PSI - Issue 32

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

307

2

Simulation of the dynamics of structures with parts in different phases of a ferroelectric is still limited. This paper investigates the effect of misfit deformation on the phase velocities of media. The results obtained can help in the creation of new devices and optimize the characteristics of old ones. 2. Constitutive relations of the medium dynamics Consider the oscillations of the n - layer of aferroelectric medium in the exchangeless approximation described by the equations of motion and quasi-static Maxwell's equations (W. Nowacki 1986).

2 2 ( ) n u



( ) n

( ) n

T

(1)

,







t



0 ( )   n D .

(2)

The equations of motion (1), (2) are supplemented by the material relations of the medium in the vector form (Levi M.O. et al. (2017a), Levi M.O. et al. (2017b)):

( ) ( ) n n

( ) n

( ) n

( ) ( ) n n

   

   

   

   

   

   

D T

c

e

E S



(3)





.

( )

( ) n

T n

e

ε

Here ( ) n T and ( ) n S are the components of the stress and strain tensors of the second order in a simplified form using the Voigt notation, ( ) n D is the electric induction vector, ( ) n E is the vector of the electric field voltage, ( ) ( ) ( ) , , n n n ε c e are the elastic, piezoelectric and dielectric coefficients, respectively, ( ) T n e – transposed piezoelectric coefficient matrix,  is the density of the medium. 3. Boundary value problem The ferroelectric 3 1 1 ; x h x    medium is a two-layer package consisting of an upper ferroelectriclayer ( n = 1) with a thickness and an underlying dielectric half-space 0 3  x ( n =2). The layer ( n =1) is assumed to be made of a material with a rhombic system, the half-space (n = 2) is assumed to be made of a material with a hexagonal system. Oscillations in the medium are initiated by an oscillating load i t x e x t    ( ) ( , ) 0 1 1 q q distributed over the area x a  1 . (   2 4 , q q  q , 2 q is the component of the mechanical load vector along, 4 q is the electrical load). Outside the area x a  1 , the surface is free from mechanical stress. We will look for the equations of motion for the upper layer in the form ( 2,4)  m :    4 1 1 (2) (1) (1) 3 (1) m ) exp( ( , ) k k mk k h y c x U   , (4)

For a semi-bounded part of the medium:

4

1   k

(2) (2) mk k

(2)

(2)

( , ) x 

exp(

);

U

y c

h

(5)





3

0

m

k

Material relations for the medium in the aa – phase take the form: