Issue 49

S.A. Bochkarev et alii, Frattura ed Integrità Strutturale, 49 (2019) 814-830; DOI: 10.3221/IGF-ESIS.49.15 819 N UMERICAL IMPLEMENTATION 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) where    ( ) ( ) ( ) 1 i i i k k k h z z is the thickness of the k -th layer;    ( ) ( ) ( ) ( ) ( 2 2) i i i i z h z h is the coordinate measured from the middle surface of the shell;    ( ) ( ) ( ) 1 i i i k k k V   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      ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , , i i i i i i i i i e e e e e  u N u F ε B u E B Φ   , (19) where F and ( ) i N are the shape functions for the velocity potential of the fluid and the vectors of the nodal displacements of the shell, ( ) i e u and e  are the vectors of the nodal values, ( ) i B are the gradient matrices which are determined by Eqn. (14) and relate the strain vectors to the shell displacements,      T ( ) ( ) ( ) ( ) 1 , , , i i i i e k N E E E E      T ( ) ( ) ( ) ( ) 1 , , , i i i i e k N V V V Φ      ( ) ( ) ( ) ( ) 1 diag 1 , ,1 , 1 . i i i i k N h h h  B Using the introduced relations the matrix ( ) i G in Eqn. (15) is generated in the following way                                        ( ) ( ) ( ) 11 1 1 ( ) ( ) ( ) 21 2 2 ( ) ( ) ( ) ( ) 11 1 1 ( ) ( ) ( ) 21 2 2 ˆ ˆ ˆ ˆ ˆ ˆ 0 0 0 0 0 0 i i i k N i i i k N i i i i k N i i i k N G G G G G G G G G G G G G , (20) where         2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 1 3 1 ˆ , , 1, , 1, 2. 2 i i i i i i i lk k l lk k k l k k G h e G z z e k N l                (21) 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 F