PSI - Issue 50

D.A. Oshmarin et al. / Procedia Structural Integrity 50 (2023) 212–219 Oshmarin D.A., Iurlova N.A., Sevodina N.V. / Structural Integrity Procedia 00 (2019) 000 – 000

214

3

where 1, 2,3 i j k l  ). The physical relations for viscoelastic part 2 V of the body is written based on the relations of linear hereditary viscoelasticity (Kligman, E.P., Matveenko, V.P., 1997). In the problems of the forced steady-state vibrations this relations are used in the form of complex dynamic moduli: ijkl C is the tensor of elastic constants of material ( , , ,

 B G G iG G i B B iB B i                Re Im Re Im Re 2 , Ge , 1 1 ij ij g b s Re



(3)

,



.

Here , G B are the complex dynamic shear and bulk moduli (in the common case these quantities are the functions of frequency  ); , g b   are the correspondent tangents of the mechanical losses;  is the volumetric strain; , ij ij s e are the components of the deviators of stress and strain tensors. Relations for determining real and imaginary parts of complex moduli are given in (Kligman, E.P., Matveenko, V.P., 1997). For the elements of the piecewise-homogeneous body made of piezoelectric materials governing relations take the following form:

ijkl kl         ijk k ijk ij ki i э E C E  





ij

(4)

D

k

In the coupled problems of the electro-viscoelasticity mechanical and electrical boundary conditions are set in the corresponding form:

:

,

: S u U 

(5)

S

n p 



ij j

i

u

i

i









nel

el

:

0,

:

,

(6)

S

n D dS 

S







3

3

nel

S

3

Here S  and u S are the parts of the surface of the V volume where the stress tensor components

ij  and the

i U are set, j n are the components of the unit normal vector to the S  surface; 3 nel S

displacement vector components is the non-electrode surface, 3 zero-valued electric potential

el S is the electrode surface of the

3 V volume. On a part of electrode surface 3

el S the

0   is set, which means the grounding condition. Thus, the value of electric

potential   set at the rest part of 3 el S surface, determines potential difference. A solution to the forced steady-state vibration problem is sought in the form (8)     0 , e i t u x t u x  

(7)





where ( ) (,,), (,,),(,,),(,,) ux uxxx uxxx uxxx xxx   is the generalized state vector, containing the components of the displacement vector 1 2 3 , , u u u as well as the electric potential  ;  is the external excitation frequency. The numerical realization of the formulated problem is done in the commercial software package ANSYS. Complete mathematical statement of the problem and algorithm of its numerical implementation are described in detail in (Iurlova et al., 2019) 0 1123 2123 3123 1 2 3