A generalised distributed-order Maxwell model

TL;DR

This paper introduces a generalized viscoelastic model based on distributed-order derivatives, extending classical models to better describe complex fluids with adjustable parameters for improved accuracy.

Contribution

It presents a novel distributed-order Maxwell model that broadens the scope of fractional viscoelastic models for complex fluid analysis.

Findings

Derived relaxation modulus, storage, and loss modulus.

Showed the model's ability to interpolate between classical and fractional models.

Enhanced description of complex fluids with adjustable parameters.

Abstract

In this work we present a generalised viscoelastic model using distributed-order derivatives. The model consists of two distributed-order elements (distributed springpots) connected in series, as in the Maxwell model. The new model generalises the fractional viscoelastic model presented in [H. Schiessel, A. Blumen, Hierarchical analogues to fractional relaxation equations, Journal of Physics A: Mathematical and General 26 (1993) 5057-50] and allows for a more broad and accurate description of complex fluids when a proper weighting function of the order of the derivatives is chosen. We discuss the connection between classical, fractional, and viscoelastic models of distributed order and highlight the fundamental concepts that support these constitutive equations. We also derive the relaxation modulus, the storage and loss modulus, and the creep compliance for specific weighting functions.