Abstract
This article focuses on fractional Maxwell model of viscoelastic materials, which are a generalization of classic Maxwell model to non-integer order derivatives. To build a fractional Maxwell model when only the noise-corrupted discrete-time measurements of the relaxation modulus are accessible for identification is a basic concern. For fitting the original measurement data an approach is suggested, which is based on approximate Scott Blair fundamental fractional non-integer models, and which means that the data are fitted by solving two dependent but simple linear least-squares problems in two separable time intervals. A complete identification algorithm is presented. The usability of the method to find the fractional Maxwell model of real biological material is shown. The parameters of the fractional Maxwell model of carrot root that approximate the experimental stress relaxation data closely are given.
Highlights
Fractional calculus is a branch of mathematical analysis that generalizes the derivative and integral of a function to non-integer order [1]
A classical manner of studying viscoelasticity is by twophase stress relaxation test, where the strain increases during the loading time interval until a predetermined strain εε0 is reached at given ramp-time, after which that strain εε0 is maintained constant at that value [11,12,25]
A certain stress relaxation test performed on the specimen of the material under investigation resulted in a set of measurements of the relaxation modulus GG(ttii) = GG(ttii) + zz(ttii) at the sampling instants ttii > 0, ii = 1, ... , NN, where zz(ttii) is measurement noise
Summary
Fractional calculus is a branch of mathematical analysis that generalizes the derivative and integral of a function to non-integer order [1]. Application of fractional calculus in classical and modern physics greatly contributed to the analysis and our understanding of physico-chemical and bio-physical complex dynamical systems, since it provides excellent instruments for the description of memory and properties of various materials and processes. During the last two decades fractional calculus has been increasingly applied to mathematical modelling in physics [2,3], engineering [4,5], and especially to rheology [6,7], where fractional calculus constitute a valuable mathematical tool to handle viscoelastic aspects of systems and materials mechanics. In recent decades fractional derivatives were found quite flexible, especially in the description of viscoelastic polymer materials [10]
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