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Fractal Fract
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Fractal and Fractional in Geotechnical Engineering
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Fractal and Fractional in Geotechnical Engineering

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

Deadline for manuscript submissions: closed (20 November 2024) | Viewed by 16442

Special Issue Editors

Dr. Shaoheng He

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Guest Editor
Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong 26680, China
Interests: experimental soil mechanics; granular material; geomechanics; microstructure
Special Issues, Collections and Topics in MDPI journals
Dr. Zhi Ding

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Guest Editor
Department of Civil Engineering, Hangzhou City University, Hangzhou 310015, China
Interests: tunnel and underground engineering; intelligent construction and maintenance of shield tunnel
Special Issues, Collections and Topics in MDPI journals
Dr. Panpan Guo

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Guest Editor
School of Civil Engineering, Hefei University of Technology, Hefei 230009, China
Interests: geotechnical engineering; tunnelling; excavation; soil mechanics; rock mechanics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractal theory offers an efficient approach to the micro-quantification of geomaterials that is envisaged to establish connections with the macro-mechanics of geomaterials and provide a new perspective for tackling challenging engineering problems in geotechnical engineering. The fractal dimension, in particular, has shown immense potential in a wide range of geotechnical applications, including the characterization and prediction of soil porosity, soil-water characteristic curve, permeability, soil strength, microscopic pore size distribution, and particle/pore shape. Moreover, fractional-order derivative has found its applications in soil constitutive models, as well as tunnel constructions, such as the forecasting of long-term deformation induced by soil creep, detection of cracks in shield tunnel segments, viscoelastic modeling of tunnel lining, and artificial intelligence framework for tunnels. These applications have proven the crucial role of fractional-order mechanics in the geotechnical engineering design field.

The aim of this Special Issue is to present Fractal and Fractional in Geotechnical Engineering. 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:

  • Fractal phenomena and characteristics of geomaterials;
  • Determination method of fractal dimension in geomaterials;
  • Microstructure testing in geomaterials;
  • Interplay between fractal dimension and macro-mechanics of geomaterials;
  • Fractional-order mechanics;
  • Fractal-and-fractional-based soil constitutive models;
  • Analysis methods for geotechnical structures based on fractal and fractional;
  • Applications of fractal and fractional in tunnel constructions;
  • Artificial intelligence and image identification techniques for tunnels;
  • Quantitative description of rock joint roughness based on fractal theory;
  • Microstructure characteristics of foamed light soil under static or dynamic loads;
  • Gradation control of bituminous mixture and cold-recycled mixture in pavement engineering;
  • Disintegration and breaking characteristics of soft rock or special soils.

Dr. Shaoheng He
Dr. Zhi Ding
Dr. Panpan Guo
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 dimension
  • fractional-order mechanics
  • micro-to-Macro analysis
  • geomaterial
  • tunnel
  • rock
  • concrete
  • foamed light soil
  • bituminous mixture

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Related Special Issue

Published Papers (12 papers)

