Fatigue
Failure
1
Materials Tetrahedron
Processing
Performance
Microstructure
2
Properties
Objective
 The objective of this lecture is to
explain the phenomenon of fatigue
and also to show how resistance to
fatigue failure depends on stress –
strain distribution, stress amplitude
and material microstructure.
3
sa :=
sm :=
R :=
e :=
Nf :=
A :=
∆pl :=
∆el :=
K’ :=
n’ :=
c :=
E :=
b :=
m :=
K :=
∆K :=
a :=
4
Alternating stress
Mean stress
Stress ratio
strain
number of cycles to failure
Amplitude ratio
Plastic strain amplitude
Elastic strain amplitude
Proportionality constant, cyclic stress-strain
Exponent in cyclic stress-strain
Exponent in Coffin-Manson Eq.;
also, crack length
Young’s modulus
exponent in Basquin Eq.
exponent in Paris Law
Stress intensity
Stress intensity amplitude
crack length
Notation
Fatigue Defined
 ASTM E206-72 Definition
The Process of PROGRESSIVE
LOCALIZED PERMANENT Structural
Change Occurring in a Material
Subjected to Conditions Which Produce
FLUCTUATING Stresses and Strains at
Some Point or Points Which May
Culminate in CRACKS or Complete
FRACTURE After a Sufficient Number
of Fluctuations
5
Fatigue
 Fatigue is the name given to failure in response
to alternating loads (as opposed to monotonic
straining).
 Instead of measuring the resistance to fatigue
failure through an upper limit to strain (as in
ductility), the typical measure of fatigue
resistance is expressed in terms of numbers of
cycles to failure. For a given number of cycles
(required in an application), sometimes the
stress (that can be safely endured by the
material) is specified.
6
Fatigue: general characteristics
 Primary design criterion in rotating parts.
 Fatigue as a name for the phenomenon based on the
notion of a material becoming “tired”, i.e. failing at
less than its nominal strength.
 Cyclical strain (stress) leads to fatigue failure.
 Occurs in metals and polymers but rarely in
ceramics.
 Also an issue for “static” parts, e.g. bridges.
 Cyclic loading stress limit<static stress capability.
7
Fatigue: general characteristics
 Most applications of structural materials involve cyclic
loading; any net tensile stress leads to fatigue.
 Fatigue failure surfaces have three characteristic
features:
• A (near-)surface defect as the origin of the crack
• Striations corresponding to slow, intermittent crack
growth
• Dull, fibrous brittle fracture surface (rapid growth).
 Life of structural components generally limited by
cyclic loading, not static strength.
 Most environmental factors shorten life.
8
Three Stages of Fatigue Failure
 Crack Initiation
 Crack Propagation
• oscillating stress… crack grows, stops
growing, grows, stops growing… with crack
growth due to tensile stresses
 Fracture
• sudden, brittle-like failure
9
Fatigue Failure

Caused by LoadCycling at <y

Brittle-Like Fracture
with Little Warning
by Plastic
Deformation
•
May take Millions of
Cycles to Failure
 Fatigue Failure
Time-Stages
10
1. Crack Initiation Site(s)
2. “Beach Marks” Indicate of
Crack Growth
3. Distinct Final
Fracture Region
Identifying Fatigue Fractures
beachmarks
11
Fatigue Crack Nucleation
 Flaws, cracks, voids can all act as crack
nucleation sites, especially at the
surface.
 Therefore, smooth surfaces increase
the time to nucleation; notches, stress
risers decrease fatigue life.
 Dislocation activity (slip) can also
nucleate fatigue cracks.
12
Role of dislocations
 Dislocations in a material provide a mechanism by
which large numbers of atomic bonds can be
broken and re-formed
• the theoretical strength of ideal, dislocation-free
materials is much higher than that measured in practice
• also, the preparation and treatment of the material
significantly influenced the measured strength
 Recall the two main types of dislocation
• edge dislocation
• screw dislocation
13
Edge dislocation
 Under an applied stress, the edge dislocation can
move in the direction of the stress
unit step
of slip
slip plane
shear stress
 This process, leading to elastic deformation, is
called slip
14
Screw dislocation
 The screw dislocation itself moves perpendicular
to the stress direction, but the deformation ends
up the same
15
Slip systems
 In crystalline materials, the anisotropy of the structure
can mean that certain slip directions are preferred
• termed slip planes
• and these move in slip directions
 Together slip planes and directions are called slip
systems
• and these slip systems act to minimise the overall
atomic distortion caused by the motion of the
dislocation
 It follows from the above that the slip planes are those
planes in the crystal which have the highest packing
density of atoms
• by keeping these densely packed atoms together,
fewer bonds are distorted
16
(111) plane of a FCC
material, showing three
<110> slip directions
 The number of slip systems depends on the
crystal structure
• each slip system for BCC and FCC has at least 12 slip
directions
• while the maximum for HCP is 6
 Hence FCC (Cu, Al, Ni, Ag, Au) and BCC (Fe)
metals tend to be ductile and exhibit large plastic
deformation
• while HCP (Ti, Zn) are more brittle
17
Dislocation Slip
Crack Nucleation
 Dislocation slip -> tendency to localize
slip in bands.
 Persistent Slip Bands (PSB’s)
characteristic of cyclic strains.
 Slip Bands -> extrusion at free surface.
 Extrusions -> intrusions and crack
nucleation.
18
Intrusions and Extrusions:
The Early Stages of Fatigue Crack Formation

