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Mathematical Problems in Rock Mechanics and Rock Engineering
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Mathematical Problems in Rock Mechanics and Rock Engineering

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (30 September 2022) | Viewed by 27148

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

Special Issue Editors

Prof. Dr. Shaofeng Wang

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Guest Editor
School of Resources and Safety Enginerring, Central South University, Changsha 410083, China
Interests: rock mechanics; geomechanics; geomaterials
Special Issues, Collections and Topics in MDPI journals
Dr. Linqi Huang

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Guest Editor
School of Resources and Safety Engineering, Central South University, Changsha 410083, China
Interests: geomechanics; microseismic monitoring
Special Issues, Collections and Topics in MDPI journals
Dr. Xin Cai

E-Mail Website
Guest Editor
School of Resources and Safety Engineering, Central South University, Changsha 410083, China
Interests: rock mechanics; rock dynamics; fracture; rock-water Interaction
Special Issues, Collections and Topics in MDPI journals
Dr. Zhengyang Song

E-Mail Website
Guest Editor
Department of Civil Engineering, School of Civil and Resource Engineering, University of Science and Technology Beijing, Beijing 100083, China
Interests: geotechnical engineering; mining engineering
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

With the increasing requirements for energy, resources and space, numerous rock engineering projects (e.g., mining, tunnelling, underground storage, geothermal energy, petroleum, water conservancy, hydropower) are more often being constructed and operated in large-scale environments with complex geology. Meanwhile, rock failures and rock instabilities (e.g., rockbursts, large-scale collapse, slabbing, zonal disintegration and microseism) occur more frequently, and severely threaten the safety and stability of rock engineering projects. It is well-recognized that rock has multi-scale structures, from minerals, particles, fractures, fissures, joints, and stratification, to fault, and involves multi-scale fracture processes. Meanwhile, rocks are commonly subjected simultaneously to complex static stress and strong dynamic disturbance, providing a hotbed for the occurrence of rock failures. In addition, there are many multi-physics coupling processes in a rock mass, such as the coupled thermo–hydromechanical interaction in fractured porous rocks. It is still difficult to understand these rock mechanics and characterize rock behavior during complex stress conditions, multi-physics processes, and multi-scale changes. Therefore, our understanding of rock mechanics and the prevention and control of failure and instability in rock engineering needs to be furthered.

The primary aim of this Special Issue is to bring together original research discussing innovative efforts regarding in situ observations, laboratory experiments and theoretical, numerical, and big-data-based methods to overcome the mathematical problems related to rock mechanics and rock engineering. Submissions showcasing the latest developments in this field are welcome.

Dr. Shaofeng Wang
Dr. Linqi Huang
Dr. Xin Cai
Dr. Zhengyang Song
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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

  • mathematical problems
  • in situ observations
  • laboratory experiments
  • theoretical methods
  • numerical methods
  • big-data-based methods
  • rock mechanics
  • multi-physics processes
  • multi-scale fracturing
  • rock engineering

Published Papers (13 papers)

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Editorial

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3 pages, 173 KiB  
Editorial
Mathematical Problems in Rock Mechanics and Rock Engineering
by Linqi Huang, Shaofeng Wang, Xin Cai and Zhengyang Song
Mathematics 2023, 11(1), 67; https://doi.org/10.3390/math11010067 - 25 Dec 2022
Cited by 1 | Viewed by 1685
Abstract
With the increasing requirements for energy, resources and space, numerous rock engineering projects (e [...] Full article
(This article belongs to the Special Issue Mathematical Problems in Rock Mechanics and Rock Engineering)

