Isotropic Rock Research Articles

Numerical Analysis of a Tunnel Subjected to Blast Loads in a Transversely Isotropic Rock Mass

Numerical Analysis of a Tunnel Subjected to Blast Loads in a Transversely Isotropic Rock Mass

The field-enriched finite element method with gravity effects for simulating the cracking behaviors of large-scale engineering rock masses

The field-enriched finite element method uses a scalar field defined as a field variable to describe cracks and characterize their impact on the displacement field and stress field of the solution model. It is capable of avoiding remeshing and employing level set functions to describe cracks when simulating the propagation of cracks. In this work, a field-enriched finite element model with gravity effects is proposed to simulate the large-scale failure process of engineering rock masses, and several numerical cases of geotechnical engineering are successfully analyzed. First, by introducing the unified tensile fracture criterion into the numerical model, the large-scale failure process of the intact slope is simulated. Second, the sliding process of rock slopes containing en echelon joints is numerically investigated. Third, the cracking process of the concrete dam is analyzed. Finally, the effects of joint and bedding plane inclination angles on the stability of tunnel chamber in transversely isotropic rock mass are studied. The numerical results indicate that the numerical method proposed in this work can accurately solve the large-scale failure process of rock masses.

Measurement of the Static Nonlinear Third‐Order Elastic Moduli of Rocks: Problems and Applicability

AbstractThe third‐order elastic (TOE) model has been used to describe the widely observed nonlinear mechanical behaviors of earth materials. In addition to linear elastic constants (λ, μ), three nonlinear elastic moduli (A, B, C) are required for isotropic rocks. Contrary to previous research on dynamic TOE moduli, this study followed the protocol to measure strain and stress under uniaxial and hydrostatic compressive tests statically, which were later used to invert for the full set of TOE moduli for four standard rock types with differing pore structures of a porous oolitic limestone, quartz‐rich sandstones, and a dense crystalline basalt. The applicability of the TOE model to characterize nonlinearity depends on the fulfillment of path‐independence and small‐strain assumptions. Using the measured static TOE moduli, the finite element model demonstrates that the stress in the vicinity of the wellbore is more amplified than the stress in the linear elastic case, which leads to a wider zone of rock failure around the wellbore. Due to the long‐standing discrepancy between static and dynamic moduli, the rarely reported full set of static TOE moduli is necessary and will benefit future research in understanding the effect of rock nonlinearity on geophysical and geomechanical applications, such as long‐term safe storage of CO2 and generating process of geohazards.

Investigation of Indirect Shear Strength of Black Shale for Urban Deep Excavation

This study thoroughly investigated the compressive and tensile strength characteristics of black shale using both experimental and analytical approaches. Uniaxial compression tests were conducted to determine the elastic constants of black shale modeled as idealized, linear elastic, homogeneous, and transversely isotropic. Additionally, Brazilian tests were carried out on shale, considering it a transversely isotropic material. Strain measurements were recorded at the center of disc specimens subjected to diametric loading. By placing strain gages at the disc centers, the five elastic constants were accurately estimated. The effects of experimental methods and diametric loading on the elastic constant determination were evaluated and analyzed, and the indirect shear strength of the black shale, considering anisotropy, was determined using the estimated stress concentration coefficient. This study revealed that the indirect tensile strength of black shale is significantly influenced by the angle between the anisotropic planes and the diametric loading direction. Moreover, it was revealed that the stress concentration coefficients for anisotropic rocks vary from those of isotropic rocks, depending on the inclination angle of the bedding planes. This study confirms that the shear (tensile) strength of anisotropic black shale is not constant but varies with the orientation of the anisotropic planes in relation to the applied load.

A Stress‐Driven Double‐Phase–Field Framework for Tensile Fracturing Processes in Transversely Isotropic Rocks

ABSTRACTWe present a double‐phase–field framework for tensile fracturing processes in transversely isotropic rocks. Two distinct phase‐field variables are introduced to represent smeared approximations of tensile fractures along the weak bedding planes and through the anisotropic rock matrix, respectively. Driving forces that control fracture propagation in the phase‐field framework are constructed as a stress‐based formula with a recently developed tensile failure criterion that distinguishes the two failure modes in transversely isotropic rocks. For numerical implementation, we adopt a staggered integration scheme and decouple the governing equations so that the displacement field and phase‐field variables can be updated in sequence for a given loading step. The finite element formulation of the proposed framework is introduced in detail in this paper and is implemented in an in‐house finite element code. The numerical implementation is then validated by reproducing the uniaxial tension test results of Lyons sandstone. After that, we conduct simulations on a pre‐notched square plate loaded in tension to demonstrate the features of the proposed framework. Finally, we conduct simulations of three‐point bending tests of Pengshui shale and show that the proposed model can reproduce the force–displacement curves and failure patterns of specimens with different bedding plane orientations observed in laboratory experiments.

