_edit_ ok: the fractal problem is related to self-affinity or memory ‘getting in the way’ aka “detrended fluctuation”

Apparently, Fractional Calculus is VERY different. It can address Fractal media* HOWEVER, in a related paper, they find fractional calculus seems LESS EFFECTIVE than using Cellular Automata**

https://royalsocietypublishing.org/doi/10.1098/rsta.2020.0050

Abstract

1. Overview

2. Material hereditariness: viscoelasticity

3. Heat conduction

4. Diffusion in porous media

5. Non-local continua

6. Fractal media

*

This topic is addressed by Zhang & Ostoja-Starzewski in the theme issue [120], focusing on Lamb-type problems. The authors discuss the theoretical limitations in developing consistent space-fractional-derivative models of fractal media as well as in finding pertinent solutions and, in view of these outstanding challenges, propose an alternative approach simulating Cauchy and Dagum natural-like random fields by a Monte Carlo cellular automata approach; the latter has the advantage to assign cell-to-cell heterogeneous material properties and ensure equivalence to the continuum elasto-dynamics equations in the limit of infinitesimal cells. Wave propagation is investigated with focus on two Lamb-type problems on an elastic half-plane, specifically under a tangential point load and a concentrated point moment.

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Impact force and moment problems on random mass density fields with fractal and Hurst effects

See the end notes on Fractional Calculus:

https://royalsocietypublishing.org/doi/full/10.1098/rsta.2019.0591#d1e4065

d) Comments on fractional calculus

A very interesting issue is the interplay of fractals with fractional calculus, in both the time and space domains. An often-cited application is viscoelasticity, where fractional temporal derivatives provide a compact constitutive model. However, finding a definite link between a spatially fractal structure of a viscoelastic material and its postulated fractional model remains an outstanding challenge [32]. The last few decades have seen much activity in partial differential equations where temporal and/or spatial derivatives are assumed to be fractional. The article [33] discusses the formulation of a continuum mechanics model smoothing a fractal porous microstructure. Here, we note that there exists no solution to a Lamb-type problem with space fractional derivatives, which would directly correspond to (at least) the anti-plane elastodynamics problem in a random, linear elastic medium [5]. The closest model relative to the focus of our paper is a diffusion-wave equation with constant coefficients in [34]. In view of the above-mentioned weak dependence of the diffusion equation on spatial randomness, we do not pursue this topic here.