re-entrant shape is required, this is formed when the punch bottoms on the lower die or counterpunch shown in Figure 4.1c.
It is important to realize that even though matching dies may be used, the sheet is not compressed between them as in a forging process, but is stretched over each convex tool
surface. Contact with the tooling is, for the most part, on only one side of the sheet; there is no through-thickness compression. In most regions the contact pressure is small compared
with the flow stress of the sheet and it is usually acceptable to neglect through-thickness stresses and assume plane stress deformation.
Stamping is the basic process for forming parts whose shape cannot be obtained simply by bending or folding. For most autobody parts, the sheet is first formed to shape in a draw
die in a double-acting press, i.e. one having separate clamping and punch actions as in Figure 4.1. Secondary forming and blanking operations may be carried out in subsequent
presses. The shapes that are formed may be quite complex and the process is a three- dimensional one. Nevertheless, in this work we shall first consider very simple geometry
as shown in Figure 4.2 and assume that this is a two-dimensional process, i.e. the strain perpendicular to the plane of the diagram is zero and the deformation is both plane stress
and plane strain.
F B
R
P
R
F
R
D
Figure 4.2 Simple draw die with the punch face having a circular profile.
4.2 Two-dimensional model of stamping
As the die in Figure 4.2 is symmetric, we consider only one half as shown in Figure 4.3. The punch has a cylindrical shape of radius R
F
; the other features are defined in Figure 4.3 and are listed below:
q q
a B
D F
h t
O A
B C D
c
b E
e f
F G
Figure 4.3 Half-section of a partially drawn strip in the die shown in Figure 4.2.
46 Mechanics of Sheet Metal Forming
a punch semi-width,
b blank semi-width
c side clearance
e land width
f width of frictional clamping
simulating a draw-bead h
punch penetration part depth t
blank thickness R
F
punch face radius R
P
punch corner radius R
D
die corner radius subscript, 0, denotes the initial value.
Initially we do not consider draw-beads, but assume that the restraint is created only by friction acting over a specified area of the binder. At some depth of punch penetration, h,
the different zones are shown in Figure 4.3 and listed below:
OB material in contact with the punch,
BC unsupported sheet in the side wall,
CD sheet in contact with the die corner radius,
DE sheet on the die land without contact pressure,
EF region over which the blank-holder force acts,
FG free edge of the blank.
We assume that all regions of the sheet from the centre-line O to the edge of the clamping area F are plastically deforming. Often this will not be true as some regions may cease
to deform even though the punch is still moving downwards. From the centre O to the point of tangency B the sheet is stretching and sliding outwards against friction and the
friction force on the sheet acts towards O. From the point of contact with the die C to the point F the sheet is sliding inwards and the friction force on the sheet acts outwards.
If either the blank-holder force B or the strain at the centre is specified, it is possible to determine all other variables in a two-dimensional process.
4.2.1 Strain of an element
At any instant during stamping, the thickness and also the stress and tension will vary over the part. If we consider an infinitesimal element as shown in Figure 4.4, the conditions at
a point will be as follows. The principal direction, 1, is in the sectioning plane and the direction 2 is perpendicular
to this. As the process is assumed to be one of plane strain, β = 0, the strain state is
ε
1
; ε
2
= βε
1
= 0; ε
3
= −1 + βε
1
= −ε
1
4.1 The effective strain in the element is, from Equation 2.19c,
ε =
4 3
1 + β + β
2
ε
1
= 2
√ 3
ε
1
4.2 Simplified stamping analysis 47
