Fractal and Fractional

Research

31 pages, 20594 KiB

Open AccessArticle

Dynamic Failure Mechanism and Fractal Features of Fractured Rocks Under Quasi-Triaxial Static Pressures and Repeated Impact Loading

by Peng Li, Yan Liu, Jie Zhang, Zhihong Dong, Xinghui Wu, Shengjun Miao and Meifeng Cai

Fractal Fract. 2025, 9(2), 71; https://doi.org/10.3390/fractalfract9020071 - 23 Jan 2025

Viewed by 471

Abstract

Mastering the dynamic mechanical behaviors of pre-stressed fractured rocks under repeated impact loads is crucial for safety management in rock engineering. To achieve this, repeated impact loading experiments were performed on produced fractured samples exposed to varying pre-applied axial and confining pressures using [...] Read more.

Mastering the dynamic mechanical behaviors of pre-stressed fractured rocks under repeated impact loads is crucial for safety management in rock engineering. To achieve this, repeated impact loading experiments were performed on produced fractured samples exposed to varying pre-applied axial and confining pressures using a split Hopkinson pressure bar test system in combination with a nuclear magnetic resonance imaging system, and the dynamic failure mechanism and fractal features were investigated. The results indicate that the dynamic stress–strain curves exemplify typical class II curves, and the strain rebound progressively diminishes with growing impact times. The impact times, axial pressure, and confining pressure all significantly affect the dynamic peak strength, average dynamic strength, dynamic deformation modulus, average dynamic deformation modulus, maximum strain, and impact resistance performance. Moreover, under low confining pressures, numerous shear cracks and tensile cracks develop, which are interconnected and converge to form large-scale macroscopic fracture surfaces. In contrast, specimens under a high confining pressure primarily experience tensile failure, accompanied by localized small-scale shear failure. Under low axial pressure, some shear cracks and tensile cracks emerge, while at high axial pressure, anti-wing cracks and secondary coplanar cracks occur, characterized predominantly by shear failure. In addition, as the confining pressure grows from 8 to 20 MPa, the fractal dimensions are 2.44, 2.32, 2.23, and 2.12, respectively. When the axial pressures are 8, 14, and 20 MPa, the fractal dimensions are 2.44, 2.46, and 2.52, respectively. Overall, the degree of fragmentation of the sample decreases with growing confining pressure and grows with rising axial pressure. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)

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27 pages, 11214 KiB

Open AccessArticle

Fractal Characteristics of the Spatial Distribution of Mine Earthquake Sources in the Vicinity of a Fault: A Case Study in the Ashele Copper Mine

by Congcong Zhao, Shigen Fu and Yinghua Huang

Fractal Fract. 2025, 9(1), 3; https://doi.org/10.3390/fractalfract9010003 - 24 Dec 2024

Viewed by 556

Abstract

Potential faults are common sensitive geological bodies that affect the safe mining of underground mines, often leading to major accidents such as rock instability and rockburst during mining. The failure mechanism of faults has been widely studied. However, due to the spatiotemporal specificity [...] Read more.

Potential faults are common sensitive geological bodies that affect the safe mining of underground mines, often leading to major accidents such as rock instability and rockburst during mining. The failure mechanism of faults has been widely studied. However, due to the spatiotemporal specificity of fault occurrence, there are few theoretical and mathematical methods suitable for effective analysis in mine safety risk management. This study aims to introduce fractal theory to characterize the spatiotemporal activity fractal characteristics of induced faults intersecting the mining site and roadway during the mining process of the Ashele copper mine in China. Using microseismic systems and fractal theory, a spatiotemporal fractal model of the fault slip process is constructed, and a fractal analysis method is proposed. The fractal dimension value is calculated based on the spatiotemporal parameters of different segments and stages. The fractal dimension is used to characterize and analyze the evolution of the fault. The physical formation process of potential faults and the relationship between fractal dimension values and multiple parameters, including spatial clustering, regional distribution characteristics, and energy-release characteristics, were analyzed based on the division of events into different time stages. Discovering fractal dimension’s temporal and spatial–temporal characteristics can provide technical references for mine disaster prevention. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)

