor plastic deformation, high strength, and a low Young modulus [3]. Owing to elastic recovery of the material, significant springback occurs after bending [4], which leads to an increase in the bending radius and a decrease in the bending angle. This has an important influence on the precision of the shape of the bent tube, and the accuracy of the bent tube often does not satisfy requirements for assembly and application. Research by Lou and Stelson revealed that springback is the major source of error in bent tube geometry and is hard to predict. Therefore, efforts should be made to improve springback prediction to reduce errors in bending angles [5], especially for the bending of a titanium alloy tube. * Corresponding author. Tel.: +86 29 88460212x805; fax: +86 29 88495632. E-mail address: [email protected] (M. Zhan). 0261-3069/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2009.10.029
For the bending of TA18 M titanium alloy tubes with the same bending radius but different values of outside diameters D and wall thickness t using an NC bending machine, it was found that the variations in springback angle with the bending angle for different tubes are complicated, as shown in Fig. 1. This results from the complicated coupling effects between material properties and the bending angle. Though these tubes have a similar geometric ratio of D/t, they have different material properties (including Young’s modulus, yield stress, strain hardening coefficient and exponent, Poisson’s ratio, and thickness anisotropy exponent) owing to the complexity of tube manufacturing process, even within the same batch. Thus, it is necessary to research the coupling effects between material properties and bending angle and determine the significance rank of these material parameters to accurately predict the springback angle and control the bending angle of the titanium alloy tube during NC bending. Research into springback during tube bending has involved experiments, theoretical analysis, numerical–analytical analysis, and numerical simulation. Lou and Stelson [6] used springback data measured online for the same batch to predict and compensate for springback. E et al. [7,8], and Murata et al. [9] investigated the springback in the draw-bending and press bending, respectively, of aluminium alloy and stainless steel tubes by experiments.
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nium alloy tube, the complexity of its bending and its high price, it is difficult to investigate coupling effects of the bending angle and the material properties using purely analytical or experimental methods. Therefore, it is essential to systematically study the bending of titanium alloy tube using FE numerical simulation. In the present study, we used a combination of explicit and implicit elastic–plastic FEMs to investigate the effects of bending angle and the material properties on the springback angle of titanium alloy tube. The material properties include Young’s modulus, yield stress, the strain hardening exponent, the strain hardening coefficient, Poisson’s ratio, and the thickness anisotropy exponent. Using multivariate and stepwise regression analysis methods, the coupling effects between material properties and the bending angle were identified and the significance of their influence was assessed. This study may serve as an important guide for predicting, compensating and controlling the springback in the NC bending of titanium alloy tubes. Fig. 1. Variation of springback angle with bending angle for tubes with similar values of D/t but different material properties.
2. Method 2.1. Analysis procedure
Tang [10] employed plastic-deformation theory to investigate the tube bending and obtained stresses and bending moment in the process, which is help for springback analysis. Using the theory of beam bending, Al-Qureshi and Russo [11,12] presented a theoretical analysis model of springback for tube bending assuming elastic–perfectly plastic material, plane strain conditions, and the absence of defects and Bauschinger effects. Using the theoretical model of tube bending, the bending radius and residual stress after springback can be obtained. Using the same method and taking the work hardening behavior of the tube into consideration, El Megharbel et al. [13] developed improved theoretical models for springback during tube bending. Their theoretical results for springback of aluminium, stainless steel, copper, and titanium tubes showed remarkable agreement with experimental data, but the springback was only dependent on the yield stress, bending radius, diameter and wall thickness of the tube, and hardening exponent. Moreover, some inevitable behavior during tube bending was not considered, such as wall thickness variations, cross-section flattening, strain neutral axis shift, and differences in stress and strain fields under various bending conditions. Considering the above deformation behaviors, Zhan et al. [14,15] proposed a numerical–analytical method for predicting the springback angle involving rigid-plastic finite element (FE) simulation of the tube bending and theoretical analysis of the springback. The method was more computationally efficient with satisfactory accuracy for analysis of the bending and springback of thin-walled tubes of aluminium alloy and stainless steel. It helped in identifying the source of springback and changes in features and in multifactorial analysis. However, the numerical–analytical method may not be suitable for the springback analysis of the bending of a titanium alloy tube. Because titanium alloys have high strength, poor plasticity, and a low Young modulus, greater elastic deformation occurs during bending [16,17], so there might be an obvious difference between the stress–strain fields obtained using a rigidplastic finite element method (FEM) and real values. In recent years, a combination of explicit and implicit FEMs has been used to investigate springback for thin-walled stainless steel and aluminium alloy tubes [18,19]. In addition, Murata et al. [9] investigated the influence of hardening exponent on the springback in the press bending of tubes by means of explicit FEM. However, much of this work has focused on the bending of stainless steel and aluminium tubes [7–9,18,19]. To date, very few reports on the springback of titanium alloy tubes during bending have been published. Owing to the material features of tita-
Fig. 2 presents the analysis procedure. Since several factors affect the NC bending process, simulations were designed using the single factorial experimental design (SFED), and multiple polynomial regression analysis (MPRA) and stepwise regression analysis (SRA) [20] were chosen to assess the effects of the parameters. The significance of the material properties was assessed using standard methods. A three-dimensional (3D) FE model was adopted to simulate the bending and springback processes and improve the analysis accuracy and efficiency and the universal applicability of the results.
Fig. 2. Analysis procedure.
Z.Q. Jiang et al. / Materials and Design 31 (2010) 2001–2010
2003
pression–torsion method and an MMX-2 friction and abrasion testing machine. In this study, two parameter types were investigated: the bending angle and the material properties (E, rs, n, K, m, and r). Variable values are listed in Table 1, where values in bold are the default values adopted in simulations when investigating the influence of other parameters on the process.
2.3. FE model Fig. 3. Tension test specimen and plug.
A 3D FE model was developed to simulate the NC bending of a titanium alloy tube using ABAQUS/Explicit and Implicit code [16], as shown in Fig. 5. The Explicit dynamic FE code was used to simulate the bending process, and the Implicit static FE code was used to simulate the unloading springback process. This approach avoids long computation times and improves the computation efficiency. In the model, the tube is defined as a 3D deformable solid body, and the bending, clamp and pressure dies are defined as discrete rigid bodies. Each rigid body is assigned a reference point to represent rigid motion for all degrees of freedom. The constitutive equation for the tube used in the models is r ¼ K en , where r is the true stress and e is the true strain. The model was validated in terms of the bent tube configuration, wall thickness and springback angle Dh [16].
800
Nominal stress / MPa
700 600 500 400
Specimen 1 Specimen 2 Specimen 3
300 200
2.4. MPRA
100 0 0.00
0.05
0.10
0.15
0.20
Nominal strain Fig. 4. Relationship between nominal stress and strain for titanium alloy tubes.
2.2. Computation condition Research was carried out on a titanium alloy tube with an initial outside diameter (D) of 8 mm and a wall thickness of 0.8 mm. The final bending angle (h) was 120°. The bending radius was 24 mm (i.e., 3D). The material properties of the tube determined by tension tests were Young’s modulus E = 94.41 GPa, yield stress rs = 569.58 MPa, hardening coefficient K = 931.0 MPa, hardening exponent n = 0.113, Poisson’s ratio m = 0.39, and thickness anisotropy exponent r = 4.0. Fig. 3 shows the specimen and tube plug used in the tests and Fig. 4 shows the curves obtained for nominal stress vs. nominal strain. The coefficient of friction between the tube and pressure die was 0.25 and that between the tube and bending die was taken as 0.1, which were obtained using a com-
After simulating NC bending of TA18 alloy tubes under different conditions, Dh was calculated. Regression equations for the parameters and indexes were identified using MPRA, which comprises both mathematical and statistical techniques and is beneficial for investigating phenomena having several independent variables. MPRA considers the influence of independent variables on a specific dependent variable (response). Initially, the independent variables of a problem are denoted by x1, x2, . . ., xk. These variables are presumed to be continuous and can be controlled with negligible error in the experiments. The response (Y) is postulated to be a random variable. For most MPRA problems, the relationship between the response and independent variables is unknown. Therefore, the primary step in MPRA is to postulate a suitable approximation for the relationship between Y and the independent variable set. If the response is a linear function of the independent variables according to the visual analysis of simulation results, then first-order equations can be used; otherwise, higher-order equations are applied.
Table 1 Parameter values for NC bending of TA18 tube. Parameter
Value 1
Value 2
Value 3
Value 4
Bending angle h (°) Young’s modulus E (GPa) Yield stress rs (MPa) Hardening coefficient K (MPa) Hardening exponent n Poisson’s ratio m Thickness anisotropy exponent r
30 84.41 539.58 831.0
60 94.41 569.58 931.0
90 104.41 599.58 1031.0
120
0.063 0.31 1.0
0.113 0.39 2.0
0.163 0.47 3.0
4.0
Value 5
5.0 Fig. 5. 3D FE model for NC bending of a titanium alloy tube.
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In this study, the bending angle and the material properties were chosen as independent process parameters, and they are denoted in Eq. (1) as xi. The coefficients b0, bi, bii and bij were determined using the least squares method, and Y is the estimated
response for Dh of the bent tubes. Once the simulation results are obtained, the coefficients of these functions can be calculated. Significance tests for the regression equations and coefficients were performed using MPRA software that we developed.
Fig. 6. Flowchart of the stepwise regression.
Z.Q. Jiang et al. / Materials and Design 31 (2010) 2001–2010
Y ¼ b0 þ
k X
b i xi þ
i¼1
k X
bii x2i þ
k X k X
i¼1
i
bijði
ð1Þ
j
3. Results and discussion 3.1. Effects of material properties and the bending angle
2.5. SRA SRA was used to identify the best expression that includes only parameters with a significant effect on Dh for the NC bending of a TA18 tube and to understand the coupling effects of the parameters. Parameters are included in each step according to their significance rank. The significance of the regression is then assessed and if some parameters have become non-significant on the inclusion of a new parameter, they are excluded from the regression equation one by one. The process is repeated for all parameters until all those that remain in the equation are significant. A flowchart of the method is shown in Fig. 6. The advantage of SRA is its shorter computation time and computation trials when compared with MPRA if a large number of regression variables are considered. The SRA regression expression is similar to the MPRA expression given as Eq. (1). In SRA, we solve the normal equation system using elimination without back substitution and with a suitable choice of significant factors. The elimination sequence is determined by the significance of the parameters, which in turn reflects their contribution in reducing the residual sum of squares. The critical values Fin and Fout for including or excluding a parameter are determined by looking up F-test tables for a given confidence level and degree of freedom.
3.1.1. Effects of Young’s modulus and the bending angle Fig. 7 shows variations in Dh with E and h. From Fig. 7a, it is evident that the increase in Dh with h decreases with increasing E, indicating that the interaction E h might have an effect on Dh. Springback after tube bending mainly depends on the stress and Young’s modulus of the tube material. For bending of tubes that only differ in E, there should be no obvious difference in the value and state of stresses before unloading. Thus, according to the elastic unloading springback principle (as shown in Fig. 8), greater springback occurs for a tube with lower E [8]. Fig. 7b shows that the decrease in Dh with increasing E is related to h and the trend is similar to a quadratic relationship. MRPA revealed that the relationship among Dh, h, E, the interaction E h, and the quadratic term E2 can be expressed as Eq. (2) with a correlation coefficient of 0.999577. The partial correlation coefficients for the terms are 0.999994, 0.999983, 0.999940, and 0.99997. All the coefficients are close to 1, which means the regression equation and regression coefficients are significant. Therefore, Eq. (2) can be used to predict and control Dh for the NC bending of a TA18 tube for a given 3D bending radius and other material properties, except for E.
Dh ¼ 11:0087 0:160791h þ 0:10899E 0:000613883Eh þ 0:000739375E2
(a) 10.0 9.5 9.0 8.5 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0
o Springback angle Δθ /
E =104.41GPa E =94.41GPa E =84.41GPa
20
40
60
80
Bending angle θ /
100
120
ð2Þ
3.1.2. Effects of yield stress and the bending angle Fig. 9 shows variations in Dh with rs and h. From Fig. 9a, it is evident that the increase in Dh with h increases slightly with rs, which means that the interaction rs h might have an effect on Dh. With increasing rs, the value of the stress at the same strain increases, and there is greater elastic deformation for the same strain [8]. Thus, according to the elastic unloading springback principle (as shown in Fig. 10), there is greater springback during unloading of a tube with greater rs. Fig. 9b shows that the increase in Dh with rs is nearly linear. MPRA revealed a relationship among h, rs and the interaction rs h according to Eq. (3) with a correlation coefficient of 0.999564. The partial correlation coefficients for the terms are 0.999598, 0.999316, and 0.999837. The correlation coefficients are close to 1, which means the regression equation is significant,
o
Springback angle Δθ /
o
(b) 9.5 9.0 8.5 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5
2005
o
θ = 30
o
θ = 60
o
θ = 90
o
θ = 120
85
90
95
100
105
Young's modulus E / GPa Fig. 7. Relationship between the springback angle and (a) the bending angle and (b) Young’s modulus.
Fig. 8. Effect of Young’s modulus on springback according to the elastic unloading principle.
2006
9.5
(a) 10.0
9.0
9.5 9.0 8.5 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0
Springback angle Δθ /
σs= 599.58MPa
8.0 7.5
σs= 569.58MPa
7.0
σs= 539.58MPa
o
o
8.5
Springback angle Δθ /
(a)
Z.Q. Jiang et al. / Materials and Design 31 (2010) 2001–2010
6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 20
40
60
80
100
Bending angle θ /
K=1031MPa K= 931MPa K= 831MPa
20
120
40
60
80
Bending angle θ /
o
100
120
o
o
(b)
θ = 30
o
8.5
θ = 60
9.0
θ = 90
8.0
8.5
θ = 120
o
9.5
Springback angle Δθ /
Springback angle Δθ /
o
(b)
9.0
7.5 7.0 6.5 6.0
o
θ = 30
5.5
o
θ = 60
5.0
o
θ = 90
4.5
o
θ = 120
4.0 3.5 530
540
550
560
8.0
o o
7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0
570
580
590
600
Yield stress σs / MPa Fig. 9. Relationship between the springback angle and (a) bending angle and (b) yield stress.
3.5 800
850
900
950
1000
1050
Strain hardening coefficient K/MPa Fig. 11. Relationship between the springback angle and (a) bending angle and (b) strain hardening coefficient.
