Intermediate Principal Stress Research Articles

Analytical study on plastic loosening zone of weak surrounding rocks tunnel in high cold areas

AbstractThe analysis of plastic loosening zone of tunnels with cold condition, high in situ stress, and weak stratum is the key scientific problem of tunnel construction in cold areas. Based on the unified strength theory and considering the coupling effect of high in situ stress and low‐temperature field, the analytical model of the plastic loosening zone of tunnel surrounding rocks was proposed, and the influence laws of physical parameters (initial ground stress, freezing temperature, excavation radius, and intermediate principal stress) was explored. The results illustrate that: (1) with the coupling of low temperature and stress fields, the in‐situ stress has a more significant effect on the radius of the plastic loosening zone of the tunnel surrounding rocks, showing a nonlinear correlation. When the in situ stress is small, the influence of the temperature field on the radius of the plastic loosening zone is more obvious. (2) The radius of the plastic loosening zone of the tunnel surrounding rocks increases synchronously with the excavation radius and gradually decreases with the increase of the intermediate principal stress. (3) The greater the temperature difference in the low‐temperature field, the tunnel stability has a more significant response to the intermediate principal stress and is positively correlated with the coefficient of the intermediate principal stress. (4) In the actual project, the proposed model can well describe the mechanical behavior and deformation characteristics of tunnel surrounding rocks in low‐temperature environment, showing applicability. The research results are expected to provide a theoretical basis for the study of tunnel stability in cold areas.

A comparative study of three types of strength criteria for rocks

Currently, the unified strength criterion (USC), the three-dimensional Hoek-Brown (3D H-B) criterion and the generalized unified strength theory (GUST) are the three types of typical rock strength criteria. In this paper, a comparative study of the three types of strength criteria is performed for rocks. Based on the nonlinear characteristics of rock strength on meridian and deviatoric planes, the USC can predict rock strength under triaxial stress state. The USC is composed of two failure functions on meridian and deviatoric planes. The failure surface of the USC in principal stress space satisfies smoothness and convexity. The predicted strength for five types of rock under the true triaxial tests were compared among the USC, the GUST, and the 3D H-B criterion. The results indicate that the USC can effectively reflect the influence of the intermediate principal stress on rock strength and accurately predict rock strength under both triaxial tension (σ1=σ2>σ3) and triaxial compression (σ1>σ2=σ3). Additionally, the conventional triaxial tests were conducted on other eight types of rock to measure the strength on meridian plane. The predicted strengths for the eight types of rock on meridian plane were compared between the USC and the original H-B, which suggests that the USC is suitable for various types of rock and provides higher accuracy.

A new three-dimensional strength criterion considering the transverse isotropy based on the α-Spatial Mobilized Plane

Natural geotechnical materials are affected by sedimentation and exhibit significant anisotropy. To study the transverse isotropy characteristics of soil, the influence of intermediate principal stress and loading direction must be considered. Currently, research on transverse isotropy primarily focuses on the modified stress space, which is cumbersome to apply in multi-yield surface constitutive models. To describe the three-dimensional mechanical properties of geomaterials in real stress space, the α-Spatial Mobilized Plane strength criterion is introduced. Then, combined with the structure tensor, the transverse isotropic three-dimensional strength criterion can account for the effect of the loading angle. Finally, the three-dimensional strengths of Fukakusa clay, unsaturated SP-SC soils, uncemented Monterey sands, Yamaguchi marble, San Francisco Bay mud, Toyoura sand, and Santa Monica Beach sand are predicted on the π-plane. The results show that the αmn-SMP criterion, in the context of transverse isotropy, can describe the three-dimensional mechanical properties reasonably, and it can provide an accurate strength criterion for geotechnical engineering practice.

