416 CHAPTER 14 Fatigue
atpointsclosetotheplaneofthecrack,
K=
Pα
(πa)1
2(14.31)
inwhichPistheload/unitthickness.Equations(14.30)and(14.31)mayberewrittenas
K=K 0 α (14.32)whereK 0 isareferencevalueofthestressintensityfactorwhichdependsontheloading.Forthesimple
caseofaremotelyloadedplateintension,
K 0 =S(πa)(^12)
(14.33)
andEqs.(14.32)and(14.30)areidenticalsothatforagivenratioofcracklengthtoplatewidthαis
thesameinbothformulations.Inmorecomplexcases,forexample,thein-planebendingofaplateof
width2bandhavingacentralcrackoflength2a,
K 0 =
3 Ma
4 b^3(πa)1(^2) (14.34)
inwhichMisthebendingmomentperunitthickness.ComparingEqs.(14.34)and(14.30),weseethat
S= 3 Ma/ 4 b^3 , which is the value of direct stress given by basic bending theory at a point a distance
±a/2fromthecentralaxis.However,ifSwasspecifiedasthebendingstressintheouterfibersofthe
plate—at±b—thenS= 3 M/2b^2 ;clearlythedifferentspecificationsofSrequiredifferentvaluesofα.
Ontheotherhand,thefinalvalueofKmustbeindependentoftheformofpresentationused.Useof
Eqs. (14.30) through (14.32) depends on the form of the solution forK 0 , and care must be taken to
ensurethattheformulausedandthewayinwhichthenominalstressisdefinedarecompatiblewith
thoseusedinthederivationofα.
ThereareanumberofmethodsavailablefordeterminingthevalueofKandα.Inonemethod,the
solutionforacomponentsubjectedtomorethanonetypeofloadingisobtainedfromavailablestandard
solutionsusingsuperposition,or,ifthegeometryisnotcovered,twoormorestandardsolutionsmay
becompounded[Ref.1].Alternatively,afiniteelementanalysismaybeused.
ThecoefficientαinEq.(14.30)has,aswehavenoted,differentvaluesdependingontheplateand
crackgeometries.Listedbelowarevaluesofαforsomeofthemorecommoncases.
(i) Asemi-infiniteplatehavinganedgecrackoflengtha;α=1.12.
(ii) Aninfiniteplatehavinganembeddedcircularcrackorasemicircularsurfacecrack,eachofradius
a,lyinginaplanenormaltotheappliedstress;α=0.64.
(iii) Aninfiniteplatehavinganembeddedellipticalcrackofaxes2aand2borasemiellipticalcrack
ofwidth2binwhichthedepthaislessthanhalftheplatethicknesseachlyinginaplanenormal
totheappliedstress;α=1.12inwhichvarieswiththeratioa/basfollows:
a/b 0 0.2 0.4 0.6 0.8
1.0 1.05 1.15 1.28 1.42
Fora/b=1,thesituationisidenticaltocase(ii).