PSI - Issue 32

M.O. Levi et al. / Procedia Structural Integrity 32 (2021) 306–312 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

309

4

3.1. Boundary conditions The boundary conditions on the surface assume the presence of mechanical stresses and electric charges 3 1 x h  : ; 2 (1) 23 T q  ; 4 (1) 3 D q  (11) Conditions at the interface between ferroelectric and dielectric assume complete mechanical adhesion and electrical grounding: ; (2) 23 (1) 23 T T  ; (2) 2 (1) 2 u u  0; (1) 4  u (12)

Applying the Fourier transform along the coordinate, we obtain the boundary conditions in the general form:

(1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

       

       

(2) 41 (2) 21 y e y e y e y e y e y e y e y e y e y e l e l e l e l e l e l e l e l e l e l e l e l e l e (1) 04 (1) 04 (1) 34 (1) 24 (1) 03 (1) 03 (1) 43 (1) 23 (1) 02 (1) 02 (1) 42 (1) 22 (1) 01 (1) 01 (1) 41 (1) 21 (2) 1 (1) 04 14 (1) (1) 24 11 (1) 03 13 (1) 11 (1) 02 12 (1) (1) 22 11 (1) 01 11 (1) 11 (1) (1) (1) 23 (1) (1) (1) 21 14 14 13 13 12 12 11 11 0 0

(1) 01

(13)

A

,



(2) 01 (2) 01

(1)

(1) (1) 2 k k 

(1) 44 c y 

(1) (1) 4 k k 

(1) 44

(2)

(2) (2) 2 k k c y   ; (2) 44

(1)

(1)

(1) 2 k l y 

(1) 33   k k ; k l (1) (1) 4

exp(

)

l

y

c

e

h

(14)







;

.

1

lk

k l

k

Here (1) k  are the roots of the characteristic equation for the ferroelectric layer:

( ) 2

α ( ) 2 66   n c

( ) 44

( ) 2 k

α σ e ( ) 2 16  n e

(n) ( ) 2 34 n k

ρ ω n

n n

c

σ

(15)

.

0



2

) α ( 16 n

( ) 2

2

σ (n) ( ) 2 34 n k

σ  n 33 k

e

e

α







11

(2) k  are the roots of the characteristic equation for a half-space:

Also

( ) 2

α ( ) 2 66   n n n c c σ ( ) 44

( ) 2 k

ρ ω n

(16)

.

0



(1) pk y are the solution to the system of constraint equations (Levi G Yu and Igumnov L A (2015)):

Unknowns

   

( ) 2

  α ( ) 2 66 n c

(1) 2 ( ) 44 k

) σ ( ) 2 k

(1) 2 k

α ( ) 2 16 n

σ ( ) 2 (n) 34 n k

(1) 4 k

ρ ω ( n

n n

(e

)

0

c

y

e

y







(17)

2

( ) 2 (n) 34 n k

) α ( 16 n

( ) 2 n

2

(1) 4 k

σ ( e

e

)

σ ( 

)

0

y

y

α











33 k

11

And for half-space:

1 (2) 2  k y .

(18)

The matrix equation for unknown coefficients ( ) n

k c takes the form (M O Levi (2019)):

A C Q   .

(19)