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.     

Strength and deformation properties of frozen sand under a true triaxial stress condition

In recent years, the mechanical properties of frozen soils under complex stress states have attracted significant attention; however, limited by the test apparatus, true triaxial tests on frozen soils have rarely been conducted. To study the strength and deformation properties of frozen sand under a true triaxial stress state, a novel frozen soil testing system, i.e., a true triaxial apparatus, was developed. The apparatus is mainly composed of a temperature control system, a servo host system, a hydraulic servo loading system, and a digital control system. Several true triaxial tests were conducted at a constant minor principal stress (σ₃) and constant intermediate principal stress ratio (b) to study the effect of intermediate principal stress (σ₂) on the mechanical properties of frozen sand. The test results showed that the stress–strain curve can be mainly divided into three stages, with evidence of strain hardening characteristics. The strength, elastic modulus, and friction angle increased with the increase in b from 0 to 0.6, but decreased when increasing b from 0.6 to 1, whereas the cohesion varied little with the variation in b. The deformation in the direction of σ₂ changed from dilative to compressive and that in the direction of σ₃ remained dilative throughout.     

Strength and failure characteristics of jointed marble under true triaxial compression

The rock structure and three-dimensional stress state play a vital role in the mechanical behaviour of rock masses. Here, a series of true triaxial compression tests (σ₁ > σ₂ > σ₃) are conducted on jointed marble (50 × 50 × 100 mm³) containing a natural stiff joint, taken from the China Jinping Underground Laboratory (CJPL-II) project. The purposes of this study are to investigate the joint effect and estimate the stress dependency of jointed marble. The test results show that jointed marble can fail in four distinct forms, namely, splitting or shearing of intact marble, opening of the joint or sliding along the joint, and these failure modes are influenced by the joint configuration and the minimum and intermediate principal stresses. Generally, jointed marble has more brittle post-peak behaviour than intact marble. The linear Mogi-Coulomb failure criterion can be modified to describe the strength of the jointed marble under true triaxial compression. The jointed marble strength is more sensitive to the minimum principal stress than to the intermediate principal stress. A maximum decline of 25% in strength is observed, which corresponds to a joint dip angle of 60° at σ₂ = 60 MPa and σ₃ = 30 MPa. The link between the experimental results and in situ fracturing at CJPL-II is also demonstrated.

     

Deformation and failure characteristics and fracture evolution of cryptocrystalline basalt

Cryptocrystalline basalt is one of the two major types of rocks exposed in the super large-scale underground powerhouse in Baihetan hydropower station in China.The rock of this type shows various sitespecific mechanical responses(e.g.fragmentation,fracturing,and relaxation) during excavation.Using conventional triaxial testing facility MTS 815.03,we obtained the stress-strain curves,macroscopic failure characteristics,and strength characteristics of cryptocrystalline basalt.On this basis,evolution of crack initiation and propagation was explored using the finite-discrete element method(FDEM) to understand the failure mechanism of cryptocrystalline basalt.The test results showed that:(1) under different confining stresses,almost all the pre-peak stress-strain curves of cryptocrystalline basalt were linear and the post-peak stresses decreased rapidly;(2) the cryptocrystalline basalt showed a failure mode in a form of fragmentation under low and medium confining stresses while fragmentation-shear coupling failure dominated at high confining stresses;and(3) the initial strength ratio(σ_(ci)/σ_f,where σ_(ci)and σ_f are the crack initiation strength and peak strength,respectively) ranged from 0.45 to 0.55 and the damage strength ratio(σ_(cd)/σ_f,where σ_(cd) is the crack damage strength) exceeded 0.9.The stress—strain curve characteristics and failure modes of cryptocrystalline basalt could be reflected numerically.For this,FDEM simulation was employed to reveal the characteristics of cryptocrystalline basalt,including high σ_(cd)/σ_f values and rapid failure after σ_(cd),with respect to the microscopic characteristics of mineral structures.The results showed that the fragmentation characteristics of cryptocrystalline basalt were closely related to the development of tensile cracks in rock samples prior to failure.Moreover,the decrease in degree of fragmentation with increasing confining stress was also correlated with the dominant effect of confining stress on the tensile cracks.     

