PSI - Issue 39

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

776

7

Fig. 5. Fringe skeletons - fringe thinned images for 95 kg (left) and 100 kg (right).

2

(

)

( ) ( 2 12 σ −

)

2

/ 2

/ (2 ) h

g

Nf

=  −

 +

11 σ σ

(6)

.





22

m

σ

m

m

m

Eq. (6) is a non-linear equation in terms of the unknown parameters m k a . Initial estimates should be made for these unknown parameters and possibly the error will not be zero since the estimates are not accurate (Ramesh (1997)). The estimates are corrected using an iterative process based on Taylor series expansion of m g . One can arrive at the solution of the incremental change by solving a simple matrix problem. Thus, the classical over-deterministic method for the determination of the Williams series expansion is applied. It should be noted that here the over-deterministic method in the form based on the stress field is used. According to numerous computational experiments performed one can conclude that this form relying on the stress field has undeniable advantages compared to the classical over 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 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.