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Effect of normal anisotropy on springback
Journal of Materials Processing Technology 190 (2007) 300–304
Effect of normal anisotropy on springback Rahul K. Verma ∗ , A. Haldar Research and Development Division, Tata Steel Ltd., Jamshedpur 831001, India Received 7 April 2006; received in revised form 23 October 2006; accepted 20 February 2007
Abstract Share of high and advanced high strength steels in automobile is increasing, however, such steels generally have poor formability and high amount of springback. One of the focus areas of research in high strength automotive steel is to increase the normal anisotropy to get better formability. Effect of strength and process parameters on springback has been studied by many researchers but that of anisotropy has not been studied by many. In the present work the effect of anisotropy on springback is predicted using finite element analysis for the benchmark problem of Numisheet-2005 [2005 Numisheet Benchmark 2, Springback prediction of a cross member, Proceedings of the 6th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, Detroit, USA, August 15–19, 2005]. An analytical model is developed to cross check the trends predicted from the finite element analysis. The effective stress has not been treated as a constant and the radial stress is considered in the present model. Both the models (FE and analytical) predict that higher anisotropy, in general, gives higher springback. Finite element analysis of the problem shows that springback is minimum for an isotropic material. © 2007 Elsevier B.V. All rights reserved. Keywords: Springback; Normal anisotropy; FEA; High strength steels
1. Introduction High and advanced high strength steels are finding wide acceptance in the automotive industry. However, one of the major problems in stamping automotive parts with high strength steel sheets is the increased levels of springback. In springback, there are two issues: • Control of springback during forming and • Prediction of springback so that it is compensated during dies design stage. Various researchers developed a variety of manufacturing techniques to control the springback. Sunseri et al. [1] showed that a variable binder force history during forming operation can reduce the springback amount while maintaining a relatively low maximum strain. In their study an initial low binder force followed by a higher binder force was used. Ruffini and Cao [2] and Cao et al. [3] proposed neural network based models to minimize the springback in a channel forming process. It was shown that even for large variations in friction condition and material ∗
Corresponding author. E-mail address: [email protected] (R.K. Verma).
0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.02.033
properties the springback is quite low. Cao et al. [3] showed that a stepped binder force can reduce the springback significantly. Two critical values, high binder force and percent punch of total punch displacement (the outputs from the neural network) are used for the process control. In their study, the material was assumed to be isotropic, elasto-plastic following the von-Mises yield criterion. Hardening was assumed to be isotropic. Though control of springback is important, the prediction of it is desirable. Both analytical and numerical models have been developed for the prediction. Zhang and Lin [4] proposed an analytical solution for springback in components stamped by a rigid punch and an elastic die whereas, Morestin et al. [5] proposed a model in which the Prandtl–Reuss plasticity equations associated with a nonlinear kinematic hardening model was solved. The calculation takes into account the change in Young’s modulus with plastic strain. Liu [6] proposed a simple model for predicting the springback and bendability. He considered the normal anisotropy and strain-hardening exponent in his model, however, the effective stress is considered to be constant through out the bending. His prediction showed that with increase in normal anisotropy the springback increases but did not give any reason. Wang et al. [7] established a mathematical solution for plane-strain bending of sheet and plate. Using their model they predicted that bending moment and, therefore, the springback increases with increase
R.K. Verma, A. Haldar / Journal of Materials Processing Technology 190 (2007) 300–304
