Reply to Discussion on “A generalized three-dimensional failure criterion for rock masses”

General:Jaiswal and Shrivastva(2012)proposed the mathematical formulations,i.e.the J–S criterion for converting generalized H-B failure criterion into 3D smooth convex failure criterion at deviatoric plane.The J–S strength criterion is in two versions:uniform and variable extension ratio.It has been observed from the analysis that at uniform extension ratio,the required strength parameters are only UCS and m(other parameters such as Ls,a,b and c are related with m).In the case of variable extension ratio,extra parameter f is required along with UCS and m.Thus,it has minimal strength parameters compared to You strength criterion.Furthermore,You strength criterion does not obey the smooth convex condition at deviatoric plane.     

Comprehensive compressive-tensile strength criterion for intact rock

This paper presents a strength criterion for intact rock,which can well describe triaxial test data under compressive or tensile stress state.The proposed criterion is defined in terms of three parameters.One parameter expresses the apparent unconfined compressive strength(AUCS),obtained from the Coulomb-Mohr criterion,as a regulated unconfined compressive strength(RUCS).Two other parameters,and,are material-dependent that can be determined by regression analysis.The proposed criterion is compared with selected applicable strength criteria separately for compressive and tensile strengths.Coefficient of determination and accordance coefficient are considered in comparisons between the proposed and selected strength criteria.     

A generalized three-dimensional failure criterion for rock masses

The smooth convex generalized failure function, which represents 1/6 part of envelope in the deviatoric plane, is proposed. The proposed function relies on four shape parameters (Ls, a, b and c), in which two parameters (a and b) are dependent on the others. The parameter L s is called extension ratio. The proposed failure function could be incorporated with any two-dimensional (2D) failure criteria to make it a three-dimensional (3D) version. In this paper, a mathematical formulation for incorporation of Hoek-Brown failure criterion with the proposed function is presented. The Hoek-Brown failure criterion is the most suited 2D failure criterion for geomaterials. Two types of analyses for best-fitting solution of published true tri-axial test data were made by considering (1) constant extension ratio and (2) variable extension ratio. The shape and strength parameters for different types of rocks have been determined by best-fitting the published true tri-axial test data for both the analyses. It is observed from the best-fitting solution by considering uniform extension ratio (Ls) that shape constants have a correlation with Hoek-Brown strength parameters. Thus, only two parameters (σc and m) are needed for representing the 3D failure criterion for intact rock. The statistical expression between shape and Hoek-Brown strength parameters is given. In the second analysis, when considering varying extension ratio, another parameter fis introduced. The modified extension ratio is related to f and extension ratio. The results at minimum mean misfit for all the nine rocks indicate that the range of f varies from 0.7 to 1.0. It is found that mean misfit by considering varying extension ratio is lower than that in the first analysis. But it requires three parameters. A statistical expression between f and Hoek-Brown strength parameters has been established. Though coefficient of correlation is not reasonable, we may eliminate it as an extra parameter. At the end of the paper, a methodology has also been given for its application to isotropic jointed rock mass, so that it can be implemented in a numerical code for stability analysis of jointed rock mass structures.     

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 σ_2.Many evidences have demonstrated that the rock strength is dependent on σ_2. Thus it is necessary to extend the HB criterion into a three-dimensional(3D) form.In this study,the effect of σ_2 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.     

Three independent parameters to describe conventional triaxial compressive strength of intact rocks

The strengths of 12 rocks cited from literatures increase in a nonlinear way with increasing confining pressure against the Coulomb criterion. The criteria with power forms like the generalized Hoek-Brown criterion are not available for describing the strength properties in the whole test range for Indiana limestone, Yamaguchi marble and Vosges sandstone, of which the differential stresses are approximately constant at high confining pressures. The exponential criterion with three parameters fits the test data of those 12 rocks well with a low misfit. The three parameters are independent of the uniaxial compressive strength (UCS), the initial increasing rate of strength with confining pressure, and the limitation of differential stress.     

