Analysis of the Viscoelastic Rod Dynamics via Models Involving Fractional Derivatives or Operators of Two different Orders

Abstract

We present an overview of articles devoted to the analysis of dynamic behavior of viscoelastic rods, whose features are described by rheological models involving fractional derivatives or operators of two different orders. We investigate ten rheological models: the generalized viscoelastic Kelvin-Voigt, Maxwell, Zener models and other models containing fractional derivatives or fractional operators of two different fractional orders. For each of the above-mentioned rheological models, we carry out a comparative analysis between the behavior of its rheological and dynamic characteristics. Vector diagrams are used as the rheological characteristics, but the roots of the characteristic equations in the problems of longitudinal vibrations of rods, the velocities and coefficients of attenuation in the problems of propagation of harmonic waves, and the stress function in the problems of transient wave propagation are considered as the dynamic characteristics. The analysis of the rheological and dynamic characteristics shows that, depending on the relative magnitudes of the orders of the fractional derivatives and fractional operators entering into the rheological model, some of the models considered may describe both the wave and diffusion processes occurring in mechanical systems, but others describe only wave or only diffusion phenomena. The parallels between the behavior of the vector diagrams, the roots of the characteristic equations, the velocities and coefficients of attenuation, and the stress functions are revealed. We also show the analogy in the behavior of viscoelastic rods, whose rheological features are described by the models with fractional time derivatives and ordinary time derivatives.