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Fractal Fract
Special Issues
Physical Phenomena on Fractals and in Fractional Dimension Spaces
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Physical Phenomena on Fractals and in Fractional Dimension Spaces

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (30 November 2023) | Viewed by 10754

Special Issue Editors

Prof. Dr. Alexander S. Balankin

E-Mail Website
Guest Editor
SEPI-ESIME Zacatenco, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos, Mexico City 07738, Mexico
Interests: fractals; fractal continuum; fractal analysis; fractional dimension space; random walks; crumpling and folding; fracture; hydrodynamics; percolation; complex systems
Dr. Didier Samayoa Ochoa

E-Mail Website
Guest Editor
SEPI-ESIME Zacatenco, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos, Mexico City 07738, Mexico
Interests: fractal analysis; fractal continuum; shear deformable beams; structural mechanics; structural health monitoring; fluid dynamics; enhanced oil recovery; elastodynamics

Special Issue Information

Dear Colleagues,

Nowadays, fractal geometry and the theory of fractional dimension spaces are powerful tools in physics and engineering. Both tools are associated with the notion of fractional dimension numbers and can be used to account for the interplay between geometry and physics in real-world systems. However, the mathematical and physical contents of the fractal and fractional dimension space frameworks are quite different. This Special Issue is a place to discuss and share new ideas and research findings concerning this matter. The discussion may be also focused on physical phenomena in the fractional dimension spaces and in the fractal media. Therefore, we invite and welcome review, expository, and original research articles dealing with these subjects.

Potential topics include, but are not limited to:

  • physics in a fractional dimension space and on fractals;
  • characterization and modeling of real-world fractals;
  • mathematical models of fractal continuum;
  • fractal kinetic models;
  • fractal dynamics and time series;
  • percolation on fractals;
  • mechanics of fractal materials;
  • fractal concepts in fracture mechanics;
  • wave phenomena in fractal media;
  • fractal hydrodynamics;
  • electromagnetic phenomena in fractional spaces and in fractal media.

We look forward to receiving your contributions.

Prof. Dr. Alexander S. Balankin
Dr. Didier Samayoa Ochoa
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fractals
  • fractional dimension space
  • fractal continuum
  • Hurst exponent
  • percolation
  • fractal dynamics
  • fractal mechanics
  • wave phenomena
  • time series
  • anomalous diffusion

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Published Papers (6 papers)

