PSI - Issue 80

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

454

4

The displacements and electric potential on each element are approximated in terms of their nodal values and shape functions

u v w

    =   =

(

) 1 2 3 , , , e u    q

u

N

u

e

V

   

e

V

(

) 1 2 3 , , , e     q

= N



(7)



e

V

e  q are vectors of nodal displacement and electric potential, respectively, and

e u q and

1 2 3 ( , , ) u    N and

where

1 2 3 ( , , )     N denote the shape function matrices.

(a) (b) Fig. 1. The brick 20 node element in the local coordinates ( ξ 1 , ξ 2 , ξ 3 ) and (b) the 3 × 3 × 3 Gaussian quadrature points in this element One can easily get strain and electric intensity vector within V e from primary field approximations (7)

0 0

11        22           33 =  

1 

        

0

0



u v w           

2

0 0



(

)

3

(8)

e

e

1 2 3    , ,

,

B

q

=

=



    

u

u

e

23       13   2 2 2 1  2

0

 

V

3

2

0



1 

3

0

 

e   

2

1

e

V

V

1     2       3 E E E

1   

2       3     

(

)

1 2 3 , ,   

e

e q . 

E

B

=

=−

=−

(9)



Ve

Ve

Ve

The additional independent approximations of strains by quadratic polynomials, and unknown constants e α :

11         22 In In    33 In

(

)

(10)

In

e

e

1 2 3 , ,   

,

A

α

=

23 In =           13 12 2 2 2 In In   





e

V

Ve