t p
R
o
t m
B 2p
R
o
2p R
o
2pR
o
B
B m
B s =
Figure 8.8 Friction arising from the blankholder force, assumed to act at the outer edge.
Both of these factors will increase the stress required to draw the flange. This stress can be determined by various numerical techniques or by approximate models that simplify
the effect of strain-hardening and thickness change in the flange Integrating Equation 8.4 for the new boundary condition, given by Equation 8.7, we
obtain σ
r
i
= σ
f av.
ln R
o
r
i
+ μB
π R
o
t 8.8
and applying Equation 8.6, the stress in the wall is σ
φ
= 1
η σ
f av.
ln R
r
i
+ μB
π Rt exp
μπ 2
8.9 Equation 8.9 is an approximate one that neglects, among other things, the energy dissipated
in bending and unbending over the die radius. For this reason, an efficiency factor η has been added; this will have a value less than unity.
8.2.3 The Limiting Drawing Ratio and anisotropy
To determine the Limiting Drawing Ratio some method of determining the average flow stress σ
f av.
, the current thickness and the maximum permissible value of wall stress in Equation 8.9 is necessary. This is beyond the scope of this work, but some comments can
be made about the influence of different properties on the Limiting Drawing Ratio. If, in the first instance, we neglect strain-hardening, then the maximum drawing stress
will be at the start of drawing; if the flow stress is σ
f
= Y = constant, the wall stress to initiate drawing is, from Equation 8.9,
σ
φ
= 1
η Y
ln R
o
r
i
+ μB
π R
o
t exp
μπ 2
8.10 As indicated, this must not exceed the load-carrying capacity of the wall. If the wall
deforms, the punch will prevent circumferential straining, so the wall must deform in plane strain. The stress at which it would deform depends on the yield criterion. Figure 8.9
illustrates yielding in plane strain for various criteria. The line OA indicates the loading path in the cup wall.
For a Tresca yield condition, Figure 8.9a, the stress in the wall, σ
φ
, will have the value Y
and substitution in Equation 8.10 gives the condition for the maximum blank size, i.e. ln
R
o
r
i
+ μB
π R
o
t Y
exp μπ
2 = η
Cylindrical deep drawing 121
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
