A quantitative analysis of strength, failure, and deformation characteristics of rocks under true-triaxial stress across brittle-ductile regime is essential for deep underground engineering construction. To this end, a series of true-triaxial tests following a common loading path were performed on green sandstone. An explicit Matsuoka-Nakai-Mohr–Coulomb strength model is proposed and validated against true-triaxial strength data for green sandstone, Sorcy limestone, and Bentheim sandstone. Based on post-peak behavior, the stress–strain curves are classified into yield softening, yield plateau, and yield hardening, and further into plastic volumetric dilation and contraction. Strain trajectories within meridian plane exhibit a clockwise rotation trend as intermediate principal stress rises, indicating that the rock tends to undergo volumetric deformation rather than shear deformation. The difference between strain and stress Lode angles decreases with increasing intermediate principal stress, reflecting Lode angle sensitivity of deformation. Failure modes are categorized into penetrating, discrete, and indistinct shear bands, primarily controlled by hydrostatic pressure. With increasing hydrostatic pressure, shear bands evolve from dilatant to contractive behavior, and the formation of penetrating shear bands inhibits sustained plastic hardening.

The reservoir rocks exist within geological environments characterized by elevated true-triaxial stress, pore pressure, and temperature. Understanding the coupled effects of temperature and true-triaxial stress on three-directional permeability is crucial for the efficient extraction of underground resources. Therefore, this study employs a self-developed true-triaxial geophysical imaging cell equipped with a temperature-controlled module to conduct three-directional helium permeability tests on purple sandstone. A stepwise loading path is adopted in this study to examine the influence of temperature (from 20 to 80 °C) and true-triaxial stress on three-dimensional permeability. As a result of inherent structural heterogeneity, the three-directional permeability exhibits significant anisotropy even under isotropic stress conditions. Under the coupled effects of thermal expansion and stress compaction, the three-directional permeability decreases with increasing temperature and principal stresses, and its sensitivity exhibits a negative correlation with both temperature and stress levels. This study introduces anisotropy ratios to quantitatively characterize the anisotropy degree in three-directional permeability ( k_y/ . -0ptk_x , k_z/ . -0ptk_x , k_z/ . -0ptk_y ). Owing to the differences in sensitivity of three-directional permeability to temperature and principal stresses, the anisotropy ratio exhibits complex nonlinear evolution with changing temperature and true-triaxial stress states. A new three-directional permeability model is developed by introducing variable temperature-stress sensitivity coefficients into exponential permeability formulation. This model can effectively characterize the evolution of three-directional permeability and corresponding anisotropy ratio under varying temperature and true-triaxial stress conditions.

Accurately describing the variation in rock strength within brittle–ductile domain is of significant importance for deep underground engineering. In this study, a novel three-dimensional strength criterion based on parameterization method is proposed, encompassing the criteria within brittle and ductile domains. The criterion within brittle domain is expressed as the multiplication of Hoek–Brown criterion under triaxial compression and a generalized brittle deviatoric function, while the criterion within ductile domain is expressed as the multiplication of Drucker–Prager cap model under triaxial compression and a generalized ductile deviatoric function. Both generalized brittle and ductile deviatoric functions are based on Matsuoka–Nakai criterion, which is controlled by deviatoric parameters. Besides, deviatoric parameters are the function of hydrostatic pressure. The brittle failure envelope intersects with ductile failure envelope. Therefore, only the portion closer to the hydrostatic pressure axis is selected. The strength data of Bentheim sandstone, green sandstone, and Sorcy limestone are used to validate the proposed strength criterion. Regardless of the rock type, the hydrostatic pressure at brittle–ductile transition increases with increasing Lode angle, indicating a positive correlation between brittleness of rock and Lode angle. During brittle–ductile transition, the deviatoric plane is jointly controlled by brittle and ductile deviatoric functions. Under low Lode angle conditions, it is controlled by the ductile deviatoric function, while under high Lode angle conditions, it is controlled by the brittle deviatoric function. Finally, a smoothing method is applied at the interface of brittle and ductile failure envelopes to form a smooth transition zone, which is regarded as the critical state.