Research

Jump to: Editorial

33 pages, 16343 KiB  
Article
Dynamic Tensile Mechanical Properties of Outburst Coal Considering Bedding Effect and Evolution Characteristics of Strain Energy Density
by Shuang Gong, Chaofei Wang, Furui Xi, Yongqiang Jia, Lei Zhou, Hansong Zhang, Jingkuo Wang, Xingyang Ren, Shuai Wang, Shibin Yao and Juan Liu
Mathematics 2022, 10(21), 4120; https://doi.org/10.3390/math10214120 - 4 Nov 2022
Cited by 3 | Viewed by 1065
Abstract
The evolution of strain energy density of outburst-prone coal is of great significance for analyzing the characteristics of energy accumulation and release in coal and rock masses. The dynamic mechanical properties of coal samples were tested by using the split Hopkinson pressure bar [...] Read more.
The evolution of strain energy density of outburst-prone coal is of great significance for analyzing the characteristics of energy accumulation and release in coal and rock masses. The dynamic mechanical properties of coal samples were tested by using the split Hopkinson pressure bar (SHPB) technique. Dynamic tensile mechanical properties, layered effect and density evolution characteristics of strain energy for coal were studied. The dynamic failure and crack propagation process of the specimen were recorded with a high-speed camera. In addition, a digital image correlation (DIC) method was used to analyze the evolution characteristics of the strain field during the deformation process of the specimen. The distribution characteristics of the particle fragments were statistically analyzed. The results show that the bedding orientation of the coal has a significant effect on its deformation and damage features. The presence of weak planes, microcracks and laminae causes its shear damage zone to behave more complex. If the crack plane coincides with the high shear stress plane, the developed shear cracks extend along the weak laminae and the shear damage zones in BD specimens are not symmetrically distributed. When the laminated surface of the coal sample is at a certain angle with the impact loading direction, the damage mode is coupled with tensile and shear damage. The percentage mass distribution of particles and fines increases with increasing bedding orientation. The effect of water on the dynamic damage of coal samples is significant. Based on the principle of pressure expansion of wing-shaped cracks, the formula for calculating the dynamic strength of water-saturated coal samples under dynamic loading was derived. Full article
(This article belongs to the Special Issue Mathematical Problems in Rock Mechanics and Rock Engineering)
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17 pages, 5115 KiB  
Article
Predicting Angle of Internal Friction and Cohesion of Rocks Based on Machine Learning Algorithms
by Niaz Muhammad Shahani, Barkat Ullah, Kausar Sultan Shah, Fawad Ul Hassan, Rashid Ali, Mohamed Abdelghany Elkotb, Mohamed E. Ghoneim and Elsayed M. Tag-Eldin
Mathematics 2022, 10(20), 3875; https://doi.org/10.3390/math10203875 - 19 Oct 2022
Cited by 8 | Viewed by 5771
Abstract
The safe and sustainable design of rock slopes, open-pit mines, tunnels, foundations, and underground excavations requires appropriate and reliable estimation of rock strength and deformation characteristics. Cohesion (𝑐) and angle of internal friction (𝜑) are the two key parameters widely used to characterize [...] Read more.
The safe and sustainable design of rock slopes, open-pit mines, tunnels, foundations, and underground excavations requires appropriate and reliable estimation of rock strength and deformation characteristics. Cohesion (𝑐) and angle of internal friction (𝜑) are the two key parameters widely used to characterize the shear strength of materials. Thus, the prediction of these parameters is essential to evaluate the deformation and stability of any rock formation. In this study, four advanced machine learning (ML)-based intelligent prediction models, namely Lasso regression (LR), ridge regression (RR), decision tree (DT), and support vector machine (SVM), were developed to predict 𝑐 in (MPa) and 𝜑 in (°), with P-wave velocity in (m/s), density in (gm/cc), UCS in (MPa), and tensile strength in (MPa) as input parameters. The actual dataset having 199 data points with no missing data was allocated identically for each model with 70% for training and 30% for testing purposes. To enhance the performance of the developed models, an iterative 5-fold cross-validation method was used. The coefficient of determination (R2), mean absolute error (MAE), mean square error (MSE), root mean square error (RMSE), and a10-index were used as performance metrics to evaluate the optimal prediction model. The results revealed the SVM to be a more efficient model in predicting 𝑐 (R2 = 0.977) and 𝜑 (R2 = 0.916) than LR (𝑐: R2 = 0.928 and 𝜑: R2 = 0.606), RR (𝑐: R2 = 0.961 and 𝜑: R2 = 0.822), and DT (𝑐: R2 = 0.934 and 𝜑: R2 = 0.607) on the testing data. Furthermore, to check the level of accuracy of the SVM model, a sensitivity analysis was performed on the testing data. The results showed that UCS and tensile strength were the most influential parameters in predicting 𝑐 and 𝜑. The findings of this study contribute to long-term stability and deformation evaluation of rock masses in surface and subsurface rock excavations. Full article
