Experimental study on failure characteristics of single-sided unloading rock under different intermediate principal stress conditions

Investigation of unloading rock failure under differentσ_(2)facilitates the control mechanism of excavation surrounding rock.This study focused on single-sided unloading tests of granite specimens under true triaxial conditions.The strength and failure characteristics were studied with micro-camera and acoustic emission(AE)monitoring.Furthermore,the choice of test path and the effect ofσ_(2)on fracture of unloading rock were discussed.Results show that the increasedσ_(2)can strengthen the stability of single-sided unloading rock.After unloading,the rock’s free surface underwent five phases,namely,inoculation,particle ejection,buckling rupture,stable failure,and unstable rockburst phases.Moreover,atσ_(2)≤30 MPa,the b value shows the following variation tendency:rising,dropping,significant fluctuation,and dropping,with dispersed damages signal.Atσ_(2)≥40 MPa,the tendency shows:a rise,a decrease,a slight fluctuation,and final drop,with concentrated damages signal.After unloading,AE energy is mainly concentrated in the micro-energy range.With the increasedσ_(2),the micro-energy ratio rises.In contrast,low,medium and large energy ratios drop gradually.The increased tensile fractures and decreased shear fractures indicate that the failure mode of the unloading rock gradually changes from tensile-shear mode to tensile-split one.The fractional dimension of the rock fragments first increases and then decreases with an inflection point at 20 MPa.The distribution of SIF on the planes changes asσ_(2)increases,resulting in strengthening and then weakening of the rock bearing capacity.

Investigation of the influence of intermediate principal stress on the dynamic responses of rocks subjected to true triaxial stress state

Precisely understanding the dynamic mechanical properties and failure modes of rocks subjected to true triaxial stress state(σ1>σ2>σ3,whereσ1,σ2,andσ3 are the major principal stress,intermediate principal stress,and minor principal stress,respectively)is essential to the safety of underground engineering.However,in the laboratory,it is difficult to maintain the constant true triaxial stress state of rocks during the dynamic testing process.Herein,a numerical servo triaxial Hopkinson bar(NSTHB)was developed to study the dynamic responses of rocks confronted with a true triaxial stress state,in which lateral stresses can maintain constant.The results indicate that the dynamic strength and elastic modulus of rocks increase with the rise of intermediate principal stressσ2,while the dynamic elastic modulus is independent of the dynamic strain rate.Simulated acoustic emission distributions indicate that the intermediate principal stressσ2 dramatically affects dynamic failure modes of triaxial confined rocks.Asσ2 increases,the failure pattern switches from a single diagonal shear zone into two parallel shear zones with a small slant.Moreover,a recent triaxial Hopkinson bar experimental system using three bar pairs is also numerically established,and the measuring discrepancies are identified between the two numerical bar systems.The proposed NSTHB system provides a controllable tool for studying the dynamic triaxial behavior of rocks.

Failure criterion for soft rocks considering intermediate principal stress

The significant differences between hard rocks(more brittle)and soft rocks(more ductile)may suggest the use of different failure criteria.A strength criterion for soft rocks that includes intermediate principal stress was proposed.The new criterion includes two independent parameters:the uniaxial compressive strength(σ_(ci)),which can be obtained from common laboratory tests or indirectly estimated by alternative index tests in the laboratory or field;and f(joint),which is used to characterize the rock mass quality and can be easily estimated.The authors compared the predictive capabilities of the new criterion with other criteria using the database of soft rocks under two conditions:with and without triaxial data.For the estimation of triaxial and true-triaxial strengths,the new criterion generally produced a better fit.The proposed criterion is practical for an approximate first estimation of rock mass strength,even without triaxial data,as it balances accuracy(lower prediction error)and simplicity(fewer independent parameters).

