and deformation becomes concentrated in necks. To illustrate the effect, we consider here a non-strain-hardening material having a tensile stress–strain rate curve
σ
1
= B ˙ε
m 1
5.15 where
˙ε
1
is strain rate, i.e. ˙ε
1
= dε
1
dt =
dl l dt
= v
l 5.16
In Equation 5.16, t is time, v the velocity or cross-head speed of the testing machine and l
is the length of the parallel reduced section of the test-piece. If we consider an imperfect strip as illustrated in Figure 5.4, then the force transmitted
through both regions is P
= σ
1
A = σ
1
+ dσ
1
A + dA
5.17 from which we obtain
dσ
1
σ
1
= − dA
A 5.18
Thus the difference in stress for the two regions is proportional to the magnitude of the imperfection. Differentiating the material law, Equation 5.15, we obtain
d ˙ε
1
˙ε
1
= 1
m dσ
1
σ
1
= − 1
m dA
A 5.19
This indicates that for a given imperfection, the difference in the strain rate between the imperfection and the uniform region is inversely proportional to the strain-rate sensitivity
index m. If m is small, as it is in most sheet metal at room temperature, the difference in strain rate will be large and the imperfection will grow rapidly. In a special class of
alloys, termed superplastic, m is unusually high, about 0.3; strips of these materials can be extended several hundred per cent as the growth of imperfections is very gradual. In some
non-metals, such as molten glass, m
∼ 1, and these can be drawn out almost indefinitely. In materials with low values of m, rate sensitivity will not greatly influence the max-
imum uniform strain, because, as shown in Figure 5.5, the strain and hence strain rate are approximately similar in both regions up to maximum load. Beyond this point, the
necking process will be affected by rate sensitivity and it is found that the post-uniform elongation is higher in materials with greater rate sensitivity.
5.4 Tensile instability in stretching continuous sheet
It is shown in the previous section that in a tensile strip, provided some imperfections exist, diffuse necking will start when the load reaches a maximum. On the other hand,
in sheet that is stretched over a punch, diffuse necking is not observed. The tension in the sheet may reach a maximum, but the punch will exert a geometric constraint on the
strain distribution that can develop. This is illustrated in Figure 5.7. If a diffuse neck did develop, the increased strain would lead to the sheet moving away from the punch and this
is implausible. It is possible that the strain may accelerate in some region compared with
Load instability and tearing 67
Non-viable shape Sheet
Punch
Figure 5.7 Diagram illustrating that a diffuse neck would lead to a non-viable strain distribution in
a continuous sheet.
another, but in a displacement controlled machine, this would not constitute an end-point in the process.
Observation shows that in continuous sheet, local necks do develop similar to those that occur within the diffuse neck of a tensile test-piece. The width of these local necks
is roughly equal to the thickness of the sheet and they will not influence the overall or global strain distribution. However, they lead very quickly to tearing of the sheet and
the end of the process. In designing sheet forming processes therefore, it is important to understand the conditions under which these local necks develop. The theory of local
necking is not fully developed, but a model of necking is given here that does predict many of the observed phenomena.
We consider a region of the sheet deforming uniformly in a proportional process as shown in Figure 5.8. The deformation in this region may be specified as
σ
1
; σ
2
= ασ
1
; σ
3
= 0 ε
1
; ε
2
= βε
1
; ε
3
= − 1 + β ε
1
5.20
T
1
T
2
1 1
s
1
s
2
t
Figure 5.8 Uniform deformation of part of a continuous sheet in a plane stress proportional process.
The principal tensions in the sheet are T
1
= σ
1
t and
T
2
= αT
1
= σ
2
t 5.21
These tensions will remain proportional during forming.
5.4.1 A condition for local necking
The condition postulated for local necking is that it will start when the major tension reaches a maximum. As the process is proportional, α and β will be constant. Differenti-
ating Equation 5.21, we obtain dT
1
T
1
= dσ
1
σ
1
+ dt
t =
dσ
1
σ
1
+ dε
3
= dσ
1
σ
1
− 1 + β dε
1
5.22 68 Mechanics of Sheet Metal Forming
