Effective Principal Stress Research Articles

In-plane biaxial fatigue life prediction model for high-cycle fatigue under synchronous sinusoidal loading

The unique advantage of in-plane biaxial loading is no rotation of the maximum principal stress during cyclic loading, which has a distinct effect on fatigue life. This study aims to propose a life prediction model for in-plane biaxial fatigue encompassing both in-phase and out-of-phase conditions. For synchronous sinusoidal loadings, an equivalent stress expression is derived using the integral method, accounting for the influences of shear and normal stresses across all planes. The equivalent stress is then compared to a reversed uniaxial constant amplitude loading to predict fatigue life. To validate and compare the model, in-phase and out-of-phase in-plane biaxial fatigue experiments were conducted using 24 nickel-based superalloy cruciform specimens at temperatures of 420℃, 550℃, and 600℃. The results show that the proposed model is superior to conventional stress-invariant based models, with most results staying within the ± 2 error bands. Using the proposed model, the effects of mean tensile stress, biaxiality and phase shift are discussed. Additionally, a novel non-proportional factor is introduced to enhance multiaxial fatigue life prediction by distinctly separating the effects of principal stress and its directional variations.

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.

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.

True triaxial unloading test on the mechanical behaviors of sandstone: Effects of the intermediate principal stress and structural plane

A series of true triaxial unloading tests are conducted on sandstone specimens with a single structural plane to investigate their mechanical behaviors and failure characteristics under different in situ stress states. The experimental results indicate that the dip angle of structural plane (θ) and the intermediate principal stress (σ2) have an important influence on the peak strength, cracking mode, and rockburst severity. The peak strength exhibits a first increase and then decrease as a function of σ2 for a constant θ. However, when σ2 is constant, the maximum peak strength is obtained at θ of 90°, and the minimum peak strength is obtained at θ of 30° or 45°. For the case of an inclined structural plane, the crack type at the tips of structural plane transforms from a mix of wing and anti-wing cracks to wing cracks with an increase in σ2, while the crack type around the tips of structural plane is always anti-wing cracks for the vertical structural plane, accompanied by a series of tensile cracks besides. The specimens with structural plane do not undergo slabbing failure regardless of θ, and always exhibit composite tensile-shear failure whatever the σ2 value is. With an increase in σ2 and θ, the intensity of the rockburst is consistent with the tendency of the peak strength. By analyzing the relationship between the cohesion (c), internal friction angle (φ), and θ in sandstone specimens, we incorporate θ into the true triaxial unloading strength criterion, and propose a modified linear Mogi-Coulomb criterion. Moreover, the crack propagation mechanism at the tips of structural plane, and closure degree of the structural plane under true triaxial unloading conditions are also discussed and summarized. This study provides theoretical guidance for stability assessment of surrounding rocks containing geological structures in deep complex stress environments.

Forming limit and failure characterization of as-received and bi-axial prestrained Cu-Cr-Zr-Ti alloy sheets in the context of multistage forming

The characterization of forming limit utilizing Cu-Cr-Zr-Ti alloy sheets during the multistage forming process is imperative for successfully fabricating thrust chamber liners used in satellite launch vehicles. Therefore, the present work studies the effect of prestrain on the forming limit and failure behaviour of this alloy under different deformation paths. A stretch-forming setup was utilized to induce 8% equi-biaxial prestrain (EBP), and subsequently, these prestrained specimens were deformed till the onset of failure using the same setup. After prestraining, a shift in the failure strains was observed in the principal strain space, whereas it was found to be absent in the polar effective plastic strain, principal stress and effective plastic strain-stress triaxiality loci. The cup height along different deformation paths was marginally decreased in the prestrained specimens because of the reduction in failure strains, and the fracture occurred in the prestrained region. Further, failure analysis revealed intergranular cracks in the specimens deformed along plane strain and uniaxial deformation paths due to the presence of micron-size Cr-rich precipitates along grain boundaries. However, intergranular and transgranular cracks were present in the prestrained specimens, which were deformed equi-biaxially similar to the as-received specimens. The presence of nano-size Cr-rich precipitates inside the grains acted as nucleation sites for voids during equi-biaxial stretching, leading to transgranular cracks. It was also observed that failure in the prestrained and as-received specimens occurred at a similar effective plastic strain value due to the combined effect of void density and void size. Based on experimental observations, it was inferred that the forming limit of Cu-Cr-Zr-Ti sheets under a two-stage forming process was primarily affected by the presence of micro and nano-size Cr-rich precipitates.

Strength characteristics and energy evolution of cement stone under true-triaxial loading conditions

Oil well cement is a commonly used cementing material in oil and gas extraction engineering. Due to the change of stress conditions, it is prone to microcracks or micro annular gaps, leading safety concerns. To study the strength characteristics and energy evolution of oil well cement stone under three-dimensional stress states, a series of true-triaxial compression tests were carried out. Based on the experimental results, the mechanical properties and energy evolution mechanisms during the true-triaxial compression process of cement stone under different stress states were explored. The research results indicate that there is an intermediate principal stress effect on the true-triaxial compressive strength of cement stone specimens. When the minimum principal stress remains constant, the strength of cement stone first increases and then decreases with the increase of intermediate principal stress. When the minimum principal stress is 5 MPa, the specimen exhibits brittle failure, with ε1 at peak stress approximately 0.015. As the minimum principal stress increases to 20 MPa, the specimen gradually transits to plastic failure, with ε1 at failure exceeding 0.04. The type of failure is determined by the energy ratio at the time of failure; when there is more elastic energy, it results in brittle failure, otherwise it leads to ductile failure. Due to the high porosity of cement stone samples, most of the input energy is converted into dissipated energy. At minimum principal stresses of 5 MPa, 10 MPa and 20 MPa, the ratio of peak dissipated energy to total input energy fluctuates around 0.55, 0.73 and 0.85, respectively. There is a strong linear relationship between the peak elastic energy, peak dissipated energy, and total input energy, with fitting correlation coefficients of 0.847 and 0.996, indicating that the cement stone sample also follows the linear energy storage law. A strength criterion for cement stone was established based on the modified Lade criterion, which is more suitable for describing the strength of high porosity materials compared to traditional strength criteria. Finally, based on the interaction mechanism between energy accumulation and energy dissipation, a nonlinear model of energy evolution in cement stone was established. The results of the theoretical model are in good agreement with the experimental data.

Performance of multi-ribbed composite wall incorporating reactive powder concrete-filled steel tubular columns under axial compression

In this study, axial compression researches of an innovative composite shear wall, named multi-ribbed composite wall incorporating reactive powder concrete-filled steel tubular columns, were conducted using theoretical analyses as well as experimental and numerical investigations. This precast hybrid system embedded reactive powder concrete-filled steel tubular columns as boundary elements of multi-ribbed composite wall. A novel approach, namely the unified strength theory, was adopted in consideration of various materials diversity. The outcomes of unified strength theory were utilized to predict the ultimate compressive capacity of multi-ribbed composite wall equipped with reactive powder concrete-filled steel tubular columns. The proposed theoretical formulas were validated against the experimental results reported in this paper. Despite the fact that the calculated formula with b= 0 was obtained to predict the closest capacity value of the test specimen infilling ordinary concrete. The calculations with different values of b clearly indicated that the intermediate principal stress effect should be rationally considered. Experimental observations suggested that the frame column significantly contributed to improving the compressive performance throughout the entire process and maintained the wall integrity. Meanwhile, finite element models were developed using the ABAQUS program, followed by the elaboration of its numerical implementation. The numerical predictions of ten shear wall specimens infilling reactive powder concrete were found to be reasonably in agreement with the calculated results corresponding to b= 0.5. These findings revealed the characteristics of multiple defensive lines of "block-ribbed column-frame column" under axial compression. Following the successful validation of finite element models, parametric studies were carried out to evaluate the influences of the filled block compressive strength, steel tube wall thickness and height-width ratio. The results show that the axial compression capacity of multi-ribbed composite wall incorporating reactive powder concrete-filled steel tubular columns can be noticeably improved with an increase in block strength and tube thickness. The proposed composite wall demonstrates favorable cooperative working performance and desirable displacement ductility. Subsequently, the primary objective of this pilot study aimed at developing a reliable theoretical calculation formula and conducting numerical simulation verification of the innovative composite wall. The aforementioned conclusions are intended to provide insights for the guidance and optimization of engineering structures.

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.

