Experimental identification of a fractional derivative linear model for viscoelastic materials

Non integer, fractional order derivative rheological models are known to be very effective in describing the linear viscoelastic dynamic behaviour of mechanical structures made of polymers. The application of fractional calculus to viscoelasticity can be physically consistent and the resulting non integer order differential stress-strain constitutive relation provides good curve fitting properties, requires only a few parameters and leads to causal behaviour. When using such models the solution of direct problems, i.e. the evaluation of time or frequency response from a known excitation can still be obtained from the equations of motion using standard tools such as modal analysis. But regarding the inverse problem, i.e. the identification from measured input-output vibrations, no general technique has so far been established, since the current methods do not seem to easily work with differential operators of non integer order. In this paper a frequency domain method is proposed for the experimental identification of a linear viscoelastic model, namely the Fractional Zener also known as Fractional Standard Linear Solid, to compute the frequency dependent complex stress-strain relationship parameters related to the material. The procedure is first checked with respect to numerically generated frequency response functions for testing its accuracy, and then to experimental inertance data from a free-free homogeneous beam made of High Density Polyethylene (HDPE) in plane flexural and axial vibration.