Generalized crack initiation and crack damage stress thresholds of brittle rock masses near underground excavations

The rock mass failure process is characterized by several distinct deformation stages which include crack initiation, crack propagation and coalescence. It is important to know the stress levels associated with these deformation stages for engineering design and practice.Extensive theoretical, experimental and numerical studies on the failure process of intact rocks exist. It is generally understood that crack initiation starts at 0.3 to 0.5 times the peak uniaxial compressive stress. In confined conditions, the constant-deviatoric stress criterion was found to describe the crack initiation stress level.Here, generalized crack initiation and crack damage thresholds of rock masses are proposed. The crack initiation threshold is defined by σ₁−σ₃=A σ_cm and the crack damage threshold is defined by σ₁−σ₃=B σ_cm for jointed rock masses, where A and B are material constants and σ_cm is the uniaxial compressive strength of the rock masses. For a massive rock mass without joints, σ_cm is equal to σ_cd, the long-term uniaxial strength of intact rock. After examining data from intact rocks and jointed rock masses, it was found that for massive to moderately jointed rock masses, the material constants A and B are in the range of 0.4 to 0.5, 0.8 to 0.9, respectively, and for moderately to highly jointed rock masses, A and B are in the range of 0.5 to 0.6, 0.9 to 1.0, respectively. The generalized crack initiation and crack damage thresholds, when combined with simple linear elastic stress analysis, assist in assessing the rock mass integrity in low confinement conditions, greatly reducing the effort needed to obtain the required material constants for engineering design of underground excavations.     

A modified Hoek–Brown failure criterion for anisotropic intact rock

The Hoek–Brown criterion parameters (σ_ci, m_i and s) are significantly influenced by the strength anisotropy of intact rock. In the present study, the criterion was modified by incorporating a new parameter (k_β) to account for the effect of strength anisotropy, thus being able to determine the strength of intact anisotropic rock under loading in different orientations of the plane of anisotropy. The range of the parameter (k_β) for the rocks tested has been analytically investigated by carrying out triaxial tests, in different orientations of the foliation plane. The proposed modification was studied for metamorphic rocks (gneiss, schist, marble), but could also be applied to other rock types exhibiting “inherent” anisotropy, e.g. sedimentary as well as igneous rocks. The proposed modified criterion is intended for use for prediction of strength of intact rock, but can also be extended to rock masses.     

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.     

Influence of intermediate principal stress on rock fracturing and strength near excavation boundaries—Insight from numerical modeling

The influence of the intermediate principal stress on rock fracturing and strength near excavation boundaries is studied using a FEM/DEM combined numerical tool. A loading condition of σ₃=0 and σ₁≠0, and σ₂≠0 exists at the tunnel boundary, where σ₁, σ₂, and σ₃, are the maximum, intermediate, and minimum principal stress components, respectively. The numerical study is based on sample loading testing that follows this type of boundary stress condition. It is seen from the simulation results that the generation of tunnel surface parallel fractures and microcracks is attributed to material heterogeneity and the existence of relatively high intermediate principal stress (σ₂), as well as zero to low minimum principal stress (σ₃) confinement. A high intermediate principal stress confines the rock in such a way that microcracks and fractures can only be developed in the direction parallel to σ₁ and σ₂. Stress-induced fracturing and microcracking in this fashion can lead to onion-skin fractures, spalling, and slabbing in shallow ground near the opening and surface parallel microcracks further away from the opening, leading to anisotropic behavior of the rock. Hence, consideration of the effect of the intermediate principal stress on rock behavior should focus on the stress-induced anisotropic strength and deformation behavior of the rocks. It is also found that the intermediate principal stress has limited influence on the peak strength of the rock near the excavation boundary.     

A statistical evaluation of intact rock failure criteria constrained by polyaxial test data for five different rocks