Non-monotonic effect of differential stress and temperature on mechanical property and rockburst proneness of granite under high-temperature true triaxial compression

Complicated geological conditions such as high geo-stress and high ground temperatures often have a significant impact on the mechanical behavior and failure potential of hard rocks in deep engineering. This study aims to investigate the mechanical behaviors and rockburst proneness of granite under high differential stress and high temperatures in deep-buried high-temperature tunnels. A comprehensive experimental study was conducted on the coupling behavior of granite under high-temperature and high differential stress by true triaxial compression tests. The strength and failure mechanism of granite as well as their relationship with high temperature and high differential stress were revealed. The rockburst proneness of granite from a deep-buried high geothermal tunnel was assessed based on the rock's ultimate energy storage characteristics. It shows the strength, failure mechanisms, and rockburst proneness of granite, are significantly correlated to the granite high-temperature high-stress conditions. The temperatures extend the strengthening range of the intermediate principal stress on the true triaxial peak strength of the granite. With increasing temperature and differential stress, secondary vertical cracks develop quickly to induce the macroscopic failure of granite. Under high temperatures, the failure of granite changes from compressive-shear failure to tensile-shear failure at low differential stress. High temperature coupled with high differential stress strengthens approximately 1.14 times the rockburst proneness of granite. The results are important for the safe construction, and disaster assessment of deep-buried high geothermal tunnels.

Experimental and simulation investigation on failure characteristic of granite considering the effect of intermediate principal stresses

This study performs one-sided lateral unloading- and three-way five-sided force-vertical continuous loading experiments with different intermediate principal stresses (IPS) to investigate the effect of the IPS on deformation, strength, failure modes, and acoustic emission (AE) signals of rock and to elucidate the role of the IPS on the rock strength. The inoculation, occurrence, development, and failure of rockburst as well as the AE evolution characteristics are discussed. The findings indicate that the rockburst ejection includes localized ejections of particles, spalling of rocks into plates, shearing of rocks into fragments, and ejections of rock fragments. The formation mechanism of rockburst involves three progressive processes, i.e., tensile, shear, and tensile-shear composite destruction. The peak stress of sample increases with increasing IPS, while the deformation modulus decreases exponentially as the unloading progresses. The sample properties under true triaxial unloading condition with different IPS can be described by the Mogi-Coulomb criterion. The evolution curves of ringing impact ratio (CH) and cumulative absolute energy can be divided into four stages. The crystal-scale refined model that considers mineral components efficiently simulate the rockburst. The simulated stress-strain (SS) curves of samples are in good agreement with those obtained from laboratory tests.

Experimental study on the deformation and failure characteristics of cement stone under true triaxial stress state

The cement stone is widely utilized to fill the gap between the formation and the casing in wells. A clear understanding of the deformation and failure mode of cement stone under true triaxial stress state is necessary to ensure the safety and stability of wells. To analyze the three-dimensional deformation and failure characteristics of cement stone, the loading paths with constant Lode angle were employed in conducting true triaxial tests on cement stone. The stress-strain curve and failure mode of cement stone are first analyzed. Then, the three-dimensional deformation characteristics are discussed in meridian and deviatoric planes. Both peak strength and deformability increase with minimum principal stress, but show a tendency to increase and then decrease with intermediate principal stress coefficient. With the increase of minimum principal stress and intermediate principal stress coefficient, the failure mode of cement stone transforms from brittle to ductile behavior. Under elastic stage, the strain path in the meridian plane is a straight line, and is under constant Lode angle within the deviatoric plane. Due to the existence of plastic strain, the strain path in the meridian plane displays a downward convex shape. The strain path in the deviatoric plane deviates from the reference of constant Lode angle for stress, directly reflecting the Lode angle dependence of deformation. Both deformation and failure characteristics of cement stone are dominated by the three-dimensional stress state. Besides, deformation of cement stone is closely related to the failure characteristics of cement stone.

A new improved 3D Hoek-Brown criterion

The Hoek-Brown strength criterion has been widely used to estimate the strength of intact rocks and rock masses, and has been continuously developed. However, in the latest version of the standard, the criterion still ignores the influence of the intermediate principal stress. To study the different effects of intermediate principal stress on rock failure under triaxial or multiaxial stress states, this paper proposes a modified three-dimensional H-B strength criterion, which can be reduced to the H-B criterion, three-dimensional Priest criterion, Jiang’s (2012) criterion, and the Cai criterion. Multiaxial test data of six intact rocks were used for validation and applicability analysis. The results show that the proposed criterion can well describe the trend of multi-axial test data of six kinds of rocks, and the average mismatch of the six types of rocks is controlled within 10 MPa, which is much smaller than the fitting error of the original H-B criterion, Mogi criterion and Jiang’s criterion, indicating that the criterion has a good prediction effect on rock strength. The proposed criteria can provide a basic theory for the future construction of in-situ rock mass strength.

