# Fractional Calculus and Linear Viscoelasticity in Structural Dynamics

Abstract: The use of linear viscoelasticity together with fractional calculus in time-domain structural modeling is studied. Both constitutive and structural aspects are investigated. It is found, that viscoelastic models based on fractional derivatives are able to model observed material behavior more accurately and in a simpler way than classical viscoelastic models based on integer-order derivatives can. Finally, a three-dimensional formulation of linear viscoelasticity based on fractional calculus is implemented into a general purpose finite element code.

In the first part of this work, basic requirements on the choice of damping models in the frequency and time domains are outlined. The fractional derivative model of viscoelasticity is studied in particular. The simplest fractional derivative model contains a single fractional derivative operator acting on stress and strain. However, when incorporated directly into the framework of structural dynamics, this model leads to higher order equations of motion, which demand fractional order initial conditions. Three alternative forms of the fractional derivative model of viscoelasticity are introduced for the use in structural modeling. The first form uses a convolution integral and a singular kernel of Mittag-Leffler function type. The second form contains fractional integral operators instead of fractional derivative operators. The third form uses internal variables. The main advantage of these three forms is that they all lead to well-posed initial value problems. Numerical procedures for the time integration of spatially discretized finite element equations for viscoelastic structures described by these models are developed. Numerical examples are given and in one case the numerical solution is compared with a time series expansion of the analytical solution. Time domain expressions for viscoelastic functions like the relaxation function and the relaxation spectrum are also presented. From these functions the parameters of the model are identified as being physically meaningful. The key effect of using a fractional derivative operator is that a whole spectrum of dissipative mechanisms can be included in a single relaxation process. As an example, the model parameters are fit to frequency domain data for a commercial high damping polymer.

In the second part of this work, a physically sound anisotropic formulation of the standard linear viscoelastic solid with integer or fractional order rate laws for a finite set of pertinent internal variables is developed. Thermodynamic admissibility is investigated. This formulation is found to be the most general and suitable one of the fractional calculus model. A rational formulation for structural finite element analysis is derived and a scheme for time integration of the constitutive response is developed. The viscoelastic formulation and the time integration scheme are implemented into a general purpose finite element code. Numerical examples are presented including the technical problem of dynamic responses of a model of a railway-track viscoelastic-ballast system subjected to loads simulating a train.

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