Series Of True Triaxial Tests Research Articles

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.

Mechanical behavior and fracture characteristics of high-temperature sandstone under true triaxial loading conditions

To understand the stability of deep underground engineering rocks at high temperatures, it is necessary to conduct comprehensive tests on the mechanical properties of the rocks. In this study, a series of true triaxial tests (TTT) were conducted on sandstones using a self-developed true triaxial testing system combined with XRD, FE-SEM and CT analyses. The study systematically examined the microstructural characteristics, physical phase behavior, deformation properties and internal fracture development of the sandstone before and after heat treatment. The results of the study show that sandstone has temperature-dependent true triaxial mechanical properties after heat treatment. The stress-strain characteristic curves of the post-peak portion from room temperature to 800 °C show a significant stress-reducing segment, indicating the presence of brittle behavior. In addition, when the temperature exceeds 1000 °C, the sandstone exhibits brittle-ductile change characteristics, with a peak stress threshold observed at 800 °C. Notably, the peak stresses corresponding to 800 °C increased by 84.5 %, 63.1 %, 47 %, and 49.8 %, respectively, compared with the room temperature intermediate principal stresses(IPS) ranging from 20 MPa to 50 MPa, respectively. Importantly, the linear Mogi-Coulomb criterion remains valid for the peak strength of the high-temperature sandstone, while the internal fracture damage pattern of the specimens shows an inverted "Y" or inverted "W" configuration. The internal fracture network of rock samples above 800 °C is more complex. Intergranular crack is the main form of microcracking. This experimental study on the mechanical properties of sandstone under the combined effects of high temperature and real triaxial stresses provides valuable guidance for the selection of sandstone parameters and safety evaluation of coal underground gasification roofs.

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.

Experimental and theoretical analysis of spalling in deep hard rock tunnels with different arch structures

To study the spalling characteristics and induction mechanism of tunnels with different arch structures, a series of true triaxial tests were conducted on three types of cubic granite specimens with a through-going arched hole and analytical stress solutions were obtained for the three arch sections. Test results show that the failure process of specimens includes four periods: a quiet period, a buckling deformation period, a gradual buckling and exfoliation period of rock fragments, and a V-shaped notch formation period. Specimens with different arch structures are listed in a descending order as the three-centered arch with f/B (arch rise/tunnel width) = 1/3 (II), three-centered arch with f/B = 1/4 (III), and semi-circular arch with f/B = 1/2 (I) according to the size of rock fragments. Before the spalling of arched tunnels, the acoustic emission (AE) count is low, while the AE count and cumulative AE count are high during spalling. Stress analysis results show that as the vertical stress rises, the arch feet of tunnels undergo initial failure at first due to stress concentration; spalling cracks appear in both side walls due to compressive stress concentration, which induce spalling failure; tensile cracks initiate in the vault and invert due to tensile stress concentration. The sequence of spalling in tunnels with different arch structures varies due to a change in section structure. Therefore, the quality of section structures can be evaluated according to the stress distribution characteristics of surrounding rocks. The vertical stress inducing the initial failure of tunnel surrounding rocks increases gradually in different arch structures from the II, III, to I; the failure severity of surrounding rocks decreases in different arch structures, also from II, III, to I. In addition, the stress analysis results also verify the accuracy of the test results. In practical engineering applications, to consider both the section utilization ratio and control of spalling in surrounding rocks, selecting tunnel sections with f/B = 1/4 is more economic and safer than that of f/B = 1/3.

Experimental study on rockburst of surrounding rock in an elliptical tunnel with different axial ratios

The effect of the ratio of long and short axes on the rockburst of an elliptical tunnel structure was investigated by performing a series of true triaxial tests on red sandstone cuboid samples containing seven types of elliptical holes with different long–short axial ratios, using a self-developed true triaxial compression equipment. The entire process of spalling was monitored and recorded using a miniature camera. According to the test results, the failure stress, geometrical characteristics of the rockburst notch, fragment characteristics, and ejection velocity field characteristics of the rockburst are summarized, and the influence of the axial ratios of the elliptical hole on the failure characteristics and stability of the tunnel are discussed in detail. The test results indicate that the rockburst process of elliptical tunnels with different axial ratios experiences four processes; namely, calm period, small particle ejection period, spalling, and rockburst. The shape of the failure area after the rockburst presents a V-shaped feature. With the increase of the axial ratio, the area, angle, and depth of the V-shaped area show a decreasing trend, the initial failure stress of the cavern and the rockburst failure stress exhibit a trend of first decreasing and then increasing; the tangential stress of the cavern shows an increasing trend; the mass percentage of micro, fine, and medium grain decreases; and the average ejection velocity first decreases and then increases. Under the same stress conditions, rockbursts are more likely to occur on circular sections. Under the same section conditions, the ellipse can bear larger loads when its major axis is parallel to the disturbance stress. These conclusions are helpful to the efficient design of the tunnel structure.