in strength, strain-hardening and normal anisotropy. They used Hill’s (1979) non-quadratic yield criteria but did not consider the radial stress and, therefore, their model may not be suitable for thick sheets or in severe bending condition. Pourboghrat and Chu [8] used a moment–curvature relationship derived for sheets undergoing plane-strain stretching, bending and unbending deformations. They used membrane finite element method with bending/unbending corrections to calculate springback. The use of membrane element in the solution reduced CPU time considerably. Though the analytical models are simple and easy to use for parametric studies, these in general do not represent the real condition. Numerical methods like FEM are frequently used to solve the real life problems. Samual [9] provided an FE program for predicting springback and sidewall curl whereas Lee and Yang [10] evaluated quantitatively the numerical factors influencing the springback prediction. Effects of contact damping parameter, penalty parameter, blank element size, number of corner elements and punch velocity were evaluated using the Taguchi method and found that the number of corner elements and the blank element size are the main factors influencing the springback prediction. It was concluded by them that kinematic hardening should be included for more realistic simulations. Gomes et al. [11] investigated the springback in simple Ushape from Numisheet-93 using three different material models; Barlat’s yield criteria, Hill’s transverse anisotropic model and von-Mises’ yield criteria. They found discrepancies between the results obtained from the different material models. Ragai et al. [12] studied the effect of sheet anisotropy on springback in draw bending of stainless steel 410 numerical methods and experiments. Though many techniques were proposed, its prediction is still a challenging issue. Many researchers have reported that poor representation of Bauschinger effect is one of the main reasons for poor springback prediction accuracy. In a recent study [13] it was shown that the stress paths of a material point moving over a radius can be quite different when using an isotropic or a kinematic hardening model. The stress magnitudes are similar at the beginning and the end of the stamping but the stress state is actually quite different. Gau and Kinzel [14] proposed a bending experiment to study the effect of Bauschinger effect on springback in aluminium and concluded that the total strain method is not sufficient to model springback when sheet undergoes cyclic deformation. In another work Gau and Kinzel [15] proposed a new hardening model in which they assumed that actuated surface both translates and expands during deformation. All surfaces within the actuated surface have a rigid body translation as described in Mroz method. Kim et al. [16] did the measurement of anisotropy, Bauschinger effect and transient behaviour of automotive dual phase steels. For the anisotropy measurement, non-quadratic anisotropic yield function Yld2000-2d has been utilized and its material parameters have been obtained using the uni-axial tension tests as well as the hydraulic bulge test. To measure the hardening behaviour including the Bauschinger and transient behaviour, they proposed a new tension and compression test. It was concluded that the Chaboche model well represented the Bauschinger effect and the transient behaviour.
301
Most of the FE simulations used for sheet metal forming prediction assume that springback is purely elastic and linear. However, Cleveland and Ghosh [17] showed that the unloading is not linear. It was demonstrated that the inelastic strain released from the formed state can be a major source of additional strain recovery, the magnitude of which depends on the forming stress state. It has been shown by them that 10–20% error in the estimation of springback is possible if the anelastic effects during loading and unloading, is ignored. Such effects arise from mobile dislocations which can move in response to the internal repulsive forces. Many works have so far been done on prediction and control of the springback. However, there are a few literatures dealing with the effects of material properties other than strength. Material scientists are working towards increasing the normal anisotropy by having a favourable texture for better formability, and therefore, it is required to know the effect of normal anisotropy on springback. In the present work the effect of normal anisotropy on the amount of springback is evaluated for the benchmark problem of Numisheet-2005 [18]. An analytical model for plane-strain bending is also developed and used for assessing the effect of various parameters, including normal anisotropy, on springback. In the present analytical model radial stress has been considered and the effective stress is not treated as a constant value (unlike [6,7]). Also this model provides a closed form solution. 2. Analysis In the present model the material is assumed to be rigid plastic and strain-hardening with normal anisotropy. Bauschinger effect is neglected and Holloman’s equation is used for modelling the hardening. The neutral axis is assumed to coincide with the geometrical mid plane of the sheet. Fig. 1 shows the stresses acting on a small element in the deforming sheet. The force equilibrium equation for the small element making an angle dθ at the centre is, dσr σθ − σr = dr r
(1)
Using the Hills’s yield criteria and associated flow rule the expressions for equivalent stress and strain can be written as
Fig. 1. State of stress on a small material element in pure bending.