Quantitatively analyzing the dependence of three-directional permeability on the true-triaxial stress state is of great significance for oil and gas resource exploitation. The three-directional permeability of purple sandstone under true-triaxial stress is measured using a self-developed geophysical imaging cell with a multi-stage monotonic loading path. The experimental results indicate that the three-directional permeability exhibits anisotropy even under isotropic stress loading due to the rock’s internal heterogeneity (intrinsic anisotropy). The three-directional permeability exhibits variable stress sensitivity arising from both true-triaxial stress-induced and intrinsic anisotropies. The anisotropy ratios are introduced to characterize the anisotropy of three-directional permeability, which exhibit three distinct trends with increasing principal stresses: fluctuating increase, fluctuating decrease, and near constancy. Based on the above-discussed permeability characteristics, a new three-directional permeability model is proposed by integrating the exponential permeability model with the anisotropic effective stress-sensitivity coefficient. The proposed model well describes the dependence of anisotropy ratios on the effective principal stresses.

A three-dimensional elastoplastic constitutive model is proposed for dilatant rocks undergoing strain hardening and softening. The yield function adopts a power-law formulation to characterize its meridian-plane features, with different power exponents selected for different rocks. In addition, the deviatoric function of MatsuokaNakai criterion is incorporated into the yield function. The parameters controlling the shapes of yield surface on the meridian and deviatoric planes are treated as functions of the internal plastic variable, and they exhibit non-monotonic variations with increasing internal plastic variable. The non-associated flow rule is adopted, and a linear plastic potential function including Lode angle effects is employed. The plastic potential function contains two parameters that govern the plastic flow directions on the meridian and deviatoric planes. The plastic potential parameters are likewise treated as functions of internal plastic variable, with their specific forms determined by the evolution of plastic flow direction. The accuracy and applicability of proposed model are validated using three sets of true-triaxial experimental data for sandstone, siltstone, and granite. To further demonstrate the importance of including Lode angle effects, the computational results obtained with and without including Lode angle effects in the plastic potential function are compared. Across all rock types, neglecting Lode angle effects in the plastic potential function consistently overpredicts the minimum principal strain while underpredicting the intermediate principal strain. Moreover, this discrepancy becomes more pronounced during softening stage.

During oil and gas extraction and the tunnel excavation, the failure characteristics of shale become highly complex due to the combined effects of inherent structural anisotropy and true-triaxial unloading stress paths. To investigate the energy evolution and failure modes of shale under true-triaxial unloading path, a series of unloading experiments were performed with emphasis on the intermediate principal stress ( σ_2 ). The peak strength and stress–strain curve of shale were first analyzed. Subsequently, the energy evolution process and failure modes under true triaxial unloading paths were discussed. The results indicate that both the strength and deformability of shale are governed by the coupled effects of the initial stress state and bedding plane inclination. With increasing σ_2 in the initial stress state, the peak elastic energy, peak dissipated energy, and total input energy increase progressively, exhibiting an approximately linear relationship. When σ_2 is relatively low, the failure modes of shale are significantly affected by the bedding plane inclinations, presenting splitting along bedding planes, shear slip along bedding planes, and shear failure across bedding planes. When σ_2 is relatively high, the influence of structural anisotropy diminishes, and the failure behavior is dominated by shear failure across bedding planes. Furthermore, pronounced differences are observed in the microstructural characteristics of shear and splitting fracture surfaces. From the perspective of energy dissipation at the fracture initiation stage, these observations provide a mechanistic interpretation for the variations in dissipated energy during unloading.

