PSI - Issue 39

O.N. Belova et al. / Procedia Structural Integrity 39 (2022) 761–769 Author name / Structural Integrity Procedia 00 (2021) 000–000

767

7

optimize provides functions for minimizing (or maximizing) objective functions. It includes solvers for nonlinear problems either. The results are given in Table 1. Having obtained the coefficients of the Williams series expansion for the stress and displacement fields experimentally one can compare the results with the numerical ones to verify the accuracy of experimentally measured coefficients. For comparison a series of finite element calculations for the same type of the cracked specimen has been performed. The verification has proved the experimental results. It is well know that the finite-element software package Simulia Abaqus allows us to find SIFs and T-stress directly. The experimental and numerical results coincide. Table 1. Coefficients of the Williams series expansion for the plate with the crack tilted at 60 o (the experimental holographic interferometry method). Coefficients of the Williams series expansion for Mode I loading Coefficients of the Williams series expansion for Mode II loading ( ) 1 1/2 1 17.66971 a Pa m = ⋅ ( ) 2 1/2 1 30.60484 a Pa mm = − ⋅

1 2 1

2 2 0.00 a Pa =

12.5000

a

Pa

= −

)

)

(

(

1/2

2 2

1/2

3 4.42140 / a Pa m = 1 4 0.00004 / a Pa m = (

7.65810 /

a

Pa m

= −

3

)

(

1 4 0.00115 / a Pa m = 5 0.95812 / a Pa m = 2

)

)

)

(

(

3/2

1 5 1

3/2

0.55317 /

a

Pa m

= −

)

)

(

(

2

2

2 6 0.00012 / a Pa m = 7 0.13841 / a Pa m = 1

0.00034 / 0.23209 /

a a

Pa m Pa m

= − = −

6

)

)

(

(

2

5/2

5/2

7

)

)

(

(

2 3 8 0.00001 / a Pa m = 9 0.07498 / a Pa m = 4 10 0.0007 / a Pa m = ( 5 12 0.00078 / a Pa m = 13 0.00985 / a Pa m = 6 14 0.00032 / a Pa m = ( ) ( 11 0.02621 / a Pa m = −

1 8 1 9 1 1 1 1 1 1

3

0.00001 / 0.043294 / 0.00005 / Pa m ( Pa m Pa m

a a a a a a

= − = −

)

)

2 2 2 2 2 2 2

7/2

7/2

) ) )

( ( (

4

= −

10

)

)

(

9/2

9/2

11 0.01516 / a Pa m =

)

( ( (

5

0.00453 / 0.00569 / 0.00058 / (

Pa m Pa m Pa m

= − = − = −

12 13 14

)

) )

11/2

11/2

)

(

6

)

(

13/2

13/2

15 0.00223 / a Pa m =

0.0022 /

a

Pa m

= −

15

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. Discussion and conclusions In this research the comprehensive experimental and numerical procedures for the evaluation of the crack-tip fields in a plate the horizontal and inclined central crack are realized. The digital image processing of the experimental data obtained in the framework of the holographic interferometry method is performed. In this study higher order coefficients of the multi-point Williams series expansion for the stress and displacement fields in the vicinity of crack tips in the rectangular plate with the central crack are obtained by the use of the digital holographic interferometry method and finite element analysis. The higher-order terms in the Williams asymptotic expansion are taken into account. It allows us to have more accurate estimation of stress, strain and displacement fields and to extend the zone 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