302
R.K. Verma, A. Haldar / Journal of Materials Processing Technology 190 (2007) 300–304 Table 1 Material properties used for the benchmark problem
below: ¯ 1+R dεeq = √ dε = C dεr , ¯ r 1 + 2R
¯ 1+R C= √ ¯ 1 + 2R
σθ − σr = Cσeq
(2) (3)
where σeq = Kεneq and for plane-strain condition dεr = −dεθ . Using Eqs. (2) and (3) in (1) and integrating from neutral line to the outer most fibre (2t is the sheet thickness), n t t R+r dr dr n+1 dσr = ln Cσeq = KC (4) r R r r r Using logarithmic series expansion and neglecting higher order terms (as the ratio r/R is small), the above expression can be written as n t r dr r2 n+1 − (5) dσr = KC 2 R 2R r r Using binomial series expansion in Eq. (5) and neglecting higher order terms, the above expression can be written as t nr r n n(n − 1)r 2 dr n+1 dσr =KC 1− + (6) R 2R 8R2 r r Integrating the above expression and using the limit σ r = 0 at r = t and σ r = σ r at r = r. Sheet thickness is 2t. n n 1 r − tn n(r n+1 − t n+1 ) K − σr = Cn+1 R n 2(n + 1)R
n(n − 1)(r n+2 − t n+2 ) + (7) 8(n + 2)R2 The moment can be calculated by integrating the following equation: t σθ r dr M = 2w 0
t r n σr + Cn+1 K ln 1 + r dr = 2w R 0 I1 = 0
t
σr r dr,
I2 = 0
t
(8)
r n Cn+1 K ln 1 + r dr R
Yield stress (MPa) Stress constant (K) (MPa) Strain-hardening exponent (n) Average normal anisotropy Coefficient of friction
403.8 1040.4 0.159 0.6–2.0 0.12
Here R is the radius before unloading and R* is the radius after unloading. Equating Eqs. (8) and (10) the change in curvature can be obtained as below: 1 3(1 − ν2 ) 1 [I1 + I2 ] (11) − ∗ = R R Et 3 where I1 and I2 are defined as above. As Rθ = R*θ* the springback ratio θ/θ can be expressed by Rx(1/R − 1/R*). 3. Results and discussions Finite element simulation for the benchmark problem of Numisheet-2005 [18] is carried out using the commercial software Pam Stamp. Reason for selecting this problem is to simulate a condition having a more realistic and severe strain path. The default numerical parameters were used in the simulation. The Hill’s material model with normal anisotropy and planer isotropy is used in this simulation. The hardening is assumed to be isotropic. The material properties used for this analysis is given in Table 1. Fig. 2 shows the comparison of the present prediction with the experimental values [18]. The springback is plotted for Section 4 of the benchmark [18]. It can be seen that there is a good agreement between the prediction and the experiment. After this validation the same set-up (for simulations) is used to predict the effect of normal anisotropy on springback. The springback is taken at the location of maximum value. Fig. 3 shows the effect of normal anisotropy on springback. It is to be noted that with increase in anisotropy the springback amount first decreases and then increases. The analysis shows that springback amount is the minimum for isotropic case (average normal anisotropy equal to 1).
The above expressions can be simplified (using binomial and logarithmic series expansions as above) as below, 1 n t t n+2 I1 + I2 = Cn+1 K n − 2R n + 2 2(n + 3) R
n(n − 1) t 2 + (9) 8(n + 4) R2 As proposed in [6], the elastic strain recovery after unloading causes the springback. On assuming that unloading moment M having the same magnitude but opposite sign to that of applied bending moment M. The unloading moment M is given as [6], 1 1 2wEt 3 (10) − M = 3(1 − ν2 ) R R∗
Fig. 2. Comparison between finite element prediction and the experiment for the benchmark problem of Numisheet-2005 [18].
R.K. Verma, A. Haldar / Journal of Materials Processing Technology 190 (2007) 300–304
Fig. 3. Effect of normal anisotropy on springback for the benchmark problem of Numisheet-2005 [18].
