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

A fractional derivative model of linear viscoelasticity based on the decomposition of the displacement _eld into an anelastic part and elastic part is developed[ The evolution equation for the anelastic part is then a di}erential equation of fractional order in time[ By using a fractional order evolution equation for the anelastic strain the present model becomes very ~exible for describing the weak frequency dependence of damping characteristics[ To illustrate the modeling capability\ the model parameters are _t 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 corresponding _nite element equations are outlined\ including the treatment of boundary conditions[ The anelastic displacement _eld is mathematically coupled to the total displacement _eld through a convolution integral with a kernel of Mittag Le/er function type[ Finally a time step algorithm for solving the _nite element equations are developed and some numerical examples are presented