Occurrence of anomalous diffusion and non-local response in highly-scattering acoustic periodic media

Abstract

We investigate the occurrence of anomalous diffusive transport associated with acoustic wave fields propagating through highly-scattering periodic media. Previous studies had correlated the occurrence of anomalous diffusion to either the random properties of the scattering medium or to the presence of localized disorder. In this study, we show that anomalous diffusive transport can occur also in perfectly periodic media and in the absence of disorder. The analysis of the fundamental physical mechanism leading to this unexpected behavior is performed via a combination of deterministic, stochastic, and fractional-order models in order to capture and illustrate the different elements contributing to this phenomenon. Results indicate that the anomalous transport can indeed occur in perfectly periodic media when the dispersion behavior is characterized by anisotropic (partial) bandgaps. In selected frequency ranges, the propagation of acoustic waves not only becomes diffusive but its intensity distribution acquires a distinctive Lévy α-stable profile having pronounced heavy-tails. In these ranges, the acoustic transport in the medium occurs according to a hybrid transport mechanism which is simultaneously propagating and anomalously diffusive. We show that such behavior is well captured by a fractional diffusive transport model whose order can be obtained by the analysis of the heavy tails.

A generalized fractional-order elastodynamic theory for non-local attenuating media

“…Hence, in practical experiment it is not immediate to discern what physical mechanism lies behind the occurrence of fractional dispersion. However, based on the mathematical formulation presented above, it is clear that space-fractional operators are indicative of attenuation in conservative media whereas time-fractional operators imply dissipation in non-conservative media [1,4,5,39]. Note also that, in the purely space-fractional case the attenuation (equation (4.7)) is a fractional-order power-law function of the wavenumber.…”

“…The effect of the length-scale on the material dynamics has been experimentally captured for several classes of materials including aluminium, copper and nickel [55]. The sources of non-locality include, among others, heterogeneity of the microstructure, presence of distributed cracks and metal plasticity [24], and even periodic material or geometric inclusions [1,39]. A specific example belonging to the latter class of systems will be shown in §6b.…”

“…Recent theoretical and experimental studies have shown that transport processes in complex media are often characterized by either hybrid or anomalous mechanisms. As an example, wave propagation in highly scattering (either periodic [1] or random [2]) media is characterized by a diffused field that coexists with the propagating component. This dual mechanism is often referred to as hybrid transport [3–5].…”

Applications of variable-order fractional operators: a review

“…The many characteristics of fractional operators have sparked, in recent years, much interest in fractional calculus and produced a plethora of applications with particular attention to the simulation of physical problems. Areas that have seen the largest number of applications include the formulation of constitutive equations for viscoelastic materials [1][2][3][4], transport processes in complex media [4][5][6][7][8][9][10][11], mechanics [12][13][14][15], non-local elasticity [16][17][18][19], plasticity [20][21][22], modelorder reduction of lumped parameter systems [23] and biomedical engineering [24][25][26]. These studies have typically used constant-order (CO) fractional operators.…”

Variable-order particle dynamics: formulation and application to the simulation of edge dislocations

“…The unique set of properties of fractional operators have determined a surge of interest in exploring their possible applications. Among the areas that have rapidly developed, we encounter the modelling of viscoelastic materials [1][2][3], transport processes in complex media [4][5][6][7][8][9][10][11][12], non-local elasticity [13][14][15][16][17][18][19] and model-order reduction of lumped parameter systems [20]. These applications mostly focused on constant-order (CO) fractional operators.…”