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28 pages, 12115 KiB

Open AccessArticle

Energy Evolution and Fractal Characteristics of Sandstones Under True Triaxial Cyclic Loading and Unloading

by Qihang Zhang, Xiangrui Meng and Guangming Zhao

Fractal Fract. 2024, 8(12), 714; https://doi.org/10.3390/fractalfract8120714 - 30 Nov 2024

Viewed by 704

Abstract

To study the fractal characteristics and energy evolution of sandstones under true three-dimensional stress states, a true triaxial compression test and a cyclic loading and unloading test of sandstone specimens under different loads were carried out using a self-developed true triaxial disturbance testing [...] Read more.

To study the fractal characteristics and energy evolution of sandstones under true three-dimensional stress states, a true triaxial compression test and a cyclic loading and unloading test of sandstone specimens under different loads were carried out using a self-developed true triaxial disturbance testing system. Based on the evolution law of true triaxial cyclic loading and unloading stress–strain, the types of loading and unloading in the cyclic loading and unloading test were delineated, and the reasons for the change in peak maximum principal stress intensity under different paths were analyzed. By analyzing the crushing characteristics of rock samples under different paths, it was found that the staged cyclic loading and unloading caused the greatest damage to the rock mass, while the equal-amplitude and unequal-lower-limit staged loading and unloading caused the least damage to the rock mass. Based on fractal theory, it was found that the rock samples under path V had the highest fractal dimension D. The elastic energy density, dissipated energy density, and input energy density of true triaxial cyclic loading and unloading under different paths were calculated by graphical area integration and superposition methods, respectively, to analyze the evolution of the three with the increase in the loading and unloading cycles and the energy distribution during the loading and unloading process. True triaxial cyclic loading and unloading tests revealed a linear relationship between the elastic energy density and total input energy density of the rock mass, and the energy storage coefficient exceeded 0.5, regardless of the loading path. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)

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15 pages, 66452 KiB

Open AccessArticle

Effect of Thermal Cracking on the Tensile Strength of Granite: Novel Insights into Numerical Simulation and Fractal Dimension

by Jiliang Pan, Leiming Zhang, Yichen Ma, Ying Zhang and Xun Xi

Fractal Fract. 2024, 8(11), 669; https://doi.org/10.3390/fractalfract8110669 - 18 Nov 2024

Viewed by 754

Abstract

This study investigates the effect of thermal cracking on the tensile strength of granite through a combination of experimental testing and numerical simulations. The primary objective is to understand how thermal stress, induced by heat treatment at various temperatures (25 °C to 600 [...] Read more.

This study investigates the effect of thermal cracking on the tensile strength of granite through a combination of experimental testing and numerical simulations. The primary objective is to understand how thermal stress, induced by heat treatment at various temperatures (25 °C to 600 °C), influences crack initiation, propagation, and tensile strength changes. The granite specimens were subjected to Brazilian splitting tests after heat treatment, and the load–displacement curves and tensile strength variations with heat treatment temperature were analyzed. A grain-based model (GBM) was developed to simulate the complex cracking behavior, incorporating the mineral compositions and thermal expansion properties of the granite. The fractal dimension of the cracks was quantified using the box-counting method, and the relationship between fractal dimension and tensile strength was discussed. The results show that the GBM can effectively simulate the microcracking behavior and tensile fracture properties of heat-treated granite, accounting for mineral composition and thermal expansion. Thermal cracks are mainly intergranular tensile cracks, which increase in number with higher temperatures, while under mechanical loading failure is primarily due to intragranular tensile cracks. Higher heat treatment temperatures lead to denser crack networks with greater fractal complexity, reducing tensile strength and creating more tortuous crack propagation paths. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)

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12 pages, 2154 KiB

Open AccessArticle

Fractal Strategy for Improving Characterization of N₂ Adsorption–Desorption in Mesopores

by Kunpeng Feng, Gaofeng Liu, Zhen Zhang, Huan Liu, Runsheng Lv, Xiaoming Wang, Ping Chang, Jia Lin and George Barakos

Fractal Fract. 2024, 8(11), 617; https://doi.org/10.3390/fractalfract8110617 - 23 Oct 2024

Viewed by 911

Abstract

The current studies primarily analyze the heterogeneity and complexity of mesopore structures based on low-temperature nitrogen (N₂) adsorption curves and the Frenkel–Halsey–Hill (FHH) fractal model. However, these studies ignore the fact that the low-temperature N₂ desorption curve can also reflect [...] Read more.

