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Applications of General Fractional Calculus Models: Insights into Viscoelasticity and...
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Applications of General Fractional Calculus Models: Insights into Viscoelasticity and Wave Propagation

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

Deadline for manuscript submissions: 20 March 2026 | Viewed by 4322

Special Issue Editors

Dr. Yiying Feng

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Guest Editor
School of Mathematics and Statistics, Suzhou University, Suzhou 234000, China
Interests: general fractional calculus; creep constitutive model; fractional viscoelastic mechanics
Special Issues, Collections and Topics in MDPI journals
Dr. Jiangen Liu

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Changshu Institute of Technology, Changshu 215500, China
Interests: fractional calculus; local fractional calculus; mathematical physics
Special Issues, Collections and Topics in MDPI journals
Dr. Yiming Wang

E-Mail Website
Guest Editor
School of Resources and Civil Engineering, Suzhou University, Suzhou 234000, China
Interests: fractal theory; fractional constitutive model; green filling materials; cemented backfill technology
Prof. Dr. Alex Elías-Zúñiga

E-Mail Website
Guest Editor
Institute of Advanced Materials for Sustainable Manufacturing, Tecnologico de Monterrey, Avenue Eugenio Garza Sada 2501, Monterrey 64849, NL, Mexico
Interests: fractal oscillators; mathematical modeling of multiscale phenomena; semi-analytical methods; fractal rheology of advanced polymer materials; two-scale fractal theory; nonlinear dynamic systems
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional derivatives and fractal theory play important roles in engineering applications, particularly in describing the mechanical behavior of complex materials and the properties of viscoelastic materials. Fractional derivatives and fractal theory can not only simulate the properties of real materials more accurately, but also explain some phenomena that traditional models cannot explain.

Firstly, the application of fractional derivatives in engineering is mainly reflected in their description of viscoelastic materials. The traditional integer order derivative equation cannot accurately describe the memory and frequency dependence of damping in viscoelastic materials, while the fractional order derivative model can better reflect these characteristics. For example, in geotechnical engineering, it is used to describe the viscoelastic plasticity of materials, and in statistical mechanics and quantum mechanics, it is used to describe anomalous diffusion phenomena in complex systems.

Secondly, the application of fractional derivatives and fractal theory in wave propagation can describe the complex behavior and phenomena of wave propagation. Fractional derivatives have characteristics such as global correlation, memory, and heritability, which enable them to better simulate physical processes with long-term memory and historical dependence. Fractal theory can describe objects or phenomena with self-similarity and complex structures, which is particularly important in wave propagation.

In summary, fractional derivatives and fractal theory have important theoretical and practical significance in engineering applications by providing more accurate mathematical tools to describe the complex behavior of materials.

Based on this, the topic of this Special Issue aims to highlight the new background and challenges of fractional calculus in the field of viscoelasticity and wave propagation. Expanding traditional integer dimensional space to fractional dimensional space involves the intersection of multiple disciplines such as mathematics and physics. Through these in-depth studies and explorations, this Special Issue will help us better understand the essence of complex phenomena in fractional dimensional space and provide more accurate mathematical expressions for its engineering applications.

Dr. Yiying Feng
Dr. Jiangen Liu
Dr. Yiming Wang
Prof. Dr. Alex Elías-Zúñiga
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

  • general fractional calculus
  • integration of mathematics and engineering
  • fractional rheological models
  • fractal dimensional space
  • nonlinear theory
  • fractional wave propagation models

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Published Papers (7 papers)

