Crack Paths 2006

,0

2 2 w TV T

VT

ww

(1)

T

0

T is shown in Fig.1.

where the polar coordinate

According to the M-criterion proposed by Kong et al. [7], the crack propagation

eq V V / H

direction is defined by the maximumvalue of the stress triaxiality ratio

M

around the crack tip (HV is the hydrostatic stress, whereas

e q V is the equivalent stress

which can be assumed equal to the Von Mises stress). Analytically, the criterion can be

stated as follows:

3 3

V

¦ i

S Q

T

T

V

ii

I

II

(plane

ª

» º

with

2) 13 ( 2

cos 2

sen 2

strain)

w T T M M

2 2 w

(2)

1

r K

K

,0

0

H

« ¬

˜

¼

Following the minimumstrain energy density criterion (S-criterion) proposed by Sih

[2, 3], crack grows in the direction of minimumof the strain energy density S around the

crack tip. Analytically, the criterion can be stated as follows:

2 T

(3)

T S

w

w

S

,0

0

2

w !

T between the initial crack and the crack growth direction can be determined

The angle

by solving the following equation:

cos3 ˜ ˜ T T II K

K

sen

0 1 .

I

According to the maximumdilatational strain energy density criterion (T-criterion)

proposed by Theocaris et al. [4-6], crack grows in the direction of maximumof the

dilatational strain energy density

V T around the crack tip. The criterion can be written as

follows:

2 2 T p I

(4)

ww T

ww

2 w V T

w

I

;0

0

V

p

w T

,0

0

w

or

2

T

where

2 2 1 2 I I I p with 2 1 , I I the first and the second stress tensor invariants,

respectively. On the other hand, the distortional strain energy

D T along the boundary of

the VonMises plastic zone is constant and equal to the yield stress and, therefore, cannot

be used to find a minimumvalue.

y

R ( ) p T

plastic region

crack

T

x

t

elastic region

F(I ,J )=0 1 2

Figure 1. Graphical representation of the R-criterion.