Time-Response Functions of Fractional-Derivative Rheological Models

Abstract

In view of the increasing attention to the time responses of complex fluids described by power-laws in association with the need to capture inertia effects that manifest in high-frequency microrheology, we compute the five basic time-response functions of in-series or in-parallel connections of two elementary fractional derivative elements known as the Scott-Blair (springpot) element. The order of fractional differentiation in each Scott-Blair element is allowed to exceed unity reaching values up to 2 and at this limit-case the Scott-Blair element becomes an inerter--a mechanical analogue of the electric capacitor that its output force is proportional only to the relative acceleration of its end-nodes. With this generalization, inertia effects may be captured beyond the traditional viscoelastic behavior. In addition to the relaxation moduli and the creep compliances, we compute closed form expressions of the memory functions, impulse fluidities (impulse response functions) and impulse strain-rate response functions of the generalized fractional derivative Maxwell fluid, the generalized fractional derivative Kelvin-Voigt element and their special cases that have been implemented in the literature. Central to these calculations is the fractional derivative of the Dirac delta function which makes possible the extraction of singularities embedded in the fractional derivatives of the two-parameter Mittag-Leffler function that emerges invariably in the time-response functions of fractional derivative rheological modes.

References in corpus (8)

• Creep, Relaxation and Viscosity Properties for Basic Fractional Models in Rheology
• Direct conversion of rheological compliance measurements into storage and loss moduli
• Accounting for inertia effects to access the high-frequency microrheology of viscoelastic fluids
• Short-time inertial response of viscoelastic fluids measured with Brownian motion and with active probes
• Time-Response Functions of Mechanical Networks with Inerters and Causality
• Space-time fractional Zener wave equation
• Viscous-Viscoelastic Correspondence Principle for Brownian Motion
• The Frequency Response Function of the Creep Compliance

Cited by in corpus (3)

• A Rheological Analogue for Brownian Motion with Hydrodynamic Memory
• The fractional derivative of the Dirac delta function and new results on the inverse Laplace transform of irrational functions
• Impulse Response Function for Brownian Motion