Intermediate Principal Stress Σ2 Research Articles

Failure characteristic of fissured rock specimens under true triaxial unloading conditions

The failure characteristics of fissured rock specimens under true triaxial unloading conditions were investigated. The effects of the intermediate principal stress (σ2) on the stress–strain responses, failure modes, specimen strength, and acoustic emission (AE) cues were evaluated. It is found that the deformation in the direction of the minimum prnicipal stress decreases as σ2 increases to a certain value and then increases during unloading. The proportion of tensile cracks decreases as σ2 increases at values lower than 30 MPa but increases at values exceeding 30 MPa. The AE cues were analyzed to identify the failure modes of the specimens under unloading conditions. The percentage of tensile cracks is much higher than that of shear cracks at a low σ2. An increase of σ2 results in a decrease in the percentage of tensile cracks. However, the percentage of tensile cracks increases as σ2 exceeds a certain value. A damage model is proposed for predict the peak stress of the fissured specimens under unloading conditions.

Progressive damage and fracture behavior of brittle rock under multi-axial prestress constraint and cyclic impact load coupling

To explore the progressive damage and fracture mechanics characteristics of brittle rock materials under combined dynamic-static loading. Taking account of the coupling effect of the constraint states of uniaxial stress (σ1 ≥ σ2 = σ3 = 0), biaxial stress (σ1 ≥ σ2 > σ3 = 0) and true triaxial stress (σ1 ≥ σ2 ≥ σ3 ≠ 0) and impact load, the strain rate effect and prestress constraint effect of dynamic mechanical characteristics of sandstone are studied. The progressive damage evolution law of sandstone under the coupling of true triaxial stress constraint and cyclic impact load is discussed. The results show that with the increase of axial stress σ1, the dynamic compressive strength and peak strain gradually decrease, and the strain rate gradually increases, resulting in crushing failure under high strain rate. When the axial stress is fixed, the lateral stress constraint reduces the damage degree of sandstone and improves the dynamic compressive strength. With the increase of strain rate, the sample changes from slight splitting failure to inclined shear failure mode. Under the true triaxial stress constraint, the intermediate principal stress σ2 obviously enhances the dynamic compressive strength of sandstone. Under the constraints of triaxial stress, biaxial stress and uniaxial stress, the enhancement effect of dynamic compressive strength and the deformation resistance of sandstone are weakened in turn. Under the coupling of true triaxial stress constraint and high strain rate, sandstone samples show obvious progressive damage evolution effect under repeated impacts, and eventually inclined shear failure occurs, resulting in complete loss of bearing capacity.

A new brittleness index based on crack characteristic stress and its engineering applications

The inherent brittleness of the surrounding rock poses a threat to the safety of underground engineering construction. One of the most straightforward methods for quantifying the degree of rock brittleness is using a rock brittleness index. Over the past several decades, a dozen rock brittleness indices based on uniaxial and conventional triaxial tests have been proposed and used. However, these indices do not reflect the impact of the intermediate principal stress σ2, which may render them unsuitable for rocks subjected to true triaxial stress. To address this limitation and explore the influence of σ2 on rock brittleness, a series of true triaxial tests were conducted. Crack characteristic stresses were introduced to investigate the different stages of deformation in the stress–strain curve of rocks. The relationship between σ2 and brittle–ductile damage of rocks was analyzed by introducing the dimensionless parameters Rci and Rcd. It is shown that a higher Rci indicates that the rock stores more elastic energy that is released in the post-peak stage, while a higher Rcd indicates that more microscopic tensile cracks are generated within the rock and indicates a shorter duration of interpenetration of tensile cracks. Thus, high values of Rci and Rcd indicate an increased likelihood of brittle failure occurring in the rock. Building upon these findings, a novel pre-peak brittleness index, Bi, is proposed. This index effectively quantifies the influence of σ2 on rock brittleness. Importantly, it overcomes the limitation posed by existing indices that are not suitable for rocks subjected to true triaxial stresses. An application of Bi to the Yebatan underground powerhouse was conducted using numerical simulation. The results indicate that the potential for brittle damage can be significantly reduced by increasing the angle between the axis of the underground cavern and the initial maximum principal stress in an appropriate manner.