In this study we examine seven different failure criteria by comparing them to published polyaxial test data (σ₁>σ₂>σ₃) for five different rock types at a variety of stress states. We employed a grid search algorithm to find the best set of parameters that describe failure for each criterion and the associated misfits. Overall, we found that the polyaxial criteria Modified Wiebols and Cook and Modified Lade achieved a good fit to most of the test data. This is especially true for rocks with a highly σ₂-dependent failure behavior (e.g. Dunham dolomite, Solenhofen limestone). However, for some rock types (e.g. Shirahama Sandstone, Yuubari shale), the intermediate stress hardly affects failure and the Mohr–Coulomb and Hoek and Brown criteria fit these test data equally well, or even better, than the more complicated polyaxial criteria. The values of C₀ yielded by the Inscribed and the Circumscribed Drucker–Prager criteria bounded the C₀ value obtained using the Mohr–Coulomb criterion as expected. In general, the Drucker–Prager failure criterion did not accurately indicate the value of σ₁ at failure. The value of the misfits achieved with the empirical 1967 and 1971 Mogi criteria were generally in between those obtained using the triaxial and the polyaxial criteria. The disadvantage of these failure criteria is that they cannot be related to strength parameters such as C₀. We also found that if only data from triaxial tests are available, it is possible to incorporate the influence of σ₂ on failure by using a polyaxial failure criterion. The results for two out of three rocks that could be analyzed in this way were encouraging.     

Effect of intermediate principal stress on strength of anisotropic rock mass

The Mohr-Coulomb criterion needs to be modified for highly anisotropic rock material and jointed rock masses. Taking σ₂ into account, a new strength criterion is suggested because both σ₂ and σ₃ would contribute to the normal stress on the existing plane of weakness. This criterion explains the enhancement of strength (σ₂ – σ3) in the underground openings because σ₂ along the tunnel axis is not relaxed significantly. Another cause of strength enhancement is less reduction in the mass modulus in tunnels due to constrained dilatancy. Empirical correlations obtained from data from block shear tests and uniaxial jacking tests have been suggested to estimate new strength parameters. A correlation for the tensile strength of the rock mass is presented. Finally, Hoek and Brown theory is extended to account for σ₂. A common strength criterion for both supported underground openings and rock slopes is suggested.     

Shaft resistance of a pile embedded in rock

A rational calculation procedure is proposed for establishing the shaft resistance of a pile embedded in rock, based upon the Hoek and Brown failure model. The state of the art of the calculation of the pile shaft resistance is analysed. Nearly all the recommendations that have appeared in the technical literature, for calculating the ultimate shear strength of a shaft embedded in rock (τ_ult) propose that τ_ult=ασ_c^k(σ_c;τ_ulten MN/m²) where the coefficient α, considered as a constant dimensional value, ranges from 0.1 to 0.8, if the unconfined compressive strength (σ_c) is expressed in MN/m². In most cases, the exponent k is 0.5.A comparison is made between the results yielded and the different empirical theories that have been put forward with respect to this shaft resistance. It can generally be stated that the results obtained with this theory are reasonable for long and deeply socketed piles (high confining pressures) but the results are on the safe side in some cases where short piles (low confining pressures) are involved.This paper is a continuation of the works developed by the same authors with piles working at the tip, socketed in rock.     

An equivalent plasticity theory for jointed rock masses

A non-representative volume element (NRVE) approach to equivalent rock mass properties shows that the form of the elastic–plastic constitutive equations is the same for homogeneous material elements and multiple-material elements, subsequently homogenized. Thus the average stress and strain increments in an arbitrary jointed rock mass volume are related by {dσ}=([C*_ep]){dε} where σ is effective stress. The equivalent elastic-plastic properties matrix [C*_ep] is the sum of an equivalent elastic moduli matrix [C^*] and a plastic ‘correction’ matrix [C*_p, as usual. However, there are no equivalent plastic potentials Y^* or yield functions, failure criteria F^* or strengths. The equivalent elastic-plastic properties are constructed from the elastic moduli and strengths of the rock mass joints, the intact rock between and strain influence functions that relate local to overall average strains. Numerical examples that simulate laboratory-like tests on jointed rock cubes illustrate the approach.     

A quantitative comparison of strength criteria for hard rock masses

Knowledge of the rock mass strength is important for the design of all types of underground excavations. A frequently applied approach for estimation of the rock mass strength is through an empirical failure criterion, often in conjunction with rock mass classification/characterisation systems. This paper presents a review of existing methods to estimate the rock mass strength using empirical failure criteria and classification/characterisation systems—in this study, commonly denoted as estimation methods. A literature review of existing methods is presented, after which a set of methods were selected for further studies. The selected methods were used in three case studies, to investigate their robustness and quantitatively compare the advantages and disadvantages of each method. A Round Robin test was used in two of the cases. The case studies revealed that the N, Yudhbir-RMR₇₆, RMi, Q-, and Hoek–Brown-GSI methods, appeared to yield a reasonable agreement with the measured strengths. These methods are thus considered the best candidates for realistic strength estimation, provided that care is taken when choosing values for each of the included parameters in each method. This study has also clearly shown the limits of presently available strength estimation methods for rock masses and further work is required to develop more precise, practical, and easy-to-use methods for determining the rock mass strength. This should be based on the mechanical behaviour and characteristics of the rock mass, which implies that parameters that consider the strength of intact rock, block size and shape, joint strength, and physical scale, are required.     

