Intermediate Principal Stress Coefficient Research Articles

A new analytical solution for deep-buried circular cavern with anchor reinforcement under seepage field

This paper presents a new method for assessing the stability of an underground cavern with anchor reinforcement under the seepage field. Firstly, the reinforcement effect of the anchor is considered as a set of clamping forces acting on both sides of the anchored rock mass, besides, whose strength parameters are also considered to be improved. Secondly, a mechanics model for a deep-buried circular cavern with anchor reinforcement under the seepage field is established, and the analytical solution incorporating the characteristics of dilatancy and strain softening of the surrounding rock are derived based on the unified strength theory. Finally, the validity and accuracy of the present solution are examined via comparison with published researches and field monitoring data. Subsequently, extensive parameter investigation is performed to systematically discuss the influence of the intermediate principal stress, dilatancy, strain softening, and seepage force on the cavern deformation and study the variation of the axial force of anchors. The results show that the surrounding rock stresses increase significantly but its radical displacement and the anchor axial force decreased as the intermediate principal stress coefficient increases. With the increase in dilatancy angle, the surrounding rock radial displacement and the anchor axial force increase sharply, but the plastic zone radius and the stresses are almost constant. Moreover, considering the seepage field, the peak value of maximum tangential stress and the stresses in the elastic zone increases but the radial stress in the plastic zone decreases, and the radial and tangential stress distributions are no longer asymptotic with the initial ground stress.

A mechanical model for the circular tunnel considering the interaction between the surrounding rock and support structure based on the unified strength criterion

It is very significant to study the stress and displacement fields in the surrounding rock induced by the tunnel excavation and its support. First of all, in order to take into account the effect of the intermediate principal stress on the tunnel surrounding rock mechanical behavior, the unified strength criterion (USC) is adopted to describe the plastic yield behavior of the surrounding rock. Secondly, the model proposed by Kastner does not consider the effect of the initial elastic displacement of the tunnel surrounding rock before the support structure is constructed and the interaction between the surrounding rock and the support structure. Therefore, a new mechanical model for the circular tunnel, which can overcome these shortcomings, is finally proposed by considering the tunnel practical construction process on basis of Kastner’s method. The effects of the calculation parameters such as the intermediate principal stress coefficient b, the initial elastic displacement u0 at the tunnel wall, the rock internal friction angle and the stiffness ks of the support structure on the surrounding rock plastic zone, and radial displacement of the tunnel wall are studied in detail. The results of the calculation examples show that the tunnel surrounding rock relative plastic zone rp/r0 and ground radial displacement us of the tunnel wall decrease with increasing b. With increasing u0, rp/r0 and us both increase, while with increasing ks and the rock internal friction angle, rp/r0 and us both decrease. In sum, it indicates that the proposed model for the circular tunnel can perfectly reflect the tunnel mechanical behavior.

Macro-Micro Mechanical Behavior of Crushable Granular Materials under Generalized Stress Conditions

This paper investigated the joint influence of particle crushing and intermediate principal stress coefficient on the macro and micro performances of granular assembles by using 3D discrete element method. The soil particle was modeled by crushable agglomerate. Numerical true triaxial tests with crushable agglomerates were performed. The mechanical behaviors of specimens with unbreakable agglomerates were also tested in this study. The results demonstrated that grain breakage can significantly affect the behaviors of granular soils in both macro and micro scales. Moreover, the present paper also showed that particle breakage did not influence the unique relationship between the intermediate principal stress coefficient b and the deviatoric fabric of strong contacts. This can provide a good basis for developing micro-parameter-based constitutive model considering particle breakage for granular soils.

An Experimental Study on the Deformation and Strength Characteristics of Q3 Loess under a Plain Strain Anisotropic Consolidation Condition

To study the strength, deformation characteristics, and intermediate principal stress of intact loess, vertical loading stress tests with different initial stress state k value were conducted under different confining pressures. Plane strain tests were carried out by the transformed true triaxial apparatus of Xi’an University of Technology. The study shows that loess tends to be in a plastic failure state in different confining pressures and k values, and the stress-strain relationship curve is of a hardening type. Results reveal that loess lateral and volume deformations have nonlinear relationships with its vertical deformation, and volume deformation shrinks in the process of shearing. The effect of confining pressure on soil deformation is greater than k value. The intermediate principal stress coefficient decreases with the increase of the confining pressure and transforms from increasing to decreasing with the increase of k value (ranging from 0.2 to 0.4). In brief, the loess failure strength is closely related to k value, confining stress, and spherical stress state. When k value increases, cohesion effect reduces, whereas internal friction angle increases linearly. The influence of k value on soil strength and deformation is closely related to confining pressure.

