PSI - Issue 80

Xinpeng Tian et al. / Procedia Structural Integrity 80 (2026) 451–461 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

453

3

normal derivative of displacement

i s

2. Basic equations for the direct flexoelectricity The direct flexoelectricity in dielectric materials with the centrosymmetric properties (no piezoelectricity) is considered. Microstructural effects are considered via strain gradients in the gradient theory of elasticity. Then, constitutive equations in the direct flexoelectricity for Cauchy stresses ij  , higher-order stresses ijk  and electric displacement i D are given as (Hu and Shen 2009)

ij ijkl kl c   = ,

ijkl i jklmni mni f E g 

,



=− +

jkl

i ij j ijkl jkl D a E f  = + , (1) where a ( a ij ) and c ( c ijkl ) denote the permittivity and elastic coefficients, respectively. The symbols f ( f ijkl ) and g ( g ijklmn ) are used for the direct flexoelectric coefficients and the higher-order elastic coefficients, respectively. The strain tensor ij  , electric intensity vector j E and strain-gradients ijk  are expressed via primary fields ( ) , , , / 2, ij i j j i j j u u E   = + =− , (2)

(

)

/ 2

, i jk j ik u u ,

  = = +

.

(3)

, ij k

ijk

where i u are displacements, and the electric potential is denoted by  . To reduce number of material coefficients in the higher-order elastic parameters parameter f ijkl , there are assumed some simplifications (Deng et al. 2017) 2 jklmni jkmn li g l c  = ,

jklmni g and in the direct flexoelectric

(4)

with li  being the Kronecker delta and ( ) 1 2 ijkl jk il ij kl ik jl f f f       = + + ,

(5) where the additional material coefficients are reduced to one internal length material parameter l (micro-stiffness length-scale parameter) and two independent parameters 1 f and 2 f . The governing equations for the coupled electro-elastic problem under static boundary conditions are given as (Sladek et al. 2017)

, ij j ijk jk   − x , ( )

( ) 0 = x

,

, ( ) 0 i i D = x .

(6)

The mechanical governing equation (6) can be also written as , ˆ ( ) 0 ij j  = x , where , ˆ ( ) ( ) ( ) ij ij

ijk k    = − x x x is the total or effective stress tensor.

3. Finite element method formulation A quadratic polynomial approximation is considered in the 3D brick element for displacement and electric potential. Due to high order derivatives in the governing equations (6) it is insufficient to apply only standard C 0 continuity approximation for primary fields, since the gradients of strains on interfaces of elements are discontinuous. Therefore, we apply the mixed FEM formulation with independent approximation of displacements and strains (Hu et al. 2025). The constraint (kinematic equation (2)) between the mechanical strains and displacement is satisfied by a collocation method at selected internal (Gaussian) points (see Fig. 1).