Crack Paths 2006

21

k k

Q

­

½

T

¨©§

¸¹·

2 2 2

2 1 2

83

s i n 8 9

)21(21[

k k

2 c o s ] 2 3

2 2 2

° ° ¯ ° ° ®

° ° ° ° ¾

T Q kk

Q Q V T k k f yld

T

T

r

k

yld

P

1(2 1 ) (

(3)

s i n ) [ 223sin4

] s i n ) 2 1 ( 2 1

2 1

¿

1

) , , ( 2 2 2 T

2

2

2

2 1

V

where rp(T) is the radius of crack-tip plastic zone for the combined mode I and II, and f is

a function of the angle T, which characterizes the shape of the plastic zone rp(T).

It is important to note that for evaluation of the above elasto-plastic boundary, only

the elastic dominant stress fields in conjunction with the von Mises yield criterion were

utilized, based on the assumption that the perturbation due to the presence of crack tip

plasticity was of no major significance. This is justifiable in the case of essentially small

scale yielding, SSY. It can be shown that the radius of crack-tip plastic zone rp is directly

proportional to the applied stress, V2. The size of the plastic region increases with

increasing loading on the body. In the present case, it is assumed that a plastic region at

the edge of a crack is subject to plane strain conditions. Towards the plate surface,

however, plane stress dominates, giving rise to a different shape of plastic region.

DIRECTIOONFF A T I G UCE R A CPKR O P A G A T I O N

In the present investigation, for simplicity, we only present the results for the special case of D =90o and Z =90o, i.e., the case of modeI and II loadings.

Minimumradius criterion

Based on the variable radius of the plastic zone rp(T) defined in Eq. 3, a minimumplastic

zone radius criterion for the two-dimensional stress state can be established. It is assumed

that the direction of fatigue crack initiation coincides with the direction of the minimum

radius of the plastic zone defined by the von Mises yield criterion. This means that the

angle To of the fatigue crack growth direction can be determined by minimizing the rp(T)

value, i.e.,

¸¸¹·¨¨©§ww

¸¹·¨©§ww f

¨©§w

¸ ¸ ·

r

r

¸ ! ¸ ¹ · o T T

¨ ¨ © § w

f

p

yld

p

yld

o

o

o

2 2

1 2

,0

1 2

0

2 2

(4)

T

V T

T

V T

T T

T T

T T

¹

By establishing w rp /wT = 0 at To, the direction of fatigue crack growth To for the

special case of D = 90o can be determined.

Minimumstrain energy density criterion

Sih [3] presented the minimumstrain energy density criterion. It states that the direction

of crack initiation coincides with the direction of minimumstrain energy density, along a