Three-dimensional cyclic behavior of saturated clays: comparison between undrained and partly drained conditions

The cyclic response of subgrade clays under traffic loadings is likely to be partly drained rather than undrained, and the traffic-induced dynamic stress field is three-dimensional rather than axisymmetric. To compare the three-dimensional deformation behaviors of saturated clays between partly drained and undrained conditions, a large number of cyclic true triaxial tests were conducted. Experimental results show that a partly drained condition leads to a remarkable increase of permanent major principal strain ([Formula: see text]) compared to an undrained condition, and the differences of [Formula: see text] between the two drainage conditions are affected greatly by the factors of cyclic stress ratio (CSR), overconsolidation ratio (OCR), and coefficient of cyclic intermediate principal stress (bcyc). The increase of bcyc induces a linear reduction of [Formula: see text] in an undrained condition, while it causes an increase first followed by a decrease of [Formula: see text] in a partly drained condition. The clays undergo stiffness softening and hardening in partly drained and undrained conditions, respectively, and the effects of bcyc and CSR on the stiffness evolution are very different between the two drainage conditions. The mechanism of the complex [Formula: see text] and resilient modulus behaviors in a three-dimensional state and partly drained condition is further discussed. In addition, two different empirical models are employed to predict [Formula: see text] in partly drained and undrained conditions.

Drained deformation characteristics of granular soil under pure principal stress axis rotation: impact of sample preparation

The distinct initial fabrics of sand may give rise to dramatically different responses to the applied loading, as evidenced by numerous laboratory tests reported in the literature. In this study, both dry deposition (DD) and moist tamping (MT) methods were employed to prepare sand specimens to generate different initial fabrics. Drained tests were conducted on these specimens under pure principal stress axis rotation while keeping the principal stress components constant. The factors that may influence the deformation characteristics of sand, such as deviatoric stress q, coefficient of intermediate principal stress b, and mean principal stress p, were also taken into consideration in the tests. Test results show that the developments of three normal strain components with number of cycles are very different between DD and MT specimens, which is also closely related to the factors of q, b, and p. It is seen that DD specimen tends to generate larger contractive volumetric strains than MT specimen, which is more significant when q and b increase or p decreases. The shear stiffness of DD specimen is greater than that of MT specimen. The shear modulus ratio, Gj/G1, increases with the increase of q and b or the decrease of p, which is more prominent in DD specimen than in MT specimen. It is also shown that DD specimen appears to exhibit stronger non-coaxiality than MT specimen, in which such difference of non-coaxiality tends to decrease with the increase in number of cycles.

Experimental Study on Influence of Intermediate Principal Stress on the Permeability of Sandstone

In engineering, it is considerably important to understand the evolution process of sandstone deformation and permeability under complex stress conditions. Although numerous scholars have reported on sandstone permeability, many of their studies are only based on uniaxial or conventional triaxial stress conditions. In practical engineering, in situ stress evidently indicates three-dimensional anisotropy. In recent years, many scholars have performed experimental studies on rock permeability under true triaxial stress. There are, however, few investigations related to the influence of intermediate principal stress and intermediate principal stress coefficient on sandstone permeability; moreover, the mechanism is unclear. In this study, the permeability of sandstone under true triaxial stress conditions is measured using a true triaxial apparatus. The results show that the intermediate principal stress has significant effects on permeability and deformation of sandstone sample. Under different principal stress conditions, the permeability is observed to decrease with the increase of the principal stress. With increasing intermediate principal stress coefficient b (from 0 to 1), the major principal strain and the minor principal strain always compressed and finally decreased by 27.31 and 36.96%, while the intermediate principal strain finally changes from expansion to compression (from − 0.045 to 0.255%). Also the normalized permeability loss increased from 9.57 to 16.59%. In addition, by fitting the experimental data with several permeability models, it was found that a true triaxial permeability model can well describe the changes in sandstone permeability. Based on the fitting parameters, the influence of intermediate principal stress and its coefficient on the stress sensitivity of sandstone was analyzed.

