1. Fatigue crack growth relies on sufficient energy absorption at the crack tip to drive the growth process. For crack growth to occur, the stress intensity factor range (ΔK) must exceed a minimum threshold (ΔKmin). 2. If ΔK is equal to ΔKmin, the smallest possible non-zero crack growth per cycle (da/dN) would correspond to the advancement of the crack tip by one atomic spacing. 3. The atomic spacing can be estimated as twice the atomic radius, which is on the order of 100-250 picometers for most metals. This provides a theoretical basis for deriving ΔKmin values from fundamental material properties.

1. The Lower Bound Threshold for Fatigue Crack Growth
“A processby which a crackcan formand then grow underfluctuating loading,in which the resulting
failure can be sudden and catastrophic”
Fatigue damage accumulationisone of several degradationmechanismsconsideredinthe designof
nuclearpowerplantcomponents.Ingeneral,the severityof fatigue damage isevaluatedby
computinga cumulative usage factor(CUF), derivedfromalineardamage rule (Miner’sRule); witha
designfatigue curve,derivedfromthe resultsof uniaxial,strain-controlledfatigue tests.Suchtests
are usuallyperformedundercontrolledlaboratoryconditions(roomtemperature,air,low-humidity)
on smooth,un-notched specimens.The general ideaisthatfatigue testdataderivedfromspecimen
testsremainvalidwhenusedforengineeringsizedcomponentssubjectedtoplantrepresentative
pressure-temperature transients. Fatigue damage assessmentmethodshave beencodifiedin
several designstandards(ASMEIII/VIII,RCC-M,EN13445 toname a few),withrelativelylittle
divergence betweenthe general proceduresapplied.Code prescribedmethodshave aproven
historyof successful implementation,withsufficientconservativism(albeittoomuchinsome cases)
to ensure componentstructural integrityoverincreasinglylongeroperatinglifetimes [1].Inthe
contextof the civil nuclearpowergeneration,operatinglicenserenewal islikelytoextendthe
operatinglifetimeof existingplantsto60 years,providedthe necessarychecksagainst
environmentallyassisteddegradationare satisfied [2].Withincreasingcomponentoperatinglives,
furtherjustificationof protectionagainstfatigue failureiswarranted;asfatigue damage assessment
isheavilydatadependent,thiswill undoubtedlyrequire agreaternumberof fatigue teststobe
carriedout infuture,includingplantrepresentative componenttesting;simulationof severe plant-
representative transients–non-isothermal,out-of-phase,thermal-mechanical loadingeventswith
significantstraingradienteffects –will alsoneedtobe investigatedfurther .
Now,itis clearthat the increasingdemandforenergycombinedwithadrive towardsgreaterenergy
efficiencywillinevitablyforce industriestoadopta more sustainable long-life approachto
componentdesignwithinreason. Withthisobjective,itbaresrememberingwhythisisactually
possible inthe firstplace!Asengineers,more oftenthannot, amacroscopicapproach isall that is
necessarytomodel a givenproblem.Withthisapproachinmind,we treatengineeringmaterialsas
an isotropic,homogenouscontinuum,andfocusourattentiononthe mechanical propertiesof the
material,andthe prescribedloadingconditions.Inthisframework,fatiguedamage canonlyoccur
underthree conditions,all of whichmustbe satisfiedconcurrently.
1. Tensile stressesmustexist.
2. Repeatedloading
3. Plasticity
The 3rd
requirementisnottrivial andwarrantsfurtherconsideration.Fatigue damage relieson
energydeposition (irreversible absorption) whichcanonlyoccur if plasticityispresent;inother
words,there mustbe a developedhysteresiscycle.We oftenonlyconsidersuchconditionsto prevail
at the tipsof visible notchesorflaws,butthe real flaw (pardonthe pun) liesinonlyconsideringthe
macroscopicpropertiesof the material.Inreality,microscopicflawswillexhibitplasticity,evenin
caseswhere the macroscopicbehaviourof the material remainselastic.Iam no metallurgist,butI
wouldhazardan informedguessthatthisphenomenonissimplyascaled-downmacroscopic
analogue;thatis,damage occurs due to plasticenergydepositionatthe tipsof microscopicflaws.

