An Orthotropic Cosserat Elasto-Plastic Model for Layered Rocks

Summary Modelling the behaviour of rock masses consisting of a large number of layers is often necessary in mining applications (e.g. coal mining). Such a modelling can be carried out in a discontinuum manner by explicit introduction of joints. When the number of rock layers is large, it is advantageous to devise a continuum-based model in which case the joints are considered to be virtually smeared across the mass. In this study, a fully elasto-plastic equivalent continuum model suitable for describing the behaviour of such layered rock masses is considered. The model is based on the Cosserat continuum theory and incorporates the moment stresses in its formulation. In contrast to the earlier Cosserat models, the possibility of rock layer plasticity is considered. The accuracy of the developed Cosserat model is verified against analytical and experimental results. Received October 15, 2000; accepted July 30, 2001 Published online August 2, 2002     

Measurement of Indirect Tensile Strength of Anisotropic Rocks by the Ring Test

Summary This paper presents a new approach, combined with the Boundary Element Method (BEM) analysis and the diametrical compression on a thin disc with a small central hole, referred to as the ring test, for determining the indirect tensile strength of anisotropic rocks. The stress distribution around the hole can be successfully obtained by the proposed single-domain BEM. The complex variable function method was used for conveniently computing the tractions and displacements of a two-dimensional anisotropic body. If we assume that the tensile strength is given by the maximum absolute value of stress in the direction perpendicular to the loaded diameter at the intersection of loaded diameter and the hole, then from the failure load recorded by laboratory testing of ring (disc), the indirect tensile strength of rocks could be obtained. A marble from Hualien (Taiwan) with clearly black-white foliation, which was assumed to be transversely isotropic, was selected to conduct both ring tests and Brazilian tests for evaluating the tensile strength. The variation of the marble tensile strength with the inclination angle of foliation and with the hole size was also investigated. In general, the tensile strength of anisotropic rocks determined by ring test is not a constant, but depends on the elastic properties of rocks, the angle between the planes of rock anisotropy and the loading direction, the diameter of the central hole, and the contact condition of loading.     

The Shear Behavior of Bedding Planes of Weakness Between Two Different Rock Types with High Strength Difference

In this article, the shear behavior of discontinuities caused by bedding planes of weakness between two different rock types with high strength difference is investigated. The effect of roughness and compressive strength of joint wall in such discontinuities are studied. The designed profiles consist of two regular and three irregular artificial joints molded by three types of plaster mortars with different uniaxial compressive strengths. Firstly, it is demonstrated that the shear behavior of discontinuities with different joint wall compressive strengths (JCS) is different from rock joints with identical wall compressive strengths by showing that Barton’s empirical criterion is not appropriate for the former discontinuities. After that, some correlation equations are proposed between the joint roughness coefficient (JRC) parameter and some surface statistical/fractal parameters, and the normal stress range of Barton’s strength criterion is also modified to be used for such discontinuities. Then, a new empirical criterion is proposed for these discontinuities in such a way that a rational function is used instead of JRC log₁₀(JCS/σ _n) as i ₀(σ _c/σ _n)^a/[b + (σ _c/σ _n) ^a ] by satisfying the peak dilation angle boundary conditions under zero and very high normal stress (physical infinite normal stress causing zero peak dilation angle). The proposed criterion has three surface parameters: i ₀, a, and b. The reason for separation of i ₀ from JRC is indicated and the method of its calculation is mentioned based on the literature. The two remaining coefficients (a and b) are discussed in detail and it is shown that a shows a power-law relationship with b, introducing the coefficient c through b = c ^a . Then, it is expressed that a is directly related to discontinuity surface topography. Finally, it is shown that the coefficient c has higher values in irregular profiles in comparison with regular profiles and is dominated by intensity of peak dilation angle reduction (majorly related to the surface irregularity and minorly related to roughness). The coefficient c is to be determined by performing regression analysis on experimental data.     

