Fatigue of Materials
• Most of the failures in machinery are due to time-varying loads rather than to
static loads.
• These failures typically occur at stress levels significantly lower than the yield
strengths of the materials.
In 1800‟s railroad-car axles made of ductile steel began failing after limited time
in service (exhibited brittle-like failures)!!
Rankine published a paper in 1843, “On the causes of unexpected breakage of
journals of railway axles” - postulated that material had crystallized and become
brittle due to a fluctuating stress
Fatigue of Materials
Fracture surface
Railway axle
Fatigue of Materials
These axles were designed with all the engineering expertise at that time,
but design was based on static loading conditions !!
Dynamic loading conditions were a new phenomenon during those times!!!
August Wohler (a German engineer), made the first systematic
investigation/testing on axles to failure in the laboratory under
fully reverse loading !! (12 yrs of work)
Found that the number of cycles of time varying stresses - real reason for
failure
- Found the existence of Endurance Limit in steels (1870)
S-N diagram / Wohler diagram
Fatigue of Materials
Time-varying stresses
The term Fatigue was first
coined by Poncelet in 1839.
Mechanism was not clearly
understood!!
Chronology of fatigue failure events
1839 Poncelet used the term Fatigue
1837 Rankine discussed the theory of crystallization of fatigue
1871 Wohler published his results on axle failures by fatigue,
developed the rotating bending test, S-N diagram, and
defined endurance limit
1903 Ewing/Humfrey discover slip lines, fatigue cracks - disproved
crystallization theory
1910 Basquin developed the exponential law of endurance tests
(Basquin equation)
1915 Smith/Wedgewood separated cyclic plastic strain from total
plastic strain
1927 Moore quantified high cycle fatigue data
1930 Goodman/Soderberg independently determined the influence
of mean stress on fatigue.
Chronology of fatigue failure events
1937 Neuber published Neuber equation for strain concentration in
notches
1955 Coffin/Manson independently published strain-based low
cycle fatigue law (Coffin-Manson law)
1961 Paris published fracture mechanics based Paris law for
fatigue crack growth
Definition of fatigue: ?
Fatigue Crack Growth
Once a crack is present in a material, it will tend to
grow under the influence of cyclic loading.
The crack may be initiated by fatigue, or may be pre-
existing from manufacture, or may be caused by an
impact, or similar event (e.g., a thermal shock.)
The crack will grow to a critical length then fracture of
the component will occur.
Cyclic Loading
max min
mean
2
range max min
max min
Rotating Machinery amplitude
2
min
Stress Ratio, R
max
Airframes, Bridges, Tanks, etc,
Three stages of fatigue failure
Crack initiation stage
At a microscopic level metals are not homogeneous and isotropic
(eg., inclusions, porosity),
There always exists regions of geometric discontinuities (eg.,
notches) at locations of significant time-varying stresses.
Local yielding at these stress concentrators (yet nominal stress is
well below the yield strength of the material).
Localized plastic yielding causes distortion and creates slip
bands (regions of intense deformation due to shear) along the
crystal boundaries of the material.
Coalescence of slip bands into microscopic cracks with continued
alternate loadings.
Three stages of fatigue failure
Crack propagation stage
After a microcrack is initiated (or could have existed even before
the loading), the sharp crack creates stress concentrations larger
than that of the original geometrical discontinuity (eg., notch).
Development of plastic zones at the tip of the crack each time a
tensile stress open the crack - blunting of the crack tip -
reduction of the effective stress concentration - small amount of
crack growth.
Crack propagation rate is quite small, 10-8 - 10-4 inches/cycle.
Observation of fatigue striations on the fractured surface of the
failed component.
Effect of corrosive environment - rapid crack propagation
(corrosion fatigue)
Three stages of fatigue failure
Sudden fracture due to unstable crack growth
Fatigue striations on the crack
surface of an Aluminum alloy
12000X
Fatigue striations/fracture
Steel keyed shaft failed in rotating Diesel engine crank shaft failed in
bending (crack started at key way) combined bending and torsion
Fatigue-failure models
Stress-life approach (S-N) - determine fatigue strength, endurance limit
Strain-life approach ( - N) - crack initiation stage
Linear elastic fracture mechanics approach (LEFM) - Crack
propagation stage
Fatigue regimes
Based on the number of stress or strain cycles that the component/machine
element is expected to undergo in its lifetime - it is relegated to either low-cycle
fatigue (LCF) or high-cycle fatigue (HCF).
N = 103 cycles
LCF HCF
Stress-life approach
Oldest of the three models - often used for high-cycle fatigue applications
(HCF) where the machine element/component is expected to last for more
than 103 cycles of stress.