2. Figure 1 illustratesthe crackgrowthprocess;sufficientenergyabsorptionisrequiredtotranslate the
crack-tipplasticzone to a newposition.Because plasticityisirreversible, regionswithsignificant
plasticstrainwill be leftbehindthe crack-tipfieldascrack growthprogresses.
Well,thisbegsthe question,if plasticityreallyisprevalentonamicroscopicscale,thenwhy doesthis
not alwaystranslate tomore damage on a macroscopicscale whenthe three above conditionsare
satisfied?Lookingbackat the openingdefinition,we know fatiguefailure canonlyoccurwhena
crack growsto a critical length,leadingtosuddenfailure byfracture.Butformicroscopicplasticityto
exist,amicroscopiccrack mustalreadybe present.Therefore,itiscrack growthwe are concerned
withhere. Since crackgrowth ultimatelyreliesonasupplyof energytothe crack tipregion(the zone
of cyclicplasticity) todrive the fatigue damage mechanism, the processcanonlyprogressif each
cycle of loadprovidesincreasinglymore energytothe region.Thus,anenergythreshold mustbe
overcome forcrack growthto occur [3].
It isuseful toconsiderhowmuchplasticenergyisdepositedpercycle of load.The fatigue crack
growthper cycle can be relatedtothe stressintensityfactorrange, ∆𝐾,underthe assumptionof
LinearElasticFracture Mechanics(LEFM),or the J-Integral range, ∆𝐽,forPost-YieldFracture
Mechanics (PYFM), where plasticityisgenerallynotconfinedtothe crack-tipregion.This empirical
relationship,knownasThe ParisLaw,isexpressedasfollows [4].
𝑑𝑎
𝑑𝑁
∝ ∆𝐾 𝑚 ⇛
𝑑𝑎
𝑑𝑁
= 𝐶∆𝐾 𝑚 (1)
If we considerthe assumptionof LEFMconditions –where the plasticzone issmall incomparisonto
the total section where LEFMfieldsprevail –we can confine ourattentiontothe crack-tipresponse
underloading. Considerthe followingscenarioassumingaconstantloadamplitude: Onfirstload,
the crack-tipplasticzone exceedsthe materialyieldstrength,but doesnotexceedtwice yieldwhen
calculatedonan elasticbasis.Hence,whenloadisremoved,the crack-tipplasticzone will unload
elastically andplasticitywill be containeddue tothe “elasticcore”behaviourof the section.
Subsequentloadcycleswill thuscycle elasticallybetweenamaximumof +𝑦𝑖𝑒𝑙𝑑 (𝑡𝑒𝑛𝑠𝑖𝑙𝑒)and
−𝑦𝑖𝑒𝑙𝑑 (𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑣𝑒), or throughthe range twice-yield;thisprocessiscalled Shakedown. Because
the crack-tipzone cycleselasticallyin thisinstance,itcannotbe responsible forabsorbingenergyto
drive fatigue crackgrowth.Onlya regionof cyclicplasticity –where the alternatingstressintensity
range exceedstwice-yield–can be responsibleforfatigue damage. Therefore,if we consideracrack
loadedinMode I, and assumingthe limitingcase of ∆𝜎 = 2𝜎 𝑦,we can derive anexpressionforthe
size of the plasticzone [5].