Seismic waves emitted by an earthquake can trigger other earthquakes over a variety of spatial and temporal scale, yet extended time delay and rupture behaviors associated with these events are not well understood. Here, we report on the experimental observation of earthquake rupture triggered by explosion-generated stress perturbations at different stress level, showing that the time delay of triggered events on gouge-filled fault is significantly higher than those on bare fault. Furthermore, we identify notable difference in rupture behaviors between the triggered events on gouge-filled and bare faults. For bare fault, the triggered events may be arrested or runaway, depending on the initial stress level at the onset of dynamic triggering. However, for gouge-filled fault, the rupture behaviors is more complex, manifested by scattered micro-ruptures, coalescent micro-ruptures, and runaway rupture. The extended time delay and complex rupture behaviors of triggered events on gouge-filled fault are attributed to the evolution of force chain within the fault gouge. Our results suggest that fault gouge plays an important role in earthquake dynamic triggering, which may provide new insights into the mechanism for extended time delay of dynamically triggered earthquakes on natural fault systems.

Stress-wave stimulation offers a promising strategy for enhancing permeability in deep reservoir rocks, yet the governing mechanisms and influences of loading characteristics remain poorly understood. To address this shortcoming, in this study, controllable stress waves were applied to green sandstone specimens using a modified triaxial split Hopkinson pressure bar system under coupled hydraulic-mechanical loading conditions. Three distinct energy-input modes, including progressively increasing, constant, and progressively decreasing, were designed to deliver identical total energy through seven impacts. In-situ permeability was measured after each impact, and the corresponding dynamic response was analyzed to clarify the mechanisms of permeability evolution. The results showed that stress wave loading substantially enhances permeability, while the evolution trend strongly depends on the energy-input patterns and coupled hydraulic-mechanical conditions. Moreover, energy-constant loadings produce the most pronounced and sustained permeability growth, whereas energy-decreasing loadings yield sharp early increases followed by reductions due to compaction. Energy-increasing loading leads to delayed permeability enhancement, governed by the eventual onset of macro failure. Hydraulic pressure promotes permeability by facilitating fracture extension, while confining pressure inhibits crack propagation. The mechanical response parameters such as peak stress, peak strain and dissipated energy cannot consistently reflect permeability evolution due to their dependence on instantaneous loading. In contrast, residual deformation correlates strongly with permeability across different loading modes, serving as a reliable indicator of damage-induced fluid transport. This work clarifies the role of energy-input patterns in enhancing permeability and provides guidance for optimizing stress wave stimulation strategies in deep geo-energy recovery.

A comprehensive understanding of mechanical behaviors of shale under the coupled effects of true-triaxial stress and structural anisotropy is crucial for reservoir development. In this study, true-triaxial tests are conducted on Changning shale with bedding planes perpendicular (region I) and parallel (region II) to the maximum principal stress. The stress–strain curves and volumetric-shear strain trajectories show strong dependence on stress conditions and bedding orientation. Under true-triaxial stress conditions, a dominant macroscopic shear band forms within the bedding-containing plane in region I, while macroscopic shear bands develop within the bedding-free plane in region II. Microscopic thin-section analysis reveals that the main shear band within the bedding-containing plane is accompanied by bedding-controlled failure bands in region I, while it within the bedding-free plane is accompanied by multiple sets of parallel failure bands (en-echelon fractures) in region II. Shale exhibits pronounced strength anisotropy, with higher strength in region II than in region I. The difference in deviatoric stress between regions I and II is positively correlated with hydrostatic pressure and negatively correlated with the Lode angle. The main difference between isotropic and anisotropic criteria lies in the deviatoric function. Accordingly, three simple methods for anisotropic deviatoric functions are proposed in this study, including corner point method, variable parameter method, and anisotropy coefficient method. The effectiveness and accuracy of these three methods are verified using the strength data of Changing shale, San Francisco Bay mud, and Santa Minica Beach sand.

