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

475 5

ˆ

where '  S g β are the compliance matrix at constant induction, the piezoelectric strain/voltage matrix, and the dielectric impermeability matrix at constant stress, respectively, evaluated under the assumption of the generalized plane strain a short circuit 3 0  = and E 3 = 0 as ' ˆ ˆ , , ' D

ˆ

3 3 i   ˆ ˆ



 

ˆ ˆ g g

ˆ g



ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

3 3 i

3 3 i

j

j

j

' D D

' ˆ S g g , D '

'

' 

 



ˆ = −

,

,

S S

 

= +

=

= −

=

ij

ij

ji

ij

ij

ij

ij

ji

ˆ

ˆ

ˆ













33

33

33

(9)

3 3 D D i j

D

S S

g S

g g

ˆ

ˆ

ˆ

ˆ

3 3

3 3 j

i

j

i

D D

D ji

 



, S g g = ˆ

,

,

S S

 



= −

=

−

= +

=

ij

ij

ij

ij

ij

ij

ji

33 D

33 D

33 D

,

S

S

S

1 1 1 , = S C C e ωeC ω eCe ω g ωeC . , D E E = − E E     − − − − = + T T E − − − 1 1 1 1

for ,

3 i j  , where

The inverse form of the constitutive relations in Eq. (5) reads

ˆ





'

'

T

ˆ C e

    

ps

      σ D

 − ε

T 



E

(10)

,

= 





'

'

E

ˆ e

ˆ −    ω 

−

 

where the in-plane elasticity and piezoelectricity matrices under the assumption of the generalized plane strain and short circuit have the following structure:

        

        

ˆ ˆ ˆ C C C C C C C C C

' ' 11 12 16 E E E E E E ' ' ' ' 12 22 26 ' 16 26 66 ˆ ˆ E E E ' ' ˆ ˆ ˆ ˆ

     

     

     

     

'

'

'

' 11 12     ' ' 21 22 ˆ ˆ ˆ ˆ     '

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

ˆ C

'

'

'

ˆ e

ˆ ω

,

,

,

=

=

=

(11)

E



with

E



ˆ ˆ   e

e

e C



3 3

3 3 i

3 3

i j

j

i j

'

'

ˆ e e ij

ˆ ˆ e e ij

,

,

,

ij e e

= −

= −

= −

ij

ij

ij

E





ˆ 

,

C



33

33

33

 

 

3 3   i ˆ ˆ

3 3 33 E e e C

3 3   i

j

i j

j

' ε

ε ij

ε ij ˆ   ε ij

'ε ˆ   ε ij ij

'ε ˆ ,  ji

ˆ = −

(12)

,

,

 

= −

= +

=

ij





ˆ 

,



33

33

ˆ

' E E E ij ij ij C C C e e e ' ' ' ˆ , = = = = ij ij

.

ij

The thermoelastic constitutive law for the isotropic substrate in plane strain is ( ) ( )( ) ( )( ) 1 0 1 1 2 1 1 2 E E         −   + − + −



     

(

)

    

    

1

T

1          2 =   6    

1 







−

+

(

)

II

1

E

−



)( E



(

)

0

1

,

T

+









−

(13)

) )( + − 1 1 2 1 1 2 0 0    + − (

)

(

2

II

    





6

E

1

 + 



where