2𝜎 𝑦 =
∆𝐾
√2𝜋𝑟
⇛ 𝑟 =
∆𝐾2
8𝜎 𝑦
2 𝜋
(2)
θ
PlasticZone ∆𝒂
Crack Growth
𝑟
Figure 1. Illustration of Crack Growth Under Cyclic Loading

3. Definingthe strainatyieldas 𝜀 𝑦, andassumingonlymechanical loading, the strainenergycanbe
expressedasfollows
𝑊 =
1
2
𝐶𝑖𝑗𝑘𝑙 𝜀𝑖, 𝑗 𝜀 𝑘,𝑙 (3)
Where 𝐶 is the fourth-orderelasticitytensor,the componentsof whichrepresentthe elasticmoduli
of the material,and 𝜀 isthe infinitesimal straintensor.The stress-strainrelation,ignoringthe effects
of temperature,canbe expressedbydifferentiatingeqn.(3) withrespecttothe infinitesimalstrain
𝜎 =
𝑑𝑊
𝑑𝜀
( 𝜀) = 𝐶𝜀 𝑜𝑟 𝜎𝑖𝑗 = 𝐶𝑖𝑗𝑘𝑙 𝜀 𝑘𝑙 𝑖𝑛 𝑖𝑛𝑑𝑖𝑐𝑖𝑎𝑙 𝑛𝑜𝑡𝑎𝑡𝑖𝑜𝑛 (4)
The energydeposited perunitvolume istherefore 2𝜎𝜀,or2𝜎 𝑦 𝜀 𝑦 inour example.
The plasticzone infront of the crack-tipis usuallyelliptical.Forsimplicity,we canassume acircular
plasticzone (like thatinFigure 1) withitscentre coincident withthe tipof the crack. Thus,the
volume of concernis 𝜋𝑟2 𝑡, where 𝑡 representsthe lengthalongthe crackfront.We can now further
expandeqn.(2) byconsideringthe energydepositedperunitvolume andrearrangingfor 𝑟.
2𝜎 𝑦 𝜀 𝑦 =
𝑊
𝜋𝑟2 𝑡
⇛ 𝑟 = √
𝑊
2𝜎 𝑦 𝜀 𝑦 𝜋𝑡
(5)
Substitute eqn.(5) intoeqn.(2) andon rearrangingwe obtainanexpressionforthe energy
depositedforeachcycle of load
√
𝑊
2𝜎 𝑦 𝜀 𝑦 𝜋𝑡
=
∆𝐾2
8𝜎 𝑦
2 𝜋
⇛ 𝑊 =
𝜀 𝑦 𝑡∆𝐾4
32𝜋𝜎 𝑦
3
(6)
Consideringeqn.(6) representsthe thresholdenergypercycle,we cansimplifythisexample further
by assumingthere existsafixedamountof strainenergy providedbythe externalloads,
correspondingtoa unitincrease incrack area – i.e.the derivativeof energywithrespecttocrack
area.We introduce the energyreleaserate, 𝐺,andfurthersimplifyeqn.(6).
𝐺 =
𝑑𝑊
𝑑𝑎
⇛
𝑑𝑎
𝑑𝑁
=
𝑑𝑊
𝐺
=
𝜀 𝑦∆𝐾4
32𝜋𝜎 𝑦
3 𝐺
(7)
The resultisthe ParisLaw with 𝐶 =
𝜀 𝑦
32𝜋𝜎 𝑦
3 𝐺
and 𝑚 = 4 whichconformswith empirical dataformost
structural steels. Obviously,we have made some fairlyonerousassumptionshere,butanimportant
observationabouteqn.(7) deservesattention:If there existsaminimum, non-zero(andfixed)
amountof energyrequiredtodrive crackgrowth,thenthismustalsocorrespondtoa minimum,
non-zero,stressintensityfactorrange, ∆𝐾 𝑚𝑖𝑛; when∆𝐾 < ∆𝐾 𝑚𝑖𝑛,there isinsufficientenergy
depositiontodrive the crackgrowthmechanism.
𝐹𝑜𝑟 ∆𝐾 < ∆𝐾 𝑚𝑖𝑛,
𝑑𝑎
𝑑𝑁
= 0
Now,forthe interestingpart!Whathappensif ∆𝐾 isexactlyequal to ∆𝐾 𝑚𝑖𝑛?
𝐹𝑜𝑟 ∆𝐾 = ∆𝐾 𝑚𝑖𝑛,
𝑑𝑎
𝑑𝑁
= ?

4. We knowunderthese conditionsthat
𝑑𝑎
𝑑𝑁
≠ 0, sowhat isthe smallest,non-zerovalue that 𝑑𝑎 can
take?A somewhatmeaningfulanswerpresentsitselfwhenwe considerthe behaviourof
engineeringmaterialsinthe quantumrealm.
For polycrystallinematerials(mostmetalsandceramics),atomsare arrangedina repeatingor
periodicarrayoverlarge atomic distances.Differentatomsandionscan be modelledassolid
spheresof fixedsize,arrangedinthe smallestpossible repeatinggroups;these groupsare knownas
Unit Cells andare illustratedin Figure 2. The grosscrystal structure of any polycrystalline material is
therefore dependentonthe configurationof atomswithinthe unitcell [6].
Figure 2. Unit Cell within a Polycrystalline Lattice Structure
Four simple crystallinestructuresaccountformostmetallicmaterials:
1. Simple Cubic(SCC)
2. Body-CentredCubic(BCC)
3. Face-CentredCubic(FCC)
4. Hexagonal Close-Packed(HCP)
Withoutdelvinginto muchdepth,the commonfeature of anyconfiguration isthatthe unitcell
correspondstothe lowestpossiblepotential energystate achievable foragivencrystal structure.If
atomsare to form a stable metallicstructure,the meanenergyof the free electronswithinthe
lattice mustbe lessthanthe energyof the electron energy level thatwouldprevailif the atom
existedfreely.The equilibriumseparationdistance of the atomsof atomswithinthe lattice isthat
for whichanyfurtherclosure wouldresultinanincrease inthe repulsive force whichexceedsthe
attendantdecrease inmeanelectronenergy [7].Thisphenomenonisdepicted inFigure 3.
UnitCell
 The smallestrepeatingarrangementof
atoms
 Usuallyprismsorparallelepipeds
 Three setsof parallel faces

5. Figure 3. Variation in Atomic Energy Levels with Inter-Atomic Closure
The inter-atomicdistance betweenatomsisdeterminedbythe inter-atomicbondlength. Asarough
approximation,the bondlengthbetweentwodifferentatomsisequal tothe sumof the individual
covalentradii,andcan be measuredusingelectrondiffraction.
Table 1 belowshowssome importantcharacteristicsof well-knownelements.
Element Symbol
Atomic
Weight
(amu)
Density
(g/cm^3)
Crystal
Structure
Atomic
Radius (pm)
Aluminium Al 26.98 2.71 FCC 143
Argon Ar 39.95
Barium Ba 137.33 3.5 BCC 217
Beryllium Be 9.012 1.85 HCP 114
Boron B 10.81 2.34 Rhombohedral
Bromine Br 79.9
Cadmium Cd 112.41 8.65 HCP 149
Calcium Ca 40.08 1.55 FCC 197
Carbon C 12.011 2.25 Hexagonal 71
Caesium Cs 132.91 1.87 BCC 265
Chlorine Cl 35.45
Chromium Cr 52 7.19 BCC 125
Cobalt Co 58.93 8.9 HCP 125
Copper Cu 63.55 8.94 FCC 128
Fluorine F 19
Gallium Ga 69.72 5.9 Orthorhombic 122
Germanium Ge 72.59 5.32
Diamond
Cubic 122
Gold Au 196.97 19.32 FCC 144
Helium He 4.003
Hydrogen H 1.008
Table 1. Selected Element Properties

6. At firstglance,we can see thatthe atomic radii donot differdrasticallybetweendifferent metals;
mostatomic radii lie inthe range of 100 − 250𝑝𝑚, withthe inter-atomicspacingcorrespondingto
twice the atomicradius. Thisanswersourquestion,atleast ina theoretical sense –the smallest
possible non-zerocrack lengthincrement, d𝒂, correspondsto one atomic spacing. Fromthis
observation,valuesof ∆𝐾 𝑚𝑖𝑛 have beenderivedformany structural materials.Of course,the
accuracy of these valuesispredicatedonthe assumptionthatthe ParisLaw still holds atthe lower
boundcrack growth limit of one atomicspacing.
