F2004/51
Mean Stress Effects in Stress-Life and Strain-Life Fatigue
Norman E. Dowling
Department of Engineering Science and Mechanics
Virginia Polytechnic Institute and State University
Blacksburg, Virginia 24061, USA
Copyright © 2004 Society of Automotive Engineers, Inc.
ABSTRACT
properties from the same sources, specifically, 0.2% offset
yield strength σo , ultimate tensile strength σu , true stress
σ , and ductility, either percent reduction in
at fracture, ~
fB
Various approaches to estimating mean stress effects on
stress-life and strain-life behavior are compared with test
data for engineering metals. The modified Goodman equation with the ultimate tensile strength is found to be highly
inaccurate, and the similar expression of Morrow using the
true fracture strength is a considerable improvement. However, the Morrow expression employing the fatigue strength
coefficient σ ′f may be grossly non-conservative for metals
other than steels. The Smith, Watson, and Topper (SWT)
method is a reasonable choice that avoids the above difficulties. Another option is the Walker approach, with an adjustable exponent γ that may be fitted to test data, allowing
superior accuracy. Handling mean stress effects for strainlife curves is also discussed, including the issue of mathematical consistency with mean stress equations expressed in
terms of stress. A new and mathematically consistent
method for incorporating the Walker approach into strainlife curves is developed. With γ = 0.5, this result gives a
new strain-based interpretation of the SWT method.
INTRODUCTION
area, or percent elongation, whichever is given. As indicated
σf B values are corrected for hoop
by the subscript B, the ~
stress due to necking according to Bridgman [16]. As noted,
σf B were unavailable and had to be estisome values of ~
mated from those for similar material by assuming proportionality with the ultimate strength.
TABLE 1 - Metals Studied, Sources of Fatigue Data, and
Tensile Properties
UltiRed
Fracture
Yield
mate
Area
Material –
~
σfB
(Elong)
σo
σu
[Data Source]
MPa
Fatigue data will be analyzed for several steels and nonferrous metals as listed in Table 1, where references to the
sources of fatigue data are given. Table 1 also lists tensile
MPa
%
SAE 1015 Steel - [4]
228
415
726
68
SAE 1045 Steel1 - [5]
1841
2248
2717
40.5
AISI 4340 Steel2 - [6]
1103
1172
1634
56
2014-T6 Al - [7]
438
2024-T3 Al - [8, 9]
359
2
Mean stress effects have long been studied, as in the
early work of Gerber [1] and Goodman [2, 3], and one might
think that all has been said on the subject that needs to be
said. Nevertheless, several methods of questionable accuracy are currently in wide use. It is the purpose of this paper
to examine the most widely used methods and to compare
their success in correlating fatigue data for engineering metals. The methods considered are those of Goodman, Morrow, Smith-Watson-Topper, and Walker. There are more
than one version of some of these, and they may be used
differently in the context of stress-life versus strain-life
curves.
MPa
494
497
4
(13.6)
4
(20.3)
580
610
2024-T4 Al - [10, 11]
303
476
631
35
7075-T6 Al - [7]
489
567
7304
(16.5)
930
978
13624
(20)
Ti-6Al-4V3 - [12, 13]
1
2
Notes: Hardness 595 HB. Fatigue specimens at longer lives
plastically strained prior to testing. 3Solution treated and vacuum
annealed. 4Values estimated from similar material in [14] or
[15] by ratioing ultimate strengths.
With the aid of Fig. 1, let us be sure that the nomenclature used herein is clear. The mean stress σm is the average level of a constant amplitude cyclic loading, and the
stress amplitude σ a is the variation about this mean. The
amplitude is also half of the overall stress range ∆σ. The
maximum and minimum values reached are, respectively,
σ max = σ m + σa and σmin = σm − σa . The ratio
R = σ min / σ max is also used to characterize the mean
stress situation. Further, note that
σa =
σ max − σ min
σm =
σ max + σ min
of the data flattens at short lives and the fit to Eq. 3 is not
very good. Table 2 gives values of σ ′f and b for the metals
to be studied here, where these values are from fitting the
zero mean stress portion of the data.
(a, b)
(1)
(a, b)
(2)
At very short lives, stress-life data tend to approach the
σf B . If the fit to
true fracture strength from a tension test, ~
Manipulating Eq. 1 into the product of σmax and an algebraic
expression, and invoking the definition of R, gives two additional useful relationships, Eq. 2.
Eq. 3 is quite good, then σ ′f , which is noted to be the intercept at one-half cycle, is approximately equal to the true
σf B . This is often the case for
fracture strength, σ ′f ≈ ~
σ max
2
(1 − R ),
,
σm =
2
σ max
2
(1 + R )
steels, as for SAE 1045 steel in Fig. 2.
σmax
σ
σm
σa
10000
∆σ
0
t
σa
CCCCC
σmin
σa, Stress Amplitude, MPa
σa =
2
SAE 1045 Steel, 595 HB
1000
2024-T4 Al, Prestrained
100
1.E-01
1.E+01
Figure 1 – Definitions for cyclic stressing
1.E+05
1.E+07
Nf, Cycles
For the special case of stress amplitude σa where the mean
stress is zero, σm = 0, the notation σ ar is employed for the
amplitude. Such a situation of zero mean stress is also called
completely reversed cycling, and corresponds to R = −1.
