arXiv subjects

Elias C. Aifantis

Publications and source records attributed to Elias C. Aifantis.

10 recordsLinked to original sources

arXiv2026

Integral Representations for Interface Problems for the Barenblatt-Sobolev-Galpern Pseudoparabolic Equation

We obtain novel integral representations, expressed as contour integrals in the complex Fourier plane, for the solution of fully nonhomogeneous initial-boundary-value as well as interface problems for the Barenblatt-Zheltov-Kochina pseudo-parabolic equation of Sobolev-Galpern type formulated on the real line, half-line and finite interval. This fundamental partial differential equation (PDE) of mathematical physics emerges in a wide variety of natural phenomena and applied sciences including continuum mechanics, thermodynamics, chemical engineering, solid-state electronics, semi-conductor devices, battery research, and nanotechnology. A suitable implementation of a modern methodology, known as Unified Transform Method, is in force throughout this study, with particular challenges arising due to the higher-order mixed-derivative term of the PDE and the generality of the problems under consideration altogether. The boundary and interface conditions appear to be non-standard but are naturally dictated by the structure of the PDE itself. Our explicit analytical formulae directly lend themselves to future explorations of the solutions' qualitative properties such as asymptotic behavior, spatio-temporal dynamics, regularity and well-posedness. This work is expected to be of utility also in the investigation of nonlinear counterparts as well as towards the study of phase-transition phenomena and free-boundary problems, where the interface evolves dynamically according to energy balance laws.

math.AP↗

arXiv2020

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

The various mathematical models developed in the past to interpret the behavior of natural and manmade materials were based on observations and experiments made at that time. Classical laws (such as Newton's for gravity, Hooke's for elasticity, Navier-Stokes for fluidity, Fick's/Fourier's for diffusion/heat transfer, Coulomb's for electricity, as well as Maxwell's for electromagnetism and Einstein's for relativity) formed the basis of current technology and shaping of our civilization. The discovery of new phenomena with the aid of recently developed experimental probes have led to various modifications of these laws across disciplines and the scale spectrum: from subatomic and elementary particle physics to cosmology and from atomistic and nano/micro to macro/giga scales. The emergence of nanotechnology and the further advancement of space technology are ultimately connected with the design of novel tools for observation and measurements, as well as the development of new methods and approaches for quantification and understanding. The paper first reviews the author's previously developed weakly nonlocal or gradient models for elasticity, diffusion and plasticity within a unifying internal length gradient (ILG) framework. It then proposes a similar extension for fluids and Maxwell's equations of electromagnetism. Finally, it ventures a gradient modification of Newton's law of gravity and examines its implications to some problems of elementary particle physics, also relevant to cosmology. Along similar lines, it suggests an analogous extension of London's quantum mechanical potential to include both an "attractive" and a "repulsive" branch. It concludes with some comments on a fractional generalization of the ILG framework.

physics.class-ph↗

arXiv2018

On Fractional and Fractal Formulations of Gradient Linear and Nonlinear Elasticity

In this paper we consider extensions of the gradient elasticity models proposed earlier by the second author to describe materials with fractional non-locality and fractality using the techniques developed recently by the first author. We derive a generalization of three-dimensional continuum gradient elasticity theory, starting from integral relations and assuming a weak non-locality of power-law type that gives constitutive relations with fractional Laplacian terms, by utilizing the fractional Taylor series in wave-vector space. In the sequel we consider non-linear field equations with fractional derivatives of non-integer order to describe nonlinear elastic effects for gradient materials with power-law long-range interactions in the framework of weak non-locality approximation. The special constitutive relationship that we elaborate on, can form the basis for developing a fractional extension of deformation theory of gradient plasticity. Using the perturbation method, we obtain corrections to the constitutive relations of linear fractional gradient elasticity, when the perturbations are caused by weak deviations from linear elasticity or by fractional gradient non-locality. Finally we discuss fractal materials described by continuum models with non-integer dimensional spaces. Using the recently suggested vector calculus for non-integer dimensional spaces, we consider problems of fractal gradient elasticity.