To confirm the effect of normal anisotropy on the amount of springback the above mathematical model was used. Fig. 4 shows the effect of normal anisotropy on springback for different bend radii. This is consistent with the prediction of Liu [6]; however, it is observed that springback is monotonously increasing with normal anisotropy and is not the minimum for isotropic material. The reason for the disagreement could be, either the strain paths are not same and/or analytical model may not be good enough for springback prediction. Parametric studies are carried out to see the effect of strain-hardening exponent, strength coefficient, bend radius and sheet thickness. The effect of bend radius for different strainhardening exponents (on absolute change in curvature) is shown in Fig. 5. It shows, with increase in bend radius springback decreases. This happens because higher bend radius means less severe bending and less plastic deformation. However, if a graph is plotted between the percent changes in curvature versus bend radius then the trend is reversed. The reason for this reversed trend is the fact that for smaller total strain the ratio of elastic to plastic strain is higher. Fig. 6 shows the effect of sheet thickness whereas Fig. 7 shows the effect of strainhardening exponent on springback for different bend radii. With increase in thickness and strain-hardening exponent springback
Fig. 4. Effect of normal anisotropy on springback. Using analytical model (σ = 662.0ε0.25 , YS = 197 MPa, t = 1.0 mm).
303
Fig. 5. Effect of bend radius on springback. Using analytical model ¯ = 1.5). (σ = 662.0ε0.25 , YS = 197 MPa, t = 1.0 mm, R
Fig. 6. Effect of sheet thickness on springback. Using analytical model ¯ = 1.5, R = 5.0 mm). (σ = 662.0ε0.25 , YS = 197 MPa, R
Fig. 7. Effect of strain-hardening exponent on springback. Using analytical ¯ = 1.5, R = 5.0 mm). model (σ = 662.0ε0.25 , YS = 197 MPa, R
304
R.K. Verma, A. Haldar / Journal of Materials Processing Technology 190 (2007) 300–304
decreases. It can be seen that for higher bend radius the effect is more. 4. Conclusions Researchers worldwide are trying to improve the anisotropy by improving on the texture in the high strength steel sheets for better formability. In the present work the effect of anisotropy on springback amount is predicted for the benchmark problem of Numisheet-2005 [18]. An analytical model is developed to cross check the prediction from the finite element analysis. Both the models predict that higher anisotropy is not good for springback. Finite element analysis of the problem shows that springback is the minimum for an isotropic material, however, it does not agree with the analytical model prediction for normal anisotropy less than 1. Acknowledgements The authors gratefully acknowledge the permission from Tata Steel Ltd. to publish the present work. They also thank Dr. N. Gope for his constant support during this work. References [1] M. Sunseri, J. Cao, A.P. Karafillis, M.C. Boyce, Accommodation of springback in channel forming using active binder force control: numerical simulations and experiments, J. Eng. Mater. Technol., Trans. ASME 118 (1996) 426–435. [2] R. Ruffini, J. Cao, Using neural for springback minimization in a channel forming process, J. Mater. Manuf. 107 (1998) 65–73. [3] J. Cao, B. Kinsey, S.A. Solla, Consistent and minimal springback using stepped binder force trajectory and neural network control, J. Eng. Mater. Tech., Trans. ASME 122 (1998) 65–73. [4] L.C. Zhang, Z. Lin, An analytical solution to springback of sheet metals stamped by a rigid punch and an elastic die, J. Mater. Process. Technol. 63 (1997) 49–54.
[5] F. Morestin, M. Boivin, C. Silva, Elasto plastic formulation using a kinematic hardening model for springback analysis in sheet metal forming, J. Mater. Process. Technol. 56 (1996) 619–630. [6] D.K. Liu, A simplified approach for evaluating bendability and springback in plastic bending of anisotropic sheet metals, J. Mater. Process. Technol. 66 (1997) 9–17. [7] C. Wang, G. Kinzel, T. Altan, Mathematical modeling of plane-strain bending of sheet and plate, J. Mater. Process. Technol. 39 (1993) 279–304. [8] F. Pourboghrat, E. Chu, Prediction of springback and side-wall curl in 2-D draw bending, J. Mater. Process. Technol. 