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A Model for the Mechanics of Jointed Rock
Abstract
The representation of discontinuities in analysis of blocky rock is discussed. A linkage type element is developed for addition of rock joint stiffness to the structural stiffness matrix describing the behavior of a system of rock blocks and joints. Several basic problems of jointed rock are studied. These examples demonstrate the marked influence joints may have on the stress distribution, displacements, and failure pattern of an underground opening or other structures in jointed rock. A new classification of joints is introduced, based on the application of the joint element to finite element analysis of structures in jointed rock. Normal stiffness, tangential stiffness, and shear strength are used as parameters in the classification system. The methods discussed in this paper allow a jointed rock mass to be treated as a system of blocks and links. Just as analysis of a reinforced concrete building requires detailed knowledge of the behavior of concrete alone and steel alone, the joint stiffness approach calls for and uses detailed description of the behavior of rock blocks and rock joints independently.
... It should be pointed out that coupling conditions associated with embedded FE models are, in many cases, defined purely displacement-based, as opposed to ground-structure modelling techniques adopted for the discrete modelling 2 Embedded Finite Elements for Deep Foundations 21 approach, such as the zero-thickness (Goodman et al. 1968;Potts 1994), thin-layer (Desai et al. 1984) and master-slave (Hallquist 1979;Wriggers et al. 2013) methodology, where the interface constitutive behaviour can be described based on stress-strain relations. This aspect has important implications for the numerical treatment of coupling constraints in embedded FE models: ...
... In the former case, geometrical details of individual D-SSE are typically discretized by means of the standard conformal FEM 24 using solid FE, thereby imposing full 3D surfaceto-surface mesh tying problems at MSE-SSE contacts; for example, see Refs. [35][36][37] With regard to the modelling of piles through D-SSE, solid FE corresponding to the pile domain may be additionally circumscribed by contact formulations, such as isoparametric contact elements [38][39][40][41][42] in order to account for SSI. In the context of computational geotechnics, this expanded D-SSE modelling approach is often termed standard FE approach (SFEA). ...
... In comparison, three-dimensional (3D) finite element analyses (FEA) have a relatively higher potential 38 to capture the complex structural behavior of pile foundation systems, particularly in the presence of 39 inhomogeneous soil conditions (Marzouk et al., 2024), non-symmetric geometries (Granitzer et al.,40 2021) and complex loading paths (Staubach et al., 2023). Nevertheless, in many cases, the use of FEA 41 is limited by the available computational resources for solving boundary value problems (BVPs). ...
... For this reason, an idealized joint element (i.e., interface element) is considered between the tool and the soil when the tool response is simulated [67]. The joint element was first proposed by Goodman et al. (1968) [89] and then adopted in FEM models that simulated interacting systems. Goodman et al. (1968) proposed a 2-D joint element with zero width, four nodes, and eight displacement degrees of freedom, enabling normal and tangential deformations [67], [89]. ...
... For this reason, an idealized joint element (i.e., interface element) is considered between the tool and the soil when the tool response is simulated [67]. The joint element was first proposed by Goodman et al. (1968) [89] and then adopted in FEM models that simulated interacting systems. Goodman et al. (1968) proposed a 2-D joint element with zero width, four nodes, and eight displacement degrees of freedom, enabling normal and tangential deformations [67], [89]. ...
... The joint element was first proposed by Goodman et al. (1968) [89] and then adopted in FEM models that simulated interacting systems. Goodman et al. (1968) proposed a 2-D joint element with zero width, four nodes, and eight displacement degrees of freedom, enabling normal and tangential deformations [67], [89]. Desai et al. (1984) [90] proposed substituting the joint at zero width with a thin-layer element. ...
... On this basis, taking a 150 m-deep cut-off wall as an example, three-dimensional numerical models were constructed for the dam by considering the cut-off wall as an integral structure and as a segmented discontinuous structure, respectively. For the discontinuous model, the Goodman elements [23] were used to describe the contact performance between wall segments. Finally, the effects of segment width and wall depth on the deformation, stress distribution, and damage behavior of the cut-off wall were discussed in detail. ...
