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Analysis of the effect of material properties on the hydroforming process of tubes
Journal of Materials Processing Technology 104 (2000) 158±166
Analysis of the effect of material properties on the hydroforming process of tubes B. Carleer, G. van der Kevie, L. de Winter*, B. van Veldhuizen Corus Research, Development and Technology, CR-YTC-PRA-AUT-MEP, PO Box 10000 3H36, Ijmuiden, CA 1970, USA Accepted 23 May 2000
Abstract The free-expansion hydroforming of tubes is discussed. Different steel grades have been studied by means of experiments, FEM simulations and a simple analytical model. The effect of different material parameters, such as the r-value and the n-value, on the hydroformability of tubes is discussed. Applying the analytical model, the favourable shape of a free-expanded tube is predicted and a ranking of different grades based on this model is proposed. # 2000 Published by Elsevier Science S.A. Keywords: Hydroforming process; FEM simulation; Analytical model
1. Introduction In tube hydroforming, internal pressure is applied to modify a tubular metal blank into a structural component having a closed cross-section. Tube hydroforming offers a way to cut material and manufacturing costs, while improving product performance in a variety of applications. Existing multi-piece stamped/welded assemblies in car body and frame structures often can be replaced with less expensive hydroformed parts that are lighter, stronger and more accurate [1±3]. In order to achieve the most economic product with the best performance, it is important to choose proper material and process selection. As tube hydroforming is a relatively new technology compared to conventional stamping, there is not much knowledge available that can be used in selecting proper material and process selection [4,5]. At Corus, research on hydroforming includes process, material and product development. Parameters that can in¯uence the process are material grade, friction, tool design and product design [6±8]. A way to determine the material parameters which are most important to a hydroform suitable steel is the free expansion of tubes. Experiments and FE simulations have been performed to gain insight into the free-expansion behaviour of various steel grades. On this basis, an analytical model has been developed to quickly
* Corresponding author. Tel.: 1-31-2514-94323. E-mail address: [email protected] (L. de Winter)
0924-0136/00/$ ± see front matter # 2000 Published by Elsevier Science S.A. PII: S 0 9 2 4 - 0 1 3 6 ( 0 0 ) 0 0 5 3 0 - 6
analyse the in¯uence of the various material parameters during free expansion. 2. Experiments Tubes made of various steel grades have been expanded applying the hydroform process together with PtU Darmstadt. The in¯uence of the material parameters on the deformation characteristics was determined. Steel grades ranging from high strength to low strength steels were used. The corresponding material parameters t (thickness), n (hardening exponent), Rm (maximum stress), Rp,0.2 (yield stress at 0.2% strain) and r (plastic anisotropy parameter) are given in Table 1. In the experiments, tubes 450 mm long and with a maximum external diameter of 55 mm were laid in a die and sealed with plungers at both ends. The tubes were ®lled with water through the plungers and as the pressure was raised, the tubes expanded until rupture. Over a length of 270 mm, the tube was able to expand freely. Only a small axial force was applied on the ends of the tubes in order to seal the ends. The axial displacements of the deformed tubes in the experiments were measured. The gridded tubes were measured at Hoogovens using the Camsys measuring system [9] to determine the strain distribution. The maximum expanded diameter of the tube was measured as well. Fig. 1 shows Camsys strain measurements and Fig. 2 shows an example of bursted tubes.
B. Carleer et al. / Journal of Materials Processing Technology 104 (2000) 158±166
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Table 1 Material parameters Material
Thickness n (mm)
Rm (MPa)
Rp,0.2 (MPa)
r
DC06 DC06 GA IF P 220 GI IF P 260 GI SG350 GD GI YMPRESS E355
1.0 0.96 0.915 0.905 0.98 1.25
295 283 365 409 431 517
150 155 243 274 369 408
2.35 2.04 2.18 1.76 1.29 1.02
0.23 0.21 0.20 0.19 0.184 0.142
The following conclusions were drawn from the experiments: The tubes made of both galvanized and galvannealed materials with low yield strength, high n- and r-values showed the largest major and minor strains. A larger axial displacement was noticed for materials with low yield stress as less pressure was required to deform the tubes resulting in lower normal forces between the tube and the die and consequently also lower frictional forces restraining the material from flowing towards the centre.
3. Simulations Next to the experiments, simulations have been performed of the free expansion using the FE programme PamStamp [10]. The simulations were veri®ed by compar-
Fig. 2. Ruptured expanded tubes (IF P 220).
ing the achieved maximum major and minimum minor strain to those measured by Camsys. Since we have not used a material model incorporating damage, the PamStamp simulation for the various steel grades was stopped when the maximum deformed diameter measured in the experiment was reached. By taking a reference material and then varying the process and/or material parameters, the in¯uence on the formability of the material was investigated. The parameters
Fig. 1. Measured strain distribution for a DC06 GA grade.
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Fig. 5. Tube pro®le for materials with different n-values.
are the r-value of the material (included in Hill's yielding model), the coef®cient of friction m between the die and the tube, and the thickness t of the tubes. Every parameter was varied and the in¯uence on the strain distribution and strain path was investigated. The ®nal crosssection of the tubes was analysed as well. The following conclusions were drawn: Fig. 3. Geometry of a free-expanded tube for different r-values.
that were varied were C (hardening coef®cient), r (anisotropy parameter), e0 (prestrain), n (hardening coef®cient), t (thickness) and m (friction coef®cient). The tubes were meshed by means of Shells and the internal pressure was increased by using ¯uid cells. The hardening model used in the simulations to represent cold worked material is a Ludwik±Nadai equation that relates the plastic equivalent strain e to the equivalent stress s: s C e0 en : Three other parameters not represented by the hardening model but affecting the deformation characteristics as well
A higher r- and n-value and a lower m given to a tube with less local deformation is shown in Fig. 3. Consequently, the strain is also more evenly distributed and will favour larger deformations. This figure also shows that the tube with r3 shortens more than the tube with r1 due to more material flow towards the middle during the forming process. The same is seen for the influence of the n-value and m. This effect is more clearly shown for the n- and rvalue in Figs. 4 and 5. The only parameters that influence the strain distribution in the forming diagram are the r-value and m. This is illustrated in Figs. 6 and 7. As a higher r-value or lower m lowers the slope of the strain path in the forming diagram, it will cross the forming limit curve at higher negative
Fig. 4. Tube pro®le for materials with different r-values.
B. Carleer et al. / Journal of Materials Processing Technology 104 (2000) 158±166
161
These simulations show that mainly a higher n- and rvalue are favourable to the ``hydroformability'' of steel and that the friction should be kept as low as possible. 4. Analytical model
Fig. 6. Strain distribution for free expansion of a tube with different rvalues.
Fig. 7. Strain distribution for free expansion of a tube with different values for the friction coef®cient.
minor and positive major strains. A straight strain path is seen in the simulations. Thus higher strains before failing may be reached. C, e0, t and n-values have hardly any influence on the strain distribution in the forming diagram and the strain path of all the forming diagrams cross the forming limit diagram at the same major and minor strain.
Fig. 8. Pro®le of half of a free-expanded tube. The local co-ordinate system is denoted by l (longitudinal) and t (thickness). The circumferential direction is perpendicular to the paper.
The results of FEM simulations showed that the strain paths during free expanding are reasonably straight (the deviation from the straight path for r1 as shown in Fig. 6 is actually due to global necking). Furthermore, the system is already on principal axes (the longitudinal, circumferential and thickness axes, Fig. 8). In the absence of friction, the ratio of the minor to major strain is given by ÿr= 1 r. In the situation where the ends are fully clamped, plane strain conditions would exist. Plane strain conditions were assumed before to study the onset of bursting during tubular hydroforming [8]. Friction has the effect of rotating the strain path from this uniaxial direction towards the planestrain direction. Therefore we introduce a friction parameter a to account for this deviation from uniaxiality. The ratio of the minor to major strain is now given by the factor ÿar= 1 r, and the smaller the a is, the higher is the friction. There is a ®xed relationship between a and the friction coef®cient m. Since the actual friction coef®cient m is unknown during the process, we prefer to use the friction parameter a. These assumptions based on the experiments/ FEM calculations will enable us to build an analytical model for free expansion hydroforming. The yield-loci are assumed to be given by the Hill-ellipse, and the hardening model by the Ludwik±Nadai relation. However, these assumptions are not crucial to the model, and a different yield surface and hardening behaviour can be considered. The basis of the model is the principle of minimisation of RR plastic deformation energy Ep s e de dV; where the integration volume is the volume occupied by the material. In a single step, the material is distributed according to this principle. In order to avoid trivial solutions, a subsidiary condition is needed. This condition is the amount of material that ¯ows in from the free ends. The material in¯ow is adjusted to give a pre-de®ned expansion height or Ð alternatively Ð is increased until necking occurs. The minimisation under the above-mentioned subsidiary conditions leads to the appropriate Euler±Lagrange equations for the shape of the tube. These equations lead to the following implicit relation for the shape y(z) of the tube (Appendix A): Z y dy q z: y0 ln1n y=y0 ÿ p y=q y=y0 ÿ 1a 2 ÿ 1 The Lagrange multiplier p and the integration constant q are determined by the amount of material in¯ow and by the symmetry condition of the tube (the equations are solved for half a tube), respectively. The initial radius of the tube is denoted by y0. It can already be concluded from this model that C and the thickness t will not in¯uence the resulting shape of the tube (note that bending terms are absent in this description).
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Fig. 9. Tube pro®les for different r-values. The tubes are expanded to 45 mm.
5. Results of analytical model In Figs. 9±12 the effects of r-value, n-value and friction parameter on the shape/thickness distribution are given. In Figs. 9±12 the shape/thickness distribution calculated for a tube that is expanded to a maximum height of 45 mm is shown. For each of these ®gures, one material parameter has
Fig. 10. Thickness distribution corresponding to the pro®les of Fig. 9.
Fig. 11. Pro®les and thickness distribution for material with different nvalues.