True triaxial strength and failure characteristics of cubic coal and sandstone under different loading paths

In deep underground coal mining, engineering activities are performed within anisotropic in-situ stress fields due to engineering disturbances and tectonic stress. Many such activities involve the development of excavations in soft rock and anisotropic coal. Accordingly, studying the mechanical properties of soft rocks is important for the stability of deep underground excavations. In this study, the deformation, strength, and failure characteristics of soft sandstone and raw coal under two different true triaxial loading paths were investigated using a self-developed true triaxial test apparatus. The results indicated that the inelastic strain in the pre-peak stage of sandstone and coal gradually increased with increasing intermediate principal stress. Also, the strength-drop in the post-peak stage increased. The crack initiation stress, crack damage stress, and peak strength of sandstone and coal first increased and then decreased with increasing intermediate principal stress for a given σ3. Moreover, with increasing intermediate principal stress, the failure mode of sandstone and coal changed from shear to tensile shear, and from brittle to semi-brittle. The linear Mogi criteria as found to characterize the true triaxial strength of coal well, while the modified Lade criteria was more applicable to soft sandstone. Owning to the symmetry assumption, the linear Mogi criteria predicted low strength when the intermediate principal stress coefficient exceeded to 0.5. In addition, the peak strength curve of rock on the π plane and the influence of weak structures on the failure mode of anisotropic coal was discussed. Weak structures have an important influence on the failure mode of coal, which depends on the strength difference between the structural plane and the coal rock mass. The strength envelope on the π plane had a significant stress Lode angle effect, which gradually decreased as the mean stress increased.

Resilient behavior of coarse granular materials in three dimensional anisotropic stress state

The coarse granular materials in the pavement base and subbase layers are generally in the stress state of three dimensional anisotropy, while most previous experimental researches regarding the base and subbase materials focused on the isotropic or axisymmetric K0 stress conditions. In this study, the resilient characteristics of unbound granular materials in initial three dimensional anisotropic stress states were investigated through an electro-mechanical true triaxial apparatus. The influences of initial shear stress, qini, coefficient of intermediate principal stress, b, amplitude of cyclic major principal stress, σ1ampl, and moisture content on the resilient responses of granular materials were investigated. Test results reveal that the resilient response of unbound granular materials is strongly affected by the three dimensional stress anisotropy in the saturated condition. With the increase of b from 0 to 0.8, a significant reduction of Mr occurs, and this reduction becomes intensified as qini increases. It is also found that Mr is increased by the increase of qini for almost all tests, and the increasing rate varies at different σ1ampl, b and moisture content values. Based on the experimental results, an empirical model to predict resilient modulus is proposed incorporating the effects of initial three dimensional stress anisotropy, which is expected to serve as a basic model to calculate the resilient modulus of granular materials in three dimensional stress state.

Non-coaxial Deformation Characteristics of Fine Sand under Principal Stress Rotation

A series of pure principal stress rotation tests with dynamic hollow cylinder apparatus were conducted on saturated fine sand. The elastic torsional shear strain was reported to account for a large proportion of the total torsional shear strain, which could not be ignored. On this basis, the non-coaxial angle is redefined and the non-coaxial deformation characteristics and influencing factors of saturated fine sand are studied. The results show that the principal stress rotation will cause obvious plastic deformation of the fine sand and that the initial shear stress has a more significant influence on the non-coaxial deformation characteristics of the sand compared to the intermediate principal stress coefficient.