(This article belongs to the Special Issue Mathematical Problems in Rock Mechanics and Rock Engineering)
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19 pages, 5493 KiB  
Article
Mechanical Properties and Strength Evolution Model of Sandstone Subjected to Freeze–Thaw Weathering Process: Considering the Confining Pressure Effect
by Xin Xiong, Feng Gao, Keping Zhou, Chun Yang and Jielin Li
Mathematics 2022, 10(20), 3841; https://doi.org/10.3390/math10203841 - 17 Oct 2022
Cited by 3 | Viewed by 1156
Abstract
Freeze-and-thaw (F&T) weathering cycles induced by day–night and seasonal temperature changes cause a large number of rock mass engineering disasters in cold areas. Investigating the impact of F&T weathering process on the strength and deformation characteristics of frozen–thawed rocks is therefore of critical [...] Read more.
Freeze-and-thaw (F&T) weathering cycles induced by day–night and seasonal temperature changes cause a large number of rock mass engineering disasters in cold areas. Investigating the impact of F&T weathering process on the strength and deformation characteristics of frozen–thawed rocks is therefore of critical scientific importance for evaluating the stability and optimizing the design of rock mass engineering in these areas. In this research, the evolution characteristics of F&T damage were analyzed based on T2 spectrum distribution curves of sandstone specimens before and after F&T weathering cycles. The coupling impact of the quantity of F&T weathering cycles and confining pressure on pre-peak and post-peak deformation behaviors of sandstone specimens were analyzed in detail. By introducing the confining pressure increase factor (CPIF), the impact of confining pressure on the triaxial compressive strength (TCS) of sandstone specimens after undergoing different quantities of F&T weathering cycles was further investigated. A novel strength evolution model was proposed that could effectively describe the coupling impact of the quantity of F&T weathering cycles and confining pressure on TCS of rocks after undergoing the F&T weathering process. The proposed strength evolution model was cross-verified with experimental data from the published literature and all correlation coefficients were above 0.95, which proved that the strength evolution model proposed in this paper was reasonable; in addition, this model has strong applicability. Full article
(This article belongs to the Special Issue Mathematical Problems in Rock Mechanics and Rock Engineering)
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19 pages, 5662 KiB  
Article
Effect of Different Tunnel Distribution on Dynamic Behavior and Damage Characteristics of Non-Adjacent Tunnel Triggered by Blasting Disturbance
by Jiadong Qiu and Fan Feng
Mathematics 2022, 10(19), 3705; https://doi.org/10.3390/math10193705 - 10 Oct 2022
Cited by 3 | Viewed by 1216
Abstract
When a blasting is executed near two tunnels, the blasting wave will trigger a dynamic response and damage to the tunnels. Depending on the tunnel distribution, the path of the blasting wave to the remote non-adjacent tunnels will change. The aim of this [...] Read more.
When a blasting is executed near two tunnels, the blasting wave will trigger a dynamic response and damage to the tunnels. Depending on the tunnel distribution, the path of the blasting wave to the remote non-adjacent tunnels will change. The aim of this study is to analyze the effect of the tunnel distribution on the dynamic response characteristics of a remote non-adjacent tunnel. Numerical models of two tunnels were established by PFC2D and three different tunnel distributions were considered. The two tunnels were divided into the adjacent tunnel and the non-adjacent tunnel according to their relative distance to the blasting source. The dynamic stress evolution, damage characteristics and the evolution of strain energy of the non-adjacent tunnel were initially analyzed. The results show that the stress wave amplitude of the non-adjacent tunnel is closely related to the tunnel distribution, but only near the sidewalls of the non-adjacent tunnel is the stress wave waveform sensitive to the tunnel distribution. The larger the tunnel dip, the more severe the damage to the non-adjacent tunnel. In addition, as the tunnel dip increases, the maximum strain energy densities (SEDs) in the roof, floor and sidewalls of the non-adjacent tunnel exhibit different trends. The influence of the wavelength of the blasting wave is further discussed. It is shown that the dynamic stress amplification factor and damage degree around the non-adjacent tunnel is usually positively correlated with the wavelength of the blasting wave. Moreover, the release of strain energy around the non-adjacent tunnel has a positive correlation with the wavelength. The SED variations in different areas around the non-adjacent tunnel also exhibit different trends with the increase of tunnel dip. Full article