Influence of intermediate principal stress ratio on strength and deformation characteristics of frozen sandy soil under different moisture contents

Frozen ground often contain non-uniform moisture content and are under true triaxial stress state and their practical engineering application are still explored. The present study is carried out to investigate the strength and deformation characteristics of frozen soil with different moisture contents under true triaxial stress state conditions. A series of true triaxial tests were conducted on frozen soil specimens having constant minimum principal stress (σ3) and constant intermediate principal stress ratio (b). The experimental test results showed that the stress-strain curves represent the strain-hardening characteristics under different test conditions. The intermediate principal stress ratio (b) has a dual effect of strengthening and weakening on the strength of frozen sandy soil. When 0 ≤ b ≤ 0.5, the strengthening effect was found to be dominant in the frozen soil specimen, whereas, the weakening effect was dominant when 0.5 < b ≤ 1. The strength of frozen soil under true triaxial stress state is 8.18% to 36.33% higher under different water content conditions, compared to that under generalized triaxial compression stress state. The difference in deformation between the direction of the intermediate principal stress and the minimum principal stress increased with the increment in the intermediate principal stress ratio value. The volumetric strain in the soil sample initially exhibited the characteristics of shear contraction and then shear dilation.

A thermodynamic damage model for 3D stress-induced mechanical characteristics and brittle–ductile transition of rock

Due to the existence of true three-dimensional high-geostress in deep underground engineering, rock shows different mechanical properties and brittle–ductile behaviours from conventional triaxial stress states, however, the different characteristics of rock are not clear. Therefore, a series of true triaxial tests were performed on deeply buried marble to investigate the effects of σ2 and σ3 on the characteristic strength (peak strength, yield strength and residual strength), post-peak deformation and brittle–ductile behaviour. Based on test results, a three-dimensional elastoplastic damage constitutive model that describes plastic hardening and damage softening of rock was established within the framework of irreversible thermodynamics, and a sensitivity analysis of key parameters ( η and ζ) was performed. A method that controls the brittle–ductile behaviour of rock through key parameters η and ζ was studied, and functions of these two parameters with σ2 and σ3 were proposed. The proposed model was implemented numerically with the cutting-plane algorithm in a finite element program. A series of numerical simulation experiments were performed, and numerical simulation and experimental results are consistent. In addition, brittle–ductile transition of marble under untested true triaxial stress levels were reasonably predicted.

An image segmentation-based method for quantifying the rock failure mechanism under true triaxial compression

Understanding the impact of geostress on the failure mechanism of surrounding rock is necessary for underground engineering applications. Here, we carried out a series of true triaxial tests on sandstone . The mechanisms of the areas on the fracture surface that appear in different colors are identified. Using the image segmentation method , the white shear fracture area on the grayscale image of the fracture surface is extracted. By calculating the proportion of the white shear fracture area, the rock failure parameter S for quantifying the rock failure mechanism is defined, and the influences of σ 2 and σ 3 on rock failure mechanisms are analyzed. Moreover, a criterion for determining the failure mechanism of rock under true triaxial stress is established. The results indicate that the shear fracture area that appears white is due to the adherence of rock debris and powder on the surface. The rock failure parameter S is appropriate to quantify the rock failure mechanism. A larger S indicates a higher proportion of shear fracture in the rock failure. The increase in σ 3 has a significant promotional effect on the development of the shear fracture. An increase in σ 2 can effectively promote the initiation of tensile fracture. According to the criterion, once the σ 2 and σ 3 of the stress environment are clarified, the failure mechanism of the rock can be predicted. This work contributes to an improved understanding of the failure mechanism of rock and a potential means by which to guide the design and construction of underground engineering.

Influence of true triaxial stress paths on mechanical properties of marble

The rock brittle failure is the result of the stress redistribution caused by the excavation of engineering construction, which can be estimated by cyclic loading tests. In this study, a series of true triaxial tests with different stress paths were conducted to investigate the influence of the variations in three principal stresses on the mechanical properties of Jinping marble. The complete stress-strain curves can be obtained due to the strain control in the monotonous loading tests whereas only the pre-peak stress-strain curves can be acquired using the stress control under cyclic loading conditions. In the true triaxial cyclic loading tests, the irreversible strains can be obtained accurately and used to describe the damage evolution of Jinping marble. Test results show that the irreversible strains have the different change trends in the different stress paths and their changes are most sensitive to the minimum principal stress. In addition, Jinping marble in the cyclic loading of the minimum principal stress test has the largest failure angle and numbers of the failure fragments, which indicates that Jinping marble becomes more brittle under this condition.

