Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

14.5 Crack Propagation 419

inwhicha 1 =a+rp,where,forplanestress

rp=

1

8 π

(

K

fy

) 2

(see[Ref.7])

ThecurvesrepresentedbyEq.(14.38)maybedividedintothreeregions.Thefirstcorrespondstoavery
slowcrackgrowthrate(< 10 −^8 m/cycle)wherethecurvesapproachathresholdvalueofstressintensity
factor Kthcorrespondingto4× 10 −^11 m/cycle—inotherwords,nocrackgrowth.Inthesecondregion
(10−^8 − 10 −^6 m/cycle), much of the crack life takes place and, for small ranges of K, Eq. (14.38)
mayberepresentedby

da

dN

=C(K)n (see[Ref.8]) (14.41)

in whichCandndepend on the material properties; over small ranges of da/dNand K,Candn
remain approximately constant. The third region corresponds to crack growth rates> 10 −^6 m/cycle,
whereinstabilityandfinalfailureoccur.
Anattempthasbeenmadetodescribethecompletesetofcurvesbytherelationship

da

dN

=

C(K)n

( 1 −R)Kc− K

(see[Ref.9]) (14.42)

in whichKcis the fracture toughness of the material obtained from toughness tests. Integration of
Eqs. (14.41) or (14.42) analytically or graphically gives an estimate of the crack growth life of the
structure,thatis,thenumberofcyclesrequiredforacracktogrowfromaninitialsizetoanunaccept-
ablelength,orthecrackgrowthrateorfailure,whicheveristhedesigncriterion.Thus,forexample,
integrationofEq.(14.41)gives,foraninfinitewidthplateforwhichα=1.0,

[N]NNfi=

1

C[(Smax−Smin)π

1

(^2) ]n

[

a(^1 −n/^2 )

1 −n/ 2

]af

ai

(14.43)

forn>2.Ananalyticalintegrationmayonlybecarriedoutifnisanintegerandαisintheformofa
polynomial;otherwisegraphicalornumericaltechniquesmustbeused.
SubstitutingthelimitsinEq.(14.43)andtakingNi=0,thenumberofcyclestofailureisgivenby

Nf=

2

C(n− 2 )[(Smax−Sm)π^1 /^2 ]n

[

1

a

(n− 2 )/ 2

i

−

1

a

(n− 2 )/ 2

f

]

(14.44)

Example 14.1
Aninfiniteplatecontainsacrackhavinganinitiallengthof0.2mmandissubjectedtoacyclicrepeated
stressrangeof175N/mm^2 .Ifthefracturetoughnessoftheplateis1708N/mm^3 /^2 andtherateofcrack
growthis40× 10 −^15 ( K)^4 mm/cycle,determinethenumberofcyclestofailure.