Torsion of end-bearing pile foundations in layered anisotropic geomaterials

ABSTRACT An analysis for the torsional dynamic response of end-bearing pile foundations embedded in a layered transversely isotropic geomaterial (soil/rock) is presented. The deformation of the transversely isotropic soil or rock is described by the method of separation of variables. The elasticity theory for a viscoelastic medium with frequency independent hysteretic material damping, and the Extended Hamilton’s Principle are utilised to derive the differential equations describing pile and soil motions. The differential equations are solved analytically in an iterative algorithm. The accuracy of the analysis is verified with existing studies reported in the literature for pile foundations embedded in a homogeneous and layered soil deposit. The effect of the degree of anisotropy on the pile-soil response – dynamic pile-head stiffness, distribution of pile rotation and torque with depth, dimensionless soil displacement function for various values of pile slenderness and pile-soil stiffness ratios in a homogenous soil deposit is investigated. Design charts of static pile-head stiffness in a homogeneous soil deposit for a wide range of pile-soil stiffness and pile slenderness ratios, and degree of anisotropy are also reported. The effect of soil layering for a pile embedded in a two-layered soil deposit is also studied.

Failure mechanism of transversely isotropic schist under Brazilian test using real-time X-ray nano tomography scanning

The study tries to address Brazilian tests on a transversely isotropic rock to investigate failure conditions by using digital volume correlation (DVC). For this purpose, an in-situ Brazilian apparatus in a nano- X-ray computed tomography (CT) scanner is applied to investigate the 3D progressive failure mechanism of the schist. Two different anisotropy angles are tested. CT scans are conducted at every selected loading stage before peak load. DVC analyzes the constructed X-ray CT images to determine the effect of schistosity orientation on failure mechanism, including crack initiation and propagation. A calibration method is presented to verify DVC parameters, including the half size of the correlation window and the space between two sub-volumes. Using DVC, 3D deviatoric strain field, strain contours, and displacement increment are determined at all stages of loading. A CT value-based method (VG Studio) is also applied to validate the DVC results. It is found that the layer boundaries affect the failure pattern, with the agreement between the DVC results and VG Studio results being observed. For the specimen with horizontal layers, the crack initiates at the center part at the lower density layer and finishes out of the center. Also, for the specimen with horizontal layers, the crack initiates and propagates at the boundary of two layers out of the middle line of the specimen. These asymmetry failures are due to the heterogeneity of layers. The study also shows the type of failure using DVC by monitoring displacement increment vectors near the crack location.

Analytical solutions of noncircular tunnels in transversely isotropic rheological rock masses

In the field of tunnelling applications, it is often found that the rock masses exhibit anisotropy and rheological properties. To optimize the utilization of underground space, the use of noncircular tunnels is often preferred. However, it is important to note that these noncircular tunnels can lead to high-stress concentrations and significant displacements.This article presents a thorough analytical study on the time-dependent ground responses induced by the excavation of noncircular tunnels in transversely isotropic viscoelastic rock masses. The study considers a comprehensive set of engineering factors, including the viscoelastic characteristics of the surrounding rock, any anisotropic angle, and arbitrary tunnel shapes.Using the generalized corresponding principle of anisotropic elasticity and anisotropic viscoelasticity, an analytical model is introduced. This model can accurately and swiftly address the problem of deformation and stresses around noncircular tunnels in anisotropic rheological rock masses. The analytical solutions are verified by their good agreement with the Finite Element Method (FEM) results under identical assumptions. Moreover, the qualitative agreement between the analytical solutions and field data further validates the practical application of the analytical solution.A parametric analysis is then performed to investigate the effects of anisotropy ratio, anisotropy angle, and coefficient of lateral pressure on stresses and displacements.The proposed analytical solutions can help reveal the particular mechanical mechanism of the time-dependent ground responses due to the combination of rock anisotropy and rheology. Furthermore, they can provide a more accurate prediction of the ground response, which may be useful to optimize the design of tunnel excavation in anisotropic rheological rock masses.

