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

729

6

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

0 -0,005

0

0,005

III

-500

I, II

-1000

ІІІ

І,ІІ

-1500

-2000

-0,005

0

0,005

Fig. 3. Crack opening (a) and electric potential (b) jump through the crack for different cross-sections of the body.

Fig. 4. Potentials jumps for different cross-sections of the body.

Crack opening at the middle point of the middle cross-section is equal to

6 3.07 10 m   , and the potential jump at

this point is equal to

3 1.74 10 В   . Calculations according to formula (2) give the value of

3 D equal to

2 0.005009 / . C m Table 2 shows the values 3 D in three cross-sections. As expected, for the middle cross-section the agreement of the obtained electric flux with the specified one is very good (the error is 0.18%), for the cross section distant from of the front end by ¼ the length of the parallelepiped 3 2 l , this error is 0.74%. For the cross sections located in close proximity to the end, this error grows and, for example, at a distance 3 /10 l from the end it is 11.8%.

Table 2. The value of the electric flux trough the crack for different cross-sections. Coordinate 2 x of the cross-section 0(center) 2 / 2 l 2 9 /10 l 3 D 0.005009 0.004963 0.004411

Since the results in Table 2 do not agree well enough with the specified values of the electric flux through the crack, the specified values were refined using the iteration method. For this, the faces of the crack were divided into strips of width 2 2 / l n , and different values of the electric flux were set on each of them. Further, the calculation was carried out according to the finite-element algorithm indicated above, and the obtained results were compared with the given ones. In case of their unsatisfactory agreement, the next iteration was carried out in a similar way, etc. Table 3 shows the result of calculations carried out at 8 n  .

8 n  .

Table 3. The value of the electric flux for different cross-sections at

Coordinate 2 x of the cross-section

2 /10 l

2 3 /10 l

2 / 2 l 4.98

2 7 /10 l

2 9 /10 l

1000 3 D 1000 3 ˆ D

5.0

4.99

4.97

4.45

5.015

5.014

5.013

4.939

4.401

3 D specified in the corresponding section of the upper

Table 3 represents the value of the electric displacement

edge of the crack, and 3 ˆ D is the value of the electric displacement obtained as a result of the finite-element solution and the application of the formula (2). Since the difference between these values for all cross-sections does not exceed 1%, the values 3 ˆ D can be considered as the required values of the electric flux through the crack. The results in Table 3 are given for the midpoints of each cross-section. But, as was confirmed in the analysis of the plane problem (Table 1), the distribution along the crack for each section is almost constant. In the 3