A study of the optimum blank for square cup drawing using the streamline method

Journal of Materials Processing Technology 110 (2001) 146±151

A study of the optimum blank for square cup drawing using the streamline method Yuung-Hwa Lua,*, Ching-Lun Lib, Sy-Wei Loc a

Department of Mechanical Engineering, National I-Lan Institute of Technology, 1 Shen-Lung Road, I-Lan 260, Taiwan, ROC b Department of Mechanical Engineering, Tamkang University, 151 Ying-Chuan Road, Tamsui, Taipei 251, Taiwan, ROC c Department of Mechanical Engineering, National Yunlin University of Science and Technology, Touliu, Yunlin 640, Taiwan, ROC Received 12 May 1999

Abstract This paper presents a theory superimposing material anisotropic property as a ``sink'' to the potential ®eld concept. An ef®cient and easily complemented streamline method is then developed to predict the optimum blank of irregular prismatic cup drawing. From the optimum blank contours calculated by the streamline method for the square cup, it is shown that material anisotropy has a signi®cant in¯uence in the prediction of the optimum blank shape. As a veri®cation of the theory, an experiment of square cup drawing was conducted, relating to the optimum blank in the experimental drawing process for a ¯at-cup head. The measured cup heights show good agreement with theoretical predictions. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Square cup drawing; Streamline method; Optimum blank

1. Introduction When an irregular cup is drawn from a sheet metal blank, trial-and-error methods are usually used to obtain the optimum blank. The aim of searching for the optimum blank not only can reduce the consumption of the material but also can save the cost of manufacture. However, the uncertainty of the trial-and-error methods mean that they require sequential modi®cation of the blank to obtain the optimum shape. Hence, how to develop a reasonable theory to design the optimum blank for the drawing of a prismatic cup is an essential problem to be solved. As material anisotropy plays an important role in sheet metal forming, special attention should be paid to the material anisotropy for the design of the optimum blank. In earlier researches, Doege [1] presented an approximate technique, which is based on the equivalent punch diameter, to evaluate the feasibility of drawing irregular shaped boxes. Karima [2] used the slip line ®eld (SLF) method to develop a blank of complex shape. However, the SLF is suitable for non-hardening material, plane-strain deformation, and convex punch contours: any violation of these restrictions will *

Corresponding author.

be tedious. An alternative potential ¯ow scheme was proposed by Liu and Sowerby [3]. They assumed that the ¯ow of the material in the ¯ange of the drawn cup is comparable to the irrotational ¯ow of the liquid. Both an electrostatic analog method and a boundary element method were introduced to predict the optimum blank for prismatic cup drawing. Zaky et al. [4] used the stress and strain relationship according to the Hill theory [5] to calculate the optimum blank shape of cylindrical cups in the deep drawing of anisotropic sheet metals. Chung et al. [6] submitted an ideal work forming theory to design the initial optimum blank. Then, an FEM simulation result was used to modify the blank to obtain a more realistic optimum blank shape. Lo [7] modi®ed the potential ¯ow scheme to predict the optimum blank shapes for prismatic cup drawing with the consideration of material anisotropy and friction. However, only the theoretical results were presented, without experimental veri®cation. In this paper, a streamline method (SLM) based on the concept that the mass in the domain bounded by equipotential contours should be balanced in the drawing process is devoted to predict the near-net shape blank with an arbitrary punch contour under the in¯uence of material anisotropy. A set of experimental dies for square cup drawing was used to verify the correctness of the SLM.

0924-0136/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 0 ) 0 0 8 5 6 - 6

Y.-H. Lu et al. / Journal of Materials Processing Technology 110 (2001) 146±151

147

2. Theory The continuity equation for an incompressible solid is @u @v ‡ ˆ ÿ_ez @x @y

(1)

in which u and v are the components of velocity in the x- and y-directions, as shown in Fig. 1, where the x±y plane is perpendicular to the drawing direction, which latter is de®ned as the z-direction. It is reasonable to assume that the material ¯ow in the ¯ange is comparable to the ideal ¯ow which can be described as a potential ®eld f. Thus, Eq. (1) can be rewritten by the potential ®eld as follows: @2f @2f ‡ ˆ ÿ_ez @x2 @y2

(2)

where

Fig. 2. Drawing with an arbitrary punch pro®le and velocity potential.