Intermediate Principal Stress Effects on the 3D Cracking Behavior of Flawed Rocks Under True Triaxial Compression

Crack initiation, growth, and coalescence in flawed rocks have been extensively studied for 2D (planar, penetrating) flaws under uniaxial/biaxial compression. However, little is known as to the mechanisms and processes of cracking from 3D flaws under true triaxial compression, where 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}) is distinguished from the major and minor principal stresses. In this work, we systematically investigate the effects 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} on the 3D cracking behavior of rock specimens with preexisting flaws, through the use of mechanistic simulations of mixed-mode fracture in rocks. We explore how two characteristics 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}, namely, (i) its orientation with respect to the flaw and (ii) its magnitude, affect two aspects of the cracking behavior, namely, (i) the cracking pattern and (ii) the peak stress. Results show that the orientation 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} exerts more control over the cracking pattern than the flaw inclination angle. The peak stress becomes highest when σ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma _1$$\\end{document} is parallel to the flaw, whereas it becomes lowest when 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} is parallel to the flaw. Also, the effects 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} magnitude are more significant when 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} becomes more oblique to the flaw plane. On the basis of our observations, we propose mechanisms underlying the cracking behavior of 3D flawed rocks under true triaxial compression.HighlightsThe effects 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}) on the 3D cracking behavior of flawed rocks under true triaxial compression are systematically investigated.The orientation 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} exerts more control over the cracking pattern than the flaw inclination angle.The effects 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} magnitude become more significant when 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} are more oblique to the flaw plane.Mechanisms underlying the cracking behavior of 3D flawed rocks under true triaxial compression are proposed based on the observations made.

Intermittent disturbance mechanical behavior and fractional deterioration mechanical model of rock under complex true triaxial stress paths

Mechanical excavation, blasting, adjacent rockburst and fracture slip that occur during mining excavation impose dynamic loads on the rock mass, leading to further fracture of damaged surrounding rock in three-dimensional high-stress and even causing disasters. Therefore, a novel complex true triaxial static-dynamic combined loading method reflecting underground excavation damage and then frequent intermittent disturbance failure is proposed. True triaxial static compression and intermittent disturbance tests are carried out on monzogabbro. The effects of intermediate principal stress and amplitude on the strength characteristics, deformation characteristics, failure characteristics, and precursors of monzogabbro are analyzed, intermediate principal stress and amplitude increase monzogabbro strength and tensile fracture mechanism. Rapid increases in microseismic parameters during rock loading can be precursors for intermittent rock disturbance. Based on the experimental result, the new damage fractional elements and method with considering crack initiation stress and crack unstable stress as initiation and acceleration condition of intermittent disturbance irreversible deformation are proposed. A novel three-dimensional disturbance fractional deterioration model considering the intermediate principal stress effect and intermittent disturbance damage effect is established, and the model predicted results align well with the experimental results. The sensitivity of stress states and model parameters is further explored, and the intermittent disturbance behaviors at different f are predicted. This study provides valuable theoretical bases for the stability analysis of deep mining engineering under dynamic loads.

A conceptual three-dimensional frictional model to predict the effect of the intermediate principal stress based on the Mohr-Coulomb and Hoek-Brown failure criteria

The intermediate effective principal stress has a considerable influence on rock strength as shown by true-triaxial data on intact rock. Therefore, a failure criterion that incorporates all three principal stresses may provide an improved prediction of failure than one that considers only the major and minor effective principal stresses. This study introduces a generalisation method, which converts two-dimensional failure criteria (i.e., Mohr-Coulomb and Hoek-Brown failure criteria) to three-dimensional failure criteria, for the prediction of this true-triaxial data. This method is based on a physical mechanism, and it assumes that the potential failure surface of a material has roughness, which provides additional frictional resistance when the intermediate effective principal stress is greater than the minor effective principal stress. This roughness is introduced theoretically by assuming a repeating diamond-shaped element over the failure surface. These roughness elements have the same dip as predicted by the Mohr-Coulomb and Hoek-Brown failure criteria but are offset by the dip direction angle which is assumed similar to that calculated traditionally from the major effective principal stress direction to the failure surface. With this roughness, it is shown that the major effective principal stress at failure can be predicted using an explicit solution that uses the intermediate and minor effective principal stresses and the Mohr-Coulomb or Hoek-Brown parameters as inputs. This generalisation predicts the same strength as the Mohr-Coulomb or Hoek-Brown failure criteria under conventional triaxial loading conditions, when the intermediate and minor effective principal stresses are equal. Under general stress conditions, this generalisation produces results that are aligned closely with the true-triaxial data as well as other previous theoretical predictions. This method importantly gives a possible physical mechanism for the strengthening influence of the intermediate effective principal stress. This may lead to a more accurate prediction of the rock strength under general stress conditions.

