PSI - Issue 28

L.V. Stepanova et al. / Procedia Structural Integrity 28 (2020) 1781–1786 Author name / Structural Integrity Procedia 00 (2019) 000–000

1784

4



(0)        (0)  , rr r

(0)

/ 8 1 / 4

1 / 2,

   

   



4

5





 / 4 ,

   

(0)

5 / 8 1 / 4

(1 / 4)    

/ 8 (1 / 2) cos 2 

   

    

5 (1 / 4)         6



5

rr



 / 4 ,



 / 4 ,

(0) 

(0)

/ 8 (1 / 2) cos 2 

(1 / 2)sin 2

    

 

    



r 

5

5



 

(0)           3 / 4

(0) 

(0)

1 / 2 3 / 4,

1 / 2,

   

 

 

6

7

rr

r



(0)

(0) 

(0)

(1 / 2) (1 / 2) cos 2 ,

(1 / 2) (1 / 2) cos 2 ,

(1 / 2)sin 2 . 

     

  

 

  

 



7

rr

r



for pure mode II problem respectively. 3. Asymptotic stress fields in the vicinity of the crack tip under plain strain conditions: mixed mode loading Generalizing formulae (10) one can obtain the following asymptotic solution for mixed mode loading      (0) (0) (0) 1 2 3 / 4 (1 / 2) 1 cos 2 , (1 / 2) 1 cos 2 , (1 / 2) sin 2 , rr r                           (0) (0) (0) 2 3 1 / 2 3 / 4, 1 / 2, rr r                         (0) 3 4 3 3 (0) (0) 3 3 3 (1 / 2) 3 / 4 (1 / 2) cos 2 3 / 4 , (1 / 2) 3 / 4 (1 / 2) cos 2 3 / 4 , (1 / 2)sin 2 3 / 4 , rr r                                           



(0)            

(0) 

(0)

p

(1 / 2) (

/ 2),

1 / 2,

tg M 

r 



(11)

4

5

rr





 / 4 ,

   

(0)          

(1 / 2) (

/ 2) (1 / 2) cos 2 

p

tg M 

5 (1 / 2)sin 2     

5

6

5

rr



 / 4 ,



 / 4 ,

(0)    5

(0)

(1 / 2) (

/ 2) (1 / 2) cos 2     

p

tg M 

  

5       

 

    

r 

5



 

(0)           3 / 4

(0) 

(0)

1 / 2 3 / 4,

1 / 2,

 

6

7

rr

r



(0) 7  The asymptotic solution (11) consists of 7 sectors. The values of angles separating different sectors are found from the continuity conditions at each boundary. However, it can be easily found that the asymptotic solution is valid for the interval 0 0.3301952 p M   . At values of the mixity parameter higher than 0.3301952 p M  the asymptotic solution consists of 6 sectors:      (0) (0) (0) 1 2 3 / 4 (1 / 2) 1 cos 2 , (1 / 2) 1 cos 2 , (1 / 2)sin 2 , rr r                           (0) (0) (0) 2 3 1 / 2 3 / 4, 1 / 2, rr r                         (0) 3 4 3 3 (0) (0) 3 3 3 (1 / 2) 3 / 4 (1 / 2) cos 2 3 / 4 , (1 / 2) 3 / 4 (1 / 2) cos 2 3 / 4 , (1 / 2)sin 2 3 / 4 , rr r                                            (1 / 2) (1 / 2) cos 2 , (1 / 2) (1 / 2) cos 2 , (1 / 2) sin 2 .  rr r                  (0)  (0)



(0)            

(0) 

(0)

(1 / 2) (

/ 2),

1 / 2,

p

tg M 

r 



(12)

4

5

rr





 / 4 ,

   

(0)          

(1 / 2) (

/ 2) (1 / 2) cos 2 

p

tg M 

5 (1 / 2)sin 2     

5

6

5

rr



 / 4 ,



 / 4 ,

(0)    5

(0)

(1 / 2) (

/ 2) (1 / 2) cos 2 

p

tg M 

  

    

 

    

r 

5

5





(0) 6  One would expect the asymptotic distribution (12) to approach the asymptotic solution which is valid for pure mode I (9) at values of the mixity parameter close to 1. However, there is no limit of the functions     (0) (0) , rr      in intervals 4 5      and 5 6      at 1 p M  in (12). Asymptotic analysis of the solution (12) has shown that the circumferential distributions of the stress tensor components approach the pure mode I solution at (1 / 2) (1 / 2) cos 2 , (1 / 2) (1 / 2) cos 2 , (1 / 2) sin 2 .  rr r                  (0)  (0)