PSI - Issue 65

Akhmetkhanov R. S. et al. / Procedia Structural Integrity 65 (2024) 1–5 Akhmetkhanov R.S. / Structural Integrity Procedia 00 (2024) 000–000

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with a certain strain level. The graph of distribution of average radius of strain zones with the same strain level shows the type of distribution. They change in the presence of defects. 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 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 (some degree of inhomogeneity of the set)

( ) N q N p i i    1

ln

1

D

q

0, 1, 2, ..., , n

lim

,





q

0 1 ln

  

q



where ε is the measure of object coverage, p i is the measure of i - th coverage. At q = 0 this expression turns into the following

0 ln ( ) lim . ln N    

D



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 ). Structure thus the set of different values of the function f ( α ) (at different values of α ) represent 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 ). In the presence of a defect the multi-fractal spectrum transforms from a mono-fractal spectrum to a multi-fractal spectrum. The structure of the thermal field (deformation) becomes complex.

Fig. 4. Characteristics of thermogram in the defect zone: (a) multifractal spectrum; (b) Minkowski connectivity μ(z); (c) size distribution of zones r with similar temperature level.