Effect of Bedding Angle on Energy and Failure Characteristics of Soft–Hard Interbedded Rock-like Specimen under Uniaxial Compression

To investigate how bedding planes affect the energy evolution and failure characteristics of transversely isotropic rock, uniaxial compression tests were conducted on soft–hard interbedded rock-like specimens with varying bedding angles (α) using the RMT-150B rock mechanics loading system. The test results indicate that throughout the loading process, the energy evolution shows obvious stage characteristics, and the change of α mainly affects the accelerating energy dissipation stage and the full energy release stage. With the increase of α, the ability of rock to resist deformation under the action of energy shows the characteristics of “strong–weak–strong”. The energy dissipation process is accelerated by medium angle bedding planes (α = 45°~60°). The precursor points of the ratios of dissipation energy to total energy (RDT) and elastic energy to dissipation energy (RED) can be used to effectively predict early failure. With the gradual increase of α, the difficulty of crack development is gradually reduced. The changes of energy storage limitation and release rate of releasable elastic energy are the immanent cause of different macroscopic failure modes of specimens with varying α.

An assumed enhanced strain finite element framework for tensile fracturing processes with dual-mechanism failure in transversely isotropic rocks

We present an assumed enhanced strain finite element framework for the simulation of tensile fracturing processes in transversely isotropic rocks. Fractures along the weak bedding planes and through the anisotropic rock matrix are treated with distinct enrichment, and a recently proposed dual-mechanism tensile failure criterion for transversely isotropic rocks is adopted to determine crack initiation for the two failure modes. The cohesive crack model is adopted to characterize the response of embedded cracks. As for the numerical implementation of the proposed framework, both algorithms for the update of local history variables at Gauss points and of the global finite element system are derived. Four boundary-value problem simulations are carried out with the proposed framework, including uniaxial tension tests of Argillite, pre-notched square loaded in tension, three-point bending tests on Longmaxi shale, and simulations of tensile cracks induced by a strip load around a tunnel in transversely isotropic rocks. Simulation results reveal that the proposed framework can properly capture the tensile strength anisotropy and the anisotropic evolution of tensile cracks in transversely isotropic rocks.

Felsic xenoliths in Triassic quartz-monzodiorite dykes from the central Jebilet (Morocco): Detailed petrography and geochemistry, with a critical assessment of the so-called “granulitic” samples

Felsic xenoliths occur locally in the Early Triassic quartz-monzodiorite dykes that crosscut the Sarhlef basin in Central Jebilet, Morocco. Irrespective of the secondary features linked to entrapment, three types of xenoliths are distinguished on the basis of textural evidence: 1) isotropic rocks of broad granitic appearance; 2) foliated samples whose planar fabric undoubtedly corresponds to a ghost foliation, 3) a specific xenolith with a blastomylonitic structure.Geochemically, samples with Na > K and moderately negative εNd values (-1.5 to - 3.8), reflecting a hybrid origin, depart from samples with K > Na and strongly negative εNd values (-9.1 to - 11.7), pointing to a stronger or even pure crustal derivation. The blastomylonitic sample differs drastically from the rest by its plagiogranitic composition and elevated Nb content, associated with a highly radiogenic Nd isotope signature (εNd240 = + 6.3). Altogether, these features and available radiometric data suggest a connection of the xenoliths with the Variscan igneous bodies that crop out in the central Jebilet. The potassic xenoliths with unradiogenic Nd are broadly similar to the granodiorite plutons. The xenoliths with Na > K and mildly negative εNd and the blastomylonitic plagiogranite are reminiscent of the bimodal association of microgranites and gabbros of the Sarhlef basin.From a structural point of view, samples with a gneissic fabric may reflect the possible occurrence of a metamorphic, migmatitic complex at somewhat deeper crustal level. However, based on detailed mineralogical and geochemical data, the samples previously known as “mid-crustal granulites” turn out to be highly differentiated, aplo-pegmatitic rocks, interpreted as early, incipient partial melts. The presence of high-grade metamorphic rocks at depth is only documented by one small, HT garnet-biotite xenolith, while peraluminous lenses (fibrolite + spinel ± corundum) dispersed in the quartz-monzodiorite dykes suggest that a metapelitic component was involved in the melting process generating these lavas. In summary, this set of felsic xenoliths does not provide much information on the crustal composition under the Jebilet, apart from fairly superficial levels.

