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.     

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 empirical criterion for strength of transversely anisotropic rocks with metamorphic origin

A modified empirical criterion is proposed to determine the strength of transversely anisotropic rocks. In this regard, mechanical properties of intact anisotropic slate obtained from three different districts of Iran were taken into consideration. Afterward, triaxial rock strength criterion introduced by Rafiai was modified for transversely anisotropic rocks. The criterion was modified by adding a new parameter $ \alpha $ for taking the influence of strength anisotropy into consideration. The results obtained have shown that the parameter $ \alpha $ can be considered as the strength reduction parameter due to rock anisotropy. The modified criterion was compared to the modified Hoek–Brown (Saroglou and Tsiambaos) and Ramamurthy criteria for different anisotropic rocks. It was concluded that the criterion proposed in this paper is a more accurate and precise criterion in predicting the strength of anisotropic rocks.     

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.     

Quantitative three-dimensional description of a rough surface and parameter evolution with shearing

The choice of a general criterion to determine the shear strength of rough rock joints is a topic that has been investigated for many years. The major problem is how to measure and then to express the roughness with a number (e.g., joint roughness coefficient) or a mathematical expression in order to introduce the morphology of the joint into a shear strength criterion. In the present research a large number of surfaces have been digitised and reconstructed using a triangulation algorithm. This approach results in a discretisation of the joint surface into a finite number of triangles, whose geometric orientations have been calculated. Furthermore, during shear tests it was observed that the common characteristic among all the contact areas is that they are located in the steepest zones facing the shear direction. Based on this observations and using the triangulated surface data, it is possible to describe the variation of the potential contact area versus the apparent dip angle with the expression A_θ^*=A₀[(θ_max^*−θ^*)/θ_max^*]^C, where A₀ is the maximum possible contact area, θ_max^* is the maximum apparent dip angle in the shear direction, and C is a “roughness” parameter, calculated using a best-fit regression function, which characterises the distribution of the apparent dip angles over the surface. The close agreement between analytical curves and measured data therefore suggests the possibility of defining the influence of roughness on shear strength by the simple knowledge of A₀, C and θ_max^*. Based on the samples studied here, the values of these parameters capture the evolution of the surface during shearing. Moreover, they tend to be characteristic for specific rock types, indicating that it might be possible to determine ranges for each rock type based on laboratory measurements on representative samples.     

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.     

The joint intersection probability

In this paper a practical method to apply block theory is presented. Block theory provides the removable joint pyramids from a given free surface regardless of the number of joints in any joint intersection. While robust, the application of the theory in real practice is hampered by the large outcome space of possibly removable joint pyramids consisting of k mutually exclusive joints in a rock mass consisting of m joint sets. In this paper, we prove that the probability that k is greater than three in a three-dimensional space is zero. Consequently, only tetrahedral blocks need to be considered in the stability analysis for the analyzed free surface. The outcome space of theoretically removable joint pyramids can be further reduced by considering “safe” joint intersections, which consist of at least one line of intersection which is sub-parallel to the free surface. The block failure likelihood of the remaining joint intersections is proportional to two independent parameters: (1) the joint intersection probability and (2) the block instability parameter. We develop here a rigorous joint intersection probability expression based on simple frequency probability considerations which predicts that the probability for x in the rock mass to fall in joint intersection L_i,j,k is inversely proportional to the volume of the parallelepiped formed by joints i,j,k with mean spacing values x_i, x_j, x_k:Using the joint intersection probability and the instability parameter associated with each removable JP the critical key blocks of the excavation can be determined. In a brittle rock mass only the critical key blocks will require reinforcement. The paper concludes with a practical example which demonstrates the application of the concepts.     

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.     

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.     

Application of critical plane approach to the prediction of strength anisotropy in transversely isotropic rock masses

A formulation describing the strength anisotropy of transversely isotropic rock masses subjected to a three-dimensional stress state is proposed based on the critical plane approach. It is assumed that the initiation of cracking is governed by the Hoek–Brown failure criterion, and the anisotropy of the strength is described through the orientation dependence of the strength parameters m and s. Using direct optimization of failure function, the direction of potential failure plane, on which the failure function reaches maximum, is determined. True triaxial compression tests as well as conventional triaxial tests are simulated in order to verify the performance of the proposed formulation.     

Compaction of flocculated material

The device used in the experiment consists of a flat-bottomed graduated cylinder and a coaxial plunger. A suspension flocculated with chemicals is sedimented after being mechanically worked within the graduate, and the supernatant water is removed with a pipette. The plunger is thrust into the sludge at a constant speed. The sludge is not only compressed but also flows into the annular gap between the plunger and the graduate, resulting in liberation of water. The liberated water is accumulated on the sludge in the annular gap. The “sludge bulkiness” β is used to describe the volumetric proportion of sludge and solids in it. The sludge bulkiness values before and after the “plunger test” are denoted as β_i and β_f, respectively. The values of β_i and β_j have been explored as a function of the time of the mechanical working. As a result, there is a definite time lag between the maximum value of β_i and the minimum value of β_f. The minimum value of β_j is obtained when the sludge consists of “pelleted flocs”.     