Failure characterization of fully grouted rock bolts under triaxial testing

Confining stresses serve as a pivotal determinant in shaping the behavior of grouted rock bolts. Nonetheless, prior investigations have oversimplified the three-dimensional stress state, primarily assuming hydrostatic stress conditions. Under these conditions, it is assumed that the intermediate principal stress (σ2) equals the minimum principal stress (σ3). This assumption overlooks the potential variations in magnitudes of in situ stress conditions along all three directions near an underground opening where a rock bolt is installed. In this study, a series of push tests was meticulously conducted under triaxial conditions. These tests involved applying non-uniform confining stresses (σ2 ≠ σ3) to cubic specimens, aiming to unveil the previously overlooked influence of intermediate principal stresses on the strength properties of rock bolts. The results show that as the confining stresses increase from zero to higher levels, the pre-failure behavior changes from linear to nonlinear forms, resulting in an increase in initial stiffness from 2.08 kN/mm to 32.51 kN/mm. The load-displacement curves further illuminate distinct post-failure behavior at elevated levels of confining stresses, characterized by enhanced stiffness. Notably, the peak load capacity ranged from 27.9 kN to 46.5 kN as confining stresses advanced from σ2 = σ3 = 0 to σ2 = 20 MPa and σ3 = 10 MPa. Additionally, the outcomes highlight an influence of confining stress on the lateral deformation of samples. Lower levels of confinement prompt overall dilation in lateral deformation, while higher confinements maintain a state of shrinkage. Furthermore, diverse failure modes have been identified, intricately tied to the arrangement of confining stresses. Lower confinements tend to induce a splitting mode of failure, whereas higher loads bring about a shift towards a pure interfacial shear-off and shear-crushed failure mechanism.

True-triaxial simulation of sandstone with full range of σ2 based on the Rigid-Body-Spring method

Accurately simulating the mechanical properties of rock under true-triaxial stress conditions (σ1 > σ2 > σ3) is crucial for understanding deep underground rock masses. Although discrete approaches have advantages in describing rock fracturing behavior, further development seems necessary for quantitatively simulating the intermediate principal stress σ2 effects on rock strength. In this paper, we study the failure mechanisms of microscopic contacts of a discrete approach, the modified Rigid-Body-Spring Method (mRBSM), in response to σ2. Numerical results indicate that the model calibrated by axisymmetric-triaxial tests underestimates the strength of Yunnan sandstone when the stress state approaches σ2 = σ1. To address this discrepancy, we propose a new microscopic contact failure criterion that takes into consideration not only the normal and shear stresses on the contact but also the average Cauchy stress tensor around the contact. Results show that the mRBSM with the new failure criterion is able to reproduce the true-triaxial strength of Yunnan sandstone more accurately than the original model, especially under high σ2 and σ3 conditions. In addition, the corresponding stress–strain curves and failure modes under the influence of σ2 still show reasonable trends under true-triaxial stress conditions. This study indicates that when simulating rock failure under three-dimensional stress conditions using the rigid-body-spring system, the contact failure criterion might be influenced by the local stress state.

Mechanical parameter evolutions and deterioration constitutive model for ductile–brittle failure of surrounding rock in high-stress underground engineering

