PSI - Issue 32

D.A. Oshmarin et al. / Procedia Structural Integrity 32 (2021) 158–165 Oshmarin D.A., Iurlova N.A., Sevodina N.V., Iurlov M.A. / Structural Integrity Procedia 00 (2021) 000 – 000

160

3

2 ,

,    B

s Ge 

ij

ij

(2)

1 3





,

s

e

 

 

ij       ij  

ij

ij

ij

ij

for the viscoelastic part of the volume 2 V

t

 

 



   ij     e d

0 

2 s G e H t    

,

0

ij

ij

(3)

t

 

  

 F t     0 B  

       d

0 

for the piezoelectric part of the volume 3 V

ij       ijk ij C D k

ijkl kl       ijk k ki i э E E

(4)

Here, , G B are the elastic shear and bulk moduli,

0 0 , G B are the instantaneous shear and bulk moduli, , H F are

/ 3 jj    is the mean stress,  is the volumetric strain, p

ijkl C is the tensor of elastic

the relaxation kernels,

ijk  and ki э are the tensors of piezoelectric and dielectric coefficients; (

constants of a piezoelectric element, , , , 1, 2, 3 i j k l  ); ij  is the Kronecker delta.

In the case of natural vibrations, under the assumption that the system vibrations occur with slowly varying amplitudes and the initial perturbations do not affect the behavior of the system (Kligman and Matveenko, 1997), the constitutive relations (3) for a viscoelastic material can be written as

Re Im Re B G G G iG G i G B B B iB B i                   Im Re 2 ,  ; 1 1 ij ij s Ge  Re

  





1 ; 

Re G i

(5)

 

Im

g

Re

  





1 . 

Re B i

 

Im

 

b

B

Re

Here , G B   are the complex dynamic shear modulus and bulk modulus of elasticity, which in the general case are the functions of frequency  ; , g b   are the corresponding mechanical loss tangents. The values of the real and imaginary parts of complex moduli Re Re Im Im , , , G B G B are determined according to the relations given in (Kligman and Matveenko, 1997). The boundary conditions for the electro-viscoelasticity problem under consideration can be divided into two types: mechanical and electrical. The mechanical boundary conditions are given as

(6)

:

0,

: S u

0

S

ij j n







u

i



where S  is the surface with specified stresses ij  , u S is the surface with specified displacement vector i u , j n is the projection of the normal unit vector to the surface S  . The corresponding boundary conditions for the electric domain can be written as follows:   3 3 3 : 0, : 0, nel nel el S S n D dS S      (7)