Some aspects of unsaturated flow in jointed rock

The applicability of Darcy's Law to two-phase flow has been discussed. Specialised triaxial equipment has been employed to separately inject two pore fluid components (air and water) into fractured rock specimens, so that two-phase flow behaviour can be studied at high axial and confining stresses. Improvements to recently developed two-phase high-pressure triaxial apparatus have enabled the authors to continue their study of air–water (i.e. unsaturated) flow in intact and fractured rock specimens under a wide range of stress conditions, similar to those encountered in underground mining operations. In this paper, a simplified stratified two-phase flow model is also presented that satisfactorily predicts flow behaviour in an inclined rock fracture over a range of linear laminar flow for particular capillary pressure relationships. The mathematical model is based upon the principles of conservation of mass and momentum, and relates the fracture aperture (e_t) to phase permeability (k_i) using Poiseuille's law and the proposed ‘phase height’, h_i(t), for water and air phases. The experimental approach used to verify the model predictions is described and the predicted results compared with the measurements. The experimental data confirmed the relationship between relative permeability and flow rate, with respect to two-phase flow conditions.     

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.     

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.     

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.     

Heterogeneous rock mass classification by means of the geological strength index: the <Emphasis Type="Italic">San Mauro</Emphasis> formation (Cilento,Italy)

This paper describes an application of the geological strength index (GSI) method to the San Mauro formation, which is characterized by sandstones alternating with argillaceous marls. The Sandstone/Pelite (S/P) ratio and structural complexity were determined. Geo-structural and geo-mechanical surveys were undertaken in situ and rock samples were tested in the laboratory. A map of the S/P ratio was produced showing the bedrock divided in four classes. Three ranges of GSI values were identified. The values of the intact UCS and of the constant m _i were appropriately reduced to reflect the variable presence of sandstone compared with the pelitic fraction. A “weighted average” of the intact strength properties of the hard and weak layers was adopted. The values for the intact materials were reduced from 20 to 60% depending on the GSI categories of the heterogeneous rock mass. In this way, seven classes of rock masses characterized by different values of GSI, reduced UCS and m _i values were identified.     

Experimental investigation of mechanical properties of bedded salt rock

Because of salt cavern utilization for liquid, gas and solid waste storage, salt rock mechanical properties are needed for assessments of facility, stability and safety. Bedded salt deposits are widespread and used as much or more than diapiric salt bodies as storage facility hosts, but experimental data on the mechanical properties of bedded salt rock with impurities are far less common than data available on relatively pure diapiric salt rocks. Through laboratory uniaxial and triaxial compression experiments on rock salt (halite), interlayers (anhydrite) and bedded composite specimens (anhydrite–halite and mudstone–halite), differences in mechanical properties of the various lithologies are explored. In the composite specimens, the weakest or the most deformable component governs the behavior. Also, the properties of bedded composite lithology specimens tend to be in between the property ranges of the “pure” lithologies. The elastic modulus of the bedded salt rock increases from 5.3 to 24.1 GPa with an increase in the confining stress from 0 to 15 MPa, with some evidence of sample damage. The ductile transition for halite at the strain rates used is at about σ₃10 MPa.With increasing σ₃, the anhydrite–halite composite lithology deformation showed strain hardening and a strong trend to ductile behavior as the halite bands tended to dominate the behavior. Strain incompatibility effects exist along interfaces between creeping and non-creeping phases in anhydrite–halite composite lithologies. Mudstone–halite rocks tended to be extremely weak, compared with all other specimens.     

A modified Activated Sludge Model to estimate solids production at low and high solids retention time

In this paper, a modified version of the IWA-ASM1 model capable of correctly simulating the production of solids over a wide range of solids retention time (SRT) is presented. The parameters of the modified model have been estimated by integrating the results of respirometric and titrimetric tests with those of studies conducted on pilot scale plants that treat industrial wastewaters of differing characteristics.On the basis of the experimental results and their subsequent processing, it appears that the production of solids may be satisfactorily estimated using the modified model in which fractions X_P and X_I are supposed to be hydrolysable with a first-order kinetic.In the cases that were examined, the constant of the aforementioned kinetics was estimated to be k_i = 0.012 d⁻¹ and k_i = 0.014 d⁻¹, for tannery and textile wastewater respectively.A reliable calibration of the parameter k_i was possible when data relative to the experiment conducted in the pilot plants for no less than 60 d and in conditions of complete solid retention was utilized.     

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.