Dh ¼ 0:664238 þ 0:00312675h þ 0:0166415rs þ 0:0000599006rs h ð3Þ 3.1.3. Effects of the strain hardening coefficient and bending angle Fig. 11 shows variations in Dh with K and h. From Fig. 11a, it is evident that the increase in Dh with h increases with K, indicating
Fig. 10. Effects of yield stress on springback according to the elastic unloading principle.
and the regression coefficient for each term is significant. Therefore, Eq. (3) can be used to predict and control Dh.
Fig. 12. Effects of the strain hardening coefficient on springback according to the elastic unloading principle.
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that the interaction K h might have an effect on Dh. The cause of the increase in K leading to an increase in Dh is similar to the case for yield stress [8], as shown in Fig. 12. Fig. 11b shows that the increase in Dh with K is related to h, and the trend is linear. MPRA revealed that the relationship among h, K and the interaction K h can be expressed as Eq. (4) with a correlation coefficient of 0.999738. The partial correlation coefficients for each term are 0.999503, 0.999248, and 0.999969. The correlation coefficients are close to 1, which means the regression equation is significant, and the regression coefficient for each term is significant. Therefore, Eq. (4) can be used to predict and control Dh.
Dh ¼ 1:11793 þ 0:00140495h 0:0129907K þ 0:0000689042Kh ð4Þ 3.1.4. Effects of the strain hardening exponent and bending angle Fig. 13 shows variations in Dh with n and h. From Fig. 13a, it is evident that the increase in Dh with h decreases with increasing n, indicating that the interaction n h has a slight influence on Dh. The variation in Dh with n during the NC bending of a TA18 M tube is different from that Murata et al. obtained for the press bending of tubes in Ref. [9]. It is found that the strain hardening exponent n has little effect on the springback of the press bending of tubes with similar properties as aluminium alloy except for varied n in Ref. [9]. This is may be result from different bending processes of tubes and material properties. It is well known that the larger
(a) 10.0 9.5 9.0 8.5 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0
Springback angle Δθ / o
n=0.163 n=0.113 n= 0.063
Fig. 14. Relationship between the hardening exponent and springback according to the elastic unloading principle.
the hardening exponent, the lower is the stress needed to achieve the same deformation during the NC bending of tubes. Thus, according to the elastic unloading springback principle (as shown in Fig. 14), there is greater springback for a tube with lower n. Fig. 13b shows that the decrease in Dh with increasing n is linear. MRPA revealed the relationship among h, n, and the interaction n h can be expressed as Eq. (5) with a correlation coefficient of 0.9997. The partial correlation coefficients for each term are 0.998043, 0.999961, and 0.998378. The correlation coefficients are close to 1, which means the regression equation is significant, and the regression coefficient for each term is significant. Therefore, Eq. (5) can be used to predict and control Dh.
Dh ¼ 3:0909 5:77295h þ 0:0596148n 0:07721nh
ð5Þ
3.1.5. Effects of Poisson’s ratio and the bending angle Fig. 15 shows the variation in Dh with h for different values of m. The results demonstrate that m has hardly any effect on Dh.
20
40
60
80
Bending angle θ /
100
3.1.6. Effects of the thickness anisotropy exponent and bending angle Fig. 16 shows variations in Dh with r and h. Fig. 16a shows that the increase in Dh with h decreases with increasing r. The value of r reflects the ratio of the hoop strain to thickness strain. Under the
120
o
o
θ = 30
o
θ = 60
(b) 9.5 9.0 8.5 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5
9.0
o
θ = 90
8.5
Springback angle Δθ /
Springback angle Δθ / o
o
o
θ = 120
8.0
ν =0.47 ν =0.39 ν =0.31
7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0
0.06
0.08
0.10
0.12
0.14
0.16
Strain hardening exponent n Fig. 13. Relationship between the springback angle and (a) bending angle and (b) hardening exponent.
3.5 20
40
60
80
Bending angle θ /
100
120
o
Fig. 15. Relation of springback angle with Poisson ratio and bending angle.
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However, the effects of the material properties and h on Dh for titanium alloy tube bending are very complicated. There are interactions between h and each material property except for m. Moreover, quadratic terms for some parameters affect Dh to some degree. Thus, we identified relations reflecting the coupling effects among these parameters to predict and control Dh during the bending of a titanium alloy tube. Considering the manufacturing effects for a tube billet, the material properties of tubes are not perfectly consistent, even within the same batch, and thus it is necessary to obtain a more universal relation to reflect the coupling effects of parameters. To avoid a large computation effort for multiple variables and their interactions, SRA was employed to identify the best regression equation including only parameters with a significant influence on Dh:
Dh ¼ b0 þ b1 h þ bj xj þ b1j hxj þ b22 x22 þ b66 x26
ðj ¼ 2; ; 6Þ
ð7Þ
where x2 is E, x3 is rs, x4 is K, x5 is n, and x6 is r. The coefficients in Eq. (7) are listed in Table 2. A coefficient with a value of zero indicates that the corresponding term has no significant effect on the target variable and does not need to be included in the regression equation. Thus, according to Table 2, only E, rs, K, n and r, as well as the interactions of these parameters with h, have a significant influence on Dh, and the final best regression equation should be as Eq. (8).
Dh ¼ b0 þ b1 h þ bj xj þ b1j hxj
ðj ¼ 2; ; 6Þ
ð8Þ
As observed from Table 2, Dh decreases with E, n, and the interactions E h, n h and r h on one hand, and increases with the other terms on the other hand. The maximum error for regression and measured values is not more than 3.5%, indicating that the regression equation obtained is of high precision and can thus be used to predict Dh for a titanium alloy tube with an outside diameter of 8 mm and thickness of 0.8 mm according to the requirements of the bending angle and variations in material properties. For a desirable bending angle of h0 , Fig. 16. Relationship between the springback angle and (a) bending angle and (b) thickness anisotropy exponent.
same deformation conditions, lower thickness deformation occurs for tubes with greater r, and thus smaller springback occurs during unloading owing to a larger constraint function. Fig. 16b shows that the decrease in Dh with increasing r is related to h and the decreasing trend is quadratic. MPRA revealed that the relationship among h, r, the interaction r h, and the quadratic term r2 can be expressed as Eq. (6) with a correlation coefficient of 0.998495. The partial correlation coefficients for each term are 0.988879, 0.999831, 0.995928, and 0.992452. All the correlation coefficients are close to 1, which means the regression equation is significant, and the regression coefficient for each term is significant. Therefore, Eq. (6) can be used to predict and control Dh.
Dh ¼ 2:62414 0:257297h þ 0:0553478r 0:00520335rh þ 0:118359r 2
Dh ¼ h h 0 :
ð9Þ
By relating Eqs. (8) and (9), expressions for the practical bending angle with springback compensation under different preconditions can be obtained (as shown in Eq. (10)) and used to control the bending angle of a titanium alloy tube of a given gauge.
h¼
h 0 þ b 0 þ b j xj 1 b1 b1j xj
ðj ¼ 2; ; 6Þ
ð10Þ
3.3. Significance analysis To assess the significance of the influence of the material parameters on Dh for NC bending of TA18 alloy tube, the rate of change in Dh as a function of standardized parameters [Eq. (11), after standardization, x0j 2 ½0; 1 and xj ¼ xjmin þ x0j Dxj ] is proposed as Eq. (12).
