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Fractal Fract
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Fractal and Fractional in Geomaterials
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Fractal and Fractional in Geomaterials

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

Deadline for manuscript submissions: closed (30 March 2022) | Viewed by 34337

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Dr. Yifei Sun

E-Mail Website
Guest Editor
Faculty of Civil and Environmental Engineering, Ruhr-Universität Bochum, 44801 Bochum, Germany
Interests: fractional plasticity; fractional viscoelasticity; fractals in particle breakage; geomechanics; heat conduction
Special Issues, Collections and Topics in MDPI journals
Dr. Cheng Chen

E-Mail Website
Guest Editor
School of Civil Engineering and Architecture, Wuhan University of Technology, Wuhan 430070, China
Interests: micromechanics; microstructure; geomaterials; discrete element methods
Dr. Meisam Goudarzy

E-Mail Website
Guest Editor
Faculty of Civil and Environmental Engineering, Ruhr-Universität, Bochum, Germany
Interests: particle breakage; micromechanics; granular soils; peats; geomaterials; discrete element methods; soil dynamic; seismic ground response; earthquake; ground vibration

Special Issue Information

Dear colleagues,

Geomaterial is one of the most common materials in the world. Based on its function, there can be different types of geomaterials, e.g., soft soils, granular aggregates, and composite material. However, no matter what kind of geomaterials are used, fractal laws can be observed in material responses. For example, the pore size or contact force network within granular aggregates can obey fractal distribution, which can be described using fractional calculus or other advanced mathematical tools. In recent years, the application of fractal theory and fractional mechanics in characterizing the micro-to-macro behavior of geomaterials has attracted worldwide attention.

The aim of this Special Issue is to present state-of-the-art research outcomes in fractal or fractional approaches for geomaterials. Therefore, high-quality review papers, full-length research articles, and technical notes from different disciplines are cordially welcome. Research topics include but are not limited to the following aspects:

  1. Fractal laws for contact networks/pore structures in geomaterial;
  2. Fractional mechanics modelling of geomaterial;
  3. Local/nonlocal deformation in geomaterial;
  4. Non-Fourier/non-Fickian/non-Darcy law for geomaterial.
  5. Other nonconventional approaches for geomaterials.

Dr. Yifei Sun
Dr. Cheng Chen
Dr. Meisam Goudarzy
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

  • fractal media
  • fractional mechanics
  • Micro-to-Macro
  • geomaterial
  • nonconventional approach

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Published Papers (17 papers)

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Editorial

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3 pages, 193 KiB  
Editorial
Editorial for Special Issue “Fractal and Fractional in Geomaterials”
by Yifei Sun, Cheng Chen and Meisam Goudarzy
Fractal Fract. 2023, 7(1), 55; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract7010055 - 01 Jan 2023
Viewed by 988
Abstract
Geomaterials, such as clay, sand, rockfill and ballast, etc [...] Full article
(This article belongs to the Special Issue Fractal and Fractional in Geomaterials)

