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.     

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.     

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.     

Crack coalescence in a rock-like material containing two cracks

This paper investigates the pattern of crack coalescence and strength of a sandstone-like material containing two parallel inclined frictional cracks under uniaxial compression, with changing values of inclination of preexisting cracks α, bridge angle β (inclination between the inner tips of the two preexisting cracks), and the frictional coefficient μ on the surfaces of the preexisting cracks. Three main modes of crack coalescence are observed: the shear (S) mode (shear cracking between the two preexisting cracks); the mixed shear/tensile (M) mode (propagation of both wing and shear cracks within the bridge area); and the wing tensile (W) mode (coalescence of wing cracks from the tips of the preexisting cracks). The M-mode and W-mode of crack coalescence can further be divided into two and six types, respectively. Simple regime classifications of coalescence in the α–β space are proposed for different values of μ (=0.6, 0.7 and 0.9). In general, the S-mode mainly occurs when α=β or when β<β*(α, μ)=a−bα, with both a and b depending on μ; the M-mode dominates when β_L>β>β*(α, μ) (where β_L≈82.5°); and the W-mode is only observed when β>β_L. However, more experiments are still required to refine the classification. The observed peak strength, in general, increases with μ. Our results show that the peak strength predicted by the Ashby and Hallam (1986) model basically agrees with experiments. A minimum occurs at about α=65° when the peak strength is plotted against α. For α>45°, the peak strength is essentially independent of the bridge angle β.     

Effect of initial water content and shear stress on immersion-induced creep deformation and strength characteristics of gravelly mudstone

Soft sedimentary rocks undergo deterioration due to weathering induced by dry–wet cycles in a process known as slaking. To investigate the effect of the initial water content and shear stress on the immersion-induced creep deformation and strength characteristics of gravelly mudstone (slaking index = 1 and 2), creep tests were conducted using a modified direct-shear-test apparatus under different creep-stress ratios. Crushed mudstone with different degrees of compaction and initial water content was subjected to immersion under the creep-shear-stress conditions. After the creep shear and vertical displacements stabilized, a monotonic shear loading was applied. During creep immersion, the increment in the creep-shear displacement increases as the creep-stress ratio increases and the initial water content and/or degree of compaction decrease. Retaining a high initial water content in crushed mudstone is recommended to maintain its shear-stress stability. Under the same density, the peak shear strength decreased with an increase in the increment of the creep-shear displacement during creep immersion. The results indicate that the occurrence of immersion-induced creep deformation in addition to slaking can result in the deterioration of slope stability.     

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.     

Time-dependent tests on intact rocks in uniaxial compression

The influence of strain history on rock specimen deformation during multi-level loading and unloading cyclic uniaxial compression creep tests is studied with a creep testing machine. An experimental data processing method for such creep tests is suggested. The correction formulas to determine the rheological model parameters are derived for the case when load relaxation is considered. Creep and relaxation tests under uniaxial compression on four types of rocks are conducted using an electronic hydraulic servo-controlled stiff testing machine. The creep and relaxation laws of the different rocks are compared. The complete stress–strain curves for red sandstone specimens are obtained at nine strain rates from 2.43×10⁻⁶ to 4.38×10⁻³/s. The effects of strain rates on rock strength and limit strain are discussed. Empirical equations to evaluate the strain rate dependence of rock mechanical properties are presented.     

A nonlinear strength criterion for rock-like materials based on fracture mechanics

A nonlinear strength criterion for rock-like materials is developed in this paper. Taking α as an angle of micro-failure orientation in rock-like materials, a formulation between α and load is derived from a mixed-mode fracture criterion based on linear elastic fracture mechanics. According to micro-failure experimental phenomena of rock-like materials, a failure characteristic parameter under triaxial compression condition is chosen, which is relevant to confining pressure and is an invariant. A theoretical nonlinear strength criterion is also derived, which is exactly in the same mathematical form as the original Hoek–Brown empirical strength criterion. In addition, it is also found that the coefficient m in the Hoek–Brown criterion has physical meaning which is related to the ratio between the uniaxial compressive strength and the uniaxial tensile strength.     