ð6Þ
3.2. Analysis of coupling effects From the above analysis, we know that the relationships between the springback angle of a TA18 M titanium alloy tube and Young’s modulus, yield stress, the hardening coefficient, and the hardening exponent are similar to those for stainless steel [7,8,15,19] and aluminium alloy tubes [7,18] for NC bending, but the springback angle of the titanium alloy tube under the same bending conditions is much greater than the springback angles of the latter tubes.
Table 2 Stepwise regression coefficients for the springback angle of titanium alloy tube bending. Parameter
Value
Parameter
Value
b0 b1 b2 b3 b4 b5 b6
1.87613 0.02391219 0.0212939 0.00306682 0.00139599 5.77669 0.205171
b12 b13 b14 b15 b16 b22, b66 Max error (%)
0.000612383 0.0000606923 0.0000690218 0.0771639 0.00552268 0.0 3.493
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Z.Q. Jiang et al. / Materials and Design 31 (2010) 2001–2010
2.0
Variation of springback angle Δθ / o
1.8 1.6 1.4
E σs
K n r
1.2 1.0 0.8 0.6 0.4 0.2 0.0 30
60
90
Bending angle θ /
120
o
Fig. 17. Significance of the influence of material properties on springback during NC bending.
xj xjmin xj xjmin ¼ xjmax xjmin Dxj @ðDhÞ ¼ ðbj þ b1j hÞ Dxj @x0j
x0j ¼
ð11Þ ð12Þ
The significance of the effect of different material parameters on Dh for NC bending of TA18 M tubes is shown in Fig. 17. For a bending angle of 30°, the significance of the effect decreases in the order n > E > K > rs > r. For a bending angle of 60°, the significance decreases in the order E > K > n > rs > r. For a bending angle of 90°, the significance decreases in the order E > K > n > r > rs. For a bending angle of 120°, the significance decreases in the order K > E > n > r > r s. 4. Conclusions Using a 3D elastic–plastic FE model and multivariate and stepwise analyses, the coupling effects of the bending angle and the material properties on the springback angle during the NC bending of a titanium alloy tube were revealed. The conclusions are as follows: (1) Compared with stainless steel and aluminium alloy tube NC bending, the springback angle of the titanium alloy under the same bending conditions is much greater. The relationships between the springback of a titanium alloy tube and Young’s modulus, yield stress, and the strain hardening coefficient and exponent are similar to those for stainless steel and aluminium alloy tubes in NC bending. The larger the thickness anisotropy exponent, the smaller the springback angle is. The Poisson ratio has hardly any influence on springback angle. (2) Young’s modulus, yield stress, the strain hardening coefficient and exponent, and the thickness anisotropy exponent, as well as coupling effects of these parameters with the bending angle, have a significant influence on the springback angle. (3) The quadratic terms of Young’s modulus and thickness anisotropy exponent affect the springback angle to some degree when only considering their individual effects on
the springback angle. However, when taking the effects of all material properties into consideration, the quadratic terms are not distinguishable. (4) The influence of the material properties of titanium alloy tubes increases with the bending angle. Regardless of the bending angle, Young’s modulus and the strain hardening coefficient and exponent are the three parameters that have the greatest influence on the springback angle. Acknowledgements This research was supported by the Program for New Century Excellent Talents in University (NCET-08-0462) and the Fund of the State Key Laboratory of Solidification Processing in NWPU (No. KP200919), for which the authors wish to express their gratitude. References [1] Hashmi MSJ. Aspects of tube pipe manufacturing processes: meter to nanometer diameter. J Mater Process Technol 2006;179:5–10. [2] Yang H, Zhan M, Liu LY. Some advanced plastic processing technologies and their numerical simulation. J Mater Process Technol 2004;151:63–9. [3] MacKenzie PM, Walker CA, Mckelvie J. A method for evaluating the mechanical performance of thin-walled titanium tubes. Thin-Walled Struct 2007;45:400–6. [4] Wang G, Zhang KF, Wu DZ, Wang JZ, Yu YD. Superplastic forming of bellows expansion joints made of titanium alloy. J Mater Process Technol 2006;178:24–8. [5] Lou HZ, Stelson KA. Three-dimensional tube geometry control for rotary draw tube bending, part 2: statistical tube tolerance analysis and adaptive bend correction. J Manuf Sci Eng 2001;123:266–71. [6] Lou HZ, Stelson KA. Three-dimensional tube geometry control for rotary draw tube bending, part 1: bend angle and overall tube geometry control. J Manuf Sci Eng 2001;123:258–65. [7] E DX, He HX, Liu XY, Ning RX. Experimental study and finite element analysis of spring-back deformation in tube bending. Int J Miner Metall Mater 2009;16:177–83. [8] E DX, Liu YF. Springback and time-dependent springback of 1Cr18Ni9Ti stainless steel tubes under bending. Mater Des 2010;31:1256–61. [9] Murata M, Kuboki T, Takahashi K, Goodarzi M, Jin Y. Effect of hardening exponent on tube bending. J Mater Process Technol 2008;201:189–92. [10] Tang NC. Plastic-deformation analysis in tube bending. Int J Press Vessels Piping 2000;77:751–9. [11] AL-Qureshi HA. Elastic–plastic analysis of tube bending. Int J Mach Tool Manuf 1999;39:87–104.
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[12] Al-Qureshi HA, Russo A. Spring-back and residual stresses in bending of thinwalled aluminum tubes. Mater Des 2002;23:217–22. [13] El Megharbel A, El Nasser GA, Domiaty Aly El. Bending of tube and section made of strain-hardening materials. J Mater Process Technol 2008;203:372–80. [14] Zhan M, Yang H, Huang L. A numerical–analytic method for quickly predicting springback of numerical control bending of thin-walled tube. J Mater Sci Technol 2006;l22(5):713–20. [15] Zhan M, Yang H, Huang L, Gu RJ. Springback analysis of numerical control bending of thin-walled tube using numerical–analytic method. J Mater Process Technol 2006;177:197–201. [16] Jiang ZQ, Yang H, Zhan M, Yue YB, Liu J, Xu XD, et al. Establishment of a 3D FE model for the bending of a titanium alloy tube. Int J Mech Sci 2009. doi:10.1016/j.ijmecsci.2009.09.029.
[17] Hur S, Park JS. The 360° cold bending of Ti–6Al–4V large-diameter seamless tube. J Miner 1999;51(6):28–30. [18] Gu RJ, Yang H, Zhan M, Li H. Springback of thin-walled tube NC precision bending and its numerical simulation. Trans Nonferr Met Soc China 2006;16:s631–8. [19] Gu RJ, Yang H, Zhan M, Li H, Li HW. Research on the springback of thin-walled tube NC bending based on the numerical simulation of the whole process. Comput Mater Sci 2008;42(4):537–49. [20] Yang H, Zhan M. Experimental modeling methods used for material forming processes. Xi’an, Northwestern Polytechnical University press; 2008 [in Chinese].