50 (1995) 361–374. [9] M. Samual, Experimental and numerical prediction of springback and side wall curl in U-bending of anisotropic sheet metals, J. Mater. Process. Technol. 105 (2000) 382–393. [10] S.W. Lee, D.Y. Yang, An influence of numerical parameters influencing springback in explicit finite element analysis of sheet metal forming process, J. Mater. Process. Technol. 80/81 (1998) 60–67. [11] C. Gomes, O. Onipede, M. Lovell, Investigation of springback in high strength anisotropic steels, J. Mater. Process. Technol. 159 (2005) 91–98. [12] I. Ragai, D. Lazim, J.A. Nemes, Anisotropy and springback in drawbending of stainless steel 410: experimental and numerical study, J. Mater. Process. Technol. 166 (2005) 116–127. [13] Gino A. Duffet, L. Neamtu, S. Oliveira, R. Weyler, The study of various parameters in the numerical simulation of springback, http://www.nafems.org (2004) 28–29. [14] J.T. Gau, G.L. Kinzel, An experimental investigation of the influence of the Bauschinger effect on springback prediction, J. Mater. Process. Technol. 108 (2001) 369–375. [15] J.T. Gau, G.L. Kinzel, A new model for springback prediction in which the Bauschinger effect is considered, Int. J. Mech. Sci. 43 (2001) 1813– 1832. [16] D. Kim, M.G. Lee, C. Kim, M.L. Wenner, R.H. Wagoner, F. Barlat, K.S. Chung, J.R. Youn, T.J. Kang, Measurement of anisotropy yielding, Bauschinger and transient behaviour of automotive dual phase steel sheets, Met. Mater. Int. 9 (2003) 561–570. [17] R.M. Cleveland, A.K. Ghosh, Inelastic effects on springback in metals, Int. J. Plasticity 18 (2001) 769–785. [18] 2005 Numisheet Benchmark 2, Springback prediction of a cross member, Proceedings of the 6th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, Detroit, USA, August 15–19, 2005.
Effect of normal anisotropy on springback Rahul K. Verma ∗ , A. Haldar Research and Development Division, Tata Steel Ltd., Jamshedpur 831001, India Received 7 April 2006; received in revised form 23 October 2006; accepted 20 February 2007
Abstract Share of high and advanced high strength steels in automobile is increasing, however, such steels generally have poor formability and high amount of springback. One of the focus areas of research in high strength automotive steel is to increase the normal anisotropy to get better formability. Effect of strength and process parameters on springback has been studied by many researchers but that of anisotropy has not been studied by many. In the present work the effect of anisotropy on springback is predicted using finite element analysis for the benchmark problem of Numisheet-2005 [2005 Numisheet Benchmark 2, Springback prediction of a cross member, Proceedings of the 6th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, Detroit, USA, August 15–19, 2005]. An analytical model is developed to cross check the trends predicted from the finite element analysis. The effective stress has not been treated as a constant and the radial stress is considered in the present model. Both the models (FE and analytical) predict that higher anisotropy, in general, gives higher springback. Finite element analysis of the problem shows that springback is minimum for an isotropic material. © 2007 Elsevier B.V. All rights reserved. Keywords: Springback; Normal anisotropy; FEA; High strength steels
1. Introduction High and advanced high strength steels are finding wide acceptance in the automotive industry. However, one of the major problems in stamping automotive parts with high strength steel sheets is the increased levels of springback. In springback, there are two issues: • Control of springback during forming and • Prediction of springback so that it is compensated during dies design stage. Various researchers developed a variety of manufacturing techniques to control the springback. Sunseri et al. [1] showed that a variable binder force history during forming operation can reduce the springback amount while maintaining a relatively low maximum strain. In their study an initial low binder force followed by a higher binder force was used. Ruffini and Cao [2] and Cao et al. [3] proposed neural network based models to minimize the springback in a channel forming process. It was shown that even for large variations in friction condition and material ∗
Corresponding author. E-mail address: [email protected] (R.K. Verma).
