Fractional Maxwell model of viscoelastic biological materials

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.

Interfacial Dilational Viscoelasticity of Adsorption Layers at the Hydrocarbon/Water Interface: The Fractional Maxwell Model

“…The constitutive equation of a (single-element consisting of one spring and one dash-pot) Maxwell model requires equilibrium between stress, σ, and strain, ε, according to the following equality [5,15]:…”

“…An alternative approach can be built on employment of the fractional Scott-Blair model. The fractional Scott-Blair element is defined by the following fractional differential equation [5,10,15]:…”

“…where τ = η/E is a characteristic time scale, α is a dimensionless positive non-integer number (0 < α < 1), assigning the order of the fractional derivative of the strain ε(t) (known as Caputo fractional derivative). This mathematical tool, represented as a ladder sequence of Maxwell spring/dash-pots, allows combining a range of multiple exponential relaxation mechanisms in a single three-parameter rheological element (see [15] for mathematical details and graphical representation).…”

“…In the present study, we will consider a fractional single Maxwell element, which consists of a fractional Scott-Blair element, defined by Equation 13with the parameters E 1 , τ 1 , and α, and a spring, E 2 , connected in a series. The constitutive equation for this model is given by [11,15]:…”

“…In recent years, the fractional Maxwell model (FMM) was fruitfully adopted for analysing the 3D-viscoelastic behavior for several classes of bulk materials, in particular for soft materials, by using only a few constitutive parameters and still retaining a good corresponding agreement between model values and the observed experimental results [5][6][7][8][9][10][11][12][13]. Lei et al [14] and Stankiewicz [15] reviewed various scientific and technological fields of successful FMM applications. Povstenko expounded the mathematical foundations of the inherent fractional-derivative tool [16].…”