Research

Jump to: Editorial

16 pages, 2815 KiB  
Article
A Fractal Entropy-Based Effective Particle Model Used to Deduce Hydraulic Conductivity of Granular Soils
by Gang Zhang, Hongyu Wang, Jahanzaib Israr, Wenguo Ma, Youzhen Yang and Keliang Ren
Fractal Fract. 2022, 6(9), 474; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6090474 - 28 Aug 2022
Cited by 5 | Viewed by 1206
Abstract
In this study, a rigorous mathematical approach used to compute an effective diameter based on particle size distribution (PSD) has been presented that can predict the hydraulic conductivity of granular soils with enhanced rigor. The PSD was discretized based on an abstract interval [...] Read more.
In this study, a rigorous mathematical approach used to compute an effective diameter based on particle size distribution (PSD) has been presented that can predict the hydraulic conductivity of granular soils with enhanced rigor. The PSD was discretized based on an abstract interval system of fractal entropy, while the effective diameter of soil was computed using the grading entropy theory. The comparisons between current entropy-based effective diameter (DE) and those computed using existing procedures show that the current DE can capture the particle size information of a given soil more accurately than others. Subsequently, the proposed DE was successfully implicated into Kozeny–Carman’s formula to deduce the saturated hydraulic conductivity of soils with enhanced accuracy. The proposed model was tested using current and previously published experimental data from literature. Not surprisingly, the results of the current model and those from previous experimental studies were found to be consistent, which can sufficiently verify the proposed entropy-based effective diameter model. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geomaterials)
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12 pages, 3966 KiB  
Article
The Fractal Characteristics of Soft Soil under Cyclic Loading Based on SEM
by Bowen Kong, Chen-Xiang Dai, Haibo Hu, Jianzhong Xia and Shao-Heng He
Fractal Fract. 2022, 6(8), 423; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6080423 - 30 Jul 2022
Cited by 9 | Viewed by 1183
Abstract
Cyclic loading always results in great damage to the pore structure and fractal characteristics of soft soil. Scanning electron microscope (SEM) can help collect data to describe the microstructure of soft soil. This paper conducted a series of SEM tests to interpret the [...] Read more.
Cyclic loading always results in great damage to the pore structure and fractal characteristics of soft soil. Scanning electron microscope (SEM) can help collect data to describe the microstructure of soft soil. This paper conducted a series of SEM tests to interpret the effect of consolidation confining pressure, circulating dynamic stress ratios and overconsolidation ratio on soil’s micro-pore structure and fractal characteristics. The results demonstrate that fractal dimension can well represent the complex characteristics of the microstructure of the soil; the larger the consolidation confining pressure, the greater the cyclic dynamic stress ratio, and the greater the overconsolidation ratio, the smaller the fractal dimension number of soil samples. Finally, an empirical fitting formula for cumulative strain considering microstructure parameters is established through data fitting. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geomaterials)
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15 pages, 6754 KiB  
Article
Discrete Element Modelling of Fractal Behavior of Particle Size Distribution and Breakage of Ballast under Monotonic Loading
by Cheng Chen, Xin Zhang, Yifei Sun, Lei Zhang, Rui Rui and Zhide Wang
Fractal Fract. 2022, 6(7), 382; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6070382 - 06 Jul 2022
Cited by 4 | Viewed by 1370
Abstract
Breakage of particles has a great influence on the particle size distribution (PSD) and the associated mechanical behavior of ballast under train loads. A discrete element method (DEM) simulation of triaxial testing under monotonic loading was carried out using FRM (fragment replacement method) [...] Read more.
Breakage of particles has a great influence on the particle size distribution (PSD) and the associated mechanical behavior of ballast under train loads. A discrete element method (DEM) simulation of triaxial testing under monotonic loading was carried out using FRM (fragment replacement method) breakable particles as ballast and a flexible shell model as membrane. The coupled model was validated by comparing the load-deformation responses with those measured in previous experiments and was then used to analyze the contact orientations and the distribution of particle breakage from a micromechanical perspective. The simulation results show that higher confining pressure and larger axial strain may increase the grain breakage (Bg) and the fractal dimension (D) of ballast. It was observed that most breakage was first-generation breakage, and that the proportions of the second- to fifth-generation breakage decreased successively. Moreover, as the axial strain or confining pressure increased, the percentage of small particle fragments increased in correspondence with the PSD curves that remained concave upwards, as the fractal dimension D of PSD increased. In addition, the evolution of D exhibited a linear correlation with grain breakage Bg. Contrarily, a quadratic curve relation between D and volumetric strain was exhibited under different axial strain stages. Therefore, D has the potential to be a key indicator to evaluate the degree of ballast crushing and PSD degradation, which may contribute to better decision making concerning track bed maintenance. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geomaterials)
