Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

14.5 Crack Propagation 417

(iv)Aplateoffinitewidthwhavingacentralcrackoflength2a,wherea≤0.3w;α=[sec(aπ/w)]^1 /^2.

(v) Foraplateoffinitewidthwhavingtwosymmetricaledgecrackseachofdepth2a,Eq.(14.30)

becomes

K=S[wtan(πa/w)+(0.1w)sin( 2 πa/w)]^1 /^2

FromEq.(14.29),itcanbeseenthatthestressintensityatapointaheadofacrackcanbeexpressed
intermsoftheparameterK.FailurewillthenoccurwhenKreachesacriticalvalueKc.Thisisknown
asthefracturetoughnessofthematerialandhasunitsMN/m^3 /^2 orN/mm^3 /^2.

14.5.2 Crack Tip Plasticity

Incertaincircumstances,itmaybenecessarytoaccountfortheeffectofplasticflowinthevicinityof
thecracktip.Thismaybeallowedforbyestimatingthesizeoftheplasticzoneandaddingthistothe
actualcracklengthtoformaneffectivecracklength2a 1 .Thus,ifrpistheradiusoftheplasticzone,
a 1 =a+rp,andEq.(14.30)becomes

Kp=S(πa 1 )

1

(^2) α 1 (14.35)
in whichKpis the stress intensity factor corrected for plasticity andα 1 corresponds toa 1. Thus, for
rp/t>0.5,thatis,aconditionofplanestress,
rp=

1

2 π

(

K

fy

) 2

or rp=

a

2

(

S

fy

) 2

α^2 (see[Ref.3]) (14.36)

inwhichfyistheyieldproofstressofthematerial.Forrp/t<0.02,aconditionofplanestrain

rp=

1

6 π

(

K

fy

) 2

(14.37)

Forintermediateconditions,thecorrectionshouldbesuchastoproduceaconservativesolution.
Dugdale [Ref. 4] showed that the fracture toughness parameterKcis highly dependent on plate
thickness.Ingeneral,sincethetoughnessofamaterialdecreaseswithdecreasingplasticity,itfollows
that the true fracture toughness is that corresponding to a plane strain condition. This lower limiting
valueisparticularlyimportanttoconsiderinhigh-strengthalloys,sincethesearepronetobrittlefailure.
Inaddition,theassumptionthattheplasticzoneiscircularisnotrepresentativeinplanestrainconditions.
Rice and Johnson [Ref. 5] showed that for a small amount of plane strain yielding, the plastic zone
extendsastwolobes(Fig.14.6)eachinclinedatanangleθtotheaxisofthecrackwhereθ= 70 ◦,and
thegreatestextentLandforwardpenetration(ryforθ=0)ofplasticityaregivenby

L=0.155(K/fy)^2

ry=0.04(K/fy)^2