Crack Paths 2012

For determiningArm,since (KIda)v )2 varies at different layers aheadof the notch, the

influence of the angle, 0, and the radial distance have to be considered simultaneously.

In order to evaluate ArFG, a layer ahead of the notch is considered in the graded

region, being X'w the distance fromthe notch tip. In Figure 1, rm”)and 6'”) indicate the vectorial radius and vectorial angle of the plastic region with respect to the ith layer. By

considering Eq. 2, rFGm can be expressedas follows:

rFG(i) :aXH(X(,i))Xf(6(i))

(8)

Hoffa):(KIC(X{i>)/0-y(X€i>))2

(9)

where K1C(X'(i) ) and o).(X’(,-)) are the fracture toughness and the yield stress correspondending to the ith layer. The relationship between the radius and the angle of

the plastic region and the position of the considered layer can be expressed as follows:

X 0 ): rmo)COS(6(1'))

(10)

By combining Eqs 8 and 10 the following expressions can be derived:

/

X 0 ) i . o. :o. HOG“) af( <,,)¢0S( 0,)

(11)

. _ Kmart.) 2_ 5 2 _ . H(x(,,)_[ 030(6)] _[DJexp|:2(D2 D4)(X d+Xm):| ._

(12)

By using Taylor’s series expansion (exp(x)=1+x+O(x2)) and neglecting the higher order

terms, X ’(,- ) assumes the following form:

X“) —1+aXB>(X* “Bl

(14)

B >20), —D.>(D1/D3)2

(15)

By substituting X ’(,-) into Eq. 8, the radius rFG( i ) is derived as follows:

rFG(6'(,-,)

=0t[g] { H a_Dg4)[X*

1+a>

(16)