PSI - Issue 32

Valery Vasiliev et al. / Procedia Structural Integrity 32 (2021) 124–130 Vasiliev and Lurie / Structural Integrity Procedia 00 (2019) 000–000

126

3

and biharmonic functions  and  , which satisfy the equations 0 s       and compensate the power-law singularities of the particular solutions of the inhomogeneous Helmholtz equations (singular solutions) at the points , 0 x l y    . Using radial multipliers properties (see Lurie at al (2019)) we can constrain functions which satisfy the Helmholtz equation and have a set of the same power singularities as classical solutions 1 2 0 r s e w     ,   3 2 1 1 r s r s e w      ,       2 5 2 2 3 1 r s r s r s e w       . As a result, we can get 2 2 s       , 0 2 2

2 2 1/ 2    

1 2 ( / 2) ( l

2 ( / 2)( 6 ( (1 / 2 2) w w l l (

)

)

1/ 2 e ie  



 

1/ 2

   1 2 4 / 2 1 15 / 16( / ) ( l s l   2

2 2 3 2 

) / ( w w w l 

)  

1/ 2   ) 

1/ 2 e ie 



(2)

   1 2 



2 2 s l w w l

2 2 5 2 

2 s l

)  

(3 / 16 2)

3/ 2 e ie 

3/ 2   



   1 2



2 s l

5/ 2 e ie  





5/ 2

where

1 2 ) , 

1 2

exp(  

)( r s w l  

exp(  

)( r s w l   )







e e

e

1/ 2

1/ 2





3 2 ) exp(

2 ) 3( 

5 2 ) exp(

(  

1)( r s w l  

), r s e 

(

1) (

)





r s 

r s 

r s 

w l 









3/ 2

5/ 2

( )( w l w l  

)



r



Nonsingular solution for the stresses near top of crack Mode I are determined by potentials  and  by formulas (1), (2). Thus, the local stresses yy  for the 0 y  can be determined as the following function of relative coordinate ˆ x x l  , ( l s   ):   2 2 2 2 5 2 1 1 2 1/ 2 1/ 2 1 2 1 2 1 3/ 2 3/ 2 5/ 2 5/ 2 ˆ ˆ ˆ ˆ ˆ ( / 2) ( 1) 6 ( 1) (8 2) (2 3 15 8 )( ) ˆ ˆ ˆ ˆ 3(8 2) ( 4 )( ) 3 (2 2) ( ) yy x x x e e e e e e                                           (3) . Note that following to AVGE an approximate solution to the same problem can be obtained if the Helmholtz equation for the local stresses yy  , 2 2 2 2 2 ( / ) / yy yy s x x x c         , will be considered on the crack axis , 0 x l y   . In this case, the solution for local stresses with condition ( ˆ 1) 0 у y x    has the form (see Vasiliev at al (2019)) where 1 2 ) exp( ( 1)), x    / 2 ˆ k e  ˆ ( 1,3,5 x l  k  

ˆ x

   

   

ˆ x  

1  

ˆ x xe dx  ˆ ˆ

ˆ x xe dx   ˆ ˆ

ˆ x xe dx   ˆ ˆ

1 

ˆ x

(2 ˆ ) x 

,

(4)

x

( ˆ) 1 2 x 









e

e

e









0 / , yy yy    

у y

ˆ x

ˆ x

ˆ x

2

2

2

1

1

1







Estimates show that equations (3) and (4) give similar results for local stresses at , 0 x l y   .

2. Let us write the stresses for semi-infinite cracks in an isotropic strip, which can be used for the realization of the concept proposed for the prediction of the failure. Gradient solution for infinite cracks Modes I, II and III were received for the AVGE model in the work Lurie at al (2019). To construct regular solutions for semi-infinite cracks in an isotropic strip, we used the Papkovich – Neuber representation and a convenient representations of the complex singular potentials , P T , Re( ) xx P T    , Re( ) yy P T    , Im xy T   , through one complex-valued function f f w w f    . It was shown that classical solutions for displacements and stresses for cracks f if   , 2 ( , ), 0 f

x

y