Bayesian data analysis to quantify the uncertainty of intact rock strength

One of the main difficulties in the geotechnical design process lies in dealing with uncertainty. Uncertainty is associated with natural variation of properties, and the imprecision and unpredictability caused by insufficient information on parameters or models. Probabilistic methods are normally used to quantify uncertainty. However, the frequentist approach commonly used for this purpose has some drawbacks.First, it lacks a formal framework for incorporating knowledge not represented by data. Second, it has limitations in providing a proper measure of the confidence of parameters inferred from data. The Bayesian approach offers a better framework for treating uncertainty in geotechnical design. The advantages of the Bayesian approach for uncertainty quantification are highlighted in this paper with the Bayesian regression analysis of laboratory test data to infer the intact rock strength parameters σ_(ci) and m_i used in the Hoek-Brown strength criterion. Two case examples are used to illustrate different aspects of the Bayesian methodology and to contrast the approach with a frequentist approach represented by the nonlinear least squares(NLLS) method. The paper discusses the use of a Student's t-distribution versus a normal distribution to handle outliers, the consideration of absolute versus relative residuals, and the comparison of quality of fitting results based on standard errors and Bayes factors. Uncertainty quantification with confidence and prediction intervals of the frequentist approach is compared with that based on scatter plots and bands of fitted envelopes of the Bayesian approach. Finally, the Bayesian method is extended to consider two improvements of the fitting analysis. The first is the case in which the Hoek-Brown parameter, a, is treated as a variable to improve the fitting in the triaxial region. The second is the incorporation of the uncertainty in the estimation of the direct tensile strength from Brazilian test results within the overall evaluation of the intact rock strength.     

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.     

Basic characteristics and development of yield criteria for geomaterials

The yield criteria of geomaterials play a crucial role in studying and designing the strength of materials and structures.The basic characteristics of yield criteria for geomaterials need to be studied under the framework of continuum mechanics.These characteristics include the effects of strength difference（SD） of materials in tension and compression,normal stress,intermediate principal stress,intermediate principal shear stress,hydrostatic stress,twin-shear stresses,and the convexity of yield surface.Most of the proposed yield criteria possess only one or some of these basic characteristics.For example,the Tresca yield criterion considers only single-shear stress effect,and ignores the effect of SD,normal stress,intermediate principal stress,intermediate principal shear stress,hydrostatic stress,and twin-shear stresses.The Mohr-Coulomb yield criterion accounts for the effect of SD,normal stress,single-shear stress and hydrostatic stress,but disregards the effect of intermediate principal stress,intermediate principal shear stress,and twin-shear stresses.The basic characteristics remain to be fully addressed in the development of yield criterion.In this paper,we propose a new yield criterion with three features,that is,newly developed,better than existing criteria and ready for application.It is shown that the proposed criterion performs better than the existing ones and is ready for application.The development of mechanical models for various yield criteria and the applications of the unified strength theory to engineering are also summarized.According to a new tetragonal mechanical model,a tension-cut condition is added to the unified strength theory.The unified strength theory is extended to the tension-tension region.     

Mechanical behavior of sandy facies of Opalinus Clay under different loadconditions

The mechanical behavior of sandy facies of Opalinus Clay at the Mont Terri underground rock laboratory(URL)in Switzerland was investigated with drained and undrained triaxial compression and extension,cyclic compression,and creep tests.Samples were taken from boreholes drilled parallel to bedding.Most of the samples were reconditioned to minimize sampling effects of desaturation and micro-cracking.The compression was accomplished by increasing axial stress at constant radial stress.The extension was carried out by increasing radial stress at constant axial stress.Moreover,extension was also achieved by simultaneously increasing radial stress and decreasing axial stress under constant mean stress.The test results showed elastoplastic stress-strain behavior with volumetric compaction until onset of dilatancy at high deviatoric stresses above 80%-90%of the peak failure strength.The strength is dependent upon load path and mean stress.The strength under triaxial compression is higher than that under extension.The respective strength increases with increasing mean stress.Desaturation enhances the stiffness and strength of the claystone.The deformation and strength of the elaystone are time-dependent.Under constant deviatoric stress,the claystone crept continuously with time,which can be characterized by a transient phase and a following stationary phase,and even a tertiary phase at high deviatoric stresses to rupture.     