... On this basis, taking a 150 m-deep cut-off wall as an example, thre dimensional numerical models were constructed for the dam by considering the cutwall as an integral structure and as a segmented discontinuous structure, respectively. F the discontinuous model, the Goodman elements [23] were used to describe the conta performance between wall segments. Finally, the effects of segment width and wall dep on the deformation, stress distribution, and damage behavior of the cut-off wall were d cussed in detail. ...
... Subsequently, method was applied to analyze the mechanical behavior of segmented disco tra-deep cut-off walls. On this basis, taking a 150 m-deep cut-off wall as an ex dimensional numerical models were constructed for the dam by considerin wall as an integral structure and as a segmented discontinuous structure, res the discontinuous model, the Goodman elements [23] were used to describ performance between wall segments. Finally, the effects of segment width an on the deformation, stress distribution, and damage behavior of the cut-off w cussed in detail. ...
Dam foundations are prone to leakage damage after being exposed to long-term water action, which seriously affects the operation safety of the dam. At present, concrete cut-off walls serve an important means of anti-seepage for dam foundations. However, due to construction challenges, the cut-off wall needs to be poured segment-by-segment during the construction process, and the joints between adjacent segments become weak parts for seepage prevention. Therefore, it is crucial to clarify the stress state of segmented discontinuous concrete cut-off walls. Based on the Lee-Fenves framework and the tension–compression constitutive relationship of fracture energy, a plastic damage calculation method was established in this paper to characterize the mechanical behavior of discontinuous cut-off walls. The method was then used to analyze the mechanical performance of discontinuous walls with segment joints containing slurry cake. The research results showed that compared to the continuous cut-off wall, the vertical settlement in the middle part of the discontinuous cut-off wall increased by 5.8%, and the displacement along the river flow direction decreased by 35.3%. As the wall segment width decreased, the joint opening and the degree of tensile damage were reduced accordingly, while the compressive damage in the middle and lower parts of the wall was intensified. As the wall depth decreased, the constraints and load on the bottom of the wall showed obvious changes, leading to a reduced stress and damage level of the wall. The findings provide reference for the design and safety control of cut-off walls.
... 45 The second method is to model the two materials as separate solids, or shells in 2D, and use zero-thickness or zero-length elements between them. These elements, initially proposed by Goodman, 46 utilize strength/stiffness parameters that also require calibration based on the materials at the interface. Unlike a full-continuum model, the use of separate solids and an interface allows for relative translation between the contacting elements. ...
... 19,45 Regarding the interface in the normal direction, a linear elastic behavior is often deemed adequate to represent the transfer of normal stresses between the contacting elements. 46 OpenSees provides the ZeroLenghtContact3D element, which defines a node-to-node, zero-length element following the Mohr-Coulomb criterion to represent friction. 35 In this element, the two nodes (slave and master) move together until the shear forces (or stresses) exceed the threshold, resulting in slippage occurring in an Elastic-Perfectly-Plastic (EPP) manner. ...
... Furthermore, K t determines the degree of tangential relative translation between the contacting elements before slippage occurs. It is related the roughness of the materials at the interface 46 (concrete and soil). In general, larger values of K t , along with a high friction coefficient, are used to represent a strong bond between the materials at the interface. ...
The recently upgraded six Degree‐of‐Freedom Shaking Table, LHPOST6, at UC San Diego, underwent a series of forced vibration tests to evaluate the post‐upgrade dynamic response of the foundation‐soil system. The resulting data were instrumental in obtaining frequency response curves of the system, which were used to determine its low‐strain natural frequencies, effective viscous damping ratios, and reaction mass displacements. The extensive experimental data motivated the creation of a detailed Soil‐Structure‐Interaction model of the reaction mass‐soil system. The structure and soil were modeled using 3D Finite Elements in STKO‐OpenSees and calibrated with the acquired data via a parametric study. The 3D continuum FE model and the calibration procedure based on a single soil parameter (shear wave velocity profile) proved to be an effective tool to reproduce the experimental results accurately. This paper describes the Finite Element model, its calibration, and validation. In addition, the paper provides suggestions to simplify continuum models and promote their use in professional practice. The objectives of this campaign, alongside the growing accessibility of high‐performance computing, may serve as a step toward using SSI continuum models in the industry.
