q q
q q
r
i
n
p
n
1
t
1
t
2
F
m
p
q m
d
q
s
f
= mY
s
f
= 0
Figure 8.13 Ironing of the wall of a cylindrical deep-drawn cup.
At exit, the axial stress must be less than the yield stress to ensure that deformation occurs only within the die. As shown in Figure 8.13, this stress is mY where m 1. The
through-thickness stress is σ
t
= − Y − mY , and at exit q
= −σ
t
= Y 1 − m 8.19
The average contact pressure is q
= Y 1
+ 1 − m 2
= Y 1
− m
2 8.20
The forces on the deformation zone are shown in Figure 8.14. The equation of equilibrium of forces in the vertical direction is
mY t
2
+ μ
p
q t
tan γ = μ
d
q t
sin γ cos λ
+ q t
sin λ sin γ
Substituting Equation 8.20 this reduces to |t|
t
2
= 2m
2 − m
. 1
1 −
μ
p
− μ
d
tan γ 8.21
The limiting condition is when m = 1, and the greatest thickness reduction is
|t| t
2 max .
= 1
1 −
μ
p
− μ
d
tan γ 8.22
126 Mechanics of Sheet Metal Forming
∆ t
t
2
mYt
2
tan g ∆
t sin g
∆ t
g tan m
∆ t
m
p
q sin g
∆ t
m
d
q
tan g ∆
t q
sin g ∆
t q
a b
Figure 8.14 a Geometry of the deforming zone. b Forces acting on the deforming zone in ironing.
These equations give the magnitude of the thickness reduction, which is a positive quantity. It is seen that the maximum reduction is when μ
p
μ
d
, i.e. when, as mentioned above, the punch is slightly rough.
Quite large reductions are possible in this process and several ironing dies may be placed in tandem. In such a case, the dies may be spaced widely apart to ensure that the
wall has left one die before entering the next. This ensures that the entry stress is zero and prevents excessive wall stresses below the die. In ironing aluminium beverage cans, three
dies are used and the wall thickness is reduced to about a third of the original thickness. This increases the cup height greatly.
8.6 EXERCISES
Ex. 8.1 A fully work-hardened aluminium blank of 100 mm diameter and 1.2 mm thick-
ness has a constant flow stress of 350 MPa. It is deep-drawn in a constant thickness process to a cylindrical cup of 50 mm mid-wall diameter. The blank-holder force is 30 kN and the
friction coefficient is 0.1.
a What is the final height?
b What is the approximate value of the maximum punch force?
[Ans : a37 .5 mm, b57 kN.]
Ex. 8.2 The typical characteristic between drawing stress and punch travel for a strain-
hardening material was shown in Figure 8.6. Assuming the same die and punch geometry, and an efficiency of 100, show the effect of work-hardening on the punch load by
sketching two curves of drawing stress vs. punch travel for n = 0.1 and n = 0.2.
[Ans: With increasing n, the curve shifts to the right, but the maximum load decreases.]
Ex. 8.3. A sheet metal part is drawn over a die corner of radius ρ, and the frictional
coefficient between the sheet and die is μ. Calculate the difference in the tensions T
1
− T
2
neglecting change in the thickness. Cylindrical deep drawing 127
r
T
1
T
2
Ans : T
2
− T
1
= T
1
exp −
π 2
μ − 1
Ex. 8.4. Some years ago, a beer can had the approximate dimensions,
Diameter, d
f
= 62 mm Height,
h
f
= 133 mm Wall thickness,
t
f
= 0.13 mm Bottom thickness,
t = 0.41 mm initial blank thickness
Material yield stress, σ
f
= 380 MPa constant a
Assuming that drawing is a constant thickness process and that the efficiency is not likely to exceed 60. Can the cup be drawn to the final diameter in one operation?
b If the actual process uses two draws of equal drawing ratio. What is the intermediate
diameter, d
i
, and cup height, h
i
? c
Determine the punch force in the first draw, assuming 65 efficiency. d
What is the cup height at the finish of the second draw? e
What is done to achieve the full height? f
Over the years, the starting thickness has been reduced to 0.33 mm. If Al alloy costs 5kg, Al density is 2800 kgm
3
and 100 billion cansyear are made. How much money is saved? Assume blank diameter is unchanged.
[Ans: a No; b 86 mm, c 21.6 kN, d 42.6 mm, e Ironing, f 1.25 Billion]
128 Mechanics of Sheet Metal Forming
