Gradient Extension of Classical Material Models: From Nuclear &amp; Condensed Matter Scales to Earth &amp; Cosmological Scales

Aifantis, Elias C.

doi:10.1007/978-3-030-63050-8_15

Cited by 18 publications

(8 citation statements)

References 127 publications

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“…In continuum mechanics non-local processes are found to be important for introducing internal length scales for localization phenomena (Aifantis, 2021). We emphasize here that consideration of the non-local cross-diffusion terms is particularly important to link the feedbacks between different self-diffusion length scales which are normally not considered in the coarse-graining homogenization approaches discussed later.…”

Section: Introductionmentioning

confidence: 95%

“…3. The approach relies on the tight coupling of the temperature reaction-self-diffusion equation with a chemical reaction-self-diffusion equation (Alevizos et al, 2014;Veveakis et al, 2014) which can lead for a highly nonlinear source term to excitation waves with a characteristic perfectly periodic oscillatory response in the temperaturepressure plot (Fig. 3).…”

Section: Application Of Two Reaction-self-diffusion Equations To Characteristic Earthquakesmentioning

confidence: 99%

“…This approach is similar to the spring-slider model discussed earlier for modelling earthquakes. A recent discussion of the PLC bands modelled by an alternative nonlocal internal length gradient approach is presented in Aifantis (2021).…”

Section: Diffusion Waves At Mechanical Scalementioning

confidence: 99%

“…Patterns in geological systems are thought to encode information on reaction-diffusion processes repeating themselves over multiple scales such that a magnified view of the structure looks like a copy of the structure itself called self-affinity or self-similarity if no affine transformation is required (Hobbs et al, 2011;Hobbs and Ord, 2015;Aifantis, 2021). The connection between these patterns as dissipative structures of reaction-diffusion systems (Ball, 2012) and their role in thermodynamic far-from-equilibrium systems was originally described by Prigogine and co-workers (Kondepudi and Prigogine, 1998).…”

Section: Introductionmentioning

confidence: 99%

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Cross-diffusion waves resulting from multiscale, multiphysics instabilities: application to earthquakes

Regenauer-Lieb

Schrank

et al. 2021

Solid Earth

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Abstract. Theoretical approaches to earthquake instabilities propose shear-dominated source mechanisms. Here we take a fresh look at the role of possible volumetric instabilities preceding a shear instability. We investigate the phenomena that may prepare earthquake instabilities using the coupling of thermo-hydro-mechano-chemical reaction–diffusion equations in a THMC diffusion matrix. We show that the off-diagonal cross-diffusivities can give rise to a new class of waves known as cross-diffusion or quasi-soliton waves. Their unique property is that for critical conditions cross-diffusion waves can funnel wave energy into a stationary wave focus from large to small scale. We show that the rich solution space of the reaction–cross-diffusion approach to earthquake instabilities can recover classical Turing instabilities (periodic in space instabilities), Hopf bifurcations (spring-slider-like earthquake models), and a new class of quasi-soliton waves. Only the quasi-soliton waves can lead to extreme focussing of the wave energy into short-wavelength instabilities of short duration. The equivalent extreme event in ocean waves and optical fibres leads to the appearance of “rogue waves” and high energy pulses of light in photonics. In the context of hydromechanical coupling, a rogue wave would appear as a sudden fluid pressure spike. This spike is likely to cause unstable slip on a pre-existing (near-critically stressed) fault acting as a trigger for the ultimate (shear) seismic moment release.