(This article belongs to the Special Issue Mathematical Problems in Rock Mechanics and Rock Engineering)
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18 pages, 4212 KiB  
Article
Prediction of Uniaxial Compressive Strength in Rocks Based on Extreme Learning Machine Improved with Metaheuristic Algorithm
by Junbo Qiu, Xin Yin, Yucong Pan, Xinyu Wang and Min Zhang
Mathematics 2022, 10(19), 3490; https://doi.org/10.3390/math10193490 - 24 Sep 2022
Cited by 11 | Viewed by 1913
Abstract
Uniaxial compressive strength (UCS) is a critical parameter in the disaster prevention of engineering projects, requiring a large budget and a long time to estimate in different rocks or the early stage of a project. If predicted accurately, the UCS of rocks significantly [...] Read more.
Uniaxial compressive strength (UCS) is a critical parameter in the disaster prevention of engineering projects, requiring a large budget and a long time to estimate in different rocks or the early stage of a project. If predicted accurately, the UCS of rocks significantly affects geotechnical applications. This paper develops a dataset of 734 samples from previous studies on different countries’ magmatic, sedimentary, and metamorphic rocks. Within the study context, three main factors, point load index, P-wave velocity, and Schmidt hammer rebound number, are utilized to estimate UCS. Moreover, it applies extreme learning machines (ELM) to map the nonlinear relationship between the UCS and the influential factors. Five metaheuristic algorithms, particle swarm optimization (PSO), grey wolf optimization (GWO), whale optimization algorithm (WOA), butterfly optimization algorithm (BOA), and sparrow search algorithm (SSA), are used to optimize the bias and weight of ELM and thus enhance its predictability. Indeed, several performance parameters are utilized to verify the proposed models’ generalization capability and predictive performance. The minimum, maximum, and average relative errors of ELM achieved by the whale optimization algorithm (WOA-ELM) are smaller than the other models, with values of 0.22%, 72.05%, and 11.48%, respectively. In contrast, the minimum and mean residual error produced by WOA-ELM are less than the other models, with values of 0.02 and 2.64 MPa, respectively. The results show that the UCS values derived from WOA-ELM are superior to those from other models. The performance indices (coefficient of determination (R2): 0.861, mean squared error (MSE): 17.61, root mean squared error (RMSE): 4.20, and value account for (VAF): 91% obtained using the WOA-ELM model indicates high accuracy and reliability, which means that it has broad application potential for estimating UCS of different rocks. Full article
(This article belongs to the Special Issue Mathematical Problems in Rock Mechanics and Rock Engineering)
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18 pages, 8091 KiB  
Article
Pollutant Migration Pattern during Open-Pit Rock Blasting Based on Digital Image Analysis Technology
by Jiangjiang Yin, Jianyou Lu, Fuchao Tian and Shaofeng Wang
Mathematics 2022, 10(17), 3205; https://doi.org/10.3390/math10173205 - 5 Sep 2022
Cited by 5 | Viewed by 1973
Abstract
Previous studies have revealed that toxic gases and dust (smoke dust) are the most common pollutants generated by the blasting operations in open-pit mines, which might lead to a threat to the environment’s condition, health and safety, and properties protection around the blasting [...] Read more.
Previous studies have revealed that toxic gases and dust (smoke dust) are the most common pollutants generated by the blasting operations in open-pit mines, which might lead to a threat to the environment’s condition, health and safety, and properties protection around the blasting site. In order to deal with the problems, a pollution evaluation system was established based on the fractal dimension theory (Dbox(P)) and grayscale average algorithm (Ga) in digital image-processing technology to recognize and analyze the distributions of the smoke-dust cloud, and subsequently determine the pollution degrees. The computation processes of Dbox(P) and Ga indicate three fitted correlations between the parameters and diffusion time of smoke dust. Then, a pollution index (Pi) is put forward to integrate the global and local features of Dbox(P) and Ga, and develop a hazard classification mechanism for the blasting pollutants. Results obviously denote three diffusion stages of the pollutants, mainly including generation stage, cloud-formation stage, and diffusion stage. In addition, it has been validated that the proposed system can also be utilized in single-point areas within a whole digital image. Besides, there are variation trends of the thresholds T1 and T2 in binarization with the diffusion of pollutants. With this identification and evaluation system, the pollution condition of smoke dust can be obviously determined and analyzed. Full article