This study investigates the influence of mean stress and Lode angle on the mechanical behavior of porous sandstone. Sandstone specimens were tested using a newly developed true-triaxial loading apparatus under five constant Lode angle conditions and seven different mean stresses, covering a transition from brittle to ductile regimes. Based on the experimental results, three types of stress-strain responses were identified, transitioning progressively from Type 1, through Type 2 to Type 3 as the mean stress increases. Type 1 response represents typical brittle behavior, characterized by prominent shear fractures. Type 2 response corresponds to the brittle-ductile transition behavior, exhibiting non-penetrating shear fractures. Type 3 response is associated with ductile behavior, characterized by no visible shear fractures. The deviatoric stress initially increases and then decreases with increasing mean stress, forming a cap surface in the meridian plane. A generalized failure criterion is subsequently developed, capable of accurately characterizing this strength response. Furthermore, the brittle-ductile transition behavior is found to be significantly dependent on the Lode angle. Finally, the brittle-ductile transition boundary is described, incorporating the dependence of Lode angle.

Understanding the anisotropic deformation behavior of geomaterials under true-triaxial stress conditions across the brittle-ductile domain is of great significance for underground engineering applications. In this study, true-triaxial tests were conducted on green sandstone with a porosity of 20% under constant Lode angle loading paths. Three levels of minimum principal stress were employed to represent the brittle domain (without a yield plateau), the brittle-ductile transition (with a distinct yield plateau), and the ductile domain (with plastic hardening), respectively. The results show that the sandstone exhibits plastic volumetric dilation in the brittle domain, whereas plastic volumetric contraction develops during the brittle-ductile transition and ductile domain, accompanied by a distinct yield plateau and continuous plastic hardening, respectively. The strain paths in the meridian and deviatoric planes reveal the evolution of the shear-volumetric strain relationship and the Lode-angle-dependent deformation behavior. Overall, increasing the intermediate principal stress coefficient and minimum principal stress promotes volumetric contraction while suppressing shear deformation. A Lode angle deviation, defined as the difference between the stress and strain Lode angles, is introduced to quantify the Lode angle dependence of deformation. The Lode angle deviation is positive in the brittle domain but becomes negative during the brittle-ductile transition and in the ductile domain. At the peak deviatoric stress, it first decreases and then increases with increasing minimum principal stress, while showing a negative correlation with the intermediate principal stress coefficient.

A parameterized three-dimensional strength criterion within brittle-ductile domain is proposed in this study. The criterion employs a power function to capture the positive correlation and an elliptic function to capture the negative correlation between deviatoric stress and hydrostatic stress. In addition, a unified deviatoric function framework is developed, enabling smooth transitions between any two deviatoric functions that satisfy the requirements of aspect ratio, curvature, and smoothness. The parameters within deviatoric function are expressed as functions of hydrostatic stress. To validate the proposed strength criterion, three sets of experimental data within the brittle-ductile domain are employed, namely Bentheim sandstone, green sandstone, and Sorcy limestone. The characteristics of three-dimensional failure envelope are analyzed in both meridian and deviatoric planes. At initial yield stage, the deviatoric stress exhibits a negative correlation with Lode angle, while the aspect ratio and curvature of deviatoric plane increase with hydrostatic stress. In contrast, the behavior during compactive yield cap stage follows an opposite trend. Furthermore, the three-dimensional failure envelope is examined in principal stress space to analyze the relationships among the principal stresses. A non-monotonic relationship is observed between the maximum and intermediate principal stresses, resulting from the combined influence of hydrostatic stress and Lode angle on deviatoric stress.

The influence of structural planes with different inclinations and locations on the instability mechanisms of tunnel surrounding rock is profound and complex, necessitating a comprehensive investigation into the underlying mechanisms. In this study, based on the mechanical properties of natural rocks and the principle of orthogonal tests, a similar material was meticulously designed to fabricate thin plate specimens that simulate underground tunnel environments. Subsequently, these thin plate specimens were subjected to biaxial loading using a specialized testing machine to replicate the in situ loading conditions of underground tunnels. The loading process was captured in real-time using a high-speed camera, enabling the application of digital image correlation (DIC) to analyze the mechanical properties of the specimens. Experimental results reveal that the presence of structural planes with different locations and inclinations around the tunnel exerts a significant impact on critical mechanical properties of the surrounding rock, including strength, energy, displacement, and strain. Specifically, when structural planes are situated at the spandrel and sidewall regions, the instability process of the surrounding rock is highly sensitive to the inclination of the structural planes. Four distinct damage modes of surrounding rock were identified and systematically analyzed, and three stages of instability of tunnel surrounding rock were delineated. These findings provide valuable theoretical insights and practical guidance for the prevention and mitigation of hazards associated with tunnel surrounding rock instability, contributing to enhanced safety and stability in tunnel engineering applications.

