PSI - Issue 52

T. Profant et al. / Procedia Structural Integrity 52 (2024) 455–471

464 10

T. Profant et al/ Structural Integrity Procedia 00 (2023) 000 – 000

(

)

2

3

2

12 3 4 l

( 12 − − + + − − + + − 3 2 3 12 13 13 12    + − − + + − 8 1 8 9   3 12 13 13 12    l )

  

1/2

4

I K l

,

C

=−

11

1/2

( ) 

3 2 El

0

(

)

2

12 3 4 l

  

1/2

2

I K l

,

C

=−

(

)

12

1/2

( ) 2 

8 1 8 9   − − +

l

El

12

0

(30)

1/2

II K l

(

) 1 ,

C

=−



+

21

1/2

( ) 2 

El

0

(

)

3

2

3

2

12 7 14512 7 10      − − + − + +

9

l



−

1/2

2 2 2 II K l El

12

C

=−

(

)

22

1/2

( ) 

8 1 8 9 l   − − +

12

0

The path independency of the -integral implies that the gradient elasticity -integrals (27) must be equal to their counterparts from the classical elasticity ( ) 2 , , 2 1 1 I II ID IIE K J E  = − (31) The comparison of the -integrals in (27) with (31) for =0.01 , 0 ⁄ =1000 , =0.1 and =0 is shown in Fig. 4. The intersections of the grey-colored curves representing (27) with the black ones, which represent the expression on the right-hand side of (31), give the solution for 0 . Therefore, the values for 0 and 0 for modes I and II can be obtained after substitution of (30) into (27).

Fig. 4. The -integral and (ii) -integral for various values of 0 and as a function of the Poisson ratio . The grey line is the values of - integral in LEFM and the black lines correspond to the values of -integral in the theory of flexoelectricity. The remaining parameters are = 0.01 , 0 ⁄ =1000 , =0.1 and =0 . In the paper (Profant et al, 2023) the matched asymptotic analysis has been performed only for mechanically loaded Mode I crack (pure insulating) and for Mode II crack (pure conducting). However, for cracks in electromechanically active materials it is very difficult to completely separate mechanical and electrical modes since they always form a mixture from the point of crack opening modes. Mao and Purohit (Mao & Purohit, 2015) introduced besides Mode I and II also pure electrical Mode D (insulating, zero tractions) and E (conducting, zero tractions) and defined the electric field intensity factor K E as