PSI - Issue 52

T. Profant et al. / Procedia Structural Integrity 52 (2024) 455–471 T. Profant et al / Structural Integrity Procedia 00 (2023) 000 – 000

467 13

    

     

(

)   2

2 5 4 7 6 

l 

 − − −

(

)( ) 4 I K

2

2

(

)

& I E

2

2

2

(46)

11 3 2 C C C − 12

12 8 8 2 + − − − +

2 2    

J

=

( 16 1

)

1

−



1

−



Observe that for the mixed modes flexoelectricity reduces the energy release rate in comparison with the SGE crack. 4. Numerical results and discussion We will employ the presented matched asymptotic analysis to estimate the contributions of direct flexoelectric effects and strain gradient effects for various combinations of flexoelectric material properties to the expected reduction of the energy release rate. The classical Griffith criterion for the crack to advance is J -integral expressions in (27) and (46) depend on flexoelectric parameters and the near tip parameters C ij , which, however, are related via matched asymptotic analysis to the classical far field stress intensity factors (SIF), cf. Eqs. (30). Thus, it is possible to examine an influence of the flexoelectricity through values of far field SIF. We start with a dependence of the J – integral & 1 I E J on the non-dimensional electric field intensity factor E I K E K for several values of the non-dimensional flexoelectric parameter α and β =0, see Fig. 5. The factor E is included for dimensional reasons. Also J – integral & 1 I E J is normalized with respect to its counterparts from the classical elasticity 2 I K E . It should be noted that there is a good qualitative agreement with results obtained using FEM in (Sladek et al, 2017a). 1 c J G = (47)

Fig. 5. 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 α . The remaining parameters are 0 ⁄ =1000 , 0 ⁄ = 0.75, = 0.3, and =0 . Fig. 6 shows the variation of the J – integral & 1 I E J with the non-dimensional electric field intensity factor for several values of the non-dimensional flexoelectric parameter β and α =0.