Time domain modeling of damping using anelastic displacement fields and fractional calculus

A fractional derivative model of linear viscoelasticity based on the decomp osition of the displacement held into an anelastic part and elastic part is developed. The evolution equation for the anelastic part is then a differe ntial equation of fractional order in time. By using a fractional order evo lution equation for the anelastic strain the present model becomes very fle xible for describing the weak frequency dependence of damping characteristi cs. To illustrate the modeling capability, the model parameters are fit to available frequency domain data for a high damping polymer. By studying the relaxation modulus and the relaxation spectrum the material parameters of the present viscoelastic model are given physical meaning. The use of this viscoelastic model in structural modeling is discussed and the correspondin g finite element equations are outlined, including the treatment of boundar y conditions. The anelastic displacement field is mathematically coupled to the total displacement field through a convolution integral with a kernel of Mittag-Leffler function type. Finally a time step algorithm for solving the finite element equations are developed and some numerical examples are presented. (C) 1999 Elsevier Science Ltd. All rights reserved.