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

478

8

     

     

i   −

e

0

0 0

r



(

) 1  

i − −

*

i   −

2 e ir  

(24)

sin

e

Z

r

=



i   −

0

0

e

r

Considering traction and charge free notch faces the following boundary conditions are imposed by: ( ) ( ) I II 1 2 0, 0.   = = T T

(25)

The displacement and traction continuity conditions are prescribed along the interface

0  = as

( ) 0

( )

( ) 0 T T = I

( )

I

II

II

0 ,

0 ,

u

u

=

(26)

The eigenvalue problem for a bi-material notch composed of a piezoelectric material and an insulator is defined in terms of the equations (17) and (21). A bi-material notch with the geometry in Fig. 1 is considered, where material I is the piezoelectric one and material II non-piezoelectric one defined by elastic constants E ij C and permittivities ij   . Let us define the following identities:

*

*

I

I

II

II

* = A A,L L,A AL L u u T T, u u T T Z Z , Z Z * * , , , . I I II II I II     = = = = = = = = =

,

(27)

,

The eigenvalue problem is introduced by the boundary conditions (25) and (26) and can be written in the matrix form as

.                     

          

          

I I

I

I

Lv

0 0

1 0 0 X X

1

I I

II X X

II

Lw L v

2

2

 =

0,

(28)

II II

I 0

I 0

II

II

0 B B B B I I I −

0

 − − 

II II

I L w

where

( ) ( )  j

( ) ( )  j L

1

−

1

−

(

)





, L X LZ =

,

1,2

= X LZ

j

=

j

j

j

j

(29)

1

1

−

−

i B AL B AL , i = =−

0

0

and 0 denotes 3 × 3 zero matrix on the left-hand side and 12 × 1 zero vector on the right-hand side of Eq.(28). After some algebraic manipulations one receives the characteristic equation for the singularity exponent values. Definitions for the asymptotic stresses and electric displacements of the piezoelectric monoclinic material and of the isotropic non-piezoelectric material, respectively, have the form: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 1 2 2 2 2 3 1 2 1 1 1 1 1 1 1 1 1, 2 2, 3 3, 1 1 1 2 1 1, 2 2, 3 3, , , , ,               − − − − − − =− − − = + + x x x x x x r H r H r H r r H r H r H r σ λ λ λ σ λ λ λ (30)