Time-domain finite element analysis of viscoelastic structures with fractional derivatives constitutive relations

Abstract

Numerical procedures for the time integration of the spatially discretized finite element equations for viscoelastic structures governed by a constitutive equation involving fractional calculus operators are presented. To avoid difficulties concerning fractional-order initial conditions, a form of the fractlonal calculus model of viscoelasticity involving a convolution integral with a singular memory kernel of Mittag-Lefler type is used. The constitutive equation is generalized to three-dimensional states for isotropic materials. A simplification of the fractional derivative of the memory kernel is used, in connection with Grunwald's definition of fractional differentiation and a backward Euler rule, for the time evolution of the convolution term. A desirable feature of this process is that no actual evaluation of the memory kernel is needed. This, together with the Newmark method for time integration, enables the direct calculation of the time evolution of the nodal degrees of freedom. To illustrate the ability of the numerical procedure a few numerical examples are presented. In one example the numerically obtained solution is compared with a time series expansion of the analytical solution.