A condition for local necking
When the tensions reach a maximum, Equation 5.22 becomes zero and the non-dimensional strain-hardening is,
1 σ
1
dσ
1
dε
1
= 1 + β 5.23
This is only valid for β −1. If β −1, the sheet will thicken and for a strain-hardening
material, the tension will never reach a maximum. For a material that obeys the generalized stress–strain law
σ = σ
f
= Kε
n
5.24 it may be shown by substituting Equations 2.18b and 2.19c in Equation 5.24 that the
relationship between principal stress and strain during proportional deformation in which α
and β are constant may be written as σ
1
= K
′
ε
n 1
5.25 where K
′
is a material constant that can be calculated from K, n, α and β. Differentiating Equation 5.25 shows that
1 σ
1
dσ
1
dε
1
= n
ε
1
and substituting in Equation 5.23, we obtain ε
∗ 1
= n
1 + β
and ε
∗ 2
= βn
1 + β
or ε
∗ 1
+ ε
∗ 2
= n 5.26
where the star indicates the strain at maximum tension. From Equation 5.26, the line in the strain diagram representing maximum tension is plotted in Figure 5.9a; this has a
slope of 45 degrees and intercepts the major strain axis at a value of n.
e
1
e
2
e
2
e
1
n Maximum tension
condition Experimental necking
strain d
T = 0 1
1 2
n
a b
− n, 2n
Figure 5.9 a Strains at the maximum tension in a continuous sheet. b Experimentally observed
necking strains in sheet.
Load instability and tearing 69
In the tensile test, β = −12, and maximum tension occurs when
1 σ
1
dσ
1
dε
1
= 1
2 =
n ε
1
or ε
∗ 1
= 2n If the maximum tension condition does signify the onset of local necking as hypothesized,
then the local necking strain in uniaxial tension is ε
∗ 1
= 2n; this is twice the strain for the load maximum and the start of diffuse necking as given by Equation 5.9. This agrees
with observation for low carbon steel sheet. As the diffuse neck develops in a tensile strip, the deformation in the neck will be approximately uniaxial tension and after some further
straining a local neck will develop in the diffusely deforming neck.
We now compare the strain at maximum tension with experimentally determined strains at the onset of local necking. These strains could be measured from grid circles close to
a local neck in a formed part. If strains are measured for many different strain paths in both quadrants of a strain diagram, such as Figure 5.9a, it is possible to establish a
‘Forming Limit Curve’ FLC that delineates the boundary of uniform straining and the onset of local necking. In practice, there is some scatter in measured necking strains and
instead of a single curve there is a band within which necking is likely to occur. Here we will consider the limit as a single curve. For materials with a similar strain-hardening
index n it is found that in the second quadrant of Figure 5.9a, i.e. for strain paths in which β 0, the experimental Forming Limit Curve is approximately coincident with the
maximum tension line. In some cases it lies a little above the line shown; the difference may be due to the effect of sheet thickness, friction, contact pressure and grid circle size on
the necking process and its measurement. As a first approximation, we conclude that the maximum tension criterion does provide a reasonable theoretical model for local necking
strains in the second quadrant β 0.
If both the major and minor strains are positive and the process is one of biaxial stretching with β 0, the experimental Forming Limit Curve does not follow the maxi-
mum tension line. A typical example is shown in Figure 5.9b. This suggests that in this quadrant there is some process that stabilizes or slows down necking after the tension has
reached a maximum, and this is examined in a later section.
We anticipate that the local neck in Figure 5.10 would occur along a line of pre-existing weakness at a limiting strain in the uniform region that is approximately that given in
Equation 5.26. If we identify the uniform region as A and the imperfection as B then certain conditions have been assumed in the analysis of the necking process. These are:
• the stress and strain ratios must remain constant, as assumed in the differentiation, both before and during the necking process;
• for the process to be a local one, the necking process should not affect the boundary conditions in Figure 5.10.
The second condition ensures that the neck must take the form of a narrow trough in the sheet, as in Figure 5.10, rather than as a patch or diffuse region that would influence
conditions away from the neck. Once the necking process becomes catastrophic, in the sense that the uniform region A ceases to strain, the strain increment parallel to the neck,
in the y direction in Figure 5.10, will be zero. Geometric constraint requires that the strain increment along the neck must be equal to that in the same direction just outside it; i.e. the
strain increment in the y direction in both regions A and B along the neck, must be zero. The first condition above requires that the strain ratio does not change, the second that it
70 Mechanics of Sheet Metal Forming
t s
2
s
1
q
A B
x y
Figure 5.10 A local neck formed in a continuous sheet oriented at an angle θ to the maximum
principal stress.
1+b 2
, 2q
1 2
dg
de
1
de
1
de
b de
1
Figure 5.11 Mohr circle of strain increment to determine the angle of zero extension.
is zero during necking, therefore the strain increment in the, y,direction must be zero at all times, i.e. the neck can develop only along a direction of zero extension. This direction
of the neck can be found from the Mohr circle of strain increment shown in Figure 5.11. The centre of the circle is at
1 + β
2 dε
1
and the radius of the circle is 1
− β 2
dε
1
The direction of zero extension, dε
y
= 0, is given by cos 2θ
= 1
+ β 1
− β 5.27
For uniaxial tension, β = −12, we find that the angle the neck makes is θ = 55
◦
and for plane strain, β
= 0, the neck is perpendicular to the maximum principal stress, θ = 90
◦
. If β 0, there is no direction in which the extension is zero.
The analysis above shows that if there is a direction in which there is no extension, local necking along the direction of zero extension is possible when the tension reaches a max-
imum and the differentiation as in Equation 5.22 where α and β are constant is valid. If
Load instability and tearing 71
there is no direction of zero extension, for example in a stretching process in which β 0, the strains at which the tension is a maximum are still given by Equation 5.26, but geomet-
ric constraints prevent the instantaneous growth of local necks. Therefore in Figure 5.9a, the line shown predicts maximum tension in both quadrants, but only indicates the onset of
local necking in the second quadrant in which the minor strain is negative. The diagrams, Figure 5.9 a and b also suggest that at plane strain where β
= 0 and ε
2
= 0, the major strain at necking is a minimum.