The Concepts and Applications of Fractional Order Differential Calculus in Modeling of Viscoelastic Systems: A Primer

Abstract

Viscoelasticity and other related phenomena are of great importance in the study of mechanical properties of materials, especially biological materials. Certain materials demonstrate some complicated behavior under mechanical tests that cannot be described by a standard linear equation (SLE), mostly due to the shape memory effect during the deformation phase. Recently, researchers have been making use of fractional calculus (FC) in order to probe viscoelasticity of such materials accurately. FC is a powerful tool for modeling complicated phenomena. In this tutorial paper, it is sought to provide clear descriptions of this powerful tool and its techniques and implementation. It is endeavored to keep the details to a minimum while still conveying a good idea of what and how can be done with this powerful tool. The reader will be provided with the basic techniques that are used to solve the fractional equations analytically and/or numerically. More specifically, simulating the shape memory phenomena with this powerful tool will be studied from different perspectives, and some physical interpretations are made in this regard. This paper is also a review of fractional order models of viscoelastic phenomena that are widespread in bioengineering. Thus, in order to show the relationship between fractional models and SLEs, a new fractal system comprising spring and damper elements is considered and the constitutive equation is approximated with a fractional element. Finally, after a brief literature review, two fractional models are utilized to investigate the viscoelasticity of the cell and a comparison is made between the findings and the experimental data from the previous models. Verification results indicate that the fractional model not only matches well with the experimental data but also can be a good substitute for previously used models.