A primer on experimental and computational rheology with fractional viscoelastic constitutive models

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. L.L. Ferris and J.M. Nobrega would like to thank the funding by FEDER through the COMPETE 2020 Programme, the National Funds through FCT - Portuguese Foundation for Science and Technology under the project UID/CTM/50025/2013. L.L. Ferias would also like to thank the funding by FCT through the scholarship SFRH/BPD/100353/2014. M.L. Morgado would like to thank the funding by FCT through Project UID/MAT/00013/2013 and M. Rebelo would also like to thank the funding by FCT through Project UID/MAT/00297/2013 (Centro de Matematica e Aplicacoes).