PSI - Issue 28

Vera Petrova et al. / Procedia Structural Integrity 28 (2020) 608–618 Author name / Structural Integrity Procedia 00 (2019) 000–000

614

7

T r are Chebyshev polynomials of the first kind.

3.3. Stress intensity factors and critical loads The stress intensity factors are obtained from the following formulas:

2    ( ) g

,

(13)

1 lim 1 n a   

nI K iK 

  





nII

n

1

2 1 ( 1) ( ) cot 4 m n m m u M  

M

 

p a 

( 1) 





In K iK 

   

,

(14)

a u 

n

n

IIn

n n

M

1

m



1

2 1 m 

M

 

M m 

( 1)

( ) tan 

p a 

u

( 1) 





,

In IIn n n K iK a u     

n

n

n m

4

M

M

1

m



n = 1, 2, …, N .

Here the signs “+” and “–“ refer to the right and left crack tips, respectively. The functions (12) written for 1   

1

2 1 m 

M

 

1

m

(1)

( 1)

( ) cot 



u

u





n

n m

4

M

M

1

m



1

2 1 m 

M

 

( 1)

( 1)

( ) tan 

M m 

u

u

 



n

n m

4

M

M

1

m



are used in the calculation of Eq. (14). For predicting the crack growth and the determination of a direction of this growth, the criterion of maximum circumferential stresses (Erdogan and Sih (1962)) is used. According to this criterion, the crack deflection angle ϕ (or the so-called fracture angle, Fig. 3) and the critical stresses are calculated as   2 2 2 arctan 8 4 n In In IIn IIn K K K K           , (15)

, n Ic tip K K  . eq





or

(16)

eq n K 

3 cos ( / 2) 

3 tan( / 2) 

K K 

K



, Ic tip

n

In

IIn

n

Using a single crack subjected to a load p normal to the crack line as a reference crack with the stress intensity factor

0 K p a   ,

(17)

the corresponding critical load is obtained as

0 1 Ic p K a   ,

(18)

where . In general, the SIFs are written as 1,..., max n n N a  a 