Fig. 12. Pro®les and thickness distribution for material with different friction parameters, a's.
been varied (as stated in the ®gure). The rest of the material parameters correspond to those of 1 mm thick DC06. In Fig. 9 the effect of different r-values on the pro®le is shown. In Fig. 10 the corresponding thickness distribution is given. Pro®le and thickness are displayed together in Fig. 11, where only the n-value is varied. The same thing has been done for a variation of friction parameter in Fig. 12. In line with the experiments and FEM simulations, the effect of a change of r-value is largest, followed by a change of friction and a change of n-value, respectively. This simple analytical model also predicts that higher r- and n-values and lower a-values result in more material in¯ow and a more homogenous thickness distribution. Another interesting phenomenon can be explained by this simple model. It is known from experiments that the ratio of diameter to free length of the tube has an effect on the ®nal shape of the tube. For tubes with a ratio of 0.2, a characteristic S-shape is observed. For tubes with a ratio of 0.4, this shape disappears. In Fig. 13 the shape is shown for a tube with a ratio of 0.4 expanded to different heights. Calculations show that for certain heights, two stable shapes exist,
Fig. 13. Stable shapes of expanded tubes as a function of axial coordinate. The full free length of the tube is 140 mm.
B. Carleer et al. / Journal of Materials Processing Technology 104 (2000) 158±166
one S-shaped and one parabola-shaped. In the initial phase of expansion, the favoured energy state is S-shaped. For later stages, however, the minimum deformation-energy shapes are parabola-like. Apparently, this has nothing to do with the bending terms but only with the height/radius ratio. The calculated shape transition is in agreement with experimental observations.
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Applying the above-mentioned model, it is possible to calculate the shape of the tube at the moment local necking occurs. A quantity that can be plotted is the height to which a tube can be expanded before local necking starts. This quantity has been plotted as a function of initial yield stress for cold-rolled grades in Fig. 14a and for hot-rolled grades in Fig. 14b. The ®gures represent the ranking of steel grades for
Fig. 14. (a) Ranking of cold-rolled materials based upon the height when necking occurs, calculated with the simple model. (b) Ranking of hot-rolled materials based upon the height when necking occurs, calculated with the simple model.
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Table 2 Half of the shortening of the tube dl, measured (exp) and calculated with both the analytical model (ana) and Pam-simulations (sim) (height and a are taken from experiment) Grade
Height (mm) exp
a exp
dl (mm) exp
dl ana a1
dl (mm) ana
dl (mm) sim
DC06 DC06 GA IF-P 220 IF-P 260 SG350 GD GI YMPRESS E355
37.5 40 45 37.5 34.5 34
0.5 0.5 0.9 0.7 0.4 0.6
9 11 19.5 10 5 4.5
25.3 19.2 27.3 14.0 12.6 6.14
10.6 11.9 24.3 13.5 3.3 4.7
15 23.5 27.5 18.5 13.5 9.5
hydroforming purposes. For a speci®c yield stress, the steel with the highest obtainable height before necking is favoured. Depending on the desired initial yield stress, an appropriate steel grade can be chosen. Alternatively, one could select materials based on the yield stress in the ®nal product. It appears that the IF-rephos steels behave above average. At this moment we are implementing a more general description of the yield surface in order to give a similar ranking for aluminium. 6. Experiment versus simulation and analytical model The results of the experiments as well as the results of the model calculations indicate that: the parameters r and n have large impact on the shape of the free-xpanded tube, the parameters r and a (parameter representative for the friction) have biggest effect on strain distribution. In order to highlight the correspondence between the results of the experiments, the Pam simulations and the analytical model, relevant observables must be compared. Examples of observables are the shortening of the tube length (Table 2) and the major and minor strains (Table 3). The height of the expanded tube and the friction parameter a are taken from the experiments. The value a1 corresponds to frictionless conditions, and can be as low as 0.4 for a steel grade with a high yield stress and under normal friction conditions.
In Table 2 the resulting (half of the total) shortening of the tube length is given. The effect in the analytical model of having an a differing from 1 is large as can be seen from the table, and improves the comparison with the experiment. The agreement between experiments and the two abovementioned methods (Pam simulations and analytical calculations) is satisfactory since it reproduces the trends observed for the different steel grades. The results of the analytical model are more close to the experiments, presumably because the friction conditions are better described. Both Pam simulation and analytical model reproduce quite nicely the observed major and minor strains (Table 3). Especially the agreement between the measured strains and the strains calculated by the analytical model is striking. 7. Conclusion Based upon the observation from experiment and simulation that strain paths are straight during free expansion hydroforming of tubes, we were able to construct an analytical model to describe this deformation process. It was shown that this model: reproduces trends observed in both experiment and FEM simulation with regard to variations in material model parameters; succeeded in explaining shape transitions during freeexpansion hydroforming of tubes; provides a simple tool to predict a ranking of steel grades suitable for hydroforming.
Table 3 Maximum deformed diameter Dmax of the deformed tube for different steel gradesa Grade
DC06 uncoated DC06 GA IF-P 220 GI IF-P 260 GI SG350 GD GI YMPRESS E355 a
Dmax (mm)
75 80 90 75 69 68
e1 (%)
e2 (%)
exp
sim (ana)
exp
sim (ana)
40 41 62 33 28 26
37 42 53 34 28 27
ÿ11 ÿ11 ÿ25 ÿ12 ÿ6 ÿ4
ÿ22 ÿ23 ÿ29 ÿ18 ÿ13 ÿ11
(36) (45) (64) (36) (25) (24)
The major strain e1 and the minor strain e2 is given as measured (exp) and as calculated with Pam (sim) and with the analytical model (ana).
