is considered in a subsequent chapter. The straight sides can be modelled approximately as two-dimensional sections, except that the deformation over the face of the punch is
no longer plane strain, but biaxial stretching with a strain ratio in the range 0 β 1. Useful information can be determined from a simple model and this is demonstrated in
the following worked example.
4.4.1 Worked example Stamping a rectangular panel
A rectangular pan as shown in Figure 4.11 is drawn in a die. The base of the panel is flat and it is specified that the base should be stretched uniformly so that a lip of 4 mm height
should be drawn up around the edge as shown in Figure 4.12. Determine the side-wall tensions T
1
and T
2
and the punch force required for the following conditions: initial sheet thickness
0.90 mm stress strain law
σ = 700 0.009 + ε
0.22
MPa
Figure 4.11
One half of a typical rectangular stamping.
800 400
400 4
1 2
4 200
4 mm
T
1
T
2
Figure 4.12 Diagram of a rectangular pan drawn to a specified strain.
Solution.
For the semi-axes in Figure 4.12b, the sheet is stretched so that ε
1
≈ ln 204
200 = 0.0198
and ε
2
≈ ln 404
400 = 0.010
4.22 the strain ratio is
β =
ε
2
ε
1
= 0.010
0.0198 = 0.505
4.23
58 Mechanics of Sheet Metal Forming
and the stress ratio is, from Equation 2.14, α
= 2β
+ 1 2
+ β = 0.802
4.24 From Equations 2.18b and 2.19, we obtain
σ = 0.917σ
1
and ε
= 1.532ε
1
4.25 The effective stress at the end of the process is
σ = 700 0.009 + 1.532 × 0.0198
0.22
= 344 MPa and hence σ
1
= 3440.917 = 375 MPa and σ
2
= α.σ
1
= 0.802 × 375 = 301 MPa. The thickness is
t = t
exp [ − 1 + β ε
1
] = 0.9 exp [− 1 + 0.505 0.0198] = 0.874 mm
and as T = σ t,
T
1
= 375 × 0.874 = 326 kNm and T
2
= 301 × 0.874 = 263 kNm Summing the tension around the side-wall and neglecting frictional effects, the punch force
is, approximately, F
= 2 326 × 0.8 + 263 × 0.4 = 732 kN = 0.73 MN
4.5 Exercises
Ex. 4.1 A material is deforming in plane strain under a major tension of 340 kNm.
The initial thickness is 0.8 mm and the material obeys an effective stress–strain relation σ
= 700ε
0.22
MPa. What is the major strain at this point? [Ans: 0.062]
Ex. 4.2 In the two-dimensional stamping operation shown in Figure 4.13, the side-walls
are vertical and the face of the punch is flat. If the blank-holder force B is increased, determine the maximum strain that can be achieved at the centre-line if the coefficient of
friction is 0.15 and the sheet obeys the stress strain law σ = 600ε
0.2
MPa. What is the blank-holder force required to reach this if the initial sheet thickness is 0.8 mm?
[Ans: 0 .026 ; 2 B = 1780 kNm]
B
B e
1 × max.
Figure 4.13 Section of a draw die with vertical sidewalls.
Simplified stamping analysis 59
Ex. 4.3 For the operation in Ex. 4.2, determine the punch force at the maximum side-wall
tension and obtain the ratio of blank-holder force to punch force. Ans: 676 kNm; 2.6]
Ex. 4.4 At a point in a stamping process the sheet that is in contact with the punch
is shown in Figure 4.3, but the sheet makes an angle of 60
◦
at the tangent point. The strain at the mid-point O is 0.025. The punch has a face radius of 2 m, semi-width of
600 mm, and corner radius of 10 mm. The material obeys an effective stress strain relation of σ
= 400ε
0.17
MPa and the initial thickness is 0.8 mm. Determine the tension at O, A and B if the coefficient of friction is 0.10.
[Ans: 197, 203 and 219 kNm]
60 Mechanics of Sheet Metal Forming
5
Load instability and tearing
5.1 Introduction
The previous chapters showed how the plastic deformation of an element can be followed during a metal forming operation. At some instant, the process may be limited or terminated
by any one of a number of events and part of the analysis of these operations includes the prediction of process limits. Each process will have its own limiting events and while it
may not be easy at first sight to anticipate which event will govern, it is likely to be one of the following.
• Inability of the sheet to transmit the required force. In the deep drawing process shown in the Introduction in Figure I.9, the force required to draw the flange inward may
exceed the strength of the cup wall. This occurs when the tension force per unit length around the circumference reaches a maximum and this will also be seen as a
maximum in the punch force. This type of limit is often termed a global instability as the whole process must be considered. A similar situation happens in the tensile
test. The object of the test is to obtain uniform deformation in the gauge length from which mechanical properties can be measured. When the load reaches a maximum,
deformation becomes concentrated in a diffuse neck and is no longer uniform. Actual failure of the strip happens at a later stage, but the uniform deformation process is
limited by the load maximum and this is also a global instability.
• Localized necking or tearing. The appearance of any local neck that rapidly leads to tearing and failure will obviously terminate a forming operation. This can be consid-
ered as local instability that can be analysed by considering a local element without involving the whole process.
• Fracture. It is possible for a plastically deforming element to fracture in almost a brittle manner. This is not common in sheet used for forming and is often preceded by
some local instability. Some instances where fracture may have to be considered will be mentioned in this chapter.
• Wrinkling. If one principal stress in an element is compressive, the sheet may buckle or wrinkle. This is a compressive instability and resembles the buckling of a column.
In sheet it is difficult to predict and will not be studied here. In a complicated process such as stamping an irregular part, the overall punch force
is made up of the loads created in forming a number of different regions. It may not be possible to identify the onset of a load maximum in any one region and even if the total
61
load reaches a maximum, this may not constitute a limit in the process. Most forming machines are displacement controlled and the motion of the punch is not determined by
the load acting on it, but by the mechanism that drives it. In other words, the machine will not go out of control because the punch force versus displacement characteristic reaches
a maximum.
Global instabilities such as in deep drawing are predicted by analysing the process as a whole and will be examined in sections dealing with these operations. On the other hand,
local instabilities, as the name implies, can be understood by studying the deformation of a single element. In sheet forming, the instability that is most likely to occur is the sudden
growth of a local neck leading to tearing. Under some conditions local necking will occur when the tension reaches a maximum, but in other conditions, a tension maximum may
not be associated with the sudden growth of a local neck or catastrophic failure.
5.2 Uniaxial tension of a perfect strip
We first consider the theoretical case of a parallel strip of metal, as in the gauge length of a tensile test-piece. We consider that the properties are uniform throughout and the
geometry is perfect. When this is stretched in tension as shown in Figure 5.1, the volume remains constant and the following relations apply. The cross-sectional area is A
= wt and the volume is
Al = A
l 5.1
P A
t w
l P
s
1
Figure 5.1
Diagram of a perfect strip deformed in uniaxial tension.
Differentiating Equation 5.1, we obtain dA
A +
dl l
= 0 or
dl l
= dε
1
= − dA
A 5.2
The strain in the strip is ε
1
= ln l
l 5.3
and the stress is σ
1
= P
A =
P A
l l
5.4 62 Mechanics of Sheet Metal Forming