Upscaling of Hooke''s Law for Imperfectly Layered Rocks

In this paper, we consider the upscaling of Hooke's law and its parameters on the fine scale, to a similar law with upscaled parameters on a larger scale. It is assumed that the fine scale material properties of the rock are imperfectly layered. In the governing equations, the deviations from perfect layering introduce a small parameter that can be used in perturbation series expansions for the stress, the strain, and the displacement. In the approximation of order zero the upscaled compliance matrix contains the well-known Backus parameters; this approximation holds exactly for a perfect layering. However, many natural rock types are imperfectly layered and in that case the approximation of order zero may not be sufficiently accurate. Therefore, we consider also the first order corrections. The derivation and results are presented both for the most general case and for the much simpler case in which the fine scale Poisson ratio may be assumed constant. From thermodynamic principles, it follows that the compliance tensor is symmetric on the fine scale. However, it is shown that the argument for symmetry cannot be extended to upscaled rigidities. One of the most important conclusions is that upscaled compliance tensors are nonsymmetric when there are trends in the deviations from perfect layering.     

Upscaling of Hooke's Law for Imperfectly Layered Rocks

In this paper, we consider the upscaling of Hooke's law and its parameters on the fine scale, to a similar law with upscaled parameters on a larger scale. It is assumed that the fine scale material properties of the rock are imperfectly layered. In the governing equations, the deviations from perfect layering introduce a small parameter that can be used in perturbation series expansions for the stress, the strain, and the displacement. In the approximation of order zero the upscaled compliance matrix contains the well-known Backus parameters; this approximation holds exactly for a perfect layering. However, many natural rock types are imperfectly layered and in that case the approximation of order zero may not be sufficiently accurate. Therefore, we consider also the first order corrections. The derivation and results are presented both for the most general case and for the much simpler case in which the fine scale Poisson ratio may be assumed constant. From thermodynamic principles, it follows that the compliance tensor is symmetric on the fine scale. However, it is shown that the argument for symmetry cannot be extended to upscaled rigidities. One of the most important conclusions is that upscaled compliance tensors are nonsymmetric when there are trends in the deviations from perfect layering.     

An Elastoplastic Model for Partially Saturated Collapsible Rocks

A unified elastoplastic model for describing the stress–strain behavior of partially saturated collapsible rocks is proposed. The elastic–plastic response due to loading and unloading is captured using bounding surface plasticity. The coupling effect of hydraulic and mechanical responses is addressed by applying the effective stress concept. Special attention is paid to the rock–fluid characteristic curve (RFCC), effective stress parameter, and suction hardening. A wide range of saturation degree is considered. The characteristics of mechanical behavior in partially saturated collapsible rocks are captured for all cases considered.     

An Anisotropic Model for Granular Material Based on Experiments

Soil is a heterogeneous material and most natural soil deposits show a definite stratification. The mechanical behaviour of such material is generally different in different directions, especially in the direction parallel and perpendicular to the stratification. A series of isotropic compression tests were carried out to study the behavior of granular material produced under controlled stratification in the laboratory. These tests were conducted both on cylindrical and square prismatic tri-axial specimens. It was observed that for hydrostatic loading, the strain response was different in different directions, especially in directions parallel and perpendicular to the direction of soil deposition. A definite trend of anisotropy was observed in the deformation pattern. The observed anisotropy is modeled in this paper by treating soil-dilatancy as a variable quantity. The equation of the plastic potential surface of the model which obeys a non-associated flow rule, is assumed to be dependent on three main variables confining pressure (\(\sigma_{3}\)), void ratio (e) and the angle of bedding plane orientation (δ) during deposition. The angle of bedding plane orientation (δ) was measured with respect to the direction of the major principal stress. The model has a cap yield surface in the isotropic stress direction, which is supplemented by a shear hardening Mohr–Coulomb surface in the deviator direction. This paper focuses on predicting the anisotropic strain response of stratified soil deposits subjected to isotropic compression. The proposed anisotropic model incorporates within an existing strain-hardening sand model, a modified cap yield surface and a modified plastic potential function related to the cap surface, to account for the anistropic response observed in isotropic compression tests. The two dimensional stress–strain model was extended to three dimensional Cartesian space. The strain anisotropy observed in the isotropic compression tests was predicted by the three dimensional anisotropic model proposed for granular materials.     

An Analytical Model for the Indentation of Rocks by Blunt Tools

Summary A methodology based on the cavity expansion model is developed to analyze the indentation of rocks by a class of blunt indenters. The analysis covers the particular self-similar case of indentation by blunt wedges or cones. As an example, the main results for the indentation of rocks by a spherical tool are presented and the analytical solution is compared with experimental results obtained by indenting a sphere in Harcourt granite.     