... Discontinuum models of rock masses (Cook, 1992;Goodman, 1975;Goodman et al., 1968;Leopold Müller, 1963;Talobre, 1957) offer critical insights into how fractured rock formations deform and conduct fluids under varying stresses and pressures. As fractures and faults substantially affect both stiffness and flow paths, they must be explicitly accounted for in any realistic geomechanical assessment. ...
... Fracture closure and hence permeability change is controlled by specific stiffness -defined as the reciprocal of fracture closure with incremented normal stress. Relations between geometric and mechanical properties (Bandis et al., 1983;Goodman, 1975;Goodman et al., 1968;Swan, 1983;Zimmerman & Bodvarsson, 1996) ultimately link these to diagnostic measurements that probe fracture stiffness, as fracture stiffness strongly impacts hydromechanical behavior. Ultrasonic methods, initially applied in the early 1990s (Myer et al., 1990;Pyrak-Nolte & Nolte, 1992;Pyrak-Nolte et al., 1990a, 1990b, have further evolved to incorporate cross-coupling modes during shearing (Choi et al., 2014;Nakagawa et al., 2003Nakagawa et al., , 2004; and to encompass both laboratory (Lubbe et al., 2008;Shokouhi et al., 2020;Wood et al., 2024) and field-scale applications (Worthington, 2007(Worthington, , 2008. ...
Fractures and faults represent planes of weakness and compliance in rock masses that serve as focal points for both microearthquakes and fluid transport, with seismicity and permeability evolution closely linked. Contact stiffness is highly stress‐sensitive and directly influences permeability. We explore the co‐evolution of specific stiffness and permeability of rough fractures under normal stress and shear offset using numerical simulations. Individual rough fractures are represented by variable amplitude (Root mean square) and wavelength (λ) using a granular mechanics model. Contacting rough surfaces are mated, offset in shear, and then compacted in displacement mode. The compacting fractures generate stress‐dependent changes in contact porosity, which govern both permeability and stiffness evolution. We establish a universal dimensionless relationship linking specific stiffness and permeability that inherently incorporates the effects of surface roughness, shear offset, and microcracking. The observed cracking effect—where local stress redistribution and pressure‐driven microcrack propagation dynamically alter the aperture field—introduces a nonlinear permeability response at high stress. Increased roughness amplitude and larger shear offsets reduce stiffness while dampening permeability sensitivity to stress, demonstrating a strong interplay between surface texture and hydro‐mechanical behavior. While the model captures this behavior effectively, deviations emerge at very low porosities due to extreme aperture sensitivity in this limit.
... Within this context, this paper proposes a Finite Element model based on elastic continuum with zero-thickness interface elements of the Goodman type. 39 These interface elements, which are placed along the contact surfaces between the continuum finite elements (equivalent to the ''cohesive'' elements used in other studies), are equipped with a recently developed fracture-based visco-plastic constitutive law. 40 The model, which also includes an excavation procedure to remove spalled blocks totally detached from the rock mass, is discussed and applied in the context of rock spalling in deep underground excavations in hard, brittle rock. ...
... Zero-thickness interface elements, sometimes referred to as ''cohesive elements'', were originally proposed by Goodman and co-workers 39 and were later used by several authors, such as Refs. 42-46. ...
Rock spalling is a brittle failure process that occurs around tunnels excavated in hard rock under high in-situ stress states. In nuclear waste disposal, spalling in fractured crystalline rock could create connected fractures, potentially providing pathways for radionuclides. Robust numerical models are therefore needed to evaluate the extent of rock spalling so that the design and layout of a prospective deep geological repository can be optimized and made fit for purpose. With this motivation in mind, this study proposes a methodology for the numerical analysis of rock spalling based on zero-thickness interface elements with a visco-plastic-fracture constitutive law, combined with a workflow for finite element removal/excavation. To simulate spalling, zero-thickness interface elements are pre-inserted along a sufficient number of mesh lines with random orientation within the rock mass. A uniform initial stress state is generated and the excavation of the circular tunnel is performed by removing the corresponding elements, which leads to stresses in excess of the elastic limit in some of the interfaces, and subsequent visco-plastic fracture openings. A criterion for excavation of the finite elements around the tunnel is established when a block which is totally surrounded by failed interfaces is totally detached or can slide off the mesh following a kinematically admissible path. The excavation of blocks causes a stress redistribution around the tunnel and this leads the need for new excavation steps, until a new equilibrium configuration is reached. The proposed methodology is applied to assess rock spalling in the Mine-by Experiment at the Atomic Energy of Canada Limited's (AECL's) Underground Research Laboratory in the massive Lac du Bonnet granite. The focus of the analysis is to understand the mechanisms and the influencing factors that lead to brittle failure, and to calibrate material properties to reproduce both the final stress state of the tunnel and its spalling depth.