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Research

17 pages, 18934 KiB  
Article
Wavefield Evolution and Arrival Behavior of Elastic Wave Propagation in Two-Dimensional Fractional Brownian Fields
by Shuaifeng Wang and Zixin Zhang
Fractal Fract. 2024, 8(12), 750; https://doi.org/10.3390/fractalfract8120750 - 20 Dec 2024
Viewed by 251
Abstract
The fractional Brownian field is often used to reproduce the fractal properties of complex heterogeneous media, which closely represent real-world geological materials. Studying elastic wave transport in this type of heterogeneous media is essential for advancing knowledge in geophysics, seismology, and rock mechanics. [...] Read more.
The fractional Brownian field is often used to reproduce the fractal properties of complex heterogeneous media, which closely represent real-world geological materials. Studying elastic wave transport in this type of heterogeneous media is essential for advancing knowledge in geophysics, seismology, and rock mechanics. In this paper, we numerically investigate the wavefield evolution and arrival behavior of elastic wave propagation in a two-dimensional fractional Brownian field characterized by the standard deviation (σ) and the Hurst exponent (H). Using a high-fidelity finite element model, we quantify the influence of these parameters on wavefront morphology, wave arrival synchronization, and energy decay. Our results reveal that increased matrix heterogeneity with higher σ and lower H values leads to pronounced wavefront roughness, asynchronous arrival phenomena, and increscent energy decay, attributed to enhanced scattering and modulus variability. For smaller H values, rougher modulus distributions scatter wave energy more intensely, producing more coda waves and distorted wavefronts, while smoother fields with larger H fields promote smoother wave propagation. Higher σ amplifies these effects by increasing modulus variability, resulting in more attenuated wave energy and substantial wavefield disturbance. This study contributes to a quantitative understanding of how fractal heterogeneity modulates wave transport and energy attenuation in random media. Our findings hold practical significance for geophysical exploration and seismic tomography, as well as aiding in subsurface imaging and structural evaluation within fractured or stratified rock formations. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)
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16 pages, 14137 KiB  
Article
Penetration Grouting Mechanism of Bingham Fluid in Porous Media Based on Fractal Theory
by Jiakun Gong, Chenxi Xu, Yuan Wang, Shan Gao, Haikuan Chen and Lu Jia
Fractal Fract. 2024, 8(12), 739; https://doi.org/10.3390/fractalfract8120739 - 15 Dec 2024
Viewed by 341
Abstract
Penetration grouting is a significant grouting technique. The pore structure has important impacts on the infiltration mechanism of slurry in porous media. In this study, based on fractal theory, a theoretical penetration grouting model for Bingham fluid is established. An experimental apparatus for [...] Read more.
Penetration grouting is a significant grouting technique. The pore structure has important impacts on the infiltration mechanism of slurry in porous media. In this study, based on fractal theory, a theoretical penetration grouting model for Bingham fluid is established. An experimental apparatus for simulating the penetration process of Bingham fluid with a constant flow rate is developed. A series of penetration-grouting experiments are conducted to validate the theoretical model established in this study and analyze the impacts of the water–cement ratio and flow rate on the slurry injection pressure. The results show that the theoretical values of the slurry pressure along the penetration direction obtained from the penetration grouting model match the experimental values well. This indicates that the proposed model can better describe the process of slurry infiltration and provide valuable support for related grouting projects. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)
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14 pages, 6158 KiB  
Article
Fractal Analysis of Particle Size and Morphology in Single-Particle Breakage Based on 3D Images
by Ruidong Li, Xiang Gao, Shao-Heng He, Dongheng Ru and Zhi Ding
Fractal Fract. 2024, 8(11), 614; https://doi.org/10.3390/fractalfract8110614 - 22 Oct 2024
Viewed by 833
Abstract
The accurate modeling of single-particle breakage based on three-dimensional (3D) images is crucial for understanding the particle-level mechanics of granular materials. This study aims to propose a systematic framework incorporating single-particle breakage experiments and numerical simulations based on a novel 3D particle reconstruction [...] Read more.
