Two-stage scheme for identification of parallel Scott-Blair fractional model of viscoelastic biological materials

Fractional calculus considers derivatives and integrals of an arbitrary order. This article focuses on fractional parallel Scott-Blair model of viscoelastic biological materials, which is a generalization of classic Kelvin-Voight model to non-integer order derivatives suggested in the previous paper. The parallel Scott-Blair model admit the closed form of analytical solution in terms of two power functions multiplied by Debye type weight function. To build a parallel Scott-Blair model when only discrete-time measurements of the relaxation modulus are accessible for identification is a basic concern. Based on asymptotic models a two-stage approach is proposed for fitting the measurement data, which means that in the first stage the data are fitted by solving two dependent, but simple, linear least-squares problems in two separate time intervals. Next, at the second stage of the identification procedure the exact parallel Scott-Blair model optimal in the least-squares sense is computed. The log-transformed relaxation modulus data is used in the first stage of identification scheme, while the original relaxation modulus data is applied for the second stage identification. A complete identification procedure is presented. The usability of the method to find the parallel Scott-Blair fractional model of real biological material is demonstrated. The parameters of the parallel Scott-Blair model of a sample of sugar beet root, which very closely approximate the experimental relaxation modulus data, are given.