PSI - Issue 37

L.V. Stepanova et al. / Procedia Structural Integrity 37 (2022) 920–925 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

922

3

Fig. 1. Holed plate under uniaxial tension.

The strain-displacement equations can be written as

U U +    

U r

1





U U r r 

r       U

1 1 2 r  

  

,

,

.

(1)



 

=

=

r

r

r  

=

+ −



rr

r

r



The compatibility condition is

2

2



r  r





  − + −  

r 

 

2

0

r

r

r

=

rr

rr





.

(2)



 

2

2

r   

r









The equilibrium equations are 1        −  r rr

1        +  r r r r r 



0

2 + =

0

,

.

(3)

rr r r +

+

=

r









At a large distance from coordinates origin stress components have a form ( ) (1 cos 2 ) 2 rr r     =  = + , ( ) (1 cos 2 ) 2 r      =  = − , ( ) r r  





sin 2



=  = −

.

(4)

2

The boundary conditions at the hole are ( ) 0 rr r a  = = , ( ) 0 r r a   = =

(5) For a material subjected to the Bailey-Norton creep power law the constitutive equations can be written as 1 1 (2 ) ( , , ), 2 n rr e rr rr rr r B F           − = − = (6) 1 1 (2 ) ( , , ), 2 n e rr rr r B F             − = − = (7) 1 3 2 ( , , ). 2 n r e r r rr r B F            − = = (8) The nonlinear boundary value problem (1)-(8) is solved using quasilinearisation technique (Boyle and Spence (1983)). Approximate solutions sequence ( ) ( ) ( ) ( , , ) k k k rr r      , k = 0,1,2 … is generated in the following manner. Linearization of the constitutive equations (6)-(8) leads to next equations , rr rr rrrr rr rr rrr r a b b b         = + + + (9) , rr rr r r a b b b             = + + + (10) , r r r rr rr r r r r a b b b             = + + + (11) where the coefficients