0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.02.033
properties the springback is quite low. Cao et al. [3] showed that a stepped binder force can reduce the springback significantly. Two critical values, high binder force and percent punch of total punch displacement (the outputs from the neural network) are used for the process control. In their study, the material was assumed to be isotropic, elasto-plastic following the von-Mises yield criterion. Hardening was assumed to be isotropic. Though control of springback is important, the prediction of it is desirable. Both analytical and numerical models have been developed for the prediction. Zhang and Lin [4] proposed an analytical solution for springback in components stamped by a rigid punch and an elastic die whereas, Morestin et al. [5] proposed a model in which the Prandtl–Reuss plasticity equations associated with a nonlinear kinematic hardening model was solved. The calculation takes into account the change in Young’s modulus with plastic strain. Liu [6] proposed a simple model for predicting the springback and bendability. He considered the normal anisotropy and strain-hardening exponent in his model, however, the effective stress is considered to be constant through out the bending. His prediction showed that with increase in normal anisotropy the springback increases but did not give any reason. Wang et al. [7] established a mathematical solution for plane-strain bending of sheet and plate. Using their model they predicted that bending moment and, therefore, the springback increases with increase
R.K. Verma, A. Haldar / Journal of Materials Processing Technology 190 (2007) 300–304
in strength, strain-hardening and normal anisotropy. They used Hill’s (1979) non-quadratic yield criteria but did not consider the radial stress and, therefore, their model may not be suitable for thick sheets or in severe bending condition. Pourboghrat and Chu [8] used a moment–curvature relationship derived for sheets undergoing plane-strain stretching, bending and unbending deformations. They used membrane finite element method with bending/unbending corrections to calculate springback. The use of membrane element in the solution reduced CPU time considerably. Though the analytical models are simple and easy to use for parametric studies, these in general do not represent the real condition. Numerical methods like FEM are frequently used to solve the real life problems. Samual [9] provided an FE program for predicting springback and sidewall curl whereas Lee and Yang [10] evaluated quantitatively the numerical factors influencing the springback prediction. Effects of contact damping parameter, penalty parameter, blank element size, number of corner elements and punch velocity were evaluated using the Taguchi method and found that the number of corner elements and the blank element size are the main factors influencing the springback prediction. It was concluded by them that kinematic hardening should be included for more realistic simulations. Gomes et al. [11] investigated the springback in simple Ushape from Numisheet-93 using three different material models; Barlat’s yield criteria, Hill’s transverse anisotropic model and von-Mises’ yield criteria. They found discrepancies between the results obtained from the different material models. Ragai et al. [12] studied the effect of sheet anisotropy on springback in draw bending of stainless steel 410 numerical methods and experiments. Though many techniques were proposed, its prediction is still a challenging issue. Many researchers have reported that poor representation of Bauschinger effect is one of the main reasons for poor springback prediction accuracy. In a recent study [13] it was shown that the stress paths of a material point moving over a radius can be quite different when using an isotropic or a kinematic hardening model. The stress magnitudes are similar at the beginning and the end of the stamping but the stress state is actually quite different. Gau and Kinzel [14] proposed a bending experiment to study the effect of Bauschinger effect on springback in aluminium and concluded that the total strain method is not sufficient to model springback when sheet undergoes cyclic deformation. In another work Gau and Kinzel [15] proposed a new hardening model in which they assumed that actuated surface both translates and expands during deformation. All surfaces within the actuated surface have a rigid body translation as described in Mroz method. Kim et al. [16] did the measurement of anisotropy, Bauschinger effect and transient behaviour of automotive dual phase steels. For the anisotropy measurement, non-quadratic anisotropic yield function Yld2000-2d has been utilized and its material parameters have been obtained using the uni-axial tension tests as well as the hydraulic bulge test. To measure the hardening behaviour including the Bauschinger and transient behaviour, they proposed a new tension and compression test. It was concluded that the Chaboche model well represented the Bauschinger effect and the transient behaviour.