In the treatment that follows, we will first briefly discuss stress-life curves. Following this, we will present various methods for estimating mean stress effects, and then we
will look at the ability of these methods to correlate stresslife data for various mean stresses. Next, we will consider
strain-life equations that include mean stress. Finally, concluding remarks are given that are intended to interpret and
summarize the earlier portions of the paper.
STRESS-LIFE CURVES
Stress-life curves are assumed to follow a power
relationship.
σ ar = σ ′f (2N f )b
1.E+03
(3)
▲
Figure 2 – Two stress-life curves with intercepts at ½ cycle
True fracture
compared to true facture strengths.
σf B and × Intercept, σ ′f .
strength, ~
However, where the data flatten at short lives, σ ′f may
σ , as for 2024-T4 aluminum in Fig.
considerably exceed ~
fB
2. Depending on how the data are fit and the range of data
available, the difference may be quite large. As an extreme
example, consider the values in Tables 1 and 2 for 7075-T6
σf B = 730 MPa. A stress-life fit
aluminum, where we have ~
over a wide range of lives for data on similar material gives
σ ′f = 1466 MPa. And a fit to the intermediate-to-long life
data to be analyzed gives σ ′f = 4402 MPa. Note that the
two σ ′f
values differ by a factor of three, and the larger
σ .
one is six times larger than ~
fB
where Nf is cycles to failure, and the stress variable is σ ar ,
σf B values in Table 1 with the correComparing the ~
as the fitting constants σ ′f and b are determined from tests
under zero mean stress, also called completely reversed
tests. The degree to which actual data closely fit Eq. 3 varies. For example, in Fig. 2, data for SAE 1045 steel fit the
expected straight line on this log-log plot fairly well. But
this is not the case for 2024-T4 aluminum, where the trend
sponding σ ′f values in Table 2, the latter are typically
higher than the former by a factor of three or four for the
nonferrous metals. This contrasts with the situation for the
three steels, where the values are similar.
AMPLITUDE-MEAN EQUATIONS
1.4
TABLE 2 - Stress-Life Fitting Constants1, 2
Fit All Data to
Fit σm = 0
Walker
Data
Material
b
bw
σ'f
σ'fw
γ
801
SAE 1045
-0.113
1.0
0.8
0.6
0.4
0.2
3050 -0.098
(Not Done)
1.0
AISI 4340
1758 -0.098
1963 -0.108 0.650
0.8
2014-T6
1120 -0.122
949 -0.108 0.480
2024-T3
1602 -0.154
1772 -0.163 0.460
0.4
2024-T43, 4
1294 -0.142
2452 -0.195 0.505
0.2
1466 -0.143
(Not Done)
6
7075-T6
4402 -0.262
(Not Done)
Ti-6Al-4V
2749 -0.144
(Not Done)
800
1200
1600
2000
2024-T3 Al
0.6
0.0
-200
σu
0
200
400
σ'f
~
σ fB
600
800 1000 1200 1400 1600
σm, MPa
Notes: 1Stress intercepts in MPa units. 2Data for Nf > 106 cycles
not included in fit if far from the log-log linear trend; runouts not
considered. 3Zero mean is overall fit for the same data from [17].
4
Walker fit from Nf > 103 data. 5Overall fit for similar material
from [18]. 6Fit to data from [7].
Note that the trend of such data is expected to pass through
σa / σ ar = 1.0 at σm = 0. A linear relationship is often assumed to occur, with the intercept along the σ a / σar = 0
axis expected to be the static strength of the material.
GOODMAN AND MORROW EQUATIONS - If the
static strength is taken as the ultimate tensile strength, the
straight line corresponds to
σa σm
+
=1
σar σu
400
σ'f
Figure 3 – Normalized stress amplitude-mean plot for AISI
4340 steel.
1.2
7075-T6
0
~
σf B
σm, MPa
799 -0.114 0.713
5
σu
0.0
-400
σa / σar
SAE 1015
AISI 4340 Steel
σu = 1172 MPa
1.2
σa/σar
Consider a set of fatigue data with cycles to failure at
various stress amplitudes and mean stresses. The number of
cycles to failure Nf for each test can be used with Eq. 3 to
calculate a value of completely reversed stress σar that is
expected to cause the same life as the actual combination of
amplitude and mean, σa and σm . One can then normalize
the amplitudes to σar , plotting the ratio σa / σ ar ,versus the
mean stress. For two of the data sets of current interest, such
normalized amplitude-mean plots are given as Figs. 3 and 4.
(4)
This is the modified Goodman relationship as formulated by
J. O. Smith [3]. It is useful to solve for σ ar .
Figure 4 – Normalized stress amplitude-mean plot for
2024-T3 aluminum.
σar =
σa
σm
1−
σu
(5)
If a given combination of stress amplitude and mean are
substituted, the calculated value of σar can be thought of as
an equivalent completely reversed stress amplitude that is
expected to cause the same life as the σ a , σm combination.
One can then estimate the life by entering Eq. 3. Since the
fitting constants σ ′f and b are obtained by testing at zero
mean stress, it is not necessary to have data at non-zero
mean stress to make a life estimate. But the life estimate
does depend on the accuracy of Eq. 5. In Figs. 3 and 4, note
that the data tend to lie above the Eq. 5 line for tensile mean
stress, causing conservative life estimates.
Morrow [19] suggested modifying the Goodman relationship by employing the true fracture strength as the intercept.
