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

482 12

where γ s denotes the spontaneous strain associated with 90 ° domain switching and 3 4 clockwise and counterclockwise 90 ° switching, respectively. Note that for 180 ° . A variation of the local stress/electric fields near the vertex induced by domain switching leads to the additional GSIFs ∆ H i which are evaluated again using Betti’s reciprocal principle using the computational procedure described in details in Hrstka et al. (2025). The computational procedure solves the boundary value problem with the prescribed spontaneous strain and polarization within the switching domain using FEM and the computed electroelastic field is employed in the Betti’s reciprocal principle . The new local GSIFs tip i H are then calculated modifying Eq.(39) as   in Eq. (43) correspond to 0 ε ij  = and   sin T 2 cos s P    =− P

tip

H

=

i

(

) ( ) θ

 

    

( ) θ

δ

−

ˆ η

r



i





(

)

(

) ( ˆ ' e

)

i

ˆ C



( ) θ

δ

FEM

'

ps

ps

−

ˆ η

,

d

u

 −  P

r

T   +

T

S











−

i

(

)

c i

E

δ

−

ˆ η

E

r

− 

i

D

i

(44)

I

=

−

(

)

( )

( ) θ

δ

δ

−

ˆ η

, θ r

η

r



i

i

i

i

(

)

α

T

E





( )

δ

−

ˆ η

I

I

d θ S

kk ε r

i

(

)

i

1 2 −

ν

D

1,2,3,

,

i

−

=

II

(

)

( ) θ

( ) θ

δ

δ

−

ˆ η

,

η

r

r



i

i

i

i

where  and  P are non-zero only within the switching zone. For a formulation of fracture mechanics criteria are the GSIFs H i rather impractical because they have an awkward physical unit due to imaginary part of the singularity exponent and contain a mixture of mechanical and electrical loading. Introducing a reference length according to the Rice concept (Rice, 1988), modified GSIFs with standard physical unit can be defined as ( ) Im , 1,2,3, i i i i H H l i  = = (45) where ( ) Im i  stands for the imaginary part of i  . Following the procedure suggested by Hwu and Ikeda (2008) and applied in Hrstka et al. (2025), the conventional stress intensity factors K I , K II , K IV can then be introduced as where the stress intensity factors K I , K II , K IV are real numbers with the units Pa √m for modes I, II and C √m for the electric mode IV and ( ) ( ) ( ) 1 1 2 3 , 1 1 1 0 0 0 x d d d dx dx dx   =       λ λ λ Λ 3. Numerical results Material parameters of the piezoelectric ceramic PZT-5H and the isotropic insulator SiO2 are listed in Tab. 1. First, a singularity exponents and electro-elastic field reconstruction of the piezoelectric/non-piezoelectric bi-material notch is calculated. Let the angle 2 = − 180 ∘ be fixed. 1 1           , Λ 2 3 2 , II I K     =   = K   x IV   K H H H  k (46)