301
Most of the FE simulations used for sheet metal forming prediction assume that springback is purely elastic and linear. However, Cleveland and Ghosh [17] showed that the unloading is not linear. It was demonstrated that the inelastic strain released from the formed state can be a major source of additional strain recovery, the magnitude of which depends on the forming stress state. It has been shown by them that 10–20% error in the estimation of springback is possible if the anelastic effects during loading and unloading, is ignored. Such effects arise from mobile dislocations which can move in response to the internal repulsive forces. Many works have so far been done on prediction and control of the springback. However, there are a few literatures dealing with the effects of material properties other than strength. Material scientists are working towards increasing the normal anisotropy by having a favourable texture for better formability, and therefore, it is required to know the effect of normal anisotropy on springback. In the present work the effect of normal anisotropy on the amount of springback is evaluated for the benchmark problem of Numisheet-2005 [18]. An analytical model for plane-strain bending is also developed and used for assessing the effect of various parameters, including normal anisotropy, on springback. In the present analytical model radial stress has been considered and the effective stress is not treated as a constant value (unlike [6,7]). Also this model provides a closed form solution. 2. Analysis In the present model the material is assumed to be rigid plastic and strain-hardening with normal anisotropy. Bauschinger effect is neglected and Holloman’s equation is used for modelling the hardening. The neutral axis is assumed to coincide with the geometrical mid plane of the sheet. Fig. 1 shows the stresses acting on a small element in the deforming sheet. The force equilibrium equation for the small element making an angle dθ at the centre is, dσr σθ − σr = dr r
(1)
Using the Hills’s yield criteria and associated flow rule the expressions for equivalent stress and strain can be written as
Fig. 1. State of stress on a small material element in pure bending.
302
R.K. Verma, A. Haldar / Journal of Materials Processing Technology 190 (2007) 300–304 Table 1 Material properties used for the benchmark problem
below: ¯ 1+R dεeq = √ dε = C dεr , ¯ r 1 + 2R
¯ 1+R C= √ ¯ 1 + 2R
σθ − σr = Cσeq
(2) (3)
where σeq = Kεneq and for plane-strain condition dεr = −dεθ . Using Eqs. (2) and (3) in (1) and integrating from neutral line to the outer most fibre (2t is the sheet thickness), n t t R+r dr dr n+1 dσr = ln Cσeq = KC (4) r R r r r Using logarithmic series expansion and neglecting higher order terms (as the ratio r/R is small), the above expression can be written as n t r dr r2 n+1 − (5) dσr = KC 2 R 2R r r Using binomial series expansion in Eq. (5) and neglecting higher order terms, the above expression can be written as t nr r n n(n − 1)r 2 dr n+1 dσr =KC 1− + (6) R 2R 8R2 r r Integrating the above expression and using the limit σ r = 0 at r = t and σ r = σ r at r = r. Sheet thickness is 2t. n n 1 r − tn n(r n+1 − t n+1 ) K − σr = Cn+1 R n 2(n + 1)R
n(n − 1)(r n+2 − t n+2 ) + (7) 8(n + 2)R2 The moment can be calculated by integrating the following equation: t σθ r dr M = 2w 0
t r n σr + Cn+1 K ln 1 + r dr = 2w R 0 I1 = 0
t
σr r dr,
I2 = 0
t
(8)
r n Cn+1 K ln 1 + r dr R
Yield stress (MPa) Stress constant (K) (MPa) Strain-hardening exponent (n) Average normal anisotropy Coefficient of friction
403.8 1040.4 0.159 0.6–2.0 0.12
Here R is the radius before unloading and R* is the radius after unloading. Equating Eqs. (8) and (10) the change in curvature can be obtained as below: 1 3(1 − ν2 ) 1 [I1 + I2 ] (11) − ∗ = R R Et 3 where I1 and I2 are defined as above. As Rθ = R*θ* the springback ratio θ/θ can be expressed by Rx(1/R − 1/R*). 3. Results and discussions Finite element simulation for the benchmark problem of Numisheet-2005 [18] is carried out using the commercial software Pam Stamp. Reason for selecting this problem is to simulate a condition having a more realistic and severe strain path. The default numerical parameters were used in the simulation. The Hill’s material model with normal anisotropy and planer isotropy is used in this simulation. The hardening is assumed to be isotropic. The material properties used for this analysis is given in Table 1. Fig. 2 shows the comparison of the present prediction with the experimental values [18]. The springback is plotted for Section 4 of the benchmark [18]. It can be seen that there is a good agreement between the prediction and the experiment. After this validation the same set-up (for simulations) is used to predict the effect of normal anisotropy on springback. The springback is taken at the location of maximum value. Fig. 3 shows the effect of normal anisotropy on springback. It is to be noted that with increase in anisotropy the springback amount first decreases and then increases. The analysis shows that springback amount is the minimum for isotropic case (average normal anisotropy equal to 1).