The accurate modeling of single-particle breakage based on three-dimensional (3D) images is crucial for understanding the particle-level mechanics of granular materials. This study aims to propose a systematic framework incorporating single-particle breakage experiments and numerical simulations based on a novel 3D particle reconstruction technique for fractal analysis of particle size and morphology in single-particle breakage. First, the vision foundation model is used to generate accurate particles from 3D images. The numerical approach is validated by simulating the single-particle breakage test with multiple Fujian sand particles. Then, the breakage processes of reconstructed sand particles under axial compression are numerically modeled. The relationship between 3D fractal dimensions and particle size, particle crushing strength, and morphology is meticulously investigated. Furthermore, the implications of these relationships on the particle breakage processes are thoroughly discussed, shedding light on the underlying mechanisms that govern particle breakage. The framework offers an effective way to investigate the breakage behavior of single sand particles, which will enhance understanding of the mechanism of the whole particle breakage process. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)
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17 pages, 7741 KiB  
Article
Research on Slope Early Warning and Displacement Prediction Based on Multifractal Characterization
by Xiaofei Sun, Ying Su, Chengtao Yang, Junzhe Tan and Dunwen Liu
Fractal Fract. 2024, 8(9), 522; https://doi.org/10.3390/fractalfract8090522 - 4 Sep 2024
Cited by 1 | Viewed by 937
Abstract
The occurrence of landslide hazards significantly induces changes in slope surface displacement. This study conducts an in-depth analysis of the multifractal characteristics and displacement prediction of highway slope surface displacement sequences. Utilizing automated monitoring devices, data are collected to analyze the deformation patterns [...] Read more.
The occurrence of landslide hazards significantly induces changes in slope surface displacement. This study conducts an in-depth analysis of the multifractal characteristics and displacement prediction of highway slope surface displacement sequences. Utilizing automated monitoring devices, data are collected to analyze the deformation patterns of the slope surface layer. Specifically, the multifractal detrended fluctuation analysis (MF-DFA) method is employed to examine the multifractal features of the monitoring data for slope surface displacement. Additionally, the Mann–Kendall (M-K) method is combined to construct the α indicator and f(α) indicator criteria, which provide early warnings for slope stability. Furthermore, the long short-term memory (LSTM) model is optimized using the particle swarm optimization (PSO) algorithm to enhance the prediction of slope surface displacement. The results indicate that the slope displacement monitoring data exhibit a distinct fractal sequence characterized by h(q), with values decreasing as the fluctuation function q decreases. Through this study, the slope landslide warning classification has been determined to be Level III. Moreover, the PSO-LSTM model demonstrates superior prediction accuracy and stability in slope displacement forecasting, achieving a root mean square error (RMSE) of 0.72 and a coefficient of determination (R2) of 91%. Finally, a joint response synthesis of the slope landslide warning levels and slope displacement predictions resulted in conclusions. Subsequent surface displacements of the slope are likely to stabilize, indicating the need for routine monitoring and inspection of the site. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)
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20 pages, 9483 KiB  
Article
Strength and Fractal Characteristics of Artificial Frozen–Thawed Sandy Soft Soil
by Bowen Kong, Yuntian Yan, Huan He, Jing Yu, Baoping Zou and Qizhi Chen
Fractal Fract. 2024, 8(7), 393; https://doi.org/10.3390/fractalfract8070393 - 29 Jun 2024
Viewed by 1036
Abstract
In regions with sandy soft soil strata, the subway foundation commonly undergoes freeze–thaw cycles during construction. This study focuses on analyzing the microstructural and fractal characteristics of frozen–thawed sandy soft soil to improve our understanding of its strength behavior and stability. Pore size [...] Read more.
In regions with sandy soft soil strata, the subway foundation commonly undergoes freeze–thaw cycles during construction. This study focuses on analyzing the microstructural and fractal characteristics of frozen–thawed sandy soft soil to improve our understanding of its strength behavior and stability. Pore size distribution curves before and after freeze–thaw cycles were examined using nuclear magnetic resonance technology. Additionally, fractal theory was applied to illustrate the soil’s fractal properties. The strength properties of frozen remolded clay under varying freezing temperatures and sand contents were investigated through uniaxial compression tests, indicating that soil strength is significantly influenced by fractal dimensions. The findings suggest that lower freezing temperatures lead to a more dispersed soil skeleton, resulting in a higher fractal dimension for the frozen–thawed soil. Likewise, an increase in sand content enlarges the soil pores and the fractal dimension of the frozen–thawed soil. Furthermore, an increase in fractal dimension caused by freezing temperatures results in increased soil strength, while an increase in fractal dimension due to changes in sand content leads to a decrease in soil strength. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)
