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

71 3

The constitutive relations referred to the Cartesian coordinate 

 1 2 3 , , x x x with 



1 2 ,0, x x coincident with the

periodic plane and 3 x -axis identical to the quasi-periodic direction have the form

0 0

0 0

0 0

13 c c c c

e e

R R

11            33 13        

  

  

    

    

    

11 13

11

31

1

1     3   E E

31       33 W W

33      

  

  

,

(1)





33

33

2

0 0 2

0

0

c

e

R

13   

44

15

3

11                        33 13    11 33 13

0 0 2

0 

0

1 D       

0 e

E e 

31        33 W

0 

  

  

1 3            E

15

11

15

,

(2)



0

3    31 33 D e e

33 e W 

33

0 0 2

0

0

31       33 H H

R

K

31 W e 

1 E      3  

  

  

  

  

33    W       

15

3

2

.

(3)





0

0

0

33 e E 

R R

K

1

2

1

The equilibrium equations and geometric equations are the following

11,1 0     , 31,1 13,3

33,3 0     ,

0 D D   ,

31,1 33,3 0 H H   ,

(4)

1,1

3,3

,

  ,

3 i i W W  , 3,

(5)

1 2 (    ij

)

, i j j i u u ,

E

,

i

i

where , 1, 3 i j  and the denotation “,” represents the derivative operation for the space variables; i u , 3 W and  are the phonon displacements, phason displacement, and electric potential, respectively; the atom arrangement is periodic in the 1 2 x x  plane and quasi-periodic in the 3 x -axis; ij  and ij  are the phonon stresses and strains, respectively; 3 i H and 3 i W are the phason stresses and strains, respectively; i D and i E are the electric displacements and electric fields, respectively; the polarization direction is along the 3 x -axis; ij c and i K are the elastic constants in the phonon and phason fields, respectively; i R represent the phonon–phason coupling elastic constants; jk e and jk e  are the piezoelectric constants in the phonon and phason fields, respectively; ii  are the permittivity constants. It is assumed that the crack is electrically permeable and uniformly distributed phonon   ,     and phason H  stresses as well as electrical displacement D  are prescribed at infinity. The interface conditions can be written in the following form

1 a x a    ,

13   , 0

33   , 0

33   , 0

0   ,

0 D  for

(6)

H





3

3 0 D  for 1 ( , ) x a a   ,

13 0   ,

33 0   ,

0  ,

0 u  ,

0 u  ,

0 W  ,

0   ,

(7)

H

33

1

3

3

where f means the jump of the function f over the material interface. In Loboda et al. (2020) the following presentations were obtained

,

(8)

(1)

(1) m H x 5 33 1 j

(1) im x  1 13 1 j

( ,0) x

( ,0)

( ,0)

1 ( ) x      j j j 

1 ( ) x









33 1







,

(9)

1 1 1 j nux inux inWx   3 3 1 5 3 1 ( ) ( ) ( ) j j

1 ( ) x   j 

1 ( ) x



j