physics.class-ph↗

arXiv2016

Existence result for a dislocation based model of single crystal gradient plasticity with isotropic or linear kinematic hardening

We consider a dislocation-based rate-independent model of single crystal gradient plasticity with isotropic or linear kinematic hardening. The model is weakly formulated through the so-called primal form of the flow rule as a variational inequality for which a result of existence and uniqueness is obtained using the functional analytical framework developed by Han-Reddy.

math.AP↗

arXiv2016

Stochastically forced dislocation density distribution in plastic deformation

The dynamical evolution of dislocations in plastically deformed metals is controlled by both deterministic factors arising out of applied loads and stochastic effects appearing due to fluctuations of internal stress. Such type of stochastic dislocation processes and the associated spatially inhomogeneous modes lead to randomness in the observed deformation structure. Previous studies have analyzed the role of randomness in such textural evolution but none of these models have considered the impact of a finite decay time (all previous models assumed instantaneous relaxation which is "unphysical") of the stochastic perturbations in the overall dynamics of the system. The present article bridges this knowledge gap by introducing a colored noise in the form of an Ornstein-Uhlenbeck noise in the analysis of a class of linear and nonlinear Wiener and Ornstein-Uhlenbeck processes that these structural dislocation dynamics could be mapped on to. Based on an analysis of the relevant Fokker-Planck model, our results show that linear Wiener processes remain unaffected by the second time scale in the problem but all nonlinear processes, both Wiener type and Ornstein-Uhlenbeck type, scale as a function of the noise decay time $τ$. The results are expected to ramify existing experimental observations and inspire new numerical and laboratory tests to gain further insight into the competition between deterministic and random effects in modeling plastically deformed samples.

cond-mat.stat-mech↗

arXiv2016

Internal Length Gradient (ILG) Material Mechanics Across Scales & Disciplines

A combined theoretical/numerical/experimental program is outlined for extending the ILG approach to consider time lags, stochasticity and multiphysics couplings. Through this extension it is possible to discuss the interplay between deformation internal lengths (ILs) and ILs induced by thermal, diffusion or electric field gradients. Size-dependent multiphysics stability diagrams are obtained, and size-dependent serrated stress-strain curves are interpreted through combined gradient-stochastic models. When differential equations are not available for describing material behavior, a Tsallis non-extensive thermodynamic formulation is employed to characterize statistical properties. A novel multiscale coarse graining technique, the equation free method (EFM), is suggested for bridging length scales, and the same is done for determining ILs through novel laboratory tests by employing specimens with fabricated gradient micro/nano structures. Three emerging research areas are discussed: (i) Plastic instabilities and size effects in nanocrystalline (NC)/ultrafine grain (UFG) and bulk metallic glass (BMG) materials; (ii) Chemomechanical damage, electromechanical degradation, and photomechanical aging in energetic materials; (iii) Brain tissue and neural cell modeling. Finally, a number of benchmark problems are considered in more detail. They include gradient chemoelasticity for Li-ion battery electrodes; gradient piezoelectric and flexoelectric materials; elimination of singularities from crack tips; derivation of size-dependent stability diagrams for shear banding in BMGs; modeling of serrated size-dependent stress-strain curves in micro/nanopillars; description of serrations and multifractal patterns through Tsallis q-statistics; and an extension of gradient elasticity/plasticity models to include fractional derivatives and fractal media.

