B
s
r
s
q
R
o
r
i
r
A C
s
r
i
Figure 8.2 Annular flange of a deep-drawn cup.
t r
d r
s
r
+ ds
r
t + dt
dq s
q
s
q
s
r
Figure 8.3 Element in the annular flange in Figure 8.2.
which reduces to, dσ
r
dr +
σ
r
t dt
dr −
σ
θ
− σ
r
r = 0
8.1 At the outer edge, point A, there is a free surface and σ
r
= 0; the stress state is therefore one of uniaxial compression in which σ
θ
= −σ
f
, where σ
f
is the current flow stress. At some intermediate radius, point B, the radial stress will be equal and opposite to the
hoop stress, while at the inner edge, point C, the radial stress will be a maximum. The stress states and the corresponding strain vectors are shown on the von Mises yield locus,
Figure 8.4. This is similar to the diagram, Figure 2.9. At the outer edge A the blank will thicken as it deforms, while at point B there will be no change in thickness. At the
inner edge C the sheet will thin. The overall effect is that in drawing, the total area of the material initially in the flange will not change greatly. This is a useful approximation
when determining blank sizes.
An incremental model can be constructed for drawing the flange using Equation 8.1 and the deformation followed using a numerical method. This is not developed here, but
we consider just the initial increment in the process. Using the Tresca yield condition, σ
θ
− σ
r
= − σ
f
8.2 118 Mechanics of Sheet Metal Forming
Thin Thicken
A B
C s
f
s
r
e
A
e
B
e
C
s
q
Figure 8.4 Stress state and strain vectors for different points on the flange.
where σ
f
is the initial flow stress and as the thickness initially is uniform, i.e. t = t
, Equation 8.1 can be integrated directly. Using the boundary conditions σ
r
= 0 at the outer radius R
o
and σ
r
= σ
r
i
, at the inner radius, r
i
, we obtain, σ
r
i
= − σ
f
ln r
i
R
o
, and
σ
θ
i
= − σ
f
− σ
r
i
8.3 For a non-strain-hardening material, the radial stress as given by Equation 8.3 is greatest
at the start and will decrease as the outer radius diminishes. The greatest stress that the wall of the cup can sustain for a material obeying the Tresca condition is σ
f
. Thus substituting σ
r
i
= σ
f
in Equation 8.3 indicates that the largest blank that can be drawn, i.e. the Limiting Drawing Ratio, has the value
R
o
r
i
= e ≈ 2.72 As indicated, this is an over-estimate, and some reasons for this are mentioned below.
8.2.1 Effect of strain-hardening
Due to strain-hardening, the stress to draw the flange may increase during the process, even though the outer radius is decreasing. As the flange is drawn inwards, the outer radius
will decrease and at any instant will be R as shown in Figure 8.5. Due to strain-hardening, the flow stress will increase and become non-uniform across the flange. If we assume an
average value σ
f av.
over the whole flange and neglect non-uniformity in thickness, then Equation 8.3 becomes
σ
r
i
= σ
f av.
ln R
r
i
8.4
R r
i
R
o
h
1
s
r
i
Figure 8.5 Part of a flange during the drawing process for frictionless conditions in which the stress
in the wall is equal to the radial stress at the inner radius σ
r
i
.
Cylindrical deep drawing 119
There are thus two opposing factors determining the drawing stress, one increasing the stress due to hardening of the material and the other reducing the stress as R becomes
smaller. Usually the drawing stress will increase initially, reach a maximum and then fall away as shown in Figure 8.6.
h s
r
i
Figure 8.6 Typical characteristic of drawing stress versus punch travel for a strain-hardening material.
8.2.2 Effect of friction on drawing stress
There are two separate ways in which friction will affect the drawing stress. One is at the die radius. Section 4.2.5 gives an analysis of an element sliding around a radius. If the
changes in thickness are neglected, Equation 4.13a can be written in terms of the stress; i.e.
dσ
φ
σ
φ
= μ dφ 8.5
At the die radius, as shown in Figure 8.7, we obtain by integration σ
φ
= σ
r
i
exp μ π
2 8.6
Friction between the blankholder and the flange will also increase the drawing stress. It is a reasonable approximation to assume that the blankholder force B will be distributed
around the edge of the flange as a line force of magnitude B2π R
o
per unit length, as shown in Figure 8.8. The friction force on the flange, per unit length around the edge, is
thus 2μB2π R
o
. This can be expressed as a stress acting on the edge of the flange, i.e. σ
r r
=R
o
= μB
π R
o
t 8.7
where t is the blank thickness.
σ
r
i
s
f
r
d
Figure 8.7 Sliding of the flange over the die radius.
120 Mechanics of Sheet Metal Forming
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
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