19

Fatigue Crack Propagation
 Crack Nucleation stress intensification at crack
tip.
 Stress intensity crack propagation (growth);
- stage I growth on shear planes (45°),
strong influence of microstructure
- stage II growth normal to tensile load (90°)
weak influence of microstructure.
 Crack propagation catastrophic, or ductile
failure at crack length dependent on boundary
conditions, fracture toughness.
20
Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding
and Transition From Mode II to Mode I
c
Δσ
Locally, the crack grows in shear;
Δσ
21
macroscopically it grows in tension.
The Process of Fatigue
The Materials Science Perspective:
• Cyclic slip,
• Fatigue crack initiation,
• Stage I fatigue crack growth,
• Stage II fatigue crack growth,
• Brittle fracture or ductile rupture
22
Features of the Fatigue Fracture Surface of a Typical Ductile Metal
Subjected to Variable Amplitude Cyclic Loading
A – fatigue crack area
B – area of the final static
failure
23
Appearance of Failure Surfaces Caused by Various
Modes of Loading (SAE Handbook)
24
Factors Influencing Fatigue Life
Applied Stresses
Stress range – The basic cause of plastic deformation and consequently the
accumulation of damage
Mean stress – Tensile mean and residual stresses aid to the formation and
growth of fatigue cracks
Stress gradients – Bending is a more favorable loading mode than axial
loading because in bending fatigue cracks propagate into the region of lower
stresses
Materials
Tensile and yield strength – Higher strength materials resist plastic
deformation and hence have a higher fatigue strength at long lives. Most
ductile materials perform better at short lives
Quality of material – Metallurgical defects such as inclusions, seams,
internal tears, and segregated elements can initiate fatigue cracks
Temperature – Temperature usually changes the yield and tensile strength
resulting in the change of fatigue resistance (high temperature decreases
fatigue resistance)
Frequency (rate of straining) – At high frequencies, the metal component
may be self-heated.







25
Strength-Fatigue Analysis Procedure
Material
Properties
Component
Geometry
Loading
History
Stress-Strain
Analysis
Damage
Analysis
Allowable Load - Fatigue Life
Information path in strength and fatigue life prediction
procedures
26
Design Philosophy:
Damage Tolerant Design
 S-N (stress-cycles) curves = basic characterization.
 Old Design Philosophy = Infinite Life design: accept
empirical information about fatigue life (S-N curves);
apply a (large!) safety factor; retire components or
assemblies at the pre-set life limit, e.g. Nf=107.
 Crack Growth Rate characterization ->
 Modern Design Philosophy (Air Force, not Navy
carriers!) = Damage Tolerant design: accept
presence of cracks in components. Determine life
based on prediction of crack growth rate.
27
Three Theories
28
Stress-Life
stress-based, for high-cycle fatigue, aims to
prevent crack initiation
Strain-Life
useful when yielding begins (i.e., during
crack initiation), for low-cycle fatigue
LEFM (Fracture
Mechanics)
best model of crack propagation, for lowcycle fatigue
Definitions: Stress Ratios
 a
Mean stress  m = (max +min)/2.
Pure sine wave Mean stress=0.
Stress ratio  R = max/min.
For m = 0, R=-1
Amplitude ratio  A = (1-R)/(1+R).
Statistical approach shows significant distribution
in Nf for given stress.
 Alternating Stress






29
Types of Fatigue Loading
Fully Reversed
Repeated
   max   min
stress range

a 
2
alternating
component
m 
30
 max   min
2
mean
component
Fluctuating
amplitude
ratio
stress ratio
a
A
m
 min
R
 max
Constant and Variable Amplitude Stress Histories; Definition of a Stress
Cycle & Stress Reversal
Constant amplitude stress history
Stress
a)
max
m
min
One
cycle
0
peak
peak
 peak   max
  min
;
 apeak
peak
  mipeak
 peak  max
n


2
2
Time
Variable amplitude stress
history

Stress
b)
One
reversal
0
31
In the case of the peak stress
history the important parameters
are:
Time
peak
m

peak
peak
 max
  min
 mipeak
R  penak
 max
2
;
Mean Stress
 Alternating stress  a = (max-min)/2.
 Raising the mean stress (m) decreases Nf.
 Various relations between R = 0 limit and the ultimate
(or yield) stress are known as Soderberg (linear to
yield stress), Goodman (linear to ultimate) and
Gerber (parabolic to ultimate).
a
endurance limit at zero mean stress


 mean

 a   fat
1 

 tensilestrength
tensile strength
mean
32
Stress amplitude, Sa
Mean Stress Effect
sm> 0
0
sm< 0
time
Stress amplitude, logSa
sm= 0
sm< 0
sm= 0
sm> 0
No. of cycles, logN
33
2*106
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or
has negligible effect on the fatigue durability.
Because most of the S – N data used in analyses was produced under zero mean stress (R = -1)
therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with
zero mean stress producing the same fatigue life.
There are several empirical methods used in practice:
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted
for. The procedure is based on a family of Sa – Sm curves obtained for various fatigue lives.
Steel AISI 4340,
Sy = 147 ksi
34
(Collins)
Mean Stress Correction for Endurance Limit
2
Gereber (1874)
Goodman (1899)
35
Sa  S m 
  1
Se  S u 
Sa S m

1
Se Su
Soderberg (1930)
Sa Sm

1
Se Sy
Morrow (1960)
Sa Sm
 ' 1
Se  f
Sa – stress amplitude
applied at the mean
stress Sm≠ 0 and fatigue
life N = 1-2x106cycles.
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
f’- true stress at fracture
Mean stress correction for arbitrary stress
amplitude applied at non-zero mean stress
Gereber (1874)
Sa
S ar
Goodman (1899)
Soderberg (1930)
Sa
 Sm 


 Su 

Sm
S ar
Su
Sa
Sm

S ar
2
1
1
Sy
1
Sa – stress amplitude
applied at the mean
stress Sm≠ 0 and
resulting in fatigue life of
N cycles.
Sm- mean stress
Sar- fully reversed stress
amplitude applied at
mean stress Sm=0 and
resulting in the same
fatigue life of N cycles
Su- ultimate strength
Morrow (1960)
Sa
S ar
36