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Research

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22 pages, 9139 KiB  
Article
Fractal Analysis and FEM Assessment of Soft Tissue Affected by Fibrosis
by Arturo Yishai Prieto-Vázquez, Alejandro Cuautle-Estrada, Mario Alberto Grave-Capistrán, Octavio Ramírez and Christopher René Torres-SanMiguel
Fractal Fract. 2023, 7(9), 661; https://doi.org/10.3390/fractalfract7090661 - 31 Aug 2023
Cited by 4 | Viewed by 1817
Abstract
This research shows an image processing method to determine the liver tissue’s mechanical behavior under physiological damage caused by fibrosis pathology. The proposed method consists of using a liver tissue CAD/CAE model obtained from a tomography of the human abdomen, where the diaphragmatic [...] Read more.
This research shows an image processing method to determine the liver tissue’s mechanical behavior under physiological damage caused by fibrosis pathology. The proposed method consists of using a liver tissue CAD/CAE model obtained from a tomography of the human abdomen, where the diaphragmatic surface of this tissue is compressed by a moving flat surface. For this work, two tools were created—the first to analyze the deformations and the second to analyze the displacements of the liver tissue. Gibbon and MATLAB® were used for numerical analysis with the FEBio computer program. Although deformation in the scenario can be treated as an orthogonal coordinate system, the relationship between the total change in height (measured) and the deformation was obtained. The outcomes show liver tissue behavior as a hyperelastic model; the Mooney–Rivlin mathematical characterization model was proposed in this case. Another method to determine the level of physiological damage caused by fibrosis is fractal analysis. This work used the Hausdorff fractal dimension (HFD) method to calculate and analyze the 2D topological surface. Full article
(This article belongs to the Special Issue Physical Phenomena on Fractals and in Fractional Dimension Spaces)
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13 pages, 9326 KiB  
Article
Creep Properties of a Viscoelastic 3D Printed Sierpinski Carpet-Based Fractal
by Juan B. Pascual-Francisco, Orlando Susarrey-Huerta, Leonardo I. Farfan-Cabrera and Rockali Flores-Hernández
Fractal Fract. 2023, 7(8), 568; https://doi.org/10.3390/fractalfract7080568 - 25 Jul 2023
Cited by 2 | Viewed by 1373
Abstract
In this paper, the phenomenon of creep compliance and the creep Poisson’s ratio of a 3D-printed Sierpinski carpet-based fractal and its bulk material (flexible resin Resione F69) was experimentally investigated, as well as the quantification of the change in the viscoelastic parameters of [...] Read more.
In this paper, the phenomenon of creep compliance and the creep Poisson’s ratio of a 3D-printed Sierpinski carpet-based fractal and its bulk material (flexible resin Resione F69) was experimentally investigated, as well as the quantification of the change in the viscoelastic parameters of the material due to the fractal structure. The samples were manufactured via a vat photopolymerization method. The fractal structure of the samples was based on the Sierpinski carpet at the fourth iteration. In order to evaluate the response of both the fractal and the bulk material under the creep phenomenon, 1 h-duration tensile creep tests at three constant temperatures (20, 30 and 40 °C) and three constant stresses (0.1, 0.2 and 0.3 MPa) were conducted. A digital image correlation (DIC) technique was implemented for strain measurement in axial and transverse directions. From the results obtained, the linear viscoelastic behavior regime of the fractal and the bulk material was identified. The linear viscoelastic parameters of both fractal and bulk materials were then estimated by fitting the creep Burgers model to the experimental data to determine the effect of the fractal geometry on the viscoelastic properties of the samples. Overall, it was found that the reduction in stiffness induced by the fractal porosity caused a more viscous behavior of the material and a reduction in its creep Poisson’s ratio, which means an increase in the compliance of the material. Full article
(This article belongs to the Special Issue Physical Phenomena on Fractals and in Fractional Dimension Spaces)
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8 pages, 759 KiB  
Article
Effects of Hausdorff Dimension on the Static and Free Vibration Response of Beams with Koch Snowflake-like Cross Section
by Didier Samayoa, Helvio Mollinedo, José Alfredo Jiménez-Bernal and Claudia del Carmen Gutiérrez-Torres
Fractal Fract. 2023, 7(2), 153; https://doi.org/10.3390/fractalfract7020153 - 4 Feb 2023
Cited by 3 | Viewed by 1286
Abstract
In this manuscript, static and free vibration responses on Euler–Bernoulli beams with a Koch snowflake cross-section are studied. By applying the finite element method, the transversal displacement in static load condition, natural frequencies, and vibration modes are solved and validated using Matlab. For [...] Read more.
In this manuscript, static and free vibration responses on Euler–Bernoulli beams with a Koch snowflake cross-section are studied. By applying the finite element method, the transversal displacement in static load condition, natural frequencies, and vibration modes are solved and validated using Matlab. For each case presented, the transversal displacement and natural frequency are analyzed as a Hausdorff dimension function. It is found that the maximum displacement increases as the Hausdorff dimension increases, with the relationship ymax=k0.79lndH+0.37, being k the iteration number of pre-fractal. The natural frequencies increase as ω∼M2.51, whereas the bending stiffness is expressed as EI=1165.4ln(dH+k). Numerical examples are given in order to discuss the mechanical implications. Full article
(This article belongs to the Special Issue Physical Phenomena on Fractals and in Fractional Dimension Spaces)
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11 pages, 694 KiB  
Article
Flow of a Self-Similar Non-Newtonian Fluid Using Fractal Dimensions
by Abdellah Bouchendouka, Zine El Abiddine Fellah, Zakaria Larbi, Nicholas O. Ongwen, Erick Ogam, Mohamed Fellah and Claude Depollier
Fractal Fract. 2022, 6(10), 582; https://doi.org/10.3390/fractalfract6100582 - 11 Oct 2022
Cited by 7 | Viewed by 1468
Abstract
In this paper, the study of the fully developed flow of a self-similar (fractal) power-law fluid is presented. The rheological way of behaving of the fluid is modeled utilizing the Ostwald–de Waele relationship (covering shear-thinning, Newtonian and shear-thickening fluids). A self-similar (fractal) fluid [...] Read more.
In this paper, the study of the fully developed flow of a self-similar (fractal) power-law fluid is presented. The rheological way of behaving of the fluid is modeled utilizing the Ostwald–de Waele relationship (covering shear-thinning, Newtonian and shear-thickening fluids). A self-similar (fractal) fluid is depicted as a continuum in a noninteger dimensional space. Involving vector calculus for the instance of a noninteger dimensional space, we determine an analytical solution of the Cauchy equation for the instance of a non-Newtonian self-similar fluid flow in a cylindrical pipe. The plot of the velocity profile obtained shows that the rheological behavior of a non-Newtonian power-law fluid is essentially impacted by its self-similar structure. A self-similar shear thinning fluid and a self-similar Newtonian fluid take on a shear-thickening way of behaving, and a self-similar shear-thickening fluid becomes more shear thickening. This approach has many useful applications in industry, for the investigation of blood flow and fractal fluid hydrology. Full article
(This article belongs to the Special Issue Physical Phenomena on Fractals and in Fractional Dimension Spaces)
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Review