Contents lists available at ScienceDirect
Materials and Design journal homepage: www.elsevier.com/locate/matdes
Coupling effects of material properties and the bending angle on the springback angle of a titanium alloy tube during numerically controlled bending Z.Q. Jiang a, H. Yang a, M. Zhan a,*, X.D. Xu b, G.J. Li b a b
School of Materials Science and Engineering, State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, China Chengdu Aircraft Industry (Group) Corporation Ltd., Chengdu 610092, China
a r t i c l e
i n f o
Article history: Received 15 June 2009 Accepted 16 October 2009 Available online 21 October 2009 Keywords: Titanium alloy tube NC bending Springback Numerical simulation Coupling effects Stepwise regression analysis
a b s t r a c t The significant springback after the numerically controlled (NC) bending of a titanium alloy tube has an important influence on the precision of the shape and size of the bent tube. This springback depends on the material properties of the tube, the bending angle, and especially their coupling effects. The influence of some material properties and the bending angle on the springback angle in the NC bending of a TA18 tube were investigated using a three-dimensional (3D) elastic–plastic finite element model. Using multivariate and stepwise analyses, the coupling effects of the bending angle and the material properties on the springback angle during NC bending were revealed. It was observed that Young’s modulus, yield stress, the strain hardening coefficient and exponent, and the thickness anisotropy exponent, as well as interactions of these parameters with the bending angle, have a significant influence on the springback angle. The bending angle, yield stress, and hardening coefficient have positive effects on the springback angle, and Young’s modulus, the hardening exponent, and the thickness anisotropy exponent have negative effects. The influence of the material properties of the titanium alloy increases with the bending angle. Young’s modulus and the strain hardening coefficient and exponent have the greatest influence on the springback angle. The results will be very useful in predicting, compensating for and controlling the springback of titanium alloy tubes during NC bending. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction The demand for bent tubes with high pressure resistance, a high strength/weight ratio and long life, as well as the demand for bending processes with high precision and manufacturing efficiency for the aviation and aerospace industries, have led to increased interest in and application of numerically controlled (NC) tube bending technology for the rotary bending of titanium alloy tubes [1,2]. Titanium alloys exhibit poor plastic deformation, high strength, and a low Young modulus [3]. Owing to elastic recovery of the material, significant springback occurs after bending [4], which leads to an increase in the bending radius and a decrease in the bending angle. This has an important influence on the precision of the shape of the bent tube, and the accuracy of the bent tube often does not satisfy requirements for assembly and application. Research by Lou and Stelson revealed that springback is the major source of error in bent tube geometry and is hard to predict. Therefore, efforts should be made to improve springback prediction to reduce errors in bending angles [5], especially for the bending of a titanium alloy tube. * Corresponding author. Tel.: +86 29 88460212x805; fax: +86 29 88495632. E-mail address: [email protected] (M. Zhan). 0261-3069/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2009.10.029
For the bending of TA18 M titanium alloy tubes with the same bending radius but different values of outside diameters D and wall thickness t using an NC bending machine, it was found that the variations in springback angle with the bending angle for different tubes are complicated, as shown in Fig. 1. This results from the complicated coupling effects between material properties and the bending angle. Though these tubes have a similar geometric ratio of D/t, they have different material properties (including Young’s modulus, yield stress, strain hardening coefficient and exponent, Poisson’s ratio, and thickness anisotropy exponent) owing to the complexity of tube manufacturing process, even within the same batch. Thus, it is necessary to research the coupling effects between material properties and bending angle and determine the significance rank of these material parameters to accurately predict the springback angle and control the bending angle of the titanium alloy tube during NC bending. Research into springback during tube bending has involved experiments, theoretical analysis, numerical–analytical analysis, and numerical simulation. Lou and Stelson [6] used springback data measured online for the same batch to predict and compensate for springback. E et al. [7,8], and Murata et al. [9] investigated the springback in the draw-bending and press bending, respectively, of aluminium alloy and stainless steel tubes by experiments.
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nium alloy tube, the complexity of its bending and its high price, it is difficult to investigate coupling effects of the bending angle and the material properties using purely analytical or experimental methods. Therefore, it is essential to systematically study the bending of titanium alloy tube using FE numerical simulation. In the present study, we used a combination of explicit and implicit elastic–plastic FEMs to investigate the effects of bending angle and the material properties on the springback angle of titanium alloy tube. The material properties include Young’s modulus, yield stress, the strain hardening exponent, the strain hardening coefficient, Poisson’s ratio, and the thickness anisotropy exponent. Using multivariate and stepwise regression analysis methods, the coupling effects between material properties and the bending angle were identified and the significance of their influence was assessed. This study may serve as an important guide for predicting, compensating and controlling the springback in the NC bending of titanium alloy tubes. Fig. 1. Variation of springback angle with bending angle for tubes with similar values of D/t but different material properties.
2. Method 2.1. Analysis procedure
Tang [10] employed plastic-deformation theory to investigate the tube bending and obtained stresses and bending moment in the process, which is help for springback analysis. Using the theory of beam bending, Al-Qureshi and Russo [11,12] presented a theoretical analysis model of springback for tube bending assuming elastic–perfectly plastic material, plane strain conditions, and the absence of defects and Bauschinger effects. Using the theoretical model of tube bending, the bending radius and residual stress after springback can be obtained. Using the same method and taking the work hardening behavior of the tube into consideration, El Megharbel et al. [13] developed improved theoretical models for springback during tube bending. Their theoretical results for springback of aluminium, stainless steel, copper, and titanium tubes showed remarkable agreement with experimental data, but the springback was only dependent on the yield stress, bending radius, diameter and wall thickness of the tube, and hardening exponent. Moreover, some inevitable behavior during tube bending was not considered, such as wall thickness variations, cross-section flattening, strain neutral axis shift, and differences in stress and strain fields under various bending conditions. Considering the above deformation behaviors, Zhan et al. [14,15] proposed a numerical–analytical method for predicting the springback angle involving rigid-plastic finite element (FE) simulation of the tube bending and theoretical analysis of the springback. The method was more computationally efficient with satisfactory accuracy for analysis of the bending and springback of thin-walled tubes of aluminium alloy and stainless steel. It helped in identifying the source of springback and changes in features and in multifactorial analysis. However, the numerical–analytical method may not be suitable for the springback analysis of the bending of a titanium alloy tube. Because titanium alloys have high strength, poor plasticity, and a low Young modulus, greater elastic deformation occurs during bending [16,17], so there might be an obvious difference between the stress–strain fields obtained using a rigidplastic finite element method (FEM) and real values. In recent years, a combination of explicit and implicit FEMs has been used to investigate springback for thin-walled stainless steel and aluminium alloy tubes [18,19]. In addition, Murata et al. [9] investigated the influence of hardening exponent on the springback in the press bending of tubes by means of explicit FEM. However, much of this work has focused on the bending of stainless steel and aluminium tubes [7–9,18,19]. To date, very few reports on the springback of titanium alloy tubes during bending have been published. Owing to the material features of tita-
Fig. 2 presents the analysis procedure. Since several factors affect the NC bending process, simulations were designed using the single factorial experimental design (SFED), and multiple polynomial regression analysis (MPRA) and stepwise regression analysis (SRA) [20] were chosen to assess the effects of the parameters. The significance of the material properties was assessed using standard methods. A three-dimensional (3D) FE model was adopted to simulate the bending and springback processes and improve the analysis accuracy and efficiency and the universal applicability of the results.
Fig. 2. Analysis procedure.