... Effective pre-cave assessment and understanding of in-situ fragmentation are vital for ensuring successful cave propagation and managing subsequent fragmentation processes (Laubscher, 2003;Pierce, 2010). The significance of in-situ rock block distribution extends to the extensive tunneling and excavation demands in cave mining development (Goodman et al., 1968). ...
... This aligns with probabilistic geometry principles, as the probability of plane intersections increases with more fracture sets distributed in various directions. This observation is supported by the findings of Goodman et al. (1968), Hatzor and Feintuch, (2005), Jimenez-Rodriguez and Sitar, (2008) and Hekmatnejad et al. (2021a,b), who demonstrated that the probability of rock block formation increases with the number of intersecting discontinuity sets. According to Fig. 5A and B, for P32 values greater than 3.25, the fluctuation in the average block volume is low and independent of both mean discontinuity size and orientation. ...
... The normal elastic response of a fracture is often quantified by the fracture normal stiffness κ (Goodman et al., 1968;L. Pyrak-Nolte & Morris, 2000;Morris et al., 2017;Jaeger et al., 2009;P. ...
... if one introduces the characteristic stiffness χ = 1/ l 0 and σ ref n as the reference normal stress, which refers to the effective normal stress level at the beginning of the test when u b = 0, and is generally a small quantity (Goodman et al., 1968). The characteristic stiffness χ denotes the slope of the log-stress against the closure curve and defines the rate of change of normal stiffness and normal stress (Evans et al., 1992). ...
We study how the normal stiffness and the permeability of a realistic rough fracture at the field scale are linked and evolve during its closure up to the percolation threshold. We base our approach on a well‐established self‐affine geometric model for fracture roughness, which has proven to be a relevant proxy from laboratory to multi‐kilometer scales. We explore its implications for fracture apertures in reservoir‐scale open channels. We build our approach on a finite element model using the MOOSE/GOLEM framework and conduct numerical flow‐through experiments in a 256×256× 256 m3 granite reservoir hosting a single, partially sealed fracture under variable normal loading conditions and undrained conditions. Navier‐Stokes flow is solved in the embedded 3‐dimensional rough fracture, and Darcy flow is solved in the surrounding poroelastic matrix. We study the evolution of the mechanical stiffness and fluid permeability of the fracture‐rock system during fracture closure including mechanisms that impact the contact surface geometry like asperity yield and deposit of fracture‐filling material in the open space of the rough fracture. The largely observed stiffness characteristic is shown to be related to the self‐affine property of the fracture surface. A strong anisotropy of the fracture permeability is evidenced when the fluid percolation thresholds are exceeded in two orthogonal directions of the imposed pressure gradient. We propose a unifying physically based law for the evolution of stiffness and permeability in the form of an exponential increase in stiffness as permeability decreases.
... Нормальное поведение трещины может быть описано гиперболической моделью, предложенной в [18,19], как: ...
... -для конкретного значения θ; -для σ ci от 50 до 100 МПа на 5 шагов (60, 70, 80, 90, 100 МПа); -для JRC от 16 до 20 на 4 шага (17,18,19,20); -для каждой пары JRC и σ ci рассчитывается E, а их среднее значение приписывается значению θ. ...