The deep surrounding rock is usually in the true triaxial stress state, and previous constitutive models based on the understanding of uniaxial and conventional triaxial test results have difficulty characterizing the degradation and fracture process of rock ductile–brittle failure under true triaxial stress state. Therefore, this study conducted a series of true triaxial tests to obtain the understanding of the ductile–brittle behaviour of rock, and then combined the test results and the Mogi–Coulomb strength criterion, and proposed calculation methods for the elastic modulus E, cohesion c and internal friction angle φ and the evolution functions of E, c and φ of rock under true triaxial stresses. With the decreasing of the minimum principal stress σ3 or increasing of the intermediate principal stress σ2, the marble post-peak stress drop rate gradually increases, the ductility gradually weakens, and the brittleness significantly strengthens. The calculation method and evolution function of rock E, c and φ under true triaxial stress were proposed. E decreased at first and then tended to remain stable with the increasing of equivalent plastic strain increment dεp. c and φ slowly increased at first and then rapidly decreased. With a method of parameter degradation rate to realize post-peak stress drop rate to reflect the ductile–brittle characteristics, a new three-dimensional ductile–brittle deterioration mechanical model (3DBDM) was established. The proposed model can accurately characterize the influence of σ2 and σ3 on mechanical parameters, the ductile–brittle behaviour of rock under true triaxial stresses, and the asymmetric failure characteristics of surrounding rock after excavation of deep underground engineering. The proposed model can be reduced to elastic–perfectly plastic, elastic–brittle, cohesion weakening friction strengthening (CWFS), Mohr–Coulomb, and Drucker–Prager models.

Failure behavior and strength deterioration model of high-performance concrete under coupled elevated temperature, biaxial constraint and impact loading

This research investigates the effects of high temperatures, multiaxial stress constraints, and impact loads on the failure behavior and strength deterioration of high-performance concrete. Combined with scanning electron microscopy and ultrasonic testing, the microscopic deterioration characteristics of thermally damaged concrete were revealed. Dynamic compression tests were performed on concrete, after exposed to elevated temperatures (from 25 °C to 750 °C), using a true triaxial Hopkinson pressure bar test system. The study analyzes the effect of temperature and strain rate on dynamic failure behaviors of concrete under biaxial stress constraints. The study establishes models for predicting the evolution of dynamic strength deterioration and strain rate. Additionally, the research discusses the mechanisms of influence of stress constraint states on the dynamic increasing factor (DIF) and failure modes of concrete. Results indicate that concrete's strength under biaxial stress constraints increases with the load rate and is hardened or weakened by changes in temperatures (taking 300 °C as a demarcation point). The strain rate effect of concrete becomes more significant with increasing temperature. Stress constraint states significantly affect the dynamic strength of concrete, and the DIF of concrete grows quasi-linearly with the logarithm of strain rate at different temperatures. Under biaxial stress constraints, concrete is dominated by oblique shear failure, and damage is less severe with the increase in the intermediate principal stress σ2. Macro-fractures are unlikely to appear under true triaxial stress constraints.

The Impact of Fractal Dimension, Stress Tensors, and Earthquake Probabilities on Seismotectonic Characterisation in the Red Sea

The frequency–magnitude statistics of 6527 earthquakes with 1.0 ≤ ml ≤ 5.7 and focal depths between 0 and 49 km in the Red Sea region between 1980 and 2021 show that the threshold magnitude, above which most of the Red Sea earthquakes are precisely located, is 1.5. The b-value, which identifies regional stress situations and associated energy release modalities, has a value of 0.75, less than in historical data, and averages between 0.4 and 0.85 as it varies over time, indicating modest stress accumulation. We utilised these instrumental data to examine dynamic stress patterns in the Red Sea region, shedding light on the region’s geodynamics and providing a foundation for estimating the region’s seismic hazard. The computed fractal dimension (Dc) has a relatively high value of 2.3, which is significant for the Red Sea’s geological complexity and structural diversity. This result indicates the regular distribution of Red Sea earthquakes, which occur in clusters or along fault lines. The low b-value and comparatively high Dc were most likely due to major earthquakes in the past and the greater stress they caused. The focal mechanisms of the big earthquakes, predominantly normal solutions, are consistent with the movement and extensional regime. The pressure and tension (P-T) axes show a compression axis trending NW-SE and a tension axis trending NE-SW. According to the stress inversion results, the maximum principal stress (σ1) is oriented vertically, the minimum stress axis (σ3) is subhorizontal and strikes in the NE-SW direction, and the intermediate principal stress (σ2) is trending in the NE-SW direction. The variance in the region that characterises the homogeneity of stress directions within the range is 0.19. The stress ratio (R), which identifies the faulting type, is 0.76, suggesting a normal faulting pattern for the region. The hazard parameters are expressed by the probability of exceedance for 1-, 10-, 50-, and 100-year return periods. The highest probability that an earthquake will occur within a 50-year period is thought to be around 6.0. The largest observed catalogue and instrumental magnitudes in the area, 5.7 and 6.7, respectively, show average recurrence intervals of 36 and 142 years.