The current studies primarily analyze the heterogeneity and complexity of mesopore structures based on low-temperature nitrogen (N₂) adsorption curves and the Frenkel–Halsey–Hill (FHH) fractal model. However, these studies ignore the fact that the low-temperature N₂ desorption curve can also reflect the desorption performance of the mesopore structure. In this research, novel fractal indicators for characterizing the adsorption–desorption performance of mesopores based on the fractal dimension from the N₂ adsorption curves and N₂ desorption curves are proposed. The novel fractal indicators I₁ and I₂ are applied to evaluate the adsorption–desorption performance of mesopores with pore size 2–5 nm and pore size 5–50 nm, respectively. The fractal indicator I₁ shows an increasing trend with coalification, reflecting that the gas adsorption performance of 2–5 nm mesopores is enhanced with coalification. The fractal indicator I₂ exhibits a trend of first increasing and then decreasing with coalification, indicating the gas desorption performance of mesopores with pore size 5–50 nm decreases first and then increases. The proposed indicators provide novel analytical parameters for further understanding the gas adsorption–desorption mechanism of porous coal-based or carbon-based materials. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)

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15 pages, 8628 KiB

Open AccessArticle

Fractal Characterization on Three-Dimensional Tortuosity of Fault Tectonic

by Runsheng Lv, Xinya Han, Gaofeng Liu, Zhen Zhang, Jia Lin, George Barakos and Ping Chang

Fractal Fract. 2024, 8(10), 572; https://doi.org/10.3390/fractalfract8100572 - 29 Sep 2024

Viewed by 1037

Abstract

Faults, as a kind of fracture tectonics, play a role in reservoir closure or provide oil and gas transportation channels. The accurate understanding of the distribution characteristics of faults is significant for oil and gas exploration. The traditional fractal dimension for fault number [...] Read more.

Faults, as a kind of fracture tectonics, play a role in reservoir closure or provide oil and gas transportation channels. The accurate understanding of the distribution characteristics of faults is significant for oil and gas exploration. The traditional fractal dimension for fault number (D_f3) cannot comprehensively characterize the complexity and heterogeneity of fault network distribution. In this paper, a fractal characterization method on three-dimensional (3D) tortuosity of fault tectonics is proposed based on 3D seismic exploration. The methodology is described in detail to establish the model on the fractal dimension for the 3D tortuosity of fault tectonics. The results show the proposed method of estimation of the D_T3 displaying high accuracy and rationality. Compared with the traditional fractal dimension D_f3, the proposed D_T3 can comprehensively characterize the fractal characteristics of faults network systems in the 3D space. This study achieves a breakthrough in the fractal characterization of the 3D tortuosity of fault tectonics. It is worth further study for establishing an analytical fractal equation based on the D_T3 and oil or gas transfer, which can provide the theoretical foundation and technical support for oil and gas exploration. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)

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21 pages, 6999 KiB

Open AccessArticle

Effects of In-Situ Stress on Damage and Fractal during Cutting Blasting Excavation

by Yongbo Wu, Xiaojun Zhang, Zhuo Li, Wenxue Gao, Zehui Xu, Yifeng Zhang and Jiguo Zhou

Fractal Fract. 2024, 8(8), 450; https://doi.org/10.3390/fractalfract8080450 - 30 Jul 2024

Viewed by 1254

Abstract

Blasting excavation of rock masses under high in-situ stress often encounters difficulties in rock fragmentation and a high boulder rate. To gain a deeper understanding of this issue, the stress distribution of rock masses under dynamic and static loads was first studied through [...] Read more.