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Section: Introductionmentioning

confidence: 95%

Section: Application Of Two Reaction-self-diffusion Equations To Characteristic Earthquakesmentioning

confidence: 99%

Section: Diffusion Waves At Mechanical Scalementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Cross-diffusion waves resulting from multiscale, multiphysics instabilities: application to earthquakes

Regenauer-Lieb

Schrank

et al. 2021

Solid Earth

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“…Over the last two decades, the D-D and W-A models have become quite popular in both the applied mathematics [11] and the material science [12] communities due to the interesting mathematical properties of the former and robust interpretation of experimental observations of the latter groups of models. This includes implementation of D-D type models in interpreting molecular and mesoscopic transport in condensed matter and cosmological systems [13], i.e. Most of such models though overlooked the contribution of stochastic forcing or spatial randomness, e.g.…”

Section: Introductionmentioning

confidence: 99%

Transmissibility in Interactive Nanocomposite Diffusion: The Nonlinear Double-Diffusion Model

Chattopadhyay,

Kundu,

Nath

et al. 2022

Preprint

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Model analogies and exchange of ideas between physics or chemistry with biology or epidemiology have often involved inter-sectoral mapping of techniques. Material mechanics has benefitted hugely from such interpolations from mathematical physics where dislocation patterning of platstically deformed metals [1, 2, 3] and mass transport in nanocomposite materials with high diffusivity paths such as dislocation and grain boundaries, have been traditionally analyzed using the paradigmatic Walgraef-Aifantis (W-A) double-diffusivity (D-D) model [4,5,6,7,8,9]. A long standing challenge in these studies has been the inherent nonlinear correlation between the diffusivity paths, making it extremely difficult to analyze their interdependence. Here, we present a novel method of approximating a closed form solution of the ensemble averaged density profiles and correlation statistics of coupled dynamical systems, drawing from a technique used in mathematical biology to calculate a quantity called the basic reproduction number R 0 , which is the average number of secondary infections generated from every infected. We show that the R 0 formulation can be used to calculate the correlation between diffusivity paths, agreeing closely with the exact numerical solution of the D-D model.

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Variations of heat equation on the half‐line via the Fokas method

Chatziafratis,

Fokas,

Aifantis

2024

Math Methods in App Sciences

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In this review paper, we discuss some of our recent results concerning the rigorous analysis of initial boundary value problems (IBVPs) and newly discovered effects for certain evolution partial differential equations (PDEs). These equations arise in the applied sciences as models of phenomena and processes pertaining, for example, to continuum mechanics, heat‐mass transfer, solid–fluid dynamics, electron physics and radiation, chemical and petroleum engineering, and nanotechnology. The mathematical problems we address include certain well‐known classical variations of the traditional heat (diffusion) equation, including (i) the Sobolev–Barenblatt pseudoparabolic PDE (or modified heat or second‐order fluid equation), (ii) a fourth‐order heat equation and the associated Cahn–Hilliard (or Kuramoto–Sivashinsky) model, and (iii) the Rubinshtein–Aifantis double‐diffusion system. Our work is based on the synergy of (i) the celebrated Fokas unified transform method (UTM) and (ii) a new approach to the rigorous analysis of this method recently introduced by one of the authors. In recent works, we considered forced versions of the aforementioned PDEs posed in a spatiotemporal quarter‐plane with arbitrary, fully non‐homogeneous initial and boundary data, and we derived formally effective solution representations, for the first time in the history of the models, justifying a posteriori their validity. This included the reconstruction of the prescribed initial and boundary conditions, which required careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula was utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain. Importantly, this analysis is indispensable for proving (non)uniqueness of solution. These works extend previous investigations. The usefulness of our closed‐form solutions will be demonstrated by studying their long‐time asymptotics. Specifically, we will briefly review some asymptotic results about Barenblatt's equation.

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Gradient Extension of Classical Material Models: From Nuclear & Condensed Matter Scales to Earth & Cosmological Scales

Cited by 18 publications

References 127 publications

Cross-diffusion waves resulting from multiscale, multiphysics instabilities: application to earthquakes

Cross-diffusion waves resulting from multiscale, multiphysics instabilities: application to earthquakes

Transmissibility in Interactive Nanocomposite Diffusion: The Nonlinear Double-Diffusion Model

Variations of heat equation on the half‐line via the Fokas method

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