Experimental study on failure evolution mechanism of clastic rock considering cementation and intermediate principal stress

The study of clastic rock failure evolution under true triaxial stress is an important research topic; however, it is rarely studied systematically due to the limitation of monitoring technology. In this study, true triaxial compression tests were conducted on clastic rock specimens to investigate the effect of cementation and intermediate principal stress (σ2) on the failure mechanism. The complete stress–strain curves were obtained, while the acoustic emission (AE) was monitored to indirectly evaluate the evolution of tensile and shear cracks, and crack evolution under true triaxial compression was imaged in real time by a high-speed camera. The results showed that the deformation and failure characteristics of clastic rock were closely related to the cementation type and intermediate principal stress. On the basis of the distribution characteristics of the ratio of rise time to amplitude (RA) and the average frequency (AF) of AE signals, tensile cracks of the contact cementation specimen propagated preferentially. Meanwhile, the enhancement of specimen cementation promoted the evolution of shear cracks, and the increase in σ2 promoted the evolution of tensile cracks. Moreover, the mesoscale cracking mechanism of clastic rock caused by cementation and σ2 under true triaxial compression was analyzed. The failure patterns of clastic rock under true triaxial compression were divided into three modes: structure-induced, structure-stress-induced and stress-induced failures. This study confirms the feasibility of high-speed camera technology in true triaxial testing, and has important implications for elucidating the disaster mechanism of deep tunnels in weak rocks.

A generalized nonlinear three-dimensional failure criterion based on fracture mechanics

Based on fracture mechanics theory and wing crack model, a three-dimensional strength criterion for hard rock was developed in detail in this paper. Although the basic expression is derived from initiation and propagation of a single crack, it can be extended to microcrack cluster so as to reflect the macroscopic failure characteristic. Besides, it can be derived as Hoek–Brown criterion when the intermediate principal stress σ2 is equal to the minimum principal stress σ3. In addition, the opening direction of the microcrack cluster decreases with the increase of the intermediate principal stress coefficient, which could be described by an empirical function and verified by 10 kinds of hard rocks. Rock strength is influenced by the coupled effect of stress level and the opening direction of the microcrack clusters related to the stress level. As the effects of these two factors on the strength are opposite, the intermediate principal stress effect is induced.

Analytical Model of Wellbore Stability Analysis of Inclined Well Based on the Advantageous Synergy among the Five Strength Criteria

The proper drilling mud density is vital for wellbore stability maintenance, which relies on both stress distribution on the borehole wall and rock strength. Thus, the selection of rock strength criterion is vital for wellbore stability analysis (WSA), and several strength criteria have been developed and employed to perform WSA, but the real collapse pressure does not comply with any single strength criterion. Therefore, to accurately predict the critical mud weight against wellbore collapse, an analytical model of WSA was proposed for inclined wells based on the advantageous synergy among the five types of strength criteria, such as the Mohr-Coulomb criterion, Mogi-Coulomb criterion, Drucker-Prager criterion, modified Wiebols-Cook criterion, and modified Lade criterion. The predicted results among these analytical and five conventional models were compared under three kinds of typical stress regimes, such as normal, strike-slip, and reverse faults. Finally, five kinds of typical oil and gas field data were collected to further verify the accuracy of this new model. The results indicated that different models predicted different collapse pressure, owing to the Mohr-Coulomb criterion ignoring the intermediate principal stress ( σ 2 ), so as to always give the greatest collapse pressure, followed by the present analytical model and the Mogi-Coulomb criterion, while the other three types of strength criteria gave different results, because of the difference in the influence of σ 2 , while the present analytical model integrated the advantageous synergy among the five strength criteria. The prediction error of the conventional analytical model ranges from 9.4 to 34.2% with an average of 22.1%, while the prediction error of this new model ranges from 2.3 to 12.5% with an average of 7.1%, so that the present analytical model has much higher accuracy than that of any single strength criterion. In addition, this new analytical model can be simplified as the conventional analytical model by adjusting the weight coefficients.

