Issue 49

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

N UMERICAL IMPLEMENTATION

F

ollowing [21], let us decompose the shells through its thickness into N layers and represent the component of the electric field 3 E for each layer as     ( ) ( ) ( ) 3 i i i k k k E V h , (18)

   ( ) ( ) ( ) 1 i i i k k k h z z

   ( ) ( ) ( ) ( ) ( 2 2) i i i i z h z h is the coordinate measured from

is the thickness of the k -th layer;

where

the middle surface of the shell;  is the difference in the electrostatic potentials between the upper and lower surfaces of the layer, which, as well as the components of the vectors of shell displacements ( ) i u and the velocity potential  , turns to be an unknown variable. The numerical solution of the problem is found using the finite element method (FEM) [33]. To describe the velocity potential ˆ ,  the basic functions n F , and membrane displacements of shells ( ) ( ) ( , ) i i u v we use the Lagrange shape functions with linear approximation, and for bending deflections of shells ( ) i w we use the Hermite non-conformal shape functions The discretization of the fluid and shell regions is made using spatial prismatic (8-node brick) and flat quadrangular (4-node plane) finite elements, respectively. Expressing the unknown variables in terms of their nodal values, we arrive at the following matrix relations    ( ) ( ) ( ) 1 i i i k k k  V

 u N u ( ) i

( ) ( ) i i

 F ε B u E B Φ   ,    ( ) ( ) i i ( ) i ( ) ( ) i i , , ( ) i

,

(19)



e

e

e

e

e

( ) i N are the shape functions for the velocity potential of the fluid and the vectors of the nodal displacements u and e  are the vectors of the nodal values, ( ) i B are the gradient matrices which are determined by Eqn.

where F and

of the shell, ( ) i e





T

( ) i

  ( ) , i k N E E E ( ) i ( ) i 1 ,



,

(14) and relate the strain vectors to the shell displacements,

E

e









T

( ) i

  ( ) 1 diag 1 , ,1 , 1 . i i i k N h h h ( ) ( )



( ) i

  ( ) , i k N V V V ( ) i ( ) i 1 ,

( ) i G in



,

Using the introduced relations the matrix

B

Φ



e

Eqn. (15) is generated in the following way

ˆ ˆ G G

ˆ

ˆ G G ˆ

         

         

( ) i

      ( ) 1 ( ) 2 ˆ 0 i k i k G G

( ) i

N

11

1

( ) 21 i

( ) i

N

2

0

0

( ) i

,

(20)



G

  G G

          ( ) 1 ( ) 2 0 i k i k G G G G

( ) i

( ) i

N

11

1

( ) 21 i

( ) i

N

2

0

0

where

1 2

  ( ) 3 i 

      2 2 ( ) ( ) ( ) 1 3 i i i k k l z e    

  

ˆ



( ) i G h e  ( ) i

( ) i G z 

1, , k N l 



,

,

1, 2.

(21)

 

lk

k l

lk

k

k

From Eqns. (5), (13) and (17) taking into account (18)–(20), we derive with the aid of the standard FEM operations a coupled system of equations, which can be used to describe the interaction of electroelastic shells with an ideal compressible fluid. The system can be represented in a matrix form as

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