It is basically a stress-based model - the interest is to determine the fatigue
strength and/or endurance limit for the material so that the cyclic stresses
are kept below that level and avoid failure.
This approach basically attempts to keep the local stresses at crack
initiation points (eg., notches) so low that the crack-initiation stage never
begins.
Desired goal - Stresses/strains remain elastic everywhere and no local
yielding occurs to initiate a crack.
S-N diagram/Wöhler diagram
Wöhler‟s approach - Loading a rotating cantilever beam in bending to
achieve variation in stresses with time.
R. R. Moore‟s approach - Loading a simply supported rotating beam in fully
reversed, pure bending.
(most of the information in literature on fatigue strength/endurance limit for various materials are
obtained from this approach)
- Highly polished specimen (0.3” dia) mounted in a fixture
- Constant magnitude, pure bending moment applied (specimen rotates at ~ 1725 rpm)
- Specimen subjected to fully reversed loading (bending stress) at any point on the
circumference
- Run the experiment at a particular stress until it fails; record the applied stress and no. of
cycles to failure
- Repeat the experiment for various specimens at various loads - generate S-N diagram
Fatigue-testing set up
S-N diagram/Wöhler diagram
doesn’t break
(Endurance limit)
Fatigue test data now can be fit by the Basquin‟s equation of the form Nap = constant,
where a = stress amplitude, p = constant (slope) - (log-log scale).
S-N curves and fatigue limit
Effect of notch & corrosive
environment on fatigue limit
S-N curves for wrought steels
S-N curve is not a smooth curve !
Relationship between fatigue strength and ultimate strengths (steels)
S-N curves and fatigue limit
Influence of
processing ?
S-N curves for aluminum alloys (wrought, die-cast, sand-cast):
Unnotched-polished specimens:Rotating beam test
Thickness Residual
Cr plating stress
(mm) (MN/m2)
0.0025 970
0.0075 510
0.025 77
0.15 77
Effect of chrome plating on the
fatigue strength of steel
Magnitude & sign of the
residual tensile stresses
will determine the fatigue
strength !
Effect of nickel plating & peening on the fatigue strength of steel
Effect of environment
Environmental effects on the fatigue strength of steel
Axial fatigue
Comparison with
axial fatigue data Difference between rotating-
bending test and axial fatigue
tests ?
10-30% lower values reported for axial fatigue
compared to rotating bending tests
Strain-life approach
Since in most of the machine elements/components the response of the
material at critical locations (eg., notches) is strain (deformation)dependent,
a strain-based approach seems more reasonable. In this approach the
plastic strain is measured directly and quantified.
A strain-based model gives an accurate picture of the crack-initiation stage
and a more reasonable approach for fatigue life prediction. Crack-growth is
not explicitly accounted for in this method and hence this approach is more
or less for crack initiation life estimates.
Combinations of fatigue loading and high temperature effects are better
handled by this method as creep effects can be included.
Often applied to low-cycle fatigue (LCF) applications where the cyclic
stresses are high enough to cause local yielding.
Cyclic stress-strain behavior/strain
controlled fatigue
If the loading process is reversed &
the specimen unloaded after yielding,
the stress-strain relationship will follow
a line parallel to the loading direction
(same slope) – AB
Further loading, in the compressive
direction leads to the curve BC.
In 1926, Massing observed that
stress-strain curve AC can be
obtained by doubling values of the
stress-strain curve OA. (assuming the
material exhibits symmetry along both
tension and compression
Cyclic stress-strain curve
If the loading process continues
from the compressive zone to the
tensile zone (i.e from -max to +
max ) – one gets a hysteresis
loop (this forms one fatigue cycle
in the strain-life approach) –
equivalent to 2 strain reversals.
Observe the strain and stress
amplitudes – they are half of the
total strain and stress ranges.
Cyclic stress-strain curve
During strain-controlled cyclic
loading, the stress-strain response
of the material can change with the
no. of cycles.
If the maximum stress keeps
increasing with each successive
cycle, then the material undergoes
cyclic hardening, & if the maximum
stress keep decreasing, then the
material undergoes cyclic softening.
If the maximum stress does not
change with number of cycles –
then the material is supposed to be
cyclically stable
Cyclic stress-strain curve
The cyclic stress-strain curve is now comprised of two components of strain,
elastic strain (e) and the plastic strain (p)
Now, the total strain can be expressed in terms of the stress (analogous to
an expression used for a non-cyclic stress strain curve)
where, K‟ = cyclic strength coefficient
n‟ = cyclic strain hardening exponent
Cyclic stress-strain curve
For a stabilized hysteresis loop (stabilization occurs say after about 100
cycles of imposed strain amplitude), the hysteresis equation would be:
The above equation is valid only for those materials which exhibit symmetry
in tension and compression.