(ÿ11) (ÿ12) (ÿ26) (ÿ13) (ÿ5) (ÿ6)
B. Carleer et al. / Journal of Materials Processing Technology 104 (2000) 158±166
Appendix A This appendix contains a short derivation of the quadrature formula for the shape of the tube. As shown in Fig. 8, the shape of the tube is described by expressing the radius y(z) as a function of the length z. The length variable z varies from 0 (the edge) to 12L (the centre). The boundary condition is that y(0)y0 (given). One additional subsidiary condition is needed; in this instance the material in¯ow is kept ®xed. The longitudinal strain el, the circumferential strain ej and the thickness strain et are assumed to be principal strains. The circumferential strain is given by ej z ln y z=y0 . By assuming that the deformation is plastic, the thickness strain can be eliminated: et ÿej ÿ el . By making use of the friction parameter a, the ratio of longitudinal to circumferential strain is given by e1 ar a0 : ÿ 1r ej The work hardening will be described by a Ludwik relation: s e Cen : The equivalent strain is a Hill's equivalent strain. Now the plastic deformation energy is given by Z Z s e de dV; Ep where the integral is taken over the volume of the tube wall. For the differential volume dV, the expression for the 0 thickness t(z) (t z t0 y=y0 ÿ 1a ) as well as for the arc length ds 1 dy=dz2 1=2 is needed to obtain 2 ÿ 1a0 s y dy 1 dz: dV 2pt0 y y0 dz In this expression, t0 is the original thickness of the tube wall. The plastic deformation energy can then be described by Z L=2 dy dz E y; Ep dz 0 2 ÿ 1a0 s Z L=2 dy y 1n y ln dz; y 1 a y0 dz y0 0 where a is a constant: a 2p
n1=2 C 1 r 0 t0 a 2 1 r 2ra0 1 r : 1 n 2r 1
It is clear that for a higher value of r, the resistance against thickness strain is larger because a large radius of the tube contributes more when r is larger (/ y1=2 for r1 and / y for large r under frictionless conditions). As stated above, the subsidiary condition is that the material in¯ow of metal is ®xed. For this in¯ow, an expression for the total amount of
165
material in the tube wall is needed, which is Z L=2 dy Mp dz M y; dz 0 s 2 ÿ 1a0 Z L=2 dy y y 1 dz: dz y0 0 The pro®le y(z) that yields minimal plastic deformation energy under the condition Mpconstant can be found by functional variation of the integrand dy dy dy E y; ÿ lM y; ; F y; dz dz dz where l is the Lagrange multiplier. The pro®le that gives a stationary integral over F is given by the Euler±Lagrange equations: @ d @ Fÿ F 0: @y dz @ dy=dz If the equation is autonomous (i.e., F is only a function of y and dy/dz), one integration can readily be performed: dy dy @ dy F y; ÿ k; F y; dz dz @ dy=dz dz where k is an integration constant. The resulting differential equation for the pro®le is then expressed by v u ÿ 1a0 !2 u dy t y y y ln1n ÿ1; ÿp dz y0 q y0 where p l=a and q k=a. This directly leads to the quadrature given in the text. Now it is clear that the resulting pro®le is independent of C and t0 as was veri®ed by FEM simulations. For the practical solution of the differential equation, the symmetry condition dy=dz L=2 0 at the middle of the tube is considered. If needed, the material in¯ow Mp can be adjusted to give a certain maximum tube radius. When the equations are solved for the pro®le of the tube, several dependent quantities can be derived, such as the average pressure P in the tube (P Ep =DV, where DV is the increase in the volume enclosed by the tube) and the shortening of the total tube dl. References [1] A. Neubauer, P.J. Bolt, Hydrostatic forming of hollow parts: state of the art and future, in: J.K. Lee, G.L. Inzle, R.H. Wagoner (Eds.), Shemet 3rd International Conference: Numerical Simulation of 3-D Sheet Metal Forming Processes ± Veri®cation of Simulations with Experiments, The Ohio State University, 1996, p. 111. [2] L. Wu, Y. Yu, Computer simulations of forming automotive structural parts by hydroforming process, in: H.J.J. Kals, B. Shiwari, U.P. Singh, M. Geiger (Eds.), Numisheet Proceedings of the International Conference (1±2 April 1996), ISBN 90-365-08045, vol. 1, 1996, p. 324.
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[3] T. Altan, K. Brewster, K. Sutter, M. Ahmetoglu, Introduction to hydroforming: fundamental aspects of processes and technology, Innovations in Hydroforming Technology, Nashville, TN, 1996. [4] S.D. Liu, D. Meuleman, K. Thompson, Limits of tubular hydroforming are examined, Automotive Body International, Fall (1998). [5] C. Kunz, P.J. Uggowitzer, M.O. Speidel, AbschaÈtzen des umformvermoÈgens beim innenhochdruckumformen, Blech Rohre Pro®le (11) 1997.
[6] F. Dohmann, W. Meyer, Tribologie der Innenhochdruckumformung, Blech Rohre Pro®le (10) 1997. [7] F. Dohmann, P. Bieling, Werkzeugparameter und prozessdaten beim aufweitenden innehochdruckumformen, Umformtechnik 26 (1992) 1. [8] Z.C. Xia, Bursting for tubular hydroforming, SAE Paper 2000-010770. [9] ASAME, The automated strain analysis and measurement environment, Reference manual version 4, Camsys Inc., May 1998. [10] Pam System International S.A., http://[email protected].