The above expressions can be simplified (using binomial and logarithmic series expansions as above) as below, 1 n t t n+2 I1 + I2 = Cn+1 K n − 2R n + 2 2(n + 3) R
n(n − 1) t 2 + (9) 8(n + 4) R2 As proposed in [6], the elastic strain recovery after unloading causes the springback. On assuming that unloading moment M having the same magnitude but opposite sign to that of applied bending moment M. The unloading moment M is given as [6], 1 1 2wEt 3 (10) − M = 3(1 − ν2 ) R R∗
Fig. 2. Comparison between finite element prediction and the experiment for the benchmark problem of Numisheet-2005 [18].
R.K. Verma, A. Haldar / Journal of Materials Processing Technology 190 (2007) 300–304
Fig. 3. Effect of normal anisotropy on springback for the benchmark problem of Numisheet-2005 [18].
To confirm the effect of normal anisotropy on the amount of springback the above mathematical model was used. Fig. 4 shows the effect of normal anisotropy on springback for different bend radii. This is consistent with the prediction of Liu [6]; however, it is observed that springback is monotonously increasing with normal anisotropy and is not the minimum for isotropic material. The reason for the disagreement could be, either the strain paths are not same and/or analytical model may not be good enough for springback prediction. Parametric studies are carried out to see the effect of strain-hardening exponent, strength coefficient, bend radius and sheet thickness. The effect of bend radius for different strainhardening exponents (on absolute change in curvature) is shown in Fig. 5. It shows, with increase in bend radius springback decreases. This happens because higher bend radius means less severe bending and less plastic deformation. However, if a graph is plotted between the percent changes in curvature versus bend radius then the trend is reversed. The reason for this reversed trend is the fact that for smaller total strain the ratio of elastic to plastic strain is higher. Fig. 6 shows the effect of sheet thickness whereas Fig. 7 shows the effect of strainhardening exponent on springback for different bend radii. With increase in thickness and strain-hardening exponent springback
Fig. 4. Effect of normal anisotropy on springback. Using analytical model (σ = 662.0ε0.25 , YS = 197 MPa, t = 1.0 mm).
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Fig. 5. Effect of bend radius on springback. Using analytical model ¯ = 1.5). (σ = 662.0ε0.25 , YS = 197 MPa, t = 1.0 mm, R
Fig. 6. Effect of sheet thickness on springback. Using analytical model ¯ = 1.5, R = 5.0 mm). (σ = 662.0ε0.25 , YS = 197 MPa, R
Fig. 7. Effect of strain-hardening exponent on springback. Using analytical ¯ = 1.5, R = 5.0 mm). model (σ = 662.0ε0.25 , YS = 197 MPa, R
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decreases. It can be seen that for higher bend radius the effect is more. 4. Conclusions Researchers worldwide are trying to improve the anisotropy by improving on the texture in the high strength steel sheets for better formability. In the present work the effect of anisotropy on springback amount is predicted for the benchmark problem of Numisheet-2005 [18]. An analytical model is developed to cross check the prediction from the finite element analysis. Both the models predict that higher anisotropy is not good for springback. Finite element analysis of the problem shows that springback is the minimum for an isotropic material, however, it does not agree with the analytical model prediction for normal anisotropy less than 1. Acknowledgements The authors gratefully acknowledge the permission from Tata Steel Ltd. to publish the present work. They also thank Dr. N. Gope for his constant support during this work. References [1] M. Sunseri, J. Cao, A.P. Karafillis, M.C. Boyce, Accommodation of springback in channel forming using active binder force control: numerical simulations and experiments, J. Eng. Mater. Technol., Trans. ASME 118 (1996) 426–435. [2] R. Ruffini, J. Cao, Using neural for springback minimization in a channel forming process, J. Mater. Manuf. 107 (1998) 65–73. [3] J. Cao, B. Kinsey, S.A. Solla, Consistent and minimal springback using stepped binder force trajectory and neural network control, J. Eng. Mater. Tech., Trans. ASME 122 (1998) 65–73. [4] L.C. Zhang, Z. Lin, An analytical solution to springback of sheet metals stamped by a rigid punch and an elastic die, J. Mater. Process. Technol. 63 (1997) 49–54.
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