PSI - Issue 40

O.N. Belova et al. / Procedia Structural Integrity 40 (2022) 46–60 O.N. Belova, L.V. Stepanova / Structural Integrity Procedia 00 (2022) 000 – 000

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3. Procedure of over-deterministic method and the Broyden – Fletcher – Goldfarb – Shanno algorithm The stress optic law relates the fringe order and the in-plane principal stresses as

1 2 / Nf h     

(4)

where f  is the material stress fringe value, N is the number of generated fringes (or fringe order) and h is the thickness of the specimen. For a plane stress problem, the stress components are related to the principal stresses as     2 2 1 2 11 22 11 22 12 , / 2 / 4             . (5) Substituting Eq. (5) in (4) one can define an error function for m th data point       2 2 2 11 22 12 / 2 / (2 ) m m m m g Nf h             . (6) 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. 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. 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 results of finite element simulations are shown in fig.11. 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. Eq. (6) is a non-linear equation in terms of the unknown parameters m