Analysis of the effect of material properties on the hydroforming process of tubes B. Carleer, G. van der Kevie, L. de Winter*, B. van Veldhuizen Corus Research, Development and Technology, CR-YTC-PRA-AUT-MEP, PO Box 10000 3H36, Ijmuiden, CA 1970, USA Accepted 23 May 2000
Abstract The free-expansion hydroforming of tubes is discussed. Different steel grades have been studied by means of experiments, FEM simulations and a simple analytical model. The effect of different material parameters, such as the r-value and the n-value, on the hydroformability of tubes is discussed. Applying the analytical model, the favourable shape of a free-expanded tube is predicted and a ranking of different grades based on this model is proposed. # 2000 Published by Elsevier Science S.A. Keywords: Hydroforming process; FEM simulation; Analytical model
1. Introduction In tube hydroforming, internal pressure is applied to modify a tubular metal blank into a structural component having a closed cross-section. Tube hydroforming offers a way to cut material and manufacturing costs, while improving product performance in a variety of applications. Existing multi-piece stamped/welded assemblies in car body and frame structures often can be replaced with less expensive hydroformed parts that are lighter, stronger and more accurate [1±3]. In order to achieve the most economic product with the best performance, it is important to choose proper material and process selection. As tube hydroforming is a relatively new technology compared to conventional stamping, there is not much knowledge available that can be used in selecting proper material and process selection [4,5]. At Corus, research on hydroforming includes process, material and product development. Parameters that can in¯uence the process are material grade, friction, tool design and product design [6±8]. A way to determine the material parameters which are most important to a hydroform suitable steel is the free expansion of tubes. Experiments and FE simulations have been performed to gain insight into the free-expansion behaviour of various steel grades. On this basis, an analytical model has been developed to quickly
* Corresponding author. Tel.: 1-31-2514-94323. E-mail address: [email protected] (L. de Winter)
0924-0136/00/$ ± see front matter # 2000 Published by Elsevier Science S.A. PII: S 0 9 2 4 - 0 1 3 6 ( 0 0 ) 0 0 5 3 0 - 6
analyse the in¯uence of the various material parameters during free expansion. 2. Experiments Tubes made of various steel grades have been expanded applying the hydroform process together with PtU Darmstadt. The in¯uence of the material parameters on the deformation characteristics was determined. Steel grades ranging from high strength to low strength steels were used. The corresponding material parameters t (thickness), n (hardening exponent), Rm (maximum stress), Rp,0.2 (yield stress at 0.2% strain) and r (plastic anisotropy parameter) are given in Table 1. In the experiments, tubes 450 mm long and with a maximum external diameter of 55 mm were laid in a die and sealed with plungers at both ends. The tubes were ®lled with water through the plungers and as the pressure was raised, the tubes expanded until rupture. Over a length of 270 mm, the tube was able to expand freely. Only a small axial force was applied on the ends of the tubes in order to seal the ends. The axial displacements of the deformed tubes in the experiments were measured. The gridded tubes were measured at Hoogovens using the Camsys measuring system [9] to determine the strain distribution. The maximum expanded diameter of the tube was measured as well. Fig. 1 shows Camsys strain measurements and Fig. 2 shows an example of bursted tubes.
B. Carleer et al. / Journal of Materials Processing Technology 104 (2000) 158±166
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Table 1 Material parameters Material
Thickness n (mm)
Rm (MPa)
Rp,0.2 (MPa)
r
DC06 DC06 GA IF P 220 GI IF P 260 GI SG350 GD GI YMPRESS E355
1.0 0.96 0.915 0.905 0.98 1.25
295 283 365 409 431 517
150 155 243 274 369 408
2.35 2.04 2.18 1.76 1.29 1.02
0.23 0.21 0.20 0.19 0.184 0.142
The following conclusions were drawn from the experiments: The tubes made of both galvanized and galvannealed materials with low yield strength, high n- and r-values showed the largest major and minor strains. A larger axial displacement was noticed for materials with low yield stress as less pressure was required to deform the tubes resulting in lower normal forces between the tube and the die and consequently also lower frictional forces restraining the material from flowing towards the centre.
3. Simulations Next to the experiments, simulations have been performed of the free expansion using the FE programme PamStamp [10]. The simulations were veri®ed by compar-
Fig. 2. Ruptured expanded tubes (IF P 220).
ing the achieved maximum major and minimum minor strain to those measured by Camsys. Since we have not used a material model incorporating damage, the PamStamp simulation for the various steel grades was stopped when the maximum deformed diameter measured in the experiment was reached. By taking a reference material and then varying the process and/or material parameters, the in¯uence on the formability of the material was investigated. The parameters
Fig. 1. Measured strain distribution for a DC06 GA grade.