Z.Q. Jiang et al. / Materials and Design 31 (2010) 2001–2010
2003
pression–torsion method and an MMX-2 friction and abrasion testing machine. In this study, two parameter types were investigated: the bending angle and the material properties (E, rs, n, K, m, and r). Variable values are listed in Table 1, where values in bold are the default values adopted in simulations when investigating the influence of other parameters on the process.
2.3. FE model Fig. 3. Tension test specimen and plug.
A 3D FE model was developed to simulate the NC bending of a titanium alloy tube using ABAQUS/Explicit and Implicit code [16], as shown in Fig. 5. The Explicit dynamic FE code was used to simulate the bending process, and the Implicit static FE code was used to simulate the unloading springback process. This approach avoids long computation times and improves the computation efficiency. In the model, the tube is defined as a 3D deformable solid body, and the bending, clamp and pressure dies are defined as discrete rigid bodies. Each rigid body is assigned a reference point to represent rigid motion for all degrees of freedom. The constitutive equation for the tube used in the models is r ¼ K en , where r is the true stress and e is the true strain. The model was validated in terms of the bent tube configuration, wall thickness and springback angle Dh [16].
800
Nominal stress / MPa
700 600 500 400
Specimen 1 Specimen 2 Specimen 3
300 200
2.4. MPRA
100 0 0.00
0.05
0.10
0.15
0.20
Nominal strain Fig. 4. Relationship between nominal stress and strain for titanium alloy tubes.
2.2. Computation condition Research was carried out on a titanium alloy tube with an initial outside diameter (D) of 8 mm and a wall thickness of 0.8 mm. The final bending angle (h) was 120°. The bending radius was 24 mm (i.e., 3D). The material properties of the tube determined by tension tests were Young’s modulus E = 94.41 GPa, yield stress rs = 569.58 MPa, hardening coefficient K = 931.0 MPa, hardening exponent n = 0.113, Poisson’s ratio m = 0.39, and thickness anisotropy exponent r = 4.0. Fig. 3 shows the specimen and tube plug used in the tests and Fig. 4 shows the curves obtained for nominal stress vs. nominal strain. The coefficient of friction between the tube and pressure die was 0.25 and that between the tube and bending die was taken as 0.1, which were obtained using a com-
After simulating NC bending of TA18 alloy tubes under different conditions, Dh was calculated. Regression equations for the parameters and indexes were identified using MPRA, which comprises both mathematical and statistical techniques and is beneficial for investigating phenomena having several independent variables. MPRA considers the influence of independent variables on a specific dependent variable (response). Initially, the independent variables of a problem are denoted by x1, x2, . . ., xk. These variables are presumed to be continuous and can be controlled with negligible error in the experiments. The response (Y) is postulated to be a random variable. For most MPRA problems, the relationship between the response and independent variables is unknown. Therefore, the primary step in MPRA is to postulate a suitable approximation for the relationship between Y and the independent variable set. If the response is a linear function of the independent variables according to the visual analysis of simulation results, then first-order equations can be used; otherwise, higher-order equations are applied.
Table 1 Parameter values for NC bending of TA18 tube. Parameter
Value 1
Value 2
Value 3
Value 4
Bending angle h (°) Young’s modulus E (GPa) Yield stress rs (MPa) Hardening coefficient K (MPa) Hardening exponent n Poisson’s ratio m Thickness anisotropy exponent r
30 84.41 539.58 831.0
60 94.41 569.58 931.0
90 104.41 599.58 1031.0
120
0.063 0.31 1.0
0.113 0.39 2.0
0.163 0.47 3.0
4.0
Value 5
5.0 Fig. 5. 3D FE model for NC bending of a titanium alloy tube.
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In this study, the bending angle and the material properties were chosen as independent process parameters, and they are denoted in Eq. (1) as xi. The coefficients b0, bi, bii and bij were determined using the least squares method, and Y is the estimated
response for Dh of the bent tubes. Once the simulation results are obtained, the coefficients of these functions can be calculated. Significance tests for the regression equations and coefficients were performed using MPRA software that we developed.
Fig. 6. Flowchart of the stepwise regression.
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Y ¼ b0 þ
k X
b i xi þ
i¼1
k X
bii x2i þ
k X k X
i¼1
i
bijði
ð1Þ
j
3. Results and discussion 3.1. Effects of material properties and the bending angle
2.5. SRA SRA was used to identify the best expression that includes only parameters with a significant effect on Dh for the NC bending of a TA18 tube and to understand the coupling effects of the parameters. Parameters are included in each step according to their significance rank. The significance of the regression is then assessed and if some parameters have become non-significant on the inclusion of a new parameter, they are excluded from the regression equation one by one. The process is repeated for all parameters until all those that remain in the equation are significant. A flowchart of the method is shown in Fig. 6. The advantage of SRA is its shorter computation time and computation trials when compared with MPRA if a large number of regression variables are considered. The SRA regression expression is similar to the MPRA expression given as Eq. (1). In SRA, we solve the normal equation system using elimination without back substitution and with a suitable choice of significant factors. The elimination sequence is determined by the significance of the parameters, which in turn reflects their contribution in reducing the residual sum of squares. The critical values Fin and Fout for including or excluding a parameter are determined by looking up F-test tables for a given confidence level and degree of freedom.
3.1.1. Effects of Young’s modulus and the bending angle Fig. 7 shows variations in Dh with E and h. From Fig. 7a, it is evident that the increase in Dh with h decreases with increasing E, indicating that the interaction E h might have an effect on Dh. Springback after tube bending mainly depends on the stress and Young’s modulus of the tube material. For bending of tubes that only differ in E, there should be no obvious difference in the value and state of stresses before unloading. Thus, according to the elastic unloading springback principle (as shown in Fig. 8), greater springback occurs for a tube with lower E [8]. Fig. 7b shows that the decrease in Dh with increasing E is related to h and the trend is similar to a quadratic relationship. MRPA revealed that the relationship among Dh, h, E, the interaction E h, and the quadratic term E2 can be expressed as Eq. (2) with a correlation coefficient of 0.999577. The partial correlation coefficients for the terms are 0.999994, 0.999983, 0.999940, and 0.99997. All the coefficients are close to 1, which means the regression equation and regression coefficients are significant. Therefore, Eq. (2) can be used to predict and control Dh for the NC bending of a TA18 tube for a given 3D bending radius and other material properties, except for E.
Dh ¼ 11:0087 0:160791h þ 0:10899E 0:000613883Eh þ 0:000739375E2
(a) 10.0 9.5 9.0 8.5 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0
o Springback angle Δθ /
E =104.41GPa E =94.41GPa E =84.41GPa
20
40
60
80
Bending angle θ /
100
120
ð2Þ
3.1.2. Effects of yield stress and the bending angle Fig. 9 shows variations in Dh with rs and h. From Fig. 9a, it is evident that the increase in Dh with h increases slightly with rs, which means that the interaction rs h might have an effect on Dh. With increasing rs, the value of the stress at the same strain increases, and there is greater elastic deformation for the same strain [8]. Thus, according to the elastic unloading springback principle (as shown in Fig. 10), there is greater springback during unloading of a tube with greater rs. Fig. 9b shows that the increase in Dh with rs is nearly linear. MPRA revealed a relationship among h, rs and the interaction rs h according to Eq. (3) with a correlation coefficient of 0.999564. The partial correlation coefficients for the terms are 0.999598, 0.999316, and 0.999837. The correlation coefficients are close to 1, which means the regression equation is significant,
o
Springback angle Δθ /
o
(b) 9.5 9.0 8.5 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5
2005
o
θ = 30
o
θ = 60
o
θ = 90
o
θ = 120
85
90
95
100
105
Young's modulus E / GPa Fig. 7. Relationship between the springback angle and (a) the bending angle and (b) Young’s modulus.