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17 pages, 3635 KiB  
Article
Effects of Relative Density and Grading on the Particle Breakage and Fractal Dimension of Granular Materials
by Gui Yang, Zhuanzhuan Chen, Yifei Sun and Yang Jiang
Fractal Fract. 2022, 6(7), 347; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6070347 - 22 Jun 2022
Cited by 4 | Viewed by 1379
Abstract
Particle breakage was reported to have great influence on the mechanical property of granular materials. However, limited studies were conducted to quantify the detailed effects of relative density and initial grading on the particle breakage behaviour of granular materials under different confining pressures. [...] Read more.
Particle breakage was reported to have great influence on the mechanical property of granular materials. However, limited studies were conducted to quantify the detailed effects of relative density and initial grading on the particle breakage behaviour of granular materials under different confining pressures. In this study, a series of monotonic drained triaxial tests were performed on isotropically consolidated granular materials with four different initial gradings and relative densities. It is observed that particle breakage increases as the confining pressure or relative density increases, whereas it decreases with the increasing coefficient of uniformity. Due to particle breakage, the grading curves of granular materials after triaxial tests can be simulated by a power-law function with fractal dimension. As the confining pressure increases, the fractal dimension approaches the limit of granular materials, i.e., 2.6. A unique normalized relation between the particle breakage extent and confining pressure by considering relative density and grading index was found. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geomaterials)
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20 pages, 6356 KiB  
Article
Investigation on Pore Structure and Permeability of Concrete–Rock Interfacial Transition Zones Based on Fractal Theory
by Juan Yue, Jinchang Sheng, Huimin Wang, Yunjin Hu, Kailai Zhang, Yulong Luo, Qing Zhou and Meili Zhan
Fractal Fract. 2022, 6(6), 329; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6060329 - 13 Jun 2022
Cited by 7 | Viewed by 1764
Abstract
The concrete–rock interfacial transition zone (ITZ) is generally considered the weak layer in hydraulic engineering, for it is more permeable than the intact concrete or rocks. The water permeability of the ITZ is a critical parameter concerned with structural safety and durability. However, [...] Read more.
The concrete–rock interfacial transition zone (ITZ) is generally considered the weak layer in hydraulic engineering, for it is more permeable than the intact concrete or rocks. The water permeability of the ITZ is a critical parameter concerned with structural safety and durability. However, the permeability and pore structure of the ITZ has not been investigated previously, and the mathematical model of ITZ permeability has not been established. This study performed multi-scale experiments on the concrete–rock ITZ with various rock types (limestone, granite, and sandstone). A series of quantitative and qualitative analysis techniques, including NMR, SEM-EDS, and XRD, characterize the ITZ pore structures. The controlled constant flow method was used to determine the permeability of the concrete, rock, and ITZ. The mathematical model of ITZ permeability was proposed using the fractal theory. The consistency between the experimental data and the proposed model indicates the reliability of this study. The results of the experiment show that ITZ permeability is between 4.08 × 10−18 m2 and 5.74 × 10−18 m2. The results of the experiment and the proposed model could determine ITZ permeability in hydraulic structure safety and durability analysis. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geomaterials)
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14 pages, 3475 KiB  
Article
Seepage–Fractal Model of Embankment Soil and Its Application
by Xiaoming Zhao, Binbin Yang, Shichong Yuan, Zhenzhou Shen and Di Feng
Fractal Fract. 2022, 6(5), 277; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6050277 - 22 May 2022
Cited by 6 | Viewed by 1842
Abstract
Over time and across space, the hydraulic conductivity, fractal dimension, and porosity of embankment soil have strong randomness, which makes analyzing seepage fields difficult, affecting embankment risk analysis and early disaster warning. This strong randomness limits the application of fractal theory in embankment [...] Read more.