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21 pages, 4691 KiB  
Review
Percolation on Fractal Networks: A Survey
by Miguel-Ángel Martínez Cruz, Julián Patiño Ortiz, Miguel Patiño Ortiz and Alexander Balankin
Fractal Fract. 2023, 7(3), 231; https://doi.org/10.3390/fractalfract7030231 - 5 Mar 2023
Cited by 9 | Viewed by 2416
Abstract
The purpose of this survey is twofold. First, we survey the studies of percolation on fractal networks. The objective is to assess the current state of the art on this topic, emphasizing the main findings, ideas and gaps in our understanding. Secondly, we [...] Read more.
The purpose of this survey is twofold. First, we survey the studies of percolation on fractal networks. The objective is to assess the current state of the art on this topic, emphasizing the main findings, ideas and gaps in our understanding. Secondly, we try to offer guidelines for future research. In particular, we focus on effects of fractal attributes on the percolation in self-similar networks. Some challenging questions are outlined. Full article
(This article belongs to the Special Issue Physical Phenomena on Fractals and in Fractional Dimension Spaces)
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Other

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12 pages, 503 KiB  
Brief Report
A Mechanical Picture of Fractal Darcy’s Law
by Lucero Damián Adame, Claudia del Carmen Gutiérrez-Torres, Bernardo Figueroa-Espinoza, Juan Gabriel Barbosa-Saldaña and José Alfredo Jiménez-Bernal
Fractal Fract. 2023, 7(9), 639; https://doi.org/10.3390/fractalfract7090639 - 22 Aug 2023
Cited by 5 | Viewed by 1140
Abstract
The main goal of this manuscript is to generalize Darcy’s law from conventional calculus to fractal calculus in order to quantify the fluid flow in subterranean heterogeneous reservoirs. For this purpose, the inherent features of fractal sets are scrutinized. A set of fractal [...] Read more.
The main goal of this manuscript is to generalize Darcy’s law from conventional calculus to fractal calculus in order to quantify the fluid flow in subterranean heterogeneous reservoirs. For this purpose, the inherent features of fractal sets are scrutinized. A set of fractal dimensions is incorporated to describe the geometry, morphology, and fractal topology of the domain under study. These characteristics are known through their Hausdorff, chemical, shortest path, and elastic backbone dimensions. Afterward, fractal continuum Darcy’s law is suggested based on the mapping of the fractal reservoir domain given in Cartesian coordinates xi into the corresponding fractal continuum domain expressed in fractal coordinates ξi by applying the relationship ξi=ϵ0(xi/ϵ0)αi−1, which possesses local fractional differential operators used in the fractal continuum calculus framework. This generalized version of Darcy’s law describes the relationship between the hydraulic gradient and flow velocity in fractal porous media at any scale including their geometry and fractal topology using the αi-parameter as the Hausdorff dimension in the fractal directions ξi, so the model captures the fractal heterogeneity and anisotropy. The equation can easily collapse to the classical Darcy’s law once we select the value of 1 for the alpha parameter. Several flow velocities are plotted to show the nonlinearity of the flow when the generalized Darcy’s law is used. These results are compared with the experimental data documented in the literature that show a good agreement in both high-velocity and low-velocity fractal Darcian flow with values of alpha equal to 0<α1<1 and 1<α1<2, respectively, whereas α1=1 represents the standard Darcy’s law. In that way, the alpha parameter describes the expected flow behavior which depends on two fractal dimensions: the Hausdorff dimension of a porous matrix and the fractal dimension of a cross-section area given by the intersection between the fractal matrix and a two-dimensional Cartesian plane. Also, some physical implications are discussed. Full article
(This article belongs to the Special Issue Physical Phenomena on Fractals and in Fractional Dimension Spaces)
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