Analysis of Rheological Equations Involving More Than One Fractional Parameters by the Use of the Simplest Mechanical Systems Based on These Equations

The behaviour of rheological models containing more than onefractional derivative or fractional operator of fractional orders areinvestigated. All rheological models discussed can be separated intothree groups depending on magnitudes of the valueα*/β* (whereα* and β* are the orders ofsenior fractional derivatives of stress and strain, respectively): themodels are thermodynamically admissible only whenα*/β* = 1 (the first group),thermodynamically compatible only forα*/β* ≤ 1 (the secondgroup) and, finally, thermodynamically well-conditioned both atα*/β* ≤ 1 andα*/β* > 1 (the third group).It is shown that, under nonstationary excitations, thebehaviour of the simplest mechanical systems (mechanical oscillators,finite and semi-infinite viscoelastic rods), based on the consideredrheological models, may be different (from the point of view ofthermodynamics) from that of the underlying rheological models. Thus,under impulse excitations, the mechanical models based on rheologicalmodels of the first and second groups become thermodynamicallyadmissible not only atα*/β* = 1 but alsowhen α*/β* < 1(mechanical models of group I), but mechanical models based onrheological models of the third group remain thermodynamicallywell-conditioned at the same magnitudes of rheological parameters as thecorresponding rheological models do (mechanical models of group II). Asthis takes place, group I mechanical models possess diffusion-wavefeatures, that is atα*/β*=1 the stress waves ina semi-infinite rod propagate at a finite speed, and the roots ofcharacteristic equations (for nonstationary vibrations of a mechanicaloscillator or a rod of finite length) as functions of the relaxation orretardation times, behave in a way similar to the characteristicequation roots of rheological models possessing instantaneous elasticity(models of the Maxwell type). Whenα*/β*<1, the stress wavesin a semi-infinite rod propagate instantaneously at infinitely largespeeds, and the roots of characteristic equations (under nonstationaryvibrations of a mechanical oscillator or a rod of finite length) asfunctions of relaxation times behave in a way similar to thecharacteristic equation roots of rheological models lackinginstantaneous elasticity (models of the Kelvin–Voigt type).Mechanical models from group II possess pure wave or pure diffusionfeatures at all magnitudes ofα*/β*.