Over time and across space, the hydraulic conductivity, fractal dimension, and porosity of embankment soil have strong randomness, which makes analyzing seepage fields difficult, affecting embankment risk analysis and early disaster warning. This strong randomness limits the application of fractal theory in embankment engineering and sometimes keeps it in the laboratory stage. Based on the capillary model of porous soil, an analytical formula of the fractal relationship between hydraulic conductivity and fractal dimension is derived herein. It is proposed that the influencing factors of hydraulic conductivity of embankment soil mainly include the capillary aperture, fractal dimension, and fluid viscosity coefficient. Based on random field theory and combined with the embankment parameters of Shijiu Lake, hydraulic conductivity is discretized, and then the soil fractal dimension is approximately solved to reveal the internal relationship between hydraulic gradient, fractal dimension, and hydraulic conductivity. The results show that an increased fractal dimension will reduce the connectivity of soil pores in a single direction, increase the hydraulic gradient, and reduce the hydraulic conductivity. A decreased fractal dimension will lead to consistency of seepage channels in the soil, increased hydraulic conductivity, and decreased hydraulic gradient. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geomaterials)
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14 pages, 3888 KiB  
Article
Macro- and Micromechanical Assessment of the Influence of Non-Plastic Fines and Stress Anisotropy on the Dynamic Shear Modulus of Binary Mixtures
by Meisam Goudarzy and Debdeep Sarkar
Fractal Fract. 2022, 6(4), 205; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6040205 - 06 Apr 2022
Cited by 3 | Viewed by 1820
Abstract
Resonant column tests were carried out on Hostun sand mixed with 5%, 10% and 20% non-plastic fines (defined as grains smaller than 0.075 mm) in order to quantify the combined influence of the void ratio (e), anisotropic stress state (defined as [...] Read more.
Resonant column tests were carried out on Hostun sand mixed with 5%, 10% and 20% non-plastic fines (defined as grains smaller than 0.075 mm) in order to quantify the combined influence of the void ratio (e), anisotropic stress state (defined as σv′/σh′) and fines content (fc) on the maximum small-strain shear modulus Gmax. A significant reduction in the Gmax with increasing fc was observed. Using the empirical model forwarded by Roesler, the influence of e and σv′/σh′ on Gmax was captured, although the model was unable to capture the influence of varying fines content using a single equation. From the micro-CT images, a qualitative observation of the initial skeletal structure of the ‘fines-in-sand’ grains was performed and the equivalent granular void ratio e* was determined. The e was henceforth replaced by e* in Roesler’s equation in order to capture the variation in fc. The new modification was quantified in terms of the mean square error R2. Furthermore, the Gmax of Hostun sand–fine mixtures was predicted with good accuracy by replacing e with e*. Additionally, a micromechanical interpretation based on the experimental observation was developed. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geomaterials)
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20 pages, 4987 KiB  
Article
Influence of Groundwater Depth on Pile–Soil Mechanical Properties and Fractal Characteristics under Cyclic Loading
by Bingxiang Yuan, Zhijie Li, Weijie Chen, Jin Zhao, Jianbing Lv, Jie Song and Xudong Cao
Fractal Fract. 2022, 6(4), 198; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6040198 - 01 Apr 2022
Cited by 68 | Viewed by 3741
Abstract
The analysis of the behavior of soil and foundations when the piles in offshore areas are subjected to long-term lateral loading (wind) is one of the major problems associated with the smooth operation of superstructure. The strength of the pile-soil system is influenced [...] Read more.
The analysis of the behavior of soil and foundations when the piles in offshore areas are subjected to long-term lateral loading (wind) is one of the major problems associated with the smooth operation of superstructure. The strength of the pile-soil system is influenced by variations in the water content of the soil. At present, there are no studies carried out analyzing the mechanical and deformational behavior of both the material of the laterally loaded piles and soil with groundwater level as a variable. In this paper, a series of 1-g model tests were conducted to explore the lateral behavior of both soil and monopile under unidirectional cyclic loading, based on the foundation of an offshore wind turbine near the island. The influence of underground water level and cyclic load magnitude on the performance of the pile–soil system was analyzed. To visualize the movements of soil particles during the experimental process, particle image velocimetry (PIV) was used to record the soil displacement field under various cyclic loading conditions. The relationship curves between pile top displacement and cyclic steps, as well as the relationship curves between cyclic stiffness and cyclic steps, were displayed. Combined with fractal theory, the fractal dimension of each curve was calculated to evaluate the sensitivity of the pile–soil interaction system. The results showed that cyclic loading conditions and groundwater depth are the main factors affecting the pile–soil interaction. The cyclic stiffness of the soil increased in all test groups as loading progressed; however, an increase in the cyclic load magnitude decreased the initial and cyclic stiffness. The initial and cyclic stiffness of dry soil was higher than that of saturated soil, but less than that of unsaturated soil. The ability of the unsaturated soil to limit the lateral displacement of the pile decreased as the depth of the groundwater level dropped. The greater the fluctuation of the pile top displacement, the larger the fractal dimension of each relationship curve, with a variation interval of roughly 1.24–1.38. The average increment of the cumulative pile top displacement between each cycle step following the cyclic loading was positively correlated with fractal dimension. Based on the PIV results, the changes in the pile–soil system were predominantly focused in the early stages of the experiment, and the short-term effects of lateral cyclic loading are greater than the long-term effects. In addition, this research was limited to a single soil layer. The pile–soil interaction under layered soil is investigated, and the results will be used in more complex ground conditions in the future. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geomaterials)
