The load in the strip is P = σ
1
A ; as the strip deforms, σ
1
will increase for a strain- hardening material and the cross-sectional area will decrease, i.e. dσ
1
will always be positive and dA will be negative. At some stage, the rate of strain-hardening will fall
below the rate of reduction in area and the load will reach a maximum. At this instant, dP
= d σ
1
A = 0
or dP
P =
dσ
1
σ
1
+ dA
A = 0
5.5 Combining with Equation 5.2, we obtain the condition for maximum load as
1 σ
1
dσ
1
dε
1
= 1 5.6
The function on the left-hand side of Equation 5.6 is a material property that is known as the non-dimensional strain-hardening characteristic and it could be determined from a
material test. If the material obeys a simple power law, σ
1
= Kε
n 1
, similar to that introduced in Section 3.5.1, this function is
1 σ
1
dσ
1
dε
1
= nK
σ
1
ε
1 n
−1
= n
ε
1
5.7 The form of this curve is illustrated in Figure 5.2.
ds
1
de
1
1 n
e
1
1 s
1
Figure 5.2 Non-dimensional strain-hardening curve for a power law material.
Combining Equations 5.6 and 5.7, 1
σ
1
dσ
1
dε
1
= n
ε
1
= 1 5.8
and the strain at maximum load is ε
∗ 1
= n 5.9
where the star denotes the maximum load condition. This point in the diagram is illustrated in Figure 5.2.
Equation 5.6 is the well-known Considere condition for maximum load in a bar or strip in tension. If the bar is part of a load-carrying structure, then this is a significant event.
It is often stated as the condition for the start of diffuse necking in a tensile strip. This may be the case in a real strip, but in a perfect strip, unlike an actual test-piece, diffuse
Load instability and tearing 63
necking in one region cannot occur as all elements in a perfect strip must behave in an identical fashion and so uniform deformation would always exist.
For a perfect strip in which the material obeys the power law expression for the stress–strain curve, the load to deform the strip may be calculated, i.e.,
P = σ
1
A = Kε
n 1
A l
l = KA
ε
n 1
exp −ε
1
5.10 The load in Equation 5.10 for a given material is a function of material properties, initial
cross-sectional area and axial strain and is plotted as a function of strain in Figure 5.3.
P
e
1
n
Figure 5.3
Variation of load with strain in a perfect strip.
If we compare the theoretical load–strain curve for a perfect strip as shown in Figure 5.3 with that from a real tensile strip, as in Figure 1.2, we see there is a significant difference.
In a real specimen, the strip starts to neck at the maximum load and the load then falls off much more rapidly than in Figure 5.3. In order to understand the actual phenomenon in a
real strip, we must include an imperfection in the strip and then analyse the deformation.
5.3 Tension of an imperfect strip
We consider a tensile strip in which a slight imperfection exists. This can be characterized by a short region having initially a slightly smaller cross-sectional area; i.e. if the initial
area of most of the strip is A , then the imperfection is initially of area A
o
+ dA where
dA is a small negative quantity. At some stage in the deformation, the strip will be as
illustrated in Figure 5.4.
A
,
s
1
,
e
1
A
+ d
A s
1
+ d
s
1
e
1
+ d
e
1
P P
Figure 5.4 Tension of an imperfect strip.
64 Mechanics of Sheet Metal Forming
e
1
+
de
1
=
n
e
1
,
e
1
+
de
1
e
1U
P F
G Uniform region
P
max.
Load Imperfection
Strain
Figure 5.5 Load, strain diagram for the uniform region and the imperfection in a tensile strip.
The same load is transmitted by the uniform region and the imperfection and, following Equation 5.10, this can be written as
P = KA
ε
n 1
exp −ε
1
= K A + dA
ε
1
+ dε
1 n
exp[ − ε
1
+ dε
1
] 5.11
The load versus strain curves for each region are illustrated in Figure 5.5, noting that the strain in the imperfection is ε
1
+ dε
1
. The imperfection will reach a maximum load, following Equation 5.9, when the strain
is ε
1
+ dε
1
= n. At this load, the uniform region is only strained to the point G and the strain is ε
1U
n ; this is known as the maximum uniform strain as it is the strain measured
in the uniform region after the test. It may be seen, from Figure 5.5 that the uniform region cannot strain beyond the point G as it would require a higher load than can be transmitted
by the imperfection. If the test is continued beyond the maximum load P
max .
, only the imperfection will deform and it will do this under a falling load. All real tension strips
will contain some imperfections even if they are very slight. The greatest imperfection will become the site of the diffuse neck and once the maximum load-carrying capacity is
reached in the imperfection or neck, all the deformation is concentrated in the neck and the uniform region will unload elastically as the load falls. The diffuse neck that is seen
in actual test-pieces usually extends for a distance approximately equal to the width of the strip and its deformation will only contribute a small amount to the overall elongation. The
curve of load versus total elongation, therefore, will fall more rapidly than for a perfect strip, as shown in Figure 5.6. It should be noted that the post-uniform elongation will also
be dependent on the ratio of the gauge length to the strip width. Other things being equal, if the gauge length is much larger than the strip width, the length of the diffuse neck will
be a small fraction of the gauge length and the post-uniform extension will be small. For tests in which the gauge length is more nearly equal to the strip width, the elongation will
be greater.
The difference between the maximum strain in the uniform region of an imperfect strip and the strain ε
1
= n at maximum load in a perfect strip can be found approximately for a material obeying the power law model σ
1
= Kε
n 1
. If, in Equation 5.11, we substitute ε
1
+ dε
1
= n for the strain at maximum load in the imperfection and ε
U
for the strain in the uniform region, then we obtain
ε
U
n
n
exp n − ε
U
= 1 + dA
A 5.12
In real cases, both n − ε
U
and dA
o
A will be small quantities. It is shown in a Note at
the end of this chapter that by taking the first terms only in the series expansions for the Load instability and tearing 65
