A Primer on Experimental and Computational Rheology with Fractional Viscoelastic Constitutive Models
Luís Lima Ferrás 1, 2, a), Neville John Ford 2, b), Maria Luísa Morgado 3, c), Magda Rebelo 4, d), Gareth Huw McKinley 5, e) and João Miguel Nóbrega 1, f)
1) IPC/i3N - Institute Polymers and Composites, Department of Polymer Engineering, University of Minho, Campus de Azurém, 4800-058 Guimarães, Portugal
2) Department of Mathematics, University of Chester, CH1 4BJ, UK
3) Centro de Matemática, Polo CMAT-UTAD, Departamento de Matemática, Universidade de Trás-os-Montes e Alto Douro, UTAD, Quinta de Prados 5001-801, Vila Real, Portugal
4) Centro de Matemática e Aplicações (CMA) and Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade NOVA de Lisboa, Quinta da Torre, 2829-516 Caparica, Portugal
5) Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 USA

a) Corresponding author: luis.ferras@dep.uminho.pt
b) njford@chester.ac.uk
c) luisam@utad.pt
d) msjr@fct.unl.pt
e) gareth@mit.edu
f) mnobrega@dep.uminho.pt

Abstract. This work presents a brief introduction to fractional calculus and its application to some problems in rheology. We present two different viscoelastic models based on fractional derivatives (the Fractional Maxwell Model – FMM and the Fractional Viscoelastic Fluid – FVF) and discuss their reduction to the classical Newtonian and Maxwell fluids. A third model is also studied (an extension of the FMM to an invariant form), being given by a combination of the K-BKZ integral model with a fractional memory function which we denote the Fractional K-BKZ model. We discuss and illustrate the ability of these models to fit experimental data, and present numerical results for simple stress relaxation following step strain and steady shearing.