Experimental study on nonlinear seepage characteristics of coal under true triaxial stress loading

The characteristics of coalbed methane flow are jointly determined by the fracture and stress environment. Previous studies conducted under conventional triaxial or uniaxial stress conditions cannot truly reflect the in situ stress environment and neglected the influence of intermediate principal stress on the nonlinear seepage characteristics of coal. For this study, coal seepage tests were performed under true triaxial stress, allowing the nonlinear seepage characteristics of coal to be analyzed under different principal stresses. This enables the applicability of nonlinear seepage characterization equations and the change laws of relevant parameters to be clarified. The results show that the nonlinear seepage characteristics of coal under true triaxial stress increase as the principal stresses and gas pressure gradient increase. Using the Forchheimer equation for characterization, the nonlinear effect factor, viscous permeability, and inertial permeability gradually decrease with increasing principal stress in all directions. Intermediate principal stress σ2 has the most significant effect on nonlinear effects and inertial permeability, whereas principal stress σ3 has the greatest effect on viscous permeability. Using the Izbash equation for characterization, the fitting coefficients λ, m increase with increasing principal stress. Principal stress σ2 has the most significant effect on m, whereas principal stress σ3 has the greatest influence on λ. Comparative analysis shows that both Forchheimer and Izbash equations accurately characterize the nonlinear seepage characteristics of coal under real triaxial stress. The Forchheimer equation requires more theoretical parameters to measure the degree of nonlinear seepage and thus better describes the evolution process of linear to nonlinear seepage.

Constitutive relations, mechanical behaviour, and failure criterion of microcapsule-based self-healing concrete under uniaxial and triaxial compression

Microcapsule-based self-healing concrete can autonomously release repairing agents to bond and seal cracks when the material is damaged. It is seen as a promising construction material from the perspective of a sustainable society. In addition, it is important to study mechanical behaviour under complex stress conditions to promote the application of self-healing concrete in engineering structures. This study investigated the constitutive relations, strength and deformation, and failure criterion of the epoxy/urea-formaldehyde microcapsule-based self-healing concrete through uniaxial compression, conventional triaxial compression and true triaxial compression tests, in which the influence of different microcapsule dosages, confinement pressures and the principal stress ratios on the mechanical behaviours was considered. The results showed that the ultimate strength of the specimen decreased with increasing microcapsule dosage under all loading conditions. The lateral pressure significantly promoted the ultimate strength and deformation capacity of the specimen, and this increase mainly depended on the maximum principal stress (σ1) rather than the intermediate principal stress (σ2). By visual checking the failure modes of the specimens, more cracks were observed under the conventional triaxial compression than those under the true triaxial compression. This multi-crack failure mode allowed more core material of microcapsules to enter the microcracks, which accordingly increased the strength of the microcapsule-based concrete.

