PSI - Issue 51

S. Zhao et al. / Procedia Structural Integrity 51 (2023) 69–75 S. Zhao et al / Structural Integrity Procedia 00 (2022) 000–000

73

5

where

2 a



 



  , 3 q

,

(19)

1  (1 )

cos

sin

sin

cos

2

5 a  

11 q iq

d

d

i d

d

  

 

1 



1 

 

1 



1 



12

1

2

1

2

  1 1 ln 2 a    .

,

1  ,

1 1 1 2    

2

1 1    

d

d

 



1

2

1



1 



  i 



 

 

, 4 q

.

(20)

cos               sin cos sin

2

2 2

5 a  

21 q iq

a

 



22

1

1

1

1

1

1 1

1

1 

The phonon and phason stresses and displacements at the crack tip a can be found from the systems (13), (14) and (15), (16), respectively. 3. The energy release rate (ERR) According to the crack closure integral (Rybicki, Kanninen, 1977), the energy release rate (ERR) at a crack tip a can be presented in the form



a l 

1

   1 33 1

  x u x l 3 1 ,0



   1 13 1

  x u x l 1 1 ,0



l   



0 lim  l

G







 

 

 

a

2

a



   1 33 1 H x W x l   3 1 ,0





 ,0   

(1) D x 3 1

( ,0)

x l

dx

.

(21)



 



1

1

The last term in the integrand is equal to zero because of zero electric potential jump in the crack region. Substituting the above mentioned phonon and phason asymptotic expressions into (21) and performing the integration we arrive to the following formula for the ERR:       2 2 1 1 2 3 5 1 1 4 8cosh a G hh a h          , (22)

where

1

15 15 m n n 

55 55 m n n 

, 2 11 22 12 21 h q q q q   ,

55 15 p m m   , 2

13 55 15 53 p n n n n   .

, 1

h

h







53

13

1

3

p p

p p

m n

1 2

11 11

1 2

4. The numerical illustration and discussion Numerical analysis was performed for the bi-material composed of the piezoelectric QCs with the characteristics given bellow (Zhang et al., 2016). The upper material: elastic constants (GPa): 11 150 c  , 12 100 c  , 13 90 c  , 33 130 c  , 44 50 c  , 1 0.18 K  , 2 0.3 K  , 1 1.50 R  , 2 1.20 R  , 3 1.20 R  ; piezoelectric constants ( 2 Cm  ): 31 15 0.160 e e    , 33 0.347 e  , 15 0.138 e  , 33 0.350 e   ; dielectric constants ( 9 2 -1 -2 10 C N m  ): 11 0.0826   , 33 0.0903   . (22)