Sm
'
f
1
f’- true stress at fracture
Comparison of various
methods of accounting
for the mean stress effect
Most of the experimental data lies between the Goodman and the yield line!
37
Approximate Goodman’s diagrams for ductile and brittle
materials
Kf
Kf
Kf
38
The following generalisations can be made when discussing
mean stress effects:
1. The Söderberg method is very conservative and seldom used.
3. Actual test data tend to fall between the Goodman and Gerber curves.
3. For hard steels (i.e., brittle), where the ultimate strength approaches the
true fracture stress, the Morrow and Goodman lines are essentially the
same. For ductile steels (of > S,,) the Morrow line predicts less
sensitivity to mean stress.
4. For most fatigue design situations, R < 1 (i.e., small mean stress in
relation to alternating stress), there is little difference in the theories.
5. In the range where the theories show a large difference (i.e., R values
approaching 1), there is little experimental data. In this region the yield
criterion may set design limits.
6. The mean stress correction methods have been developed mainly for the
cases of tensile mean stress.
For finite-life calculations the endurance limit in any of the equations can be
replaced with a fully reversed alternating stress level corresponding to that
finite-life value!
39
Cyclic strain vs. cyclic stress
 Cyclic strain control complements cyclic stress
characterization: applicable to thermal fatigue, or
fixed displacement conditions.
 Cyclic stress-strain testing defined by a controlled
strain range, ∆pl.
 Soft, annealed metals tend to harden;
strengthened metals tend to soften.
 Thus, many materials tend towards a fixed cycle,
i.e. constant stress, strain amplitudes.
40
Cyclic stress-strain curve
[Courtney]
• Large number of cycles typically needed to reach
asymptotic hysteresis loop (~100).
• Softening or hardening possible.
41
Cyclic stress-strain
 Wavy-slip materials generally
reach asymptote in cyclic
stress-strain: planar slip
materials (e.g. brass) exhibit
history dependence.
 Cyclic stress-strain curve
defined by the extrema, i.e.
the “tips” of the hysteresis
loops. Cyclic stress-strain
curves tend to lie below those
for monotonic tensile tests.
 Polymers tend to soften in
cyclic straining.
42
Cyclic Strain Control
 Strain is a more logical independent variable for
characterization of fatigue.
 Define an elastic strain range as ∆el = ∆/E.
 Define a plastic strain range, ∆pl.
 Typically observe a change in slope between the
elastic and plastic regimes.
 Low cycle fatigue (small Nf) dominated by plastic
strain: high cycle fatigue (large Nf) dominated by
elastic strain.
43
Cyclic Strain control: low cycle
 Constitutive relation
n
  K  
for cyclic stress-strain:
 n’ ≈ 0.1-0.2
 Fatigue life: Coffin Manson relation:
 p
c
 f 2N f
2
 f ~ true fracture strain; close to tensile
ductility
 c ≈ -0.5 to -0.7
 c = -1/(1+5n’); large n’  longer life.

44

Cyclic Strain control: high cycle
 For elastic-dominated strains
 e
a  E
 f 2N b
at high cycles, adapt
2
Basquin’s equation:
 Intercept on strain axis of extrapolated elastic line =
f/E.
 High cycle = elastic strain control:
slope (in elastic regime) = b = -n’/(1+5n’)
 The high cycle fatigue strength, f, scales with the
yield stress  high strength good in high-cycle
45
Low vs. High Cycle
>103 cycles, high cycle fatigue
car crank shaft – ~2.5 E8 Rev/105 miles
manufacturing equipment @ 100 rpm – 1.25 E8 Rev/year
<103 cycles, low cycle fatigue
ships, planes, vehicle chassis
46
Strain amplitude - cycles
[Courtney]
47
Total strain (plastic+elastic) life
 Low cycle = plastic control: slope = c
 Add the elastic and plastic strains.
  el  pl f



2N f
2
2
2
E


b
 
 f 2N f
c
 Cross-over between elastic and plastic control is
typically at Nf = 103 cycles.
 Ductility useful for low-cycle; strength for high cycle
 Examples of Maraging steel for high cycle
endurance, annealed 4340 for low cycle fatigue
strength.
48
The Fatigue S-N method
(Nominal Stress Approach)

The principles of the S-N approach (the nominal stress method)

Fatigue damage accumulation

Significance of geometry (notches) and stress analysis in fatigue
evaluations of engineering structures


