PSI - Issue 80

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

455

5

1 2 3 ( , , ) e     A is the shape function matrix for the independent approximation of ε In :

where

0 0 0 0 0

P

        

         

0

0 0 0 0

P

0 0

0 0 0

P

(

) 1 2 3 , ,

e    

A

= 

,

(11)

0 0 0

0 0

P

0 0 0 0

0

P

0 0 0 0 0

P

6 120 

with





2 2 2 2 1 2 3 12 23 13 1 2 3 12 12 23 23 13 13 123 123 123 123120 1, , , , , , , , , , , , , , , , , , ,         = P . (12) The coincidence of two independent approximations (8) and (10) at selected Gaussian points 1 2 3 ( , , ) c c c c    = ξ , is leading to get unknown coefficients  1 ( ) ( ) . e e u u  − = G G α A ξ B ξ q (13) After substituting the equation (13) into (10), one gets the final expression G L ξ A ξ B ξ The governing equations (6) are replaced by the weak-form of the boundary-value problem to reduce order of derivatives. The corresponding weak formulation is given by (Sladek et al. 2017) ( ) , , , t R Q ij i j ijk i jk i i i i i i V u u D dV t u d R s d Q d            + + = + +      . (15) where the higher order traction vector and traction vector are defined as : i k j ijk R n n  = , (16) ( ) , ( ) ( ) c c i i j ij ijk k i c t n       = − − + −   x x x π , (17) : i k j ijk n    = , and the normal derivative of displacement as : / ( / ) i i i j j s u u x n =  =   n in which i n and i  are the Cartesian components of the unit normal and tangent vector on boundary, respectively. The electric flux is defined as : i i Q nD = . Substituting approximations (7) and (14) into the weak form (15), we obtain the system of algebraic equations for nodal values of displacements and electric potential. 3. The far-field expression of the J integral for 3D crack problems Recently, the authors have derived the J-integral for a 2D crack problem described by the continuum theory of flexoelectricity (Sladek et al. 2017, Tian et al. 2019). The J -integral is defined there as 2 2 2 2 2 2 2 2 1 2 3 ( , , ) ( ) . e u     G L ξ q | e In e V = ε A (14) where 1 ( ) ( ) ( ). u G  − = G

 

  

u x

s x

 

 





U

i − −

i R n D d −

J

1 n t

,

(18)

=



 

i

i

j

j

x







1

1

1