The Behavior of a Coarse Granular Material under Complex Stress Conditions

In recent years, dozens of high rockfill dams are under construction or planning for hydropower exploration in western China. In dam construction, the mechanical behavior of coarse granular material greatly affects the compatible deformation of dam body. In this article, an indirect in situ density prediction approach for coarse granular material is firstly proposed to solve the technical obstacle on prediction of the material density in thick overburden layer of a dam site in southwest China. Adopting a self-developed large-scale true triaxial apparatus with a special friction-reduction technique, four series of true triaxial tests were then performed to investigate the behavior of a coarse granular material with a maximum particle diameter of 60 mm. Test results show that the peak strength of the material increases together with the increasing confining stress and the increasing intermediate principal stress ratio. The material dilatancy is restricted by both the confining stress and the intermediate principal stress ratio. With the increase in intermediate principal stress ratio, the internal friction angle increases firstly and then decreases slightly, but the slope of stress path reduces gradually. The tested peak states were compared with several well-known strength criteria under the framework of generalized stress, showing a good fitness with the Lade–Duncan criterion and underestimation by the Mohr–Coulomb criterion and the Matsuoka–Nakai criterion. The strength envelope in the π plane shrinks with the increasing confining stress.

Mechanical Properties of Layered Composite Coal–Rock Subjected to True Triaxial Stress

Underground mining or tunnelling activity is always associated with the composite geological formations. The mechanical properties of layered composite coal–rock subjected to true triaxial stress conditions are significantly different from those under conventional triaxial or uniaxial stress conditions. In this work, we conducted a series of true triaxial tests using the self-developed true-triaxial apparatus to investigate the mechanical response (e.g., deformation, strength, and failure characteristics) of the layered composite coal–rock (CCR). The results show that the uniaxial strength of CCR lies between the strength of pure sandstone and coal, and the direction of the bedding affects the overall strength of the samples. The true triaxial strength of both the pure rock and CCR increases first and then decreases with the increase of the intermediate principal stress. Moreover, for a given loading direction, as the thickness of the sandstone layer increased, the strength of the CCR increases. The deformation of the CCR shows more obvious plasticity than that of the pure sandstone due to the coordinated deformation of the coal and sandstone layers. In addition, a new true triaxial strength criterion expressed by the first and third equivalent principal stress invariants was proposed, which can well describe the strength characteristics of different coal rocks. The stress states, weak structural planes, and localized stress have a great influence on the failure modes of CCR. The local stress concentration near the contact surface promotes the development of the secondary failure fractures. These findings are of great significance in stability designing in deep underground engineering.

Strength and deformability of a low-porosity sandstone under true triaxial compression conditions

True triaxial tests (σ1≥σ2≥σ3) were conducted on the Zigong sandstone with a relatively low porosity of ~6.5% with the aim of assessing the influence of the intermediate principal stress σ2 on the strength and deformability. Three series of true triaxial tests (σ3= 0, 20, and 60 MPa) were performed. Within each series, σ2 was planned to be varied from σ2=σ3 to σ2=σ1 from test to test. For each test, σ1 was raised monotonically to failure while σ2 and σ3 were kept constant and the post-peak behavior has been captured. It is found that the strain in the σ3-direction is always larger than that in the σ2-direction under true triaxial stresses, which is attributed to the stress-induced tensile microcracks aligned with the σ1-σ2 plane. For a constant σ3, the dilatant strain in the σ2-direction is progressively suppressed and becomes compactive as σ2 raises from σ2=σ3 to σ2=σ1. Based on a modified Hoek-Brown failure criterion with three stress invariants, the best fitting brittle strength envelope reveals that the σ2 effect on rock strength, i.e., the ascending-then-descending trend for a given σ3, is the combined effect of mean stress and Lode angle. In the series of biaxial tests (σ3=0 MPa), macro tensile fractures that parallel to the σ1-σ2 plane develop regardless of σ2, displaying a failure plane angle of approximately 90°. As σ3 is raised, failure plane angle generally decreases. When σ3 = 20 MPa, the failure plane angle generally increases with σ2 while there is no evident dependence on σ2 when σ3 = 60 MPa. Comparisons with the other two sandstones with higher porosity reveal that porosity exerts critical control on the brittle strength and that the σ2 effect on deformability is also the result of two competing effects induced by mean stress and Lode angle. It is also found that the residual strength data facilitate the prediction of brittle-ductile transition pressure when σ2=σ3 in spite of relatively low σ3 level, which matches well with the empirical relation proposed by Mogi.1

