Page 145 - Mechanical Behavior of Materials

146 Chapter 4 Mechanical Testing: Tension Test and Other Basic Tests

arbitrary lower limit of twice the strain that accompanies the offset yield strength, 2ε o , is suggested
in Fig. 4.19. (Note that ε o is defined in Fig. 4.11(a).) Below this limit, the difference between true
and engineering stress is generally so small that it can be neglected, so that no conversion is needed.
A similar limitation is encountered by Eqs. 4.19 and 4.20, which are otherwise valid at any strain.

4.5.4 Bridgman Correction for Hoop Stress

A complication exists in interpreting tensile results near the end of a test where there is a large
amount of necking. As pointed out by P. W. Bridgman in 1944, large amounts of necking result in a
tensile hoop stress being generated around the circumference in the necked region. Thus, the state of
stress is no longer uniaxial as assumed, and the behavior of the material is affected. In particular, the
axial stress is increased above what it would otherwise be. (This is caused by plastic deformation;
see Chapter 12.)
A correction for steel can be made on the basis of the empirical curve developed by Bridgman,
which is shown in Fig. 4.20. The curve is entered with the true strain based on area, and it gives a
value of the correction factor, B, which is used as follows:
˜ σ B = B ˜σ (4.21)

σ
Here, ˜σ is true stress simply computed from area by using Eq. 4.12, and ˜ B is the corrected value
of true stress. Values of B for steels may be estimated from the following equation that closely
approximates the curve of Fig. 4.20:
3 2
B = 0.0684x + 0.0461x − 0.205x + 0.825, where x = log ε ˜ (0.12 ≤˜ε ≤ 3)
10
(4.22)
The correction is not needed for ˜ε< 0.12. Note that a 10% correction (B = 0.9) corresponds to a
true strain of ˜ε = 0.44. By Eq. 4.20, this gives a ratio of initial to necked diameter of 1.25. Hence,
fairly large strains must occur for the correction to be significant.
Similar correction curves are not generally available for other metals, but a correction can still
be done if the radius of the neck profile is measured. For details, see the references Bridgman (1952)

1.0
B = σ /σ, Correction Factor 0.9

0.8
~
~ B 0.7

0.6
0 1 2 3
ε = ln (A /A), True Strain
i
Figure 4.20 Curve of [Bridgman 44] giving correction factors on true stress for the effect of
hoop stress due to necking in steels.