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

481 11

H

=

i

(

) ( ) θ

   

    

( ) θ

δ

−

ˆ η

r



i

)  − 



(

(

)

E

II α T 

i



ˆ C

( ) θ

( )

δ

δ

FEM

'

ps

' ps 

−

−

ˆ η

ˆ e

ˆ η

,

d

d θ S

u

r

T

T

S

kk ε r









−

i

i

(

)

(

)

c i

E

i

1 2 −

ν

δ

−

ˆ η

E

r

−

i

D

D

i

I

II

,

=

(39)

(

)

( )

( ) θ

δ

δ

−

ˆ η

, θ r

η

r



i

i

i

i

(

) , , 3

1,2

i

=

where

ω

1

(

)

( ) FEM u

  

  

T

T



( ) 

( ) 

( )  η t

δ

δ 1

δ

'

FEM

FEM

−

− −

−

ˆ η

,

d

u

λ

r

r

r

r

=

+





(40)

i

i

i

c i

c

i

c i

c

ω

2

with FEM u and FEM t standing for the FEM approximation to the vectors u and t , respectively, and with r c denoting the radius of the circular path remote from the notch singularity. Elements of the vector FEM t along the integrating contour have to be computed from the stresses using the Cauchy formula t i = σ ij n j , in the matrix form written as FEM FEM = t σ n , where FEM σ is the two-dimensional generalized stress tensor and n is the outer normal to the domain enclosed by the circular integrating path of the radius r c defined as

     

     

11     21 FEM FEM FEM FEM FEM = 12 22

( ) ( )  

    

    

cos sin

,

.

σ

n

(41)

=

FEM FEM D D

1

2

For the prediction of the domain switching zone in the piezoelectric material I, the energy-based criterion proposed in Hwang et al. (1995) is applied

(42)

ε i s c E P PE    +  2 , ij ij i

where ε ij  and i P  are the changes in the spontaneous strain and the spontaneous polarization during switching, respectively. P s is the magnitude of the spontaneous polarization (remanent polarization); and E c the coercive electric field. As a first approximation it is assumed that stresses σ ij and electrical fields E i remain unchanged during switching. That means that the linear asymptotic field in Eq.(17) dominates at the notch tip and the loading is thus controlled by the GSIFs H i . This case is referred to as small scale switching when the size of the switching zone is much smaller than other specimen dimensions. Due to electric and/or stress loading the spontaneous polarization of a domain near the notch tip can rotate by 180 ° , +90 ° or − 90 ° . Considering that a ferroelectric domain forms an angle α with x 1 axis the changes in spontaneous strain ε ij  and polarization i P  for 90 ° domain switchings can be expressed in matrix form as (Hwang et al. (1995, Zeng and Rajapakse (2001))

     

   

3



  

  

cos

cos2 cos2 ,     





1  −           11 2  = =           22 12 −    2  s   6   

4

2

  = ε

 = P

s P

(43)

3







 

2sin2

sin





4       