_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. __ 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.

_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.
 
__
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.
 

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