PSI - Issue 42

Jan Sladek et al. / Procedia Structural Integrity 42 (2022) 1584–1590 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

1586

3

(

)

1 (4) Derivation of governing equations for 2D problems in the direct flexoelectricity can be found in works (Sladek et al., 2017, Tian et al. 2020): , , ( ) ( ) 0 ij j ijk jk   − = x x , , ( ) 0 i i D = x . (5) In boundary conditions there are occurred normal derivatives of displacement, traction vector of higher-order of stresses, electric charge 2 ijkl jk il ij kl ik jl f f f       = + + .

: s u

, j i j n u =   = n , / i

: i k j ijk R n n  = , :

i i Q n D = ,

(6)

i

and the traction vector ( ) , i j ij ijk k t n   = − −

 

i 

( ) ( c    + 

c

)

− x x x

,

(7)

j

i

x

c

j

with

: i k j ijk n    = ,

(8)

and the jump at a corner on the oriented boundary contour  is defined as , and i n and i  are the Cartesian components of the unit normal and tangent vector on boundary, respectively. 3. Finite element methods and results The principle of virtual work can be applied to get the weak-form of governing equations (5). The variation of the deformation energy has to equal to the variation of external forces (Sladek et al. 2017, Tian et al. 2020) ( ) , , , ij i j ijk i jk i i i i i i V t R Q u u D dV t u d R s d Q d            + + = + +      . (9) The mixed FEM with independent C 0 continuous interpolation for both elastic displacements and strains is developed here. The displacement vector and electric potential in each element are approximated by ( ) 1 2 [ , ] u u   = u N q ( ) 1 2 [ , ]      = N q , (10) ( ) : c x ( ) (  ) c c i  i  i = − - 0 +0 x x

where u q and  q are vectors of nodal displacements and electric potential, respectively. Above expressions can be utilized for approximation of strains and electric intensity vector 11 1 1 22 2 1 2 2 12 2 1 0 0 [ ( , )] 2 u u u u                 = =    =                 ε B q ,

1 2 =          2       E E 1

1 2 [ ( , )]    B

− = − = E

 q .

(11)

In the mixed FEM, we apply also an independent approximation of strains ( ) 1 2 ˆ ( ) [ ] , In   = ε x α p ,

(12)

where [  ] are unknown coefficients, and (

) 1 2 ,   p is the polynomial function vector

)  =



(

1 2 T   . Since two independent approximations of strains given by (11) and (12) have to be equal at selected internal points 1 2 ( , ) c c c   = ξ , one can derive the final approximation formula 1 2 1 2     , 1 p