Elastoplastic analysis on deformation and failure characteristics of surrounding rock of soft-coal roadway based on true triaxial loading and unloading tests

Since accidents such as roof caving, rock fragmentation, and severe deformation are particularly likely to occur during roadway excavation in soft and thick coal seams, grasping the range and distribution of deformation and fracturing of surrounding rock is of crucial for evaluating roadway stability and optimizing support design in such coal seams. In this study, based on the stress paths encountered during roadway excavation, true triaxial loading and unloading tests were carried out on soft coal, and the deformation and strength evolutions of soft coal under different intermediate principal stress conditions were analyzed. The test results show that the stress–strain relationship in the pre-peak plasticity-strengthening and post-peak plasticity-weakening stages follows a quadratic function, and the strengeth evolution conforms to the Mogi–Coulomb criterion. Moreover, analytical solutions for the displacement of surrounding rock, the radius of the broken zone, and the radius of the plastic zone of soft-coal roadways under excavation stress paths were derived after taking the nonlinear hardening and softening characteristics of the strain of soft coal, the Mogi–Coulomb criterion, the intermediate principal stress, and the dilatancy characteristics of surrounding rock into comprehensive consideration. Finally, in accordance with a practical engineering case, the influences of the intermediate principal stress coefficient, the lateral pressure coefficient, and the support force on the deformation and failure characteristics of the soft-coal roadway were analyzed. The analysis reveals that an increase in intermediate principal stress aggravates the deformation of surrounding rock and enlarges the plastic and broken zones; variations in the lateral pressure coefficient alter the shape of the broken zone and the distribution of surface displacement; and an increase in the support force effectively reduces the plastic zone, broken zone, and surface displacement of the roadway. The research results can provide valuable theoretical basis for the stability evaluation and support design of soft-coal roadways.

An improved dual shear unified strength model (IDSUSM) considering strain softening effect

This study proposes an improved dual shear unified strength model by introducing the plastic internal variable which reflects the collective effects of strain softening, intermediate principal stress and unequal strength under tension and compression. The improved model is then simplified into simple forms for typical stress states, including uniaxial tension and compression, plane stress pure shear and tri-axial stress states. The smooth method and conjugate gradient method are utilized to facilitate its numerical implementation, avoiding numerical singularity and non-convergence in the solution process. The physical meanings of the parameters are further clarified and their values for self-compacting concrete are determined from the results of triaxial compression tests through a combination of direct determination, equation solution and back propagation (BP) neural network optimization. Validated against the test results, the improved model gives a more accurate prediction than the traditional dual shear unified strength model and Mohr-Coulomb model, in terms of both the overall trend and representative values. Validation results show that the improved model is applicable to materials for which the compressive strength is greater than the tensile strength and the tensile strength is greater than the shear strength.

A novel three dimensional failure criterion for quasi-brittle materials based on multi-scale damage approach

In this paper, we propose a novel three-dimensional micromechanics-based failure criterion to assess the load-bearing capacity of quasi-brittle materials under complex multiaxial stress conditions. This criterion not only inherits benefits of the multi-scale friction-damage coupling modeling approach but also accounts for the effect of the intermediate principal stress. Physically, the initiation and propagation of microcracks contribute to the damage, and the failure of the material ultimately occurs due to the unstable growth of microcracks. Simultaneously, plastic deformation, which results from frictional sliding along microcracks, is intimately coupled with the damage process. Employing friction-damage coupling up-scale analyses and introducing a novel parabolic local frictional law, we derive a new nonlinear compression meridian criterion within the upscaling framework. Moreover, by incorporating a Lode dependence function, this criterion effectively addresses variations in strength induced by the intermediate principal stress. To validate this criterion, we utilize data from triaxial compression, triaxial extension, and true triaxial tests conducted on various rock materials and concrete, all of which demonstrate excellent agreement.

