sa := Alternating stress sm := Mean stress R := Stress ratio

1

• sa Alternating stresssm Mean stressR
  Stress ratioe strainNf number of
  cycles to failureA Amplitude ratio??pl
  Plastic strain amplitude??el Elastic
  strain amplitudeK Proportionality constant,
  cyclic stress-strainn Exponent in cyclic
  stress-strainc Exponent in Coffin-Manson
  Eq. also, crack lengthE Youngs modulusb
  exponent in Basquin Eq.m exponent in
  Paris LawK Stress intensity
• ?K Stress intensity amplitude
• a crack length

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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.

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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 limitltstatic stress
  capability.

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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.

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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 lt 200Hz, metals are insensitive
  to frequency fatigue life in polymers is
  frequency dependent.

Hertzberg
6
Fatigue testing, S-N curve
smean 3 gt smean 2 gt smean 1
The greater the number ofcycles in the loading
history,the smaller the stress thatthe material
can withstandwithout failure.
sa
smean 1
smean 2
smean 3
log Nf
Note the presence of afatigue limit in
manysteels and its absencein aluminum alloys.
Dieter
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Endurance Limits

• Some materials exhibit endurance limits, i.e. a
  stress below which the life is infinite fig.
  12.8
• 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.

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Fatigue fracture surface
Hertzberg
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Fatigue crack stages
Stage 1
Dieter
Stage 2
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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 Courtney
  fig.12.3a- stage II growth normal to tensile
  load (90)weak influence of microstructure
  Courtney fig.12.3b.
• Crack propagation ??catastrophic, or ductile
  failure at crack length dependent on boundary
  conditions, fracture toughness.

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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.

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Dislocation Slip Crack Nucleation

• Dislocation slip -gt tendency to localize slip in
  bands.
• Persistent Slip Bands (PSBs) characteristic of
  cyclic strains.
• Slip Bands -gt extrusion at free surface.
• Extrusions -gt intrusions and crack nucleation.

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Slip steps and the stress-strain loop
14
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. Nf107.
• Crack Growth Rate characterization -gt
• 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.

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Definitions Stress Ratios

• Alternating Stress
• Mean stress ? ?m (?max ?min)/2.
• Pure sine wave ??Mean stress0.
• Stress ratio ? R ?max/?min.
• For ?m 0, R-1
• Amplitude ratio ? A (1-R)/(1R).
• Statistical approach shows significant
  distribution in Nf for given stress.

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Alternating Stress Diagrams
Dieter
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Mean Stress

• Alternating stress ? ?a (?max-?min)/2.
• Raising the mean stress (?m) decreases Nf. see
  slide 19, also Courtney fig. 12.9
• 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).
  Courtney, fig. 12.10, problem 12.3

endurance limit at zero mean stress
sa
tensile strength
smean
18
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.

19
Cyclic stress-strain curve
Courtney
Large number of cycles typically needed to
reach asymptotic hysteresis loop (100).
Softening or hardening possible.
20
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.
  Courtney fig. 12.27
• Cyclic stress-strain curves tend to lie below
  those for monotonic tensile tests.
• Polymers tend to soften in cyclic straining.

Courtney
21
Cyclic Strain Control

• Strain is a more logical independent variable for
  characterization of fatigue.
• Define an elastic strain range as ?eel ?s/E.
• Define a plastic strain range, ?epl.
• 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.

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Strain control of fatigue
Courtney
23
Cyclic Strain control low cycle

• Constitutive relation for cyclic stress-strain
• n 0.1-0.2
• Fatigue life Coffin Manson relation
• ?f true fracture strain close to tensile
  ductility
• c -0.5 to -0.7
• c -1/(15n) large n ? longer life.

24
Cyclic Strain control high cycle

• For elastic-dominated strains at high cycles,
  adapt Basquins equation
• Intercept on strain axis of extrapolated elastic
  line sf/E.
• High cycle elastic strain control slope (in
  elastic regime) b -n/(15n)
• The high cycle fatigue strength, sf, scales with
  the yield stress ? high strength good in
  high-cycle

25
Strain amplitude - cycles
Courtney
26
Total strain (plasticelastic) life

• Low cycle plastic control slope c
• Add the elastic and plastic strains.
• 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.

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Fatigue Crack Propagation

• Crack Length a.Number of cycles NCrack
  Growth Rate da/dNAmplitude of Stress
  Intensity ?K ?svc.
• 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
  Rlt0) in Stage II fig. 12.16, 12.18b
• Environmental effects can be dramatic, e.g. H in
  Fe, in increasing crack growth rates.

46
Fatigue Crack Propagation
da/dN

• 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.

I
?Kc
II
III
?K
?Kth
47
Paris Law

• Paris Law
• 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 Rgt0).
  fig. 12.18a

48
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. Eqs.
  12.6-12.8

49
Striations, contd.

• Provided that mgt2 and a is constant, can
  integrate.
• If the initial crack length is much less than the
  final length, c0ltcf, then approximate thus
• Can use this to predict fatigue life based on
  known crack

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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.

65
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

66
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.

67
Microstructure affects Crack Nucleation
da/dN

• 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.

I
?Kc
II
III
?K
?Kth
68
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).

69
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
  dispersoids, vs. only shearable ppts.

graph courtesy of J. Staley, Alcoa
70
Coarse particle effect on fatigue

• Inclusions nucleate cracks ??cleanliness (w.r.t.
  coarse particles) improves fatigue life, e.g.
  7475 improved by lower FeSi compared to 7075
  0.12Fe in 7475, compared to 0.5Fe in 7075
  0.1Si in 7475, compared to 0.4Si in 7075.

graph courtesy of J. Staley, Alcoa
71
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
Dieter
72
Casting porosity affects fatigue
Gravity cast versussqueeze castversuswroughtA
l-7010
Polmear

• 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.

73
Titanium alloys
Polmear

• 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 two-phase structure, as
  Polmears example, 6.7a. Rapid cooling from
  above the transus in the single phase (beta)
  region results in a two-phase microstructure with
  Widmanstätten laths of (martensitic) alpha in a
  beta matrix, 6.7b.
• 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.

74
Design Considerations

• If crack growth rates are normalized by the
  elastic modulus, then material dependence is
  mostly removed! Courtney fig. 12.20
• Can distinguish between intrinsic fatigue use
  Eq. 12.4 for combined elastic, plastic strain
  range for small crack sizes and extrinsic
  fatigue use Eq. 12.6 for crack growth rate
  controlled at longer crack lengths. fig.
  12.21.
• Inspection of design charts, fig. 12.22, shows
  that ceramics sensitive to crack propagation
  (high endurance limit in relation to fatigue
  threshold).

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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. Courtney fig. 12.22
• Also interesting to compare fracture toughness
  with fatigue threshold. Courtney fig. 12.23
• Note that ceramics are almost on ratio1 line,
  whereas metals tend to lie well below, i.e.
  fatigue is more significant criterion.

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Fatigue property map
Courtney
77
Fatigue property map
Courtney
78
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. Ex. Problem
  12.2

Courtneys Eq. 12.9 is confusing he has Nf in
the numerator also
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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, figs. 12.34, 12.25.
• S-N curves exhibit three regions, with steeply
  decreasing region II, fig. 12.31.
• Nearness to Tg results in strong temperature
  sensitivity, fig. 12.42

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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!