160
B. Carleer et al. / Journal of Materials Processing Technology 104 (2000) 158±166
Fig. 5. Tube pro®le for materials with different n-values.
are the r-value of the material (included in Hill's yielding model), the coef®cient of friction m between the die and the tube, and the thickness t of the tubes. Every parameter was varied and the in¯uence on the strain distribution and strain path was investigated. The ®nal crosssection of the tubes was analysed as well. The following conclusions were drawn: Fig. 3. Geometry of a free-expanded tube for different r-values.
that were varied were C (hardening coef®cient), r (anisotropy parameter), e0 (prestrain), n (hardening coef®cient), t (thickness) and m (friction coef®cient). The tubes were meshed by means of Shells and the internal pressure was increased by using ¯uid cells. The hardening model used in the simulations to represent cold worked material is a Ludwik±Nadai equation that relates the plastic equivalent strain e to the equivalent stress s: s C e0 en : Three other parameters not represented by the hardening model but affecting the deformation characteristics as well
A higher r- and n-value and a lower m given to a tube with less local deformation is shown in Fig. 3. Consequently, the strain is also more evenly distributed and will favour larger deformations. This figure also shows that the tube with r3 shortens more than the tube with r1 due to more material flow towards the middle during the forming process. The same is seen for the influence of the n-value and m. This effect is more clearly shown for the n- and rvalue in Figs. 4 and 5. The only parameters that influence the strain distribution in the forming diagram are the r-value and m. This is illustrated in Figs. 6 and 7. As a higher r-value or lower m lowers the slope of the strain path in the forming diagram, it will cross the forming limit curve at higher negative
Fig. 4. Tube pro®le for materials with different r-values.
B. Carleer et al. / Journal of Materials Processing Technology 104 (2000) 158±166
161
These simulations show that mainly a higher n- and rvalue are favourable to the ``hydroformability'' of steel and that the friction should be kept as low as possible. 4. Analytical model
Fig. 6. Strain distribution for free expansion of a tube with different rvalues.
Fig. 7. Strain distribution for free expansion of a tube with different values for the friction coef®cient.
minor and positive major strains. A straight strain path is seen in the simulations. Thus higher strains before failing may be reached. C, e0, t and n-values have hardly any influence on the strain distribution in the forming diagram and the strain path of all the forming diagrams cross the forming limit diagram at the same major and minor strain.
Fig. 8. Pro®le of half of a free-expanded tube. The local co-ordinate system is denoted by l (longitudinal) and t (thickness). The circumferential direction is perpendicular to the paper.
The results of FEM simulations showed that the strain paths during free expanding are reasonably straight (the deviation from the straight path for r1 as shown in Fig. 6 is actually due to global necking). Furthermore, the system is already on principal axes (the longitudinal, circumferential and thickness axes, Fig. 8). In the absence of friction, the ratio of the minor to major strain is given by ÿr= 1 r. In the situation where the ends are fully clamped, plane strain conditions would exist. Plane strain conditions were assumed before to study the onset of bursting during tubular hydroforming [8]. Friction has the effect of rotating the strain path from this uniaxial direction towards the planestrain direction. Therefore we introduce a friction parameter a to account for this deviation from uniaxiality. The ratio of the minor to major strain is now given by the factor ÿar= 1 r, and the smaller the a is, the higher is the friction. There is a ®xed relationship between a and the friction coef®cient m. Since the actual friction coef®cient m is unknown during the process, we prefer to use the friction parameter a. These assumptions based on the experiments/ FEM calculations will enable us to build an analytical model for free expansion hydroforming. The yield-loci are assumed to be given by the Hill-ellipse, and the hardening model by the Ludwik±Nadai relation. However, these assumptions are not crucial to the model, and a different yield surface and hardening behaviour can be considered. The basis of the model is the principle of minimisation of RR plastic deformation energy Ep s e de dV; where the integration volume is the volume occupied by the material. In a single step, the material is distributed according to this principle. In order to avoid trivial solutions, a subsidiary condition is needed. This condition is the amount of material that ¯ows in from the free ends. The material in¯ow is adjusted to give a pre-de®ned expansion height or Ð alternatively Ð is increased until necking occurs. The minimisation under the above-mentioned subsidiary conditions leads to the appropriate Euler±Lagrange equations for the shape of the tube. These equations lead to the following implicit relation for the shape y(z) of the tube (Appendix A): Z y dy q z: y0 ln1n y=y0 ÿ p y=q y=y0 ÿ 1a 2 ÿ 1 The Lagrange multiplier p and the integration constant q are determined by the amount of material in¯ow and by the symmetry condition of the tube (the equations are solved for half a tube), respectively. The initial radius of the tube is denoted by y0. It can already be concluded from this model that C and the thickness t will not in¯uence the resulting shape of the tube (note that bending terms are absent in this description).
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Fig. 9. Tube pro®les for different r-values. The tubes are expanded to 45 mm.
5. Results of analytical model In Figs. 9±12 the effects of r-value, n-value and friction parameter on the shape/thickness distribution are given. In Figs. 9±12 the shape/thickness distribution calculated for a tube that is expanded to a maximum height of 45 mm is shown. For each of these ®gures, one material parameter has
Fig. 10. Thickness distribution corresponding to the pro®les of Fig. 9.
Fig. 11. Pro®les and thickness distribution for material with different nvalues.