σ ar
σa
=
,
σ
1− m
σ fB
σ ar
σa
=
σ
1− m
σ ′f
(a,b)
(6)
σf B as being equal to
where form (b) arises from estimating ~
σ ′f . In Figs. 3 and 4, Eq. 6(a) fits the data quite well. Equation 6(b) seems to work equally well for the AISI 4340 steel
σf B works quite
of Fig. 3, as the approximation σ ′f ≈ ~
well. But for 2024-T3 aluminum in Fig. 4, Eq. 6(b) completely misses most of the data due to σ ′f being far larger
σ . This of course arises from a stress-life curve that
than ~
fB
lustrated in Figs. 5 and 6 for the same sets of data. In Fig. 5,
the SWT equation ( γ = 0.50) deviates from the data for
AISI 4340 steel somewhat, but Walker with γ = 0.65 fits it
quite well. For 2024-T3 aluminum in Fig. 6, the curves for
SWT and Walker with γ = 0.46 are close together, the difference being less than the considerable scatter in the data.
The Walker equation has the obvious advantage of having
an adjustable parameter γ to aid in fitting data. But this is
of no benefit unless the value is known from mean stress
data for at least similar material.
WALKER EQUATION FITTING - To fit a set of amplitude-mean-life data to the Walker equation, first write Eq. 3
in the following convenient form:
does not fit Eq. 3 very well in the manner of the similar
aluminum alloy in Fig. 2.
SMITH-WATSON-TOPPER (SWT) AND WALKER
EQUATIONS - Numerous additional relationships have
been proposed, including the widely used one of Smith,
Watson, and Topper [4].
σ ar = σ max σa
σ ar = σmax
σ ar = σa
σ ar = AN bf , where A = σ ′f 2b
Combine this with the Walker form of Eq. 8(b) and then
solve for Nf .
(b)
2
1− R
Nf =
(7)
(10)
(c)
1.6
where forms (b) and (c) are equivalent to (a) and are obtained from (a) by making substitutions from Eq. 2(a).
1.4
Another proposal is that of Walker [20], which may be
written
1.0
−γ
σ ar = σ1max
σ aγ
σa / σar
0.8
0.6
0.4
γ
(b)
AISI 4340 Steel
σu = 1172 MPa
1.2
(a)
⎛ 1− R ⎞
⎜
⎟
⎝ 2 ⎠
1− γ
⎛ 2 ⎞
σ ar = σa ⎜
⎟
⎝1− R ⎠
σ ar = σmax
γ
⎛ 1− R ⎞
⎟
⎝ 2 ⎠
1/b
γ
⎡
⎤
⎛1− R ⎞ 1
⎢σ max ⎜
⎥
⎟
⎝ 2 ⎠ A⎥
⎢
⎣
⎦
σ ar = AN bf = σ max ⎜
(a)
1− R
2
(9)
(8)
(c)
where equivalent forms (b) and (c) are obtained from (a) by
making substitutions from Eq. 2(a). The quantity γ is a
fitting constant that may be considered to be a materials
property. Obviously, all of the above forms of the Walker
equation reduce to the corresponding SWT forms for the
special case γ = 0.5.
Neither the SWT nor the Walker equations give a single
trend on a plot of the type of Figs. 3 and 4, forming instead
a family of curves. However, both do form a single curve if
the mean stress axis is also normalized by σar . This is il-
0.2
γ = 0.650
Data
SWT
Walker
0.0
-1.2 -0.8 -0.4
γ = 0.500
0.0
0.4
0.8
1.2
1.6
2.0
2.4
σm / σar
Figure 5 – SWT and Walker amplitude-mean curves for
AISI 4340 steel.
analysis. From Table 2 and other similar fitting for engineering metals, values of γ seem to generally be around
0.50 or above, with steels typically having higher values
than nonferrous metals. Note that lower values of γ correspond to greater sensitivity to mean stress, there being no
effect for γ = 1, in which case σar = σ a
1.4
Data
SWT
Walker
1.2
0.8
0.6
0.4
2024-T3 Al
0.2
0.0
-0.4
0.0
0.4
0.8
1.2
1.6
2.0
γ = 0.500
CORRELATION OF STRESS-LIFE DATA
γ = 0.460
For a set of amplitude-mean-life data, it is useful to calculate equivalent completely reversed stress amplitudes for
each test and plot these versus life. This is shown in Fig.
7(a-e) for AISI 4340 steel and in Fig. 8(a-e) for 2024-T3
aluminum. For each material, the σar values plotted are from
Eqs. 5, 6(a), 6(b), 7, or 8, respectively, for the (a) to (e) parts
of each of Figs. 7 and 8. In these plots, the line fitted to Eq.
3 for the σm = 0 data is shown, except for the Walker
2.4
2.8
σm / σar
Figure 6 – SWT and Walker amplitude-mean curves for
2024-T3 aluminum.
correlations, where the line corresponds to σ fw and
Next, take the logarithm to the base 10 of both sides.
b w used with Eq. 3, as fitted to the entire set of data with γ
as listed in Table 2.
γ
1
1
⎛1− R ⎞
log ⎜
log A
log N f =
log σmax +
⎟ −
b
b
b
⎝ 2 ⎠
(11)
Based on Eq. 11, we can now do a multiple linear regression
with independent variables x1 and x2 and dependent variable y.
y = m1 x1 + m 2 x 2 + d
(12)
where
⎛ 1− R ⎞
⎟
⎝ 2 ⎠
y = log N f , x1 = log σ max , x2 = log ⎜
γ
1
m1 = , m2 = ,
b
b
1
d = − log A
b
(13)
Once the fitting constants m1, m2, and d are known, the desired values are easily determined.
b=
1
,
m1
A = 10
− db
γ = bm 2 =
= 10
− d / m1
,
m2
m1
The correlation is favorable to the extent that the data
points fall very near the line. Points to the right of the line
correspond to lives longer than expected for the particular
σar equation, that is, conservative "predictions". Conversely,
points to the left of the line indicate lives shorter than expected, or nonconservative "predictions". Similar plots were
prepared for all of the metals listed in Table 2, except that
Walker correlations were not done for all of them, as noted.
