A note on deep drawing process: numerical simulation andexperimental validation

S. Natarajana,*, S. Venkataswamyb, P. Bagavathiperumala

aDepartment of Mechanical Engineering, Anna University, Chennai 600 025, IndiabDepartment of Production Technology, Anna University, MIT Campus, Chennai 600 044, India

Abstract

Sheet metal working occupies the most important position in the area of metal forming. Without a good knowledge of the process and

material variables, it would be difficult to prevent the defects and optimize the process. This paper deals with the analysis of deep drawing of

circular blanks into axi-symmetric cylindrical cups using numerical modelling at different draw conditions. A finite element program to

simulate the process and analyse axi-symmetric components is developed. A rigid plastic material model with the variational approach is used

in the finite element analysis. The amount of draw obtainable in the drawing process has been related both theoretically and experimentally

with the initial diameter of the blank. The surface strains in the radial and circumferential directions have been computed and measured. A

correlation on the flange thickness variation with the analytical and experimental values also has been arrived at.

# 2002 Elsevier Science B.V. All rights reserved.

Keywords: Deep drawing; Sheet metal; Strain; Simulation

1. Introduction

Despite the fact that sheet metal forming technologies

are extensively used in modern industry, the tooling and

production process is still largely based on empirical

results. The development of numerical simulation deter-

mines, the capability of an objective assessment of form-

ability, the strain distribution at different stages of draw

and the possibility of reducing stamping trials during

assessment process. Theoretical analysis of deep drawing

of cups was first reported by Hessenberg [1], and Danckert

[2] studied the effect of residual stress in deep drawing

of cylindrical cups by process modelling the die profile.

The results of the parametric variation of the numerical

simulation by Kobayashi and co-workers [3,4] compared

reasonably well with experimental work of Swift and

Chung [5], Yamada and Oshimura [6], introduced plasti-

city matrices with elasto-plasto model for analysing cup

drawing.

2. Process simulation

A rigid plastic material model is considered and a

general purpose finite element algorithm is developed

for the analysis of hemispherical cup drawing with Alu-

minium 1100-O grade material. The finite element for-

mulation is based on variational principle with penalty

function approach. The formulation takes into account the

contact friction conditions. A general set-up for tooling for

the analysis of cup drawing is shown in Fig. 1. Four noded,

two-dimensional, iso-parametric elements are used in the

modelling. The incremental computation with geometry

updating is used to simulate the sliding of sheet over the

die surface confirming to the shape of tool. The program

consists of processing, pre-processing and post processing

segments. Simple key points data regarding the geometry

of the punch, die, blank holder and the sheet metal are

given. The required number of elements at three zones of

sheet namely: (i) material under the punch; (ii) material

zone without any initial contact with the tool; (iii) the

material between the die and blank holder are to be

mentioned. This helps to create a mesh during finite

element discretization. The profile of the tool geometry

is developed by arc and line with prescribed key points,

which will enable the determination of the nodes that have

initial contact with the tool. The velocity of the punch is

applied if the nodes are in contact with the punch. The

material under the blank holder is allowed to move over

the die in the drawing process. The frictional boundary

condition is applied for the nodes contacting the tool.

The input data includes material properties, time increment,

Journal of Materials Processing Technology 127 (2002) 64–67

* Corresponding author.

0924-0136/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 4 - 0 1 3 6 ( 0 2 ) 0 0 2 5 8 - 3

tool geometry, key locations, punch velocity, nodal co-

ordinates and boundary conditions. The program simulates

the strain distribution at different draw conditions. The

thickness strain compares favourably well in the flange

portion of the cup with the bench marking data reported

by Kobayashi and Alton [3].

3. Experimental validation

Draw quality Aluminium 1100-O grade thickness 0.8 mm

was blanked out and the uniformity of blank thickness and

flatness were tested. The experiment was carried out on a

double action hydraulic press after standardizing the press in

terms of press variables. A draw ratio of 2 was considered for

all the draws, maintaining a constant holding force for all the

set of draws. The test blanks were photo-grided with 2.5 mm

diameter grid circles and inspected for uniformity of the grid

pattern (Fig. 2). The distortion of the grid measured after

each single stage draw provided a means for evaluating the

strain (Fig. 3).

4. Results and discussions

The algorithm used in developing the finite element

program allows visualizing the deformation zones at differ-

ent stages of draw. The program also evaluates the strain for

the different stages of draw.

4.1. Thickness strain

The thickness strain distribution for both the experimental

and analytical values for different punch travel steps are

shown in Fig. 4. The thickness strain is positive near the

flange portion of the cup indicating thickening of material

under blank holder and die flange. The thinning is more

pronounced near the punch envelope at full draw condition.

The thickness strain distribution indicates the failure zone,

which is around the punch radius. The simulated results

compare reasonably well with experimental work with an

error about 8%.

4.2. Radial strain

The radial strain indicates the thinning effect of the sheet

metal during cup formation. Fig. 5 shows the strain dis-

tribution in the radial directions. The stretching is more in

the inner elements and less in the outer elements between die

and blank holder. As the punch travels down, the peak radial

strain is shifting outward indicating the drawing of the cup

and the lesser radial strain in the outer elements indicate

thickening of flange.

4.3. Circumferential strain

The circumferential strain for different draw conditions is

shown in Fig. 6. The maximum circumferential strain occurs

near the die corner. The circumferential strain is negligible at

the bottom of the cup indicating the metal flow resulting in

Fig. 1. Hemispherical punch configuration.

Fig. 2. Photo-grided test blank.

Fig. 3. View of hemispherical cup with grid marking.

S. Natarajan et al. / Journal of Materials Processing Technology 127 (2002) 64–67 65

Fig. 4. Thickness strain distribution.

Fig. 5. Radial strain distribution.

Fig. 6. Circumferential strain distribution.

66 S. Natarajan et al. / Journal of Materials Processing Technology 127 (2002) 64–67

cup formation. The strain distribution predicted by rigid

plastic finite element method shows good agreement with

the experimental results over the flange portion of the cup.

5. Conclusion

The following conclusions could be made based on the

above investigation:

(i) The rigid plastic finite element formulation gives

strain distribution without compromising the numerical

efficiency and accuracy.

(ii) The simulation allows to try out different solutions for

the manufacturing of the product and the process can

be optimized. The choice of optimum process can be

made in an objective manner with numerical support

data and reduces the unnecessary trials.

(iii) The lesser radial strain in the outer elements indicate

thickening of the flange.

(iv) The thickness strain compares reasonably well at

flange portion of the cup with the experimental results.

References

[1] W.C. Hessenberg, A Simple Account of Prof. Swift Work on Deep

Drawing, BISRA Report, 1954.

[2] J. Danckert, Reduction of residual stress in deep drawing

by modelling the draw die profile, Ann. CIRP 44 (1) (1995) 259–

262.

[3] S. Kobayashi, T. Alton, Metal Forming and Finite Element Method,

Oxford University Press, Oxford, 1989.

[4] S. Kobayashi, Shah, The matrix method for the analysis of metal

forming, Advances in Deformation Process, Plenum Press, New

York, 1978.

[5] W. Swift, S.Y. Chung, Cup drawing from a flat blank—experimental

investigation, Proc. Int. Mech. Eng. (1951) 199–211.

[6] Y. Yamada, N. Oshimura, Plastic stress strain matrix and application

for the solution of elasto-plastic problem by FEM, Int. J. Mech. Sci.

10 (1968) 343–354.

S. Natarajan et al. / Journal of Materials Processing Technology 127 (2002) 64–67 67