A practical procedure for the back analysis of slope failures in closely jointed rock masses

Where closely jointed rock masses are encountered in slopes, failure can occur both through the rock mass, as a result of combination of macro and micro jointing, and through the rock substance. Determination of the strength of this category of rock mass is extraordinarily difficult since the size of representative specimens is too large for laboratory testing. This difficulty can be overcome by using a non-linear rock mass failure criterion or by back analysis of such slopes to estimate the rock mass strength. In this paper, a practical procedure and a computer program are presented for the back determination of shear strength parameters mobilized in slopes cut in closely jointed rock masses which obey a non-linear failure criterion rather than a linear one. The procedure shows that the constants to derive normal stress dependent shear strength parameters of the failed rock masses can be determined by utilizing a main cross-section and without a pre-determined value of rock mass rating (RMR). Trials are made for different RMR_m and RMR_s values corresponding to various possible combinations of the constant m and s, which are used in the Hoek–Brown failure criterion, satisfying the limit equilibrium condition. It is also noted that the procedure provides a quick check for the rock mass rating obtained from the site investigations. The method is used in conjunction with the Bishop's method of analysis based on circular slip surfaces. The procedure outlined in this paper has also been satisfactorily applied to documented slope failure case histories in three open pit mines in Turkey.     

Stress–strain–electrical resistance effects and associated state equations for uniaxial rock compression

This paper reports stress–strain–electric resistance experiments for diabase, limestone and marble containing NaCl solution during the whole process of uniaxial compression. We obtained the complete testing data for the stress–strain curve and the associated electrical resistance–strain curve. The change caused by internal cracking of the rock causes the corresponding variation of rock electrical resistance. There is a minimum value for all the electric resistance–strain curves, corresponding to the cracking stress σ_c or the initial cohesion c_i. Based on the experimental results and stochastic property analyses of the rock fracture variation, we put forward a group of state equations for rock in sections to express the characteristics of rock during the whole process of uniaxial compression. The three variables, stress, strain and electrical resistance, together with data-fitted parameters, α₁,α₂ and β, are contained in the equations. The equations are used to express the inelastic response which intensifies with the propagation of cracking.     

Quantifying progressive pre-peak brittle fracture damage in rock during uniaxial compression

This paper presents the findings of an extensive laboratory investigation into the identification and quantification of stress-induced brittle fracture damage in rock. By integrating the use of strain gauge measurements and acoustic emission monitoring, a rigorous methodology has been developed to aid in the identification and characterization of brittle fracture processes induced through uniaxial compressive loading. Results derived from monocyclic loading tests demonstrate that damage and the subsequent deformation characteristics of the damaged rock can be easily quantified by normalizing the stresses and strains observed in progression from one stage of crack development to another. Results of this analysis show that the crack initiation, σ_ci, and crack damage, σ_cd, thresholds for pink Lac du Bonnet granite occur at 0.39σ_UCS and 0.75σ_UCS, respectively. Acoustic emissions from these tests were found to provide a direct measure of the rapid release of energy associated with damage-related mechanisms. Simplified models describing the loss of cohesion and the subsequent development of microfractures leading up to unstable crack propagation were derived using normalized acoustic emission rates. Damage-controlled cyclic loading tests were subsequently used to examine the effects of accumulating fracture damage and its influence on altering the deformation characteristics of the rock. These tests revealed that two distinct failure processes involving the progressive development of the microfracture network, may occur depending on whether the applied cyclic loads exceed or are restrained by the crack damage stress threshold.     