Mechanical properties and constitutive model of artificial frozen sandy soils under true triaxial stress state conditions

Triaxial compression tests were conducted on frozen sandy soils under a constant minimum principal stress (σ3 = 1.6 MPa) and various intermediate principal stresses (σ2 = 1.6, 3.4, 5.2, 7.0, 8.8, 9.8 MPa). The purpose of the research was to investigate the influence of intermediate principal stress on mechanical properties of frozen soil, and to establish a constitutive model to predict the stress-strain relationship of frozen soil under complex stress state. The test results obtained demonstrated that the crack damage stress and failure stress initially increase and then decrease with an increase in the intermediate principal stress. However, the crack initiation stress exhibits an initial increase up to a specific value, after which it stabilizes. From the perspective of crack propagation, the influence mechanism of intermediate principal stress on the strength was discussed. With the variation of stress state, the difference between intermediate principal strain and minimum principal strain gradually increases. The deformation difference was revealed using the stress superposition principle and Poisson's effect. Finally, the constitutive model based on the Weibull distribution and Drucker-Prager strength criterion can accurately represent the stress-strain relationships of frozen sandy soils under true triaxial stress states.

Fatigue damage evolution behaviors and fractional fatigue mechanical model of monzogabbro under true triaxial disturbance test

The disturbance wave caused by excavation or blasting of underground surrounding rock causes fatigue degradation effect of rock and eventually leads to disasters. However, the fatigue damage characteristics and fatigue models of rock under true triaxial disturbance are scare. Therefore, a series of true triaxial disturbance tests were conducted to investigate the rock fatigue deformation, strength and damage behaviors under different conditions. The evolutions of static damage and fatigue damage are separated and investigated respectively. Fatigue deformation and damage of rock under true triaxial stress undergoes three stages: attenuation, constant velocity and acceleration stage. The crack initiation stress can be as the initial condition of the fatigue deformation; the fatigue critical stress σdc of rock entering the acceleration failure stage was proposed and explored, with increasing frequency, σdc increase slightly and with increasing σ2, σdc increase obviously. Then, a novel fractional fatigue mechanical model considering the fatigue damage and intermediate principal stress effects of rock under true triaxial disturbance was proposed. The theoretical results of the model agree well with the results of the tests. Finally, the sensitivity analysis of stresses and model parameters and the model predictions under other untesting conditions were carried out to improve the understanding and prediction level of fatigue failure in underground engineering.

Factor of safety analysis for mine pillar considering the influence of the intermediate principal stress component

Failure of mine pillars, especially in deep underground mines, significantly threatens the safety of miners and equipment. Previous studies on mine pillar stability design have used classical constitutive models that ignore the intermediate principal stress component when determining the factor of safety. In this study, we develop and implement a three-dimensional modified Hoek–Brown (HB) constitutive model that incorporates the intermediate principal stress component into the numerical simulation tool FLAC3D. Furthermore, we propose and apply a strength-reduction technique to determine a more accurate factor of safety for mine pillars. This novel approach provides a more comprehensive and realistic method for geomechanical analysis and pillar design, enhancing our understanding of pillar stability. Through numerical analysis, we illustrate the impact of the intermediate principal stress component on mine pillar plasticity. The factor of safety is calculated via the strength reduction method, revealing a substantial improvement from 1.7 with the classical HB model to 2.0 with the 3D HB model. Including the intermediate principal stress component reduces the evolution of plasticity in the mine pillar. For instance, the volume of plastic zones diminishes, and the factor of safety increases as the width-to-height ratio increases. Exemplary simulations show that ignoring the effect of the intermediate principal stress component, including underestimating safety levels, designing suboptimal pillar design, and misinterpreting in situ observations and measurements, can lead to severe consequences.

Strength and fabric anisotropy of granular materials under true triaxial configurations using DEM

This paper investigates the strength and fabric anisotropy of granular materials under true triaxial configurations using the discrete element method. A clump logic based on the multi-sphere approach was utilized to replicate realistic particle shapes. The evolutions of deviatoric stress and volumetric strains were found to be independent of mean effective stress, and intermediate principal stress parameter (b). From a macroscopic viewpoint, all samples exhibited initial strain hardening followed by softening behaviour, stabilizing at large deviatoric strains due to the dense state of assemblies. The peak state friction angles showed dependency on the b-value, aligning closely with experimental findings. Microscopic parameters such as mechanical coordination number and sliding fraction were found to be approximately independent of b-values. Additionally, non-coaxiality was observed for various b-values except for specific shear modes, aligning with previous findings using spherical particles. These results highlight the capability of clumped particles to simulate the true mechanical behaviour of granular materials and contribute to our understanding of their complex response under different loading conditions.

