Department of Materials Science and Engineering
Massachusetts Institute of Technology
Cambridge, MA 02139
May 1, 2001
The concept of “fatigue” arose several times in the Module on Fracture (Module 23), as in the
growth of cracks in the Comet aircraft that led to disaster when they became large enough
to propagate catastrophically as predicted by the Griﬃth criterion. Fatigue, as understood
by materials technologists, is a process in which damage accumulates due to the repetitive
application of loads that may be well below the yield point. The process is dangerous because a
single application of the load would not produce any ill eﬀects, and a conventional stress analysis
might lead to a assumption of safety that does not exist.
In one popular view of fatigue in metals, the fatigue process is thought to begin at an
internal or surface ﬂaw where the stresses are concentrated, and consists initially of shear ﬂow
along slip planes. Over a number of cycles, this slip generates intrusions and extrusions that
begin to resemble a crack. A true crack running inward from an intrusion region may propagate
initially along one of the original slip planes, but eventually turns to propagate transversely to
the principal normal stress as seen in Fig. 1.
Figure 1: Intrusion-extrusion model of fatigue crack initiation.
When the failure surface of a fatigued specimen is examined, a region of slow crack growth
is usually evident in the form of a “clamshell” concentric around the location of the initial ﬂaw.
(See Fig. 2.) The clamshell region often contains concentric “beach marks” at which the crack
was arrested for some number of cycles before resuming its growth. Eventually, the crack may
become large enough to satisfy the energy or stress intensity criteria for rapid propagation,
following the previous expressions for fracture mechanics. This ﬁnal phase produces the rough
surface typical of fast fracture. In postmortem examination of failed parts, it is often possible to
correlate the beach marks with speciﬁc instances of overstress, and to estimate the applied stress
at failure from the size of the crack just before rapid propagation and the fracture toughness of
Figure 2: Typical fatigue-failure surfaces. From B. Chalmers, Physical Metallurgy, Wiley, p. 212,
The modern study of fatigue is generally dated from the work of A. W¨hler, a technologist in
the German railroad system in the mid-nineteenth century. Wohler was concerned by the failure
of axles after various times in service, at loads considerably less than expected. A railcar axle
is essentially a round beam in four-point bending, which produces a compressive stress along
the top surface and a tensile stress along the bottom (see Fig. 3). After the axle has rotated a
half turn, the bottom becomes the top and vice versa, so the stresses on a particular region of
material at the surface varies sinusoidally from tension to compression and back again. This is
now known as fully reversed fatigue loading.
Figure 3: Fatigue in a railcar axle.
Well before a microstructural understanding of fatigue processes was developed, engineers had
developed empirical means of quantifying the fatigue process and designing against it. Perhaps
the most important concept is the S-N diagram, such as those shown in Fig. 41 , in which a
constant cyclic stress amplitude S is applied to a specimen and the number of loading cycles N
until the specimen fails is determined. Millions of cycles might be required to cause failure at
lower loading levels, so the abscissa in usually plotted logarithmically.
Figure 4: S − N curves for aluminum and low-carbon steel.
In some materials, notably ferrous alloys, the S − N curve ﬂattens out eventually, so that
below a certain endurance limit σe failure does not occur no matter how long the loads are
cycled. Obviously, the designer will size the structure to keep the stresses below σe by a suitable
safety factor if cyclic loads are to be withstood. For some other materials such as aluminum, no
endurance limit exists and the designer must arrange for the planned lifetime of the structure
to be less than the failure point on the S − N curve.
Statistical variability is troublesome in fatigue testing; it is necessary to measure the lifetimes
of perhaps twenty specimens at each of ten or so load levels to deﬁne the S − N curve with
statistical conﬁdence2 . It is generally impossible to cycle the specimen at more than approximately 10Hz (inertia in components of the testing machine and heating of the specimen often
become problematic at higher speeds) and at that speed it takes 11.6 days to reach 107 cycles
of loading. Obtaining a full S − N curve is obviously a tedious and expensive procedure.
Figure 5: Variability in fatigue lifetimes and fracture strengths.
H.W. Hayden, W.G. Moﬀatt, and J. Wulﬀ, The Structure and Properties of Materials, Vol. III, John Wiley
& Sons, 1965.