Blasting excavation of rock masses under high in-situ stress often encounters difficulties in rock fragmentation and a high boulder rate. To gain a deeper understanding of this issue, the stress distribution of rock masses under dynamic and static loads was first studied through theoretical analysis. Then, the ANSYS/LS-DYNA software was employed to simulate the blasting crack propagation in rock masses under various in-situ stress conditions. The fractal dimension was introduced to quantitatively analyze the influence of in-situ stress on the distribution of blasting cracks. The results indicate that in-situ stress primarily affects crack propagation in the later stages of the explosion, while crack initiation and propagation in the early stages are mainly driven by the explosion load. In-situ stress significantly influences the damage area and fractal dimension of cut blasting. Under hydrostatic in-situ stress, as the in-situ stress increases, the damage area and fractal dimension of blasting cracks gradually decrease. Under non-hydrostatic in-situ stress, when the principal stress difference is small, in-situ stress promotes the damage area and fractal dimension of the surrounding rock, enhancing rock fragmentation. However, when the principal stress difference is large, in-situ stress inhibits the damage area and fractal dimension of the surrounding rock, hindering effective rock breaking. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)

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18 pages, 6003 KiB

Open AccessArticle

Evolution of Pore Structure and Fractal Characteristics in Red Sandstone under Cyclic Impact Loading

by Huanhuan Qiao, Peng Wang, Zhen Jiang, Yao Liu, Guanglin Tian and Bokun Zhao

Fractal Fract. 2024, 8(8), 437; https://doi.org/10.3390/fractalfract8080437 - 24 Jul 2024

Viewed by 1457

Abstract

Fatigue damage can occur in surface rock engineering due to various factors, including earthquakes, blasting, and impacts. The underlying cause for the variations in physical and mechanical properties of the rock resulting from impact loading is the alteration in the internal pore structure. [...] Read more.

Fatigue damage can occur in surface rock engineering due to various factors, including earthquakes, blasting, and impacts. The underlying cause for the variations in physical and mechanical properties of the rock resulting from impact loading is the alteration in the internal pore structure. To investigate the evolution characteristics of the pore structure under impact fatigue damage, red sandstone subjected to cyclic impact compression by split Hopkinson pressure bar (SHPB) was analyzed using nuclear magnetic resonance (NMR) technology. The parameters describing the evolution of pore structure were obtained and quantified using fractal methods. The development of the pore structure in rocks subjected to cyclic impact was quantitatively analyzed, and two fractal evolution models based on pore size and pore connectivity were constructed. The results indicate that with an increasing number of impact loading cycles, the porosity of the red sandstone gradually increases, the T₂ cutoff (T_2c) value decreases, the most probable gray value of magnetic resonance imaging (MRI) increases, the pores’ connectivity is enhanced, and the fractal dimension decreases gradually. Moreover, the pore distribution space tends to transition from three-dimensional to two-dimensional, suggesting the expansion of dominant pores into clusters, forming microfractures or even macroscopic fissures. The findings provide valuable insights into the impact fatigue characteristics of rocks from a microscopic perspective and contribute to the evaluation of time-varying stability and the assessment of progressive damage in rock engineering. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)

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16 pages, 25034 KiB

Open AccessArticle

Predicting the Deformation of a Slope Using a Random Coefficient Panel Data Model

by Zhenxia Yuan, Yadong Bian, Weijian Liu, Fuzhou Qi, Haohao Ma, Sen Zheng and Zhenzhu Meng

Fractal Fract. 2024, 8(7), 429; https://doi.org/10.3390/fractalfract8070429 - 22 Jul 2024

Viewed by 1047

Abstract

Engineering constructions in coastal areas not only affect existing landslides, but also induce new landslides. Variation of the water level makes the coastal area a geological hazard-prone. Prediction of the slope displacement based on monitoring data plays an important role in early warning [...] Read more.

