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AnelasticDeformation at theCrack Tip & Irwin’sModel

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Appro% 'hape and si(e of theplastic (one

• )nl* appro%imate sol+tions can ,e determined d+eto the comple%it* of the pro,lem and ri oro+sanal*sis is to ,e done

• It is to ,e noted that material in plastic (one nolon er takes hi h stresses predicted ,* elasticanal*sis and therefore the elastic .eld s+rro+ndinthe plastic (one sho+ld ,e corrected for a moreri oro+s anal*sis

• Two commonl* +sed *ield criterionare +sed namel* Mises *ield criteriaand Tresca *ield criteria

• /efore we appl* a *ield criterion0 it is+sef+l to determine the principal

stresses• Mohr circle is +sed

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Plastic (one 'hape for Plane'tress

Mises *ield

criterion Tresca *ieldcriterion

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Plastic (one 'hape for Plane'train

Mises *ieldcriterion

Tresca *ieldcriterion

1or θ 2

34 5 ° &67 83

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Appro%imate Approach

1i Appro%imate plastic (onesi(e r

The len th of theplastic (one r alon % direction is o,tained,* the relation0

σ :: 7 k σ *s

We know that0

σ::

7

'ol6in ,oth e;n weet r0

r 7

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We ha6e principal stresses asσ 7 σ ::σ

: 7 σ ::σ 3 7 $ =for plane stress>σ

3 7 6 = σ ? σ : > 7 :6 σ :: =for plane strain>

And we o,tain the 6al+e of k ,* s+,s σ 0 σ : and σ 3 inMises criterion@

=σ - σ : >: ? = σ : - σ 3>: ? = σ 3 - σ >: 2 : σ : *sand comparin with

σ :: 7 k σ *s

we et k 7 for plane stress

k 7 8= - :6> for plane strain

'+,s The 6al+e of k in r0 we et

r 7 for plane stress

r 7 for plane strain

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Irwin’s Model ="lasto-PlasticAnal*sis>

• Irwin presented a simpli.ed model whichconsiders the redistri,+tion of load d+e to plasticdeformation

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Irwin proposed that δ is so chosen s+ch that the load nottaken ,e*ond the point M0 i6en ,* the area P c on len th 0is e;+al to the load s+stained on len th δ 0 i6en ,* area P A

Then0 the load not s+stained on len th ,ecomesPc 7 /

where / is the plate thickness The load s+stained =P A> on len th δ is

PA 7 / *s δ

Irwin’s correction =P A 7 P / > leads to

*sδ 7

We know that0σ

:: 7

where I is ,ased on the e9ecti6e crack len th =a? δ >0 andis determined ,* p+ttin % 7 and 7 for plane stress=k7 > i e 0

σ*s 7

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Rearran in 0 we ha6e I 7 =: > 8: *s

'+,s in a,o6e e;n

*sδ 7 *s

)n sol6in 0 we o,tainδ 7

The o6erall plastic (one si(e ,ecomes0r p 7 : 7

The e9ecti6e crack len th is i6en ,*0

a e9 7 a ? δ 7 a ? 7 a ?

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----------------------------------------• Irwin’s correction to the plane strain case is +sef+l todetermine the plastic (one si(e

• D+e to the plastic deformation the crack tip ,ecomes ro+ndedor ,l+nts 'ince0 the crack tip acts as a free s+rface0 isreleased to (ero

• The e9ect of release of is felt for some distance on B-a%is,e*ond the crack tip

• Irwin fo+nd that fail+re stress is no lon er 3 ,+t is closer towhich ma* ,e taken as Then0 the plastic (one si(e for theplane strain ,ecomes

r 7 7 σ *s

• 1or an e%perimental determination of IC of material0 planestrain conditions are ass+red ,* takin plate thickness to ,em+ch thicker than the plastic (one In fact0 it is chosen to ,emore than :! times of the plastic (one si(e and therefore0 theA'M" codes re;+ire

/ 2 : !

•

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Thank *o+