Strain evolution of saturated clays under cyclic loadings in three-dimensional stress condition

The strain evolution of saturated clays under cyclic loadings has been studied by means of various laboratory testing methods, aiming for the accurate simulation of engineering conditions such as the consolidation states of clays, cyclic stress paths of external loadings and boundary conditions in situ. Considering that the subgrade clays are usually overconsolidated rather than normally consolidated, the drainage condition is normally partially drained instead of undrained under long-term cyclic loadings, and the cyclic stress paths are in three-dimensional stress state, this study explores the three-dimensional strain characteristics of both normally and overconsolidated clays in partially drained condition based on a true triaxial apparatus. Emphasis are put on the effects of cyclic stress ratio (CSR), overconsolidation ratio (OCR) and coefficient of cyclic intermediate principal stress (bcyc) on the accumulation of permanent strain components including major, intermediate, minor principal strains, and volumetric strain. Test results show that the relative magnitudes of strain components are strongly influenced by both bcyc and OCR, while the accumulated directions of strain paths tend to be identical for various bcyc values. The increase of bcyc leads to a linear growth of permanent volumetric strain for both normally and overconsolidated clays, while the permanent major principal strain shows a firstly increasing and then decreasing trend versus the increase of bcyc. For the normally consolidated and overconsolidated clays, the largest permanent major principal strains are achieved at around bcyc = 0.4 and 0.2, respectively. The accumulation of permanent intermediate principal strain is always promoted by the increase of bcyc. In addition, the development of permanent major principal and volumetric strains with cycle number is analyzed in the double logarithmic plane, which are found to be combined by two linear parts.

A long-term strength criterion for frozen clay under complex stress states

To study long-term strength characteristics under complex stress states, a series of hollow cylinder creep tests were conducted on frozen clay under different stress levels at a temperature of −6 °C. Experimental results showed that the creep characteristics of frozen clay were significantly affected by the mean principal stress and the intermediate principal stress coefficient. Most generalized shear creep curves of frozen clay presented typical non-attenuation creep under axial load and torque. A long-term strength criterion for frozen soil, exhibiting the strength characteristics of multiple stress components under complex stress states, was obtained based on activation energy theory. In addition, a modified elliptical function and a generalized nonlinear strength theory (GNST) were introduced to describe the evolution of the strength envelope with time. Finally, a new strength criterion for frozen clay in three-dimensional principal stress space, considering the effect of time, was established by combining the modified elliptical equation and the GNST shape function. The proposed strength criterion has great significance for predicting the strength characteristics of frozen clay under complex stress states.

A bounding surface model for frozen sulfate saline silty clay considering rotation of principal stress axes

Research results show that rotation of principal stress axes (RPSA) has a large influence on the mechanical properties of soil. To study the mechanical and deformation properties of frozen saline soil under RPSA, in this paper, a constitutive model considering RPSA is proposed for frozen saline soil by considering the plastic deformation caused by the increase of principal stress amplitude and the plastic deformation caused by RPSA separately. The proposed model has the following properties: (1) For the part of principal stress amplitude increase, the critical stress ratio Mc(s)g(θ) changing with hydrostatic pressure is adopted to describe pressure melting and crushing phenomenon characteristic of frozen saline soil under high confining pressure. A function of plastic shear strain and plastic volumetric strain is taken as hardening parameter to control the size of the bounding surface under plastic volumetric compression mechanism. (2) Based on the experimental results, a new shape function is presented for frozen soil, and the proposed shape function is introduced into bounding surface function and plastic potential function to control the shapes change of bounding surface and plastic potential surface. (3) For the part of RPSA, a more accurate method for determining loading direction and a formula for solving non-coaxial angle are given. A plastic modulus formula considering the effects of hydrostatic pressure, generalized shear stress, and anisotropy is proposed. A dilatancy equation which can reflect the effect of intermediate principal stress coefficient is presented. By comparing the simulated results of the proposed model with the experimental results, it is found that the proposed model not only can reflect the effect of the increase of principal stress amplitude on the mechanical and deformation behaviors of frozen saline soil, but also can describe the effect of RPSA on the mechanical and deformation behaviors of frozen soil.

Analysis of Three‐Dimensional Damage Constitutive Model of Frozen Sand Based on the Influence of Intermediate Principal Stress and Temperature