The trends observed were similar to those seen for AISI
4340 steel and 2024-T3 aluminum in Figs. 7 and 8.
In particular, the Goodman relationship, Eq. 5, gives
poor results, usually being excessively conservative for tensile mean stresses, but nonconservative for the limited data
available at compressive mean stresses. An exception is
SAE 1045 steel, where the results are quite good, as might
σf B .
be expected form σu being only 17% below ~
1000
Mean Stress, MPa
(14)
σ′f =
A
2b
Table 2 gives the resulting three values, σ ′fw , b w , and γ ,
for five of the data sets, where subscripts w, for Walker, are
added to avoid confusion with values fitted to only zero
mean stress data.
σar , MPa
σa / σar
1.0
100
1.E+02
621
414
207
-207
0
0 Fit
1.E+03
Goodman
AISI 4340 Steel
σu = 1172 MPa
1.E+04
1.E+05
1.E+06
Nf , Cycles
Figure 7(a) – Goodman life correlation for AISI 4340 steel.
It is significant that all of the stress-life data at all mean
stresses are now involved in the fit. Treating the data as a
single, larger set enhances the possibility of statistical
1000
Mean Stress, MPa
621
414
207
-207
0
0 Fit
100
1.E+02
1.E+03
σar, MPa
σar , MPa
1000
Morrow, Fracture
Mean Stress, MPa
621
414
207
-207
0
Fit
AISI 4340 Steel
σu = 1172 MPa
1.E+04
1.E+05
100
1.0E+02
1.E+06
Walker
AISI 4340 Steel
σu= 1172 MPa
1.0E+03
1.0E+04
1.0E+05
1.0E+06
Nf, Cycles
Nf , Cycles
σf B life correlation for AISI
Figure 7(b) - Morrow ~
4340 steel
Figure 7(e) – Walker life correlation for AISI 4340 steel
with γ = 0.650.
1000
1000
Goodman
2024-T3 Al
621
414
207
-207
0
0 Fit
100
1.E+02
1.E+03
σar, MPa
σar , MPa
Mean Stress, MPa
Morrow, Intercept
AISI 4340 Steel
σu = 1172 MPa
1.E+04
1.E+05
1.E+06
100
1.E+02
0.6
0.4
0.02
-0.3
-0.6 R-ratio
-1
-1 Fit
1.E+03
Nf , Cycles
1.E+04
1.E+05
1.E+06
1.E+07
Nf, Cycles
Figure 7(c) - Morrow σ ′f life correlation for AISI 4340
steel.
Figure 8(a) – Goodman life correlation for 2024-T3 aluminum.
1000
1000
Mean Stress, MPa
100
1.E+02
621
414
207
-207
0
0 Fit
1.E+03
σar , MPa
σar , MPa
Morrow, Fracture
2024-T3 Al
SWT
AISI 4340 Steel
σu = 1172 MPa
100
1.E+02
1.E+04
1.E+05
1.E+06
Nf , Cycles
Figure 7(d) – SWT life correlation for AISI 4340 steel.
0.6
0.4
0.02
-0.3
-0.6 R-ratio
-1
-1 Fit
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
Nf , Cycles
σf B life correlation for 2024-T3 aluFigure 8(b) - Morrow ~
minum.
1000
1000
0.6
0.4
0.02
-0.3
-0.6
-1
-1 Fit
Walker
2024-T3 Al
Morrow, Intercept
2024-T3 Al
100
1.E+02
1.E+03
0.6
0.4
0.02
-0.3
-0.6
-1
Fit
R-ratio
σar, MPa
σar, MPa
R-ratio
1.E+04
1.E+05
1.E+06
100
1.0E+02
1.E+07
1.0E+03
Nf, Cycles
1.0E+04
1.0E+05
1.0E+06
1.0E+07
Nf, Cycles
Figure 8(c) – Morrow σ ′f life correlation for 2024-T3 alu-
minum.
Figure 8(e) – Walker life correlation for 2024-T3 aluminum
with γ = 0.460.
1000
0.6
0.4
0.02
-0.3
-0.6
-1
-1 Fit
1000
Morrow, Fracture
SAE 1015 Steel
SWT
2024-T3 Al
100
1.E+02
1.E+03
Mean Stress, MPa
σar , MPa
σar , MPa
R-ratio
1.E+04
1.E+05
1.E+06
0
34.5
69
103
-34.5
-69
0 Fit
1.E+07
Nf , Cycles
100
1.E+04
Figure 8(d) – SWT life correlation for 2024-T3 aluminum.
1.E+05
1.E+06
Nf , Cycles
man in all cases, except for SAE 1045 steel, where the results are similar. For the three steels, the Morrow form of
Eq. 6(b) with the intercept constant σ ′f gives essentially the
σ and σ ′
same result as Eq. 6(a), as expected due to ~
fB
f
having similar values. However, Eq. 6(b) gives very poor
and nonconservative values for the nonferrous metals due to
the high values of σ ′f .