New artificial neural networks for true triaxial stress state analysis and demonstration of intermediate principal stress effects on intact rock strength

Simulations are conducted using five new artificial neural networks developed herein to demonstrate and investigate the behavior of rock material under polyaxial loading. The effects of the intermediate principal stress on the intact rock strength are investigated and compared with laboratory results from the literature. To normalize differences in laboratory testing conditions, the stress state is used as the objective parameter in the artificial neural network model predictions. The variations of major principal stress of rock material with intermediate principal stress, minor principal stress and stress state are investigated. The artificial neural network simulations show that for the rock types examined, none were independent of intermediate principal stress effects. In addition, the results of the artificial neural network models, in general agreement with observations made by others, show （a） a general trend of strength increasing and reaching a peak at some intermediate stress state factor, followed by a decline in strength for most rock types; （b） a post-peak strength behavior dependent on the minor principal stress, with respect to rock type; （c） sensitivity to the stress state, and to the interaction between the stress state and uniaxial compressive strength of the test data by the artificial neural networks models （two-way analysis of variance; 95% confidence interval）. Artificial neural network modeling, a self-learning approach to polyaxial stress simulation, can thus complement the commonly observed difficult task of conducting true triaxial laboratory tests, and/or other methods that attempt to improve two-dimensional （2D） failure criteria by incorporating intermediate principal stress effects.

Retrospective and prospective review of the generalized nonlinear strength theory for geomaterials

Strength theory is the basic theory for calculating and designing the strength of engineering materials in civil,hydraulic,mechanical,aerospace,military,and other engineering disciplines.Therefore,the comprehensive study of the generalized nonlinear strength theory(GNST)of geomaterials has significance for the construction of engineering rock strength.This paper reviews the GNST of geomaterials to demonstrate the research status of nonlinear strength characteristics of geomaterials under complex stress paths.First,it systematically summarizes the research progress of GNST(classical and empirical criteria).Then,the latest research the authors conducted over the past five years on the GNST is introduced,and a generalized three-dimensional(3D)nonlinear Hoek‒Brown(HB)criterion(NGHB criterion)is proposed for practical applications.This criterion can be degenerated into the existing three modified HB criteria and has a better prediction performance.The strength prediction errors for six rocks and two in-situ rock masses are 2.0724%-3.5091%and 1.0144%-3.2321%,respectively.Finally,the development and outlook of the GNST are expounded,and a new topic about the building strength index of rock mass and determining the strength of in-situ engineering rock mass is proposed.The summarization of the GNST provides theoretical traceability and optimization for constructing in-situ engineering rock mass strength.

A generalized nonlinear three-dimensional Hoek‒Brown failure criterion

To understand the strengths of rocks under complex stress states,a generalized nonlinear threedimensional(3D)Hoek‒Brown failure(NGHB)criterion was proposed in this study.This criterion shares the same parameters with the generalized HB(GHB)criterion and inherits the parameter advantages of GHB.Two new parameters,b,and n,were introduced into the NGHB criterion that primarily controls the deviatoric plane shape of the NGHB criterion under triaxial tension and compression,respectively.The NGHB criterion can consider the influence of intermediate principal stress(IPS),where the deviatoric plane shape satisfies the smoothness requirements,while the HB criterion not.This criterion can degenerate into the two modified 3D HB criteria,the Priest criterion under triaxial compression condition and the HB criterion under triaxial compression and tension condition.This criterion was verified using true triaxial test data for different parameters,six types of rocks,and two kinds of in situ rock masses.For comparison,three existing 3D HB criteria were selected for performance comparison research.The result showed that the NGHB criterion gave better prediction performance than other criteria.The prediction errors of the strength of six types of rocks and two kinds of in situ rock masses were in the range of 2.0724%-3.5091%and 1.0144%-3.2321%,respectively.The proposed criterion lays a preliminary theoretical foundation for prediction of engineering rock mass strength under complex in situ stress conditions.

Analysis of strength characteristics of loess before and after freezing using a hollow cylinder torsional shear apparatus

This paper aims to comprehensively analyze the influence of the principal stress angle rotation and intermediate principal stress on loess's strength and deformation characteristics.A hollow cylinder torsional shear apparatus was utilized to conduct tests on remolded samples under both normal and frozen conditions to investigate the mechanical properties and deformation behavior of loess under complex stress conditions.The results indicate significant differences in the internal changes of soil particles,unfrozen water,and relative positions in soil samples under normal and frozen conditions,leading to noticeable variations in strength and strain development.In frozen state,loess experiences primarily compressive failure with a slow growth of cracks,while at normal temperature,it predominantly exhibits shear failure.With the increase in the principal stress angle,the deformation patterns of the soil samples under different conditions become essentially consistent,gradually transitioning from compression to extension,accompanied by a reduction in axial strength.The gradual increase in the principal stress axis angle(α)reduces the strength of the generalized shear stress and shear strain curves.Under an increasingα,frozen soil exhibits strain-hardening characteristics,with the maximum shear strength occurring atα¼45.The intermediate principal stress coefficient(b)also significantly impacts the strength of frozen soil,with an increasing b resulting in a gradual decrease in generalized shear stress strength.This study provides a reference for comprehensively exploring the mechanical properties of soil under traffic load and a reliable theoretical basis for the design and maintenance of roadbeds.