An experimental investigation of the failure mechanism of simulated transversely isotropic rocks

This paper presents the results of an experimental investigation of the failure mechanism, including failure process and failure modes, of transversely isotropic rock. This paper employs a rotary scanner to obtain the “unrolled” images of rock specimens at different stress levels during the uniaxial compressive tests. The unrolled image constitutes a circumferential surface image of the cylindrical specimen in a single picture and facilitates the study of failure processes and failure modes. Based on the experimental results, the failure of simulated transversely isotropic rock with varied orientations at different confining pressures is classified into one of two modes: (a) sliding failure along the discontinuities and (b) non-sliding failure along the discontinuities. The latter can be further classified into one of the following three sub-failure modes: (1) tensile fracture across the discontinuities, (2) tensile-split along the discontinuities, and (3) sliding failure across the discontinuities. The failure processes of these modes are also examined in this study. Failure criterion proposed by Tien and Kuo is found to predict accurately the strength and failure modes of simulated transversely isotropic rocks.     

A geo-engineering classification for rocks and rock masses

In this article, an attempt is made to assess the reliability of predicting the uniaxial compressive strength and the corresponding modulus of a rock mass by current approaches. These two basic engineering properties, when estimated from rock mass rating (RMR), Q and geological strength index (GSI), indicate hardly any change in the modulus ratio with the change in the quality of the rock mass from very good to very poor. However, the modulus ratio obtained from the relations involving the joint factor, J_f, indicate a definite decrease in the modulus ratio with a decrease in the quality of the rock mass. The strength and modulus in the unconfined and confined states, the modulus ratio and failure strain in the unconfined case were linked to J_f in earlier publications based on a large experimental database. Some of these relations were adopted to verify the response of jointed test specimens, the response of the rock mass during excavations for mining and civil underground chambers, in establishing ground reaction curves including the extent of the broken zone, and the bearing capacity of shallow foundations.The joint factor is now linked to RMR, Q and GSI. The prediction of compressive strength and modulus of the rock mass appears to be more suitable. For classifying the rock, based on these properties, the Deere and Miller engineering classification, applicable to intact rocks, has been suitably modified and adopted. The results of different modes of failure of jointed specimens establish definite trends of changes in the modulus ratio originating from the intact rock value on the modified Deere and Miller plot. A geo-engineering classification is evolved by considering strength, modulus, quantifiable weathering index and lithological aspects of the rock.     

Estimation of rock modulus: For intact rocks with an artificial neural network and for rock masses with a new empirical equation

The elastic modulus of intact rock is used for many rock engineering projects, such as tunnels, slopes, and foundations, but due to the requirements of high-quality core samples and associated sophisticated test equipment, instead the use of empirical models to obtain this parameter has been an attractive research topic. In the rock mechanics literature, some empirical relations exist between the elastic modulus of intact rock and other rock properties, such as the uniaxial compressive strength (σ_ci), unit weight (γ), Schmidt hammer rebound number, point load index and petrographic composition. However, the past use of specific rock types is the main limitation of the existing empirical equations. In other words, they are not open to the general purpose use. To eliminate this deficiency, a total of 529 datasets, including uniaxial compressive strength, unit weight and elastic modulus of intact rock (E_i), were collected via an extensive literature review. In addition to these datasets, a further total of 80 datasets was obtained from laboratory tests performed on greywacke and agglomerate core samples for this study. To prepare a chart for the prediction of the elastic modulus of intact rock, an artificial neural network was constructed using the large database. In addition, after a brief overview of existing empirical equations, a new empirical equation, which considers RMR and the elastic modulus of intact rock (E_i) as input parameters, is also proposed using worldwide data.     

Characterizing rock masses by the RMi for use in practical rock engineering: Part 1: The development of the Rock Mass index (RMi)

The Rock Mass index, RMi, has been developed to satisfy a need for a strength characterization of rock masses for use in rock engineering and design. The method gives a measure of the reduction of intact rock strength caused by discontinuities given by RMi = σ · JP. Here, σ is the uniaxial compressive strength of the intact rock measured on 50 mm diameter samples, and JP is the jointing parameter which is a combined measure of block size (or intensity of jointing) and joint characteristics as measured by joint roughness, alteration and size. This paper describes the method of determining the RMi for a rock mass using various common field observations. The determination of a meaningful equivalent block size is a key issue which is discussed in detail. Several areas of application of the RMi are presented, among others for design of rock support. Discussion of these applications will be developed in Part 2 of this paper.     