The anisotropy in the deformational behavior of blocky rock masses has been comprehensively investigated. The uniaxial deformation modulus was selected as the key parameter. This modulus is generally anisotropic and depends on the loading direction, as well as on the properties of the intact rock, joints, and joint setting. Representative volumes of blocky rock masses were numerically simulated using the discrete element method and were loaded uniaxially in various directions. Subsequently, the failure mode and the deformation modulus were studied for different loading directions and various relative joint settings. A new nonlinear, stress- dependent stiffness matrix for joints was introduced, incorporating the surface conditions of the joints in terms of the Joint Roughness Coefficient (JRC) and the properties of the intact rock materials in terms of the Uniaxial Compressive Strength (UCS). The results of the assessments are presented in the form of rose diagrams, showing variations in the deformation modulus of the blocky rock mass that depend on the joint’s JRC, the intact rock’s UCS, and the structure of the rock mass in term of the relative joint angle. Also, the expected degree of anisotropy for various joint surface conditions and uniaxial compressive strengths of intact rock were introduced. In the Geological Strength Index (GSI) table, results are classified such that assigning a value to the JRC for each class of joint surface conditions allows for the corresponding deformation modulus and degree of anisotropy. According to this chart, it is deduced that the effect of joint roughness on the deformation modulus of blocky rock masses is greater than that of the intact rock UCS. The results support the hypothesis that a blocky rock mass has a critical strain that is independent of the loading angle (θ) and the orientation of the third joint set (α).
... Based on centrifugal model tests, Weng et al. [35] reproduced the entire process of buckling instability of stratified rock slopes and revealed the impact of scale effects on the buckling instability and failure mechanism of stratified rock slopes. The scale effect refers to the phenomenon where the size of a slope influences its deformation and failure [63,64]. Recently, based on shaking table tests, Li et al. [65] physically modeled the failure process and acceleration responses of two interbedded dip slopes under different seismic intensities, and they found that buckling failure tends to occur when the dip of rock layer is less than 65 • . ...
The flexural buckling failure is a relatively common instability phenomenon in rock slopes both at the small and large scale. It poses a serious threat to the normal operation and maintenance of nearby infrastructure and the life and property safety of surrounding residents. A profound understanding of the deformation and failure mechanism of flexural buckling, as well as the development of quantitative approaches for buckling stability analyses, are of great significance for the risk identification and control of buckling landslides. This study combines the literature relevant to flexural buckling of rock slopes by systematically reviewing the research progress and trends from 1970 to 2023, following the research path of “triggering mechanism → analysis methods”. Based on this proposition, the intrinsic and triggering factors that influence the deformation process of flexural buckling failure in dip rock slopes are detailed and clarified. Then, the main progress achieved in physical, analytical, and numerical modelling regarding the stability and run-out analysis of buckling landslides is comprehensively introduced. Finally, this study provides some outlooks for future research and practice in the field of buckling landslides.
... Although the frictional contact relationship is conceptually straightforward and physically meaningful, it does not fully capture the normal embedment and tangential shear plastic failure between the soil and pile, potentially introducing discrepancies when simulating cyclic loading. The Goodman element, a zero-thickness, four-node contact element with eight degrees of freedom, was proposed by Goodman to better represent the development of tangential stress and deformation at the contact surface, considering nonlinearity and with simple parameter determination (Goodman et al. 1968). Goodman also suggested that the contact surface has no thickness, only length, before being subjected to force, which is consistent with the surface-to-surface contact approach in ABAQUS (Fei and Zhang 2010). ...
This paper presents a study on model tests of single energy piles subjected to cyclic axial loads in sand and the development and validation of a 3D thermo-mechanical finite element model. The model accurately simulated the behavior of the pile–soil interface under cyclic shear loads. A subsequent parametric analysis examined the effects of the number of loading cycles and the loading amplitude on the vertical dynamic response characteristics of energy piles. The results showed that under heating conditions, the maximum variation in compressive thermal stress in the energy pile gradually decreased, with its location shifting upward along the pile shaft. A critical cyclic amplitude ratio was identified: below this threshold, the rate of increase in pile tip resistance continuously increased while the average pile side resistance weakened progressively. The presence of a static load accelerated the weakening of the average pile side resistance to some extent. As the number of loading cycles increased, the settlement rate of the energy pile gradually degraded. The cumulative settlement rate at the pile top increased with the cyclic amplitude ratio, peaking before slightly declining. In comparison, the static load ratio had a relatively minor influence on cumulative settlement.
... The interaction between deformable blocks is carried out using joint finite elements (FEs) [46,47] that require (i) a small-displacement hypothesis; (ii) that each block's finite element boundary discretization is compatible with the boundary finite element discretization of the neighboring blocks. When compared with the DEM, which handles large displacements [42], the FEM-based joint element approach makes it easier to incorporate nonlinear elastoplastic and damage models. ...