Shear banding in torsion shear tests on cross-anisotropic deposits of fine Nevada sand

A series of torsion shear experiments was performed on large hollow cylinder specimens of Fine Nevada sand with major principal stress directions relative to vertical, α, varying from 0° to 90° and with the intermediate principal stress, σ₂, varying from σ₃ to σ₁ as indicated by b=(σ₂–σ₃)/(σ₁–σ₃). The Fine Nevada sand was deposited by dry pluviation, thus producing a sand fabric with horizontal bedding planes and cross-anisotropic characteristics. The various stress conditions were achieved by varying the pressures inside and outside the hollow cylinder specimen relative to the shear stress and the vertical deviator stress according to a pre-calculated pattern. All stresses and all strains were determined from careful measurements so that analysis of the soil behavior could be made reliably. The soil behavior was determined for a pattern of combinations of α varying with increments of 22.5° from 0° to 90° and b varying with increments of 0.25 from 0.0 to 1.0. Thus, 25 test locations were established, but many tests were repeated to study the consistency of the results. The friction angles varied considerably with α and b, thus indicating the importance of the intermediate principal stress and the principal stress directions relative to the horizontal bedding planes. The observed shear bands essentially followed the expected directions, but due to the cross-anisotropy shear bands were also observed in the direction of the major principal stress in regions with high b-values. The strength variation was also influenced by the flexibility of the boundaries in these regions.     

Effects of principal stress rotation on stress–strain behaviors of saturated clay under traffic–load–induced stress path

A series of cyclic torsional shear tests using hollow cylinder apparatus (HCA) were performed to investigate the effect of principal stress rotation (PSR) on the stress–strain behaviors of saturated soft clay. The traffic–load–induced shear stress path was used in the cyclic test and the investigation mainly concerned the influence of PSR on the shear stiffness and non-coaxiality. It indicated that the effects of PSR substantially depends on the magnitude of deviatoric stress (q?=?{[(σ₁???σ₂)²?+?(σ₂???σ₃)²?+?(σ₁???σ₃)²]/2}^1/2) as well as the intermediate principal stress ratio (b?=?(σ₂???σ₃)/(σ₁???σ₃)). At low deviatoric stress, the trajectory envelope of deviatoric strain path translates with a nearly constant size, showing constant shear stiffness and strong non-coaxiality. However, at high deviatoric stress, the trajectory envelope of deviatoric strain rapidly expands towards instability, showing degenerating shear stiffness and weak non-coaxiality. Moreover, the excess pore water pressure increases and the shear stiffness decreases more rapidly as b value increases. The results can provide an experimental basis for constitutive modelling of clays under traffic–induced loadings.     

Effect of shear displacement on the hydraulic conductivity of a fracture

The effect of shear displacement inclined relative to macroscopic water flow on the hydraulic conductivity of a rock fracture was estimated, using synthetic fractures that reproduce a tensile fracture in granite. The results showed that the hydraulic aperture normalized by the mean aperture increased with the angle between the directions of shear displacement and macroscopic water flow, according to a sinusoidal function of twice the angle. Formulae were established to estimate the hydraulic aperture of the fracture as a function of the mean aperture, the standard deviation of the initial aperture, the shear displacement, and the angle between the shear displacement and macroscopic water flow, based on results obtained in both this work and previous work, but neglecting scale effects. By assuming the mechanical properties of the fracture based on experimental results for granite, but neglecting scale effects, the hydraulic conductivity of the fracture with an arbitrary direction under a given state of stress (σ₁=29 MPa, σ₂=25 MPa and σ₃=13.5 MPa) was estimated for macroscopic water flow in the directions of both σ₁ and σ₂. When the contour map of the transmissivity of the fracture is plotted on a stereonet of the normal direction of the fracture in the principal axes of stress, there is a ridge (line of the local maximum) of transmissivity in the circumferential direction, and the inclination angle of the ridge from the σ₃-axis decreases with shear displacement, since shear dilation increases with both a decrease in normal stress and an increase in shear displacement. Furthermore, for the condition of stress given in this study, the transmissivity for macroscopic water flow in the direction of σ₁ is maximum for a fracture with a normal direction within the σ₂–σ₃ plane, while that in the direction of σ₂ is maximum for a fracture with a normal direction within the σ₁–σ₃ plane.     

