PSI - Issue 32

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

310

5

Here is the load vector of the medium in the space of Fourier images. The coefficients participating in (4) and (5) are found from the solution of the matrix equation (19).The dispersion equation of the medium is written in the form:

(20)

0 det  A .

By solving the matrix equation (20) numerically with respect to  , the phase velocities can be found in the form:

/ e  f s V V

  / .

(21)

4. Computations BaTiO 3 in the aa -phase was chosen as the layer material, the half-space material was MgO. For the convenience of calculations, dimensionless constants were used:

( ) n se n h V     , ( )

( ) ( ) c c c n ij n ij  

(2) 44

( ) ( ) e e k c e n ij n ij  

(2) 44

( ) n    ij

( ) 2 n

(2) 44

(22)

,

,

.

k c e



ij

Here ( ) n se V is the shear wave velocity in the n -th layer of the ferroelectric medium, e k and m k are special multipliers. The linear parameters of the problem are related to the thickness of the upper layer; density is related to the density of the half-space. Material properties are provided in table 1 and table 2 (Shirokov V B et al. (2013), V B Shirokov et al. (2014), V B Shirokov et al. (2015a), V B Shirokov et al. (2015b), V B Shirokov et al. (2015c), V B Shirokov et al. (2016)).

Table 1. Material properties of BaTiO 3 with different misfit strains.

66 c (GPa)

11  (C

33  (C

44 c (GPa)

34 e (C/m

2 )

2 /Nm 2 )

2 /Nm 2 )

 (kg/m 3 )

-9

3.02 ×10 -8 2.04 ×10 -8 7.88 ×10 -9 4.92 ×10 -9 2.83 ×10 -9

BaTiO 3 (1.6) BaTiO 3 (2) BaTiO 3 (4) BaTiO 3 (6) BaTiO 3 (10)

67.12 78.80 94.12 98.01

108.25 108.22 108.22 108.22 108.22

35.24 24.51 10.54

3.72 ×10 3.34 ×10 2.17 ×10 1.58 ×10 1.01 ×10

5800 5800 5800 5800 5800

-9

-9

-9

7.09

-9

101.07

4.5

0.8

Vf/Vs

0.75

0.7

0.65



0.6

0

2

4

6

8

10

Fig. 1. Dimensionless phase velocities for BaTiO 3 (1.6)/MgO.