A true triaxial strength criterion for rocks by gene expression programming

Rock strength is a crucial factor to consider when designing and constructing underground projects.This study utilizes a gene expression programming(GEP)algorithm-based model to predict the true triaxial strength of rocks,taking into account the influence of rock genesis on their mechanical behavior during the model building process.A true triaxial strength criterion based on the GEP model for igneous,metamorphic and magmatic rocks was obtained by training the model using collected data.Compared to the modified Weibols-Cook criterion,the modified Mohr-Coulomb criterion,and the modified Lade criterion,the strength criterion based on the GEP model exhibits superior prediction accuracy performance.The strength criterion based on the GEP model has better performance in R2,RMSE and MAPE for the data set used in this study.Furthermore,the strength criterion based on the GEP model shows greater stability in predicting the true triaxial strength of rocks across different types.Compared to the existing strength criterion based on the genetic programming(GP)model,the proposed criterion based on GEP model achieves more accurate predictions of the variation of true triaxial strength(s1)with intermediate principal stress(s2).Finally,based on the Sobol sensitivity analysis technique,the effects of the parameters of the three obtained strength criteria on the true triaxial strength of the rock are analysed.In general,the proposed strength criterion exhibits superior performance in terms of both accuracy and stability of prediction results.

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 HoekeBrown criterion when the intermediate principal stress σ_(2) is equal to the minimum principal stress σ_(3)(Zuo et al.,2015).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.

Elasto-plastic analysis of the surrounding rock mass in circular tunnel based on the generalized nonlinear unified strength theory

The present paper aims to establish a versatile strength theory suitable for elasto-plastic analysis of underground tunnel surrounding rock. In order to analyze the effects of intermediate principal stress and the rock properties on its deformation and failure of rock mass, the generalized nonlinear unified strength theory and elasto-plastic mechanics are used to deduce analytic solution of the radius and stress of tunnel plastic zone and the periphery displacement of tunnel under uniform ground stress field. The results show that: intermediate principal stress coefficient b has significant effect on the plastic range,the magnitude of stress and surrounding rock pressure. Then, the results are compared with the unified strength criterion solution and Mohr–Coulomb criterion solution, and concluded that the generalized nonlinear unified strength criterion is more applicable to elasto-plastic analysis of underground tunnel surrounding rock.

A simplified three-dimensional extension of Hoek-Brown strength criterion

The Hoek-Brown(HB)strength criterion has been applied widely in a large number of projects around the world.However,this criterion ignores the intermediate principal stress s2.Many evidences have demonstrated that the rock strength is dependent on s2.Thus it is necessary to extend the HB criterion into a three-dimensional(3D)form.In this study,the effect of s2 on the strength of rocks is identified by reviewing the true triaxial tests of various rock types reported in the literature.A simple 3D strength criterion is developed.The modified criterion is verified by the true triaxial tests of 13 rock types.The results indicate that the modified criterion can achieve a good fit to most of rock types.It can represent a series of criteria as b varies.For comparisons,several existing 3D versions of the HB criterion are selected to predict the strengths of these rock types.It is indicated that the proposed criterion works better than other criteria.A substantial relationship between parameter b and the unconfined compressive strength is established,which guarantees that the proposed criterion can still work well even in the absence of true triaxial test data.

Comparative analysis of deformation and failure mechanisms of underground powerhouses on the left and right banks of Baihetan hydropower station

The stability of the surrounding rocks of large underground powerhouses is always emphasized during the construction process,especially in large-scale underground projects under construction,such as the Baihetan hydropower station in China.According to field investigations,numerical simulations and monitoring data analysis,we present a comparative analysis of the deformation and failure characteristics of the surrounding rocks of underground powerhouses on the left and right banks of the Baihetan hydropower station.The failure characteristics and deformation magnitude of the underground powerhouses on the left and right banks are quite different.Under the disadvantageous condition where the maximum principal stress intersects the axis of the powerhouse at a large angle,the left bank underground powerhouse shows prominent stress-controlled failure characteristics such as spalling,slack collapse and concrete cracking.Although the maximum principal stress is in the favorable condition which intersects the right bank powerhouse at a small angle,the relatively high intermediate principal stress with an angle subvertical to the right bank powerhouse plays an essential role in its deformation and failure,indicating that the influence of high intermediate principal stress cannot be ignored.In addition,structural plane-controlled failure and large deformation are also more evident on the right bank due to the extensive distribution of weak structural planes and complex surrounding rock properties.

Polyaxial strength criterion and closed-form solution for squeezing rock conditions

In this paper,a nonlinear strength criterion is proposed using the average of intermediate(σ2)and minor(σ3)principal stresses in place of σ3 in Ramamurthy(1994)’s strength criterion.The proposed criterion has the main advantages of negligible variation of strength parameters with confining stress and ability to link with conventional strength parameters.Additionally,a new closed-form solution based on the proposed criterion is derived and validated for Chhibro Khodri tunnel.Further,analytical solutions including Singh’s elastoplastic theory,Scussel’s approach,and closed-form solutions based on conventional and modified Ramamurthy(2007)criteria are compared with the results of proposed approach.It is shown that the in situ squeezing pressure predictions made by the proposed approach are more accurate.Also,a parametric study of the present analytical solution is carried out,which displays explicit dependency of tunnel stability on internal support pressure and tunnel depth.The influence of tunnel geometry is observed to be dependent on the applied support pressure.

Three-dimensional FDEM numerical simulation of failure processes observed in Opalinus Clay laboratory samples

This study presents the first step of a research project that aims at using a three-dimensional （3D） hybridfinite-discrete element method （FDEM） to investigate the development of an excavation damaged zone（EDZ） around tunnels in a clay shale formation known as Opalinus Clay. The 3D FDEM was first calibratedagainst standard laboratory experiments, including Brazilian disc test and uniaxial compression test. Theeffect of increasing confining pressure on the mechanical response and fracture propagation of the rockwas quantified under triaxial compression tests. Polyaxial （or true triaxial） simulations highlighted theeffect of the intermediate principal stress （s2） on fracture directions in the model： as the intermediateprincipal stress increased, fractures tended to align in the direction parallel to the plane defined by themajor and intermediate principal stresses. The peak strength was also shown to vary with changing s2. 2014 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting byElsevier B.V. All rights reserved.

Practical application of failure criteria in determining safe mud weight windows in drilling operations

Wellbore instability is reported frequently as one of the most significant incidents during drilling operations.Analysis of wellbore instability includes estimation of formation mechanical properties and the state of in situ stresses.In this analysis,the only controllable parameter during drilling operation is the mud weight.If the mud weight is larger than anticipated,the mud will invade into the formation,causing tensile failure of the formation.On the other hand,a lower mud weight can result in shear failures of rock,which is known as borehole breakouts.To predict the potential for failures around the wellbore during drilling,one should use a failure criterion to compare the rock strength against induced tangential stresses around the wellbore at a given mud pressure.The Mohr–Coulomb failure criterion is one of the commonly accepted criteria for estimation of rock strength at a given state of stress.However,the use of other criteria has been debated in the literature.In this paper,Mohr–Coulomb,Hoek–Brown and Mogi–Coulomb failure criteria were used to estimate the potential rock failure around a wellbore located in an onshorefield of Iran.The log based analysis was used to estimate rock mechanical properties of formations and state of stresses.The results indicated that amongst different failure criteria,the Mohr–Coulomb criterion underestimates the highest mud pressure required to avoid breakouts around the wellbore.It also predicts a lower fracture gradient pressure.In addition,it was found that the results obtained from Mogi–Coulomb criterion yield a better comparison with breakouts observed from the caliper logs than that of Hoek–Brown criterion.It was concluded that the Mogi–Coulomb criterion is a better failure criterion as it considers the effect of the intermediate principal stress component in the failure analysis.

