Issue 49

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

          (1) (1) (2) u               (2)     u    

                     (1) (1) (2) (2) u     

       

 

(1)

       

0 0 0 0

(1)

u C

0 0 0 0

M

sf

s

       

0 0 0 0 0

0 0 0 0 0

(2)



  

0 0 0 0

(2)

0 0

0 0

C

M

sf

s

0 0 0 0 0

0 0 0 0 0 0 0 0 0 M

(1)

(2)



0

0 0

C C

f

fs

fs

(22)

         

  

 K K (1)

(1)

0

0 0

                      (1) (1) (2) (2) u u   



s

  s  K K T (1)

(1)

0

0 0







0,

 K K (2)

(2)

0

0

0



s

s

  (1) s  K K T

(2)

0 0

0 0

0





 

0

0

K

f

where ( ) , i u  and ( ) i  are the generalized vectors of the shell displacements and rotation angles, the perturbation velocity potential of the fluid and the difference in the electrostatic potentials of the shells. The typical stiffness ( K ), mass ( M ) and damping ( C ) matrices for individual finite elements are generated as follows:

T                   F F F F F F T T x x y y z z  

1







( ) i

T ( ) i

T



d , V M F F C 

 

d , V

d , S

K

F N

f

f

fs

w

2

c

( ) i

V

V

S

f

f



  ( ) i B D B  T

  ( ) i ( ) i d , V N N C T

  ( ) i w

T







( ) i

( ) ( ) i i

( ) i

( ) i

( ) i





s 



f 

d , V

d , S

K

M

N F

s

s

sf

(23)

( ) i

( ) i

( ) i

V

V

S



s

s

  ( ) i

  ( ) i   B H B T

T



( ) i

( ) ( ) i i B G B K d , V

i

( ) ( ) i i

(

)





d , V

K









s

( ) i

( ) i

V

V

s

s





  ( ) 33 i 

  ( ) 33 i    ( ) i h d , , , , k

  ( ) 33 i N k N 

( ) i

( ) i h d 1

( ) i h d



diag

.

H

1

Here ( ) i w N are the shape functions for the normal component of the nodal displacement vector of the shells. Since some matrices in (23) are written in the Cartesian coordinates   , , x y z , the derivation of the solution in the global coordinates, which have been used to specify the problem geometry and to write the constitutive equations for the fluid, involves the transformation of the displacements and rotation angles using the matrix of the directional cosines γ of size  3 3 in the following way:

0 

0 

 

  

( ) i

( ) i

.

(24)

 

u

u

During the numerical implementation, at each node of the finite element of the shell it is necessary to determine six unknowns (three displacements and three angles of rotation). Therefore, the transformation of conventional matrices is performed using the diagonal matrix            diag , , , , , , L :              T T T ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) T ( ) ( ) ( ) ( ) ( ) ( ) , , , , . i i i i i i i i i i i s s s s sf sf i i i i i i s s fs fs   K L K L M L M L C L C K L K C C L (25)

Expressing the quantities associated with the electrostatic potential in terms of the shell displacements

820