Engineering constructions in coastal areas not only affect existing landslides, but also induce new landslides. Variation of the water level makes the coastal area a geological hazard-prone. Prediction of the slope displacement based on monitoring data plays an important role in early warning of potential landslide and slope failure, and supports the risk management of hazards. Given the complex characteristic of the slope deformation, we proposed a prediction model using random coefficient model under the frame of panel data analysis, so as to take the correlation among monitoring points into consideration. In addition, we classified the monitoring data using Gaussian mixture model, to take the temporal-spatial characteristics into consideration. Monitoring data of Guobu slope was used to validate the model. Results indicated that the proposed model have a better performance in prediction accuracy. We also compared the proposed model with the BP neural network model and temporal – temperature model, and found that the prediction accuracy of the proposed model is better than those of the two control models. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)

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20 pages, 42171 KiB

Open AccessArticle

Fractal Characteristics and Energy Evolution Analysis of Rocks under True Triaxial Unloading Conditions

by Cheng Pan, Chongyan Liu, Guangming Zhao, Wei Yuan, Xiao Wang and Xiangrui Meng

Fractal Fract. 2024, 8(7), 387; https://doi.org/10.3390/fractalfract8070387 - 28 Jun 2024

Cited by 4 | Viewed by 986

Abstract

To investigate the mechanical properties and energy evolution laws of rocks under true triaxial unloading conditions, a study was conducted using a true triaxial rock testing system on three different types of rocks: coal, sandy mudstone, and siltstone. The study examined the mechanical [...] Read more.

To investigate the mechanical properties and energy evolution laws of rocks under true triaxial unloading conditions, a study was conducted using a true triaxial rock testing system on three different types of rocks: coal, sandy mudstone, and siltstone. The study examined the mechanical behavior, failure patterns, and fractal dimensions of these rocks under true triaxial unloading conditions. The tests revealed significant variations in stress–strain curves and failure patterns among the different rock types. Observation indicated that rocks with lower peak strength exhibited higher fractal dimensions and increased fragmentation upon failure. Subsequently, based on the experimental data of siltstone, the impact of the unloading rate and particle size distribution on the energy evolution under true triaxial single-sided unloading paths was further investigated using the three-dimensional particle flow software PFC3D 6.0, revealing the micro-mechanisms of rock energy evolution. The study revealed that when the initial stress unloading level was low, the total energy and strain energy at the peak strength exhibited a strong linear relationship with the unloading rate. Before the stress peak, the dissipative energy was mainly composed of frictional energy. After the stress peak, the dissipative energy consisted of frictional energy, damping energy, and kinetic energy. The heterogeneity of rock significantly affected the distribution of dissipative energy, with an increase in rock heterogeneity leading to a decrease in frictional energy and an increase in kinetic energy. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)

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21 pages, 5168 KiB

Open AccessArticle

Quantifying the Pore Heterogeneity of Alkaline Lake Shale during Hydrous Pyrolysis by Using the Multifractal Method

by Yanxin Liu, Hong Zhang, Zhengchen Zhang, Luda Jing and Kouqi Liu

Fractal Fract. 2024, 8(6), 335; https://doi.org/10.3390/fractalfract8060335 - 4 Jun 2024

Cited by 1 | Viewed by 1218

Abstract

Distinguishing itself from marine shale formations, alkaline lake shale, as a significant hydrocarbon source rock and petroleum reservoir, exhibits distinct multifractal characteristics and evolutionary patterns. This study employs a combination of hydrous pyrolysis experimentation, nitrogen adsorption analysis, and multifractal theory to investigate the [...] Read more.

Distinguishing itself from marine shale formations, alkaline lake shale, as a significant hydrocarbon source rock and petroleum reservoir, exhibits distinct multifractal characteristics and evolutionary patterns. This study employs a combination of hydrous pyrolysis experimentation, nitrogen adsorption analysis, and multifractal theory to investigate the factors influencing pore heterogeneity and multifractal dimension during the maturation process of shale with abundant rich alkaline minerals. Utilizing partial least squares (PLS) analysis, a comparative examination is conducted, elucidating the disparate influence of mineralogical composition on their respective multifractal dimensions. The findings reveal a dynamic evolution of pore characteristics throughout the maturation process of alkaline lake shale, delineated into three distinct stages. Initially, in Stage 1 (200 °C to 300 °C), both ΔD and H demonstrate an incremental trend, rising from 1.2699 to 1.3 and from 0.8615 to 0.8636, respectively. Subsequently, in Stages 2 and 3, fluctuations are observed in the values of ΔD and D, while the H value undergoes a pronounced decline to 0.85. Additionally, the parameter D₁ exhibits a diminishing trajectory across all stages, decreasing from 0.859 to 0.829, indicative of evolving pore structure characteristics throughout the maturation process. The distinct alkaline environment and mineral composition of alkaline lake shale engender disparate diagenetic effects during its maturation process compared with other shale varieties. Consequently, this disparity results in contrasting evolutionary trajectories in pore heterogeneity and multifractal characteristics. Specifically, multifractal characteristics of alkaline lake shale are primarily influenced by quartz, potassium feldspar, clay minerals, and alkaline minerals. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)