(This article belongs to the Special Issue Mathematical Problems in Rock Mechanics and Rock Engineering)
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17 pages, 6412 KiB  
Article
Research on b Value Estimation Based on Apparent Amplitude-Frequency Distribution in Rock Acoustic Emission Tests
by Daolong Chen, Changgen Xia, Huini Liu, Xiling Liu and Kun Du
Mathematics 2022, 10(17), 3202; https://doi.org/10.3390/math10173202 - 5 Sep 2022
Cited by 9 | Viewed by 1775
Abstract
The rock acoustic emission (AE) technique has often been used to study rock destruction properties and has also been considered an important measure for simulating earthquake foreshock sequences. Among them, the AE b value is an essential parameter for the size distribution characteristics [...] Read more.
The rock acoustic emission (AE) technique has often been used to study rock destruction properties and has also been considered an important measure for simulating earthquake foreshock sequences. Among them, the AE b value is an essential parameter for the size distribution characteristics and probabilistic hazard analysis of rock fractures. Variations in b values obtained in rock AE tests and earthquakes are often compared to establish analogies in the damage process and precursory analysis. Nevertheless, because the amplitudes measured on the sample boundary by an acoustic sensor (apparent amplitude) are often used to estimate the b value, which cannot descript the source size distribution, it is necessary to develop a method to obtain the size distribution characteristics of the real source from the apparent amplitude in doubly truncated distribution. In this study, we obtain AE apparent amplitudes by applying an attenuation operator to source amplitudes generated by a computer with an underlying exponential distribution and then use these simulated apparent amplitudes to perform a comparative analysis of various b value estimation methods that are used in earthquakes and propose an optimal b value estimation procedure for rock AE tests through apparent amplitudes. To further verify the reliability of the newly proposed procedure, a b value characteristics analysis was carried out on a non-explosive expansion agent rock AE test and transparent refractive index experiment with red sandstone, marble, granite, and limestone. The results indicate that mineral grains of different sizes and compositions and different types of discontinuities of rock specimens determine the rock fracture characteristics, as well as the b value. The dynamic b values decreased linearly during the loading process, which confirms that variations in the b value also depend on the stress. These results indicate that the newly proposed procedure for estimating the b value in rock AE tests based on apparent amplitudes has high reliability. Full article
(This article belongs to the Special Issue Mathematical Problems in Rock Mechanics and Rock Engineering)
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22 pages, 3679 KiB  
Article
Development of Predictive Models for Determination of the Extent of Damage in Granite Caused by Thermal Treatment and Cooling Conditions Using Artificial Intelligence
by Naseer Muhammad Khan, Kewang Cao, Muhammad Zaka Emad, Sajjad Hussain, Hafeezur Rehman, Kausar Sultan Shah, Faheem Ur Rehman and Aamir Muhammad
Mathematics 2022, 10(16), 2883; https://doi.org/10.3390/math10162883 - 11 Aug 2022
Cited by 11 | Viewed by 1615
Abstract
Thermal treatment followed by subsequent cooling conditions (slow and rapid) can induce damage to the rock surface and internal structure, which may lead to the instability and failure of the rock. The extent of the damage is measured by the damage factor ( [...] Read more.
Thermal treatment followed by subsequent cooling conditions (slow and rapid) can induce damage to the rock surface and internal structure, which may lead to the instability and failure of the rock. The extent of the damage is measured by the damage factor (DT), which can be quantified in a laboratory by evaluating the changes in porosity, elastic modulus, ultrasonic velocities, acoustic emission signals, etc. However, the execution process for quantifying the damage factor necessitates laborious procedures and sophisticated equipment, which are time-consuming, costly, and may require technical expertise. Therefore, it is essential to quantify the extent of damage to the rock via alternate computer simulations. In this research, a new predictive model is proposed to quantify the damage factor. Three predictive models for quantifying the damage factors were developed based on multilinear regression (MLR), artificial neural networks (ANNs), and the adoptive neural-fuzzy inference system (ANFIS). The temperature (T), porosity (ρ), density (D), and P-waves were used as input variables in the development of predictive models