The SWT relationship of Eq. 7 gives good results in all
σf B
cases. For steels, it is not quite as good as Morrow with ~
or σ ′f , and it tends to be nonconservative for compressive
mean stresses. But SWT is consistently better than Morrow
σf B for the nonferrous metals. The correlations for
with ~
σ and SWT, Eqs. 6(a) and 7, are given in
Morrow with ~
fB
Figs. 9(a,b) through 13(a,b) for SAE 1015 steel, SAE 1045
steel, 2014-T6 aluminum, 7075-T6 aluminum, and titanium
6Al-4V, respectively.
σf B life correlation for SAE 1015
Figure 9(a) - Morrow ~
steel.
1000
SWT
SAE 1015 Steel
Mean Stress, MPa
σar , MPa
The Morrow expression of Eq. 6(a) with the true fracture
σf B gives considerably better results than Goodstrength ~
100
1.E+04
1.E+05
0
34.5
69
103
-34.5
-69
0 Fit
1.E+06
Nf , Cycles
Figure 9(b) - SWT life correlation for SAE 1015 steel.
The Walker expression of Eq. 8 always gives an excellent correlation, as might be expected from its ability to adjust the value of γ . Correlations have already been presented in Fig. 7(e) for AISI 4340 steel and Fig. 8(e) for
2024-T3 aluminum. In addition, Fig. 14 gives the plot for
SAE 1015 steel, and Fig. 15 for 2014-T6 aluminum.
10,000
1,000
0 and -138 to +138
690
-345
Mean Stress, MPa
0 Fit
100
1.E+00
1.E+01
1.E+02
1.E+03
1000
Approx. R
1.E+04
1.E+05
0.45
0.05
-0.37
-1.0
-1.0 Fit
1.E+06
Nf , Cycles
σf B life correlation for SAE 1045
Figure 10(a) - Morrow ~
steel.
σar, MPa
σar , MPa
Morrow, Fracture
SAE 1045 Steel
595 HB
Real data obey the above mathematical forms only imperfectly. To aid with accurately fitting each curve under
these circumstances, it is recommended that the above two
equations be fitted separately to stress-strain-life test data,
with theoretical relationships among the six fitting constants
not being invoked. See Landgraf [21] for discussion of Eqs.
15 and 16, and see Dowling [22, 23] for descriptions and
discussion of the overall strain-based approach for making
life estimates for notched components.
10,000
σar , MPa
SWT
SAE 1045 Steel
595 HB
Morrow, Fracture
2014-T6 Al
100
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
Nf, Cycles
1,000
σf B life correlation for 2014-T6
Figure 11(a) - Morrow ~
aluminum.
0 and -138 to +138
690
-345
Mean Stress, MPa
0 Fit
100
1.E+00
1000
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
Approx. R
0.45
0.05
-0.37
-1.0
-1.0 Fit
Figure 10(b) – SWT life correlation for SAE 1045 steel.
STRAIN-LIFE EQUATIONS WITH MEAN STRESS
Mean stress adjustments are needed in making strainbased fatigue life estimates. The materials properties needed
to apply a strain-based approach are obtained from tests
under completely reversed controlled strain, so that mean
stresses in the tests are at or near zero. Such test results provide a cyclic stress-strain curve and a strain-life curve,
which are usually represented by
ε ar =
σ′f
E
(2 N f ) b + ε′f (2 N f ) c
SWT
2014-T6 Al
100
1.E+03
1.E+04
1.E+05
Nf, Cycles
1.E+06
1.E+07
Figure 11(b) – SWT life correlation for 2014-T6 aluminum.
1000
Morrow, Fracture
7075-T6 Al
(15)
(16)
The quantity E is the elastic modulus. Both of these equations represent summation of elastic strain (σ/E) and plastic
strain terms. Equation 15 gives the cyclic stress-strain curve,
in which H' and n' are fitting constants. For the strain-life
curve of Eq. 16, the quantities σ ′f and b are the same as in
Eq. 3, and ε'f and c are additional fitting constants for the
plastic strain term.
σar, MPa
σa ⎛ σa ⎞1/ n′
εa =
+⎜ ⎟
E ⎝ H′⎠
σar, MPa
Nf , Cycles
100
Approx. R
0.45
0.05
-0.37
-1.0
-1.0 Fit
10
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
Nf, Cycles
σf B life correlation for 7075-T6
Figure 12(a) - Morrow ~
aluminum.
1000
1000
σar, MPa
σar, MPa
SWT
7075-T6 Al
Approx. R
100
0.45
0.05
-0.37
-1.0
-1.0 Fit
10
1.E+03
1.E+04
1.E+05
1.E+06
Nf, Cycles
1.E+07
Approx. R
-1.0
0 to -0.78
0.1 to 0.2
0.5
-1.0 Fit
100
1.E+02
1.E+08
SWT
Ti-6Al-4V
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
Nf, Cycles
Figure 12(b) – SWT life correlation for 7075-T6 aluminum.
Figure 13(b) – SWT life correlation for titanium 6A1-4V.
STRAIN-LIFE EQUATIONS FOR NONZERO MEAN
STRESS - Since Eq. 16 gives only the life for zero mean
stress, it needs to be generalized to include mean stress effects. One equation that is often used for this purpose is an
extension of the Morrow equation used with σ ′f , Eq. 6(b).
MATHEMATICALLY CONSISTENT FORMS - We
can extend the σar equations such that the strain-life curve
is generalized in a mathematically consistent manner. Consider the general case of an amplitude-mean equation expressed in terms of stress.