Effect of temperature and pressure on the thermal conductivity of sandstone

Effective thermal conductivity (ETC) of dry sandstone was measured over a temperature range from 275 to 523 K and at pressures up to 400 MPa with a guarded parallel-plate apparatus. The estimated uncertainty of the ETC measurements is 2%. The porosity of the sample was 13%. A rapid increase of ETC was found for dry sandstone at low pressures between 0.1 and 100 MPa along various isotherms. At high-pressure range (P>100 MPa) a weak linear dependence of the ETC with pressure was observed. The pressure effect is negligibly small after first 80–100 MPa where bridging of microcracks or improvement of grain contacts takes place. We interpreted the measured ETC data using a various theoretical and semi-empirical models in order to check their accuracy and predictive capability. The effect of structure (size, shape, and distribution of the pores), porosity, and mineralogical composition on temperature and pressure dependences of the ETC of sandstone was discussed. To estimate the effect of temperature and pressure on the ETC of sandstone the pressure, β_P, and temperature, β_T, coefficients of ETC were calculated from the measured values of ETC. The measured values of the ETC were also used to calculate the values of the isothermal compressibility, χ_T, and thermal expansion coefficient, α. The equation of state of sandstone was developed using the measured ETC data.     

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.     

The flattened Brazilian disc specimen used for testing elastic modulus, tensile strength and fracture toughness of brittle rocks: analytical and numerical results

The flattened Brazilian disc specimen is proposed for determination of the elastic modulus E, tensile strength σ_t and opening mode fracture toughness K_IC for brittle rocks in just one test. This paper is concerned with the theoretical analysis as well as analytical and numerical results for the formulas. According to the results of stress analysis and Griffith's strength criteria, in order to guarantee crack initiation at the centre of the specimen, which is considered to be crucial for the test validity, the loading angle corresponding to the flat end width must be greater than a critical value (2α20°). The analysis shows that, based on the recorded complete load–displacement curve of the specimen (the curve should include the ‘fluctuation’ section after the maximum load), E can be determined by the slope of the section before the maximum load, σ_t by the maximum load, and K_IC by the local minimum load immediately subsequent to the maximum load. The relevant formulas for the calculation of E, σ_t, K_IC are obtained, and the key coefficients in these formulas are calibrated by finite-element analysis. In addition, some approximate closed-form formulas based on elasticity are provided, and their accuracy is shown to be adequate by comparison with the finite-element results.     

Anisotropic strength and deformational behavior of Himalayan schists

Anisotropy, which is characteristic of metamorphic rocks such as schists, is due to a process of metamorphic differentiation. Preferred orientation of minerals like mica and chlorite in response to tectonic stresses makes schistose rocks foliated. As a result their engineering properties vary with the direction of loading.The influence of transverse anisotropy on strength and deformational responses of four schistose rocks obtained from the foundation of two underground powerhouse sites in the Himalayas has been critically examined. Specimens at different orientation (β) of the foliations varying from 0° to 90° with respect to the axial stress (σ₁) in the unconfined state and also in the confined states up to 100 MPa of confining pressure were tested to evaluate the applicability of the non-linear strength criterion for the prediction of triaxial compressive strength and modulus. Based on the analysis of large experimental results it has been possible to predict strength and modulus with minimum pre-evaluation experimental data, i.e. only with three uniaxial compressive strength tests at 0°, 30° and 90° and two triaxial compression tests conducted at convenient confining pressures at β=90°orientation. Predicted non-linear stress–strain curves, using predicted values of strength and modulus have been found to match well with the experimental stress–strain curves even at higher confining pressures.     