A novel meso-damage constitutive model of rock under true triaxial stress with three-dimensional cracking strength, threshold and closure effect

Deep underground engineering is in a true three-dimensional stress state, and the adjustment of the three-dimensional stress state caused by engineering excavation will induce the fracture or even instability of the surrounding rock. However, three-dimensional mechanical model research suitable for the stability analysis of deep surrounding rock is very scarce. Therefore, a series of tests under different true triaxial stresses on two rocks (rhyodacite and marble) were conducted, and the characteristic strength (crack stable propagation initiation stress, crack unstable propagation initiation stress and peak strength) and deformation characteristics were further analyzed. After that, using the Lemaitre strain equivalence hypothesis and rock statistical damage theory, a new statistical damage constitutive model at true triaxial stress states was proposed, which introduced the three-dimensional strength criterion Modified Wiebols Cook to characterize the three-dimensional strength of the rock microelement. Therefore, the intermediate principal stress can be reasonably considered. The damage threshold, initial compaction effect and residual strength of the rock microelement at different true triaxial stress conditions were also considered. Then the relationships between the proposed model parameters and σ2 and σ3 were analyzed. Furthermore, sensitivity analysis of the influence of parameters m and F0 in proposed model on the shape of rock stress–strain curve and peak strength was also investigated. The comparison between the results predicted by proposed model and the experimental data shows that the new model established in this study can well simulate the prepeak and postpeak deformation characteristics of rock and the intermediate principal stress effect under true triaxial stress conditions.

DEM Analysis of Fabric Evolution and Behavior of Granular Geomaterials in True Triaxial Test with Flexible Boundary

The fabric of granular geomaterials used in roads and embankments is related to properties such as shear strength, permeability and liquefaction resistance. To evaluate the fabric evolution and behavior of geomaterials, a series of the true triaxial test simulations with flexible boundaries under general stress states in this study were performed using the three-dimensional discrete element method (DEM). First, the stress states at the axial strain of 20%, which indicates that the stress under the critical state conforms to the Matsuoka-Nakai criterion, were examined. Then the uniqueness of the coordination number and non-uniqueness of void ratio and invariant of anisotropic fabric tensor under the critical state were investigated based on the contact density, which indicates the relative number of contacts in different orientations. In particular, the contact density at the critical state under different mean effective principal stresses with its effects on the invariant of anisotropic fabric tensor were discussed. It was found that the contribution of the contact density to the fabric anisotropy decreased as the mean effective principal stress increased due to geometric limitations.

Triaxial stress and failure modes in hydrothermal mineral systems

Rock mechanics experiments show that the magnitude of the intermediate principal stress significantly affects rock failure. Since triaxial stress states (no principal stress is zero) are ubiquitous in the crust, and polyaxial axial states (all three principal stresses are different) are general, the magnitude of the intermediate principal stress should have an important effect on hydrothermal mineralisation. For example, extensional veins or dykes in vein-hosted gold or porphyry deposits may have multiple orientations when the intermediate and least principal stresses have similar magnitudes, or single orientations when the intermediate and maximum principal stresses are similar. The Griffith-Murrell triaxial fracture criterion with a tensile cutoff can be used to illustrate the effects of the intermediate principal stress on failure. At the lowest values of mean stress, the criterion suggests that only extensional failure can occur. At low–intermediate values of mean stress, either extensional or shear failure may occur: extensional failure is favoured when the intermediate and maximum principal stresses have similar magnitudes. At higher mean stresses, shear failure will occur at lower values of pore fluid pressure and differential stress when the magnitudes of the intermediate and minimum principal stresses are similar.

A Simple Solution and Dilatancy Characterization for a Circular Tunnel Excavated in the Nonlinear Strain-Softening and Nonlinear Dilatancy Rock Mass

Tunnel opening or excavation in the strain-softening rock masses primarily leads to initial stress release and rock deformation. The increasing plastic shear strain induces a nonlinear variation of dilatancy in rock masses. Therefore, tunnel stability is strongly dependent on the stress state and the strain-softening and dilatancy behaviors of rock masses. In view of this, this paper proposes an elastoplastic finite difference solution based on the triple-shear element (TS) unified strength criterion (USC) and the nonassociative flow law; thereafter, some typical examples are referred to verify its validity. This solution allows for the description of the intermediate principal stress effect, the nonlinear strain-softening and dilatancy behaviors of the rock mass. Parametric analysis finally determines the influence characteristics of the intermediate principal stress effect (b), the critical softening parameter ( η ∗ ), softening factor (δ), and support force (pi) on the dilation angle (ψ) of the rock mass in the plastic zone. The findings confirm that the intermediate principal stress mainly affects peak values of ψ, which increases as b grows. The rate of change of ψ mainly depends on η ∗ , which tends to be slower with increasing η ∗ . The variation pattern of ψ in the residual zone under different δ is generally consistent, while the variability of ψ in the softening zone is significant. For a given intermediate principal stress state and strain-softening behavior of the rock mass, the values and the developed form of ψ in the plastic zone is independent of pi.

