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

727

4

  1

  1 F x m D    4 j j j

, 

 1,3,4 j  , where





1 , x b b   .

F x 



(8)

j

The solution of the problem (8) satisfying the required conditions at infinity is obtained by use of Muskhelishvili (1975) in     43 1 1 13 2 44 1 1 14 2 a n H x n H D n H x n H     for 1 ( , ) x b b   , (9)

where

1 1  

1



 4





0 1        , 1 ˆ ( cos ˆ sin )

4 ˆ 2   ,

( )

H x

H





,

,

1    ,

,

ˆ j r    

ˆ

1 j j m r  /

m d

r

j 



0

1 1

2

j

j

j

1 

j

ln



1 b x b x    1    

j

,

.

ln  

 



j 

1

2



For each value 1 x the relation (9) is a transcendental equation with respect to electric flux D . Its solution can be easily found numerically. It is important to note that for real piezoelectric materials the value 1  is very small, therefore cos( )  і sin( )  practically do not change for 1 ( , ) x b b   . This means that 1 1 ( ) H x almost does not change in this interval. Thus the magnitude of the electric displacement D is practically a constant for every a  . For example, Table 1 shows the values of the roots of equation (9) for different 1 ( , ) x b b   at 0 a    and 10 b  mm. The bimaterial PZT-4/ PZT-5 was used. Table 1. The value of the roots of equation (10) for different 1 (0, ) x b  at 0 a    and 10 b  mm. 1 x , mm 0 1 2 3 4 5 6 7 8 9 9.5 This table shows that in most part of the interval (0, ) b the value of D remains almost constant and only near the point b there is a slight deviation in magnitude 0.067%  . This confirms the validity of the assumption regarding the constant value of the electric displacement through the crack and allows of using, say, the middle of the crack 1 0 x  for its finding. 4. Numerical analysis for 3-D case Let us now consider the 3- dimensional case formulated above. We’ll assume 1 3 25 l l   mm, 2 40 l  mm, 5 b  mm. The solution was carried out by using the finite element method. There was little refining of the mesh when approaching the crack, especially to its left and right fronts. The structure of the mesh as well as the distribution of the electric potential in the whole region and at the crack tip are shown in Fig. 2. 3 2 10 , / D C m 5.0709 5.0709 5.0710 5.0710 5.0710 5.0711 5.0712 5.0713 5.0715 5.0710 5.0720