Finally,asanaside,itisworth consideringthe engineeringconsequences thatwould arise –inthe
contextof very-highcycle fatigue(VHCF)–if the thresholdSIFrange, ∆𝐾 𝑚𝑖𝑛,didnotexist.VHCF
range correspondsroughlytoover 108 cyclesto failure.Componentsoperatinginthisregime
include internal combustionengine cylinderheads,gasturbine disks,andball bearings [8].
Conventionalfatiguetestingisusuallylimitedto100Hz, correspondingtoaround 107cyclesperday,
or 3 × 109 cyclesper year.Thisof course isimpractical andcostlydue to time constraints,hence
mostVHCF testingisdone using ultrasonicfatiguetestingrangesof 15 to 30 kHz, corresponding
roughlyto107 cyclesperminute or1.6 × 109 cyclesperday!
Let usconsiderboththe conventional andultrasonictestingscenarios,underthe assumptionof no
fatigue crack growththreshold,withthe smallestpossible crackgrowthincrementof asingle atomic
displacement.Forsimplicity,we willassume anaverage inter-atomicspacingof 200𝑝𝑚 (or
2 𝐴𝑛𝑔𝑠𝑡𝑟𝑜𝑚𝑠, 𝐴̇).Afterone day, the resultingcrackgrowthisas follows:
𝑎 𝑐𝑜𝑛𝑣𝑒𝑛𝑡𝑖𝑜𝑛𝑎𝑙 = 107 × 2𝑒−10 = 2 𝑚𝑚
𝑎 𝑢𝑙𝑡 𝑟𝑎𝑠𝑜𝑛𝑖𝑐 = 1.6𝑒9 × 2𝑒−10 = 320 𝑚𝑚
“Failure”– or the terminationpoint –of most fatigue testsisusuallyestablishedusingthe 25% load-
drop criterion;thiscorrespondstothe initiationandpropagationof afatigue crackto “engineering
size”,orroughly 3 𝑚𝑚. The worryingconclusionisclearfromthese results:
In absenceof a fatiguethreshold belowwhich zero crackgrowth occurs,mostengineering structures
would incur fatiguefailurewithin a matter of weeks at most.
Alas, we can be thankful thatrealityisat leasta little more forgiving!

7. References
[1] ASME, “ASME III,Division1,SubsectionNB,”in ASMEBoiler & PressureVesselCode,2015.
[2] NuclearRegulatoryCommision,“RegulatoryGuide1.207: GUIDELINES FOR EVALUATING
FATIGUE ANALYSESINCORPORATINGTHELIFE REDUCTION OF METAL COMPONENTSDUE TO
THE EFFECTS OFTHE LIGHT-WATER REACTOR ENVIRONMENTFORNEW REACTORS,”no.
March, pp.1–7, 2007.
[3] R. Bradford,“The Effectsof PlasticIrreversibilityonLow Temperature Fracture,”2015.
[4] P. C.Paris,M. P.Gomez,and W. E. Anderson,“A Rational AnalyticTheoryof Fatigue,” Trend
Eng.,no.13, pp. 9–14, 1961.
[5] R. J. Roark,W. C. Young,and R. Plunkett, Formulas forStressand Strain,vol.43,no.3. 1976.
[6] J. Datsko, MaterialsSelection forDesign and Manufacturing,3rded.AnnArbor,Michigan:
MichiganUniversityPress,1997.
[7] R. E. SmallmanandR. J. BiShop,“ModernPhysical MetallurgyandMaterialsEngineering,”
Mod.Phys.Metall.Mater.Eng.,pp.320–350, 1999.
[8] V.Kazymyrovych, Very high cycle fatigueof engineering materials - A literature review.2009.