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13 pages, 12921 KiB

Open AccessArticle

Fractal Evolution Characteristics on the Three-Dimensional Fractures in Coal Induced by CO₂ Phase Transition Fracturing

by Zhen Zhang, Gaofeng Liu, Jia Lin, George Barakos and Ping Chang

Fractal Fract. 2024, 8(5), 273; https://doi.org/10.3390/fractalfract8050273 - 4 May 2024

Cited by 5 | Viewed by 1566

Abstract

To analyze the transformed effect of three-dimensional (3D) fracture in coal by CO₂ phase transition fracturing (CO₂-PTF), the CO₂-PTF experiment under a fracturing pressure of 185 MPa was carried out. Computed Tomography (CT) scanning and fractal theory were [...] Read more.

To analyze the transformed effect of three-dimensional (3D) fracture in coal by CO₂ phase transition fracturing (CO₂-PTF), the CO₂-PTF experiment under a fracturing pressure of 185 MPa was carried out. Computed Tomography (CT) scanning and fractal theory were used to analyze the 3D fracture structure parameters. The fractal evolution characteristics of the 3D fractures in coal induced by CO₂-PTF were analyzed. The results indicate that the CO₂ phase transition fracturing coal has the fracture generation effect and fracture expansion-transformation effect, causing the maximum fracture length, fracture number, fracture volume and fracture surface area to be increased by 71.25%, 161.94%, 3970.88% and 1330.03%. The fractal dimension (D_N) for fracture number increases from 2.3523 to 2.3668, and the fractal dimension (D_V) for fracture volume increases from 2.8440 to 2.9040. The early dynamic high-pressure gas jet stage of CO₂-PTF coal influences the fracture generation effect and promotes the generation of 3D fractures with a length greater than 140 μm. The subsequent quasi-static high-pressure gas stage influences the fracture expansion-transformation effect, which promotes the expansion transformation of 3D fractures with a length of less than 140 μm. The 140 μm is the critical value for the fracture expansion-transformation effect and fracture generation effect. Five indicators are proposed to evaluate the 3D fracture evolution in coal caused by CO₂-PTF, which can provide theoretical and methodological references for the study of fracture evolution characteristics of other unconventional natural gas reservoirs and their reservoir stimulation. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)

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18 pages, 7062 KiB

Open AccessArticle

Compressive Failure Characteristics of a Coal–Rock Combination at Different Angles: Experimental Study and Fractal Analysis

by Long Tang, Shihao Tu, Hongsheng Tu, Kaijun Miao, Wenlong Li, Hongbin Zhao, Jieyang Ma and Lei Zhang

Fractal Fract. 2024, 8(4), 240; https://doi.org/10.3390/fractalfract8040240 - 20 Apr 2024

Cited by 2 | Viewed by 1206

Abstract

In order to explore the influence of dip angles on the deformation and failure of a coal–rock combination, uniaxial compression experiments were carried out on a coal–rock combination with different dip angles, and the acoustic emissions (hereinafter referred to as AE) responses during [...] Read more.

In order to explore the influence of dip angles on the deformation and failure of a coal–rock combination, uniaxial compression experiments were carried out on a coal–rock combination with different dip angles, and the acoustic emissions (hereinafter referred to as AE) responses during loading were collected. Based on the damage mechanics theory and fractal theory, the fractal dimensions of different damage degrees were calculated. The results show that, with the increase in the inclination angle, the compressive strength and elastic modulus of the coal–rock combination gradually decreased, while the AE ringing count gradually increased first and then decreased. At the initial loading stage of the specimen, the greater the damage degree of the coal–rock combination under the same strain condition, the larger the value of its overall fractal dimension. The AE fractal dimension of the coal–rock combination increases gradually between 10% and 20% of the damage degree. It suddenly decreased between 50% and 60%, then increased slightly before gradually decreasing to the minimum between 80% and 100%. The sudden decrease in fractal dimension, a slight increase, and then a continuous decrease can be used as the precursor information for the instability and failure of the coal–rock combination. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)

