PSI - Issue 13

M.R. Ayatollahi et al. / Procedia Structural Integrity 13 (2018) 735–740 Author name / Structural Integrity Procedia 00 (2018) 000–000

737

3

3. Fracture assessment criterion The elastic stress field around the crack tip can be expressed as a series expansion according to the following form:

1

3 2







2 cos [ (1 sin )  K

2

1/ 2 ( )

( cos 2 tan )] cos    T

K

O r











rr

I

II

2

2

2

2

r



1

3





(1)

2 cos [ cos 2 1 cos [ sin   I K r 2

2

1/ 2 ( )

sin ] sin   T

K

O r











II



2 2

1/ 2 ( )

(3cos 1)] sin cos       T

K

K

O r

 r







I

II



2

2 2

r



where ( r, θ ) are the crack tip coordinates, σ ij ( i, j = r , θ ) are the stress components, K I and K II are the Mode-I and Mode-II stress intensity factors, respectively and T is a non-singular stress term, which is usually called T -stress. The higher order non-singular terms O( r 1/2 ) are often negligible in the close vicinity of the crack tip. Considering the plane strain condition for the tested specimens, the minimum strain energy density function stored in an element of material in three-dimensional stress systems can be written as follows (Ayatollahi, et al, 2015):

2            rr rr r 2 [ 1 ( )  

]

 S r

(2)

2 2 G

where ν is the Poisson’s ratio and G is the shear modulus of elasticity. The minimum strain energy density criterion states that the direction of fracture initiation ( θ 0 ) coincides with the direction of minimum strain energy density factor S along a constant radius r c around the crack tip (Ayatollahi et al., 2016). This can be written mathematically as:

2

S

S

 

 

0

0

(3)





2 



0   

0   

By substituting the stress components for the plane conditions (Eq. (1)) into Eq. (2) and using Eq. (3), the fracture initiation angle θ 0 can be determined by solving the following equation 2 2 2 1 2 3 4 5 6 2 ( ) ( ) ( ) 0          I II I II eff I eff II eff b K b K b K K b B K K b B K K b B K (4) where

0 2sin (cos 2 1) 2sin (3cos 2 1) (2cos(2 ) 2(1 2 ) cos ) sin (5(cos(2 ) cos ) 4(1 )) 2 cos (5(cos(2 ) cos ) 2(3 2 )) 2 0                                       0 0 0 0 0 0 0 0 0 0 0

b b b b b b

1

2

3

(5)

4

5

6

The dimensionless parameters B and α in Eq. (4) are defined as:

2

r

2 II T a T a B K K K 2      eff I

 

and

(6)

c

a

The dimensionless parameter, B is often called the biaxiality ratio that shows the ratio of the T -stress relative to the stress intensity factors. a is the crack length and r c is the critical distance that can be calculated using Eq. (7) 2 1 2          Ic c t r K (7) in which K Ic is the fracture toughness of material obtained from the specimens with negligible T-stress values (here CT specimen). By substituting the fracture toughness, K Ic = 1.4 MPa.m 0.5 and ultimate strength σ t = 55 MPa in Eq. (7), the calculated value of r c for the tested PMMA is found to be 0.1 mm. Eqs. (4) and (5) show that the crack growth angle depends not only on the singular terms tied to SIFs ( K I and K II ) but also on the magnitude and sign of the T -stress and on the material properties ( ν and r c ) as well. For pure mode-I loading when K II is equal to zero, Eq. (4) can be simplified as follows: