Page 456 - Mechanical Behavior of Materials

456 Chapter 9 Fatigue of Materials: Introduction and Stress-Based Approach

Fitting of data for more than one mean stress or R-ratio is needed to obtain γ , which will be
described in Chapter 10. Note that the SWT relationship may be thought of as a special case of
the Walker one where γ = 0.5. Neither the SWT nor the Walker equations can be shown as a single
curve on a plot of σ a /σ ar versus σ m , as in Fig. 9.39. But they both do form a single curve on a plot
of σ a /σ ar versus σ m /σ ar , so that comparisons to test data can be done on this basis.
Considering all of the mean stress equations given, neither the Goodman nor the Gerber
equations are very accurate, with the former often being overly conservative, and the latter often
nonconservative. The Morrow relationship in the form of Eq. 9.17(a) is usually reasonably accurate,
but suffers from the value of the true fracture strength ˜σ fB not always being known. Equation 9.17(b)

with σ fits data very well for steels, but should be avoided for aluminum alloys and perhaps for
f
other nonferrous metals. The SWT expression of Eq. 9.18 is a good choice for general use and fits
data particularly well for aluminum alloys. The Walker relationship, Eq. 9.19, is the best choice
where data exist for fitting the value of γ .
A recent paper (Dowling, 2009) provides a comparison of these mean stress equations for a
number of sets of fatigue test data. In that study, it is noted that the Walker constant γ decreases
for higher strength metals, indicating an increasing sensitivity to mean stress. For steels, the data
analyzed give the following equation for estimating γ from the ultimate tensile strength:

γ =−0.000200 σ u + 0.8818 (σ u in MPa) (9.20)

9.7.4 Life Estimates with Mean Stress
Let the equation representing the amplitude-mean behavior, such as Eq. 9.17(b), be solved for the
completely reversed stress σ ar :
σ a
σ ar = (9.21)
σ m
1 −
σ
f
Substituting values of stress amplitude σ a and mean stress σ m gives a stress amplitude σ ar that is
expected to produce the same life at zero mean stress as the (σ a ,σ m ) combination. Hence, σ ar can be
thought of as an equivalent completely reversed stress amplitude. Substituting σ ar into a stress–life
curve for zero mean stress thus provides a life estimate for the (σ a ,σ m ) combination.
For example, assume that the S-N curve for completely reversed loading is known and has the
form of Eq. 9.7. Since tests at σ m = 0 are employed to obtain the constants σ and b, the stress

f
amplitude σ a corresponds to the special case denoted σ ar , so that, for our present purposes, the
equation needs to be written as

σ ar = σ (2N f ) b (9.22)
f
Combining this with Eq. 9.21 yields a more general stress–life equation that applies for nonzero
mean stress:

σ a = (σ − σ m )(2N f ) b (9.23)
f
Note that this reduces to Eq. 9.7 for the special case of σ m = 0. On a log–log plot, Eq. 9.23 produces
a family of σ a -N f curves for different values of mean stress, which are all parallel straight lines.