Crack Paths 2006

In the presence of a crack in the domain [2, the boundary of a two-dimensional body, F I 6(Q), can be divided into parts: F I 178 + it} + rt}, where 172+} and I"?

represent the upper and lower crack surfaces and 17,, represents the remaining

boundary. From fundamental solutions 3, the displacement and traction on the lower

and upper crack surfaces have the properties that

U,J(r,R'C{+})I U,j(r,R'C{‘}) ,

(5a)

Ty‘ (r, Ric/H) : _Tlj (r, Ric/B) -

(5b)

The change in sign in tractions in expression 5b is because the direction of the normal is

opposite on the two crack surfaces. A simple description of a crack is two coplanar

surfaces that are closed, e.g., r5} —> PF} (see Ref. [2]). Because of Pt} I FF} and

expressions 5, Eq. 4 becomes

u,(r)+ [T,,(r,R')u,(R')dr(R')

+

MILE’;tramway) I IU,(.,Rg-});t,(Rg-})dr(Rg-})

. (6)

rg-l

Fé-l

IU,,(r,R')tj(R’)dF(R')

FE

where

Aui(Rii_))=ui(Rii+))_”i( 'C(‘—))’

(73‘)

ZtlRi‘U-tl til-ti 2*’).

(7b)

For traction free cracks, or whenthe crack is loaded by equal and opposite tractions Z tj (R'éii) I 0 . Hence, Eq. 6 can be rewritten in the form:

u,(r)+ [T,,(r,R')u,(R')dr(R')

+ [TI,(r,R'CH)Au,(R'C{-})dr(

'C{-})

r,

Fig}

:

lUI-(RRUIAR'MMR'),

(8)

F B

Equation 8 has the form as the standard integral equation 4 with an additional integral

along the lower crack surface, Fifi. The boundary form of this equation is indetermi

nate (i.e., when R e 172*} the number of unknownvariables in the system of linear

equations is greater than the numberof linear equations, which represent the boundary