for the damage factor. The performance of each predictive model was evaluated by the coefficient of determination (R2), the A20 index, the mean absolute percentage error (MAPE), the root mean square error (RMSE), and the variance accounted for (VAF). The comparative analysis of predictive models revealed that ANN models used for predicting the rock damage factor based on porosity in slow conditions give an R2 of 0.99, A20 index of 0.99, RMSE of 0.01, MAPE of 0.14, and a VAF of 100%, while rapid cooling gives an R2 of 0.99, A20 index of 0.99, RMSE of 0.02, MAPE of 0.36%, and a VAF of 99.99%. It has been proposed that an ANN-based predictive model is the most efficient model for quantifying the rock damage factor based on porosity compared to other models. The findings of this study will facilitate the rapid quantification of damage factors induced by thermal treatment and cooling conditions for effective and successful engineering project execution in high-temperature rock mechanics environments. Full article
(This article belongs to the Special Issue Mathematical Problems in Rock Mechanics and Rock Engineering)
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16 pages, 4982 KiB  
Article
Research on Dynamic Properties of Deep Marble Influenced by High Temperature
by Xianglong Li, Yongbo Wu, Lihua He, Xiaohua Zhang and Jianguo Wang
Mathematics 2022, 10(15), 2603; https://doi.org/10.3390/math10152603 - 26 Jul 2022
Cited by 7 | Viewed by 1681
Abstract
Deep rock will be influenced by the excavation disturbances of different degrees, which seriously affects the safety production of underground mines. Considering that deep rock will be impacted by different temperatures and varied disturbance degrees, this work analyzes the effect of temperature on [...] Read more.
Deep rock will be influenced by the excavation disturbances of different degrees, which seriously affects the safety production of underground mines. Considering that deep rock will be impacted by different temperatures and varied disturbance degrees, this work analyzes the effect of temperature on the dynamic properties of marble by means of the dynamic and static combined SHPB test device. The results reveal that as the temperature climbed, the diameter and height of the specimen increased and the mass and longitudinal wave velocity dropped. The variation laws of total stress–strain curves after varied high temperatures are substantially the same; the peak stress was negatively correlated with the action temperature. At 25 °C~400 °C, the failure mode of specimens is less affected by temperature. When the temperature is higher than 400 °C, the failure degree of specimens increases with the growth of temperature. At 25~400 °C, the above energy varies minimally. At 400~800 °C, with the increase in temperature, the incident energy, transmitted energy and absorption energy decrease, and the reflection energy increases gradually. Full article
(This article belongs to the Special Issue Mathematical Problems in Rock Mechanics and Rock Engineering)
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16 pages, 8533 KiB  
Article
Characterization of 3D Displacement and Stress Fields in Coal Based on CT Scans
by Qi Li, Zhen Li, Peng Li, Ruikai Pan and Qingqing Zhang
Mathematics 2022, 10(14), 2512; https://doi.org/10.3390/math10142512 - 19 Jul 2022
Cited by 1 | Viewed by 1349
Abstract
Computed tomography (CT) scans were performed on samples of an outburst-prone coal seam at different loading stages. The area and roundness of the CT images were used to quantify the degree of the coal macroscopic deformation under different loads. A spatial matching algorithm [...] Read more.
Computed tomography (CT) scans were performed on samples of an outburst-prone coal seam at different loading stages. The area and roundness of the CT images were used to quantify the degree of the coal macroscopic deformation under different loads. A spatial matching algorithm was used to calculate the three-dimensional (3D) displacement fields of different regions of interest (ROIs, containing primary fractures, minerals, and only coal) under different loads. The presence of fractures and minerals were found to promote and inhibit displacement, respectively, and the 3D displacement field data followed a normal distribution. A meshfree numerical simulation was used to determine the 3D maximum principal stress, shear stress and displacement fields under different loads. The following results were obtained: fractures and minerals significantly affect the stress state and displacement field distribution features, the maximum principal stresses and shear stresses in different matrices differ significantly, and the presence of minerals and fractures induce a prevalent shear stress in coal and make coal prone to stress concentration. Full article
(This article belongs to the Special Issue Mathematical Problems in Rock Mechanics and Rock Engineering)
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28 pages, 15130 KiB  
Article