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) hybrid finite-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 calibrated against standard laboratory experiments, including Brazilian disc test and uniaxial compression test. The effect of increasing confining pressure on the mechanical response and fracture propagation of the rock was quantified under triaxial compression tests. Polyaxial(or true triaxial) simulations highlighted the effect of the intermediate principal stress(s2) on fracture directions in the model: as the intermediate principal stress increased, fractures tended to align in the direction parallel to the plane defined by the major and intermediate principal stresses. The peak strength was also shown to vary with changing σ2.     

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.     

ANALYSIS OF SIZE EFFECT，SHEAR DEFORMATION AND DILATION IN DIRECT SHEAR TEST BASED ON GRADIENT- DEPENDENT PLASTICITY

1 INTRODUCTION The direct shear test is a widely used soil or rock test that provides essential design data in stability analysis of slope,foundation and rockburst and so on. The test is inevitably subject to criticisms because of the non-uniformity of stress and strains,which may facilitate the occurrence of progressive failure along the potential shear plane[1]. However,the test is the standard method for the measurement of plane strain strength parameters. To obtain a full understan…     

A simplified approach to directly consider intact rock anisotropy inHoekeBrown failure criterion

Many rock types have naturally occurring inherent anisotropic planes, such as bedding planes, foliation,or flow structures. Such characteristic induces directional features and anisotropy in rocks＇ strength anddeformational properties. The HoekeBrown （HeB） failure criterion is an empirical strength criterionwidely applied to rock mechanics and engineering. A direct modification to HeB failure criterion toaccount for rock anisotropy is considered as the base of the research. Such modification introduced a newdefinition of the anisotropy as direct parameter named the anisotropic parameter （Kb）. However, thecomputation of this parameter takes much experimental work and cannot be calculated in a simple way.The aim of this paper is to study the trend of the relation between the degree of anisotropy （Rc） and theminimum value of anisotropic parameter （Kmin）, and to predict the Kmin directly from the uniaxialcompression tests instead of triaxial tests, and also to decrease the amount of experimental work.
2014 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting byElsevier B.V. All rights reserved.     

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(σ_2) and minor(σ_3) principal stresses in place of σ_3 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.     

Excavation-induced deep hard rock fracturing: Methodology and applications

To analyze and predict the mechanical behaviors of deep hard rocks,some key issues concerning rock fracturing mechanics for deep hard rock excavations are discussed.First,a series of apparatuses and methods have been developed to test the mechanical properties and fracturing behaviors of hard rocks under high true triaxial stress paths.Evolution mechanisms of stress-induced disasters in deep hard rock excavations,such as spalling,deep cracking,massive roof collapse,large deformation and rockbursts,have been recognized.The analytical theory for the fracturing process of hard rock masses,including the three-dimensional failure criterion,stress-induced mechanical model,fracturing degree index,energy release index and numerical method,has been established.The cracking-restraint method is developed for mitigating or controlling rock spalling,deep cracking and massive collapse of deep hard rocks.An energy-controlled method is also proposed for the prevention of rockbursts.Finally,two typical cases are used to illustrate the application of the proposed methodology in the Baihetan caverns and Bayu tunnels of China.     

A modified maximum tangential tensile stress criterion for three-dimensional crack propagation

The three-dimensional(3D) crack propagation is a hot issue in rock mechanics.To properly simulate 3D crack propagation,a modified maximum tangential tensile stress criterion is proposed.In this modified criterion,it is supposed that cracks propagate only at crack front in the principal normal plane.The tangential tensile stress at crack front in the principal normal plane in local coordinates is employed to determine crack propagation,which is calculated through coordinate transformation from global to local coordinates.New cracks will propagate when the maximum tangential tensile stress at crack front in the principal normal plane reaches the tensile strength of rock-like materials.Compared with the previous crack propagation criteria,the modified crack propagation criterion is helpful in calculating 3D crack stress intensity factor,and can overcome the limitations of propagation step determined by individual experiences in previous studies.Finally,the 3D crack propagation process is traced by element-free Galerkin method.The numerical results agree well with the experimental ones for a frozen resin sample with prefabricated 3D cracks.     