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15 pages, 5652 KiB

Open AccessArticle

Experimental Study on the Butterfly Shape of the Plastic Zone around a Hole near Rock Failure

by Wenlong Zhang, Jiajia Yu, Jianju Ren, Chen Li and Ji Ma

Fractal Fract. 2024, 8(4), 215; https://doi.org/10.3390/fractalfract8040215 - 7 Apr 2024

Cited by 1 | Viewed by 1187

Abstract

The precursor of rock failure around a hole has always been one of the research hotspots in the field of rock mechanics, and the distribution of the plastic zone is often adopted to reflect the location and form of rock failure. The shape [...] Read more.

The precursor of rock failure around a hole has always been one of the research hotspots in the field of rock mechanics, and the distribution of the plastic zone is often adopted to reflect the location and form of rock failure. The shape of the plastic zone around a hole before rock failure can guide the mechanism of and early warning methods for geotechnical engineering disasters, while previous theoretical research and numerical simulation results show that the shape of the plastic zone around the hole is butterfly shaped under specific stress, which is referred to as butterfly failure theory. Studies also indicate that the butterfly shape of the plastic zone around a hole is considered to be the main cause of many disasters, which signifies the importance of studying the morphology of the plastic zone near rock failure. Therefore, this study is committed to finding the specific shape of the plastic zone near rock failure through relatively accurate and a high number of AE event location results, and the final experimental results show that the plastic zone around the hole is basically a butterfly shape near rock failure. This study verifies the correctness of the butterfly failure theory and provides an important reference for the study of geotechnical engineering disaster mechanisms and monitoring methods. The fact that the plastic zone in the early stage of rock failure in this study tends to be butterfly shaped preliminarily indicates the fractal law of rock failure. In the moment before rock failure, the distribution of AE events is more regular, which leads to large-scale collapse type failure. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)

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14 pages, 5769 KiB

Open AccessArticle

Influences of Different Acid Solutions on Pore Structures and Fractal Features of Coal

by Jingshuo Zhang, Xiaoming Ni, Xiaolei Liu and Erlei Su

Fractal Fract. 2024, 8(2), 82; https://doi.org/10.3390/fractalfract8020082 - 26 Jan 2024

Cited by 4 | Viewed by 1429

Abstract

The effect of different acids on the pore structure and fractal characteristics of micropores and mesopores was determined with the help of low-temperature liquid nitrogen adsorption, X-ray diffraction, and the Frenkel–Halsey–Hill (FHH) model by using Yuwu coal as a sample and placing it [...] Read more.

The effect of different acids on the pore structure and fractal characteristics of micropores and mesopores was determined with the help of low-temperature liquid nitrogen adsorption, X-ray diffraction, and the Frenkel–Halsey–Hill (FHH) model by using Yuwu coal as a sample and placing it in acidic environments, such as HF, HCl, HNO₃, and CH₃COOH. The results show that the acidization effects of HF and CH₃COOH are separately dominated by the micropore and mesopore formation effects, while HCl and HNO₃ mainly play their roles in expanding mesopores. After acidization, the surface fractal dimensions D₁ and D₁′ of micropores and mesopores in coal are always negatively correlated with the total specific surface area S_BET, specific surface area S_mic of micropores, and specific surface area S_mes of mesopores. After being acidized by HF, D₂ is negatively correlated with the total volume V_tot and the corresponding micropore volume V_mic, while acidization with HCl and HNO₃ leads to the opposite result. After being acidized by CH₃COOH, D₂ has a negative correlation with V_tot and a positive correlation with V_mic. The structural fractal dimensions D₂′ of mesopores in samples acidized by HF and CH₃COOH are positively correlated with both the volume V_tot and mesopore volume V_mes, while it is the opposite for samples acidized by HNO₃. D₂′ of coal samples acidized by HCl is negatively correlated with V_tot while positively correlated with V_mes. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)