Experimental Studies on Rock Thin-Section Image Classification by Deep Learning-Based Approaches
by Diyuan Li, Junjie Zhao and Jinyin Ma
Mathematics 2022, 10(13), 2317; https://doi.org/10.3390/math10132317 - 2 Jul 2022
Cited by 12 | Viewed by 2956
Abstract
Experimental studies were carried out to analyze the impact of optimizers and learning rate on the performance of deep learning-based algorithms for rock thin-section image classification. A total of 2634 rock thin-section images including three rock types—metamorphic, sedimentary, and volcanic rocks—were acquired from [...] Read more.
Experimental studies were carried out to analyze the impact of optimizers and learning rate on the performance of deep learning-based algorithms for rock thin-section image classification. A total of 2634 rock thin-section images including three rock types—metamorphic, sedimentary, and volcanic rocks—were acquired from an online open-source science data bank. Four CNNs using three different optimizer algorithms (Adam, SGD, RMSprop) under two learning-rate decay schedules (lambda and cosine decay modes) were trained and validated. Then, a systematic comparison was conducted based on the performance of the trained model. Precision, f1-scores, and confusion matrix were adopted as the evaluation indicators. Trials revealed that deep learning-based approaches for rock thin-section image classification were highly effective and stable. Meanwhile, the experimental results showed that the cosine learning-rate decay mode was the better option for learning-rate adjustment during the training process. In addition, the performance of the four neural networks was confirmed and ranked as VGG16, GoogLeNet, MobileNetV2, and ShuffleNetV2. In the last step, the influence of optimization algorithms was evaluated based on VGG16 and GoogLeNet, and the results demonstrated that the capabilities of the model using Adam and RMSprop optimizers were more robust than that of SGD. The experimental study in this paper provides important practical value for training a high-precision rock thin-section image classification model, which can also be transferred to other similar image classification tasks. Full article
(This article belongs to the Special Issue Mathematical Problems in Rock Mechanics and Rock Engineering)
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21 pages, 7050 KiB  
Article
Effect of Particle Size Distribution on the Dynamic Mechanical Properties and Fractal Characteristics of Cemented Rock Strata
by Jiajun Wang, Linqi Huang, Xibing Li, Yangchun Wu and Huilin Liu
Mathematics 2022, 10(12), 2078; https://doi.org/10.3390/math10122078 - 15 Jun 2022
Cited by 3 | Viewed by 1580
Abstract
To investigate the dynamic mechanics and post-failure characteristics of fault-cemented rock strata, broken rock particles were reshaped to obtain cemented rock samples with various particle size distributions (PSDs). Split Hopkinson pressure bar (SHPB) dynamic impact tests were performed on the cemented rock samples [...] Read more.
To investigate the dynamic mechanics and post-failure characteristics of fault-cemented rock strata, broken rock particles were reshaped to obtain cemented rock samples with various particle size distributions (PSDs). Split Hopkinson pressure bar (SHPB) dynamic impact tests were performed on the cemented rock samples under different strain rates. The test results show that plastic deformation occurs in the cemented rock sample as a result of its porous structure. Therefore, there is no linear phase in the dynamic stress–strain curves. With an increase in the Talbot index and mixture type, more large particles were contained inside the cemented rock sample, and the dynamic strength gradually increased. A power function can effectively describe the relationship between the strain rate and dynamic strength for various Talbot indices. After dynamic impact, the fragments of the cemented rock samples exhibit evident fractal laws, and the breakage of the samples includes breakage of the original rock particle itself and breakage between the rock particles and cementations. The breakage ratio and fractal dimension both decrease with the increase in the number of mixture type and Talbot index but increase with the increase in strain rate. It is worth noting that the breakage ratio and fractal dimension have a linear relationship regardless of the PSD or strain. The relationship between the dynamic strength and fractal dimension has different response laws for the PSD and strain rate effects. The dynamic strength is negatively linearly related to the fractal dimension under the PSD effect but positively linearly related to the fractal dimension under the strain rate effect. This research work can provide foundation support for investigating the instability mechanism of fault cemented rock strata under dynamic stress. Full article
(This article belongs to the Special Issue Mathematical Problems in Rock Mechanics and Rock Engineering)
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