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Research

10 pages, 1750 KiB  
Article
Local Fractional Modeling of Microorganism Physiology Arising in Wastewater Treatment: Lawrence–McCarty Model in Cantor Sets
by Yiming Wang, Yiying Feng, Xiurong Xu and Shoubo Jin
Fractal Fract. 2025, 9(7), 413; https://doi.org/10.3390/fractalfract9070413 - 25 Jun 2025
Viewed by 394
Abstract
Water pollution from industrial and domestic sewage demands the accurate modeling of wastewater treatment processes. While the Lawrence–McCarty model is widely used for activated sludge systems, its integer-order formulation cannot fully capture the fractal characteristics of microbial aggregation. This study proposed a fractal [...] Read more.
Water pollution from industrial and domestic sewage demands the accurate modeling of wastewater treatment processes. While the Lawrence–McCarty model is widely used for activated sludge systems, its integer-order formulation cannot fully capture the fractal characteristics of microbial aggregation. This study proposed a fractal Lawrence–McCarty model (FLMM) by incorporating local fractional derivatives (α = ln2/ln3) to describe microbial growth dynamics on Cantor sets. Theoretical analysis reveals that the FLMM exhibits Mittag-Leffler-type solutions, which naturally generate step-wise growth curves—consistent with the phased behavior (lag, rapid growth, and stabilization) observed in real sludge systems. Compared with classical models, the FLMM’s fractional-order structure provides a more flexible framework to represent memory effects and spatial heterogeneity in microbial communities. These advances establish a mathematical foundation for future experimental validation and suggest potential improvements in predicting nonlinear biomass accumulation patterns. Full article
(This article belongs to the Special Issue Applications of General Fractional Calculus Models: Insights into Viscoelasticity and Wave Propagation)
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14 pages, 2132 KiB  
Article
Using He’s Two-Scale Fractal Transform to Predict the Dynamic Response of Viscohyperelastic Elastomers with Fractal Damping
by Alex Elías-Zúñiga, Oscar Martínez-Romero, Daniel Olvera-Trejo and Luis Manuel Palacios-Pineda
Fractal Fract. 2025, 9(6), 357; https://doi.org/10.3390/fractalfract9060357 - 29 May 2025
Viewed by 368
Abstract
This article aims to clarify the applicability of He’s two-scale fractal dimension transform by replacing tα with τ. It demonstrates the potential to capture the influence of the fractal parameter on the system’s damping frequency, particularly when the viscoelastic term (damping) [...] Read more.
This article aims to clarify the applicability of He’s two-scale fractal dimension transform by replacing tα with τ. It demonstrates the potential to capture the influence of the fractal parameter on the system’s damping frequency, particularly when the viscoelastic term (damping) does not equal half of the fractional inertia force term. The analysis examines the elastomer materials’ dynamic fractal amplitude–time response, considering the viscohyperelastic effects related to the material’s energy dissipation capacity. To determine the amplitude of oscillations for the nonlinear equation of motion of a body supported by a viscohyperelastic elastomer subjected to uniaxial stretching, the harmonic balance perturbation method, combined with the two-scale fractal dimension transform and Ross’s formula, is employed. Numerical calculations demonstrate the effectiveness of He’s two-scale fractal transformation in capturing fractal phenomena associated with the fractional time derivative of deformation. This is due to a correlation between the fractional rate of viscoelasticity and the fractal structure of media in elastomer materials, which is reflected in the oscillation amplitude decay. Furthermore, the approach introduced by El-Dib to replace the original fractional equation of motion with an equivalent linear oscillator with integer derivatives is used to further assess the qualitative and quantitative performance of our derived solution. The proposed approach elucidates the applicability of He’s two-scale fractal calculus for determining the amplitude of oscillations in viscohyperelastic systems, where the fractal derivative order of the inertia and damping terms varies. Full article
(This article belongs to the Special Issue Applications of General Fractional Calculus Models: Insights into Viscoelasticity and Wave Propagation)
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11 pages, 712 KiB  
Article
Qualitative Analysis and Traveling Wave Solutions of a (3 + 1)- Dimensional Generalized Nonlinear Konopelchenko-Dubrovsky-Kaup-Kupershmidt System
by Zhao Li and Ejaz Hussain
Fractal Fract. 2025, 9(5), 285; https://doi.org/10.3390/fractalfract9050285 - 27 Apr 2025
Cited by 10 | Viewed by 401
Abstract
This article investigates the qualitative analysis and traveling wave solutions of a (3 + 1)-dimensional generalized nonlinear Konopelchenko-Dubrovsky-Kaup-Kupershmidt system. This equation is commonly used to simulate nonlinear wave problems in the fields of fluid mechanics, plasma physics, and nonlinear optics, as well as [...] Read more.
This article investigates the qualitative analysis and traveling wave solutions of a (3 + 1)-dimensional generalized nonlinear Konopelchenko-Dubrovsky-Kaup-Kupershmidt system. This equation is commonly used to simulate nonlinear wave problems in the fields of fluid mechanics, plasma physics, and nonlinear optics, as well as to transform nonlinear partial differential equations into nonlinear ordinary differential equations through wave transformations. Based on the analysis of planar dynamical systems, a nonlinear ordinary differential equation is transformed into a two-dimensional dynamical system, and the qualitative behavior of the two-dimensional dynamical system and its periodic disturbance system is studied. A two-dimensional phase portrait, three-dimensional phase portrait, sensitivity analysis diagrams, Poincaré section diagrams, and Lyapunov exponent diagrams are provided to illustrate the dynamic behavior of two-dimensional dynamical systems with disturbances. The traveling wave solution of a Konopelchenko-Dubrovsky-Kaup-Kupershmidt system is studied based on the complete discriminant system method, and its three-dimensional, two-dimensional graphs and contour plots are plotted. These works can provide a deeper understanding of the dynamic behavior of Konopelchenko-Dubrovsky-Kaup-Kupershmidt systems and the propagation process of waves. Full article