Laboratory and numerical investigation of anisotropic creep behavior of transversely isotropic composite rock

Laboratory and numerical investigation of anisotropic creep behavior of transversely isotropic composite rock

Electrical conductivity model for transversely isotropic rocks with interconnected cracks

The electrical conductivity of subsurface rocks is generally anisotropic. The anisotropy of the subsurface electrical conductivity provides important information on the stress-strain state and geodynamics. To quantitatively interpret anisotropic conductivity structures revealed by electromagnetic surveys, it is essential to use a mixing model considering the anisotropy. Although there exists a mixing model for transversely isotropic rocks with crack-shaped pores, the previous model seems inappropriate in interpreting conductive anomalies revealed by electromagnetic exploration because cracks are assumed to be isolated in the model. Therefore, this study develops a theoretical mixing model for transversely isotropic rocks with mutually interconnected cracks by a statistical approach. The derived mixing model considers the macroscopic tortuosity of a collection of cracks as well as the tortuosity of each crack. The derived model can represent general transverse isotropy and includes the isotropic and parallel models as special cases. I compare the developed model to previously proposed mixing models, showing that the developed model can reproduce a much wider range of anisotropy than the already-existing anisotropic mixing model. By applying the developed model to an example of the anisotropic conductivity in the oceanic upper crust inferred by electromagnetic exploration, I demonstrate that the developed mixing model enables us to quantitatively infer the crack orientation and fluid volume fraction that reproduce significant anisotropic conductivity found by field observations. Furthermore, I compare the developed model to the anisotropic seismic velocity model for fluid-filled cracks.

A Procedure for Assessing the Orientation of Failure Plane in Transversely Isotropic Rocks

A Procedure for Assessing the Orientation of Failure Plane in Transversely Isotropic Rocks

Time-dependent analytical solutions for tunnels excavated in anisotropic rheological rock masses

The anisotropy combined with the time-dependency of rock mass is commonly observed in tunnel construction. In this study, a comprehensive analytical investigation is performed on analysing time-dependent ground responses induced by tunnel excavation in an anisotropic rheological rock mass.An innovative analytical solution for the anisotropic viscoelastic problems is first proposed, which can be applied to the general cases that the rheological behaviours in anisotropic principal directions are independent and arbitrary. Using the generalized corresponding principle, the analytical model, which can precisely and rapidly address the problem, is then presented for the deformation and stresses around tunnels in transversely isotropic viscoelastic rock mass, considering different viscoelastic characteristics, different anisotropic angles, and circular/elliptical tunnel shapes. The analytical solutions agree very well with the numerical predictions for the consistent models, and can also be reduced to the isotropic viscoelastic case. Furthermore, the application of analytical solutions in practical engineering is validated by the good agreements between analytical solutions and field measurements. A parametric analysis is then performed to investigate the time-dependency of stress, and the effects of anisotropy ratios and anisotropy angles on displacements and stresses.The proposed analytical solution has a valuable contribution to the field of rock rheological mechanics, as it offers a benchmark for anisotropic viscoelastic problems, insights into the complex behaviour of anisotropic rock masses and provides a more accurate prediction of the ground response that may be useful to optimize the design of tunnel excavation in anisotropic rock masses.

Anisotropic Mechanical Properties and Fracture Mechanism of Transversely Isotropic Rocks under Uniaxial Cyclic Loading

Transversely isotropic rocks, which are special anisotropic materials, are widely encountered in civil, mining, petroleum, geothermal, and radioactive waste-disposal engineering. Rock is frequently subject to cyclic loads resulting from natural and human-caused events. However, to date, the fracture mechanism of transversely isotropic rocks under cyclic loading remains poorly understood. To address this gap, uniaxial monotonic-loading and cyclic-loading tests were performed on slate specimens by the MTS815 system, during which acoustic emission (AE) signals inside the rock were monitored, and finally the fracture surfaces of the tested rock were scanned by scanning electron microscopy (SEM). Through these tests, the anisotropic mechanical properties, damage evolution, AE characteristics and fracture pattern of slate as a transversely isotropic rock were studied. The results show that the peak strength of specimens varies with the loading–foliation angle under monotonic and cyclic loading, following a U-shaped trend. The deformation modulus during unloading is more capable of characterizing the damage inside the specimen than that during loading. By defining the damage degree based on dissipation energy, it is found that the damage variable is influenced by the loading–foliation angle and the cyclic stress step. The AE characteristics of specimens exhibit significant anisotropy, closely correlated to the loading condition and loading–foliation angle. Regardless of cyclic stress step, the AE counts of specimens with a loading–foliation angle of 0° are mainly distributed near the peak region, whereas those of specimens with other loading–foliation angles occur primarily in the early stage of each cyclic loading. Finally, it is revealed that the fracture mechanism of slate specimens is determined by the loading–foliation angle, loading condition, and cyclic stress step.