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15 pages, 5795 KiB  
Article
Pore Structure and Fractal Characteristics of Frozen–Thawed Soft Soil
by Bowen Kong, Shao-Heng He, Yanli Tao and Jianzhong Xia
Fractal Fract. 2022, 6(4), 183; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6040183 - 25 Mar 2022
Cited by 9 | Viewed by 1901
Abstract
Freezing action always results in great damage to the pore structure and fractal characteristics of freezing–thawing soft soil. Nuclear magnetic resonance (NMR) can help collect data to describe the microstructure of frozen–thawed soft soil. This paper conducted a series of nuclear magnetic resonance [...] Read more.
Freezing action always results in great damage to the pore structure and fractal characteristics of freezing–thawing soft soil. Nuclear magnetic resonance (NMR) can help collect data to describe the microstructure of frozen–thawed soft soil. This paper conducted a series of nuclear magnetic resonance (NMR) tests to interpret the effect of freezing duration, freezing–thawing pressure and freezing temperature on soil’s micro-pore structure and fractal characteristics. The pore size distributions (PSDs) of the tested materials were obtained from the NMR T2 spectra, and fractal theory was introduced to describe the fractal properties of PSD. The results demonstrate that the soil assembly with a larger pore structure tends to be a skeleton pore structure with strong fractal characteristics; the shorter the freezing duration is, the less damage caused by the freezing action; a larger pressure during freezing and thawing results in a lower fractal dimension value for thawing soil’s structure, while a lower freezing temperature causes a larger fractal dimension of frozen-thawed soil. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geomaterials)
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15 pages, 3478 KiB  
Article
Influence of the Fractal Distribution of Particle Size on the Critical State Characteristics of Calcareous Sand
by Xue Shen, Yang Shen, Junhong Xu and Hanlong Liu
Fractal Fract. 2022, 6(3), 165; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6030165 - 17 Mar 2022
Cited by 5 | Viewed by 1853
Abstract
To study the influence of the fractal distribution of particle size on the critical state characteristics of calcareous sand, a type of calcareous sand from a certain reef of the South China Sea was used in this study. For comparison, standard quartz sand [...] Read more.
To study the influence of the fractal distribution of particle size on the critical state characteristics of calcareous sand, a type of calcareous sand from a certain reef of the South China Sea was used in this study. For comparison, standard quartz sand was also used. A series of drained shear tests on the two sands were then conducted to investigate their critical state characteristics. It was demonstrated that the fractal dimension is suitable for characterizing the particle size distribution (PSD) of calcareous sand with different fine sand content. The critical state equation of sand proposed by Li and Wang (1998) is suitable for fitting the critical state line of calcareous sand. In the plane of deviatoric stress versus the effective confining pressure (q–p′ plane) and the plane of void ratio versus (p′/pa)α, the critical state lines of calcareous sand are always above those of quartz sand. The critical state lines of calcareous sand with different fractal dimensions in the q–p′ plane are unique. However, in the e–(p′/pa)α plane, the critical state lines appear to rotate anticlockwise as the fractal dimension increases. In addition, there is an “intersection” in the e–(p′/pa)α plane. Considering the influence of the fractal distribution of particle size, an expression for the critical state line of calcareous sand in the e–(p′/pa)α plane was proposed. The related constitutive model was also revised, where a complete set of model parameters suitable for modeling calcareous sand was provided. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geomaterials)
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17 pages, 4343 KiB  
Article
Fractal Analysis of Particle Distribution and Scale Effect in a Soil–Rock Mixture
by Xiaodong Fu, Haifeng Ding, Qian Sheng, Zhenping Zhang, Dawei Yin and Fei Chen
Fractal Fract. 2022, 6(2), 120; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6020120 - 19 Feb 2022
Cited by 23 | Viewed by 2501
Abstract
A soil–rock mixture (SRM) is a type of heterogeneous geomaterial, and the particle distribution of SRM can be described by fractal theory. At present, it is difficult to quantify the fractal dimension of a particle size distribution and understand the scale effect in [...] Read more.