49
Fatigue life prediction in the design process
S-N Curves
 S-N [stress-number of cycles to failure] curve defines
locus of cycles-to-failure for given cyclic stress.
 Rotating-beam fatigue test is standard; also alternating
tension-compression.
 Plot stress versus the log (number of cycles to failure),
log(Nf).
 For frequencies < 200Hz, metals are insensitive to
frequency; fatigue life in polymers is frequency
dependent.
50
Testing Fatigue Properties
 Rotating Beam – most data is from this type
 Axial
• lower or higher? Why?
 Cantilever
 Torsion
51
Wöhler’s Fatigue Test
Note! In the case of smooth
components such as the
railway axle the nominal stress
and the local peak stress are
the same!
S 
peak
Smin
Smax
52
B
A
Fatigue testing, S-N curve
a
mean 3 > mean 2 > mean 1
mean 1
mean 2
mean 3
The greater the number of
cycles in the loading history,
the smaller the stress that
the material can withstand
without failure.
log Nf
Note the presence of a
fatigue limit in many
steels and its absence
in aluminum alloys.
53
Infinite life
Part 1
Part 2
Stress amplitude, Sa (ksi)
Su
Part 3
Fatigue S-N curve
S103
Sy
Se
Number of cycles, N
Fully reversed axial S-N curve for AISI 4130 steel. Note the break at the LCF/HCF transition and
the endurance limit
Characteristic parameters of the S - N curve are:
Se - fatigue limit corresponding to N = 1 or 2106 cycles for
steels and N = 108 cycles for aluminum alloys,
m
A
3
3
S10 - fully reversed stress amplitude corresponding to N = 10
a
cycles
m - slope of the high cycle regime curve (Part 2)
S CN
54
 10  N m
Fully Reversed Empirical Data
An S-N Curve
Wrought Steel
55
Fully Reversed Empirical Data
Aluminum
56
Relative stress amplitude, Sa/Su
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests.
However, material behavior and the resultant S - N curves are different for different types of loading.
It concerns in particular the fatigue limit Se.
S103
Se
1.0
0.5
Bending
Axial
0.3
Torsion
0.1
103
10
105
106
4
10
7
Number of cycles, Log(N)
The stress endurance limit, Se, of steels (at 106 cycles) and the fatigue strength, S103 corresponding
to 103 cycles for three types of loading can be approximated as (ref. 1, 23, 24):
57
S103 = 0.90Su
and
Se = S106 = 0.5 Su
- bending
S103 = 0.75Su
and
Se = S106 = 0.35 - 0.45Su - axial
S103 = 0.72Su
and
Se = S106 = 0.29 Su
- torsion
Endurance Limit
S e
A stress level below which a material can be cycle
infinitely without failure
Many materials have an endurance limit:
low-strength carbon and alloy steels, some stainless steels, irons,
molybdenum alloys, titanium alloys, and some polymers
Many other materials DO NOT have an endurance limit:
aluminum, magnesium, copper, nickel alloys, some stainless steels,
high-strength carbon and alloy steels
Sf
58
for these, we use a FATIGUE STRENGTH defined for a certain
number of cycles (5E8 is typical)
Endurance Limits
 Some materials exhibit endurance limits, i.e. a stress
below which the life is infinite:
• Steels typically show an endurance limit, = 40% of
yield; this is typically associated with the presence
of a solute (carbon, nitrogen) that pines
dislocations and prevents dislocation motion at
small displacements or strains (which is apparent
in an upper yield point).
• Aluminum alloys do not show endurance limits;
this is related to the absence of dislocation-pinning
solutes.
 At large Nf, the lifetime is dominated by nucleation.
• Therefore strengthening the surface (shot
peening) is beneficial to delay crack nucleation
and extend life.
59
Approximate endurance limit for various materials:
Magnesium alloys (at 108 cycles) Se = 0.35Su
Copper alloys (at 108 cycles)
0.25Su< Se <0.50Su
Nickel alloys (at 108 cycles)
0.35Su <Se < 0.50Su
Titanium alloys (at 107 cycles)
0.45Su <Se< 0.65Su
Al alloys (at 5x108 cycles) Se = 0.45Su (if Su ≤ 48 ksi) or Se = 19 ksi (if Su> 48 ksi)
Steels (at 106 cycles)
Se = 0.5Su (if Su ≤ 200 ksi) or Se = 100 ksi (if Su>200 ksi)
Irons
Se = 0.4Su (if Su ≤ 60 ksi)
(at 106 cycles)
or Se = 24 ksi (if Su> 60 ksi)
S – N curve
S a  C  N m  10 A  N m
or N  C

1
m
 Sa 

 S
S
 103 
1
103

m   log 
 and A  log 
3
Se
 Se 

60
1
m

2
C





A
m
 Sa 
1
m
Fatigue Limit – Modifying Factors
For many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on
the base-line S-N curves for ferrous alloys in the intermediate
to long life ranges. The variables investigated include:
- Rotational bending fatigue limit, Se’,
- Surface conditions, ka,
Fatigue limit of a machine
part, Se
- Size, kb,
- Mode of loading, kc,
- Temperature, kd
- Reliability factor, ke
- Miscellaneous effects (notch), kf
61
Se = ka kb kc kd ke kf·Se’
Surface Finish Effects on Fatigue Endurance Limit
The scratches, pits and machining marks on the surface of a material add stress concentrations to the
ones already present due to component geometry. The correction factor for surface finish is sometimes
presented on graphs that use a qualitative description of surface finish such as “polished” or “machined”.
Ca
ka
Below a generalized empirical graph
is shown which can be used to
estimate the effect of surface finish
in comparison with mirror-polished
specimens [Shigley (23), Juvinal
(24), Bannantine (1) and other
textbooks].
Effect of various surface finishes
on the fatigue limit of steel.
Shown are values of the ka, the
ratio of the fatigue limit to that
for polished specimens.
(from J. Bannantine, ref.1)
62
Size Effects on Endurance Limit
Fatigue is controlled by the weakest link of the material, with the probability of existence (or density) of a
weak link increasing with material volume. The size effect has been correlated with the thin layer of
surface material subjected to 95% or more of the maximum surface stress.
There are many empirical fits to the size effect data. A fairly conservative one is:
kb 
Se 1

'
Se  0.869d 0.097
or
Se 1.0
kb  '  
Se 1.189d 0.097
d  0.3 in 

if 0.3 in  d  10.0 in 
if
if
if
d  8 mm


8  d  250 mm 
• The size effect is seen mainly at very long lives.
• The effect is small in diameters up to 2.0 in (even in bending and torsion).
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter, de.
The effective diameter, de, for non-circular cross sections is obtained by equating the volume of material
stressed at and above 95% of the maximum stress to the same volume in the rotating-bending
specimen.
63
max
The effective diameter, de, for members
with non-circular cross sections
0.95max
+
The material volume subjected to stresses
  0.95max is concentrated in the ring of
0.05d/2 thick.
0.05d/2
The surface area of such a ring is:
A0.95 max 
-
 2
2
d   0.95d    0.0766d 2
4