@f uˆ @x

(3)

and vˆ

@f @y

(4)

It is easy to discover that the above equation will become a Laplace's equation for plane-strain cup drawing in which the strain rate e_ z is equal to zero. Under other conditions, the existence of the strain rate e_ z will give rise to the inhomogeneous term in Eq. (2). As in plane-stress drawing, the major factor which causes the variation of the strain rate e_ z is the planar anisotropy of the sheet metal. Hence, in order to combine the anisotropic property with Eq. (2), consider the drawing problem with an arbitrary punch pro®le, as shown in Fig. 2. G1 and G2 represent the inner and outer boundaries in which the velocity potential is uniformly distributed. A line element initially located on polar coordinates (r, y) moves to the new position …r ÿ dr; y ÿ dy† during the time interval dt. It is apparent that the line element should be con®ned by these two adjacent streamlines. If

the orientation, the local radius of curvature r, and the angle do do not vary signi®cantly during the time interval, then the principal strain rate e_ 2 can be estimated from e_ 2 ˆ

jqj r_ ˆÿ r r

(5)

where q is the velocity component normal to the boundary, i.e. @f (6) jqj ˆ @n and n is the unit normal vector. The determination of the sign of q in Eq. (6) depends on the direction of motion and the position of the material. In addition, r is de®ned to be positive for a convex contour and negative for a concave contour. Also, e1 e1 (7) Rl‡90 ˆ ˆ ÿ ez e1 ‡ e2 where Rl‡90 is the R-value measured in a tensile test normal to l, refer to Fig. 2. Thus, Rl‡90 e2 (8) e1 ˆ ÿ 1 ‡ Rl‡90 and ez ˆ ÿ

1 e2 1 ‡ Rl‡90

(9)

After substituting Eqs. (5) and (9) into Eq. (2), a Possion's equation which considers the anisotropy property of sheet metal is derived as jqj r…1 ‡ Rl‡90 †

(10)

@2f jqj ˆÿ 2 @n r…1 ‡ Rl‡90 †

(11)

r2 f ˆ ÿ or Fig. 1. Potential ®eld and velocities.

148

Y.-H. Lu et al. / Journal of Materials Processing Technology 110 (2001) 146±151

Fig. 3. Control volume and material ¯uxes.

The R-value of a material element having an angle l with respect to the rolling direction of the sheets can be expressed in terms of R0, R45, and R90 from Hill's anisotropic plasticity theory [5]: Rl ˆ

R0 R90 cos2 …2l† ‡ ‰R45 …R0 ‡ R90 †sin2 …2l†Š 2…R0 sin2 l ‡ R90 cos2 l†

(12)

Fig. 4. Optimum blanks for various punch strokes.

Eq. (11) also can be replaced by the normal anisotropic value with

where s is the boundary of the domain D. Integrating the above equation,

 ˆ 1 …R0 ‡ R90 ‡ 2R45 † R 4

q1 s1 ÿ q2 s2 ˆ fa

(13)

where a is the area of domain D, and s1 and s2 are the lengths of the sides ij and kl, respectively.

and @2f jqj ˆÿ  @n2 r…1 ‡ R†

(16)

(14)

An alternative ``streamline method'' named by the authors is used to integrate Eq. (11) or (14). The concept is that the mass in the domain D consisting of the streamlines and equipotential contours must be balanced. Hence, with reference to Fig. 3, if the velocities q1 and q2 on sides ij and kl, and the intensity of the source/sink f, are assumed constant in the small area `ijkl', then Z Z @q ds ˆ f dD (15) s @n D

3. SLM blank and experiments 3.1. SLM blank The optimum blanks for the square cup were predicted by the streamline method (SLM). One hundred and sixty line elements were used to mesh the square punch head contour. The optimum blank shape was then calculated from this initial position inversely to the last position when the punch stroke reached 15 and 20 mm. Fig. 4 depicts the optimum blanks for the square cups. The dashed contours were the

Fig. 5. Schematic representation of the experimental die assembly.

Y.-H. Lu et al. / Journal of Materials Processing Technology 110 (2001) 146±151

149

drawing process. The experimental apparatus involves a 500 kN hydraulic press forming machine and a computer controller. The latter displays the relationship between the punch stroke and the punch load. The experimental die sets comprised the punch head, die, blank-holder, and spacer ring. The purpose of the spacer ring was not only to provide 0.1 mm clearance between the sheet blank and the blankholder, but also to reduce the in¯uence of the friction effect. Figs. 5 and 6 show the experimental apparatus and the dimensions of the punch head and die, respectively. The sheet thickness of the blank used in the experiment is 1 mm. With this experimental apparatus, the experimental procedure was as follows:

Fig. 6. Dimensions (mm) of the dies and the optimum blank at a punch stroke of 20 mm.

plane-strain cases, whereas the solid contours were the normal anisotropic conditions. The anisotropic values, proposed by the China Steel, of R0, R45, and R90 are 2.39, 2.53, and 2.70, respectively. The normal anisotropic value was then able to be determined as 2.70. It is observed that the discrepancy of the optimum blank contour at the corners was signi®cant between the plane-strain and anisotropic cases. This means that the effect of the material anisotropic property should not be omitted in the prediction of the optimum blank for irregular prismatic cup drawing. 3.2. Experimental work As a veri®cation of the theory of SLM and computer code, experiments were conducted to examine the square cup