Shear strength of an unsaturated weakly expansive soil

To study the weakly expansive clay obtained from a slope along Wuhan—Shiyan expressway in Hubei Province,soil-water property tests and some unsaturated triaxial tests with suction control were conducted,and the soil-water retention curve（SWRC） and unsaturated shear strength of this soil were obtained.Results show that the air-entry suction and the residual degree of saturation of the tested soil are 106 kPa and 8%,respectively.The boundary effect zone and the transition zone can be identified on the desorption curve,but the residual zone is not so obvious.The unsaturated shear strength increases as suction increases within the range of controlled suction in the test,and friction angle,b,in the triaxial shear test is 17.6°.Based on the results,constitutive models for predicting the unsaturated shear strength using the SWRC were evaluated,and comparisons between prediction and measurement were made.It is concluded that for engineering purpose,the constitutive model should be carefully selected based on soil properties when predicting the unsaturated shear strength using the SWRC.     

Effect of insoluble materials on the volumetric behavior of rock salt

This paper focuses on the presence of nodules of insoluble materials within salt specimens,and their effect on the volumetric strain measurements and the dilatancy phenomenon.We analyzed experimental results of over 120 conventional triaxial compression tests,and found that in 20%of the cases,the volumetric strain measurements were atypical.We also noted that the natural variability of the specimens can lead to a non-negligible data scattering in the volumetric strain measurements when different specimens are subjected to the same test.This is expected given the small magnitude of those strains,but it occasionally implies that the corresponding specimens are not representative of the volumetric behavior of the studied rock.In order to understand these results,we numerically investigated salt specimens modeled as halite matrices with inclusions of impurities.Simulations of triaxial compression tests on these structures proved that such heterogeneities can induce dilatancy,and their presence can lead to the appearance of tensile zones which is physically translated into a micro-cracking activity.The modeling approach is validated as the patterns displayed in the numerical results are identical to that in the laboratory.It was then employed to explain the observed irregularities in experimental results.We studied the natural variability effect as well and proposed a methodology to overcome the issue of specimen representativity from both deviatoric and volumetric perspectives.     

Fracture initiation and propagation in intact rock—A review

The initiation and propagation of failure in intact rock are a matter of fundamental importance in rock engineering. At low confining pressures, tensile fracturing initiates in samples at 40%-60% of the uniaxial compressive strength and as loading continues, and these tensile fractures increase in density, ultimately coalescing and leading to strain localization and macro-scale shear failure of the samples. The Griffith theory of brittle failure provides a simplified model and a useful basis for discussion of this process. The Hoek-Brown failure criterion provides an acceptable estimate of the peak strength for shear failure but a cutoff has been added for tensile conditions. However, neither of these criteria adequately explains the progressive coalition of tensile cracks and the final shearing of the specimens at higher confining stresses. Grain-based numerical models, in which the grain size distributions as well as the physical properties of the component grains of the rock are incorporated, have proved to be very useful in studying these more complex fracture processes.     

基于非线性破坏包络线的三维边坡稳定性分析

The effects of nonlinearity of strength envelopes on 3D slope stability analysis are investigated.A power relation for the nonlinear envelope is employed to derive the 3D factor of safety equations of an extended Spencer method hich satisfies boty force equilibrium and moment equilibrium.Then,a search procedure is presented based on dynamic programming to determine the 3D critical slip surface for a general slope,Linear and nonlinear strength envelopes used for slope stability computations are obtained by fitting curves to the 103 strength data of consolidated-undrained(CU)triaxial compression tests for compacted Israeli clay.Results of a typical 3D problem show that a linear approximation of the nonlinear strength envelope may lead to a significant overestimation of calculated safety factors.