* rectangular cross section under bending
t
0.95t
A  Ft
A0.95 max  Ft  0.95t  0.05 Ft
Equivalent diameter
0.0766d e2  0.05Ft
F
64
d e  0.808 Ft
Loading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating
bending tests ranges from 0.6 to 0.9.
Se ( axial )  (0.7  0.9) Se ( bending )
kc  0.7  0.9 ( suggested by Shigley kc  0.85)
The ratio of endurance limits found using torsion and rotating bending tests
ranges from 0.5 to 0.6. A theoretical value obtained from von Mises-HuberHencky failure criterion is been used as the most popular estimate.
 e ( torsion )  0.577 Se ( bending )
kc  0.57( suggested by Shigley kc  0.59)
65
Temperature Effect
From: Shigley and Mischke, Mechanical Engineering Design, 2001
Se ,T  Se , RT kd  Se , RT
66
Su,T
Su, RT
;
kd 
Su,T
Su, RT
Reliability factor ke
The reliability factor accounts for the scatter of reference data such
as the rotational bending fatigue limit Se’.
The estimation of the reliability factor is based on the assumption that
the scatter can be approximated by the normal statistical probability
density distribution.
ke  1  0.08  za
The values of parameter za associated with various levels of
reliability can be found in Table 7-7 in the textbook by Shigley et.al.
67
S-N curves for assigned probability of failure; P - S - N curves
68
Fatigue Parameters
 Recall Fatigue Testing (RR Moore Tester)
specimen
compression on top
motor
counter
flex coupling
tension on bottom
 Stress Varies with Time;
Key Parameters
• m  Mean Stress (MPa)
• S  Stress Amplitude (MPa)
 Failure Even though
max < c
69
max
m

 m   max   min  2
S   max   min  2
S
min
 Cause of ~90% of
Mechanical Failures
time
Fatigue Design Parameter
 Fatigue (Endurance)
Limit, Se’ in MPa
• Unlimited Cycles if S
< Se’
S = stress amplitude
unsafe
S e’
case for
steel (typ.)
safe
103 105
107
109
N = Cycles to failure
 Some Materials will
NOT permit
Limitless Cycling
• i.e.; Se’ = ZERO
S = stress amplitude
unsafe
safe
103 105
107
109
N = Cycles to failure
70
case for
Al (typ.)
Fatigue Parameters
 Fatigue Cracks Grow INCREMENTALLY
during the TENSION part of the Cycle
 Math Model for Incremental Crack Extension
typ. 1 to 6


da
m
K I ~  a
 K
dN
Opening-Mode (Mode-I) Stress Intensity Factor
 
increase in crack length per loading cycle
 Example: Austenitic Stainless Steel


da
12
m / cyc   5.6 10  K Mpa m
dN
71

3.25
Fatigue Crack Propagation
 Crack Length := a.
Number of cycles := N
Crack Growth Rate := da/dN
Amplitude of Stress Intensity := ∆K = ∆√c.
 Define three stages of crack growth, I, II and III,
in a plot of da/dN versus ∆K.
 Stage II crack growth: application of linear elastic fracture mechanics.
 Can consider the crack growth rate to be related to the applied stress
intensity.
 Crack growth rate somewhat insensitive to R (if R<0) in Stage II
 Environmental effects can be dramatic, e.g. H in Fe, in increasing
crack growth rates.
72
Fatigue Crack Propagation
 Three stages of crack
growth, I, II and III.
 Stage I: transition to a
finite crack growth rate
from no propagation
below a threshold value
of ∆K.
 Stage II: “power law”
dependence of crack
growth rate on ∆K.
 Stage III: acceleration of
growth rate with ∆K,
approaching catastrophic
fracture.
73
da/dN
I
II
∆Kc
III
∆Kth
∆K
Paris Law
 Paris Law:






74
dc
 A(K)m
dN
m ~ 3 (steel); m ~ 4 (aluminum).
Crack nucleation ignored!
Threshold ~ Stage I
The threshold represents an endurance limit.
For ceramics, threshold is close to KIC.
Crack growth rate increases with R (for R>0).
Striations- mechanism
 Striations occur by development of slip bands
in each cycle, followed by tip blunting,
followed by closure.
 Can integrate the growth rate to obtain cycles
as related to cyclic stress-strain behavior.
cf
dc
N II  
c dc / dN
0
75
cf
N II  
c0
dc
A
m

c
m
Striations, contd.
 Provided that m>2 and is constant, can integrate.
A1  m 1m / 2  1m / 2 
N II 
c0
cf
(m / 2)  1


 If the initial crack length is much less than the final
length, c0<cf, then approximate thus:
A1  m 1m / 2 
N II 
c0
(m / 2)  1
 Can use this to predict fatigue life based on known
crack
76
Damage Tolerant Design
 Calculate expected growth rates from
dc/dN data.
 Perform NDE on all flight-critical
components.
 If crack is found, calculate the expected
life of the component.
 Replace, rebuild if too close to life limit.
 Endurance limits.
77
Geometrical effects
 Notches decrease fatigue life through stress
concentration.
 Increasing specimen size lowers fatigue life.
 Surface roughness lowers life, again through stress
concentration.
 Moderate compressive stress at the surface
increases life (shot peening); it is harder to nucleate a
crack when the local stress state opposes crack
opening.
 Corrosive environment lowers life; corrosion either
increases the rate at which material is removed from
the crack tip and/or it produces material on the crack
surfaces that forces the crack open (e.g. oxidation).
 Failure mechanisms
78
Stress concentration factor, Kt, and the
notch factor effect, kf
Fatigue notch factor effect kf depends on the stress
concentration factor Kt (geometry), scale and material
properties and it is expressed in terms of the Fatigue Notch
Factor Kf.
1
kf 
Kf
79
F
2
2
2
C
peak
A, B
3
3
D
n
3
3
11
A
2
2
C
2
2
2
2
D
3
11
3
3
80
Stresses in axisymmetric
notched body
F
B
1
F
n  S 
A
and
 peak  K t n
Stresses in prismatic notched body
2
2
F
A,
B, C
2
peak
2
2
C
2
2
D
11
F
n  S 
A
and
 peak  K t n
n
11
 Kt S
E
2
2
A
D
B
E
3
3
3
81
F
3
3
1
Stress concentration factors used in fatigue analysis
M
S
n, S
r
0
x
dn
W
82
 peak
Kt 
n

