Dynamics of a Rod Made of Generalized Kelvin–Voigt Visco-elastic Material

In this paper we develop a new mathematical model for the lateral vibration of an axially compressed visco-elastic rod. As the basis for this model we use a fractional derivative type of stress–strain relation. We show that the dynamics of the lateral vibration is governed by two coupled linear differential equations with fractional derivatives. For a special case of the generalized Kelvin–Voigt body, this system is reduced to a single fractional derivative differential equation (Eq. (19)). For a class of problems to which (19) belongs the questions of the existence of a solution and its regularity are analyzed. Both continuous and impulsive loading are treated.