A Guide for Fatigue Testing and the Statistical Analysis of Fatigue Data, ASTM STP-91-A, 1963.
At ﬁrst glance, the scatter in measured lifetimes seems enormous, especially given the logarithmic scale of the abscissa. If the coeﬃcient of variability in conventional tensile testing is
usually only a few percent, why do the fatigue lifetimes vary over orders of magnitude? It must
be remembered that in tensile testing, we are measuring the variability in stress at a given
number of cycles (one), while in fatigue we are measuring the variability in cycles at a given
stress. Stated diﬀerently, in tensile testing we are generating vertical scatter bars, but in fatigue
they are horizontal (see Fig. 5). Note that we must expect more variability in the lifetimes as
the S − N curve becomes ﬂatter, so that materials that are less prone to fatigue damage require
more specimens to provide a given conﬁdence limit on lifetime.
Eﬀect of mean load
Of course, not all actual loading applications involve fully reversed stress cycling. A more general
sort of fatigue testing adds a mean stress σm on which a sinusoidal cycle is superimposed, as
shown in Fig. 6. Such a cycle can be phrased in several ways, a common one being to state the
alternating stress σalt and the stress ratio R = σmin /σmax . For fully reversed loading, R = −1.
A stress cycle of R = 0.1 is often used in aircraft component testing, and corresponds to a
tension-tension cycle in which σmin = 0.1σmax .
Figure 6: Simultaneous mean and cyclic loading.
A very substantial amount of testing is required to obtain an S − N curve for the simple
case of fully reversed loading, and it will usually be impractical to determine whole families of
curves for every combination of mean and alternating stress. There are a number of strategems
for ﬁnessing this diﬃculty, one common one being the Goodman diagram. shown in Fig. 7. Here
a graph is constructed with mean stress as the abscissa and alternating stress as the ordinate,
and a straight “lifeline” is drawn from σe on the σalt axis to the ultimate tensile stress σf on the
σm axis. Then for any given mean stress, the endurance limit — the value of alternating stress
at which fatigue fracture never occurs — can be read directly as the ordinate of the lifeline at
that value of σm . Alternatively, if the design application dictates a given ratio of σe to σalt , a
line is drawn from the origin with a slope equal to that ratio. Its intersection with the lifeline
then gives the eﬀective endurance limit for that combination of σf and σm .
Miner’s law for cumulative damage
When the cyclic load level varies during the fatigue process, a cumulative damage model is often
hypothesized. To illustrate, take the lifetime to be N1 cycles at a stress level σ1 and N2 at σ2 .
If damage is assumed to accumulate at a constant rate during fatigue and a number of cycles n1
is applied at stress σ1 , where n1 < N1 as shown in Fig. 8, then the fraction of lifetime consumed
Figure 7: The Goodman diagram.
Figure 8: The concept of fractional lifetime.
will be n1 /N1 . To determine how many additional cycles the specimen will survive at stress σ2 ,
an additional fraction of life will be available such that the sum of the two fractions equals one:
Note that absolute cycles and not log cycles are used here. Solving for the remaining cycles
permissible at σ2 :
The generalization of this approach is called Miner’s Law, and can be written
n2 = N2 1 −
where nj is the number of cycles applied at a load corresponding to a lifetime of Nj .
Consider a hypothetical material in which the S-N curve is linear from a value equal to the fracture stress
σf at one cycle (log N = 0), falling to a value of σf /2 at log N = 7 as shown in Fig. 9. This behavior can
be described by the relation
log N = 14 1 −
The material has been subjected to n1 = 105 load cycles at a level S = 0.6σf , and we wish to estimate
how many cycles n2 the material can now withstand if we raise the load to S = 0.7σf . From the S-N
relationship, we know the lifetime at S = 0.6σf = constant would be N1 = 3.98 × 105 and the lifetime at
S = 0.7σf = constant would be N2 = 1.58 × 104 . Now applying Eqn. 1:
Figure 9: Linear S-N curve.