n
r
 peak
S
peak
Stress
Stress
peak
x
0
dn
W
Stress concentration
factors, Kt, in shafts
S=
Bending load
S=
Axial load
83
S=
84
Similarities and differences between the stress field near the notch
and in a smooth specimen
P
S
Stress
peak
n=S
peak= n=S
r
x
0
dn
W
85
The Notch Effect in Terms of the Nominal Stress
S
Smax
n1
N (S)m = C
S
n2
S2
Sesmooth
Stress range
n3
n4
np
Kf 
86
Fatigue notch factor!
Senotched
N2
N0
Sesmooth

Kf
cycles
K f  K t !!!
Definition of the fatigue notch factor Kf
 22   peak
S
M
  esmooth

2
notched
e
Se
1
3
Stress
22= peak
Nsmooth
Stress
2
 enotched
2
87
1
1
Nnotched
Nnotched
 esmooth
K f  notched
e
peak
for
N smooth  N notched
PETERSON's approach
Kt  1
K f  1
 1  q  K t  1
1 a r
1
q
;
1 a r
1.8
a – constant,
r – notch tip radius;
 300 
3
a
  10  in.
 Su 
for S u in  in.
NEUBER’s approach
Kt  1
K f  1
1 r r
88
ρ – constant,
r – notch tip radius
The Neuber constant ‘ρ’ for steels and aluminium alloys
89
Curves of notch sensitivity index ‘q’ versus notch radius
(McGraw Hill Book Co, from ref. 1)
90
Illustration of the notch/scale effect
Plate 1
W1 = 5.0 in
d1 = 0.5 in.
pea
k
pea
k
m1
m2
Su = 100 ksi
Kt = 2.7
q = 0.97
Kf1 = 2.65
Plate 2
d2
d1
r
r
W2= 0.5 in
d2 = 0.5 i
W1
W2
Su = 100 ksi
Kt = 2.7
q = 0.78
Kf1 = 2.32
91
Procedures for construction of approximate fully reversed S-N curves for
smooth and notched components
f ’
Collins method
Juvinal/Shigley method
Nf (logartmic)
Sar, ar – nominal/local
stress amplitude at zero
mean stress m=0 (fully
reversed cycle)!
Nf (logartmic)
92
Procedures for construction of approximate fully reversed S-N curves for
smooth and notched components
Manson method
Sar (logartmic)
0.9Su
Se’kakckbkdke
Se’kakckbkdkekf
Se
100
101
102
103
104
105
106
2·106
Nf (logartmic)
Sar, ar – nominal/local stress amplitude at zero mean stress m=0
(fully reversed cycle)!
93
NOTE!
• The empirical relationships concerning the S –N curve data are
only estimates! Depending on the acceptable level of uncertainty
in the fatigue design, actual test data may be necessary.
• The most useful concept of the S - N method is the endurance
limit, which is used in “infinite-life”, or “safe stress” design
philosophy.
• In general, the S – N approach should not be used to estimate
lives below 1000 cycles (N < 1000).
94
Microstructure-Fatigue Relationships
 What are the important issues in microstructure-fatigue
relationships?
 Answer: three major factors.
1: geometry of the specimen (previous slide); anything on the
surface that is a site of stress concentration will promote
crack formation (shorten the time required for nucleation of
cracks).
2: defects in the material; anything inside the material that can
reduce the stress and/or strain required to nucleate a crack
(shorten the time required for nucleation of cracks).
3: dislocation slip characteristics; if dislocation glide is confined
to particular slip planes (called planar slip) then dislocations
can pile up at any grain boundary or phase boundary. The
head of the pile-up is a stress concentration which can
initiate a crack.
95
Microstructure affects Crack Nucleation
 The main effect of
microstructure (defects,
surface treatment, etc.) is
almost all in the low stress
intensity regime, i.e. Stage I.
Defects, for example, make
it easier to nucleate a crack,
which translates into a lower
threshold for crack
propagation (∆Kth).
 Microstructure also affects
fracture toughness and
therefore Stage III.
96
da/dN
I
II
∆Kc
III
∆Kth
∆K
Defects in Materials
 Descriptions of defects in materials at the sophomore level
focuses, appropriately on intrinsic defects (vacancies,
dislocations). For the materials engineer, however, defects
include extrinsic defects such as voids, inclusions, grain
boundary films, and other types of undesirable second phases.
 Voids are introduced either by gas evolution in solidification or
by incomplete sintering in powder consolidation.
 Inclusions are second phases entrained in a material during
solidification. In metals, inclusions are generally oxides from the
surface of the metal melt, or a slag.
 Grain boundary films are common in ceramics as glassy films
from impurities.
 In aluminum alloys, there is a hierachy of names for second
phase particles; inclusions are unwanted oxides (e.g. Al2O3);
dispersoids are intermetallic particles that, once precipitated, are
thermodynamically stable (e.g. AlFeSi compounds); precipitates
are intermetallic particles that can be dissolved or precipiated
depending on temperature (e.g. AlCu compounds).
97
Metallurgical Control: fine particles
 Tendency to localization of flow is deleterious to the initiation of
fatigue cracks, e.g. Al-7050 with non-shearable vs. shearable
precipitates (Stage I in a da/dN plot). Also Al-Cu-Mg with
shearable precipitates but non-shearable dispersions, vs. only
shearable ppts.
98
Coarse particle effect on fatigue
 Inclusions nucleate cracks cleanliness (w.r.t. coarse particles)
improves fatigue life, e.g.
 7475 improved by lower Fe+Si compared to 7075:
0.12Fe in 7475, compared to 0.5Fe in 7075;
0.1Si in 7475, compared to 0.4Si in 7075.
99
Alloy steel heat treatment
 Increasing hardness tends to raise the endurance limit for high
cycle fatigue. This is largely a function of the resistance to
fatigue crack formation (Stage I in a plot of da/dN).
Mobile solutes that pin
dislocations fatigue
limit, e.g. carbon in steel
100
Casting porosity affects fatigue
Gravity cast
versus
squeeze cast
versus
wrought
Al-7010
 Casting tends to result in porosity. Pores are effective sites
for nucleation of fatigue cracks. Castings thus tend to have
lower fatigue resistance (as measured by S-N curves) than
wrought materials.
 Casting technologies, such as squeeze casting, that reduce
porosity tend to eliminate this difference.
101
Titanium alloys