Petrofabric Analyses of Rhum and Skaergaard Layered Rocks

Petrofabric analyses of layered rocks from Rhum have revealeda preferred orientation for felspar in the allivalites and forolivine in the peridotites; a regional petrofabric map of felsparorientation contains a radial pattern which suggests the presenceof convection currents during crystal settling. An orientedspecimen of Skaergaard ferrogabbro from the margin of a troughband has allowed comparison to be made between a known magmacurrent direction and the preferred orientation of felspar,olivine, clinopyroxene, and apatite crystals in the rock.     

A Neuro-Genetic Network for Predicting Uniaxial Compressive Strength of Rocks

Uniaxial Compressive Strength (UCS) is considered as one of the most important parameters in designing rock structures. Determination of this parameter requires preparation of rock samples which is costly and time consuming. Moreover discrepancy of laboratory test results is often observed. To overcome the drawbacks of traditional method of UCS measurement, in this paper, predictive models based on neuro-genetic approach and multivariable regression analysis have been developed for predicting compressive strength of different type of rocks. Coefficient of determinatoin (R2) and the Mean Square Error (MSE) were calculated for comparison of the models’ efficiency. It was observed that accuracy of the neuro-genetic model is significantly better than regression model. For the neuro-genetic and regression models, R2 and MSE were equal to 95.89 % and 0.0045 and 77.4 % and 1.61, respectively. According to sensitivity analysis for neuro-genetic model, Schmidt rebound number is the most effective parameter in predicting UCS.     

The Significance of the Mesostasis of Basic Layered Igneous Rocks

Many layered igneous rocks consist of two main parts: the cumulus(plus adcumulus and heteradcumulus) material and the pore material,or mesostasis, produced by the crystallization of the trappedliquid. Knowledge of the chemistry of the mesostasis will beuseful in deducing magma fractionation trends and in the determinationof solid/liquid trace element partition coefficients, whichin turn will be significant in helping to elucidate the petrogeneticrelationship between certain basalts and layered gabbros. Anew geochemical method for the determination of both the compositionand amount of mesostasis in layered rocks is proposed. Thismethod uses elements of contrasting behaviour and is best appliedto rhythmically layered rocks rich in mafic or felsic minerals.Data on strontium, cobalt, and uranium are used to apply themethod to two rocks from the Skaergaard intrusion. The resultsobtained are in agreement with published data.     

Micromechanical Modeling of Anisotropic Damage-Induced Permeability Variation in Crystalline Rocks

This paper presents a study on the initiation and progress of anisotropic damage and its impact on the permeability variation of crystalline rocks of low porosity. This work was based on an existing micromechanical model considering the frictional sliding and dilatancy behaviors of microcracks and the recovery of degraded stiffness when the microcracks are closed. By virtue of an analytical ellipsoidal inclusion solution, lower bound estimates were formulated through a rigorous homogenization procedure for the damage-induced effective permeability of the microcracks-matrix system, and their predictive limitations were discussed with superconducting penny-shaped microcracks, in which the greatest lower bounds were obtained for each homogenization scheme. On this basis, an empirical upper bound estimation model was suggested to account for the influences of anisotropic damage growth, connectivity, frictional sliding, dilatancy, and normal stiffness recovery of closed microcracks, as well as tensile stress-induced microcrack opening on the permeability variation, with a small number of material parameters. The developed model was calibrated and validated by a series of existing laboratory triaxial compression tests with permeability measurements on crystalline rocks, and applied for characterizing the excavation-induced damage zone and permeability variation in the surrounding granitic rock of the TSX tunnel at the Atomic Energy of Canada Limited’s (AECL) Underground Research Laboratory (URL) in Canada, with an acceptable agreement between the predicted and measured data.     

