Page 457 - Mechanical Behavior of Materials

Section 9.7 Mean Stresses 457

Any of the preceding mean stress equations can be similarly employed to generalize the stress–
life equation. As a further example, combining Eq. 9.22 with the SWT relationship of Eq. 9.18
gives

√ b

σ max σ a = σ (2N f ) (σ max > 0) (a)
f
!
1 − R b

σ max = σ (2N f ) (σ max > 0) (b) (9.24)
f
2
N f =∞ (σ max ≤ 0) (c)
where either of the two forms (a) and (b) may be used, as convenient. Note that (c) is necessary,
as σ ar from Eq. 9.18 is zero if σ max is zero and is undefined if σ max is negative. Hence, the
SWT equation predicts that fatigue failure is not possible unless the cyclic stress ranges into
tension.
The equivalent completely reversed stress σ ar is also useful as a means of assessing the
success of any given mean stress equation in a way that makes the accuracy of life estimates
apparent. Assume that fatigue life data N f are available for various combinations of stress amplitude
and mean, σ a and σ m , or for various combinations of σ max and R. Values of σ ar can then be
calculated for each test, and then these all plotted versus the corresponding N f values. If the mean
stress equation is successful, then all of the σ ar data will agree closely with the stress–life curve
for zero mean stress, such as Eq. 9.22. This will be demonstrated by one of the examples that
follow.

Example 9.3
The AISI 4340 steel of Table 9.1 is subjected to cyclic loading with a tensile mean stress of
σ m = 200 MPa.

(a) What life is expected if the stress amplitude is σ a = 450 MPa?
(b) Also estimate the σ a versus N f curve for this σ m value.

First Solution (a) The S-N curve from Table 9.1 for zero mean stress is given by constants
σ = 1758 MPa and b =−0.0977 for Eq. 9.7. Life estimates may be made for nonzero σ m by

f
entering Eq. 9.7 with values of equivalent completely reversed stress σ ar :
b −0.0977

σ ar = σ f 2N f = 1758 2N f MPa
Calculating σ ar from Eq. 9.21 gives
σ a 450
σ ar = = = 507.8MPa
σ m 200
1 − 1 −
σ
f 1758