Fig. 12. Pro®les and thickness distribution for material with different friction parameters, a's.
been varied (as stated in the ®gure). The rest of the material parameters correspond to those of 1 mm thick DC06. In Fig. 9 the effect of different r-values on the pro®le is shown. In Fig. 10 the corresponding thickness distribution is given. Pro®le and thickness are displayed together in Fig. 11, where only the n-value is varied. The same thing has been done for a variation of friction parameter in Fig. 12. In line with the experiments and FEM simulations, the effect of a change of r-value is largest, followed by a change of friction and a change of n-value, respectively. This simple analytical model also predicts that higher r- and n-values and lower a-values result in more material in¯ow and a more homogenous thickness distribution. Another interesting phenomenon can be explained by this simple model. It is known from experiments that the ratio of diameter to free length of the tube has an effect on the ®nal shape of the tube. For tubes with a ratio of 0.2, a characteristic S-shape is observed. For tubes with a ratio of 0.4, this shape disappears. In Fig. 13 the shape is shown for a tube with a ratio of 0.4 expanded to different heights. Calculations show that for certain heights, two stable shapes exist,
Fig. 13. Stable shapes of expanded tubes as a function of axial coordinate. The full free length of the tube is 140 mm.
B. Carleer et al. / Journal of Materials Processing Technology 104 (2000) 158±166
one S-shaped and one parabola-shaped. In the initial phase of expansion, the favoured energy state is S-shaped. For later stages, however, the minimum deformation-energy shapes are parabola-like. Apparently, this has nothing to do with the bending terms but only with the height/radius ratio. The calculated shape transition is in agreement with experimental observations.
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Applying the above-mentioned model, it is possible to calculate the shape of the tube at the moment local necking occurs. A quantity that can be plotted is the height to which a tube can be expanded before local necking starts. This quantity has been plotted as a function of initial yield stress for cold-rolled grades in Fig. 14a and for hot-rolled grades in Fig. 14b. The ®gures represent the ranking of steel grades for
Fig. 14. (a) Ranking of cold-rolled materials based upon the height when necking occurs, calculated with the simple model. (b) Ranking of hot-rolled materials based upon the height when necking occurs, calculated with the simple model.
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Table 2 Half of the shortening of the tube dl, measured (exp) and calculated with both the analytical model (ana) and Pam-simulations (sim) (height and a are taken from experiment) Grade
Height (mm) exp
a exp
dl (mm) exp
dl ana a1
dl (mm) ana
dl (mm) sim
DC06 DC06 GA IF-P 220 IF-P 260 SG350 GD GI YMPRESS E355
37.5 40 45 37.5 34.5 34
0.5 0.5 0.9 0.7 0.4 0.6
9 11 19.5 10 5 4.5
25.3 19.2 27.3 14.0 12.6 6.14
10.6 11.9 24.3 13.5 3.3 4.7
15 23.5 27.5 18.5 13.5 9.5
hydroforming purposes. For a speci®c yield stress, the steel with the highest obtainable height before necking is favoured. Depending on the desired initial yield stress, an appropriate steel grade can be chosen. Alternatively, one could select materials based on the yield stress in the ®nal product. It appears that the IF-rephos steels behave above average. At this moment we are implementing a more general description of the yield surface in order to give a similar ranking for aluminium. 6. Experiment versus simulation and analytical model The results of the experiments as well as the results of the model calculations indicate that: the parameters r and n have large impact on the shape of the free-xpanded tube, the parameters r and a (parameter representative for the friction) have biggest effect on strain distribution. In order to highlight the correspondence between the results of the experiments, the Pam simulations and the analytical model, relevant observables must be compared. Examples of observables are the shortening of the tube length (Table 2) and the major and minor strains (Table 3). The height of the expanded tube and the friction parameter a are taken from the experiments. The value a1 corresponds to frictionless conditions, and can be as low as 0.4 for a steel grade with a high yield stress and under normal friction conditions.
In Table 2 the resulting (half of the total) shortening of the tube length is given. The effect in the analytical model of having an a differing from 1 is large as can be seen from the table, and improves the comparison with the experiment. The agreement between experiments and the two abovementioned methods (Pam simulations and analytical calculations) is satisfactory since it reproduces the trends observed for the different steel grades. The results of the analytical model are more close to the experiments, presumably because the friction conditions are better described. Both Pam simulation and analytical model reproduce quite nicely the observed major and minor strains (Table 3). Especially the agreement between the measured strains and the strains calculated by the analytical model is striking. 7. Conclusion Based upon the observation from experiment and simulation that strain paths are straight during free expansion hydroforming of tubes, we were able to construct an analytical model to describe this deformation process. It was shown that this model: reproduces trends observed in both experiment and FEM simulation with regard to variations in material model parameters; succeeded in explaining shape transitions during freeexpansion hydroforming of tubes; provides a simple tool to predict a ranking of steel grades suitable for hydroforming.
Table 3 Maximum deformed diameter Dmax of the deformed tube for different steel gradesa Grade
DC06 uncoated DC06 GA IF-P 220 GI IF-P 260 GI SG350 GD GI YMPRESS E355 a
Dmax (mm)
75 80 90 75 69 68
e1 (%)
e2 (%)
exp
sim (ana)
exp
sim (ana)
40 41 62 33 28 26
37 42 53 34 28 27
ÿ11 ÿ11 ÿ25 ÿ12 ÿ6 ÿ4
ÿ22 ÿ23 ÿ29 ÿ18 ÿ13 ÿ11
(36) (45) (64) (36) (25) (24)
The major strain e1 and the minor strain e2 is given as measured (exp) and as calculated with Pam (sim) and with the analytical model (ana).
