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

9

4

At q = 0 this expression turns into the following

0 ln N

ln ( ) 

D

lim  

.



0



This expression gives the definition of the fractal dimension, D 0 . The physical meaning of the function f ( α ) is that it represents the Hausdorff dimension of some homogeneous fractal subset ζα of the original set ζ characterized by the same cell filling probabilities p i . In this case of uniform distribution of the measure on the set, the spectrum of singularities is a single point on the plane ( α, f ). In the case of non-uniform distribution of the measure, the function f (α) has a more complex (bell-shaped) form. In the absence of multifractality we have: D q = D 0 = α max = α min = f ( α ). Thus the set of different values of the function f (α) (at different values of α ) represents the spectrum of fractal dimensions of homogeneous subsets of ζ α into which the original set ζ can be partitioned. Hence the term multifractal becomes clear. It can be understood as a certain union of different homogeneous fractal subsets of ζ α , each of which has its own value of fractal dimension f ( α ). The third variant of the shock is shown in Fig. 3. The maximum temperatures were shifted relative to the point of impact. The impact led to the formation of delamination or the delamination was prior to it.

Fig. 3. Thermogram and characteristics of the thermal field: (a) thermogram; (b) multifractal spectrum; (c) Minkowski connectivity μ(z) ; (d) distribution of thermal zones n (clusters with average radius r) with similar values of temperatures (strains).

3. Conclusion

Quantitative differences ranging from 10% (coefficient of variation) and up to 355% (kurtosis) in the case of impact on the given thermograms for cases without and in the presence of a defect were revealed. The most informative criteria are peak height (temperature), excess and coefficient of variation. The change in Minkowski connectivity shows the change in homogeneity of material properties. And multi-fractal spectra reflect the properties of the material structure and the presence of defects. These methods of thermogram processing allow to identify defects in the material structure and can be used in non-destructive thermal inspection to assess the nature of loading and the moment of defects occurrence in the material under beyond-design loads. The work was funded by the RNF grant №20-19-00769-P.