An improved three-dimensional extension of Hoek–Brown criterion for rocks

The Hoek–Brown (H–B) criterion has found widespread application in numerous rock engineering projects. However, its efficacy is compromised by an underestimation of rock strength due to its neglect of the influence of the intermediate principal stress (σ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma_{2}$$\\end{document}). Experimental evidence underscores the significant impact of σ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma_{2}$$\\end{document}. Consequently, there exists an imperative to formulate a three-dimensional (3D) criterion. In this study, a new deviatoric function with two additional parameters (k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k$$\\end{document} and A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A$$\\end{document}) is developed firstly, which ensure compliance with the prerequisites of smoothness and convexity. In addition, the parameters k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k$$\\end{document} and A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A$$\\end{document} are bonded with the weakening effect of the Lode angle (θσ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta_{\\sigma }$$\\end{document}) and the strengthening effect of the mean stress (σm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma_{m}$$\\end{document}), respectively. Then a new 3D strength criterion for rocks is proposed by combining this new deviatoric function and the triaxial compression meridian function of the original H–B criterion. Four distinct sets of test data encompassing various rock types are employed to validate the proposed criterion. The results demonstrate that the proposed criterion adeptly captures the strength characteristics of the four rock types, providing a good depiction of failure surfaces within the 3D principal stress space. Comparative analyses involve the utilization of several existing 3D H–B criteria for strength predictions. The proposed criterion exhibits superior fitting performance for all the selected rocks.

A unified nonlinear yield criterion for fracture and fault slip in quasi-brittle rocks

The micromechanical damage model has been developed to better describe multiscale fracture behavior in quasi-brittle rocks. While the deduced yield criterion under tension and compression has been carefully investigated in different stress spaces, it fails to capture a smooth transition from extension to shear fracture and overestimates the tensile strength of the rock. This paper introduces a unified nonlinear yield criterion based on a stress-dependent fracture energy parameter. In particular, the effect of the intermediate principal stress is encapsulated. The proposed yield criterion is then validated in the principal stress space based on a detailed parameter calibration procedure. Subsequently, four verification examples corresponding to Beishan granite, Lac du Bonnect granite, Carrara marble and Berea sandstone are carried out to show a desirable predictive capacity of the proposed yield criterion on progressive failure, extension to compressive-shear fracture transition and fault slip in quasi-brittle rocks.

The mechanism of low strain rate and moderate principal stress effects in triaxial prestressed marble under lateral impulsive load

Stress waves have a significant impact on the strength and deformation characteristics of deep rock masses at a certain distance from the epicenter or explosion source. In order to study the dynamic responses of deep-buried marble under the disturbance condition, the true triaxial compression test (TTC), the true triaxial disturbed compression test with lateral impulsive load (TTDC) and the synchronous acoustic emission monitoring on the marble were respectively conducted to simulate the influencing process of the seismic wave. The test results show that under lateral impulsive load, a ‘short-range plateau’ phenomenon occurs in the stress-strain curves of the marble samples. With the increase of intermediate principal stress σ2, the ‘short-range plateau’ of ε2 and ε3 curves gradually shrinks and even disappears, and the anisotropy of sample deformation becomes more obvious. The σ2 significantly affects the microfracture activity and the evolution of energy inside the samples under the impulsive load in the whole process of true triaxial disturbed compression. The lateral impulsive load weakens the marble strength which shows an obvious low strain-rate disturbance effect. The marble samples under the impulsive load have a high sensitivity to axial load and are liable to form local brittle fracture. This fracture induced by local stress concentration and strain energy release becomes more pronounced when the axial load is further applied to the samples. By contrast, the impulsive load in σ2 direction has a protective effect on limiting the lateral deformation of the samples and improving its axial deformation capacity, which greatly changes the action mechanism of σ2 on the sample lateral deformation. The variation of marble mechanical properties under lateral impulsive load is the result of the interaction between the low strain-rate disturbance and the intermediate principal stress effect.