Experimental and Analytical Investigations of the Effect of Hole Size on Borehole Breakout Geometries for Estimation of In Situ Stresses

Borehole breakout is a natural phenomenon in boreholes drilled in rock due to the induced stress concentration. Many researchers have attempted to correlate this phenomenon with in situ stress magnitudes. In this paper, a series of true triaxial tests on sandstone blocks (120 × 120 × 120 mm3) with different diameter pre-drilled holes have been carried out. Results confirmed that breakout geometries (angular span and depth) are dependent on the relative stress magnitudes. It is also noticed that a larger hole size (hole radius) yielded a wider angular span and deeper normalised depth (breakout depth/hole size), which indicates that hole size is an important parameter for breakout geometries. In addition, the analysis on previous experimental studies suggested that the relationship between two breakout geometries is not unique and is heavily influenced by the horizontal stress magnitudes. The analysis of the existing model also revealed the angular span may be narrowed with increasing horizontal stress ratio under a certain stress–strength condition. Both analyses indicate the breakout geometries are not only dependent on each other but also on the horizontal stress magnitudes. This leads to a tentative conclusion that breakout geometries may not be redundant factors and might be used for horizontal stress estimation.

Anisotropic Strength Characteristics of Loess under Three-Dimensional Stress Conditions

Abstract To study the failure conditions of anisotropic loess in the principal stress space, a series of true triaxial tests were performed on loess sampled in the Bailu Plateau of Xi’an, as loess deposits have a strong cross-anisotropic structure. The cuboid specimens of the prepared samples were cut along the vertical and horizontal directions to enable testing in any sector of the three-dimensional principal stress space. The shear strength laws of the p–q planes with different intermediate principal stress ratios corresponding to different principal stress spaces were revealed. The failure surface was assumed to be symmetric with respect to the vertical axis of the principal stress space octahedral plane, which varies as a function of the Lode angle. On the same octahedral plane, the shear strength of a specimen, as the major stress, is applied along the vertical direction and is larger than that along the horizontal direction. Failure surfaces in all sectors are ellipses, which are symmetric with respect to the vertical axis, and the loess structure is not damaged under the stress conditions of true triaxial compression. Once the structure was damaged, the failure surface of the cross-anisotropic loess on the octahedral planes presented as a circle or a curved triangle as the average principal stress increased.

CONTACT FABRIC CHARACTERISTICS OF GRANULAR MATERIALS UNDER THREE DIMENSIONAL STRESS PATHS

Macroscopic mechanical characteristics of granular materials are closely related to the microscopic contact force and fabric. Generally speaking, the strong contact system contributes to the force transmission of internal granular system, and then its corresponding fabric tensor has important influence on the macroscopic stress. Microscopic numerical methods, such as discrete element method, can reproduce the laboratory tests with reasonable macroscopic responses and extract macro- and micro-data conveniently for investigating the underlying mechanism of the granular system. Based on discrete element method (DEM), a series of true triaxial tests for granular materials under constant $p$ and constant $b$ stress paths are carried out, and the evolutions of macro- and micro-mechanical parameters of granular materials, the multiple relationship between three-dimensional fabric tensor and stress tensor and the macro-stress characteristics reflected by strong contact system are studied. The results demonstrate that some macro- and microscopic parameters at the stress peak and critical state in the granular system are independent on the loading path. Non-coaxiality between fabric tensor and stress tensor is observed under three-dimensional stress path, but the evolution of the joint invariant of the two tensors is independent on the 3D loading path. Compared to fabric tensor of weak contact system, the fabric tensor of strong contact system reflects better the characteristics of macroscopic stress tensor. Fabric tensors of strong and weak contact systems contribute differently to the granular macroscopic response. To divide the strong and weak contact system, there is a range for the threshold, however adopting the average contact force is relatively simple and reasonable.

Experimental Study of Failure Differences in Hard Rock Under True Triaxial Compression

In view of a previous study of the intermediate principal stress effect at a limited σ2 range, a series of true triaxial tests, covering a full range of intermediate principal stresses that vary from the generalized triaxial compression stress state (σ2 = σ3) to the generalized triaxial tensile stress state (σ1 = σ2), was carried out on sandstone and granite samples. The experimental results revealed that the deformation, failure strength and failure mode have a significant dependence on the stress state. As an effect of the intermediate principal stress on crack evolution, the deformation difference known as stress-induced deformation anisotropy occurred and should be considered when developing the mechanical model. Moreover, a post-peak deformation with a step-shaped stress drop is observed and illustrates that there will be a multi-stage bearing capacity after the rock failure. The peak strength is non-symmetrical with the increasing σ2 and is closely related to the Lode angle. Based on the final fracture surface and SEM analysis under true triaxial compression, three failure modes and failure zones, including tension failure, shear failure and mixed failure, are delineated and discussed. Combining the failure mode and the strength under true triaxial compression, it is found that the strength variation exhibited a close relationship to the failure mechanism.

Constitutive Modeling of Coarse-Grained Materials Incorporating the Effect of Particle Breakage on Critical State Behavior in a Framework of Generalized Plasticity

AbstractCoarse-grained materials (CGMs), including gravel, ballast, and rockfill material, exhibit a complicated stress-strain-volume change behavior, which is state dependent and influenced by considerable particle breakage even under relatively low pressure. A generalized plasticity model with a multiaxial formulation is developed for CGMs based on the critical state concept. The effect of particle breakage on their critical state behavior, including the nonlinear variation of both shear strength and void ratio with the mean effective stress, is fully incorporated with an implicit form in the current model. Two state functions and the corresponding virtual stress ratios are proposed to construct the new formulation of dilatancy, plastic flow, loading direction, and plastic modulus in the present model. The numerical analyses are performed for a series of true triaxial tests on CGMs, and model predictions are in good agreement with experimental results of true triaxial tests over a wide range of pressure...

Brittleness of gas shale reservoirs: A case study from the north Perth basin, Australia

Shale reservoirs have gained the attention of many in recent years due to their potential as a major gas resource. Production from this kind of formation, however, requires an accurate estimation of brittleness and employments of hydraulic fracturing. There have been many studies as to how brittleness can be estimated, but few research works were carried out so far indicating how brittleness indices vary in gas shale formations. The aim of this paper is to evaluate the variation of brittleness in one of the gas shale reservoirs located in the north Perth Basin of Australia. The results obtained indicated that the lower part of the Carynginia shale should be selected for a hydraulic fracturing job due to a high brittleness index, although a careful analysis of Total Organic Content (TOC) might be required before initiating any plans. The mineralogical report and interpretations revealed that the space created by cross-plotting the elastic parameters is able to identify dominant minerals contributing into brittleness. Performing a series of true triaxial tests, which are capable of simulating the real field condition by applying three independent principal stresses, implied that as the stress anisotropy increases, a transition takes place from brittle towards the ductile behaviours. However, when this anisotropy becomes significant, samples regain their strength. This study, therefore, recommends more studies to get a practical conclusion on brittleness under true triaxial conditions.

Study on the Prediction Method for Brittle Failure of Hard Rock Based on Acoustic Emission Test

In this paper, a series of true triaxial tests indoor with acoustic emission mornitoring were conducted and the characteristics of acoustic emission rate and energy releasing rate in the section adjacent to failure were gained. According to the different characteristics of acoustic emission rate, we divided the events rate into three types which were main shock, foreshock-main shock and cluster shocks. And then, a prediction method for hard rock was put forward according to different events rate types based on the trends of AE signals in the section adjacent to failure for hard rock.

The Onset of Strain Localization in Cross-Anisotropic Soils Under True Triaxial Condition

ABSTRACT The influences of inherent cross-anisotropy on soil strength on the homogeneous deformation and bifurcation characteristic of the stress-strain relationship are studied. By neglecting the cohesion term and employing an elliptical shape function in Mohr-Coulomb failure criterion, a 3D anisotropic failure criterion is proposed to describe the strength of cross-anisotropic sands under true triaxial conditions. Based on the proposed failure criterion, the influence of the anisotropic parameter on the failure curve on the deviatoric plane and the relationship between the peak friction angle and the intermediate principal stress ratio are obtained. The suitability of the criterion is justified by comparing with a series of true triaxial tests on sands without strain localization. The proposed failure criterion is adopted to build a three dimensional anisotropic elasto-plastic model, which allows the bifurcation analysis to be incorporated with the non-coaxial flow rule for the purpose of studying the onset of strain localization. Compared with the true triaxial tests under several intermediate principal stress ratios conditions in the literature, the proposed model and bifurcation analysis is shown to be capable of predicting the influence of inherent cross-anisotropy on the onset of strain localization.