1 × 105
3.98 × 10
1.58 × 104
n2 = 1.18 × 104
Miner’s “law” should be viewed like many other material “laws,” a useful approximation,
quite easy to apply, that might be accurate enough to use in design. But damage accumulation
in fatigue is usually a complicated mixture of several diﬀerent mechanisms, and the assumption
of linear damage accumulation inherent in Miner’s law should be viewed skeptically. If portions
of the material’s microstructure become unable to bear load as fatigue progresses, the stress
must be carried by the surviving microstructural elements. The rate of damage accumulation
in these elements then increases, so that the material suﬀers damage much more rapidly in
the last portions of its fatigue lifetime. If on the other hand cyclic loads induce strengthening
mechanisms such as molecular orientation or crack blunting, the rate of damage accumulation
could drop during some part of the material’s lifetime. Miner’s law ignores such eﬀects, and
often fails to capture the essential physics of the fatigue process.
Crack growth rates
Certainly in aircraft, but also in other structures as well, it is vital that engineers be able to
predict the rate of crack growth during load cycling, so that the part in question be replaced
or repaired before the crack reaches a critical length. A great deal of experimental evidence
supports the view that the crack growth rate can be correlated with the cyclic variation in the
stress intensity factor3 :
= A ∆K m
where da/dN is the fatigue crack growth rate per cycle, ∆K = Kmax − Kmin is the stress
intensity factor range during the cycle, and A and m are parameters that depend the material,
environment, frequency, temperature and stress ratio. This is sometimes known as the “Paris
law,” and leads to plots similar to that shown in Fig. 10.
See Module 23.
Figure 10: The Paris law for fatigue crack growth rates.
The exponent m is often near 4 for metallic systems, which might be rationalized as the
damage accumulation being related to the volume Vp of the plastic zone: since the volume Vp of
the zone scales with rp and rp ∝ KI , then da/dn ∝ ∆K 4 . Some speciﬁc values of the constants
m and A for various alloys in given in Table 1.
Table 1: Numerical parameters in the Paris equation.
4 × 10−12
1. A steel has an ultimate tensile strength of 110 kpsi and a fatigue endurance limit of 50
kpsi. The load is such that the alternating stress is 0.4 of the mean stress. Using the
Goodman method with a safety factor of 1.5, ﬁnd the magnitude of alternating stress that
gives safe operation.
2. A titanium alloy has an ultimate tensile strength of 120 kpsi and a fatigue endurance limit
of 60 kpsi. The alternating stress is 20 kpsi. Find the allowable mean stress, using a safety
factor of 2.
3. A material has an S-N curve that is linear from a value equal to the fracture stress σf
at one cycle (log N = 0), falling to a value of σf /3 at log N = 7. The material has been
subjected to n1 = 1000 load cycles at a level S = 0.7σf . Estimate how many cycles n2 the
material can withstand if the stress amplitude is now raised to S = 0.8σf .
4. A steel alloy has an S-N curve that falls linearly from 240 kpsi at 104 cycles to 135 kpsi at
106 cycles. A specimen is loaded at 160 kpsi alternating stress for 10 5 cycles, after which
the alternating stress is raised to 180 kpsi. How many additional cycles at this higher
stress would the specimen be expected to survive?
5. Consider a body, large enough to be considered inﬁnite in lateral dimension, containing a
central through-thickness crack initially of length 2a0 and subjected to a cyclic stress of
amplitude ∆σ. Using the Paris Law (Eqn. 2), show that the number of cycles Nf needed
for the crack to grow to a length 2af is given by the relation
when m = 2, and for other values of m
= A (∆σ)2 πNf
A (∆σ)m π m/2 Nf
6. Use the expression obtained in Prob. 5 to compute the number of cycles a steel component
can sustain before failure, where the initial crack halﬂength is 0.1 mm and the critical
crack halﬂength to cause fracture is 2.5 mm. The stress amplitude per cycle is 950 MPa.
Take the crack to be that of a central crack in an inﬁnite plate.
7. Use the expression developed in Prob. 5 to investigate whether it is better to limit the
size a0 of initial ﬂaws or to extend the size af of the ﬂaw at which fast fracture occurs.
Limiting a0 might be done with improved manufacturing or better inspection methods,
and increasing af could be done by selecting a material with greater fracture toughness.
For the “baseline” case, take m = 3.5, a0 = 2 mm, af = mm. Compute the percentage
increase in Nf by letting (a) the initial ﬂaw size to be reduced to a0 = 1 mm, and (b)
increasing the ﬁnal ﬂaw size to Nf = 10 mm.