Wave Propagation in Viscoelastic Rods, Fractional Operator Models

Synonyms

Damping features of which are modelled by fractional operators; Wave propagation in viscoelastic rods

Definitions

The propagation of harmonic waves in viscoelastic rods of finite length and stress waves in viscoelastic semi-infinite rods have been considered for the cases, when the viscoelastic features of rods are described by ten rheological models involving several fractional operators of different orders.

Backgrounds

It is known (Bland, 1960) that the viscoelastic models containing derivatives of integer order may describe both diffusion and wave phenomena occurring in solids. Thus, the wave processes may occur in the material whose behavior is described by the equation

$$\displaystyle \begin{aligned} \sigma +\sum^n_{i=1}\tau_\varepsilon^i \;\frac{d^i \sigma}{dt^i}=E_0\left(\varepsilon+\sum^n_{i=1} \tau_\sigma^i \;\frac{d^i\varepsilon}{dt^i} \right), \end{aligned} $$

(1)

where σ is the stress; ε is the strain; t is the time; τ_ε and τ_σare the relaxation and retardation...