For many Ti alloys, the proportion of hcp (alpha) and bcc (beta) phases depends
strongly on the heat treatment. Cooling from the two-phase region results in a twophase structure

Rapid cooling from above the transformation in the single phase (beta) region
results in a two-phase microstructure with Widmanstätten laths of (martensitic)
alpha in a beta matrix.

The fatigue properties of the two-phase structure are significantly better than the
Widmanstätten structure (more resistance to fatigue crack formation).

The alloy in this example is IM834, Ti-5.5Al-4Sn-4Zr-0.3Mo-1Nb-0.35Si-0.6C.
102
Design Considerations
 If crack growth rates are normalized by the elastic
modulus, then material dependence is mostly
removed!
 Can distinguish between intrinsic fatigue for
combined elastic, plastic strain range for small crack
sizes and extrinsic fatigue and for crack growth rate
controlled at longer crack lengths.
 Inspection of design charts, shows that ceramics
sensitive to crack propagation (high endurance limit
in relation to fatigue threshold).
103
Design Considerations: 2
 Metals show a higher fatigue threshold in relation
to their endurance limit. PMMA and Mg are at
the lower end of the toughness range in their
class.
 Also interesting to compare fracture toughness
with fatigue threshold.
 Note that ceramics are almost on ratio=1 line,
whereas metals tend to lie well below, i.e. fatigue
is more significant criterion.
104
Fatigue property map
105
Fatigue property map
106
*Variable Stress/Strain Histories
 When the stress/strain history is stochastically
varying, a rule for combining portions of fatigue life is
needed.
 Palmgren-Miner Rule is useful: ni is the number of
cycles at each stress level, and Nfi is the failure point
for that stress.
ni
1

i N fi
107
*Fatigue in Polymers
 Many differences from metals
 Cyclic stress-strain behavior often exhibits
softening; also affected by visco-elastic effects;
crazing in the tensile portion produces
asymmetries.
 S-N curves exhibit three regions, with steeply
decreasing region II.
 Nearness to Tg results in strong temperature
sensitivity.
108
Fatigue: summary
 Critical to practical use of structural
materials.
 Fatigue affects most structural
components, even apparently statically
loaded ones.
 Well characterized empirically.
 Connection between dislocation
behavior and fatigue life offers exciting
research opportunities, i.e. physically
based models are lacking!
109
Improving Fatigue Performance
S = stress amplitude
1. Impose a
Compressive
Surface Stress (to
Suppress Surface
cracks from growing)
near
zero ortensile,
compressive, m
moderate
larger tensile,
m
m
N = Cycles to failure
• Method 1: shot peening
shot
put
surface
into
compression
2. Remove Stress
Concentrating sharp
corners
110
• Method 2: carburizing (interstitial)
C-rich gas
bad
better
bad
better
S-N Data for 2014-T6 Al
19.5 ksi
111
Procedure for Fatigue Damage Calculation
n1
1
Ni(i)m = a
Stress range, 
n2
2
3
n3
n4
'e
NT
D
e
n1 cycles applied at  1
N 1 cycles to failure at  1

N1
N2
n2 cycles applied at  2
N 2 cycles to failure at  2
N3 No
....  ...
cycles
ni cycles applied at  i
N i cycles to failure at  i
i
ni
n
n1
n2
D  Dn1  Dn 2  .....  Dni 

 ... 
 i
N1 N 2
Ni
1 Ni
LR 
112
1
1

D n1 N1  n2 N 2  .... ni N i
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2
ni - number of cycles of stress range i,
D1 
1
N1
- damage induced by one cycle of stress range 1,
n1
Dn1 
N1
D2 
1
N2
- damage induced by one cycle of stress range 2,
Dn 2 
1
Di 
Ni
n2
N2
- damage induced by n2 cycles of stress range 2,
- damage induced by one cycle of stress range i,
Dni 
113
- damage induced by n1 cycles of stress range 1,
ni
Ni
- damage induced by ni cycles of stress range i,
Total Damage Induced by the Stress History
D
n cycles applied at  i
n1 cycles applied at  1
n cycles applied at  2
 2
....  ... i
N 1 cycles to failure at  1 N 2 cycles to failure at  2
N i cycles to failure at  i
i
ni
n
n1
n2
D  Dn1  Dn 2  .....  Dni 