(This article belongs to the Special Issue Applications of General Fractional Calculus Models: Insights into Viscoelasticity and Wave Propagation)
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30 pages, 29741 KiB  
Article
Evolution Characteristics of Pore–Fractures and Mechanical Response of Dehydrated Lignite Based on In Situ Computed Tomography (CT) Scanning
by Shuai Yan, Lijun Han, Shasha Zhang, Weisheng Zhao and Lingdong Meng
Fractal Fract. 2025, 9(4), 220; https://doi.org/10.3390/fractalfract9040220 - 31 Mar 2025
Viewed by 425
Abstract
Based on the uniaxial compression tests and in situ CT scanning experiments of lignite with different dehydration times and the fractal theory, this paper qualitatively and quantitatively investigated the influence of the dehydration effect on the evolution of pore–fractures and the mechanical behavior [...] Read more.
Based on the uniaxial compression tests and in situ CT scanning experiments of lignite with different dehydration times and the fractal theory, this paper qualitatively and quantitatively investigated the influence of the dehydration effect on the evolution of pore–fractures and the mechanical behavior of lignite under uniaxial compression conditions. The results show that the dehydration effect significantly affects the pre-peak deformation and post-peak failure behavior of lignite but has no significant impact on its peak strength. The pore–fracture parameters, such as the fractal dimension, surface porosity, and fracture volume, of three samples all exhibit an evolutionary pattern of “continuous decrease in the compaction and elastic stages–gradual increase in the plastic stage–sharp growth in the post-peak stage” with the dynamic evolution of the pore–fractures. However, the dehydration effect leads to an increase in the intensity of pore–crack evolution and a nonlinear rise in all the parameters characterizing the pore–crack complexity during uniaxial compression, which, in turn, leads to an increment in the fluctuation of the above evolutionary trends. The mechanism underlying the differential influence of the dehydration effect on the macroscopic mechanical behavior of lignite is follows: The dehydration effect non-linearly and positively affects the initial pore–fracture structure of lignite, thereby non-linearly and positively promoting the evolution of pore–fractures during the loading process. Nevertheless, since it fails to weaken the micro-mechanical properties of lignite and cannot form effective through-going fractures, it has no significant impact on the uniaxial compressive strength of the coal samples. The findings of this study can provide some references for the support design and deformation control of underground lignite roadways. Full article
(This article belongs to the Special Issue Applications of General Fractional Calculus Models: Insights into Viscoelasticity and Wave Propagation)
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23 pages, 4764 KiB  
Article
Effects of Different Aggregate Gradations and CO2 Nanobubble Water Concentrations on Mechanical Properties and Damage Behavior of Cemented Backfill Materials
by Xiaoxiao Cao, Meimei Feng, Haoran Bai and Taifeng Wu
Fractal Fract. 2025, 9(4), 217; https://doi.org/10.3390/fractalfract9040217 - 30 Mar 2025
Viewed by 431
Abstract
Against the backdrop of increasingly severe global climate challenges, various industries are in urgent need of developing materials that can both improve performance and reduce carbon emissions. In this study, carbon dioxide nanobubble water (CO2NBW) was evaluated as an innovative additive [...] Read more.
Against the backdrop of increasingly severe global climate challenges, various industries are in urgent need of developing materials that can both improve performance and reduce carbon emissions. In this study, carbon dioxide nanobubble water (CO2NBW) was evaluated as an innovative additive for cemented backfill materials (CBMs), and its optimization effect on the mechanical properties and microstructure of the materials was explored. The effects of different concentrations of CO2NBW on stress–strain behavior, compressive strength, and microstructure were studied by uniaxial compression tests and scanning electron microscopy (SEM) analysis. The results show that with changes in CO2NBW concentration and fractal dimension, the uniaxial compressive strength (UCS), peak strain, and elastic modulus of the specimens first increase and then decrease. At the optimal concentration level (C = 3) and fractal dimension (2.4150–2.6084), UCS reaches a peak value of 24.88 MPa, which is significantly higher than the initial value (C = 1). The peak strain and elastic modulus also reach maximum values of 0.01231 and 3.005 GPa, respectively. When the fractal dimension was between 2.4150 and 2.6084, the microstructural optimization effect of CO2NBW on CBM was most significant, which was reflected in the compactness of the internal pore structure and the thoroughness of the hydration degree. In addition, based on the close correlation between peak strain and elastic modulus and UCS, a damage constitutive model of CBM specimens considering the influence of CO2NBW concentration and fractal dimension was constructed. The study also found that the damage of CBM specimens is normally distributed with strain, and the accumulated damage in the plastic deformation stage dominates the total damage. Full article