Fig. 8. Effect of Young’s modulus on springback according to the elastic unloading principle.
2006
9.5
(a) 10.0
9.0
9.5 9.0 8.5 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0
Springback angle Δθ /
σs= 599.58MPa
8.0 7.5
σs= 569.58MPa
7.0
σs= 539.58MPa
o
o
8.5
Springback angle Δθ /
(a)
Z.Q. Jiang et al. / Materials and Design 31 (2010) 2001–2010
6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 20
40
60
80
100
Bending angle θ /
K=1031MPa K= 931MPa K= 831MPa
20
120
40
60
80
Bending angle θ /
o
100
120
o
o
(b)
θ = 30
o
8.5
θ = 60
9.0
θ = 90
8.0
8.5
θ = 120
o
9.5
Springback angle Δθ /
Springback angle Δθ /
o
(b)
9.0
7.5 7.0 6.5 6.0
o
θ = 30
5.5
o
θ = 60
5.0
o
θ = 90
4.5
o
θ = 120
4.0 3.5 530
540
550
560
8.0
o o
7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0
570
580
590
600
Yield stress σs / MPa Fig. 9. Relationship between the springback angle and (a) bending angle and (b) yield stress.
3.5 800
850
900
950
1000
1050
Strain hardening coefficient K/MPa Fig. 11. Relationship between the springback angle and (a) bending angle and (b) strain hardening coefficient.
Dh ¼ 0:664238 þ 0:00312675h þ 0:0166415rs þ 0:0000599006rs h ð3Þ 3.1.3. Effects of the strain hardening coefficient and bending angle Fig. 11 shows variations in Dh with K and h. From Fig. 11a, it is evident that the increase in Dh with h increases with K, indicating
Fig. 10. Effects of yield stress on springback according to the elastic unloading principle.
and the regression coefficient for each term is significant. Therefore, Eq. (3) can be used to predict and control Dh.
Fig. 12. Effects of the strain hardening coefficient on springback according to the elastic unloading principle.
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Z.Q. Jiang et al. / Materials and Design 31 (2010) 2001–2010
that the interaction K h might have an effect on Dh. The cause of the increase in K leading to an increase in Dh is similar to the case for yield stress [8], as shown in Fig. 12. Fig. 11b shows that the increase in Dh with K is related to h, and the trend is linear. MPRA revealed that the relationship among h, K and the interaction K h can be expressed as Eq. (4) with a correlation coefficient of 0.999738. The partial correlation coefficients for each term are 0.999503, 0.999248, and 0.999969. The correlation coefficients are close to 1, which means the regression equation is significant, and the regression coefficient for each term is significant. Therefore, Eq. (4) can be used to predict and control Dh.
Dh ¼ 1:11793 þ 0:00140495h 0:0129907K þ 0:0000689042Kh ð4Þ 3.1.4. Effects of the strain hardening exponent and bending angle Fig. 13 shows variations in Dh with n and h. From Fig. 13a, it is evident that the increase in Dh with h decreases with increasing n, indicating that the interaction n h has a slight influence on Dh. The variation in Dh with n during the NC bending of a TA18 M tube is different from that Murata et al. obtained for the press bending of tubes in Ref. [9]. It is found that the strain hardening exponent n has little effect on the springback of the press bending of tubes with similar properties as aluminium alloy except for varied n in Ref. [9]. This is may be result from different bending processes of tubes and material properties. It is well known that the larger
(a) 10.0 9.5 9.0 8.5 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0
Springback angle Δθ / o
n=0.163 n=0.113 n= 0.063
Fig. 14. Relationship between the hardening exponent and springback according to the elastic unloading principle.
the hardening exponent, the lower is the stress needed to achieve the same deformation during the NC bending of tubes. Thus, according to the elastic unloading springback principle (as shown in Fig. 14), there is greater springback for a tube with lower n. Fig. 13b shows that the decrease in Dh with increasing n is linear. MRPA revealed the relationship among h, n, and the interaction n h can be expressed as Eq. (5) with a correlation coefficient of 0.9997. The partial correlation coefficients for each term are 0.998043, 0.999961, and 0.998378. The correlation coefficients are close to 1, which means the regression equation is significant, and the regression coefficient for each term is significant. Therefore, Eq. (5) can be used to predict and control Dh.
Dh ¼ 3:0909 5:77295h þ 0:0596148n 0:07721nh
ð5Þ
3.1.5. Effects of Poisson’s ratio and the bending angle Fig. 15 shows the variation in Dh with h for different values of m. The results demonstrate that m has hardly any effect on Dh.
20
40
60
80
Bending angle θ /
100
3.1.6. Effects of the thickness anisotropy exponent and bending angle Fig. 16 shows variations in Dh with r and h. Fig. 16a shows that the increase in Dh with h decreases with increasing r. The value of r reflects the ratio of the hoop strain to thickness strain. Under the
120
o
o
θ = 30
o
θ = 60
(b) 9.5 9.0 8.5 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5
9.0
o
θ = 90
8.5
Springback angle Δθ /
Springback angle Δθ / o
o
o
θ = 120
8.0
ν =0.47 ν =0.39 ν =0.31
7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0
0.06
0.08
0.10
0.12
0.14
0.16
Strain hardening exponent n Fig. 13. Relationship between the springback angle and (a) bending angle and (b) hardening exponent.
3.5 20
40
60
80
Bending angle θ /
100
120
o
Fig. 15. Relation of springback angle with Poisson ratio and bending angle.
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Z.Q. Jiang et al. / Materials and Design 31 (2010) 2001–2010
However, the effects of the material properties and h on Dh for titanium alloy tube bending are very complicated. There are interactions between h and each material property except for m. Moreover, quadratic terms for some parameters affect Dh to some degree. Thus, we identified relations reflecting the coupling effects among these parameters to predict and control Dh during the bending of a titanium alloy tube. Considering the manufacturing effects for a tube billet, the material properties of tubes are not perfectly consistent, even within the same batch, and thus it is necessary to obtain a more universal relation to reflect the coupling effects of parameters. To avoid a large computation effort for multiple variables and their interactions, SRA was employed to identify the best regression equation including only parameters with a significant influence on Dh:
Dh ¼ b0 þ b1 h þ bj xj þ b1j hxj þ b22 x22 þ b66 x26
ðj ¼ 2; ; 6Þ
ð7Þ
where x2 is E, x3 is rs, x4 is K, x5 is n, and x6 is r. The coefficients in Eq. (7) are listed in Table 2. A coefficient with a value of zero indicates that the corresponding term has no significant effect on the target variable and does not need to be included in the regression equation. Thus, according to Table 2, only E, rs, K, n and r, as well as the interactions of these parameters with h, have a significant influence on Dh, and the final best regression equation should be as Eq. (8).
