PSI - Issue 42

4

Author name / Structural Integrity Procedia 00 (2019) 000 – 000

Jan Sladek et al. / Procedia Structural Integrity 42 (2022) 1584–1590

1587

1 2 ( , )   = ε p Lq ˆ In

(13)

u

where

( ) ( c c u     B . Then, the derivatives of strains can be obtained from (13) ( ) ( ) ( ) 1 1 1 2 * 1 2 2 1 2 2 ˆ , ˆ , , ˆ Ïn In u Ïn            = = =            ε p α η p Lq p α ε . ) 1 1 2 c c 1 2 , , − = L p

(14)

Substituting above approximations into variational form (9) and taking into account arbitrariness of the variations   u  q and     q , we obtain the system of algebraic equations for unknown nodal quantities

  [ ( )] [ ][ ( )] T u u u V  B ξ C B ξ q

dV +

(

)

  

  s

  u

2

T

[ ][ ( )] G B ξ q F B ξ q [ ] [ ( )] T +

T N T

[ ( )] B ξ

[ ] u

B R

l

dV

d

d

+

=

 +



,

(15)













V





t

R





(

)   u

  

T

T

T

[ ( )] [ ] [ ( )] [ ][ ( )] + B ξ (16) where C , G , Π and F represent elastic, higher-order elastic, dielectric and flexoexoelectric matrix coefficients, respectively. A straight interface crack between two dissimilar dielectric materials is solved numerically (Fig. 1). A stationary boundary conditions with a pure tension load 7 0 1 10 ( ) p Pa =  are applied on the top and bottom surfaces. Following geometry is considered: 8 8 125 10 , 10 10 m cr w m l − − =  =  , 8 20 10 h m − =  , 8 10 10 cr h m − =  . Impermeable electric boundary conditions are considered on the crack surfaces with a reference value of electric potential at the crack tip, 0  = . All other surfaces have vanishing electric charge 0 i i n D = . The higher-order traction 0 i R = is also vanishing on all surfaces. [ ][ ( )] P B ξ F B ξ q Π B ξ q [ ]  N u V Q dV Qd     − =    ,

Fig. 1. A symmetric part of the cracked strip under a uniform axial tension

,

0 11

1 0 13.9*10

0 12

1 0 7.4*10

c

Pa

c

Pa

A standard dielectric ceramics (SDC) with following parameters:

,

=

=

,

0 22

1 0 11.5*10

0 44

1 0 2.56*10

c

Pa

c

Pa

, a

1 =15.1*10

-9 C(VM) -1 , a

2 =13.0*10

-9 C(VM) -1 is considered for domain I

=

=

(upper layer). Three various materials are considered in the lower layer (domain II) to investigate influence of dissimilar elastic properties on behaviour of the interface crack: