PSI - Issue 28

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

1783

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case. Nevertheless, some peculiarities of the problem have not been studied in detail. The main goal of the present study is to obtain the analytical formulae for the angular distributions of the stress tensor components in the vicinity of the crack tip in a full range of the mixity parameter, namely, for values of the mixity parameter close to 1 p M  in perfect plastic materials when the plastic flow zone surrounds the crack tip. 2. Asymptotic stress fields in the vicinity of the crack tip under plain strain conditions: mathematical statement of the problem The mixity parameter is defined as (Shih (1974))         2 / lim , 0 / , 0 p r r M arctg r r                . (1) Equilibrium equations for 2D problem in polar coordinates are written in the form , , , , 0, 2 0. rr r r rr r r r r r                      (2) The von Misses yield criterion for plane strain and plane stress conditions is expressed as   2 2 2 2 2 2 2 4 4 , 3 3 rr r rr rr r k k                     , (3) where / 3 Y k   , Y  is the yield stress under tensile loading. The traction free boundary conditions take the form     , 0, , 0. r r r               (4) The asymptotic solution to the problem can be presented in form       (0) (1) , , 0. ij ij ij r r r           (5) Substituting the asymptotic solution (5) in governing equations (2), (3) and neglecting small quantities as 0 r  one can obtain the following system of ordinary differential equations (0) (0) (0) (0) (0) , , 0, 2 0 r rr r                 (6) and the yield condition in the form     2 2 (0) (0) (0) 2 4 4 . rr r k         (7) Equation (7) will be satisfied identically if one can introduce new functions   (0) (0) , ( )     such that     (0) (0) (0) (0) (0) (0) (0) (0) cos2 ( ), cos2 ( ), sin 2 ( ), rr r k k k                     (8) The asymptotic solutions for pure mode I and for pure mode II are well-known and can be presented as      (0) (0) (0) 3 / 4 (1 / 2) 1 cos 2 , (1 / 2) 1 cos 2 , (1 / 2)sin 2 , rr r                       (0) (0) (0) 3 / 4 / 4 (1 / 2) 3 / 4, (1 / 2), rr r                     (0)

  

/ 4      / 4

/ 2 (1 / 2) (1 / 2) cos 2 , 

  

  

rr

(9)

(0)   

(0)

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

(1 / 2)sin 2 , 

  



 

r



   

(0)         / 4 3 / 4

(0) 

(0)

(1 / 2) 3 / 4,

(1 / 2),

    

 



rr

r











(0)

(0) 

(0)

3 / 4   

(1 / 2) 1 cos 2 , 

(1 / 2) 1 cos 2 , 

(1 / 2) sin 2 , 

   



 



 

 

rr

r



for pure mode I problem and









(0)

(0) 

(0)

3 / 4            

(1 / 2) 1 cos 2 ,     (0) (0)  rr

(1 / 2) 1 cos 2 , 

(1 / 2)sin 2 , 



 



 

 

1

2

rr

r



(0)

(5 / 8 1 / 4)  

1 / 2 3 / 4,

1 / 2,

  

   

   

 

 

2

3

r







   

(0)            / 2

(1 / 2) (3 / 4) (1 / 2) cos 2  

3 / 4 ,

  

    

3

4

3

3

3

rr

(10)









(0)    3

(0)

(1 / 2) 3 / 4 (1 / 2) cos 2 

3 / 4 ,

(1 / 2)sin 2

3 / 4 ,

   

    

 

    

r 

3

3

