Resonance structure and inelastic strain and defect localization in loaded media

The paper provides a review of basic concepts of nonlinear dynamics and certain results relevant to the physical mechanisms of strain localization and increase in localization scales in loaded media. It is shown that the mechanisms of the processes can be associated with stable resonances occurring in loaded solids. A loaded solid is considered as a system of many interacting atoms perturbed by an external force with a continuous frequency spectrum. The question put in the work is what frequencies from this continuous spectrum are selected by the perturbed nonlinear system of interacting atoms, i.e., what frequencies are found to be most stable. The structure of resonances is analyzed using Hamilton's approach and Kolmogorov-Arnold-Moser theory. It is demonstrated that nonlinear dynamics gives results that fully explain the experimentally found regularity of increasing localization scales and fits it in the universal principle of fractal divisibility of solids and media. According to this principle, the minimum scale is the lattice parameter of a loaded solid and each subsequent scale is the sum of the two previous scales. The results obtained by Kolmogorov-Arnold-Moser theory show that invariant tori with an irrational angular coefficient to ω = (1, (√5−1)/2) are most stable, and this gives the experimental regularity Ln+1 = Ln/Ln−1, where Ln ∼ 1/ωn, and Ln+l/Ln ≈ (√5+1)/2 for the corresponding wavelengths (scales).