Low-cycle fatigue (Strain-life approach)
For low-cycle fatigue applications, using the test
data, a plot of the plastic strain range (p) is
plotted as a function of number of cycles to failure.
LCF data for stainless
steel exhibiting a linear
dependence on a double
log scale
The above data can be fit using, Where, P/2 = plastic strain amplitude,
P ‟f = regression intercept (fatigue ductility
'f (2 N) C coefficient) and
2
C = regression slope (fatigue ductility
Coffin-Manson relation exponent)
High cycle fatigue (low strains)/Modified
Basquin‟s equation
For high cycle fatigue – low strain cases, the Basquin‟s equation that is used
to fit S-N data can be modified as follows:
e
a E 'f (2 N ) b Strain-life equation
2
Where, a = stress amplitude
e/2 = strain amplitude (elastic)
‟f = fatigue strength coefficient
b = fatigue strength exponent
Strain life curve
Now one can combine the Coffin-Manson equation (plastic strain amplitude)
and the modified Basquin‟s equation (elastic strain amplitude) to get the total
strain amplitude.
Strain life curve
Influence of plastic Influence of elastic
strain on fatigue life strain on fatigue life The transition life (2Nt) represent the
life at which the elastic and plastic
strain ranges are equivalent, which
can be expressed as
Elastic strains have dominant
influence on fatigue lives above the
transition life and plastic strains
below the transition life (observe
figure)
Clear dilineation between low and high cycle fatigue
Recap
For long fatigue lives the fatigue strength controls the
fatigue performance and the Stress-life and Strain-life
approaches essentially yield the same results.
For short fatigue lives, the fatigue ductility controls the
fatigue performance
Linear elastic fracture mechanics(LEFM)
approach
This is the best model for understanding the crack propagation stage during
fatigue process.
Normally applied to low cycle fatigue (LCF), finite life problems where the
cyclic loadings are known to cause formation of cracks and is useful in
predicting the remaining life of cracked parts in service.
This approach heavily relies on the expression for the geometric factor ()
of the stress-intensity factor and on the estimate of the initial crack size „a‟
K = a
Now, the crack propagation life can be determined from the assumed initial
crack size to a final crack size.
Driving force for crack growth
Driving force for crack growth is the range in the stress intensity
factor during cycling:
K f (a/W) a
max min for R 0
max for R 0
Where, K = stress intensity factor range (Kmax – Kmin)
= stress range
a = crack length
R = min/max (stress ratio)
Crack growth rate (da/dN)
1 < 2 1
ac1 X
Crack Length, a
2
ac2 X
da/dN2
da/dN1
ao
Initial crack
length, ao
Cycles, N
Paris Law
Paris Region
Sigmoidal curve
Stable Growth
Threshold Region
da
A(K) m
dN
Rapid-unstable Growth
log(da/dN)
Slow Growth
Fast Fracture Region
m
A II
I III
Kth log(K)
KC
Paris law
Applied stress intensity factor (K) < Threshold (Kth) implies no fatigue crack growth.
Fatigue crack growth life is found by integrating the Paris equation between a known
(or assumed) initial crack length and a maximum acceptable crack length.
da
A(K) m
dN
ac
da
Np m
a o A(K)
Initial crack size (ao) is usually found by inspection/assume a min. crack
length,
2
Kc
Critical crack size (ac) is found from a c 1
f (a/W) max
Problem
When subjected to fatigue under = 140 MPa, an alloy showed the
following Paris-type fatigue crack propagation relationship:
where K is in MPa.m1/2. Estimate the number of cycles required for the
crack to grow from 2 mm to 8.8 mm. The geometric factor can be taken as
1.12.
Answer: Nf = 1413 cycles
Fatigue fracture surfaces
• Crack origin:
• Fatigue zone:
• Progression marks:
Macroscopic surface features
• Overload zone:
Fatigue fracture surfaces
Progression marks
Ratchet mark - boundary
• indicative of direction of between two failure planes
propagation of crack
• indicative of multiple origins
• indicative of substantial
• relatively high stresses
changes in load
• not to be confused with
fatigue striations
Origin of failures & river marks
• multiple ratchet marks indicative River marks - indicative of
of multiple origins of cracks direction of fatigue crack
• angle between ratchet marks • appear at fast-growing
indicative of primary origin sections of the fatigue zones
Torsional fatigue failures
Plane bending, reversed bending &
rotational bending fatigue failures