For the safety assessment of concrete dam–foundation systems, this study used an explicit time-stepping small-displacement algorithm, which simulates the hydromechanical interaction and considers the discrete representation of the foundation discontinuities. The proposed innovative methodology allows for the definition of more reliable safety factors and the identification of more realistic failure modes by integrating (i) softening-based constitutive laws that are closer to the real behavior identified experimentally in concrete–concrete and concrete–rock interfaces; (ii) a water height increase that can be considered in both hydraulic and mechanical models; and (iii) fracture propagation along the dam–foundation interface. Parametric studies were conducted to assess the impact of the mechanical properties on the global safety factors of three gravity dams with different heights. The results obtained using a coupled/fracture propagation model were compared with those from the strength reduction method and the overtopping scenario not considering the hydraulic pressure increase. The results show that the safety assessment should be conducted using the proposed methodology. It is shown that the concrete–rock interface should preferably have a high value of fracture energy or, ideally, higher tensile and cohesion strengths and high associated fracture energy. The results also indicate that with a brittle concrete–rock model, the predicted safety factors are always conservative when compared with those that consider the fracture energy.
... Modeling of rocks and rock masses during that period was carried out by methods of mechanics of both continuous [10, 11] and discrete medium [12,13,14,15]. To account the fracture disturbance, methods of mechanics of porous and fractured media were used, which believed the properties averaging of the geomedium representative volume The application of these methods showed the presence of one contour zone of the host rock discontinuity for the majority of mine openings and underground structures, the construction of which was carried out in typical conditions for that period at average depths up to 500 meters. ...
... Goodman et al. [12] proposed a zero-thickness interface four-node element. Subsequent studies revealed that a shear band was observed at the interface between sand and the rough steel plate [13,14], with the shear band thickness ranging from 5 to 14 D 50 [15][16][17]. ...
The entire deformation and overturning of numerous engineering structures commence from the failure of the interface between engineering structures and environmental soils, and the shear band formed by such failure results in variations in the water transfer law within the soil. In this study, a direct shear test was carried out to analyze the alterations in dry density of the soil both inside and outside the shear band before and after the disruption of the interface between lean clay and structure bodies, and the effect of the shear band on water migration in lean clay in the interface area under different shear displacements and normal stress values was examined. A numerical model of water transfer in lean clay with the shear band was constructed to predict the soil water distribution in the interfacial band across various temporal and spatial conditions. The results indicated that the existence of the shear band in the interface delayed the water migration; shear displacement and normal stress substantially affected the rate of water migration and volumetric water content in the interface region. The established water migration model could effectively simulate the migration patterns of water in the interface region and model the entire process of changes in free water in the soil under different spatial and temporal conditions. The research findings can provide a reference for the evaluation of structural permeability stability in hydraulic engineering.
... It should be noted that both the tensile and shear strength of interlayers inside brittle materials are commonly affected by strain rate. However, many conventional cohesive zone models in numerical simulations have not taken rate effect and friction into account [22][23][24][25]. Although several coupling cohesion-friction models were established in the FEM [26], for most of them, friction can only act when the cohesive forces completely vanish, which is therefore unreasonable compared with the actual failure process of brittle materials. ...
To characterize the rate‐dependency and frictional behavior of quasi‐brittle interface material, a coupling rate‐dependent and friction interface model improved from the Park‐Paulino‐Roesler (PPR) cohesive model, is proposed and validated in this paper. Based on the potential function, this novel coupling model forms the basic relationship of traction‐displacement within the interfaces, in which rate effect and friction behavior are considered by constructing a rate‐sensitive item and smooth friction term, respectively. Specifically, governing equations for typical mode I and mode II crack formation, as well as for normal and tangential directions, are established, and the model includes a complete unloading/reloading mode for the complex loading situations. To validate this model, the 3D simplified shear test of the anchor rod and mortar block model and a three‐point bend test of the composite concrete‐FRP beam with different loading rates are established to verify the engineering availability, considering different loading rates and friction coefficients. The results show that shear and tensile behaviors of brittle material in numerical models and laboratory tests are similar in the fracture initiation and propagation characteristics. The proposed model can reflect not only the elastic, softening, and residual stages, but also the strength rate‐related effects and friction effects of interface materials. This provides a comprehensive solution for describing the complex mechanical behavior of quasi‐brittle materials subjected to tensile and shear loads.
