F r
1
r
2
t
Die Punch
Retainer r
Figure 8.11 Forward redrawing of a deep-drawn cup.
Assuming that the yield tension T remains constant, then from Equation 7.10, the wall tension is
T
φ
= T ln r
1
r
2
= σ
f
t ln
r
1
r
2
8.13 It is shown later, in Section 10.5.1, that in plane strain bending or unbending under tension,
there will be an increase in the tension given by Equation 10.20. As an approximation here, the flow stress will be substituted for the plane strain yield stress, the efficiency η
taken as unity, and the term T T
y
neglected as the tension in redrawing is usually not very high. Thus for either a bend or unbend, the tension increase is
T
φ
≈ σ
f
t
2
4ρ 8.14
It may be seen from Figure 8.11, that there are two bend and two unbend operations in forward redrawing. Combining Equations 8.12–8.14, the redrawing force is
F = 2πt
T
φ
+ 4T = 2πr
2
tσ
f
ln r
1
r
2
+ t
ρ 8.15
This shows that the redrawing force increases with larger reductions and with smaller bend ratios ρt.
Another form of redrawing is shown in Figure 8.12. This is reverse redrawing and the cup is turned inside out. An advantage is that there is only one bend and one unbend
operation and the force is reduced to
F = 2πr
2
tσ
f
ln r
1
r
2
+ t
2ρ 8.16
124 Mechanics of Sheet Metal Forming
r
1
r
2
F
t r
Figure 8.12 Reverse redrawing of a cylindrical cup.
Reverse redrawing can require a smaller punch force if the difference between the initial radius and the final radius is large compared with the wall thickness. If it is small, the die
radius ratio ρt will also be small increasing the tension increase due to bending. With forward redrawing, the radii ρ can be greater than r
2
− r
1
2.
8.5 Wall ironing of deep-drawn cups
The wall of a deep-drawn cup can be reduced by passing the cup through a die, as shown in Figure 8.13. The clearance between the die and the punch is less than the initial thickness
of the cup wall. As the punch must remain in contact with the base of the cup, the velocity of the material as it exits the die, v
p
, is the same as the punch velocity. During ironing, there is no change in volume and the rate at which material enters the die equals the rate
leaving, therefore, 2π r
i
t
1
v
1
= 2πr
i
t
2
v
p
or v
1
= v
p
t
2
t
1
8.17 The punch is thus moving faster than the incoming material and the friction force between
the punch and the sheet is downwards. This assists in the process. The friction force between the die and the sheet opposes the process. It is an advantage in ironing to have a
high punch-side friction μ
p
, and for this reason the punch is often roughened slightly. On the other hand, the die-side friction μ
d
should be low and usually the outside of the cup is flooded with lubricant.
An approximate model can be created for ironing a rigid, perfectly plastic material with a flow stress σ
f
= Y = constant. Assuming a Tresca yield condition, the through-thickness stress at entry, where the axial stress is zero, is,
−Y , and q
= −σ
t
= Y 8.18
Cylindrical deep drawing 125
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
