PSI - Issue 37

O.N. Belova et al. / Procedia Structural Integrity 37 (2022) 888–899 Author name / Structural Integrity Procedia 00 (2021) 000 – 000

896

9

deterministic method based on the displacement field (Stepanova (2020), Stepanova (2021)). This advantage is elucidated by the simplicity of the approach because it is not necessary to exclude the displacements of the rigid body from the analysis. Along with the conventional over-deterministic method the Broyden – Fletcher – Goldfarb – Shanno (BFGS) algorithm which is an iterative method for solving nonlinear optimization problems has been used. The BFGS method is regarded as the most popular and efficient quasi-Newton algorithm. The optimization problem is to minimize the function m g . The algorithm was realized by the use of package scipy.optimize of Python. SciPy optimize provides functions for minimizing (or maximizing) objective functions. It includes solvers for nonlinear problems. Thus, we can minimize the function m g directly without Taylor series expansion of m g . The results are given in Tables 1 and 2. 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 two inclined parallel cracks (the experimental photoelasticity method). The crack tip C Coefficients of the Williams series expansion for Mode I loading Coefficients of the Williams series expansion for Mode II loading ( ) 1 1/2 1 4.553 a Pa mm =  ( ) 2 1/2 1 1.604 a Pa mm = 

1 2 1 1 4 1 1 7 1 8 1 1 1 1 1 1 1

2 2 0.00 a Pa =

0.4213

a

Pa

= −

)

)

( ( (

(

2

1/2

1/2

3 0.6532 / a Pa mm =

3 0.1476 / a Pa mm = 2 4 0.1584 / a Pa mm = (

)

(

)

0.3425 /

a

Pa mm

= −

)

) )

( (

3/2

2

3/2

0.0895 / 0.3017 /

5 0.1825 / a Pa mm = 2 6 0.6515 / a Pa mm = 1

a a

Pa mm Pa mm

= −

5

)

)

2

2

= −

6

)

(

(

5/2

2

5/2

0.4288 / 0.31832 /

7 0.23209 / a Pa mm = 3 8 0.1591 / a Pa mm = 2

a a

Pa mm

= −

)

)

(

(

3

Pa mm

= −

) )

)

(

(

2 2 2 2 2 2

7/2

7/2

9 0.1046 / a Pa mm =

0.0542 /

a

Pa mm

= −

9

)

(

(

4

4 10 0.0077 / a Pa mm = 11 0.1137 / a Pa mm = 5 12 0.0078 / a Pa mm = 6 14 0.0032 / a Pa mm = 15 0.0022 / a Pa mm = 2 ( ( ( 13 0.0287 / a Pa mm = −

0.0172 / 0.3389 / 0.0150 /

a a a

Pa mm Pa mm Pa mm

= − = − = −

10 11 12

)

)

( (

9/2

9/2

)

)

5

)

)

(

11/2

11/2

13 0.0419 / a Pa mm =

)

)

( (

( (

6

0.0058 / 0.0321 /

a a

Pa mm Pa mm

= − = −

14 15

)

)

13/2

13/2

Table 2. Coefficients of the Williams series expansion for the plate with two inclined parallel cracks (the experimental photoelasticity method). The crack tip D Coefficients of the Williams series expansion for Mode I loading Coefficients of the Williams series expansion for Mode II loading ( ) 1 1/2 1 5.357 a Pa mm =  ( ) 2 1/2 1 2.399 a Pa mm = 

1 2 1 1 4

2 2 0.00 a Pa =

0.7623

a

Pa

= −

)

)

(

(

1/2

2

1/2

3 0.1956 / a Pa mm = 2 4 0.1822 / a Pa mm = (

3 0.6744 / a Pa mm =

)

(

)

0.3640 /

a

Pa mm

= −