E
⎛
⎜1 −
⎜
⎝
σar = f (σa , σm )
σm ⎞⎟
( 2 N f ) b + ε′f ( 2 N f ) c
σ′f ⎟⎠
(17)
Another is an extension of the SWT relationship, obtained
by replacing σa in Eq. 7(a) with εa . Combining this with
Eq. 16 gives the following relationship for determining life.
σ max
εa =
(σ ′f )2
E
(2 N f )2b + σ ′f ε ′f (2 N f )b
fB
σar, MPa
100
1.0E+04
1.0E+05
0
34.5
69
103
-34.5
-69
Fit
1.0E+06
Figure 14 – Walker life correlation for SAE 1015 steel with
γ = 0.713.
1000
We can now manipulate Eq. 3 to obtain a relationship
between stress amplitude and life, with the mean stress effect included on the Nf side of the equation. To proceed,
first combine Eqs. 3 and 19 to obtain
Approx. R
-1.0
0 to -0.78
0.1 to 0.2
0.5
-1.0 Fit
1.E+03
Morrow, Fracture
Ti-6Al-4V
1.E+04
1.E+05
1.E+06
σ ar = f (σa , σm ) = σ a
f (σ a , σ m )
σa
= σ ′f (2 N f )b
(20)
1.E+07
Nf, Cycles
σf B life correlation for titanium
Figure 13(a) - Morrow ~
6Al-4V.
Mean Stress, MPa
Nf, Cycles
similar to Eq. 6(a).
100
1.E+02
1000
(18)
Although Eqs. 17 and 18 may give reasonable life estimates,
neither is mathematically consistent with their parent σar
equations expressed in terms of stress. Also, in view of the
discussion above, the quantity σm / σ ′f in Eq. 17 should be
σ for nonferrous metals, so that it is more
replaced by σ / ~
m
Simple substitutions based on the definitions of the various
stress variables of Fig. 1, such as Eq. 2, allow this relationship to be expressed in terms of σmax and R, or in terms of
σ a and R, so that Eq. 19 can be any of Eqs. 5 to 8.
Walker
SAE 1015 Steel
+ c
(19)
σar, MPa
εa =
σ′f
Then solve for stress amplitude σa and manipulate the
stress quantities on the right side of the equation to be
within brackets with Nf , allowing us to define an equivalent
life N*.
1000
σar, MPa
Approx. R
0.45
0.05
-0.37
-1
Fit
*
N mf
σm
~
σf B
1/ b
⎞
⎟
⎟
⎠
(25)
0.100
1.0E+04
1.0E+05
1.0E+06
1.0E+07
⎡
⎛
σ ′f ⎢ 2 N f ⎜
⎢
⎝
⎣
1/ b ⎤
⎞
⎥
⎟
f (σ a , σ m ) ⎠ ⎥
⎦
σa
b
= σ ′f (2 N *)b
(21)
εa, Strain Amplitude
Figure 15 – Walker life correlation for 2014-T6
aluminum with γ = 0.480.
1/ b
⎞
σa
⎟
f (σ a , σ m ) ⎠
Nf = N*
⎞
σa
⎟
f (σ a , σ m ) ⎠
−1/ b
(23)
The effect on life must be the same regardless of
whether one employs a stress-life or a strain-life curve. This
permits Eq. 16 to be generalized to
εa =
σ′f
E
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
Figure 16 – Strain amplitude versus Walker equivalent life
(22)
Hence, one can determine the life N* that is expected for a
given stress amplitude σa under zero mean stress, and then
estimate the life Nf , as affected by a nonzero mean stress,
by solving Eq. 22 for Nf .
⎛
⎜
⎝
γ = 0.650
N*w , Walker Equivalent Cycles
N*w for AISI 4340 steel with γ = 0.650.
0.100
γ = 0.505
Mean Stress, MPa
εa, Strain Amplitude
N* =
⎛
Nf ⎜
⎝
0.010
0.001
1.E+02
An explicit expression for N* is thus
Mean Stress, MPa
621
414
207
-207
0
Fit
AISI 4340 Steel
σu = 1172 MPa
Nf, Cycles
σa =
=N
⎛
⎜
f ⎜1 −
⎝
Note that subscripts mf are added to N* to specify the Morrow equation based on the true fracture strength.
Walker
2014-T6 Al
100
1.0E+03
PARTICULAR CASES - Now let us consider particular
cases of σ ar = f ( σ a , σm ) . For the Morrow form of Eq.
6(a), substitution into Eq. 22 gives
0.010
0
72
140 to 230
290
-70 to -135
Fit
2024-T4 Al
Prestrained
0.001
1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08
N*w, Walker Equivalent Cycles
Figure 17 – Strain amplitude versus Walker equivalent life
(2 N *)b + ε′f (2 N *)c
(24)
where N* is the life calculated from the strain amplitude εa
as if the mean stress were zero, and then Nf as affected by
the nonzero mean stress is obtained from Eq. 23. Also, on a
strain-life plot, data plotted as εa versus the equivalent life
N* are expected all fall together along the curve for zero
mean stress, Eq. 16. This is demonstrated in Figs. 16 and 17
for AISI 4340 steel and 2024-T4 aluminum, respectively,
where the f ( σ a , σm ) is in this case based on the Walker
expression, Eq. 8.
N*w for 2024-T4 aluminum with γ = 0.505.
