Finite element formulations for transient dynamic analysis in structural systems with viscoelastic treatments containing fractional derivative models

This paper presents finite element formulations for transient dynamic analysis in structural systems with damping treatments in which viscoelastic materials are characterized by means of fractional derivative models. In contrast to other formulations, such as that of Padovan employing the principle of virtual work, the proposed formulations begin from the local equation of linear momentum and make use of the weighted residual method, providing a matrix equation of motion involving fractional operators. The numerical approximations of these are developed through the Gr¨unwald–Letnikov definition, which allows to formulate explicit and implicit numerical schemes. The principal advantage of the proposed formulations is that the history of the displacements and external forces must be stored, but not that of the stress, which reduces computational time and storage needs. Numerical applications are presented for a cantilever beam, where a viscoelastic treatment has been applied using damping material modelled by a five-parameter fractional derivative model. The results of the proposed formulation are compared among them and with those of Padovan for different damping values and for different load cases. The influence of the truncation of Gr¨unwald coefficients and of the integration time-step is also investigated. Copyright q 2006 John Wiley & Sons, Ltd