Gapped Momentum States and Dispersion Analysis of Mechanical Behavior of Viscoelastic Media

The occurrence of gapped momentum states at different scales, characterized by wavenumber intervals with zero frequency values in the dispersion relation, determines qualitative changes in the momentum transfer mechanism during the interaction of collective modes in non-equilibrium critical systems. To describe the formation of gaps in dispersion curves one needs specialized forms of the dispersion relation. The investigation of dispersion relations with the gap in momentum space can facilitate the establishment of universal viscoelastic properties in condensed matter under specific conditions, where fluids exhibit shear elasticity and solids demonstrate flow behavior. The paper focuses on identification of gapped momentum states in the analysis of dispersion relations obtained using viscoelastic models, specifically the Kelvin-Voigt, Maxwell, standard linear solid, and fractional derivative Kelvin-Voigt models. To derive wave equations corresponding to the presented models, a modification of the wave equation for non-decayed transverse waves in solids was performed to account for viscosity and dissipation. Using the plane wave hypothesis, the general form of the dispersion equations was determined for each model, and analytical (numerical) solutions were obtained. Criteria for a qualitative change in the form of the dispersion relations accompanied by the appearance of a gap in momentum space (k-space) have been formulated. Frequency-wavenumber dispersion curves were constructed for various relaxation and retardation times, considering classical viscoelastic models. The phenomenological significance of fractional models for describing the mechanical behavior of polymeric, composite, and biological systems with a broad spectrum of relaxation mechanisms is highlighted. A numerical solution for the fractional derivative Kelvin-Voigt model was obtained for various values of the fractional derivative order. It is shown that the dispersion equations of the fractional derivative Kelvin-Voigt model and the standard linear solid model transform into the dispersion relations of the Kelvin-Voigt and Maxwell models, respectively, under specific conditions, which indicate the adequacy of the derived relations.