(ÿ11) (ÿ12) (ÿ26) (ÿ13) (ÿ5) (ÿ6)
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Appendix A This appendix contains a short derivation of the quadrature formula for the shape of the tube. As shown in Fig. 8, the shape of the tube is described by expressing the radius y(z) as a function of the length z. The length variable z varies from 0 (the edge) to 12L (the centre). The boundary condition is that y(0)y0 (given). One additional subsidiary condition is needed; in this instance the material in¯ow is kept ®xed. The longitudinal strain el, the circumferential strain ej and the thickness strain et are assumed to be principal strains. The circumferential strain is given by ej z ln y z=y0 . By assuming that the deformation is plastic, the thickness strain can be eliminated: et ÿej ÿ el . By making use of the friction parameter a, the ratio of longitudinal to circumferential strain is given by e1 ar a0 : ÿ 1r ej The work hardening will be described by a Ludwik relation: s e Cen : The equivalent strain is a Hill's equivalent strain. Now the plastic deformation energy is given by Z Z s e de dV; Ep where the integral is taken over the volume of the tube wall. For the differential volume dV, the expression for the 0 thickness t(z) (t z t0 y=y0 ÿ 1a ) as well as for the arc length ds 1 dy=dz2 1=2 is needed to obtain 2 ÿ 1a0 s y dy 1 dz: dV 2pt0 y y0 dz In this expression, t0 is the original thickness of the tube wall. The plastic deformation energy can then be described by Z L=2 dy dz E y; Ep dz 0 2 ÿ 1a0 s Z L=2 dy y 1n y ln dz; y 1 a y0 dz y0 0 where a is a constant: a 2p
n1=2 C 1 r 0 t0 a 2 1 r 2ra0 1 r : 1 n 2r 1
It is clear that for a higher value of r, the resistance against thickness strain is larger because a large radius of the tube contributes more when r is larger (/ y1=2 for r1 and / y for large r under frictionless conditions). As stated above, the subsidiary condition is that the material in¯ow of metal is ®xed. For this in¯ow, an expression for the total amount of
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material in the tube wall is needed, which is Z L=2 dy Mp dz M y; dz 0 s 2 ÿ 1a0 Z L=2 dy y y 1 dz: dz y0 0 The pro®le y(z) that yields minimal plastic deformation energy under the condition Mpconstant can be found by functional variation of the integrand dy dy dy E y; ÿ lM y; ; F y; dz dz dz where l is the Lagrange multiplier. The pro®le that gives a stationary integral over F is given by the Euler±Lagrange equations: @ d @ Fÿ F 0: @y dz @ dy=dz If the equation is autonomous (i.e., F is only a function of y and dy/dz), one integration can readily be performed: dy dy @ dy F y; ÿ k; F y; dz dz @ dy=dz dz where k is an integration constant. The resulting differential equation for the pro®le is then expressed by v u ÿ 1a0 !2 u dy t y y y ln1n ÿ1; ÿp dz y0 q y0 where p l=a and q k=a. This directly leads to the quadrature given in the text. Now it is clear that the resulting pro®le is independent of C and t0 as was veri®ed by FEM simulations. For the practical solution of the differential equation, the symmetry condition dy=dz L=2 0 at the middle of the tube is considered. If needed, the material in¯ow Mp can be adjusted to give a certain maximum tube radius. When the equations are solved for the pro®le of the tube, several dependent quantities can be derived, such as the average pressure P in the tube (P Ep =DV, where DV is the increase in the volume enclosed by the tube) and the shortening of the total tube dl. References [1] A. Neubauer, P.J. Bolt, Hydrostatic forming of hollow parts: state of the art and future, in: J.K. Lee, G.L. Inzle, R.H. Wagoner (Eds.), Shemet 3rd International Conference: Numerical Simulation of 3-D Sheet Metal Forming Processes ± Veri®cation of Simulations with Experiments, The Ohio State University, 1996, p. 111. [2] L. Wu, Y. Yu, Computer simulations of forming automotive structural parts by hydroforming process, in: H.J.J. Kals, B. Shiwari, U.P. Singh, M. Geiger (Eds.), Numisheet Proceedings of the International Conference (1±2 April 1996), ISBN 90-365-08045, vol. 1, 1996, p. 324.
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[3] T. Altan, K. Brewster, K. Sutter, M. Ahmetoglu, Introduction to hydroforming: fundamental aspects of processes and technology, Innovations in Hydroforming Technology, Nashville, TN, 1996. [4] S.D. Liu, D. Meuleman, K. Thompson, Limits of tubular hydroforming are examined, Automotive Body International, Fall (1998). [5] C. Kunz, P.J. Uggowitzer, M.O. Speidel, AbschaÈtzen des umformvermoÈgens beim innenhochdruckumformen, Blech Rohre Pro®le (11) 1997.
[6] F. Dohmann, W. Meyer, Tribologie der Innenhochdruckumformung, Blech Rohre Pro®le (10) 1997. [7] F. Dohmann, P. Bieling, Werkzeugparameter und prozessdaten beim aufweitenden innehochdruckumformen, Umformtechnik 26 (1992) 1. [8] Z.C. Xia, Bursting for tubular hydroforming, SAE Paper 2000-010770. [9] ASAME, The automated strain analysis and measurement environment, Reference manual version 4, Camsys Inc., May 1998. [10] Pam System International S.A., http://[email protected].