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19 pages, 27821 KiB  
Article
Multi-Scale Research on the Mechanisms of Soil Arching Development and Degradation in Granular Materials with Different Relative Density
by Luju Liang, Yi Pik Cheng, Xiaozhen Fan, Zhi Ding and Changjie Xu
Fractal Fract. 2024, 8(5), 247; https://doi.org/10.3390/fractalfract8050247 - 24 Apr 2024
Viewed by 1476
Abstract
Soil arching is significantly influenced by relative density, while its mechanisms have barely been analyzed. A series of DEM numerical simulations of the classical trapdoor test were carried out to investigate the multi-scale mechanisms of arching development and degradation in granular materials with [...] Read more.
Soil arching is significantly influenced by relative density, while its mechanisms have barely been analyzed. A series of DEM numerical simulations of the classical trapdoor test were carried out to investigate the multi-scale mechanisms of arching development and degradation in granular materials with different relative density. For analysis, the granular assembly was divided into three zones according to the particle vertical displacement normalized by the trapdoor displacement δ. The results show that before the maximum arching state (corresponding to the minimum arching ratio), contact forces between particles in a specific zone (where the vertical displacement of particles is larger than 0.1δ but less than 0.9δ) increase rapidly and robust arched force chains with large particle contact forces are generated. The variation in contact forces and force chains becomes more obvious as the sample porosity decreases. As a result, soil arching generated in a denser particle assembly is stronger, and the minimum value of the arching ratio is increased with the sample porosity. After the maximum arching state, the force chains in this zone are degenerated gradually, leading to a decrease in particle contact forces in microscale and an increase in the arching ratio in macroscale. The recovery of the arching ratio after the minimum value is also more significant in simulations with a larger relative density, as the degeneration of contact force chains is more obvious in denser samples. These results indicate the importance of contact force chain stabilities in specific zones for improving soil arching in engineering practice. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)
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18 pages, 1072 KiB  
Article
Fractal Operators and Convergence Analysis in Fractional Viscoelastic Theory
by Xiaobin Yu and Yajun Yin
Fractal Fract. 2024, 8(4), 200; https://doi.org/10.3390/fractalfract8040200 - 29 Mar 2024
Cited by 1 | Viewed by 1081
Abstract
This study delves into the convergence of operators and the viscoelastic properties of fractal ladder and tree structures. It proves the convergence of fractal stiffness operators through operator algebra, revealing a fundamental connection between operator sequence limits and fractal operator algebraic equations. Our [...] Read more.
This study delves into the convergence of operators and the viscoelastic properties of fractal ladder and tree structures. It proves the convergence of fractal stiffness operators through operator algebra, revealing a fundamental connection between operator sequence limits and fractal operator algebraic equations. Our findings demonstrate that, as the hierarchical levels of these structures increase, their viscoelastic responses increasingly align with the fractional viscoelastic behavior observed in infinite-level fractal structures. We explore the similarity in creep and relaxation behaviors between fractal ladders and trees, emphasizing the emergence of ultra-long characteristic times in steady-state creep and pronounced tailing effects in relaxation curves. This research provides novel insights into the design of fractional-order viscoelastic structures, presenting significant implications for materials science and mechanical engineering. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)
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18 pages, 3066 KiB  
Article
A Fractal-Based Quantitative Method for the Study of Fracture Evolution of Coal under Different Confining Pressures
by Ancheng Wang and Lei Wang
Fractal Fract. 2024, 8(3), 159; https://doi.org/10.3390/fractalfract8030159 - 11 Mar 2024
Cited by 2 | Viewed by 1480
Abstract
To study the dynamic crack evolution process of loaded coal from the perspective of fractals, we carried out in situ industrial CT scanning tests of loaded coal under different confining pressures, visualizing loaded coal fracturing. Combined with fractal theory, the temporal and spatial [...] Read more.