... The normal behavior of a joint can be described by the hyperbolic model proposed by Goodman et al. (1968) and Bandis et al. (1983) as: ...
... The same mathematical procedure used for the porous medium is applied to fractures, which are represented in the model as 2D geometries with zero thickness. This approach simplifies the mesh design and enhances computational efficiency by mathematically incorporating the fracture apertures into the numerical equations without any physical thickness within the mesh [67][68][69]. ...
Although Aquifer Thermal Energy Storage (ATES) systems are widely researched, Fractured Thermal Energy Storage (FTES) systems are comparatively underexplored. This study presents a detailed numerical model of a fractured granitic reservoir at the Bedretto underground laboratory in Switzerland, developed using COMSOL Multiphysics. Energy efficiency was evaluated across different flow rates and well configurations, including single-well and doublet systems, as well as for two different temperatures, namely 60 °C and 120 °C. The doublet configuration at an injection temperature of 60 °C with a flow rate of 2 kg/s demonstrated the highest energy efficiency among the cases studied. Potential applications for the stored heat are discussed, with scenarios including district heating for the nearby village and greenhouse heating. The results show that although FTES is associated with unique challenges, it has significant potential as a reliable thermal energy storage method, particularly in regions without suitable aquifers. It can also be considered as a cost-effective and competitive approach for climate mitigation (assuming the system is solely powered by solar-PV). This study provides insights into the viability and optimization of FTES systems and highlights the role of fracture/fault properties in enhancing energy efficiency.
... The FEM with interface elements is a type of embedded discontinuity method widely used in geotechnical engineering applications for simulating the coupled HM process in fractured porous media. [45][46][47][48][49] The lower-dimensional interface element approach in the finite element method (LIE-FEM) 50 is a modified adaptation of XFEM to include pre-existing fractures. It combines features of both XFEM and the embedded discontinuity methods, representing fractures with lower-dimensional elements, applying local enrichment only at the fracture elements for a discontinuous displacement function, and using a local enrichment FEM approximation for a fracture relative displacement function. ...
... 42 The Goodman element is one of the commonly used interface elements, which possesses a simple unit form and clear physical significance. 43 Its superiority in addressing a variety of problems with ease of adjustment has led to its widespread application in simulating rock fractures, artificial block structures, and the contact interfaces between soil and structures, among other issues. The stress-strain relationship of Goodman element is expressed as ...
The propagation of waterflood-induced fractures (WIFs) occurs during prolonged water injection and is influenced by the distribution and properties of natural fractures (NFs). Available numerical models rarely consider fracture activation and rupture in an integrated manner, which makes it difficult to reflect complex fracture morphology. In this paper, we propose a hydraulic-mechanical model with strain-dependent damage variables to describe the dynamic expansion characteristics of WIFs. There are discrete filled NFs in the matrix with non-equal-thickness joint elements, for which we derive the constitutive equations to calculate fracture widths during water injection and production. Damage variables for the matrix and fractures are calculated according to the maximum tensile stress criterion and the Mohr–Coulomb criterion. A comparison between the coupled model and experimental results is conducted to demonstrate its validity. Finally, we simulated and analyzed four influencing factors of the pressure response and fracture evolution. The study demonstrates that fracture behavior and damage area evolution are highly sensitive to injection rate, communication sequence, NF density, and orientation. The activation, cross, and capture interactions between NFs and WIFs complicate the fracture-damage network and enhance seepage efficiency. High injection rates promote crack tip propagation, while lower rates facilitate the evolution of secondary fractures at low pressure. For high NF density reservoirs, low-pressure injection fully activates NFs, aiding damage evolution. In low NF density reservoirs, excessive pressure induces simpler fracture morphologies, making unstable water injection more effective than continuous injection. This work guides appropriately induced fractures to improve water absorption in tight reservoirs.
... A multitude of scholars has investigated the rocks' mechanical properties and failure mechanisms or the materials as rock-like which belong to F-T cycles [8,9]. Indoor experiments are commonly utilized to assess their physical and mechanical characteristics. ...
