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

480 10

To determine generalized stress intensity factors in Eqs. (30) and (31), a conservative line integral for the bi material notch developed from Betti’s reciprocal principle is applied. However, because of thermal stresses developed due to a uniform temperature change ∆ T , this line integral referred to as the  -integral is no longer path independent and can be written for a bimaterial notch characterized by angles ω 1 and ω 2, see Fig.1, as

( ) ˆ i u

  

 







) ( = 

)

(

)

ˆ C

C 



(

ˆ ˆ u t u t T T −

ˆ ˆ u t u t T T −

'

ps

' ps 

, u u ˆ

ˆ e

d

d

ds

s

T

T

S



=

−

−







( ) ˆ i  −    u E

i

i

i

i

i

E

(36)

C

D

ε

I

E

II α T 

) ( ) * ˆ kk i ε u



d

S

−

(

2

1

ν

−

D

I

I

where C is an arbitrary remote contour circumventing the crack tip and connecting the traction free crack faces, whereas C ε is a path shrinking to the crack tip and D I , D II are domains encircled by the contour C in the Material I and II, respectively. The vectors ˆ T i u and ˆ i t are the auxiliary solutions to the displacements, tractions, electric potential and the charge and correspond to the exponent ˆ i i   =− . The auxiliary solutions are defined as ( ) ( ) ( ) ( ) ( ) ( ) 1 ' , , , 1 , , 1,2,3 ˆ ˆ ˆ ˆ ˆ i i i i i i i r r r r r i r         − − − =  =− =− =  u η T t λ (37) where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) δ δ ' ' ' ˆ ˆ ˆ , , ˆ ˆ ˆ i i i i i i i i i i         − − − − = + = + η AZ v AZ w λ L Z v L Z w where (.)’ denotes the differentiation with respect to θ . Observe that the vectors u and t in Eq. (36) represent either the regular asymptotic or a full field solution obtained numerically e.g. by FEM. In the first case, the vector u is given by Eq. (17) 1 for the piezoelectric material, or by Eq. (21) 1 for the isotropic substrate, the vector t is given by the derivative with respect to θ of Eq.(17) 2 for the piezoelectric material or by the derivative of Eq. (21) 2 for the isotropic substrate, similarly as ˆ i t in Eq.(37) 2 . The full-field solution reduces to the asymptotic solution if the integration contour shrinks to the notch tip. Since the regular and corresponding auxiliary solutions are orthogonal with respect to the “scalar product“ defined by the integral on the left-hand side of Eq.(36) i.e.

)

(

const 0 for 

, ,

i j i j = 

 = 

( )   η δ j r

( ) 

δ

−

ˆ η

,

r

(38)

i

j

i

0 for



an important result for the GSIFs evaluation follows as