To study the dynamic crack evolution process of loaded coal from the perspective of fractals, we carried out in situ industrial CT scanning tests of loaded coal under different confining pressures, visualizing loaded coal fracturing. Combined with fractal theory, the temporal and spatial evolution law of coal cracks is described quantitatively. The results provide two findings: (1) from the perspective of two-dimensional images and three-dimensional space, the evolution characteristics of cracks in coal under different confining pressures were basically the same in each loading stage. During the loading stages, the cracks exhibited a change rule of a slow reduction, initiation/development, rapid increase, expansion, and penetration. (2) The fractal dimension of coal was calculated by introducing fractal theory, and its change law was in good agreement with the dynamic changes of the cracks, which can explain the influence of the confining pressure on the loaded coal. The fractal dimension showed three stages: a slight decrease, a stable increase, and then a significant increase. The larger the confining pressure, the more obvious the limiting effect. Thus, our approach provides a more accurate method for evaluating the spatial and temporal evolution of cracks in loaded coal. This study can be used to predict the instability failure of loaded coal samples. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)
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21 pages, 5481 KiB  
Article
Analysis and Warning Prediction of Tunnel Deformation Based on Multifractal Theory
by Chengtao Yang, Rendong Huang, Dunwen Liu, Weichao Qiu, Ruiping Zhang and Yu Tang
Fractal Fract. 2024, 8(2), 108; https://doi.org/10.3390/fractalfract8020108 - 12 Feb 2024
Cited by 7 | Viewed by 1757
Abstract
To better analyze the fluctuation characteristics and development law of tunnel deformation data, multifractal theory is applied to tunnel deformation analysis. That is, the multifractal detrended fluctuation analysis (MF-DFA) model is first utilized to carry out the multifractal characterization of tunnel deformation data. [...] Read more.
To better analyze the fluctuation characteristics and development law of tunnel deformation data, multifractal theory is applied to tunnel deformation analysis. That is, the multifractal detrended fluctuation analysis (MF-DFA) model is first utilized to carry out the multifractal characterization of tunnel deformation data. Further, Mann–Kendall (M–K) analysis is utilized to construct the dual criterion (∆α indicator criterion and ∆f(α) indicator criterion) for the tunnel deformation early warning study. In addition, the particle swarm optimization long-short-term memory (PSO-LSTM) prediction model is used for predicting tunnel settlement. The results show that, in reference to the tunnel warning level criteria and based on the Z-value results of the indicator criterion, the warning level of all four sections is class II. At the same time, through the analysis of tunnel settlement predictions, the PSO-LSTM model has a better prediction effect and stability for tunnel settlement. The predicted results show a slow increase in tunnel settlement over the next 5 days. Finally, the tunnel warning level and the predicted results of tunnel settlement are analyzed in a comprehensive manner. The deformation will increase slowly in the future. Therefore, monitoring and measurement should be strengthened, and disaster preparedness plans should be prepared. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)
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23 pages, 14115 KiB  
Article
Multifractal Characteristics of Smooth Blasting Overbreak in Extra-Long Hard Rock Tunnel
by Wanmao Zhang, Dunwen Liu, Yu Tang, Weichao Qiu and Ruiping Zhang
Fractal Fract. 2023, 7(12), 842; https://doi.org/10.3390/fractalfract7120842 - 27 Nov 2023
Cited by 4 | Viewed by 1895
Abstract
With the development of infrastructure construction in mountainous areas, the number of new extra-long tunnels is increasing. However, these tunnels often face the challenge of complex and variable surrounding rock grades, resulting in a large number of overbreak and underbreak due to the [...] Read more.
With the development of infrastructure construction in mountainous areas, the number of new extra-long tunnels is increasing. However, these tunnels often face the challenge of complex and variable surrounding rock grades, resulting in a large number of overbreak and underbreak due to the untimely adjustment of smooth blasting parameters. This study focuses on the optimization of the peripheral hole charging structure and blasting parameters for extra-long hard rock tunnels, aiming to improve the effectiveness of smooth blasting technology. The results of this study demonstrate a significant improvement in the effect of smooth blasting after implementing bidirectional polymerization blasting in the tunnel. A comparison between the bidirectional shaped charge and spaced decoupled charge blasting reveals that the former yields better results. To obtain accurate data on the tunnel section profile during excavation, a laser cross-section meter is used for measurement. Furthermore, this study quantitatively compares the optimization effect of smooth blasting parameters. The multifractal characteristics of the tunnel profile overbreak point sequences are analyzed under different smooth blasting schemes using the multifractal detrended fluctuation analysis (MF-DFA) method. It is found that both the spaced decoupled charge and the bidirectional shaped charge blasting exhibit multifractal features in the overbreak measurement point sequences. The calculation results of the multifractal features of the tunnel profile under different smooth blasting plans are in line with the actual situation. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)
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16 pages, 4550 KiB  
Article
The Formative Factors of a Rock Burst Based on Energy Calculations and the Experimental Verification of Butterfly-Shaped Plastic Zones
by Wenlong Zhang, Jicheng Feng, Jianju Ren, Ji Ma, Jianjun Shi and Junfeng Zhang
Fractal Fract. 2023, 7(11), 829; https://doi.org/10.3390/fractalfract7110829 - 20 Nov 2023
Cited by 2 | Viewed by 1361
Abstract
The research on the formation factors of rock burst is one of the main research directions of rock mechanics in recent years, which is helpful to solve the problem of rock burst accidents. So, in this study, the calculation method of energy released [...] Read more.
The research on the formation factors of rock burst is one of the main research directions of rock mechanics in recent years, which is helpful to solve the problem of rock burst accidents. So, in this study, the calculation method of energy released during rock burst is first obtained by using different medium models, and then, the formation factors of rock bursts are obtained by comparing the calculation energy with the actual accident energy. The method of energy calculation utilizes the difference between elastoplastic and pure elastic models to innovatively quantify the specific values of energy released before and after the occurrence of the rock burst. It is considered that the stress and plastic zone state before the occurrence of rock burst have an important influence on the occurrence of the accident and are one of the formation factors, while the deviatoric stress field and butterfly-shaped plastic zone create conditions for greater energy release. In addition, the trigger stress constitutes another formation factor. The plastic zone state before rock failure is verified by the experimental test; the location distribution shape of acoustic emission (AE) events during the later stage of compression failure is approximately the same as theoretical result. The results also preliminarily indicated the fractal characteristics of acoustic emission events distribution before sample failure. The study obtained the formative factors of rock burst accident, which provides a new ideas and references for the research on the formation of rock bursts. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)
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11 pages, 1036 KiB  
Article
Dilatancy Equation Based on the Property-Dependent Plastic Potential Theory for Geomaterials
by Xuefeng Li, Houying Zhu and Qi Yuan
Fractal Fract. 2023, 7(11), 824; https://doi.org/10.3390/fractalfract7110824 - 15 Nov 2023
Cited by 18 | Viewed by 2158
Abstract
The dilatancy equation ignores the noncoaxiality of granular soil for the coaxial assumption of the direction of the stress and strain rate in conventional plastic potential theory, which is inconsistent with extensive laboratory tests. To reasonably describe the noncoaxial effects on dilatancy, the [...] Read more.
The dilatancy equation ignores the noncoaxiality of granular soil for the coaxial assumption of the direction of the stress and strain rate in conventional plastic potential theory, which is inconsistent with extensive laboratory tests. To reasonably describe the noncoaxial effects on dilatancy, the energy dissipation of plastic flow is derived based on the property-dependent plastic potential theory for geomaterials and integrates the noncoaxiality, the potential theory links the plastic strain of granular materials with its fabric, and the noncoaxiality is naturally related to the mesoscopic properties of materials. When the fabric is isotropic, the dilatancy equation degenerates into the form of the critical state theory, and when the fabric is anisotropic, it naturally describes the effects of noncoaxiality. In the plane stress state, a comparison between a simple shear test and prediction of the dilatancy equation shows that the equation can reasonably describe the effect of noncoaxiality on dilatancy with the introduction of microscopic fabric parameters, and its physical significance is clear. This paper can provide a reference for the theoretical description of the macro and micro mechanical properties of geomaterials. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)
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