A soil–rock mixture (SRM) is a type of heterogeneous geomaterial, and the particle distribution of SRM can be described by fractal theory. At present, it is difficult to quantify the fractal dimension of a particle size distribution and understand the scale effect in SRMs. In this study, the fractal theory and discrete element method (DEM) were introduced to solve this problem. First, the particle gradation of SRM was dealt with by using fractal theory. The fractal structure of particle distribution was studied, and a method of calculation of the fractal dimension is presented in this paper. Second, based on the fractal dimension and relative threshold, the particle gradations of SRMs at different scales were predicted. Third, numerical direct shear tests of SRM at different scales were simulated by using the DEM. The scale effects of shear displacement, shear zone, and shear strength parameters were revealed. Last, taking the maximum particle size of 60 mm as the standard value, the piece-wise functional relationship between shear strength parameters and particle size was established. The results are as follows: for SRM in a representative engineering area, by plotting the relationship between particle cumulative mass percentage and particle size, we can judge whether the SRM has a fractal structure; in Southwest China, the frequency of the fractal dimension of the SRM is in the normal distribution, and the median fractal dimension is 2.62; the particle gradations of SRMs at different scales calculated by fractal dimension and relative threshold can expand the study scope of particle size analysis; when the particle size is less than 70 mm, the strength parameters show a parabolic trend with the particle size increases, and if not, a nearly linear trend is found. The proposed method can describe the fractal characteristics of SRM in a representative engineering area and provides a quantitative estimation of shear strength parameters of SRM at different scales. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geomaterials)
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9 pages, 1214 KiB  
Article
Cyclic Mobilisation of Soil–Structure Interface in the Framework of Fractional Plasticity
by Junhong Xu, Yang Shen and Yifei Sun
Fractal Fract. 2022, 6(2), 76; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6020076 - 31 Jan 2022
Cited by 4 | Viewed by 1998
Abstract
The strength of the soil–structure interface can be mobilised when subjected to cyclic loading. To capture the cyclic mobilisation of the soil–structure interface, an advanced elastoplastic constitutive model is developed within the framework of fractional plasticity, where no additional use of an additional [...] Read more.
The strength of the soil–structure interface can be mobilised when subjected to cyclic loading. To capture the cyclic mobilisation of the soil–structure interface, an advanced elastoplastic constitutive model is developed within the framework of fractional plasticity, where no additional use of an additional plastic potential is required. Considering the influence of material state and soil fabric on the plastic response of the soil–structure interface, the state-dependent fractional order and hardening modulus are proposed. Further numerical simulation of the developed model shows that it can reasonably capture the mobilised strength and deformation of the soil–structure interface under cyclic loads. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geomaterials)
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9 pages, 2006 KiB  
Article
Numerical Solutions of Space-Fractional Advection–Diffusion–Reaction Equations
by Valentina Anna Lia Salomoni and Nico De Marchi
Fractal Fract. 2022, 6(1), 21; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6010021 - 31 Dec 2021
Cited by 7 | Viewed by 2255
Abstract
Background: solute transport in highly heterogeneous media and even neutron diffusion in nuclear environments are among the numerous applications of fractional differential equations (FDEs), being demonstrated by field experiments that solute concentration profiles exhibit anomalous non-Fickian growth rates and so-called “heavy tails”. Methods: [...] Read more.
Background: solute transport in highly heterogeneous media and even neutron diffusion in nuclear environments are among the numerous applications of fractional differential equations (FDEs), being demonstrated by field experiments that solute concentration profiles exhibit anomalous non-Fickian growth rates and so-called “heavy tails”. Methods: a nonlinear-coupled 3D fractional hydro-mechanical model accounting for anomalous diffusion (FD) and advection–dispersion (FAD) for solute flux is described, accounting for a Riesz derivative treated through the Grünwald–Letnikow definition. Results: a long-tailed solute contaminant distribution is displayed due to the variation of flow velocity in both time and distance. Conclusions: a finite difference approximation is proposed to solve the problem in 1D domains, and subsequently, two scenarios are considered for numerical computations. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geomaterials)
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16 pages, 3419 KiB  
Article
Simplified Relation Model of Soil Saturation Permeability Coefficient and Air-Entry Value and Its Application
by Gaoliang Tao, Zhijia Wu, Wentao Li, Yi Li and Heming Dong
Fractal Fract. 2021, 5(4), 180; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5040180 - 23 Oct 2021
Cited by 6 | Viewed by 2093
Abstract
Based on the Tao and Kong (TK) model and the fractal model of the soil–water characteristic curve, a simplified model of the relationship between the saturated permeability coefficient and the air-entry value is established in this study: ks = k0ψ [...] Read more.
