PSI - Issue 39

O.N. Belova et al. / Procedia Structural Integrity 39 (2022) 770–785 Author name / Structural Integrity Procedia 00 (2021) 000–000

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Table 1. Coefficients of the Williams series expansion for the plate with two cracks. The crack tip A, B, C, D.

Coefficients of the Williams series expansion, crack tip А

Coefficients of the Williams series expansion, crack tip B

Coefficients of the Williams series expansion, crack tip C

Coefficients of the Williams series expansion, crack tip D

1 222.9869 1 a = 2 9.1228 1 a = − 1 25.8818 2 a = − 2 6.9421 1 0.9840 3 a = − 2 0.3305 3 a = − 1 0.1290 4 a = 2 0.0049 4 a = − 1 0.0393 2 a = 1 0.0172 6 a = − 2 0.012 6 a = − 1 0.0011 7 a = 2 0.0003 5 a = 5 a = 2 0.0010

1 229.6520 1 a = 2 16.6262 1 28.3924 2 a = − 2 7.8099 1 a =

1 178.5776 1 a = 2 76.4100 1 17.1055 2 a = − 2 5.6921 1 2.6836 3 a = − 2 0.0888 3 a = − 1 0.2619 4 a = 2 0.0938 4 a = − 1 0.0231 5 a = − 1 a = − 2 a = 1 0.0050 6 a = 2 0.0008 1 0.0008 7 a = 2 0.0002 1 0.0004 8 a = − 2 0.00004 6 a = − 7 a = − 2 0.0123 5 a =

1 145.0405 1 a = 2 88.9855 1 5.9816 2 a = − 2 3.1423 1 4.8492 3 a = − 2 0.0820 3 a = − 1 1.0561 1 a = − 2 a = 1 0.1003 5 a = − 2 0.0427 5 a = − 1 0.0018 6 a = − 2 0.0115 1 0.0043 7 a = 2 0.0008 1 0.0028 8 a = − 2 0.0002 8 a = − 6 a = 7 a = 4 a = 4 a = 2 0.1256

2 a = 3 a =

1 0.6703

2 0.4055

3 a = −

1 0.1777 4 a = 2 0.0362

4 a = −

1 0.0973

5 a = 5 a =

2 0.0134

1 0.0305 6 a = 2 0.0067 1 0.0011 7 a = − 2 0.0012 1 0.0012 8 a = − 2 0.0001 8 a = − 6 a = − 7 a =

7 a = 8 a =

1 0.0003

2 0.00007

8 a = −

8 a =

Here the values of the higher-order coefficients of the Williams series expansion are not given since they coincide with the coefficients obtained by the digital photoelasticity method. Thus, one can conclude that the coefficients are determined by both the approaches with good accuracy. In all crack tips there is good agreement between experimental and numerical results. The percentage error ranged from zero to five percent. Therefore, one can conclude that the experimental approach used here is reliable in determination of mixed mode parameters for plane problems. Angular distributions of the stress tensor components in the vicinity of the crack tips A, B, C and D generated with the theoretical solution (red line) and by finite element solution (black points) are shown in figs. 6-9. The theoretical solution is constructed with the help of truncated Williams series expansion: 1) the first row shows the four-term asymptotic solution; 2) the second row shows the six-term asymptotic solution; 3) the third row shows the eight-term asymptotic solution; 4) the fourth row shows the ten-term asymptotic solution5) the fifth row shows the twenty-term asymptotic solution. One can see that the four-term asymptotic expansion don’t provide accurate description of the stress field. Figs. 6-19 clearly show that the twenty-term asymptotic expansion gives the angular distributions that coincide with the finite element solution.