the maximum load, using the fact that plastic deformation in metals and alloys takes place without any appreciable change in volume. The volume of the gauge section is constant,
i.e.
A l
= Al 1.8
and the true stress is calculated as σ
= P
A l
l 1.9
If, during deformation of the test-piece, the gauge length increases by a small amount, dl, a suitable definition of strain is that the strain increment is the extension per unit current
length, i.e.
dε =
dl l
1.10 For very small strains, where l
≈ l , the strain increment is very similar to the engineering
strain, but for larger strains there is a significant difference. If the straining process con- tinues uniformly in the one direction, as it does in the tensile test, the strain increment can
be integrated to give the true strain, i.e.
ε =
dε =
l l
dl l
= ln l
l 1.11
The true stress–strain curve calculated from the load–extension diagram above is shown in Figure 1.5. This could also be calculated from the engineering stress–strain diagram
using the relationships
σ =
P A
= P
A A
A = σ
eng.
l l
= σ
eng.
1 +
e
eng.
100 1.12
50 100
150 200
250 300
350 400
0.000 0.050
0.100 0.150
0.200 0.250
True strain True stress, MPa
s
f
e
u
Figure 1.5 The true stress–strain curve calculated from the load–extension diagram for drawing
quality sheet steel.
6 Mechanics of Sheet Metal Forming
and ε
= ln 1
+ e
eng.
100 1.13
It can be seen that the true stress–strain curve does not reach a maximum as strain- hardening is continuous although it occurs at a diminishing rate with deformation. When
necking starts, deformation in the gauge length is no longer uniform so that Equation 1.11 is no longer valid. The curve in Figure 1.5 cannot be calculated beyond a strain corre-
sponding to maximum load; this strain is called the maximum uniform strain:
ε
u
= ln 1
+ E
u
100 1.14
If the true stress and strain are plotted on logarithmic scales, as in Figure 1.6, many samples of sheet metal in the soft, annealed condition will show the characteristics of this
diagram. At low strains in the elastic range, the curve is approximately linear with a slope of unity; this corresponds to an equation for the elastic regime of
σ = Eε or log σ = log E + log ε
1.15
1 1.5
2 2.5
3 3.5
−3.5 −3
−2.5 −2
−1.5 −1
−0.5 Log strain
Log stress 1
1 n
log K
Elastic Plastic
Figure 1.6 True stress–strain from the above diagram plotted in a logarithmic diagram.
At higher strains, the curve shown can be fitted by an equation of the form σ
= Kε
n
1.16a or
log σ = log K + n log ε
1.16b The fitted curve has a slope of n, which is known as the strain-hardening index, and an
intercept of log K at a strain of unity, i.e. when ε = 1, or log ε = 0; K is the strength
coefficient. The empirical equation or power law Equation 1.16a is often used to describe the plastic properties of annealed low carbon steel sheet. As may be seen from Figure 1.6,
Material properties 7
it provides an accurate description, except for the elastic regime and during the first few per cent of plastic strain. Empirical equations of this form are often used to extrapolate
the material property description to strains greater than those that can be obtained in the tensile test; this may or may not be valid, depending on the nature of the material.
1.1.4 Worked example tensile test properties
The initial gauge length, width and thickness of a tensile test-piece are, 50, 12.5 and 0.80 mm respectively. The initial yield load is 1.791 kN. At a point, A, the load is 2.059 kN
and the extension is 1.22 mm. The maximum load is 2.94 kN and this occurs at an extension of 13.55 mm. The test-piece fails at an extension of 22.69 mm.
Determine the following: initial cross-sectional area,
= 12.5 × 0.80 = 10 mm
2
= 10
−5
m
2
initial yield stress, =
1.791 × 10
3
10
−5
= 179 × 10
6
P a = 179 MPa
tensile strength, = 2.94 × 10
3
÷ 10
−5
= 294 MPa total elongation,
= 22.6950 × 100 = 45.4 true stress at maximum load,
= 29450 + 13.5550 = 374 MPa maximum uniform strain,
= ln 50
+ 13.55 50
= 0.24 true stress at A,
= 2.059
× 10
3
10
−5
× 50
+ 1.22 50
= 211 MPa true strain at A,
= ln 50
+ 1.22 50
= 0.024 By fitting a power law to two points, point A, and the maximum load point, determine an
approximate value of the strain-hardening index and the value of K. n
= log σ
max.
− log σ
A
log ε
u
− log ε
A
= log 374
− log 211 log 0.24
− log 0.024 = 0.25
By substitution, 211 = K × 0.024
0.25
, ∴K = 536 MPa.
Note that the maximum uniform strain, 0.24, is close to the value of the strain-hardening index. This can be anticipated, as shown in a later chapter.
1.1.5 Anisotropy
Material in which the same properties are measured in any direction is termed isotropic, but most industrial sheet will show a difference in properties measured in test-pieces aligned,
for example, with the rolling, transverse and 45
◦
directions of the coil. This variation is known as planar anisotropy. In addition, there can be a difference between the average of
properties in the plane of the sheet and those in the through-thickness direction. In tensile tests of a material in which the properties are the same in all directions, one would expect,
by symmetry, that the width and thickness strains would be equal; if they are different, this suggests that some anisotropy exists.
8 Mechanics of Sheet Metal Forming
In materials in which the properties depend on direction, the state of anisotropy is usually indicated by the R-value. This is defined as the ratio of width strain, ε
w
= lnww , to
thickness strain, ε
t
= lntt . In some cases, the thickness strain is measured directly,
but it may be calculated also from the length and width measurements using the constant volume assumption, i.e.
wtl = w
t l
or t
t =
w l
wl The R-value is therefore,
R =
ln w
w ln
w l
wl 1.17
If the change in width is measured during the test, the R-value can be determined con- tinuously and some variation with strain may be observed. Often measurements are taken
at a particular value of strain, e.g. at e
eng.
= 15. The direction in which the R-value is measured is indicated by a suffix, i.e. R
, R
45
and R
90
for tests in the rolling, diagonal and transverse directions respectively. If, for a given material, these values are different,
the sheet is said to display planar anisotropy and the most common description of this is R
= R
+ R
90
− 2R
45
2 1.18
which may be positive or negative, although in steels it is usually positive. If the measured R-value differs from unity, this shows a difference between average
in-plane and through-thickness properties which is usually characterized by the normal plastic anisotropy ratio, defined as
R =
R + 2R
45
+ R
90
4 1.19
The term ‘normal’ is used here in the sense of properties ‘perpendicular’ to the plane of the sheet.
1.1.6 Rate sensitivity
For many materials at room temperature, the properties measured will not vary greatly with small changes in the speed at which the test is performed. The property most sensitive
to rate of deformation is the lower yield stress and therefore it is customary to specify the cross-head speed of the testing machine – typically about 25 mmminute.
If the cross-head speed, v, is suddenly changed by a factor of 10 or more during the uniform deformation region of a tensile test, a small jump in the load may be observed
as shown in Figure 1.7. This indicates some strain-rate sensitivity in the material that can be described by the exponent, m, in the equation
σ = Kε
n
˙ε
m
1.20 Material properties 9
