Crack Paths 2006

V

In Eq. 1 the displacements,

iu , tractions,

(i.e., the stress vectors,

t

n

, related to

j j i

it

i

the outward normal vector,

j n ), and body forces,

ib , are respectively determined on the

boundary

w and in the domain .

For the i-direction at any field point

r

yx,

due to the unit force

j e in the j

ycxc

direction applied at the load point

r

c

,

, fields of u , t and b corresponding to

the governing solution of elasticity theory can be expressed as:

r r r

(2a)

* i u U r

,

c c j e

,

fieldnt displaceme

ij

r r r

(2b)

field

t

r

T

,

c c j e

,

traction

ij

i

*

e r

(2c)

body

G

b

r

r

c

,

field force

i

*

i

where rrc, ij U and rrc, ijT

are the fundamental solutions for linear elastic problems,

r G is the Dirac delta function and r rc ris the distance between r and rc . The

lower case of r or rc represents a point located in I n t w,hile the upper case of R

or Rc represents a point placed on . For two-dimensional elastostatic problems,

rrc,ijU and rrc,ijT are given by

@ji

>

F

ij G Q rr

r r

(3a)

C P

,

c

U

2

r l n 4 3

,

ji

, ,

G Q

n r n r

, ,

j i

F

2 1 Q

T

,

c

r

2 1

2

ww n r r r

¯ ® ­

¿ ¾ ½

ji

ij

j i

i j

C

,

,

r r

>

(3b)

,

@

@ Q S

>

F

where

1 4 1

, ijG is the Kronecker delta function, P is the shear modulus

C

of elasticity and Q

is the Poisson’s ratio. Note that components of the gradient of the

one-form Ad are denoted by the commaderivative:

k k x A A w w , .

According to the expressions 2 in the absence of body forces,

ib, Eq. 1 can be

rewritten for two-dimensional elastostatic problems as:

³ i d t U d u T u r : R R R r R R R r r (4) ³ w w c c c c c c ji Int j i j j i

,

,



2 , 1 ,

.

M O D E L I CN OG P L A N CA R A CSKU R F A C E S

The straightforward application of Eq. 4 to crack problems leads to mathematical

degeneration when upper and lower crack surfaces of a body occupy the same location.