Earthquake source process governs seismic wave radiation and resultant ground shaking. The peak fault slip velocity and acceleration (Vpeak and apeak) are important source kinematic inputs for ground-motion prediction and seismic hazard assessment for future earthquakes. However, there are different ideas about the relationship between source kinematics and earthquake magnitude. Here, through integrated analysis of earthquakes ranging from Mw -4.6 to Mw 7.8, we demonstrate that both Vpeak and apeak do not increase with earthquake magnitude. Furthermore, by invoking dynamic shear rupture model coupled with cohesive force, we unify Vpeak and apeak of laboratory and natural earthquakes. Our results show that Vpeak increases with the product of breakdown stress drop and rupture speed, and apeak is related to breakdown stress drop, rupture speed, and critical slip distance. This study confirms fundamental self-similarity in earthquake source processes, with direct implications for physics-based seismic hazard assessment.

Machine learning models can predict laboratory earthquakes using Acoustic emission, the lab equivalent of microseismicity, and changes in fault zone elastic properties during the lab seismic cycle. Applying them to natural earthquakes requires testing their generalizability across lab settings and stress conditions. Here, we show a fine-tuned convolutional neural network (CNN) model effectively transfer across different conditions. Our model employs techniques from natural language processing, including decoder techniques, to capture the relationship between AE and fault stress. We fine-tune the regression head of a deep CNN while fixing the decoder's weights and successfully predict lab seismic events for a range of conditions. With fine-tuning, CNN models trained on one lab fault configuration predict time to failure and shear stress for another configuration at varying fault slip rates. These results demonstrate the potential of extending lab-based methods to different conditions that could eventually include tectonic earthquakes and seismic forecasting.

Due to the development of plastic strains, the strain path within the meridian plane deviates from the reference line corresponding to elastic state. Similarly, under true triaxial stress conditions, the strain path within the deviatoric plane deviates from the reference line corresponding to the constant Lode angle. This deviation is attributed to the plastic shear strain associated with the Lode angle. To account for these phenomena, a novel three-dimensional elastoplastic constitutive model incorporating Lode angle is proposed to characterize the deformation behavior of sandstone. The yield and potential functions within this model incorporate parameters that vary with the plastic internal variable, enabling the evolution of the yield and plastic potential surfaces in both the meridian and deviatoric planes. The comparison between experimental data and the analytic solution derived from the constitutive model validates its reliability and accuracy. To examine the differences between yield surface and plastic potential surface, a comparison between the associated and non-associated flow rules is conducted. The results indicate that the associated flow rule tends to overestimate the dilatancy of sandstone. Furthermore, the role of Lode angle dependence in the potential function is explored, highlighting its importance in accurately describing the rock's deformation.