Based on the Tao and Kong (TK) model and the fractal model of the soil–water characteristic curve, a simplified model of the relationship between the saturated permeability coefficient and the air-entry value is established in this study: ks = k0ψa−2. It is shown that the saturated permeability coefficient of soil is determined by its maximum pore size. In order to facilitate the mutual prediction of saturation permeability coefficient and air-entry value, based on the data of five types of soil in the UNSODA database, the comprehensive proportionality constant k0 of the five types of soil were obtained: sand k0 = 0.03051; clay k0 = 0.001878; loam k0 = 0.001426; sandy loam k0 = 0.009301; and silty clay loam k0 = 0.0007055. Based on the obtained comprehensive proportionality constant k0 and the relationship model between saturated permeability coefficient and air intake value, the air-entry value of five kinds of soils in the existing literature and the SoilVision database were calculated. Comparing the calculated air-entry value with the measured one, the results showed that the model simplifies the traditional air-entry value prediction method to some extent and can effectively predict the air-entry value of different types of soil. On the whole, the model better predicts the air-entry value for sandy, clay, and silty clay loam than loam and sandy loam. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geomaterials)
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14 pages, 1632 KiB  
Article
Simple Graphical Prediction of Relative Permeability of Unsaturated Soils under Deformations
by Gaoliang Tao, Qing Wang, Qingsheng Chen, Sanjay Nimbalkar, Yinjie Peng and Heming Dong
Fractal Fract. 2021, 5(4), 153; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5040153 - 05 Oct 2021
Cited by 4 | Viewed by 1612
Abstract
At present, there are only a few existing models that can be used to predict the relative permeability of unsaturated soil under deformations, and the calculation process is relatively complex. In order to fit the measured value of the relative permeability coefficient of [...] Read more.
At present, there are only a few existing models that can be used to predict the relative permeability of unsaturated soil under deformations, and the calculation process is relatively complex. In order to fit the measured value of the relative permeability coefficient of unsaturated soil before deformation, this work employs the simplified unified model of the relative permeability coefficient of unsaturated soil, and it obtains the index λ before deformation of the soil. In addition, the value of index λ remains unchanged before and after deformation. Based on the actual measured value of the soil–water characteristic curve before deformation, the air-entry value prediction model is used to predict the air-entry value of soil with different initial void ratios. The relative permeability coefficient of unsaturated soil is then conveniently predicted using the graphical method in combination with the simplified unified model. The method is validated by using the test data of silt loam, sandy loam, and unconsoildated sand. The results show that the predicted results are consistent with the measured values. The prediction method in this paper is simple and overcomes the limitations associated with the determination of the index λ. It expands the application range of the unsaturated relative permeability coefficient model while improving the accuracy of predictions. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geomaterials)
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17 pages, 4470 KiB  
Article
Effect of Grain Size on Microscopic Pore Structure and Fractal Characteristics of Carbonate-Based Sand and Silicate-Based Sand
by Shao-Heng He, Zhi Ding, Hai-Bo Hu and Min Gao
Fractal Fract. 2021, 5(4), 152; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5040152 - 04 Oct 2021
Cited by 16 | Viewed by 1941
Abstract
In this study, a series of nuclear magnetic resonance (NMR) tests was conducted on calcareous sand, quartz sand, and glass bead with a wide range of grain sizes, to understand the effect of grain size on the micro-pore structure and fractal characteristics of [...] Read more.
In this study, a series of nuclear magnetic resonance (NMR) tests was conducted on calcareous sand, quartz sand, and glass bead with a wide range of grain sizes, to understand the effect of grain size on the micro-pore structure and fractal characteristics of the carbonate-based sand and silicate-based sand. The pore size distribution (PSD) of the tested materials were obtained from the NMR T2 spectra, and fractal theory was introduced to describe the fractal properties of PSD. Results demonstrate that grain size has a significant effect on the PSD of carbonate-based sand and silicate-based sand. As grain size increases, the PSD of sands evolves from a binary structure with two peaks to a ternary structure with three peaks. The increase in the grain size can cause a remarkable increase in the maximum pore size. It is also found that the more irregular the particle shape, the better the continuity between the large and medium pores. In addition, grain size has a considerable effect on the fractal dimension of the micro-pore structure. The increase of grain size can lead to a significant increase in the heterogeneity and fractal dimension in PSD for calcareous sand, quartz sand and glass bead. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geomaterials)
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