岩石拉压特征的相似性

通过近年来岩石力学的试验结果，其中包括单轴拉应力和压应力作用下不同岩石的完全应力－应变曲线、受拉强度和受压强度的载荷速度效应以及蠕变寿命随蠕变应力水平的变化等，研究了岩石在拉应力和压应力作用下上述力学特征的相似性和规律性。     

Coefficient of restitution and rotational motions of rockfall impacts

This paper presents experimentally obtained results for the coefficient of restitution for spherical boulders impacting on rock slopes. Plaster modeling material is used for casting both the boulders and slopes. It is observed that the normal component of the coefficient of restitution (R_n) increases with the slope angle α, which agrees with Wu's observations (Trans. Res. Rec. 1–5 (1985) 1031). However, there appears to be no clear correlation between the tangential component of the coefficient of restitution (R_t) and the slope angle α. When the ratio of the resultant velocities and the ratio of the kinetic energies before and after impacts are used to define the coefficient of restitution (i.e. R_V and R_E), a very clear increasing trend in the coefficient of restitution with α is observed. When all data are plotted onto the R_t−R_n space, our laboratory data fall into the rock slope regime proposed by Fornaro et al. (In: D.G. Price (Ed.), Proceedings of the Sixth International Congress IAEG, Amsterdam, Balkema, Rotterdam, 1990, p. 2173) and also agree with those data gleaned from literature. In addition, the rotational kinetic energy E_r, induced at each impact, increases with the slope angle α, achieves a maximum at about α=40°, before decreasing again to a negligible value at α=70°. A simple theoretical model is proposed to explain this observation based on the locking between the boulder and the slope during impact. The α-dependence of E_r differs from the recommendation by the Japanese Railway Association that the induced rotational energy is about 10% of that of the translational kinetic energy.     

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.     

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.     

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.     

Simplified P–M interaction curve for square steel tube filled with high-strength concrete

The purpose of this study is to propose simplified strength equations that can be conveniently used to establish a P–M interaction curve of square concrete filled tubes (CFTs) with concrete strength of up to 100 MPa. The method presented in the author's previous study [Choi Y-H, Foutch DA, LaFave JM. New approach to AISC P–M interaction curve for concrete filled tube beam-columns. Eng Struct 2006;28(11):1586–98] was used as a basic unified formula for pure steel members and CFT ones, and a parametric study was performed to determine the contribution of the concrete, which were expressed by two variables: a normalized maximum moment, α, and the axial load ratio at the maximum moment, β. The two variables were formulated with respect to tube width-to-thickness ratio (b/t) and relative concrete compressive strength to yield strength of the steel tube (f′_c/F_y). Then, the proposed method were compared to experimental data found in literatures, which showed greatly improved results in terms of accuracy and amount of computation, when compared to the current AISC design methods.     

Strength of ferric hydroxide flocs

Strength and break-up of flocs produced in flocculation of water and effluent with ferric chloride were studied. Floc size was determined by a photographing technique and area and perimeter measurements by a Quantimet apparatus.The experimental results obeyed the expression d_max = GC⁻²γ with γ = 0.3 for effluent and γ = 0.5 for water indicating both viscous and inertial effects in break-up. Floc strength parameter C increased with polymeric coagulant aid addition, higher C values were obtained in water than in effluent.Floc break-up seems to take place by erosion of particles and flocs once disrupted do not grow again.     

The effect of length to diameter ratio of test specimens on the uniaxial compressive strength of rock

One of the parameters which affect the uniaxial compressive strength (UCS) of rock materials is the length to diameter ratio (L/D) of test cores. ASTM recommends a ratio of between 2 and 2.5, and ISRM suggests 2.5–3:1. Research has shown that high UCS values are obtained for L/D ratios <2, a very slight difference in values between 2 and 2.5, and they remain effectively constant with a L/D ratio >2.5:1. In this study, the shape effect on the UCS of seven rocks was investigated by testing dry cores with L/D ratios from 1 to 2.5:1. Based on the results, a decrease in UCS with increasing L/D ratios up to 2.5:1 was measured for all the rocks except the tuff. By omitting the data from the tuff sample, a correction formula was determined for the rocks tested in this study and results obtained from the literature. However, further work is required on different types of rock to verify or modify this formula.