A A
A O
O O
a b
c de
de de
Y Y
Y s
f
s
f
s
f
s
f
s
q
s
q
s
q
s
f
s
f
Figure 8.9 Loading path for the cup wall for different yield criteria. a The Tresca condition. b The
von Mises condition. c An anisotropic yield locus for a material with an R-value 1.
For the von Mises yield condition b, the limiting stress in the wall is σ
φ
= 2 √
3Y and the predicted Limiting Drawing Ratio will be greater. If the material exhibits anisotropy,
the yield surface will be distorted. For the case in which the strength of the sheet is higher in the through-thickness direction compared with that in the plane of the sheet, i.e. the
R-value is greater than unity, a quadratic yield locus will be elongated along the right-hand diagonal as shown in Figure 8.9c. The effect is to strengthen the wall, so that a higher
stress is required to yield it and therefore the Limiting Drawing ratio will be greater. The increase in LDR predicted from a high exponent yield criterion for a high R value as
shown in Section 5.5.5 would be less.
It has been assumed in Figure 8.7 and Equation 8.6 that the thickness of the sheet will not change as it is drawn over the die corner radius. This is not true, and as shown in
Section 10.5.2, when sheet is bent or unbent under tension there will be a reduction in thickness. From Equation 10.21, at each bend or unbend, the thickness reduction is
t t
= − 1
2ρt T
T
y
where T
y
is the tension to yield the sheet and ρt the bend ratio at the die corner. Thus a small bend ratio will increase the thickness reduction, reducing the load-carrying capacity
of the side-wall and reducing the Limiting Drawing Ratio. The largest size blank that can be drawn is therefore significantly less than that predicted
by the simple analysis and is usually in the range of 2.0 to 2.2. The relations above indicate that, qualitatively, the Limiting Drawing Ratio is:
reduced by • a higher blank-holder force B;
• greater strain-hardening, because the rate of increase in the average flow stress in the flange will be greater than the strengthening of the cup wall;
and increased by • better lubrication reducing the friction coefficient μ.;
• a more ample die corner radius, increasing the bend ratio ρt; • anisotropy characterized by R1.
122 Mechanics of Sheet Metal Forming
8.3 Cup height
If a disc is drawn to a cylindrical cup, the height of the cup wall will be determined principally by the diameter of the disc. As indicated above, during drawing the flange, the
outer region will tend to thicken and the top of the cup could be greater than the initial blank thickness, as illustrated in an exaggerated way in Figure 8.10.
The thinnest region will be near the base at point E where the sheet is bent and unbent over the punch corner radius. At some point mid-way up the wall, the thickness will
be the same as the initial thickness. An approximate estimate of the final cup height is obtained by assuming that it consists of a circular base and cylindrical wall as shown on
the right-hand side of the cup diagram and that the thickness is everywhere the same as the initial value. By equating volumes,
π R
2
t = πr
i
t + 2πr
i
t h
r
i
E h
R
t
Figure 8.10 A disc of initial radius, R
, and thickness, t , drawn to a cylindrical cup of height h.
and the cup height is given by h
≈ r
i
2 R
o
r
i 2
− 1 8.11
As indicated in Section 8.2.3, the drawing ratio R
o
r
i
, is usually less than about 2.2; Equation 8.11 shows that the cup height for this ratio is nearly twice the wall radius, or
the height to diameter ratio of the cup is just less than unity. Deeper cups can be obtained by redrawing as described below.
8.4 Redrawing cylindrical cups
In Figure 8.11, a cup of radius r
1
and thickness t is redrawn without change in wall thickness to a smaller radius r
2
. If the tension in the wall between the bottom of the punch and the die is T
φ
, the force exerted by the punch is F
= 2πr
2
T
φ
8.12 Cylindrical deep drawing 123
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
