Incorporating Rheological Nonlinearity into Fractional Calculus Descriptions of Fractal Matter and Multi-Scale Complex Fluids

In this paper, we use ideas from fractional calculus to study the rheological response of soft materials under steady-shearing flow conditions. The linear viscoelastic properties many multi-scale complex fluids exhibit a power-law behavior that spans over orders magnitude in time or frequency, and can accurately describe rheology using constitutive models. By measuring non-linear during large step strain deformations, also demonstrate class often follows time-strain separability principle, which enables us characterize their nonlinear through an experimentally determined damping function. To model these materials, incorporate function with integral formulation develop analytical framework calculation material such as rate-dependent shear viscosity measured steady-state shearing flows. We focus on general subclass equations, known Fractional Maxwell Model, consider several different forms for Through computational evaluations viscosity, show sufficiently strong functions, example, exponential decay fluid memory strain, observed shear-thinning exponents are set by indices model. For weak however, index high rate is terminal itself at strains. limit very function, theoretical predicts unbounded growth stress continuously growing transient does not converge meaningful value. determining leading terms our solution both low rates, construct approximate analytic expression viscosity. An error analysis shows that, each functions considered, closed-form accurate wide range rates.