Rheological model of viscoelastic body with memory and differential equations of fractional oscillator

E. N. Ogorodnikov ; V. P. Radchenko ; N. S. Yashagin

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 1 (2011), pp. 255-268

Résumé

One-dimensional generalized rheologic model of viscoelastic body with Riemann-Liouville derivatives is considered. Instead of derivatives of order $\alpha>1$ there are employed in defining relations derivatives of order $0\alpha1$ from integer derivatives. It’s shown, that the differential equation for the deformation with given dependence of the tension from the time with classical initial conditions of Cauchy are reduced to the Volterra integral equations. Some variants of the generalized fractional Voigt’s model are considered. Explicit solutions for corresponding differential equation for the deformation are found out. It’s indicated, that these solutions coincide with the classical ones when the fractional parameter vanishes.

Détail

Keywords: rheological model of viscoelastic body, differential equations with fractional Riemann–Liouville derivatives Mittag–Leffler type special functions.