Crack Paths 2006

time, d, r) I i ][h,, (x, b, r) -K,S(b, r) + h,,,S(x,b,r)

-K,,S(b,r)]db

H X

(7a)

2 a

rH ][h,, (x, b, r) - K,, (b, r) + h,, (x, b, r) -K,,, (b, r)]db

Where, as for eqn. (1), A/B refers to the right and left crack tips, H is equal to E (Young

modulus) for plane stress and E/(l-v2) for plane strain (v is the Poisson’s ratio). The values

(K1S,K11S) and (K1A,K11A ) are produced respectively by the symmetric and anti-symmetric

load cases and, by recalling eqn. (1), can be evaluated as follows:

12.0w)

hrstxur)

h’tstxur)

(“(20) .

1

(8a)

[Knswfl’ij _6I]:[I1HGS(XI,F) l’lHTs(XI,V)

r(x') S dx

who rated r6101) (60)] . 1

(8b)

(Kn/107279]

_6II(hII”A(x',r)

hH’A(x',r)

r(x') A dx

By indicating the W Pwith a matrix notation as follows:

h m(x, b, r)

h”(x, b, r) ]

(9)

[W(x’b’r):IS/A = [

h l l o ' ( x fl b g r ) h 1 1 T ( X , b , I ” )

S / A

After introducing the expressions of eqns. (8) into eqns. 7, and changing the order of

integration [10], the following expression is obtained:

IVWII]

Ila/(moi;-IW("I’I”'I)ISIIII60].IdxI

max(x,x‘)

Z - ( X I )

a i i j[W(r,b,r)],T-[W(r',b,r)],db]-(U(x)] a

-dx'

I

H 0 max(x,x‘)

2 ( X I )

Where [ ]T is the transpose matrix. By introducing the following 2X2 matrices:

I

_ Gw(x,x',r) Gv,(x,x',r) _ a

T.

I

GUIXIII)S _ IGue(x,x',r)

G1,.(x, x',r)IS _ maXlIfI/(x’b’IOIS

III/(x ’b’r)SIdb (Ila)

I

_ Gvc,(x,x',r)

Gv,(x,x',r) _ ‘I

T i

I

GUIXIIOA _ IGue(x,x',r)

G11.(x, x',r)IA _ meX(I,II/:/(x’b’r)IA

III/(x ,1)’ II)AIdb (11b)

Equation (10) can be rewritten as:

re. a. r)

2 a

air)

‘1

eix')

I — - _|I[G(x,x',r)S]-

I

dx'i][G(x,x',r)A]-

I

-dx‘ (12)

u(x, a, r)

H 0

2'(x ) S

O

2'(x ) A

Thus demonstrating that [G(x,x ’,r)] represents the Green’s functions (GF) as it relates the

load applied on the crack faces to the local displacement. Considering the power law

expansion proposed for the W P(eqns. 3-4) and taking into account of the combination