Influence of stress state on dynamic behaviors of concrete under true triaxial confinements

The strain rate sensitivity and stress state are two important factors affecting the mechanical behavior of concrete under dynamic load. However, few studies involved high strain rates and different stress states. Herein, the strain rate effect and intermediate principal stress effect of concrete are studied by using the true triaxial Hopkinson test system, and the influence of heterogeneity under dynamic load is also discussed. A series of experiments, including the static biaxial and static triaxial confinements, are conducted to investigate the intermediate principal stress effects by a dynamic testing system for cubic concrete specimens under static triaxial confinements. The results show that the triaxial experimental data show good consistency with the dynamic M-C strength, but the biaxial experimental data deviate greatly from the dynamic M-C strength, which indicates evident the intermediate principal stress effect. The Drucker-Prager (D-P) model is then modified to describe the intermediate principal stress dependence and loading rate effect by introducing the Lode angle (θ) and the strain rate, and better simulation results are obtained. The meso finite element model (meso‑FEM) is employed to further explain the effects of non-uniformity in concrete specimens, and a Weibull-type modification is introduced to describe the transversal inertia effect due to concrete heterogeneity. Although the relative dynamic stress (η=(σx−σe)/σx) in the x-axis is rate-dependent, hydrostatic pressure-dependent, and size-dependent, the relation of η to the stress triaxiality is insensitive to these effects. This work would provide experimental and theoretical guidance for the effect of stress state on the mechanical behavior of concrete under dynamic load.

Characteristics of solid–liquid phase transformation of saturated coral sand subjected to various patterns of cyclic loading

This study conducts a series of undrained cyclic shear tests on saturated coral sand with different initial relative densities ( Dr) subjected to a 90° jump rotation of the principal stress with various initial orientations. Considering liquefiable coral sand as a fluid, an important finding is that the variation in the apparent viscosity with the number of cycles is significantly affected by Dr, the effective mean principal stress ( [Formula: see text] ), cyclic stress ratio, cyclic loading path, and cyclic loading frequency. The average flow coefficient (κ) is introduced to describe the fluidity of saturated coral sand under cyclic loading by considering the variation in the shape of the stress–strain rate curve. A positive exponential correlation is observed between κ and the excess pore water pressure (EPWP) ratio ( ru) under the given cyclic loading modes. Another significant finding is that the solid–liquid phase change time is the turning point of the apparent viscosity and the average flow coefficient gradients with ru. The inversion point’s EPWP ratio is approximately 0.9, defined as the EPWP ratio during the solid–liquid phase transformation ( ruth). The ruth value is not affected by cyclic loading mode, [Formula: see text] , and Dr.

Influence of moisture content and intermediate principal stress on cracking behavior of sandstone subjected to true triaxial unloading conditions

The presence of water has been confirmed to play an important role in affecting the failure properties of underground surrounding rock when subjected to complicated stress circumstances. In the current study, the crack propagation behavior of red sandstone affected by various moisture contents and intermediate principal stresses were investigated by using the QKX-YB200 true triaxial test system with unloading minimum principal stress. The experimental results demonstrate that the failure mode can be divided into shear, tension with shear, and slabbing. A clear observation of the fractured specimen shows that the extensional cracks tend to increase while shear cracks gradually diminish with the increase of moisture content and intermediate principal stress. For natural sandstone (with lower moisture content), the transition of failure mode from shear to slabbing requires higher intermediate principal stress. However, with the increase of moisture content, the occurrence of slabbing failure requires lower intermediate principal stress. The analysis of failure process indicates that the presence of water may restrain the possibility of rockburst, and the intermediate principal stress exhibits distinct functions in the failure behavior of natural and saturated sandstone. The surface microscopic morphological characteristics and acoustic emission signal feature were also analyzed to validate the transformation of the cracking mode. The current research indicated that the water-induced micro defect and damage effect, “water wedge” effect, intermediate principal stress effect are the key factors that contribute to the final slabbing failure in immersed hard rocks under true‑triaxial unloading conditions.

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.