σf B might be a good approximation, as for
Where σ′f ≈ ~
steels, Eq. 6(b) applies, and Eq. 22 yields
⎛
*
= N f ⎜⎜1 −
N mi
⎝
σm
σ′f
1/ b
⎞
⎟
⎟
⎠
(26)
The added subscripts now indicate Morrow and intercept.
Note that Eqs. 24 and 26 combined are not the same as Eq. 17,
as the latter has no mean stress adjustment for the second
term.
For the Walker relationship, the form of Eq. 8(c) and Eq.
22 give
*
Nw
= Nf
⎛1− R ⎞
⎜
⎟
⎝ 2 ⎠
(1− γ ) / b
(27)
where subscript w of course specifies Walker. Letting γ =
0.5 gives us the special case of this for the SWT equation.
1/( 2b)
⎛ 1− R ⎞
⎜
⎟
⎝ 2 ⎠
*
= Nf
N swt
(28)
It is useful to write as a single equation the form provided by
Eqs. 24 and 27 for the Walker relationship.
εa =
σ ′f
E
(1− γ ) / b ⎤
⎡
⎛1− R ⎞
⎢2 N f ⎜
⎥
⎟
⎝ 2 ⎠
⎢
⎥
⎣
⎦
b
⎡
+ ε ′f ⎢ 2 N f
⎢
⎣
⎛ 1− R ⎞
⎜
⎟
⎝ 2 ⎠
(1− γ ) / b ⎤ c
⎥
⎥
⎦
εa =
σ ′f
E
⎡
⎛ 1− R ⎞
⎢2N f ⎜
⎟
⎝ 2 ⎠
⎢⎣
⎥
⎥⎦
+
⎡
⎛1− R ⎞
ε ′f ⎢ 2 N f ⎜
⎟
⎝ 2 ⎠
⎢⎣
⎥
⎥⎦
ε'fw
AISI 4340
Steel3
2024-T4 Al4
Cyclic σ-ε Curve1, 2
Ε
cw
H'
n'
0.624 -0.620
207,000 1655 0.131
0.632 -0.858
73,100 738
0.080
Notes: 1Units are MPa for E and H'. 2Values from [6] for AISI 4340
and [17] for 2024-T4. 3Fit to P > 1.5 × 10−4 and Nw < 5 × 105. 4Fit to
P > 1.5 × 10−4 and Nw < 104.
First, employ the cyclic stress-strain curve, Eq. 15, to
estimate strain amplitudes εa from stress amplitudes σ a for
any tests in the data set where strain was not measured.
(This is often the case for tests run in stress control at relatively long lives.) Next, using b w and γ , calculate values
*
Nw
to calculate values of the second (plastic strain) term of
Eq. 24 by subtracting the first (elastic strain) term from εa .
Similarly, Eqs. 24 and 28 give the corresponding form for
the SWT equation.
1/(2b ) ⎤ c
*
Fit N w
vs. P
Material
*
of N w
from Eq. 27 for each data point. Then use εa and
(29)
1/(2b) ⎤ b
TABLE 3 - Additional Constants for Strain-Life Curve
* cw
P = ε ′fw (2 N w
)
= εa −
σ ′fw
(30)
Note that the latter is not the same as Eq. 18, but instead
represents a different generalization of the strain-life equation that is consistent with SWT expressed in terms of stress,
Eq. 7.
*
lives N w
Walker equivalent
from Eq. 27, in which b w and
γ from Table 2 are employed.
The curves shown in Figs. 16 and 17 correspond to the
usual form of strain-life equation for zero mean stress, Eq.
24, but new values of the fitting constants are employed.
This is done to take advantage of the ability of the Walker
equation to include all of the data at all mean stresses in the
fit, as previously described. The constants used with Eq. 24
are the Walker stress-life fitted values, σ ′fw and b w from
Table 2, along with ε ′fw and c w as listed in Table 3. Fitting
of ε ′fw and c w to the test data needs to be described.
* bw
(2 N w
)
(31)
where the values of the plastic strain term are denoted P as a
convenience. Now do a least squares fit using these P and
*
*
values by first solving for 2 N w
.
Nw
*
2N w
CORRELATION OF STRAIN-LIFE DATA
As already noted, strain-life correlations based on the
Walker mean stress equation are given in Figs. 16 and 17 for
AISI 4340 steel and 2024-T4 aluminum, respectively. In
each case, the data points are seen to agree closely with the
strain-life curve for zero mean stress, Eq. 24, indicating success for Eq. 27. Note that the strain values plotted are simply
strain amplitudes, ε a . Correlation of the data for various
mean stresses is achieved by plotting on the horizontal axis
E
⎛ P
= ⎜⎜
⎝ ε′fw
1/ c
⎞ w
⎟
⎟
⎠
(32)
Taking logarithms of both sides of this equation gives
*
log (2 N w
)=
1
1
log P −
log ε′fw
cw
cw
(33)
A linear regression can now proceed using
y = mx + d
(34)
where
*
y = log (2 N w
),
m=
1
,
cw
d =−
x = log P
1
log ε ′fw
cw
(35)
Once the fitting constants m and d have been determined, the
desired values are easily obtained.
cw =
1
,
m
ε′fw = 10 − dcw = 10 − d / m
(36)
In performing these fits, it was found that the data exhibited extreme scatter at low P values corresponding to
*
. Accordingly, each fit was relong equivalent lives N w
stricted to the region of well behaved data by a dual criterion, specifically, a lower limit on P and an upper limit on
*
, both of which had to be satisfied for a data point to be
Nw
used. See the notes to Table 3 for the actual values chosen
for these limits.
