An approach to generalized one-dimensional self-similar elasticity

Michelitsch, Thomas; Maugin, Gérard A.; Mujibur, Rahman; Derogar, Shahram; Nowakowski, Andrzej; Nicolleau, F.

doi:10.1016/j.ijengsci.2012.06.014

Cited by 15 publications

(22 citation statements)

References 9 publications

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“…To this end we should elaborate first of all the notion of "self-similarity" employed in this paper. The notion of "self-similarity" which is employed in this paper as well as in [21,22,23,24,25] corresponds to the notion "self-similarity at a point" which is commonly used in the mathematical literature [30]. Generally, an object is exactly self-similar in the strict sense if it can be decomposed into parts which are exact rescaled copies of the entire object.…”

Section: Preliminariesmentioning

confidence: 99%

“…In contrast to the "classical" non-local elasticity theory as outlined by Eringen [7] the self-similar case is characterized by long-range (heavy-tailed) power law kernels decaying critically slowly and without a characteristic length scale. Further details on 1D cases can be found in [22,24,25]. The following section is devoted to the generalization of the approach to n-dimensions.…”

Section: One-dimensional Casementioning

confidence: 99%

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The fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion

Michelitsch

Maugin

Nowakowski

et al. 2013

fcaa

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We analyze the "fractional continuum limit" and its generalization to n dimensions of a self-similar discrete spring model which we introduced recently [21]. Application of Hamilton's (variational) principle determines in rigorous manner a self-similar and as consequence non-local Laplacian operator. In the fractional continuum limit the discrete self-similar Laplacian takes the form of the fractional Laplacian −(−Δ) α 2 with 0 < α < 2. We analyze the fundamental link of fractal vibrational features of the discrete self-similar spring model and the smooth regular ones of the corresponding fractional continuum limit model in n dimensions: We find a characteristic scaling law for the density of normal modes ∼ ω 828of Lévi flights in n-dimensions. In the limit of "large scaled times" ∼ t/r α >> 1 we show that all distributions exhibit the same asymptotically algebraic decay ∼ t −n/α → 0 independent from the initial distribution and spatial position. This universal scaling depends only on the ratio n/α of the dimension n of the physical space and the Lévi parameter α.MSC 2010: 28A80, 35Q84, 35R11, 60E07, 26A33, 82C31 Keywords and phrases: fractional Laplacian, self-similarity, power laws, scale invariance, fractals, Weierstrass-Mandelbrot function, continuum limit, fractional calculus, Fokker-Planck equation, anomalous diffusion, Lévi flights, Lévi (stable) distributions, non-locality IntroductionSelf-similarity (scaling-invariance) can be found in many problems of physics. There is a need for understanding the dynamics that leads to fractal and self-similarity properties in domains of the physics as varied as flow turbulence and complex materials. In "traditional modelling" the continuum is assumed to have a characteristic length scale which determines the wavelengths where wave fields interact with the microstructure. However, there are numerous materials in nature which are constituted by a scale hierarchy of recurrent microstructure which can be conceived in a good approximation as self-similar. This is true for solids and porous media but also in fluid mechanics, or multicomponent and multiphase flows. As a consequence, there are many areas of modelling that could benefit from a better understanding of the mechanisms underlying the dynamics of objects exhibiting self-similarity properties.In fluid mechanics, synthetic turbulence models are one of many examples of tempering with the input spectrum in order to understand better the physics underlying Lagrangian diffusion [26] and such spectra can be traced to the fractal distributions of velocity accumulation points in the synthetic flow. Fractal approaches are developing rapidly in fluid mechanics. Such approaches consist in either experimentally or numerically interfering with the flow, forcing it through self-similar objects (or through numerical forcing), [36,11,27].On the other hand, there is a large area of research devoted to the so called Fractional Laplacian and a huge number of references exists, employing this operator in various contexts of ...

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Section: Preliminariesmentioning

confidence: 99%

Section: One-dimensional Casementioning

confidence: 99%

The fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion

Michelitsch

Maugin

Nowakowski

et al. 2013

fcaa

Self Cite

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“…In a series of works [54][55][56] a new approach to self-similar harmonic interparticle interactions has been advanced. First, starting with a linear chain model, this has led to a self-similar Laplacian which, in the continuum limit, takes the form of a combination of fractional integrals.…”

Section: Other Fractal and Fractional Calculus Mechanics Modelsmentioning

confidence: 99%

From fractal media to continuum mechanics

Ostoja–Starzewski

Joumaa

et al. 2013

Z Angew Math Mech

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This paper presents an overview of modeling fractal media by continuum mechanics using the method of dimensional regularization. The basis of this method is to express the balance laws for fractal media in terms of fractional integrals and, then, convert them to integer-order integrals in conventional (Euclidean) space. Following an account of this method, we develop balance laws of fractal media (continuity, linear and angular momenta, energy, and second law) and discuss wave equations in several settings (1d and 3d wave motions, fractal Timoshenko beam, and elastodynamics under finite strains). We then discuss extremum and variational principles, fracture mechanics, and equations of turbulent flow in fractal media. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions and reduce to conventional forms for continuous media with Euclidean geometries upon setting the dimensions to integers. We also point out relations and potential extensions of dimensional regularization to other models of microscopically heterogeneous physical systems.

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“…Recently, a continuum mechanics model based on the Riesz potential [38,46] is proposed in [16], where the mapping of field variables to Euclidean space is done via the Riesz potential. In addition to modeling fractal media, inelastic models and nonlocal elastic models are proposed or reformulated in various studies [11,13,29,33,40,47].…”

Section: Introductionmentioning

confidence: 99%

“…It has been known that a material point in a linear elastic body gets information from either side of the point with left and right running characteristics [17] which makes the symmetry properties of the operators used in modeling the problem important [33]. Local fractional derivatives, which are used in modeling continuum problems [39,50], are based on one sided fractional derivatives and do not have the symmetry property.…”

Section: Introductionmentioning

confidence: 99%

Wave Propagation in Heterogeneous Media with Local and Nonlocal Material Behavior

Aksoy¹

2015

J Elast

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Wave propagation in heterogeneous solids has been an interest of researchers due to industrial applications. Some of the heterogeneous materials can exhibit power law scaling in material behavior which can be characterized by the fractal dimension of the microstructure. In this study, wave propagation in heterogeneous media with self-similar structure is investigated via fractional calculus along with space-time discontinuous Galerkin method. One and two dimensional problems are studied to demonstrate the capability of the proposed model in modeling heterogeneous media. The results show that the proposed model is a good candidate for modeling the mechanical behavior of disordered materials.

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An approach to generalized one-dimensional self-similar elasticity

Cited by 15 publications

References 9 publications

The fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion

The fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion

From fractal media to continuum mechanics

Wave Propagation in Heterogeneous Media with Local and Nonlocal Material Behavior

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