Geological carbon storage, particularly within coal seams, is recognized as a viable strategy for achieving net-zero emissions. However, following CO₂ injection into the coal seam, limited studies have addressed the competitive sorption dynamics of CH₄ and CO₂ on coal, despite the natural presence of CH₄ in these formations. In this work, sorption experiments were conducted using two types of coal: sub-bituminous coal and anthracite. Initially, pure CH4 and CO2 gases were employed to conduct individual sorption tests. Subsequently, the competitive sorption of CH4 and CO2 was evaluated using pre-mixed binary gas mixtures with varying CH4/CO2 ratios. Further, the displacement effect of CO2 on CH4 was investigated by injecting CO2 into coal samples that had been pre-adsorbed with CH4. The data reveal that, for both coals, the ideal selectivity calculated from pure gas measurements underestimates the corresponding values from real multicomponent systems. Anthracite demonstrates a higher selectivity for CO₂ over CH₄ during both adsorption and desorption processes when compared to the ideal selectivity calculated from pure gas sorption data. Conversely, the sub-bituminous coal initially shows lower selectivity for CO₂ than for CH₄ during adsorption, but this trend reverses and intensifies during desorption. If competitive sorption effects are neglected, coal selectivity for CO₂ over CH₄ under the examined conditions would be underestimated by a factor of 1.5 to 2.5. Due to the competitive sorption effects between CH₄ and CO₂, the Langmuir equilibrium constants for gas mixtures are influenced by compositional changes, leading to dynamic deviations from those observed under pure gas conditions. This finding contrasts with the traditional extended Langmuir model, which presumes that the equilibrium constants remain unchanged regardless of gas composition. In addition, the displacement study underscores the efficacy of CO2 in displacing CH4, with higher CO2 ratios intensifying displacement effects. The study highlights the substantial impact of real multicomponent scenarios on coal selectivity for CO₂ and CH₄, offering deeper insights into predicting competitive sorption behavior between CO₂ and CH₄ during CO₂-enhanced coalbed methane recovery and carbon storage processes.

Dynamic shear fracture is a common failure mode in deep rock excavation. To enhance the efficiency of deep excavation and drilling operations, microwave-assisted mechanical rock crushing has emerged as an extremely promising technique. Therefore, understanding the influence of microwave irradiation (MI) on the dynamic mode II (shear) fracture responses of rocks is significant for improving excavation efficiency. However, the effects of different MI durations and water content on mode II fracture parameters have not yet been comprehensively investigated. In this study, dynamic fracture tests were conducted on red sandstone specimens with short-core compression (SCC) configuration subjected to various MI durations and saturation levels. Prior to the experiments, the temperature of the specimen surface after MI was recorded, and internal moisture changes were analyzed using nuclear magnetic resonance (NMR) techniques. The dynamic SCC tests were performed using a split Hopkinson pressure bar (SHPB) system, equipped with a momentum trap to ensure accurate measurement of the dynamic energy absorbed by the specimens. The results indicate that the dynamic mode II fracture toughness of specimens exhibits a significant loading-rate dependence. Microwave irradiation was found to significantly weaken the rock, with water content further exacerbating the damage under microwave exposure. The dynamic fracture energy of dry red sandstone remains largely unaffected by irradiation duration. However, under water-bearing conditions, it significantly decreases with prolonged microwave exposure. Moreover, at irradiation durations of 40 s and 60 s, higher saturation levels correspond to lower fracture energy.

During deep underground engineering construction, rocks transition into ductile domain under the influence of high three-dimensional geostress. Therefore, this study proposed a three-dimensional elastoplastic constitutive model incorporating Lode angle dependence within ductile domain. Besides, this study conducts a series of experiments on green sandstone within ductile domain, including hydrostatic compression test and true-triaxial test adopting constant Lode angle loading path. Based on the strength and plastic deformation characteristics of rock within ductile domain, both yield function and potential function are expressed as the product of elliptical equation and deviatoric plane. Both yield function and potential function incorporate parameters that evolve with the plastic internal variable. This enables the yield surface and plastic potential surface to evolve in the deviatoric and meridian planes, providing a more accurate depiction of the stress state and plastic flow direction during hardening. The comparison between proposed model and experimental data of green sandstone validates its applicability and accuracy. A comparison between the associated (yield surface) and non-associated (plastic potential surface) flow rules indicates that the plastic shear strain predicted by the associated flow rule is smaller than that predicted by the non-associated flow rule. To demonstrate the significance of Lode angle dependence in the potential function, a comparison is made between potential functions with and without Lode angle dependence. The comparison results indicate that the potential function without Lode angle dependence overestimates the intermediate principal strain under true-triaxial stress state. The parameter sensitivity analysis reveals that the intermediate principal strain is mainly controlled by deviatoric parameter within potential function. This study provides a theoretical foundation for upcoming numerical simulations of underground engineering within the ductile domain.