Experimental and numerical failure mechanism evaluation of anisotropic rocks using extended finite element method

Rocks often show an anisotropic behavior due to microcracks, mineral-based structure, and unequal in-situ stresses. The crack tip's stress intensity factors (SIFs) that control rock failure heavily depend on anisotropy. Given the significance of extensive research to identify the effects of anisotropy on the mechanical behavior and strength of rocks, numerical modeling is considered an essential method in designing structures and analyzing their stability. This study experimentally and numerically evaluated the effects of the bedding angle, initial crack angle with the loading axis, crack length, and anisotropy ratio on the stress intensity factor of transversely isotropic rocks via the extended finite element method (XFEM). Cracked chevron-notched Brazilian disks (CCNBDs) were fabricated from phyllite under the International Society for Rock Mechanics (ISRM) standards and tested at different initial crack lengths and angles for validation. The XFEM approach was then used to predict toughness and crack propagation path in the tested and new specimens with varying ratios of anisotropy and bedding angles (ψ = 0, 30, 45, 60, and 90°). The results demonstrated that with the increase of the anisotropy ratio, in equal crack length and initial angle, the absolute value of the SIFs have larger values. In addition, with the increase of crack length, the first and second modes of SIF increased, which indicates the failure of the specimen at lower loads. Also, bedding angle is effective on the stress intensity factors and this effect becomes more intense as the crack length increases. Finally, the anisotropy properties not only affect the values of SIFs, but also can change the angle at which the specimen experiences pure mode I or mode II.

Elastic Behavior of Transversely Isotropic Cylindrical Rock Samples under Uniaxial Compression Considering Ideal and Frictional Boundary Conditions

Conceptualizing sophisticated measurement set-ups as well as testing and evaluation procedures for laboratory experiments on anisotropic rocks requires a basic understanding of the potential specimen behavior. The focus of the present work was therefore to investigate the influence of different transversely isotropic parameters and their ratios on the elastic behavior of cylindrical rock samples in uniaxial compression tests. Parameter sets corresponding to soft anisotropic rocks were chosen based on naturally observed ranges for the five elastic transversely isotropic constants. Analytical results for the radial and vertical strain distributions around the sample circumference and a comparison with finite element simulations are presented. Further, the effect of interface friction between samples and loading platens was analyzed within the numerical models. The results suggest that radial strains around cylindrical anisotropic samples are rarely uniform except for specific combinations of parameters and isotropy plane inclinations. The effect of interface friction was found to have a clear influence on the developing elastic stress and strain distributions for samples with inclined isotropy planes. Nevertheless, no significant influence of frictional boundary conditions on the back-calibrated values of the elastic parameters could be identified, suggesting that friction-reducing measures in uniaxial compression tests on transversely isotropic samples with predominantly linear behavior are not required.

An Anisotropic Peridynamic Model for Simulating Crack Propagation in Isotropic and Anisotropic Rocks.

In this work, we present a peridynamic-based simulation method for modeling quasi-static fracture propagation in isotropic and anisotropic rock within the framework of peridynamic least square minimization (PDLSM). The original isotropic elastic PDLSM is further extended to investigate fracture propagation in anisotropic materials in this study. The proposed AN-PDLSM model integrates an anisotropic model in fracture mechanics to analyze the failure process of anisotropic rocks. An important advancement in this research lies in the incorporation of the maximum energy release rate criterion (MERR) into the PDLSM framework for the first time. This enhancement enables accurately determining crack propagation and the associated crack angles. The proposed model utilizes the energy release rate calculated through the J-integral method to assess bond breakage, and it employs a mesh-independent, piecewise linear fracture model to describe crack propagation. The proposed method fully combines the merits of traditional fracture mechanics with the unique capabilities of peridynamics. To demonstrate the effectiveness of the proposed model, a simulation of fracture evolution in isotropic plates subjected to semi-circular bending tests is presented and compared with experimental results. It is shown that the proposed model accurately replicates fracture trajectories in isotropic specimens. In the context of anisotropic rock, the effect of a weak coefficient on crack morphology is discussed in order to obtain a suitable value. Additionally, the impact of bedding angles on fracture paths through our proposed model is also explored, revealing excellent agreement with experimental results. The findings in this research demonstrate that the proposed AN-PDLSM model is exceptionally proficient at capturing the intricate, oscillating crack paths observed in anisotropic rock materials.

A nonlinear creep model for transversely isotropic rock based on fractional calculus theory

A nonlinear creep model based on fractional calculus theory is proposed to better describe the creep characteristics of transversely isotropic rock. The model includes a Hooke elastomer, a fractional Abel dashpot, a Kelvin body, and a nonlinear viscoplastic body, which effectively capture the three stages of creep (including the primary, steady-state, and accelerating creep stages) while also reflect the influence of different bedding angles on creep behavior. The parameters of the model are identified by the Levenberg–Marquardt algorithm. Compared to previous models assuming constant Poisson’s ratio for transversely isotropic rock, our nonlinear creep model demonstrates improved accuracy and rationality.