Crack Paths 2006

¦ ¦ N j B

¦ ¦ N j 1 2 C

^ `

u

u

u

g

Fj

i

c

Fij

L j L i j

c

j i j

IH

(20a)

F

F

H

H

DE

D

E

E

E D E

J

E D J

1 2 1

1

B N j

2

FP

t

E D E

¦ ¦ G

c

j i j

2

E

1

1

i F

u

u

g

Fj

Lj

ij

c

¦ ¦ B N j 1 21 E

Fij

Lij

c

¦ ¦ C N j 1 2

j

C H

C H

CIH

c

DE F

2

2

PF

PF

DE

DJ

D

E

E

J

1

E

(20b)

B N j

C G

^`

^`

C

D E

E

E

¦ ¦ 1 2

t

ij

j

i

1

u u

where

u

,

Li u u E

,

LNi Fi B u u 1and

E

Fi

Li

Fi

for

,,2,1 !

1

21

i

D

D

D

E

1

i C

B N

are arbitrary constants of integration. On each boundary contour segment,

E B , displacement components:

and

D E [ u

D E K u , are approximated by the linear interpolation function. However, traction compo

and

D E K t , are constant along

like dislocation density components:

nents:

E B

DE[t

and

and resultant force components:

^` DE[ F and

^` DE[ F , along any crack

^` DE[g

^` DEKg,

contour segment,

E C . Therefore, integrations of integrand functions of the discrete

version of the integral equations 20 can be performed exactly. Exact integration is

generally faster than numerical integration for a level of reasonable numerical accuracy.

The transformation of integration results from the local co-ordinates system

D E D E K [ ,

to the global one yx, is straightforward.

Due to proper shape functions for the displacement field, the strain field and the

stress field along each contour segment a special treatment, used to circumvent the well

knowncorner problem of the boundary element method, is not required. The matrices: D E F HD,E L HD, E F C DH E, L C HD ,E G D, E C GD E,I H and D E L C I H , in Eqs 20 are

assessed by integrating the fundamental solutions analytically (see Ref [5]) without the

necessity to use numerical integration over each contour segment.

R E G U L A R I Z A TBIYOANN O N L O CCAHLA R A C T E R I Z A TFIUONNC T I O N

According to the nonlocal theory of Eringen [6], the stress is computed by averaging the

local stress that would be obtained from the local model. Thus, the nonlocal approach of

Eringen can be characterized as averaging of the stress. To analyze the nonlocal

mechanical behavior, the expressions that contain stress components in Eqs 20 are

regularized by