True-triaxial strength criteria for rock

This paper reviews some strength criteria which include the role of the intermediate principal stress, and proposes a new criterion. Strength criteria of the form σ_oct=f_N(σ_oct), such as Drucker–Prager, represent a rotation surface in the principal stress space, symmetric to the line σ₁=σ₂=σ₃ in the meridian plane. Because σ_oct=f_N(σ_oct) must fit the pseudo-triaxial compressive strength, it will have a non-physical outcome for triaxial extension. Mogi's criteria, σ_oct=g₁(σ_m,2) and σ_max=g₂(σ_β) are able to fit experimental data reasonably well, but the prediction of strength is not good and sometimes problematic. Strength criterion with the form λ(σ₁, σ₂, σ₃)=F[η(σ₁, σ₂, σ₃)], or a curve of two variables which can be decided by fitting pseudo-triaxial experimental data, is not expected to describe the strength under various stress states, no matter how high the correlation coefficient of λ and η is, or how low the misfit of the equation λ=F(η) is, as these seemingly good correlations usually result from the dominant influence of the maximum principal stress in the metrics of λ and η. The intermediate principal stress may improve the strength of rock specimen, but its influence will be restricted by σ₃. Also when σ₂ is high enough to cause failure in the σ₂–σ₃ direction, the strength will decrease with the increasing σ₂. The new strength criterion with exponent form has just such a character, and gives much lower misfits than do all seven criteria discussed by Colmenares and Zoback [Colmenares LB, Zoback MD. A statistical evaluation of intact rock failure criteria constrained by polyaxial test data for five different rocks. Int J Rock Mech Min Sci 2002;39:695–729].     

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

There are many methods to construct true triaxial strength criteria for rocks.Jaiswal and Shrivastva(2012)proposed a strength criterion,named J–S criterion,in the deviatoric plane,which provides nearly the same misfts for true triaxial test data as the exponential criterion.It is diffcult to calculate the strength at given2and3using the J–S criterion,and the multiple solutions to the nonlinear equation may induce confusion and mistake.Strength envelopes in deviatoric planes are not geometric similar;therefore,true triaxial test data cannot be grouped in the mean stress to check strength criteria in the deviatoric plane.     

Experimental study on strength and deformation of plain concrete under triaxial compression after freeze-thaw cycles

The behavior of strength and deformation of plain concrete under triaxial compression after 0, 25, 50 and 75 cycles of freeze-thaw are experimentally studied using the static and dynamic triaxial experimental machine. The compressive strength, strain at the peak of stress and stress–strain relationship under triaxial compression were measured. The failure modes of concrete specimens are also described. The experimental results showed that the triaxial compressive strength decreased as the freeze-thaw cycles were repeated for plain concrete. The influence of the number of freeze-thaw cycles and the stress ratio on the principal compressive stresses_σ3${\sigma }_3$and corresponding strain_ε3${\epsilon }_3$, stress–strain relationship was analyzed. On this basis, the failure criterion of concrete under triaxial compression after freeze-thaw cycles is proposed. It can serve as a reference for the maintenance, design and the life prediction of ocean structures, hydraulic structures, marine structures and offshore platform in cold regions.     

