Crack Paths 2012

Expressions ofthe stress intensity factor andfundamentalkernel at order 1

The expression of K1(a;z1) is obtained by applying Rice‘s first formula (1) to the

straight configuration of the front using Eq. (6), and integrating by parts:

dK° K0(a) +~¢'(Z) da (a)¢(z1)+ 2” PVLOZ_ZI dz.

(7)

K1(a;Z1) =

The expression of Z1(z1,z2) m a ybe obtained in a similar way from Rice's second

formula (3) and Eq. (6). A n arbitrary perturbation 549(2) generally violates the

conditions e¢(z1) : e¢(z2) : 0 , necessary for Eq. (3) to be applicable, but this difficulty

maybe overcome by using Rice’s suggestion [4] to decomposethis perturbation in the

form e¢(z) : [e¢(z) —e¢,.(z)] +e¢..(z) where e¢..(z) is a suitable combination of a

translatory motion and a rotation (having no effect on the FK) such that eo,(zl) : £¢(z1)

and e¢,.(z2) :e¢(z2). One thus gets:

ZI(ZI,ZZ)IZ;PVI:|:( 1 +21 ]¢'(Z)

4n‘ (zl—z2)2

z—z

Expression ofthe stress intensity factor at order 2

The expression of K2(a;z1) maybe obtained by applying Rice's first formula (1) to a

configuration of the front deduced from the straight one through the perturbation

8¢(Z), and further perturbed by the amount 55¢(z) where 55 is an infinitesimal

quantity. The formula provides an integral expression of the derivative 8K(a,e;z1)/ 8e

accurate to first-order in e if the first-order expressions of Z(e;z1,z) and K(a,e;z) are

employed; and the second-order expression of K(a,€;z1) follows through integration.

One thus gets:

K2(Zi)=%dTIi(a)[¢(Z1)]2+ 1 d i ( a ) p j / [ j % d z

; da

K°(a) +0 +9

1

2

'

I

+ 8”, PI/[M [_M [[Z,_Z1+Z,_Zj¢(z)

(9)

+ 2 [ 1 _ ; ] ¢ ( Z 1 ) : | ¢ ( z ) _ ¢ ( 2 z l ) d z d z i .

z—z1 z'—z1 z'—z

(z—zl

754