Issue 49

S.A. Bochkarev et alii, Frattura ed Integrità Strutturale, 49 (2019) 814-830; DOI: 10.3221/IGF-ESIS.49.15





  ( ) i

f 

p

,

(4)



t

where f  is the fluid density and the sign of the right-hand parts of Eqns. (3)–(4) depends on the direction of the vectors of the normal to the external surfaces of the shells. Eqn. (1) together with the boundary conditions (2)–(3) are converted by the Bubnov – Galerkin method to a weak form [26]

 2 2 ˆ 1 c t  2

(1)

(2)

 w ˆ

ˆ w







ˆ d









   

 V F

 S F

 

F V F

d 0, S

1, n m

d

d

,

(5)

n

n

n

n

f



t

t

(1)

(2)

V

V

S

S

f

f





where  ˆ and ( ) ˆ i w are the approximations of the velocity potential and the normal components of the vectors of shell displacements; n F and f m are the basis functions and their number, respectively. In the general case, the behavior of the electroelastic body is described by the equations of state, piezoelectric effect, and the electrostatic relations [29, 30]

T cε e E ,

  

(6)

  D eε dE ,

(7)

   0 D ,

(8)

  

.

(9)

E

Here: , ε  are the stress and linear strain tensors; E and D are the vectors of electric field intensity and electrical induction; c , e , d are the matrices of elastic constants, piezoelectric and dielectric coefficients;  is the electrostatic potential. In the case when a thin-walled structure with a radially polarized field is in the plane stress state, Eqns. (6)–(9) can be simplified. In particular, of all components the vectors of the electric field strength and nonzero electrical induction retain only 3 E and 3 D , and Eqns. (6) and (7) are written as follows [31, 32]:

                          11 11 22 22 12 12 c        

  e e

    c c c c

0 0 0 0

             3 0 E  

0 0

    

    

31

11 12



0 ,

,

(10)

 c

32

21 22 0 0

c

0 0 0



66

    

           3 0 0 D e     

0 0 0 0 0 0

0 0 0 0 0 0 0 0

              33 3 d E  

11

      22  

  

  



,

(11)

  e

0

0 0

    12

31 32

where



2

  lm lm l m c c c  c



  e

  d d e c



c

c

c

e

e c

c

, l m

,

,

,

,

1, 2

(12)





3 m m

33 3 33 m

3 3 33

66 66

3

33

33 33 33

Here we proceed from the assumption that the shell surface is covered with thin weightless electrodes, which can either be shorted, which corresponds to the boundary condition  3 0 E (short circuited) or disconnected  3 ( 0, D open circuited). From Eqns. (6)–(9) the following integral relation can be obtained [21]

817