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28 pages, 11882 KiB

Open AccessArticle

A Binary Medium Constitutive Model for Frozen Solidified Saline Soil in Cold Regions and Its Fractal Characteristics Analysis

by Xinrui Kang, Hongbo Li, Gang Zhang, Sheng Li, Long Shan, Jing Zhao and Zhe Zhang

Fractal Fract. 2024, 8(1), 33; https://doi.org/10.3390/fractalfract8010033 - 2 Jan 2024

Cited by 2 | Viewed by 1727

Abstract

In addressing the issue of strength degradation in saline soil foundations under the salt-freeze coupling effects, a binary medium constitutive model suitable for un-solidified and solidified frozen saline soil is proposed considering both bonding and friction effects. To verify the validity of the [...] Read more.

In addressing the issue of strength degradation in saline soil foundations under the salt-freeze coupling effects, a binary medium constitutive model suitable for un-solidified and solidified frozen saline soil is proposed considering both bonding and friction effects. To verify the validity of the constitutive model, freezing triaxial tests are carried out under different negative temperatures, confining pressures, and water contents. The pore structure and fractal characteristics of saline soil are analyzed using mercury intrusion porosimetry (MIP) and the fractal dimension D qualitatively and quantitatively, which shed light on the strength enhancement mechanism during the solidification of frozen saline soils. The results show that the constitutive model for frozen solidified saline soil based on binary medium theory aptly captures the stress–strain relationship before and after the solidification of frozen saline soil. The stress–strain relationship of frozen saline soil before and after solidification can be delineated into linear elasticity, elastoplasticity, and strain-hardening or -softening phases. Each of these phases can be coherently interpreted through the binary medium constitutive model. The un-solidified and solidified frozen both show pronounced fractal characteristics in fractal analysis. Notably, the fractal dimension D of the solidified saline soil exhibits a significant increase compared to that of un-solidified ones. In Regions I and III, the values of D for solidified saline soil are lower than those for untreated saline soil, which is attributed to the filling effect of hydration products and un-hydrated solidifying agent particles. In Region II, the fractal dimensions D_M_II and D_N_II of the solidified saline soil exhibit a “non-physical state”, which is mainly caused by the formation of a significant number of inkpot-type pores due to the binding of soil particles by hydration products. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)

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25 pages, 7143 KiB

Open AccessArticle

Energy Transfer and Destabilizing Impulse Inducing Mechanism of Coal–Rock System in Roadway through Coal Seam in Deep Zone

by Jiazhuo Li, Wei Zhou, Jiaqi Chu, Wentao Ren, Linming Dou and Shikang Song

Fractal Fract. 2023, 7(7), 550; https://doi.org/10.3390/fractalfract7070550 - 16 Jul 2023

Cited by 1 | Viewed by 1442

Abstract

Aiming at the problem of the unclear rock burst generation mechanism and anti-impact measures of a large roadway through a coal seam in a deep panel area, taking the rock burst of a large roadway in the first panel area of Gaojiapu Coal [...] Read more.

Aiming at the problem of the unclear rock burst generation mechanism and anti-impact measures of a large roadway through a coal seam in a deep panel area, taking the rock burst of a large roadway in the first panel area of Gaojiapu Coal Mine as the engineering background, this paper adopts the comprehensive research methods of theoretical analysis, experiments, numerical simulation, fragmentation fractal analysis, and field monitoring, to discuss the mechanical characteristics of the loading process of the assemblage and the energy transfer law and its difference in the deformation and failure process. The possibility and strength of the impact failure of coal under the grip of rock masses with different stiffness are related to the γ value. The smaller the γ value is, the higher the impact possibility is, and the more severe the impact degree is. The assemblage under the grip of soft rock is more prone to system instability. Energy relief and impact reduction are adopted to reduce the post-peak stiffness and elastic strain energy of the coal body in a short distance, avoid the energy transfer and concentration of the roadway surrounding the rock system under the disturbance of a long-distance dynamic load, and reduce the likelihood of impact pressure occurring and the extent to which an impact manifests. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Rock Engineering)

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