PSI - Issue 52

T. Profant et al. / Procedia Structural Integrity 52 (2024) 455–471

458

4

T. Profant et al/ Structural Integrity Procedia 00 (2023) 000 – 000

−1 = − 0 . u u ν n u ν t n σ τ r n n τ r ( ) ( ( ) , = =   = =  =  

(4)

The governing equations admit the following boundary conditions

, ,

on , on

V V

 

u



) ( ) ( )   

)

(



,

, on

− −  +  n τ

= n n n τ t

V



t

(5)

,

on

V

V =



r

,

, on

   = = 





(

)

n

,

, o

n E P

V

+ =





0

D

where is the external unit normal to the smooth boundary . The boundary is composed such a way, that u t r D V V V V V V V   =  =  =  and u t r D V V V V V V    == = . The operator   is defined as ( ) = −   I nn (6) and the symbols u , ν , t , r ,  and  are the boundary values of displacements u , their gradients ν , auxiliary force tractions t , double force tractions r , electric potential  and surface charge ω . The Navier-type equation for the displacement field u and the electric potential  derives from the linearized theory mentioned above. Substituting (1)into the Maxwell equation (2) 2 and into the divergences of (2) 3 -(3) 2 one gets ∇ 2 ( + ∇ ⋅ ) = 0, (7) where f = f 1 + 2 f 2 . Using this relation between the electric potential  and the displacements u , the second Navier type equation can be derived from the rest of the equilibrium and constitutive equations (2) 1 , (2) 3 and (3). This equation can be written in the form ( ) ( ) ( ) ( ) 2 2 2 2 2 1 2 1 1 0, l l    + − + − = u u (8)

where l 1 and l 2 are given by

2

2

2

f

f

f



2 2

2 2

0

2

2

2 , l

.

l

= − l

= − l

+

(9)

(

) (     + + a )

1

a

a



These relations enable to set up non-dimensional parameters (Mao & Purohit, 2015)

f

f

2

(10)

and



=



=

l a

l a





A polar coordinate system with the basis ( e r , e θ ) is practical to use in the asymptotic analysis at the tip of the crack. For this reason, the vectors s 1 and s 2 are introduced to represent the following differential expressions ( ) 1 1 1 2 2 2 2 , , r r r r s s s s       = +  = + s u e e s u e e (11)