Mechanical characteristics of the exponential strength criterion under conventional triaxial stresses

The minor principal stress is the key factor influencing the mechanical property of rock, and the strength criterion of conventional triaxial stresses is the basis of any true triaxial criterion. This paper examines the Coulomb criterion, Hoek–Brown criterion, the generalized Hoek–Brown criterion, and the exponential criterion proposed by the author. The procedure to determine the parameters in criteria is studied. Parameters in the generalized Hoek–Brown criterion are sensitive to the power n, thus Coulomb criterion with n=1.0 and the original form of Hoek–Brown criterion with n=0.5, especially that determined by linear regression, are not the best choice for the fitting solution of test data. The exponential criterion determined on the least absolute deviation will approximate a great number of normal test data, and expose the oddity data. It can describe the relation between strength and confining pressure in the entire stress range, with low misfit. The uniaxial compressive strengths predicted by the exponential criterion are nearly the same as the real magnitudes for all rocks. The deviator of strengths under conventional triaxial extension and compression is larger than half of the maximum strength increase from the intermediate principal stress. Therefore, the parameters in the exponential criterion may be completely determined from the strengths under conventional triaxial compression and extension.     

Dynamic characterization of EPS geofoam

This paper attempts to describe the dynamic behavior of expanded polystyrene EPS geofoam, and shows the dependence of shear modulus, G, and damping ratio, λ, on shear strain, γ, density, ρ, and confining stress, σ₃, through the results of a series of resonant column and strain- and stress-controlled cyclic compression tests. Shear modulus and damping ratio versus shear strain curves were obtained and a series of equations were developed to model the dynamic behavior of EPS. From stress-controlled cyclic compression tests the effect of the number of cyclic load applications, N, on the maximum axial strain ?_max (for a specific static deviator stress, σ_e, plus the amplitude of the loading cyclic stress, σ_c) and on the dynamic modulus of elasticity E_dyn was evaluated as a function of the EPS density, confining stress, and the applied cyclic stress amplitude σ_c.     

Effect of shearing history on stress wave velocities of sands observed in triaxial compression tests

For the accurate design of geotechnical structures subjected to static and dynamic loadings, precise estimation of elastic wave velocities and hence, small strain stiffness of soil is essential. However, the interpretation of elastic wave velocities propagating in deformed/sheared soil has not been comprehensively explored. In this research, shear (V_s) and compression wave velocity (V_p) measurements have been undertaken on three kinds of uniformly graded sands during drained triaxial compression by keeping minor principal stress constant. Planar piezoelectric transducers have been used to overcome the limitations associated with the more commonly used bender elements, such as distortion of transducers during specimen shearing. This technical note reveals that the increase of major principal stress in the wave propagation direction has a more significant influence on V_p in comparison to V_s. The axial strain (ε_a) at which peak V_s is noted is comparable to the ε_a at which specimen dilation or phase transformation takes place. The V_s values show a substantial drop after phase transformation, although there is an increase in the mean stress level. However, V_p increases even after specimen dilation takes place, and it is the major principal stress that dictates its evolution during triaxial compression. Moreover, for a given material and initial stress level, elastic wave velocities of specimens prepared at different initial densities approach one another during shearing and later merge at a large ε_a.     

Non-dilatant deformation and failure mechanism in two Long Valley Caldera rocks under true triaxial compression

We conducted laboratory rock strength experiments in two ultra-fine-grained brittle rocks, hornfels and metapelite, which together are the major constituent of the Long Valley Caldera (California, USA) basement in the 2025–2996 m depth range. Both rocks are banded, and have very low porosity. Uniaxial compression tests at different orientations with respect to banding planes reveal that while the hornfels compressive strength is nearly isotropic, the metapelite possesses distinct anisotropy. Conventional triaxial tests in these rocks reveal that their respective strengths in a specific orientation increase approximately linearly with confining pressure. True triaxial compression experiments in specimens oriented at a consistent angle to banding, in which the magnitudes of the least (σ₃) and the intermediate (σ₂) principal stresses are different but kept constant during testing while the maximum principal stress is increased until failure, exhibit a behavior unlike that previously observed in other rocks under similar testing conditions. For a given magnitude of σ₃, compressive strength σ₁ does not vary significantly in both Long Valley rock types, regardless of the applied σ₂, suggesting little or no intermediate principal stress effect. Strains measured in all three principal directions during loading were used to obtain plots of σ₁ versus volumetric strain. These are consistently linear almost to the point of rock failure, suggesting no dilatancy. The phenomenon was corroborated by SEM inspection of failed specimens that showed no microcrack development prior to the emergence of one through-going shear failure plane steeply dipping in the σ₃ direction. The strong dependency of compressive strength on the intermediate principal stress in other crystalline rocks was found to be related to microcrack initiation upon dilatancy onset, which rises with increased σ₂ and retards the failure process. We infer that strength independence of σ₂ in the Long Valley rocks derives directly from their non-dilatant deformation.     

指数型岩石真三轴强度准则

岩石强度是工程岩体稳定性评价和结构优化设计的前提和基础,通过岩石真三轴试验分析了岩石强度演变特征:(1)随着最小主应力的增加,岩石强度逐渐增大,但增幅逐渐减小并趋于零;(2)岩石广义压拉强度比随静水压力的增大呈先增大再减小的规律,并最终趋于1,即在π平面上呈三角形向圆形转变的过程;(3)岩石强度随中间主应力增大表现为先增大再减小的过程。三参数型指数函数完全符合岩石强度在子午面上的基本特点,而L_(WW),L_(MN)和L_(YMH) 3个罗德函数准确反映了中间主应力对岩石强度的影响,且无条件满足拉压子午面区间光滑、连续、外凸性要求。利用π平面广义拉压强度比和罗德函数将指数型拉压子午面结合,构建了指数型真三轴强度准则,分析了强度参数对岩石强度的影响及其空间包络特征。最后采用14种岩石的真三轴试验数据对指数型真三轴强度准则进行了最优拟合误差分析,结果表明指数型真三轴强度准则均具有良好的拟合精度,可正确描述软–硬不同性质岩石的强度特征。     

统一强度理论应用于岩石的讨论

 具有多种形式的双剪理论已得到广泛的研究和应用,但没有基于试验数据的拟合与其他真三轴强度准则进行直接比较。以主应力形式直接构建线性和非线性统一强度理论,利用岩石的真三轴压缩、常规三轴压缩和伸长的试验结果,确定多个强度准则中的材料参数。统一强度理论对试验数据的拟合偏差较大,不能描述岩石常规三轴压缩和伸长强度随最小主应力的非线性增加,难以描述强度随中间主应力的变化趋势;而非线性理论使用2个隐式函数分段表示,实际计算时需进行多次比较判断。在最小主应力较低时,统一强度理论给出的岩石强度显著偏高,似乎不宜应用于孔壁崩落确定地应力、钻孔稳定性分析等。     

基于非共轴本构模型的砂土真三轴试验分叉分析

现有的弹塑性本构模型多数是基于共轴理论建立的,且只在三轴压缩应力条件下得到验证,因而不能合理描述真三轴状态下砂土的力学特性。为此,改进三维本构模型屈服函数的π面形态,运用非共轴分叉理论对真三轴状态下砂土的分叉强度与分叉前的应力应变关系进行分析。理论分析表明,真三轴条件下中主应力对分叉前应力应变曲线及分叉特性有着显著的影响,其中中主应力比超过0.2时应变局部化产生并决定了土体的峰值强度。真三轴试验结果与传统的共轴分叉理论的预测结果进行对比分析,验证了非共轴分叉理论预测的合理性。     

Space decomposition based parallelization solutions for the combinedfiniteediscrete element method in 2D

The combined finiteediscrete element method （FDEM） belongs to a family of methods of computationalmechanics of discontinua. The method is suitable for problems of discontinua, where particles aredeformable and can fracture or fragment. The applications of FDEM have spread over a number of disciplinesincluding rock mechanics, where problems like mining, mineral processing or rock blasting canbe solved by employing FDEM. In this work, a novel approach for the parallelization of two-dimensional（2D） FDEM aiming at clusters and desktop computers is developed. Dynamic domain decompositionbased parallelization solvers covering all aspects of FDEM have been developed. These have beenimplemented into the open source Y2D software package and have been tested on a PC cluster. Theoverall performance and scalability of the parallel code have been studied using numerical examples. Theresults obtained confirm the suitability of the parallel implementation for solving large scale problems.
2014 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting byElsevier B.V. All rights reserved.     

