Issue 68

Z. Moqadaszadeh et alii, Frattura ed Integrità Strutturale, 68 (2024) 186-196; DOI: 10.3221/IGF-ESIS.68.12

Figure 1: Crack tip stress components in polar coordinates

GSED criterion considers the effect of c r and T besides singular terms derived from preceding series expansion. The strain energy density function ( / ) dW dV supplied within an element related to the plane problems is defined as:



1 dW K dV G  [

1

2   )        rr rr r   2

(

]

(2)

2 8

2(1 ) E 

)  is the rigidity modulus of elasticity and K is an elastic invariable which takes the value of 3 4   and



where ( G



 

3 1

 associated to the plane strain and stress problems, respectively. The strength in elastic energy field close to tip of crack is called strain energy density factor ( S ), and it can be explained as:



dW r K

1

2   )        rr rr r   2

 

S r

[

(

]

(3)

dV G

2 8

By replacing polar stress field components in elastic region from Eqn. (1) into Eqn. (2) and streamlining the aforementioned Eqn. (3), S is outlined below:

1

2

2 II

2 (2 ) ] 



2 a K a K a K K a   

( 2 ) 

r K T a 

( 2 ) 

r K T a r T 

S

[

(4)

I

I II

I

II

1

2

3

4

5

6



G

16

i a i  

( 1:6)

where

coefficients are shown below:

1 a k a k a  2 (       cos )(1 cos ) [ (1 cos ) cos (1 3cos )] sin (2cos ( 1)) cos (cos(2 ) cos ( 1)) 2 sin (cos(2 ) cos ( 1)) 2 ( 1) 2 k k k k                           a a a 3 4 5 6

(5)

188