To study the mechanical properties of frozen soil, it is necessary to understand the damage characteristics of frozen soil. Four types of three‐dimensional indoor tests of frozen sand were carried out at −5°C, −10°C, and −15°C to study the mechanical damage properties. These include different stress path tests with the principal stress coefficients of 0, 0.25, 0.5, and 0.75 while analyzing the entire failure process. First, the three‐dimensional compression test of frozen sand was studied to analyze the influence of temperature and intermediate principal stress coefficient on the large principal stress of frozen soil. The damage cost of frozen sand under the influence of different temperatures and intermediate principal stress coefficients was also established. Second, using the characteristics of discreteness and randomness of the distribution of the microelements inside the frozen soil and assuming that the failure of the microelement of the frozen soil obeys the Weibull distribution, the Drucker–Prager strength criterion was used as the statistical distribution variable of the microelement of the frozen soil based on the strain equivalence hypothesis, statistical theory, and continuous damage mechanics. This allows for a constitutive model of frozen sand damage under the three‐dimensional stress state to be established. Finally, the model parameter values through low‐temperature three‐dimensional test data were able to be determined. This model allows for the physical meaning of Weibull distribution parameters F0 and m to be analyzed, and the distribution parameters with temperature and intermediate principal stress coefficient can be modified to obtain a modified frozen sand damage constitutive model. The results show that the modified damage constitutive model can simulate the entire process curve of the large principal stress‐strain of frozen sand. It shows that the large principal stress of frozen sand increases with the increase of temperature and intermediate principal stress coefficient. Concurrently, the frozen sand damage constitutive model proposed in this paper can describe the deformation behavior of frozen soil under different temperature and stress paths and can be adapted to various other sediment types.

Experimental Study on Noncoaxial Characteristics of Saturated Remolded Loess

In practical engineering, if the influence of noncoaxial stress and strain is not considered, part of soil deformation will be ignored, resulting in the structural design which is not safe enough. A series of undrained tests was performed on remolded loess specimens using a hollow cylinder apparatus to examine the coupling between principal stress magnitude and direction in these specimens. First, the elastic parameters of remolded loess were obtained, and these parameters were used as the basis for investigating the noncoaxiality of the soil body under principal stress axis rotation (PSAR). The effects of elastic strain, intermediate principal stress coefficient, and magnitude of the deviatoric stress on the noncoaxiality of remolded loess were also investigated. The results of these experiments show that remolded loess exhibits significant noncoaxial behavior during PSAR. The noncoaxiality angle of remolded loess cyclically fluctuates with increases in the principal stress angle. It was also observed that the noncoaxiality angle will be overestimated if the effects of elastic strain are overlooked. Reversals in the direction of PSAR cause dramatic changes in the noncoaxiality angle. Increases in the intermediate principal stress coefficient are accompanied by increases in the noncoaxiality angle, up to a certain degree; however, these changes do not affect the development of the noncoaxiality angle. In coupled rotational tests with a range of deviatoric stress amplitudes, it was observed that changes in the deviatoric stress amplitude will affect the development of the noncoaxiality angle; increases in the deviatoric stress amplitude cause the noncoaxiality angle versus principle stress angle plot to shift to the left gradually, thus accelerating the trends of the noncoaxiality angle. Increases in the cycle number also increase the noncoaxiality of remolded loess.

Stability Analysis of Surrounding Rock in Circular Tunnels Based on Critical Support Pressure

Accurate calculation for the critical support pressure of tunnels plays an important role in tunnel stability evaluation and support design. In this study, a mechanical model for circular tunnels is developed. Considering the intermediate principal stress and strain‐softening characteristic of rock mass, the critical support pressure when the plastic zone and damage zone begin to occur is determined based on the unified strength criterion and strain‐softening model. Through the example study, the critical support pressure under different intermediate principal stress coefficient is solved. Furthermore, the effect of initial field stress, softening coefficient, and maximum damage variable on the critical support pressure are also discussed. The results show that the critical support pressure and radii of plastic and damage zones all decrease with the increase of the intermediate principal stress coefficient. The larger the initial field stress, the larger the critical support pressure. The softening coefficient and maximum damage variable of rock mass has no influence on the critical support pressure when the plastic zone begins to form, but has a significant effect on the critical support pressure when the damage zone begins to form. As softening coefficient increases and maximum damage variable decreases, the critical support pressure when the damage zone which begins to form increases. Data presented in this contribution provide significant theoretical insights into evaluating tunnel stability and support system reliability.

