PSI - Issue 52

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

468 14

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

Fig. 6. The & -integral as a function of the normalized ratio of the electric field intensity factor and the classical far field SIF for several values of the parameter . The remaining parameters are 0 ⁄ =1000 , 0 ⁄ = 0.75, = 0.3, and =0 . Both figures indicate that for larger K E we need to supply more energy in order to achieve the critical energy release rate G c . Next, we will quantify the effect of flexoelectricity on the critical load for crack growth compared to the case where the material obeys only the gradient theory of elasticity. Special case K E = 0 is considered for this purpose. The classical Griffith criterion for the crack to advance (47) is applied and the critical energy release rate G c is considered constant, i.e. it does not depend on flexoelectric material properties. Remind that the J - integral od SGE is given by

l 

) 2

  

2 8 .  12

(

)(

I

5 4 3 2  − −

J

C C C +

=

 

( 16 1

)

1

11

12

−



(48)

Hence, from the criterion (47) it follows

& I E crit I crit J J 1, 1,

1

=

(49) Observe that the matched asymptotic analysis gives the expressions for C ij in Eq. (30), which linearly depend on the far field SIF. Hence, one can write 11 = ̃ 11 , C 12 =K I C̃ 12 , where ̃ 11 , ̃ 12 stand for the rest of the expressions in (30). Denote by the SIF in the flexoelectric case and by the SIF, when material is governed only by SGE. Also note that for K E = 0 it follows from (45)

1 2 2 2

 −  −    

 

C

  −

12

4 I

,

K

=

1 2 2 2 − −



2

1

−





(50)

which means that 4 can also be put in the form 4 = ̃ 4 . Eq. (49) can be rewritten as ( ) ( ) ( )( ) ( ) ( ) 2 2 2 2 2 2 11 12 12 4 2 2 2 11 1 2 1 , 2 , 2 5 4 7 6 3 2 8 8 2 2 2 1 5 4 3 , 2 8 1 f I cr I it s I crit C C C K C K C C K          − − − − + − − − + − − − +     =           

(51)