PSI - Issue 59

M. Levchenko et al. / Procedia Structural Integrity 59 (2024) 724–730 M. Levchenko et al. / Structural Integrity Procedia 00 (2019) 000 – 000

726

3

 D x

 







,

(4)

,

,

1 2 x x    ,

x

u x

x

3 1 2 3 1 2 where { } ,  function when crossing the plane 3 0 x  . a

 13 23 33 3 , , , T D     t and the angle brackets mean the jump of the corresponding

Fig. 1. Piezoelectric bimaterial plane with an interfacial crack having finite electrical permeability.

If the size of the parallelepiped in the direction of the axis 2 x is much larger than in other coordinate directions, then in the cross-section 2 0 x  and around it a stress-strain state close to plane strain will be realized. In this case, based on the results of Govorukha et al (2006) for the piezoelectric bimaterial plane, the following representations will be valid (in the following formulas, the coordinate 2 x is temporarily removed):             1 1 1 33 1 4 3 1 1 13 1 ,0 ,0 ,0 j j x m D x іm x        1 1 ( ) j j j F x F x      , (5)





  1 j j F x F x   

  1

  j j n u x i n u x n       1 1 1 j 3 3 1

  1

x  



.

(6)



4





, , jl j m n j l   jl

, 1,3,4

1 3 z x ix   ,

are defined by material constants and have real values for certain   j F z is analytical throughout the cross-section 2 0 x  .

where

classes of piezoceramics. In addition, the function,

3. Analytical solution of the plane problem for the central cross-section Suppose that in the section 2 0 x  the electric displacement is constant along the crack faces, i.e     3 1 3 1 ,0 ,0 D x D x D     for   1 , x b b   .  , b b  much smaller than the cross-sectional size 2 0 x  . Then this section can be considered infinitely large and the conditions on its sides can be interpreted as conditions at infinity. Equations (5) and (7) together with conditions on the interface (4) lead to the following Hilbert-Riemann problem of linear relationship (7) We will also assume that the size of the crack 