Necking in biaxial tension
The same tension in the 1 direction is transmitted across both regions, therefore T
1
= σ
1A
t
A
= σ
1B
t
B
5.31 and consequently
t
B
t
A
= σ
1A
σ
1B
= f 5.32
For given initial conditions and stress strain curve, the deformation of both regions in Figure 5.12 can be analysed numerically. This will not be done here, but the salient
features of such an analysis will be illustrated. We consider the initial yielding as shown in Figure 5.13. If the material has a definite initial yield point σ
f
, then on loading, the groove will reach yield first as, from Equation 5.32, σ
1B
σ
1A
for f 1. The material in the groove cannot deform because of the geometric constraint, Equation 5.29, therefore
as the stress in A increases to reach the yield locus, the point representing the region B must move around the yield locus to B
as shown.
A B
1 1
s
f
s
1
a a
s
2
s
1A
s
1B
Figure 5.13
Initial yielding conditions for the uniform region and the imperfection.
1 1 de
1 A
de
2
b b
1 1
A B
de
B
de
A
a a
de
1 B
Figure 5.14 Strain vectors for the imperfection and the uniform region.
We now consider some increment in deformation, for which, from Equation 5.29, the increments parallel to the groove must be the same, i.e. dε
2A
= dε
2B
. The strain vectors for both regions are illustrated in the magnified view of the yield locus, Figure 5.14, noting
that these strain vectors are perpendicular to the yield surface. Because each region is now deforming under different stress and strain ratios, we note that the strain vector for the
groove has rotated to the left and for the same strain increment parallel to the groove, the strain increment across the groove will be greater than that in the uniform region A and
Load instability and tearing 73
the inhomogeneity will become greater, i.e. f will diminish. As shown in the insert on the right, dε
1B
dε
1A
. A numerical analysis will show that the strain in the groove will run ahead of that in
the uniform region, but only slightly while the tension is increasing. The effect gradually accelerates after the tension maximum and continues until the groove reaches a state of
plane strain as shown in Figure 5.15.
a b
s
1
a
s
2
s
f
1 1
B
f
A
f
A B
Initial yield locus
12 e
1
e
2
e
∗ 1B
e
∗ 1A
e
∗ 2A,B
b B
A 1
Figure 5.15 Growth of an imperfection from initial yielding to the limit at plane strain in a the
stress diagram and b the strain diagram.
When the stress state in the groove reaches plane strain, at B
f
in Figure 5.15a, the strain parallel to the groove ceases. The groove will then continue until failure tearing
and the strain in the uniform region ceases. This strain state, just outside the neck, is the maximum strain that can be achieved in this process for these values of α
and β and
the strains, ε
∗ 1A
and ε
∗ 2A
are known as the limit strains. If the analysis is repeated for different values of α
and β , a diagram can be established
in the biaxial strain region as illustrated in Figure 5.16. Combining this curve with the maximum tension line in the second quadrant, we obtain a curve indicating the onset
of local necking in both regions. This is known as the Forming Limit Curve and is a valuable material property curve; it is used frequently in failure diagnosis of sheet metal
forming. The shape of the curve depends on a number of different material properties and on the initial inhomogeneity chosen, i.e. f
o
. The homogeneity factor cannot be determined independently, but for small values of 1
− f
o
, the curve will intersect the major strain axis at about the value of the strain-hardening index n.
The forming limit curve describes a local process, necking and tearing, that is a material property curve dependent on the strain state, but not on the boundary conditions. The
object of sheet metal process design, therefore, is to ensure that strains in the sheet do not approach this limit curve. For the process to be robust and able to tolerate small changes in
material or process conditions, a safe forming region can be identified that has a suitable
74 Mechanics of Sheet Metal Forming
e
1
e
1
n 1
−1
Figure 5.16 The forming limit diagram satisfying the maximum tension criterion on the left-hand
side of the diagram and that derived from the imperfection analysis on the right.
margin between it and the limit curve. Different materials will have different forming limit curves and in the following section the effect of individual properties will be outlined.