PSI - Issue 32

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

127

4

Mode I are written through one harmonic function:

/  

(1 / 4 (1 )) (

2 ) / ,      w f w f w P ) /     w f w f w

(1 / 2(1 ))( / 

/ ) f w f w

U f

  

     

(5)

2 2( / ) (1 / 2(1 )) (      T f w

here ( , ) f w w is the harmonic complex potential which for cracks of Mode I, has respectively the following form:   1 2 1 2 2 / / (5 8 ) (1 ) I I f f K w w         , (Lurie et al (2019). For the AVGE model local regular fields of stresses and displacements are found as solutions of the inhomogeneous Helmholtz equations: 2 2 u s u U    , 2 2 p s p P    , 2 2 t s t T    . The right-hand sides of two last equations

contain classical singular complex-valued potentials , P T given by (5). Regular local solution i.e. potentials , p t , displacements x

y u u iu   and stresses

,

Re( p t    )

xx

Re( p t    , )

are constructed using the radial factor method (see Lurie at al (2019). For Mode I crack,

Im

t

xy  

yy

such a regular solution has the form





ˆ





 

/ ( 2 ) 3 / 2 2    

1 2 w w s   1 2

2 / 4 ( / 2) 1 ( ) / 2 2 ( ) r s h r w   2

3 2



u K 





1

I



 



ˆ / 2 1 ( )  

1 2 h r w w  

1 2

/ 2,



p K 

(6)

0

I









ˆ

ˆ

 

 

1 2 h r w s 

2 6 1 ( ) / 6 4 ( ) / 3 r s h r    2

5 2

/ (4 2 ) 1 ( )  



t

K

w

 

0

2

I

1 ˆ ( ) h r

0 ˆ ( ) r s h r e   ,

2 2 ˆ ( ) [( ) 3( h r r s 

/ 2 r s

( r s e     1)

,

where

/ 4 r s

1)]

e 

r s  

Note that for sufficiently long cracks solutions (3) and (6) give practically the same results. 3. At last, it is easy to write the nonsingular solutions for the orthotropic plates withthe finite cracks for which following to AVGE model the generalized stresses are written as   [ / (1 )] , ( , ) xx x xy yx xx xy yy E E E x y        and yy xy E E G are generalized deformations. Using AVGE model and classical solution for the considered problem (see Rabotnov (1988)) we can get the following equations defining the local stresses at , 0 x l y   . xy   xy xy G E , where , , , x xy y E E G  are physical moduli and , , xx

2   , 1)

2 2     2 x  ( / s

2 E E x x c 

2 2       2 ( / ) s x

2 x x c  /

2

) ( / ) ( /





xx

xx

x

y

yy

yy

yy  with condition

is defined by equation

The regular solutions for the local stresses

( 1) 0,  

/    

у y x 

yy

yy

(4) and for local stresses xx 

1/ 2

( ) x      

( / ) E E

(7)

x 





0

x

y

y

3. Stress concentration concept Nonsingular solutions (3), (4), (6), (7) are the basis for implementing the concept of stress concentration in crack mechanics proposed in Vasiliev and Lurie (2018), Vasiliev at al (2019). The peculiarity of the proposed approach is that, on the basis of nonsingular solutions, it is possible to use the concept of stress concentration known in classical elasticity and predict the maximum failure loads for the materials samples with cracks Mode I with high accuracy.In the proposed concept of a fracture mechanic, the scale parameter s is determined from the experiment. This parameter