PSI - Issue 32

O.N. Belova et al. / Procedia Structural Integrity 32 (2021) 32–41 Author name / Structural Integrity Procedia 00 (2021) 000 – 000

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The algorithm was realized by the use of package scipy.optimize of Python. SciPyoptimizeprovides 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. 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.6. The verification has proved the experimental results. It is well-know that the multi-purpose program Simulia Abaqus allows us to find SIFs and T-stress directly. The experimental and numerical results coincide.

Fig. 6. The finite element simulation: distribution of the von Mises stress and the stress tensor component 22  .

Fig. 7. The finite element simulation: distribution of stress tensor component 11  .

One can compare the reconstructed theoretical solution (1)-(3) with the coefficients m k a given by Table 1. Figure 8 shows the comparison between the theoretical solution (1) with the coefficients computed by the BFGS algorithm and the finite element solution at two different nondimensional distances from the crack tip. The leading order term is shown by the red curve. The two-term series expansion is shown by green curve. Each curve of fig. 8 (left) corresponds to different number of terms of the Williams series expansion. The finite element solution is shown by blue points. It is clear that in order to accurately describe the crack-tip stress field it is necessary to keep higher order terms in the Williams series solution.