Deformation characteristics of saturated clay in three-dimensional cyclic stress state

The dynamic stress field induced by moving loads is three dimensional, involving the cyclic variation of major, intermediate, and minor principal stresses, while so far very limited laboratory studies have been undertaken on the one-way deformation behavior of saturated clays in three-dimensional stress state. In this study, an advanced true triaxial apparatus, which can apply cyclic major and intermediate principal stresses simultaneously, is employed to carry out a total of 65 one-way cyclic tests on both normally and overconsolidated clays. Four values of overconsolidation ratio (OCR), five values of bcyc, which is termed the coefficient of cyclic intermediate principal stress, and a wide range of cyclic stress ratios (CSR) are tested. Emphasis is put on the effects of bcyc and OCR on the characteristics of permanent major and intermediate principal strains. Test results show that the increase of bcyc significantly reduces the accumulation of major principal strain, and linear relationships are observed between the permanent major principal strain and bcyc for test data with the same CSR, OCR, and cycle number. A critical value of bcyc ≈ 0.5, at which the permanent intermediate principal strain changes from tension to compression, is observed for the remolded clay. Furthermore, an empirical model is proposed, allowing the long-term deformation of saturated clays to be predicted in three-dimensional cyclic stress state.

Resilient behavior of coarse granular materials in three-dimensional stress state

The traffic-induced cyclic stresses on the road base and subbase courses are usually in three-dimensional stress state, while so far most laboratory studies have focused on the deformation behavior of base and subbase layers in axisymmetric stress state. This study investigates the three-dimensional resilient behavior of coarse granular base and subbase materials based on a true triaxial apparatus. The factors of effective confining pressure, [Formula: see text], amplitude of cyclic shear stress, qampl, coefficient of cyclic intermediate principal stress, bcyc, and fines content are involved. Test results indicate that the increase of either [Formula: see text] or qamplleads to a nonlinear growth of resilient modulus with a decreasing growth rate. bcyc, which is proposed to represent the coupling of cyclic major and intermediate principal stresses, is found to have a promotion effect on the resilient stiffness, and the promotion effect tends to be enhanced by the increase of qampl. A modified model is established to predict the resilient modulus of coarse granular materials in three-dimensional stress state. There appears to exist a critical value of fines content, at which the variation trend of resilient modulus is changed from increasing to decreasing, and the critical fines content is related to both [Formula: see text] and bcyc.

Analysis of Deformation of Circular Roadway Considering Effects of Intermediate Principal Stress and Dilatancy

Based on the unified strength theory and non-associated flow rule, with the consideration of the effects of intermediate principal stress and dilation characteristics, the analytical expressions of surrounding rock stress, displacement and plastic zone radius were obtained. Then the influences of the intermediate principal stresses and dilation angles on the surrounding rock stress, displacement and the plastic zone radius were analyzed by engineering example. The analysis shows that surrounding rock stress distribution and plastic zone radius have close relations with the intermediate principal stress. With the increase of the intermediate principal stress coefficient, the ultimate strength of surrounding rock increased, but the plastic zone radius shows a decreasing trend. The displacement of plastic zone is not only related to the intermediate principal stress coefficient, but also to the dilation angle. The displacement of plastic zone increased with the increase of dilation angle, and the displacement of plastic zone decreased with the increase of the intermediate principal stress coefficient.

A novel transversely isotropic strength criterion for soils based on a mobilised plane approach

The peak shear strength rules of transversely isotropic soils are stress state dependent and dependent on relative orientation between bedding plane and principal stress. Accordingly, the shear strength of transversely isotropic soils exhibits two primary characteristics: (a) the strength curve on the deviatoric plane is asymmetrical with respect to three principal stress axes; (b) the shear strength changes with the direction angle of the bedding plane when the intermediate principal stress coefficient is a constant. In this paper, the mobilised plane is introduced and used to reveal the failure mechanism of soils. By projecting the microstructure tensor of transversely isotropic soils onto the normal of the mobilised plane, the directionality of the transversely isotropic soils is introduced into the friction rules on the mobilised plane, and a transversely isotropic strength parameter is proposed. The proposed strength parameter can extend isotropic strength criteria into transversely isotropic strength criteria. This mobilised plane approach is used to establish a novel transversely isotropic non-linear unified strength criterion (TI-NUSC). The difficulty in establishing a unified description of the asymmetrical strength curve and its evolution with direction angle is overcome by the established criterion. Comparisons between available test results and the TI-NUSC show that the TI-NUSC can successfully describe these two primary peak strength characteristics.