Selecting Equivalent Strength for Intact Rocks in Heterogeneous Rock Masses

Many surface and underground structures are constructed in heterogeneous rock formations. These formations have a combination of weak and strong rock layers. Due to the alternation of the weak and strong layers, selecting the equivalent and appropriate geomechanical parameters for these formations is challenging. One of these problems is choosing the equivalent strength (i.e., uniaxial compressive strength) of intact rock for a group of rocks. Based on the volume of weak and strong parts and their strength, the equivalent strength of heterogeneous rocks changes. Marinos and Hoek (Bull Eng Geol Environ 60(2):85–92, 2001) presented the “weighted average method” for defining the uniaxial compressive strength (UCS) of heterogeneous rock masses based on the volume of weak and strong parts. Laubscher (1977) used the volume ratio of the strength of a weak part to a strong part (UCS weak/UCS strong) to determine the equivalent strength. In this study, the two methods are compared and their validity is evaluated by experimental data and numerical analyses. The geomechanical parameters of two heterogeneous formations (Aghajari and Lahbari) in the west of Iran were estimated using these methods. The results of the present study obtained through numerical analyses using particle flow code are compared with those of previous studies and discussed. Laboratory and numerical results show UCS decrease and approach to weak strength with an increasing in volume of weak part. When strength ratio of strong to weak rock increase, equivalent strength decrease more severely. The findings show that Laubscher’s method gives more appropriate results than the weighted average method.     

Alternative Quantification of the Geological Strength Index Chart for Jointed Rocks

The paper suggests an alternative quantification of the Geological Strength Index (GSI) Chart published by Hoek et al. in Paper prepared for presentation at the 47th US Rock mechanics/geomechanics symposium held in San Francisco (2013). The engineering parameters proposed for the horizontal and vertical axes of the Chart are the most commonly used in routine field investigations on rock masses by means of standard exposure surveys and borehole logging. They mainly include parameters traditionally used for block size characterization (i.e., volumetric joint count Jv, joint spacing S and RQD/Jn factor) as well as other parameters purposely defined in the present study and based on combinations of the traditional ratings of the Bieniawski’s RMR₈₉ classification. Adopting these options, new empirical equations for calculating the GSI have been fine tuned and tested on a real rock mass dataset. The comparison between the GSI values calculated using the new equations and those directly mapped during underground excavations has demonstrated the potentiality of the proposed methodology. The new quantification of the GSI Chart can be regarded as a useful and practical tool suitable to integrate the Hoek et al. (2013) approach. The complementary use of different and independent methods represents an effective practice to properly check and validate the final estimation of the GSI.     

Determination of Fracture Toughness of Anisotropic Rocks by Boundary Element Method

Summary  This paper presents a systematic procedure for determining fracture toughness of an anisotropic marble using the diametral compression test (Brazilian test) with a central crack on the discs. Additionally, a novel formulation to increase the accuracy in Stress Intensity Factor (SIF) calculations using Boundary Element Method (BEM) is applied to determine the stress intensity factors and the fracture toughness of anisotropic rocks under mixed-mode loading. The numerical results show that the SIFs for both isotropic and anisotropic problems are in good agreement with those reported by previous authors. The marble with clear white-black foliation from Hualien (in eastern Taiwan), was selected for the Brazilian tests. Diametral loading was conducted on the Cracked Straight Through Brazilian Disc (CSTBD) specimens to evaluate their fracture toughness. In addition, a new fracture criterion was developed to predict pure mode I, pure mode II or mixed mode (I–II) fracture toughness of the anisotropic marble. The new fracture criterion is based on the examination of mode I, mode II and mixed mode (I–II) fracture toughness for different crack angles and anisotropic orientation. Authors’ address: Associate Professor Chao-Shi Chen, Department of Resources Engineering, National Cheng Kung University, No. 1 Dasyue Rd., East District, Tainan 701, Taiwan     

Modeling of SH-Wave Propagation in a Pre-stressed Highly Anisotropic Layered Structure

Mathematical Geosciences - The present study presents the propagation of horizontally polarized shear waves (SH-waves) in highly anisotropic layered media under the effect of horizontal and...     

深部各向异性硬岩钻进用新型金刚石钻头试验研究

我国正在进行深部地质找矿和深部矿产资源开发,“十三五”计划规定了对地下空间的利用和开发进行研究,这些都离不开深部钻探。深部钻探中遇到的岩石多为各向异性岩石,在这种岩石中钻进时,钻孔容易偏斜,影响钻探效率和质量。俄罗斯博士Скоромных В.В.教授等对此进行了研究,提出了钻头水口大小变化的设计思想。对比试验结果表明,这种新钻头钻进技术指标较好,值得研究和探讨。     

K_o固结粘土各向异性不排水剪强度研究

在三维各向异性弹塑性本构关系基础上，求解不排水条件和破坏条件，导出各种不同试验条件下K0团结粘土的各向异性不排水剪强度表达式。对常规超固结比定义下的不排水剪强度进行研究，将理论计算结果与试验结果进行比较，验证所提理论的合理性。