Discrete Spectral Modelling of Continuous Structures With Fractional Derivative Viscoelastic Behaviour

Fractional derivative rheological models were recognised to be very effective in describing the viscoelastic behaviour of materials, especially of polymers, and when applied to dynamic problems the resulting equations of motion, after a fractional state-space expansion, can still be studied in terms of modal analysis. But the growth in matrix dimensions carried by this expansion is in general so fast to make the calculations too cumbersome, especially for finite element applications. In this paper a discretization technique for continuous structures is presented, adopting the Rayleigh-Ritz method, aimed to reduce the computational effort. The solution of the equation of motion is approximated on the basis of a linear combination of of shape-functions selected among the analytical eigenfunctions of standard known structures. The resulting condensed eigen- problem is then expanded in a low dimension fractional state-space. The Fractional Standard Linear Solid is the adopted rheological model, but the same methodology could be applied to problems involving different fractional derivative linear models. Examples regarding different continuos structures are proposed and discussed in detail.