Mathematical Problems in Rock Mechanics and Rock Engineering

With the increasing requirements for energy, resources and space, numerous rock engineering projects (e.g., mining, tunnelling, underground storage, geothermal energy, petroleum, water conservancy and 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, severely threatening 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. This Special Issue, “Mathematical Problems in Rock Mechanics and Rock Engineering”, aims 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. It includes 12 manuscripts that illustrate the valuable efforts for addressing mathematical problems in rock mechanics and rock engineering.

The article by Wang et al. [1] aims to investigate the dynamic mechanics and post-failure characteristics of fault-cemented rock strata by Split Hopkinson pressure bar (SHPB) dynamic impact tests on cemented rock samples with various particle size distributions (PSDs). The results show that the breakage ratio and fractal dimension have a linear relationship regardless of the PSD or strain, and 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.

The article by Li et al. [2] aims to analyze the impact of optimizers (Adam, SGD, RMSprop) and learning rate (lambda and cosine decay modes) on the performance of deep learning-based algorithms for rock thin-section image classification by using 2634 rock thin-section images including three rock types—metamorphic, sedimentary and volcanic rocks. The investigation shows that the cosine learning-rate decay mode is the better option for learning-rate adjustment during the training process, and the capabilities of the model using Adam and RMSprop optimizers were more robust than that of SGD.

The study proposed by Li et al. [3] aimed to quantify the degree of coal macroscopic deformation under different loads using Computed tomography (CT) scans. The results illustrate that 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 prevalent shear stress in coal and make coal prone to stress concentration.

Li et al. [4] investigated the effect of temperature on the dynamic properties of marble using the dynamic and static combined SHPB test device, considering that deep rock will be impacted by different temperatures and varied disturbance degrees. The results revealed that the diameter and height of the specimen increased, and the mass and longitudinal wave velocity dropped as the temperature climbed. The variation laws of the total stress–strain curves after varied high temperatures are substantially the same, and the peak stress was negatively correlated with the action temperature.

Khan et al. [5] propose a new predictive model based on artificial intelligence to quantify the damage factor of rock by thermal treatment followed by subsequent cooling conditions (slow and rapid). The results show that an ANN-based predictive model is the most efficient model for quantifying the rock damage factor based on porosity compared to other models based on multilinear regression (MLR) and the adoptive neural-fuzzy inference system (ANFIS).

Chen et al. [6] developed a method to obtain the size distribution characteristics of the real source from the apparent amplitude in doubly truncated distribution using rock acoustic emission (AE) tests. The results indicate that mineral grains of different sizes and compositions and different types of discontinuities of rock specimens determine the rock fracture characteristics and the AE 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.

Yin et al. [7] proposed a pollution evaluation system based on the fractal dimension theory (Dbox(P)) and the 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 results obviously denote three diffusion stages of the pollutants, mainly including the generation stage, cloud-formation stage and the diffusion stage during open-pit rock blasting.

Qiu et al. [8] developed a dataset of 734 samples from previous studies on different countries’ magmatic, sedimentary and metamorphic rocks to estimate uniaxial compressive strength (UCS) using three main factors of point load index, P-wave velocity and the Schmidt hammer rebound number based on an extreme learning machine improved with a metaheuristic algorithm. The results show that the extreme learning machine with the whale optimization algorithm (WOA-ELM model) has high accuracy and reliability, which means that it has broad application potential for estimating the UCS of different rocks.

Qiu et al. [9] aimed to analyze the effect of tunnel distribution on the dynamic response characteristics of a remote non-adjacent tunnel. 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.

Xiong et al. [10] analyze the evolution characteristics of freeze-and-thaw (F&T) damage based on the T₂ spectrum distribution curves of sandstone specimens before and after F&T weathering cycles. The results show that the quantity of F&T weathering cycles and confining pressure can significantly influence the pre-peak and post-peak deformation behaviors of sandstone specimens.

Shahani et al. [11] developed four advanced machine learning (ML)-based intelligent prediction models, namely Lasso regression (LR), ridge regression (RR), decision tree (DT) and support vector machine (SVM), to predict 𝑐 in (MPa) and 𝜑 in (°) of rock, with P-wave velocity in (m/s), density in (gm/cc), UCS in (MPa) and tensile strength in (MPa) as the input parameters. The results show that UCS and tensile strength were the most influential parameters in predicting 𝑐 and 𝜑.

Gong et al. [12] studied the dynamic tensile mechanical properties, layered effect and density evolution characteristics of strain energy for coal using the split Hopkinson pressure bar (SHPB) technique. 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 with more complexity.

The guest editors hope that the selected papers will help scholars and researchers to push forward the progress in dealing with the mathematical problems in rock mechanics and rock engineering.