Crack Paths 2006

T R E A T M EO NFTTH ES T R O N GSLIYN G U LIANRT E G R A L S

To avoid the numerical difficulties with the strongly singular integrals in Eq. 8 the idea

of integration by parts developed by Ghoshet al. [3] and Zang [4] is adopted.

A single integration by parts starts with

%I—W.u.-I=%I-mature)-n

<13)

and integrates both sides of the equality 13,

j % VPVij(r’R')uj(R')]dS(R') : j 66_Si_ VVzj(r’R')]uj(R')dS(R') +

—

' I rnwfgimdsm'i

F A B

F

AB

S

Rearranging gives

jTly(r,R')uj-(R’)ds(R’)=

[- W].(r,R’)uj(R’)]j

+ I (rgRqauéim')

ds(R’) ,

(15)

S

F A B

F A B

where TU(r,R')= 6i[—W,j.(r,R')]. Applying integration by parts for the upper and

s

lower crack surfaces gives

[T,(r, R'C{*})uj(R;,i+})dF(R§+})

Fifi

fr-j-(nR'tat-(Ravlij

+ lVK.(r»R’é*})Lfi+})dr(R§+}).

I

[1,-(r.R§*})u.(Ré-})dF(R2§-})

FI-I

I—r.-u.-<

or: + IwrnRwwdrIRt-u

For the internal crack problems whenthe displacements of the u per and lower crack (1)121‘? ) = u,(Ré‘iT}) and

surfaces at both crack tips A and B coincide (i.e., u,

u,(R§{5} ) = %(Réiig) )), the lower-order singular integral can be expressed by