DISCUSSION AND SUMMARY
We have compared various stress amplitude-mean equations as to their ability to correlate fatigue data. It is clear
that the Goodman equation employing the ultimate tensile
strength σu is inaccurate. The Morrow equation using the
σ works well, but has the disadvantrue fracture strength ~
fB
σf B are not always available. An empiritage that values of ~
cal study to develop estimates of this quantity from other
tensile properties would enhance its usefulness.
If the Morrow equation is instead employed with the intercept constant σ ′f , the results are still quite good for
steels. However, this is not the case for aluminum alloys or
for the one titanium alloy studied, where Morrow with σ ′f
is seen to be highly inaccurate and nonconservative. This
difficulty is associated with stress-life behavior that does not
fit a power law very well, in particular, with stress-life
curves that tend to flatten at short lives. As a result, σ ′f
σ , and causing
values may be quite large, far exceeding ~
fB
the highly inaccurate behavior. Hence, Morrow with σ ′f
should not be used for aluminum alloys or for other metals
where such stress-life behavior occurs.
The Smith, Watson, and Topper (SWT) equation gave
good results for all cases studied. If it is desired to choose
one simple stress amplitude-mean equation for all metals,
SWT would be the preferred choice. Another option would
be to use SWT except where Morrow with σ ′f is known to
give good results, as for steels.
The Walker equation has the advantage of enhanced
ability to fit data by its use of the adjustable parameter γ ,
which may be considered to be an additional materials property. However, there is the accompanying disadvantage in
that γ values are not generally known unless fatigue data at
various mean stresses are available for a given material.
Where γ is known, the Walker equation appears to be quite
accurate and is the best mean stress equation of those studied.
Additional study may allow generic values of γ to be
developed for various classes of alloy, so that any disadvantage of the Walker equation is removed. For example, based
on the limited study done so far, γ ≈ 0.50 may be a good
choice for aluminum alloys. (Note that if γ = 0.50 exactly,
this method is equivalent to SWT.) For steels, a higher generic value appears to be appropriate, perhaps γ ≈ 0.65.
Also, higher strength steels are generally more sensitive to
mean stress that lower strength ones, so that it may be possible to develop a correlation between γ and say ultimate
tensile strength.
Stress amplitude-mean equations may be incorporated
into strain-life equations in a mathematically consistent
manner. (Some strain-life or related equations in common
use are not mathematically consistent with their parent
stress-based equations.) The procedure for doing so for any
stress amplitude-mean relation is given, as well as the particulars for those studied. This logic leads to Eq. 30, which
is a new strain-life extension of the SWT mean stress equation.
Similarly extending the Walker mean stress relation
gives an entirely new version of the strain-life equation, Eq,
29. This strain-life equation includes SWT as the special
case for γ = 0.50. It should be further evaluated and compared to experimental data. Data fits for a steel and for an
aluminum alloy give excellent results.
CONCLUSIONS
The following conclusions are drawn from the discussion above:
1) The Goodman mean stress equation employing the ultimate tensile strength σu is inaccurate, being excessively conservative for tensile mean stresses.
2) The Morrow mean stress equation using the true fracσf B works well for various metals, but has
ture strength ~
σ are not always
the disadvantage that values of ~
fB
available.
3) The Morrow mean stress equation with σ ′f works well
for steels. But it is highly inaccurate and nonconservative for materials with log-log stress-life behavior that
flattens at short lives. Thus, it should not be used for
aluminum alloys.
4) The Smith, Watson, and Topper (SWT) mean stress
equation is a good choice for general use. It is quite accurate for aluminum alloys, and for steels it is acceptable, although not quite as good as Morrow with σ ′f .
5) The Walker mean stress equation with adjustable constant γ gives superior results where γ is known or can
be estimated.
Steel," NACA TN 2324, National Advisory Committee for Aeronautics, Washington, DC, March 1951.
[9]
Illg, W., "Fatigue Tests on Notched and Unnotched
Sheet Specimens of 2024-T3 and 7075-T6 Aluminum
Alloys and of SAE 4130 Steel with Special Consideration of the Life Range from 2 to 10,000 Cycles,"
NACA TN 3866, National Advisory Committee for
Aeronautics, Washington, DC, Dec. 1956.
[10]
Topper, T. H., and B. I. Sandor, “Effects of Mean
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Effects of Environment and Complex Load History on
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and Materials, West Conshohocken, PA, 1970, pp.
93-104.
Mann, J. Y., “The Historical Development of Research on the Fatigue of Materials and Structures,”
The Journal of the Australian Institute of Metals,
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[11]
Endo, T., and J. Morrow, “Cyclic Stress-Strain and
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[12]
Smith, J. O., “The Effect of Range of Stress on the
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Urbana, IL, Feb. 1942. See also Bulletin No. 316,
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[13]
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Tech, Blacksburg, VA, Jan. 2003.
[14]
Conle, F. A., R. W. Landgraf and F. D. Richards,
Materials Data Book: Monotonic and Cyclic Properties of Engineering Materials, Ford Motor Co., Scientific Research Staff, Dearborn, MI, 1984.
[15]
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[16]
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[17]
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PA, 1989, pp. 150-171.
6) Any future work on mean stress equations should concentrate on the Walker relationship, such as identifying
generic values of γ for various classes of metal, or developing correlations for estimating γ from tensile
properties.
7) The incorporation of the Walker equation into the
strain-life curve, Eq. 29, is a promising approach that
should be further evaluated and employed.
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[2]
[3]
[4]
[5]
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