 ... 
 i
N1 N 2
Ni
1 Ni
It is usually assumed that fatigue failure occurs when the cumulative damage
exceeds some critical value such as D =1,
i.e.
if
D>1
- fatigue failure occurs!
For D < 1 we can determine the remaining fatigue life:
LR 
1
1

D n1 N 1  n2 N 2  .... ni N i
N  LR  n1  n2  n3  .....  ni 
114
LR - number of repetitions of
the stress history to failure
N - total number of cycles to failure

N j = a  j

m
if
j > e .
It is assumed that stress cycles lower than the fatigue limit, j < e, produce no damage (Nj=) in
the case of constant amplitude loading however in the case of variable amplitude loading the
extension of the S-N curve with the slope ‘m+2” is recommended. The total damage produced by
the entire stress spectrum is equal to:
j
D   Dj
j 1
It is assumed that the component fails if the damage is equal to or exceeds unity, i.e. when D  1.
This may happen after a certain number of repetitions, BL (blocks), of the stress spectrum, which
can be calculated as:
BL = 1/D.
Hence, the fatigue life of a component in cycles can be calculated as:
N = BLNT,
where, NT is the spectrum volume or the number of cycles extracted from given stress history.
NT = (NOP - 1)/2
If the record time of the stress history or the stress spectrum is equal to Tr, the fatigue life can be
expressed in working hours as:
T = BL Tr.
115
Main Steps in the S-N Fatigue Life Estimation Procedure
 Analysis of external forces acting on the structure and the component
in question,
 Analysis of internal loads in chosen cross section of a component,
 Selection of individual notched component in the structure,
 Selection (from ready made family of S-N curves) or construction of SN curve adequate for given notched element (corrected for all effects),
 Identification of the stress parameter used for the determination of the
S-N curve (nominal/reference stress),
 Determination of analogous stress parameter for the actual element in
the structure, as described above,
 Identification of appropriate stress history,
 Extraction of stress cycles (rainflow counting) from the stress history,
 Calculation of fatigue damage,
 Fatigue damage summation (Miner- Palmgren hypothesis),
 Determination of fatigue life in terms of number of stress history
repetitions, Nblck, (No. of blocks) or the number of cycles to failure, N.
 The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or
structure.
116
Example #2
An unnotched machine component undergoes a variable amplitude
stress history Si given below. The component is made from a steel
with the ultimate strength Suts=150 ksi, the endurance limit
Se=60 ksi and the fully reversed stress amplitude at N1000=1000
cycles given as S1000=110 ksi.
Determine the expected fatigue life of the component.
Data: Kt=1, SY=100 ksi Suts=150 ksi, Se=60 ksi, S1000=110 ksi
The stress history:
Si = 0, 20, -10, 50, 10, 60, 30, 100, -70, -20, -60, -40, -80, 70, -30,
20, -10, 90, -40, 10, -30, -10, -70, -40, -90, 80, -20, 10, -20, 10, 0
Stress History
117
Si = 0, 20, -10, 50, 10, 60, 30, 100, -70, -20, -60, -40, -80, 70, -30,
20, -10, 90, -40, 10, -30, -10, -70, -40, -90, 80, -20, 10, -20, 10, 0
Stress History
100
80
60
Stress (ksi)
40
20
0
-20 0
5
10
15
20
-40
-60
-80
-100
Reversal point No.
118
25
30
35
N S a   C
m
log N  m log S a  log C
log1000  m log110  log C

6
log
10
 m log 60  log C

3  m log110  log C

6  m log 60  log C
C  1.886  1026 m  11.4
119
S-N Curve
Stress amplitude (ksi)
1000
100
10
1000
10000
100000
No. of cycles
120
1000000
10000000
Goodman Diagram
SY
100
Stress amplitude (ksi)
90
80
70
60
50
40
30
20
10
Suts
0
-100
-50
0
50
100
150
Mean stress (ksi)
Sa Sm

1
Se Suts
1
for fatigue endurance
Sa
S
 m  1 for any stress amplitude
S a ,r Suts
121
S a ,r  at Sm 0

S 
 1  m  S a
 Suts 
S a ,r  at Sm 0 
Sa
S
1 m
Suts
Calculations of Fatigue Damage
a) Cycle No.11
Sa,r=87.93 ksi
N11= C( Sa,N)-m= 1.866102687.93-11.4=12805 cycles
D11=0.000078093
b) Cycle No. 14
Sa,r=75.0 ksi
N14= C( Sa,N)-m= 1.866102675.0-11.4=78561 cycles
D14=0.000012729
c) Cycle no. 15
Sa,r=98.28 ksi
N14= C( Sa,N)-m= 1.866102698.28-11.4=3606 cycles
D14=0.00027732
122
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
n0=15
S
30
40
30
20
50
30
100
20
30
30
170
30
30
140
190
Results of "rainflow" counting
Sm
Sa
Sa,r (Sm=0)
5
15
15.52
30
20
25.00
45
15
21.43
-50
10
10.00
-45
25
25.00
5
15
15.52
20
50
57.69
-20
10
10.00
-25
15
15.00
-55
15
15.00
5
85
87.93
-5
15
15.00
-5
15
15.00
10
70
75.00
5
95
98.28
D=
D=0.000370
LR = 1/D =2712.03
N=n0*LR=15*2712.03=40680
123
Damage
Di=1/Ni=1/C*Sa-m
2.0155E-13
0
4.6303E-11
0
7.9875E-12
0
1.3461E-15
0
4.6303E-11
0
2.0155E-13
0
6.3949E-07
0
1.3461E-15
0
1.3694E-13
0
1.3694E-13
0
7.8039E-05 7.80E-05
1.3694E-13
0
1.3694E-13
0
1.2729E-05 1.27E-05
0.00027732 0.000277
0.00036873 3.677E-04
D=3.677E-04
L R = 1/D =2719.61
N=n0*LR=15*2719.61=40794