Dh ¼ b0 þ b1 h þ bj xj þ b1j hxj
ðj ¼ 2; ; 6Þ
ð8Þ
As observed from Table 2, Dh decreases with E, n, and the interactions E h, n h and r h on one hand, and increases with the other terms on the other hand. The maximum error for regression and measured values is not more than 3.5%, indicating that the regression equation obtained is of high precision and can thus be used to predict Dh for a titanium alloy tube with an outside diameter of 8 mm and thickness of 0.8 mm according to the requirements of the bending angle and variations in material properties. For a desirable bending angle of h0 , Fig. 16. Relationship between the springback angle and (a) bending angle and (b) thickness anisotropy exponent.
same deformation conditions, lower thickness deformation occurs for tubes with greater r, and thus smaller springback occurs during unloading owing to a larger constraint function. Fig. 16b shows that the decrease in Dh with increasing r is related to h and the decreasing trend is quadratic. MPRA revealed that the relationship among h, r, the interaction r h, and the quadratic term r2 can be expressed as Eq. (6) with a correlation coefficient of 0.998495. The partial correlation coefficients for each term are 0.988879, 0.999831, 0.995928, and 0.992452. All the correlation coefficients are close to 1, which means the regression equation is significant, and the regression coefficient for each term is significant. Therefore, Eq. (6) can be used to predict and control Dh.
Dh ¼ 2:62414 0:257297h þ 0:0553478r 0:00520335rh þ 0:118359r 2
Dh ¼ h h 0 :
ð9Þ
By relating Eqs. (8) and (9), expressions for the practical bending angle with springback compensation under different preconditions can be obtained (as shown in Eq. (10)) and used to control the bending angle of a titanium alloy tube of a given gauge.
h¼
h 0 þ b 0 þ b j xj 1 b1 b1j xj
ðj ¼ 2; ; 6Þ
ð10Þ
3.3. Significance analysis To assess the significance of the influence of the material parameters on Dh for NC bending of TA18 alloy tube, the rate of change in Dh as a function of standardized parameters [Eq. (11), after standardization, x0j 2 ½0; 1 and xj ¼ xjmin þ x0j Dxj ] is proposed as Eq. (12).
ð6Þ
3.2. Analysis of coupling effects From the above analysis, we know that the relationships between the springback angle of a TA18 M titanium alloy tube and Young’s modulus, yield stress, the hardening coefficient, and the hardening exponent are similar to those for stainless steel [7,8,15,19] and aluminium alloy tubes [7,18] for NC bending, but the springback angle of the titanium alloy tube under the same bending conditions is much greater than the springback angles of the latter tubes.
Table 2 Stepwise regression coefficients for the springback angle of titanium alloy tube bending. Parameter
Value
Parameter
Value
b0 b1 b2 b3 b4 b5 b6
1.87613 0.02391219 0.0212939 0.00306682 0.00139599 5.77669 0.205171
b12 b13 b14 b15 b16 b22, b66 Max error (%)
0.000612383 0.0000606923 0.0000690218 0.0771639 0.00552268 0.0 3.493
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Z.Q. Jiang et al. / Materials and Design 31 (2010) 2001–2010
2.0
Variation of springback angle Δθ / o
1.8 1.6 1.4
E σs
K n r
1.2 1.0 0.8 0.6 0.4 0.2 0.0 30
60
90
Bending angle θ /
120
o
Fig. 17. Significance of the influence of material properties on springback during NC bending.
xj xjmin xj xjmin ¼ xjmax xjmin Dxj @ðDhÞ ¼ ðbj þ b1j hÞ Dxj @x0j
x0j ¼
ð11Þ ð12Þ
The significance of the effect of different material parameters on Dh for NC bending of TA18 M tubes is shown in Fig. 17. For a bending angle of 30°, the significance of the effect decreases in the order n > E > K > rs > r. For a bending angle of 60°, the significance decreases in the order E > K > n > rs > r. For a bending angle of 90°, the significance decreases in the order E > K > n > r > rs. For a bending angle of 120°, the significance decreases in the order K > E > n > r > r s. 4. Conclusions Using a 3D elastic–plastic FE model and multivariate and stepwise analyses, the coupling effects of the bending angle and the material properties on the springback angle during the NC bending of a titanium alloy tube were revealed. The conclusions are as follows: (1) Compared with stainless steel and aluminium alloy tube NC bending, the springback angle of the titanium alloy under the same bending conditions is much greater. The relationships between the springback of a titanium alloy tube and Young’s modulus, yield stress, and the strain hardening coefficient and exponent are similar to those for stainless steel and aluminium alloy tubes in NC bending. The larger the thickness anisotropy exponent, the smaller the springback angle is. The Poisson ratio has hardly any influence on springback angle. (2) Young’s modulus, yield stress, the strain hardening coefficient and exponent, and the thickness anisotropy exponent, as well as coupling effects of these parameters with the bending angle, have a significant influence on the springback angle. (3) The quadratic terms of Young’s modulus and thickness anisotropy exponent affect the springback angle to some degree when only considering their individual effects on
the springback angle. However, when taking the effects of all material properties into consideration, the quadratic terms are not distinguishable. (4) The influence of the material properties of titanium alloy tubes increases with the bending angle. Regardless of the bending angle, Young’s modulus and the strain hardening coefficient and exponent are the three parameters that have the greatest influence on the springback angle. Acknowledgements This research was supported by the Program for New Century Excellent Talents in University (NCET-08-0462) and the Fund of the State Key Laboratory of Solidification Processing in NWPU (No. KP200919), for which the authors wish to express their gratitude. References [1] Hashmi MSJ. Aspects of tube pipe manufacturing processes: meter to nanometer diameter. J Mater Process Technol 2006;179:5–10. [2] Yang H, Zhan M, Liu LY. Some advanced plastic processing technologies and their numerical simulation. J Mater Process Technol 2004;151:63–9. [3] MacKenzie PM, Walker CA, Mckelvie J. A method for evaluating the mechanical performance of thin-walled titanium tubes. Thin-Walled Struct 2007;45:400–6. [4] Wang G, Zhang KF, Wu DZ, Wang JZ, Yu YD. Superplastic forming of bellows expansion joints made of titanium alloy. J Mater Process Technol 2006;178:24–8. [5] Lou HZ, Stelson KA. Three-dimensional tube geometry control for rotary draw tube bending, part 2: statistical tube tolerance analysis and adaptive bend correction. J Manuf Sci Eng 2001;123:266–71. [6] Lou HZ, Stelson KA. Three-dimensional tube geometry control for rotary draw tube bending, part 1: bend angle and overall tube geometry control. J Manuf Sci Eng 2001;123:258–65. [7] E DX, He HX, Liu XY, Ning RX. Experimental study and finite element analysis of spring-back deformation in tube bending. Int J Miner Metall Mater 2009;16:177–83. [8] E DX, Liu YF. Springback and time-dependent springback of 1Cr18Ni9Ti stainless steel tubes under bending. Mater Des 2010;31:1256–61. [9] Murata M, Kuboki T, Takahashi K, Goodarzi M, Jin Y. Effect of hardening exponent on tube bending. J Mater Process Technol 2008;201:189–92. [10] Tang NC. Plastic-deformation analysis in tube bending. Int J Press Vessels Piping 2000;77:751–9. [11] AL-Qureshi HA. Elastic–plastic analysis of tube bending. Int J Mach Tool Manuf 1999;39:87–104.
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