PSI - Issue 28

D.A. Oshmarin et al. / Procedia Structural Integrity 28 (2020) 1438–1448 Author name / Structural Integrity Procedia 00 (2019) 000–000

1440

3

are the components of the symmetric Cauchy stress tensor, ij  and i u are the components of the linear strain tensor and displacement vector, respectively; 1 2 3 , ,    are the specific densities of elastic, viscoelastic and piezoelectric parts,  is the electrical potential; , , L R C       are the potential difference on the inductive, resistive and

capacitive elements, respectively;  is the variation of the corresponding variable. For the electric field, the potentiality condition is fulfilled and can be written as:

(2)

i E

  

, i

The constitutive relations for each of the considered parts of piecewise homogeneous body can take the following form: - for the elastic isotropic part of the volume 1 V ;

2 , Ge

,    B

s



ij

ij

(3)

1 3





,

s

e

 

ij       ij  

 

ij

ij

ij

ij

- the mechanical behavior of the viscoelastic part of the volume 2 V is described by the model of linear hereditary viscoelasticity with complex dynamic moduli (Kligman, Matveenko, 1997). This model is well-fitted to describing the mechanical behavior of a wide class of structural materials, including composite materials

2 , Ge 

;    B 

s



ij

ij

Re Im Re G G G iG G i G B B B iB B i               Im Re 1 1 Re

  





(4)

1  

;

Re G i



Im

g

Re

  





1 . 

Re B i

 

Im

 

b

B

Re

- for the piezoelectric part of the volume 3 V .

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





ij

(5)

D

k

Here, , G B are the elastic shear and bulk moduli, / 3 jj    is the mean stress,  is the volumetric strain, p ijkl C is the tensor of elastic constants of a piezoelectric element, ijk  and ki e are the tensors of piezoelectric and dielectric coefficients; ( , , , 1, 2, 3 i j k l  ); ij  is the Kronecker delta. , 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). For the coupled electro-viscoelastic problem under study, the boundary conditions can be divided into two types: mechanical and electrical. The mechanical boundary conditions are written by analogy with problems in the theory of elasticity and viscoelasticity: : 0, : 0 ij j u i S n S u     (6)