Unified elastoplastic finite difference and its application

Two elastoplastic constitutive models based on the unified strength the- ory （UST） are established and implemented in an explicit finite difference code, fast Lagrangian analysis of continua （FLAC/FLAC3D）, which includes an associated/non- associated flow rule, strain-hardening/softening, and solutions of singularities. Those two constitutive models are appropriate for metallic and strength-different （SD） materials, respectively. Two verification examples are used to compare the computation results and test data using the two-dimensional finite difference code FLAC and the finite element code ANSYS, and the two constitutive models proposed in this paper are verified. Two application examples, the large deformation of a prismatic bar and the strain-softening be- havior of soft rock under a complex stress state, are analyzed using the three-dimensional code FLAC3D. The two new elastoplastic constitutive models proposed in this paper can be used in bearing capacity evaluation or stability analysis of structures built of metallic or SD materials. The effect of the intermediate principal stress on metallic or SD mate- rial structures under complex stress states, including large deformation, three-dimensional and non-association problems, can be analyzed easily using the two constitutive models proposed in this paper.

Definition of failure criterion for frozen soil under directional shear-stress path

A series of directional shear tests on remolded frozen soil was carried out at 10°C by using a hollow cylinder apparatus to study failure criterion under a directional shear-stress path.Directional shear tests were conducted at five shear rates(10,20,30,40,and 50 kPa/min)and five intermediate principal stress coefficients(b=0,0.25,0.5,0.75,and 1),with the mean principal stress(p=4.5 MPa)kept constant.The results show that the torsional strength and the generalized strength both increase with the increase of the shear rates.According to the failure modes of frozen soil under different shear rates,the specimens present obvious plastic failure and shear band;and the torsional shear component dominates the failure modes of hollow cylindrical specimens.A shear rate of 30 kPa/min is chosen as the loading rate in the directional shear tests of frozen soil.The shape of the failure curve in theπplane is dependent on the directional anglesαof the major prin cipal stress.It is reasonable to use the strain-hardening curves to define the deviatoric stress value atγg=15%(generalized shear strain)as the failure criterion of frozen soil under a directional shear-stress path.

A STRESS VECTOR-BASED CONSTITUTIVE MODEL FOR COHESIONLESS SOIL (Ⅰ)-THEORY

On the basis of the sufficient consideration of vectorial characteristics of stress, a new nonlinear constitutive model for cohesionless soil under plane strain and 3-D conditions was presented in a way that the action effects of stress vector are decomposed into the action effect of mean effective stress and that of the stress ratio vector (ratio of deviatoric stress vector to mean effective stress). The constitutive model can take account of the influence of both numerical and directional changes of stress vector on deformation of soil simultaneously, and is applicable of both static and dynamic loading.

Analytical solutions for deep tunnels in strain-softening rocks modeled by different elastic strain definitions with the unified strength theory

This paper presents the analytical solutions for the responses of tunnels excavated in rock masses exhibiting strain-softening behavior. Since previous analyses give little consideration to the effect of the intermediate principal stress on the strain-softening rock behavior, the unified strength theory was introduced to analyze the tunnel response. Four cases of different definitions of the elastic strain in the softening and residual regions, used in the existing solutions, were considered. The tunnel displacements,stresses, radii of the softening and residual zones and critical stresses were deduced. The proposed solutions were verified by comparing with numerical simulations, model tests and existing solutions. Furthermore, the solutions of the four cases were compared with each other to investigate the influence of the elastic strain expressions on the tunnel responses. The results showed that the intermediate principal stress coefficient b has a significant effect on the tunnel displacements, stress fields, and plastic radii. Parametric studies were performed to analyze the influences of the softening and residual dilatancy coefficients,softening modulus and residual strength on the tunnel responses. The parametric analysis indicated that the existing models should be carefully evaluated in the analysis of tunnels constructed through average-quality rocks;the proposed solutions outperformed the existing models in solving the mentioned problem.

Three-dimensional DEM investigation of the stress-dilatancy relation of grain-cementing type methane hydrate-bearing sediment

In this study,the Discrete Element Method(DEM)was employed to investigate numerically the effects of hydrate cementation and intermediate principal stress on the stress-dilatancy relation of graincementing type methane hydrate-bearing sediment(MHBS)by conducting a series of conventional and true triaxial tests.A novel 3D thermo-hydro-mechanical-chemical(THMC)contact model for MHBS was employed.The numerical results show that with increasing hydrate saturation and back pressure,or decreasing confining pressure,temperature and salinity,the stress-dilation relation of grain-cementing type MHBS evolves from dilation-dominant to bond-dominant.For the clean sand samples,the relationship between the normalized stress ratio h/Mcr and the dilatancy rate d is close under different intermediate principal stress coefficients.However,for the MHBS samples,this relationship is still affected by the intermediate principal stress coefficient b,due to the effect of hydrate cementation.