1. The die assembly was set carefully onto the hydraulic press forming machine. The punch head and die were aligned to be concentric by means of an inscribed line on the die and the head. 2. A thin ®lm of zinc stearate [Zn(C18H35O2)] was sprayed on the sheet, punch head, and the die. The purpose of zinc stearate is not only to reduce the friction between the sheet±metal interface, but also to increase the life of the die. 3. A force of 5 kN was chosen for the blank-holder. The velocity of punch head is 6 mm/min. 4. The relationship of the punch stroke to the punch load and time was recorded by the computer. The laser printer was used to draw the relationship between the punch stroke and the punch load.

4. Experimental results and discussion Fig. 7 shows a photograph of drawn square cups with normal anisotropy consideration for various punch strokes. It is observed that both of the cups have a regular smooth ¯at

Fig. 7. Photograph of drawn cups for the optimum blank.

150

Y.-H. Lu et al. / Journal of Materials Processing Technology 110 (2001) 146±151

Fig. 8. Experimental cup height distribution for a predicted blank at a punch stroke of 15 mm.

Fig. 9. Experimental cup height distribution for a predicted blank at a punch stroke of 20 mm.

Fig. 10. Photograph of the cup for a predicted plane-strain blank at a punch stroke of 20 mm.

head on the top. Figs. 8 and 9 show the cup height distribution measured from the square cup using the normal anisotropic optimum blanks predicted in Fig. 4. As the variation of the height was not signi®cant in the two cases, this means that adopting the normal anisotropic consideration can secure a good prediction of the optimum blank for the sheet used in the experiment. Due to a radius of 6 mm over the punch head, the height of the cups is approximately 3 mm higher than the punch strokes used in the prediction of the blanks. In addition, the plane-strain optimum blanks, which have a ``sink'' value equal to zero, were also examined. A realization that the thickness should not be changed in the

¯ange for the plane-strain condition meant that the sheet material in the four corners should slide into the die even though the 1.1 mm spacer ring was still used in the experiment. This was demonstrated in Fig. 10 which shows a photograph of the drawn cups for the plane-strain optimum blank predicted at a punch stroke of 20 mm. 5. Conclusions After considering the effect of material anisotropy as a ``sink'' term in a potential ¯ow concept, a theory used to

Y.-H. Lu et al. / Journal of Materials Processing Technology 110 (2001) 146±151

predict the near-net optimum blank of irregular prismatic cup drawing was developed. An ef®cient SLM, which can be easily implemented in industrial application, was also developed. The optimum blank shapes for square cup drawing predicted by the SLM were studied. The measured cup height shows a good agreement between theoretical and experimental results for different punch strokes. It is also found that the discrepancy of the optimum blank pro®les is signi®cant when the material anisotropic property ``sink'' is considered. However, the effect of friction between the sheet±metal interface may also play an important factor in prismatic cup drawing. A direct solution is to superimpose frictional ``sink'' in the potential ¯ow theory. The modelling of this frictional ``sink'' is a topic that is in current progress. Acknowledgements The authors wish to thank research assistants B.H. Lai, J.H. Huang, J.S. Wu, and S.S. Jan for their efforts in performing the experimental works.

151

References [1] E. Doege, Prediction of failure in deep drawing, computer modelling of sheet metal forming process: theory, veri®cation and application, in: N.M. Wang, S.C. Tang (Eds.), Proceedings of the 12th Automotive Material Symposium, TMS-AIME, University of Michigan, Ann Arbor, MI, 1985, pp. 209±224. [2] M. Karima, Blank development and tooling design for drawn parts using a modi®ed slip line ®eld based on approach, ASME J. Eng. Ind. 111 (4) (1989) 345±350. [3] F. Liu, R. Sowerby, The determination of optimum blank shapes when deep drawing prismatic cups, J. Mater. Shaping Technol. 9 (3) (1991) 153±159. [4] A.M. Zaky, A.B. Nassr, M.G. El-Sebaie, Optimum blank shape of cylindrical cups in deep drawing of anisotropic sheet metals, J. Mater. Process. Technol. 76 (1998) 203±211. [5] R. Hill, The Mathematical Theory of Plasticity, Oxford University Press, Oxford, 1983. [6] K. Chung, F. Barlat, J.C. Brem, D.J. Lege, O. Richmond, Blank shape design for a planar anisotropic sheet based on ideal forming design theory and FEM analysis, Int. J. Mech. Sci. 39 (1) (1997) 105±120. [7] S.W. Lo, Optimum blank shapes for prismatic cup drawing Ð consideration of friction and material anisotropy, ASME J. Manuf. Sci. Eng. 120 (1999).