cond-mat.mtrl-sci↗

arXiv2016

On the role of micro-inertia in enriched continuum mechanics

In this paper the role of gradient micro-inertia terms $\barη\lVert\nabla u_{,t}\rVert^{2}$ and free micro-inertia terms $η\lVert p_{,t}\rVert^{2}$ is investigated to unveil their respective effect on the dynamical behavior of band-gap metamaterials. We show that the term $\barη\lVert\nabla u_{,t}\rVert^{2}$ alone is only able to disclose relatively simplified dispersive behaviors. On the other hand, the term $η\lVert p_{,t}\rVert^{2}$ is in charge of the description of the full complex behavior of band-gap metamaterials. A suitable mixing of the two micro-inertia terms allows to describe a new feature of the relaxed-micromorphic model, i.e. the description of a second band-gap occurring for higher frequencies. We also show that a split of the gradient micro-inertia $\barη\lVert\nabla u_{,t}\rVert^{2}$, in the sense of Cartan-Lie decomposition of matrices, allows to flatten separately longitudinal and transverse optic branches thus giving the possibility of a second band-gap. Finally, we investigate the effect of the gradient inertia $\barη\lVert\nabla u_{,t}\rVert^{2}$ on more classical enriched models as the Mindlin-Eringen and the internal variable ones. We find that the addition of such gradient micro-inertia allows for the onset of one band-gap in the Mindlin-Eringen model and of three band-gaps in the internal variable model. In this last case, however, non-local effects cannot be accounted for which is a too drastic simplification for most metamaterials. We conclude that, even when adding gradient micro-inertia terms, the relaxed micromorphic model remains the most performing one, among the considered enriched model, for the description of non-local band-gap metamaterials.

physics.class-ph↗

arXiv2014

Non-Standard Extensions of Gradient Elasticity: Fractional Non-Locality, Memory and Fractality

Derivatives and integrals of non-integer order may have a wide application in describing complex properties of materials including long-term memory, non-locality of power-law type and fractality. In this paper we consider extensions of elasticity theory that allow us to describe elasticity of materials with fractional non-locality, memory and fractality. The basis of our consideration is an extension of the usual variational principle for fractional non-locality and fractality. For materials with power-law non-locality described by Riesz derivatives of non-integer order, we suggest a fractional variational equation. Equations for fractal materials are derived by a generalization of the variational principle for fractal media. We demonstrate the suggested approaches to derive corresponding generalizations of the Euler-Bernoulli beam and the Timoshenko beam equations for the considered fractional non-local and fractal models. Various equations for materials with fractional non-locality, fractality and fractional acceleration are considered.

cond-mat.mtrl-sci↗

arXiv2013

Towards Fractional Gradient Elasticity

An extension of gradient elasticity through the inclusion of spatial derivatives of fractional order to describe power-law type of non-locality is discussed. Two phenomenological possibilities are explored. The first is based on the Caputo fractional derivatives in one-dimension. The second involves the Riesz fractional derivative in three-dimensions. Explicit solutions of the corresponding fractional differential equations are obtained in both cases. In the first case it is shown that stress equilibrium in a Caputo elastic bar requires the existence of a non-zero internal body force to equilibrate it. In the second case, it is shown that in a Riesz type gradient elastic continuum under the action of a point load, the displacement may or may not be singular depending on the order of the fractional derivative assumed.

physics.class-ph↗

arXiv2004

Statistical Analysis for Long Term Correlations in the Stress Time Series of Jerky Flow

Stress time series from the PLC effect typically exhibit stick-slips of upload and download type. These data contain strong short-term correlations of a nonlinear type. We investigate whether there are also long term correlations, i.e. the successive up-down patterns are generated by a deterministic mechanism. A statistical test is conducted for the null hypothesis that the sequence of the up-down patterns is totally random. The test is constructed by means of surrogate data, suitably generated to represent the null hypothesis. Linear and nonlinear estimates are used as test statistics, namely autocorrelation, mutual information and Lyapunov exponents, which are found to have proper performance for the test. The test is then applied to three stress time series under different experimental conditions. Rejections are obtained for one of them and not with all statistics. From the overall results we cannot conclude that the underlying mechanism to the PLC effect has long memory.

physics.data-an↗