Three-Parameter Twin τ2 Strength Theory for Brittle Materials under Multiaxial Stress State and Its Application

The strength failure of brittle materials under complex stress is an important problem. Herein, we propose a novel three-parameter twin τ2 strength theory considering the influence of hydrostatic pressure and normal stress on the principal shear-stress surface, derive a mathematical expression for the strength theory, and compare the theoretical predictions under several stress states with existing experimental data. The results show that different ultimate stress ratios, α and β, correspond to different strength theories for brittle materials. The principal stress σ1 increases gradually with an increase in σ2 (=σ3) under the stress state σ1 > σ2 = σ3; σ1 (=σ2) increases gradually with an increase in σ3 under the stress state σ1 = σ2 > σ3. Furthermore, the biaxial compressive strength is considerably higher than the uniaxial compressive strength under the biaxial compressive stress for σ1 > 0, σ2 > 0, and σ3 = 0. When σ3 is fixed and σ2 is relatively small under the stress states of σ1 > 0, σ2 > 0, and σ3 > 0, the maximum principal stress σ1 increases with the increasing σ2. When σ2 is relatively large and as σ1 gradually decreases with the increasing σ2, the effect law of intermediate principal stress σ2 is obtained.

A Three-Dimensional Constitutive Model for Rocks Based on a Strain-Dependent Elastic Modulus

Abstract Our research group previously proposed a simple two-dimensional (2D) constitutive model for rocks to simulate not only the axial stress–axial strain relationship, but also the axial stress–lateral strain relationship, with few complicated equations. However, the failure envelope that it predicted was linear, and it did not consider the effect of the intermediate principal stress (σ2). In the present study, the authors modify this simple 2D model to have a convex failure criterion. Then, the model is extended to a simple three-dimensional (3D) model that well approximates true triaxial stress–strain curves for real rocks under specific values of σ2 and σ3 and uses only four parameters. However, the predicted peak stress–σ2 relationship is linear. Finally, a modified 3D model was developed, which exhibited the true triaxial convex failure criterion. The equations in this model are simpler than the conventional true triaxial failure criteria. The proposed models can be implemented with a finite element method to improve the design of rock structures.

Strength characteristics of brittle rocks under multi-stage intermediate principal stress loading

A series of single-stage true triaxial compression (TTC) tests and multi-stage intermediate principal stress (σ2) loading tests were carried out on three types of brittle rock specimens in this study to examine the influence of σ2 and the validity of obtaining strength parameters by using a multi-stage loading method under true triaxial stress conditions. There exists a critical σ2 at which the rock strength attains its ultimate value under both loading conditions. Under the same σ3 level, the critical σ2 is much smaller under the multi-stage σ2 loading compared with under the single-stage TTC. The test results also show that the strength determined under multi-stage σ2 loading is much smaller than that determined under single-stage TTC. The strength difference can attain 39%. A lower strength is measured under multi-stage σ2 loading because the local fracture plane induced by true triaxial stress is parallel to the σ2, and the strength of the rock specimen containing this type of local fracture plane is not affected by σ2. On the contrary, rock damage accumulates during multiple loading and unloading cycles. The current testing results indicate that the multi-stage loading test is not suitable for determining the strength envelope of brittle rock under true triaxial conditions.

Numerical Simulation of Pressure Distribution Changes in Porous Rocks under the Action of High Pressure Fluid

The rock medium has the characteristics of heterogeneity, opacity and porosity. Fluid injection (such as hydraulic fracturing technology) can make the tight rock mass form a gas permeability enhancement zone near the borehole, which is conducive to unconventional natural gas extraction. Compared with experimental research, numerical simulation research has lower cost and higher efficiency, and has become an important method for research on fluid interaction and pore pressure. Based on this, this article mainly conducts a numerical simulation study on the changes in the internal pressure distribution of porous rocks under the action of high-pressure fluid (HPF). This article outlines the mechanism of quasi-static HPF fracturing coal and rock, and describes the classification of tensile fracture and shear fracture. In this paper, a numerical simulation method is used to construct a model of a porous rock sample, and a test of the internal pressure of the porous rock under the action of a HPF is carried out on the rock simulation sample. The experiment shows that when the intermediate principal stress σ2 is 20 MPa, the peak pressure of fluid is 12.5 MPa; when σ 2 is 25 MPa, the peak fluid pressure of is 12 MPa. This shows that the intermediate principal stress (or horizontal stress difference) is negatively related to the maximum fluid pressure.

