PSI - Issue 40

13

O.N. Belova, L.V. Stepanova / Structural Integrity Procedia 00 (2022) 000 – 000

O.N. Belova et al. / Procedia Structural Integrity 40 (2022) 46–60

58

Table 2. Coefficients of the Williams series expansion for the plate with two inclined parallel cracks (the experimental photoelasticity method). The crack tip B

Coefficients of the Williams series expansion for Mode I loading

Coefficients of the Williams series expansion for Mode II loading









2

1/2

1

1/2

1 2.410 a Pa mm  

1 5.358 a Pa mm  

1 2 1 1 4 1

2 2 0.00 a Pa 

0.7640

a

Pa

 



 





2

1/2

1/2

3 0.1976 / a Pa mm  2 4 0.1828 / a Pa mm 

3 0.6755 / a Pa mm 







0.3645 /

a

Pa mm

 









2

3/2

3/2

0.1043 /

a

Pa mm

 

5 0.2103 / a Pa mm  2 6 0.7615 / a Pa mm  1

5







 

2

2

0.3501 /

a

Pa mm

 

6



 



2

5/2

1 7

5/2

7 0.2425 / a Pa mm  3 8 0.1845 / a Pa mm  2

0.4986 / 0.3356 /

a a

Pa mm

 





 

1 8

3

Pa mm

 







2

7/2

1

7/2

0.075 /

a

Pa mm

 

9 0.1101 / a Pa mm 

9









2 4 10 0.0094 / a kg mm  11 0.1295 / a Pa mm  5 12 0.0109 / a Pa mm   2 2

1 10 1 11 1 12

4

0.0215 / 0.3921 / 0.0218 /

a a

kg mm

 



 



9/2

9/2

Pa mm Pa mm

 







5

a

 





 



2 13

11/2

1

11/2

0.0483 /

a

Pa mm

 

13 0.0567 / a Pa mm 





 

1 14 1 15

6

2 6 14 0.0061 / a Pa mm  15 0.0048 / a Pa mm  2

0.0087 / 0.0629 /

a

Pa mm

 







13/2

13/2

a

Pa mm

 

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. 4. Conclusions

In this research the complete experimental and numerical procedures for determination of the crack-tip fields in new type of cracked specimen are realized. The digital image processing of the experimental isochromatic and isoclinic data obtained in the framework of the photoelasticity method is performed. In this study higher order coefficients of the multi-parameter Williams series expansion for the stress and displacement fields in the vicinity of crack tips in the rectangular plate with two inclined parallel cracks are obtained by the use of the digital photoelasticity method and finite element analysis. The higher-order terms in the Williams asymptotic expansion are considered. It allows us to have more accurate estimation of stress, strain and displacement fields and to extend the domain of validity for the Williams series expansion. The program is specially developed for the interpretation and processing of experimental data from the photoelasticity measurement experiments. The developed tool allows us to find points that belong to isochromatic fringes with minimal light intensity. The analysis of the experimental data uses a Java application programmed for the advanced determination of the fracture characteristics: coefficients of the Williams series expansion for the stress field in the vicinity of the crack tip. The tool allows us to collect experimental points from the photoelasticity tests on the cracked specimens. An automatic routine implemented as a Java application permits to determine the values of coefficients of higher order terms of the WE that describe crack-