5.5.2 Rate sensitivity
It was shown above that in the tensile test, rate sensitivity will not affect the strain at which the tension reaches a maximum, but it will influence the rate of growth of a neck.
In biaxial stretching, it has been shown that necking is a gradual process beyond the maximum tension condition and is controlled by the shape of the yield locus. In this
region, rate sensitivity will delay growth of the neck as shown in Figure 5.18a. As shown in Figure 5.18b, the forming limit curve for a material with a high rate sensitivity
could intercept the major strain axis at a strain greater than n.
e
1
e
1
e
2
e
2
e
1
, e
2
m m
A B
B
a b
n High
m High
Low m
Low
Figure 5.18 Diagram showing the effect of rate sensitivity on a the rate of growth of a neck and
b on the forming limit curve.
5.5.3 Ductile fracture
In the discussion above, it was assumed that tearing in the sheet came about after the necked region B had reached a state of plane strain and that this neck would then proceed
to failure without further straining in the uniform region A. In many ductile materials, this is the case and the actual strain at which the neck fractures will not influence the limit strain
in A. In less ductile materials, the material within the neck may fracture before plane strain is reached as shown in Figure 5.19; this will reduce the limit strains ε
∗ 1A
and ε
∗ 2A
. Fracture in ductile materials often results from intense localization of strain on planes of maximum
shear. It is possible to measure these fracture strains and plot them on a strain diagram similar to the forming limit. If these fracture curves are well away from the forming limit
curve, it may reasonably be assumed that they will not influence the limit strains.
5.5.4 Inhomogeneity
As mentioned, inhomogeneity has not been well characterized in typical sheet. It may be expected that the greater the imperfection, the lower will be the limit strain Figure 5.20,
so that with large imperfections, the plane strain limit strain may be less than the strain- hardening index n. In this work, the imperfection has been expressed in terms of a local
76 Mechanics of Sheet Metal Forming
e
1
e
2
e
2
e
1
High e
f
Low e
f
Figure 5.19 Diagram showing the effect of fracture strain ε
f
on the limit strains.
e
1
e
2
Small imperfection
Large imperfection
Figure 5.20
Effect of the magnitude of the imperfections on the forming limit curves.
reduction in thickness, but other forms of imperfection are possible, such as inclusions, local reductions in strength due to segregation of strengthening elements or texture vari-
ation. Surface roughness may also be a factor. Whatever the form of the imperfection, it will also have a distribution both spatially and in size population; as the critically strained
regions may only occupy a small area of the sheet, there is also a probabilistic aspect. The critical region may, or may not contain a large defect and therefore there is likely to
be some scatter in measured limit strains and the forming limit curve is more properly a region of increasing probability of failure.
5.5.5 Anisotropy
The shape of the yield locus is shown to influence the forming limit in biaxial tension. This locus changes if the material becomes anisotropic. If a quadratic yield function is used, as
in Equation 2.11, anisotropy in the sheet, characterized by an R-value 1, will cause the locus to be extended along the biaxial stress axis as shown in Figure 5.21a. The effect
of this in a numerical analysis of the forming limit would be to reduce the biaxial strain limit. This is not observed experimentally and it appears that a different yield function
employing higher exponents, 6–8, is more realistic for certain materials. The shape of a yield locus for a high exponent law is shown in Figure 5.21b and for such a model it
Load instability and tearing 77
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