The modified Mohr-Coulomb model considering softening effect and intermediate principal stress

Based on the Mohr-Coulomb model, a modified Mohr-Coulomb model that can consider both softening effect and intermediate principal stress effect was formed by introducing plastic internal variables that characterize material damage and a modified stress Rhodes angle. The modified model is numerically integrated by implicit algorithm, the physical meaning of the model parameters of the modified model is clarified, and the multivariate analysis and sensitivity analysis of the model parameters are carried out. Combined with the results of the gypsum triaxial test, the model parameters of the modified model were determined by the equation solving and RBF neural network optimization method, and the validity of the model was verified. The research results showed that: The modified model established can consider both the material softening effect and the influence of the intermediate principal stress. Compared with the traditional Mohr-Coulomb model, the calculation results of the modified model are more consistent with the test results qualitatively and quantitatively. The comprehensive use of direct determination, equation solving and RBF neural network optimization methods can effectively determine the model parameters of the modified model, and the determination process is relatively simple. It is suggested that this combined method can be used for subsequent determination of more complex constitutive model parameters. Compared with the traditional Mohr-Coulomb model, the modified model only needs one more true-triaxial test to determine the model parameters. In addition, the experience of using the parameters of the traditional Mohr-Coulomb model in a large number of engineering practices can be used to improve the model, so the modified model has great potential for engineering applications.

A true triaxial experiment investigation of the mechanical and deformation failure behaviors of flawed granite after exposure to high-temperature treatment

Understanding the mechanical and deformation failure behaviors of heated flawed granite under true triaxial stresses is of great significance for evaluating the wellbore stability in developing hot dry rock resources. In this study, true triaxial experiments are conducted on flawed granite specimens of a varying flaw overlapping length lfo with and without high temperature (HT) treatment, with the emphases on the crack development under true triaxial stresses and its correlations with the strength, plastic anisotropy and failure forecast. Investigations demonstrate that HT, σ2 (intermediate principal stress) and lfo, significantly affect the mechanical properties of flawed granite. Due to strong 3D confinement, the loop crack, the secondary transverse crack and the far-field crack are frequently induced in flawed granite, unlike experiments of uniaxial, biaxial and conventional triaxial compression. When increasing lfo, the rock bridge undergoes the stress amplification which tendentiously drives the internal crack coalescence and typically causes a reduced strength. The PSIRs (Plastic Strain Increments Ratios) analysis, first extended to the true-triaxial experimental data, reveals that the 3D stress-induced extensional/shear fracturing from the flaw tips in σ2-direction is the mechanism of the plastic anisotropy generated. Besides, an attempt to anticipate the failure time in the framework of the time‐reversed Omori’s law suggests a poor predictability by using the acoustic emission but a higher-quality forecast (being more precise and less biased) by using the dissipated energy.

An adaptive phase field approach to 3D internal crack growth in rocks

Modeling the 3D internal crack under compression entails complex fracture mechanics (mode I-II-III fracture), resulting in substantial computational costs and challenges in characterizing fracture morphology characterization for Phase Field Method (PFM) simulations. In the present paper, we developed a 3D adaptive crack propagation model integrated into the PFM to study the internal cracks in rocks. The proposed locally refined solution was implemented within the isogeometric analysis (IGA) framework, utilizing PHT-splines and employing the phase field variable as an error indicator to accurately capture non-smooth crack surfaces. The simulation begins with a coarser mesh, and the spatial discretization is adaptively refined in real-time as the computation progresses and the phase variable threshold is reached. By comparing our results with previous numerical tests, we validated the effectiveness of our model in simulating crack growth. Notably, a smaller length scale parameter led to a higher peak force in notched square plates under tension, and our adaptive PFM successfully reduced errors caused by length scale parameters. We further validated our model by comparing it with uniaxial compression experiments on 3D-printed resin samples with internal penny-shaped cracks, achieving good agreement with the experimental data. Using the 3D adaptive PFM, we predicted changes in crack morphology and fracturing mechanisms by varying the inclination angles of internal penny-shaped cracks. Our numerical results showed that as the inclination angle increased, wing cracks initiated closer to the end tips of the original crack, and their length decreased due to the wings wrapping effect. The corresponding fracturing mechanisms transition from mode I-III fracture (0°) to mode I-II-III fracture (30°, 45°, 60°), and eventually to mode I fracture (90°) as inclination angles increase. Comparing 2D and 3D models, we found that the growth of wing cracks in 3D was significantly influenced by the intermediate principal stress. We conclude that the extended 3D adaptive PFM-based model accurately predicted complex mechanical behavior and crack growth patterns in rock.