PSI - Issue 65

Akhmetkhanov R. S. et al. / Procedia Structural Integrity 65 (2024) 6–10 Akhmetkhanov R. S. / Structural Integrity Procedia 00 (2024) 000–000

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Figure 1a shows a variant of impact without defect occurrence. In Fig.1b there is a delamination defect. In this case, the thermal field is not symmetrical with respect to the impact point. The table shows statistical data of thermal fields in the case without material defect and in the presence of defect. The data were calculated in gray scale of the thermogram image.

Table. Statistical data of thermograms at point impact.

Thermograms

Asymmetry

Excess 1.77545 5.32403

Peak height

Coefficient of variation, %

Fig 1a Fig 1b

-1.76655 -2.30518

0.062893 0.041840

34.29 30.79

Figure 2 shows images of the multifractal spectrum, Minkowski connectivity, and the distribution of thermal zones.

Fig.2. Thermogram and characteristics of the thermal field: (a) multifractal spectrum; (b) Minkowski connectivity μ(z) ; (c) distribution of thermal zones n (clusters with average radius r ) with similar values of temperatures (deformations).

Multifractals are inhomogeneous fractal objects, for a complete description of which, unlike ordinary fractals, it is not enough to introduce only one quantity, its fractal dimension D , but it requires a whole spectrum of such dimensions, the number of which, generally speaking, is infinite. The multifractality of a process is usually represented by a multifractal spectrum (singularity spectrum) f (α). Multifractal spectra are characterized by spectrum width S , asymmetry A and curvature. An increase in the width of the spectrum corresponds to a non uniformity measure. Multifractal spectra are used in the works of the following authors Poyarkova E.V. (2016), Pavlov A.N., Anishchenko V.S.(2007), Schroeder M.(2001) and Akhmetkhanov R.S.(2023). In the theory of fractals, to reveal the structure and features of the limit set, it was proposed to use a set of dimensions D q ( q = 0,1,2,..., n ) characterizing the statistical structure (i.e., some degree of inhomogeneity of the set) ( ) ln 1 1 lim , 0, 1, 2, ..., , 0 1 ln N q N p i i D q n q q         

Where ε is the measure of object coverage, p i is the measure of i -th coverage.