PSI - Issue 80

Miroslav Hrstka et al. / Procedia Structural Integrity 80 (2026) 471–492 M. Hrstka et al./ Structural Integrity Procedia 00 (2025) 000 – 000

474

4

       

        

1 2 3       E E E       = 

       

       





34 35 0 0 0 0

11 12 13 e e e e e e 21 22 23 0 0 0

e e

D D D

0 0

11 12     12 22 0 0  

    =  

16

1 2 3

e

E

D

=

ε ω

= 

,

,

,

26

   

0

e e





33

The structure of the matrices S D , g and  β is similar to the structure of their inverse counterparts and is not stated here for the sake of brevity. The directional properties of the matrices depend on the poling axis, which can attain two limit configurations, either it coincides with x 1 -axis or with x 2 -axis. Between these states their structure corresponds to the above mentioned monoclinic one. The material constants in the general poling direction are obtained by using transformation relations ( ) 1 1 1 1 , , T D D   − − − − = = = * * * S K S K g Ωg K β Ωβ Ω . (4) σ β are material coefficient matrices with the poling direction aligned with certain principal direction (e.g. in which the properties were measured). The form of the transformation matrices K and Ω are given in Appendix A. External loads are assumed to be parallel to the plane defined by x 3 = 0 and plane strain deformations characterized by linear piezoelectricity are considered. These assumptions allow to decouple the in-plane and anti-plane relations and to solve the in-plane and anti-plane problem separately. In the following only the in-plane problem is considered. The g -type constitutive relations for the in-plane problem can be expressed as (Hrstka (2019), Hrstka et al. (2019), Hwu and Ikeda (2008)) where D * S , * g and *

' ˆ S g D ˆ



   

'

T

  − ε

   

ps

    σ

T 



(5)

,

= 

  

' σ ˆ β −   D

'

E

ˆ g

−

  

ps  for isotropic thermal expansion α (see Table1) are

where the components of the effective plane strain CTE

T

   

    

D

23 D 33 D

36 D 33 D

S S

S S

S S

(6)

ps

13 1 , 1 , −

α = − 



−

I

33 D

and

x x     −      −     

1             11   2 22         6   12  

1   

11   

2   E E   = =

1     2   D D

2 22             6 12 2        

1       1

(7)

,

,

,

σ

ε

E

D

= =

= =

=

2

are the stress vector, the strain vector, the electric field vector, the electric potential, and the electric displacement vector, respectively,

' ' 11 12 16 D D D D D D S S S S S S S S S ' ' ' ' 12 22 26 ' 16 26 66 ˆ ˆ D D D ' ' ˆ ˆ ˆ ˆ ˆ ˆ ˆ

     

     

' '   11 12     ' '   12 22 ˆ ˆ ˆ ˆ



   

'

'

'

   

   

11 12 16 g g g g g g ' ' ' 21 22 26 ˆ ˆ ˆ ˆ ˆ ˆ

ˆ S

ˆ β

'

'

'

ˆ g

,

,

,

=

=

= 

(8)

D



 