统一强度理论的试验数据拟合及评价

 基于中间主应力对岩石强度影响,统一强度理论可作为Coulomb准则的推广,并已得到广泛的研究和使用。利用已有的8种岩石真三轴试验结果,研究强度准则参数的确定方法以及相应的拟合误差。依据常规三轴压缩强度与围压的关系确定Q和K,再依据真三轴的强度确定参数b,统一强度理论所给出的双折线难以描述强度变化的整体趋势,对部分岩石试验数据的拟合误差超过Coulomb准则;如果以最小拟合误差为目标,对参数Q,K和b进行协同搜索,那么统一强度理论可以有较小的拟合误差,但其给出的部分岩石常规三轴压缩强度明显偏离实际结果。此外,统一强度理论认为岩石三轴伸长强度与常规三轴压缩强度相同,在最小主应力恒定时中间主应力引起的强度最大增加值随最小主应力线性增加,且强度关于中间主应力的一阶导数在峰值点是不连续的,这些都与试验结果不能完全协调。由于真三轴压缩试验结果还较少,中间主应力的试验范围也较小,而数据的离散性又较大,以最小拟合误差确定的强度准则能否适用于实际工程的分析计算,需要更深入的研究。     

Risk of shear failure and extensional failure around over-stressed excavations in brittle rock

The authors investigate the failure modes surrounding over-stressed tunnels in rock.Three lines of investigation are employed:failure in over-stressed three-dimensional(3D) models of tunnels bored under 3D stress,failure modes in two-dimensional(2D) numerical simulations of 1000 m and 2000 m deep tunnels using FRACOD,both in intact rock and in rock masses with one or two joint sets,and finally,observations in TBM(tunnel boring machine) tunnels in hard and medium hard massive rocks.The reason for 'stress-induced' failure to initiate,when the assumed maximum tangential stress is approximately(0.4-0.5)σ_c(UCS,uniaxial compressive strength) in massive rock,is now known to be due to exceedance of a critical extensional strain which is generated by a Poisson's ratio effect.However,because similar 'stress/strength' failure limits are found in mining,nuclear waste research excavations,and deep road tunnels in Norway,one is easily misled into thinking of compressive stress induced failure.Because of this,the empirical SRF(stress reduction factor in the Q-system) is set to accelerate as the estimated ratio σ_(θmax)/σ_c 0.4.In mining,similar 'stress/strength' ratios are used to suggest depth of break-out.The reality behind the fracture initiation stress/strength ratio of '0.4' is actually because of combinations of familiar tensile and compressive strength ratios(such as 10) with Poisson's ratio(say0.25).We exceed the extensional strain limits and start to see acoustic emission(AE) when tangential stress σθ≈ 0.4σc,due to simple arithmetic.The combination of 2D theoretical FRACOD models and actual tunnelling suggests frequent initiation of failure by 'stable' extensional strain fracturing,but propagation in 'unstable' and therefore dynamic shearing.In the case of very deep tunnels(and 3D physical simulations),compressive stresses may be too high for extensional strain fracturing,and shearing will dominate,both ahead of the face and following the face.When shallower,the concept of 'extensional strain initiation but propagation' in shear is suggested.The various failure modes are richly illustrated,and the inability of conventional continuum modelling is emphasized,unless cohesion weakening and friction mobilization at different strain levels are used to reach a pseudo state of yield,but still considering a continuum.