Application of Weibull's theory to estimating in situ maximum stress σH by hydrofracturing

Hydrofracturing is a widely used and established method for rock stress measurement and is especially valuable at great depths. In conventional hydrofracturing (Haimson, Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 15 (1978) 167), dealing with an axi-parallel fracture, the horizontal minimum stress σ_h is obtained as the shut-in pressure and the maximum stress σ_H is calculated from the breakdown pressure or reopening pressure. It has been pointed out, however, that σ_H is not as reliable as σ_h. This paper therefore presents a new approach for estimating σ_H. In this approach the probabilistic aspects of tensile failure are considered as new sources of information, because the probability density of fracture direction may provide valuable information concerning the stress difference σ_H−σ_h. As the basic theory to describe the tensile failure of rock, we adopted the Weibull’s weakest link theory. The applicability of the theory is first verified via tensile tests on rock specimens of different shape and size, then the probabilistic approach is applied to hydrofracturing to give the probability function of breakdown and the probability density function for the fracture direction. The applicability of the proposed method is presented through numerical calculations and an example in which σ_H−σ_h is estimated from the probabilistic variability of the fracture direction.     

Testing of rock joints filled with gouge using a triaxial apparatus

The rock mass contains discontinuities and its instability depends on the geometry of these discontinuities and the slope and orientation of the excavated face. One of the most important factors is the shear strength of potential failure planes. The characterisation of a discontinuity or a shear zone is not possible merely by visual examination of a specimen nor by subjecting it to conventional laboratory testing. The combined effects of the shear zone, its stress–strain history and the resulting strength deformation relation modulate the behaviour of rock mass, particularly when it approaches the state of limit equilibrium.This paper presents a testing technique for rock joints filled with gouge of various thicknesses (t=5–30 mm), dip angle (β=5–50°) and at strain rate (e=5–80 mm/h) in a triaxial testing system. The results of unconsolidated undrained tests carried out in triaxial conditions both for undulating and planar types of joints filled with gouge are reported. Extensive experimental results provided an insight into the development of a constitutive relation to predict strength criteria of discontinuous rock masses.     

Numerical simulation of the excavation damaged zone around an opening in brittle rock

The micromechanics-based damage model proposed by Golshani et al. [A micromechanical model for brittle failure of rock and its relation to crack growth observed in triaxial compression tests of granite. Mech Mater 2006;38:287–303] is extended so that time-dependent behavior of brittle material can be taken into account, with special attention to the numerical analysis of an excavation damaged zone (EDZ) around an opening, which is a major concern in assessing the safety of underground repositories. The present model is capable of reproducing the three characteristic stages of creep behavior (i.e., primary, secondary, and tertiary creep) commonly observed in the laboratory creep tests. The sub-critical microcrack growth parameters (i.e., n and A) can be determined for Inada granite by fitting the numerical results of elapse time to failure versus the creep stress ratio curve with the experimental data under both dry and wet conditions. It is found that moisture has a significant influence on the parameter A rather than the parameter n. Use of the extended model makes it possible to analyze not only the extension of microcrack length, but also the development of EDZ around an opening as a function of time. The damaged zones mainly develop in the sidewalls of the opening in the case that the vertical stress σ₂₂ is larger than the horizontal stress σ₁₁.     

断续节理岩体强度与破坏特征的数值模拟研究

基于细观统计损伤数值模型,通过改变包含单组节理岩体的节理倾角、节理台阶角、层距d和岩桥长度l_r,建立不同节理分布的断续节理岩体数值试样,展开系列数值试验,模拟了节理岩体的破坏过程,探讨了节理结构几何参数和应力水平对破坏模式以及岩体力学参数的影响规律。研究结果表明,断续节理岩体破坏模式共分为4种：沿节理面破坏、转动块体破坏、台阶状破坏和混合破坏。沿节理面破坏与台阶状破坏的岩体峰值强度高、破坏应变大,转动破坏的岩体峰值强度低、破坏应变小。随着节理倾角的增大,岩体力学行为表现出脆性破坏—渐进破坏—脆性破坏的循环过程。随着应力水平的增加,岩体破坏区域由中间向端部扩展,并且对于强度的提高有显著作用,但提高水平随围压增加而降低。节理台阶角对于=90°时的破坏形式影响较大,由台阶状破坏转变为转动块体破坏,层距d对阶梯状破坏模式影响较小,对转动破坏模式影响较大,岩桥长度l_r不影响破坏模式,但对面破坏与台阶状破坏模式的峰值强度、破坏应变影响较大。通过对比,模拟结果与物理试验规律一致,但数值模拟结果可以清晰获得节理岩体中应力场分布、裂纹起裂点与扩展方向、破坏图像等,有利于分析其内在破坏规律与机理。