(This article belongs to the Special Issue Applications of General Fractional Calculus Models: Insights into Viscoelasticity and Wave Propagation)
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12 pages, 3048 KiB  
Article
A Fractional Hybrid Staggered-Grid Grünwald–Letnikov Method for Numerical Simulation of Viscoelastic Seismic Wave Propagation
by Xinmin Zhang, Guojie Song, Puchun Chen and Dan Wang
Fractal Fract. 2025, 9(3), 153; https://doi.org/10.3390/fractalfract9030153 - 28 Feb 2025
Viewed by 495
Abstract
The accurate and efficient simulation of seismic wave energy dissipation and phase dispersion during propagation in subsurface media due to inelastic attenuation is critical for the hydrocarbon-bearing distinction and improving the quality of seismic imaging in strongly attenuating geological media. The fractional viscoelastic [...] Read more.
The accurate and efficient simulation of seismic wave energy dissipation and phase dispersion during propagation in subsurface media due to inelastic attenuation is critical for the hydrocarbon-bearing distinction and improving the quality of seismic imaging in strongly attenuating geological media. The fractional viscoelastic equation, which quantifies frequency-independent anelastic effects, has recently become a focal point in seismic exploration. We have developed a novel hybrid staggered-grid Grünwald–Letnikov (HSGGL) finite difference method for solving the fractional viscoelastic equation in the time domain. The proposed method achieves accurate and computationally efficient solutions by using a staggered grid to discretize the first-order partial derivatives of the velocity–stress equations, combined with Grünwald–Letnikov finite difference discretization for the fractional-order terms. To improve the computational efficiency, we employ a preset accuracy to truncate the difference stencil, resulting in a compact fractional-order difference scheme. A stability analysis using the eigenvalue method reveals that the proposed method confers a relaxed stability condition, providing greater flexibility in the selection of sampling intervals. The numerical experiments indicate that the HSGGL method achieves a maximum relative error of no more than 0.17% compared to the reference solution (on a finely meshed domain) while being significantly faster than the conventional global FD method (GFD). In a 500 × 500 computational domain, the computation times for the proposed methods, which meet the specified accuracy levels used, are only approximately 4.67%, 4.47%, 4.44%, and 4.42% of that of the GFD method. This indicates that the novel HSGGL method has the potential as an effective forward modeling tool for understanding complex subsurface structures by employing a fractional viscoelastic equation. Full article
(This article belongs to the Special Issue Applications of General Fractional Calculus Models: Insights into Viscoelasticity and Wave Propagation)
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27 pages, 16020 KiB  
Article
Pore Structure and Its Fractal Dimension: A Case Study of the Marine Shales of the Niutitang Formation in Northwest Hunan, South China
by Wei Jiang, Yang Zhang, Tianran Ma, Song Chen, Yang Hu, Qiang Wei and Dingxiang Zhuang
Fractal Fract. 2025, 9(1), 49; https://doi.org/10.3390/fractalfract9010049 - 17 Jan 2025
Cited by 2 | Viewed by 957
Abstract
To analyze the pore structure and fractal characteristics of marine shale in the lower Cambrian Niutitang Formation in northwestern Hunan Province, China, the pore characteristics of shale were characterized using total organic carbon (TOC) content, field emission scanning electron microscopy (FESEM), X-ray diffraction [...] Read more.
To analyze the pore structure and fractal characteristics of marine shale in the lower Cambrian Niutitang Formation in northwestern Hunan Province, China, the pore characteristics of shale were characterized using total organic carbon (TOC) content, field emission scanning electron microscopy (FESEM), X-ray diffraction (XRD), low temperature nitrogen adsorption (LT-N2GA) and methane adsorption experiments. The pore surface and pore space fractal dimensions of samples were calculated, respectively. The influencing factors of fractal dimensions and their impact on the adsorption of shale reservoirs were discussed. The results indicate the Niutitang Formation shale mainly develops four types of pores: organic pores, intragranular pores, intergranular pores and microcracks. The pores have a large specific surface area (SSA), primarily consisting of mesopores. The fractal dimensions are calculated using the FHH model and the XS model. The fractal dimensions (D2 and Df) are greater than D1, indicating that the pore surface with larger pore size is rougher, and the pore structure of shale is complex. The pore volume (PV), SSA, and TOC show positive correlations with the fractal dimensions but negative correlations with APS. There is no obvious correlation between fractal dimensions and quartz content, while clay minerals show a negative correlation with D2 and Df. This is mainly because clay mineral particles are small in size and have weak resistance to compaction. The pyrite content is positively correlated with the fractal dimensions because pyrite promotes the development of organic, intergranular, and mold pores. According to Pearson correlation analysis, the main influencing factors of the pore surface fractal dimension are PV, SSA, and APS. The main influencing factors of the pore space fractal dimension are APS and the content of clay minerals. Further analysis of the influence of the fractal dimension on the adsorption capacity of shale reveals that the fractal dimensions are positively correlated with Langmuir volume, indicating that fractal dimensions can be used as a quantitative target for evaluating shale gas reservoirs. Full article
(This article belongs to the Special Issue Applications of General Fractional Calculus Models: Insights into Viscoelasticity and Wave Propagation)
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