Strength and deformation properties of frozen sand under a true triaxial stress condition

In recent years, the mechanical properties of frozen soils under complex stress states have attracted significant attention; however, limited by the test apparatus, true triaxial tests on frozen soils have rarely been conducted. To study the strength and deformation properties of frozen sand under a true triaxial stress state, a novel frozen soil testing system, i.e., a true triaxial apparatus, was developed. The apparatus is mainly composed of a temperature control system, a servo host system, a hydraulic servo loading system, and a digital control system. Several true triaxial tests were conducted at a constant minor principal stress (σ3) and constant intermediate principal stress ratio (b) to study the effect of intermediate principal stress (σ2) on the mechanical properties of frozen sand. The test results showed that the stress–strain curve can be mainly divided into three stages, with evidence of strain hardening characteristics. The strength, elastic modulus, and friction angle increased with the increase in b from 0 to 0.6, but decreased when increasing b from 0.6 to 1, whereas the cohesion varied little with the variation in b. The deformation in the direction of σ2 changed from dilative to compressive and that in the direction of σ3 remained dilative throughout.

Experimental study on mechanical and damage characteristics of coal under true triaxial cyclic disturbance

A true triaxial test system was used to study the mechanical and damage characteristics of coal under tiered cyclic maximum principal stress (σ1), intermediate principal stress (σ2), and minimum principal stress (σ3). The influence of each principal stress cycle on the deformation, energy evolution and damage characteristics of coal was systematically analyzed, and the mechanical evolution characteristics of coal under the disturbance action of true triaxial excavation were revealed. In the process of true triaxial tiered cyclic loading, it is found that: the stress-strain curves of different principal stress cycles processes are quite different, but they will all cause plastic deformation or plastic hysteresis loop of principal strain in each principal stress directions of coal, and the principal strain corresponding to the cyclic principal stress direction is the dominant deformation, and its deformation is relatively large. According to the true triaxial strength curve and experimental results, the cyclic loading or unloading σ2 may cause instability and failure of coal. The variation trend of residual deformation caused by σ1 and σ3 is the same: with the increase of the amplitude of true triaxial tiered cyclic loading, the absolute value of the cumulative maximum and minimum residual principal strains increase exponentially and quadratic parabola, respectively. And the absolute value of the relative maximum and minimum residual principal strains decrease first and then increase, showing U-shaped and arch shaped changes, respectively. While the changes of cumulative and relative residual intermediate principal strains are all small. The total dissipation energy of coal increases exponentially with the increase of the amplitude of true triaxial tiered cyclic loading. Combined with the energy dissipation characteristics, the damage variable equation of coal under true triaxial cyclic loading is newly defined. With the increase of the amplitude of true triaxial tiered cyclic loading, the damage variable presents an obvious S-type change trend of deceleration acceleration re-deceleration.

Effect of Unloading Direction on Rock Failure under True Triaxial Stress Conditions

Abstract Rock failures induced by excavation unloading in the direction of the minimum principal stress σ3 and the intermediate principal stress σ2 were numerically simulated and analyzed. The damage evolution process, the failure mode, and the failure mechanism under the two conditions were investigated. The results show that the unloading direction of the principal stress has a significant impact on the rock failure mode. Under the same triaxial stress level, when unloading σ3, rock failure is mainly caused by localized tensile-shear composite failure; when unloading σ2, rock failure is mainly caused by tensile buckling failure. When unloading σ3, the damage is mainly concentrated near the free surface; however, when unloading σ2, the damage range increases and develops toward the interior of the rock. The greater the intermediate principal stress is before unloading, the larger the number of splitting fractures and the magnitude of dilatation are, and the smaller the thickness of the splitting plate is. These results may shed light on the prevention and mitigation of rock instability in deep underground engineering.