Investigating the mechanical properties and microscopic damage behavior of fissured rock masses subjected to freeze–thaw (F-T) cycles is essential for informing stability evaluation and disaster prevention strategies in geotechnical engineering within cold regions. In this study, a numerical simulation model of rock mass specimens with double cruciform fissures was developed applied PFC2D, and uniaxial compression strength (UCS) tests were performed until the F-T cycles of 10, 20, 30, and 40 values for strength variation and damage characteristics assessed. The results indicate a distinct trend in damage evolution: Tensile cracks predominate during the early F-T cycles, while the proportion of shear cracks increases significantly with the number of cycles, rising from 5.89% at 0 cycles to 17.95% subjected to 40 cycles. A comparison of the cracks evolution in rock specimens between 0 and 40 F-T cycles at various inclinations revealed that the damage initially occurs at the tips of prefabricated fissures. After 40 F-T cycles, damage at these tips intensified markedly, accompanied by numerous surface cracks on rock specimens experiencing freeze–thaw deterioration. UCS tests on the models demonstrated that when only one fissure is altered, peak stress exhibits an N-shaped variation relative to changes in the fissure angle, with relatively small variances (4.77 for peak stress variance and 0.04 for modulus variance). In contrast, when both fissures are adjusted simultaneously, variances increase sharply to 12.9 and 0.16, respectively; maximum strength occurs at angles of 30° and 75°, while minimum values are observed at angles of 15° and 60°. Finally, force chains and stress distribution within the rock samples were predominantly concentrated around the fissures and shifted responsively according to alterations in loading stress and fissure angle; following damage occurrence, a low-stress zone developed near the fissures which expanded as the angle increased.
... Goodman [35] examines the discontinuity of contact elements in relation to slippage and swelling, and develops a thick-less element featuring four nodes. Desai [36] proposed a thin-shell element that utilizes shear stiffness and normal stiffness determined through shear testing. ...
This paper systematically investigates the mechanical properties of contact surfaces between soil and piles under static loading conditions across various scenarios through comprehensive laboratory experiments. The study offers an overview of current experimental equipment and testing protocols, succinctly describes the mechanical characteristics of pile-soil contact surfaces influenced by multiple factors, and summarizes existing constitutive models related to these interfaces. The objective of this research is to provide a more accurate and reliable laboratory methodology for analyzing the static mechanical properties of pile-soil contact surfaces, thereby fostering innovative advancements in the safety of pile foundations.
... At this point, shearing leads to breakage of asperities and thus reduces the effective resistance available. Once the failure plane is created, the shear force starts decreasing until it becomes constant which is denoted on the figure as residual shear stress (Goodman et al. 1968; Barton and Choubey 1977;Martin and Chandler 1994). ...
Interfaces formed by rock surfaces against concrete are usually encountered in various geotechnical and civil engineering applications such as dam's foundations, rock piles and rock bolts. The shear behavior of discontinuities at the rock-concrete interface is a key issue for ensuring the stability and durability of the rock engineering structures. The shear behavior of such interfaces has been studied in the literature and few empirical formulae have also been proposed to represent the peak shear stress at the discontinuities considering the surface morphology and boundary condition. Nevertheless, the pre-peak and post peak stress profiles of loaded rock-concrete interfaces is of great importance in dynamic environments prone to time dependent failure and rock burst events. The complete stress-displacement surface model, CSDS, was recently updated by the authors for representing the progressive shear behavior at rock-rock interfaces. The model is used in this study specifically focusing on the behavior of rock to concrete interaction for joint interface instability problems. First, the key factors influencing deformability and shear strength at rock-concrete interfaces are recalled. Then, the application of the CSDS model is validated for unbonded rock-concrete interfaces with different roughness surface and boundary conditions. Conclusions and suggestions are provided at the end. RÉSUMÉ Les interfaces formées par les surfaces rocheuses contre le béton sont généralement rencontrées dans diverses applications géotechniques et de génie civil telles que les fondations de barrages, les pieux rocheux et les boulons d'ancrage. Le comportement en cisaillement des discontinuités à l'interface roche-béton est un enjeu clé pour assurer la stabilité et la durabilité des ouvrages d'art en roche. Le comportement en cisaillement de telles interfaces a été étudié dans la littérature et peu de formules empiriques ont également été proposées pour représenter la contrainte de cisaillement maximale au niveau des discontinuités en tenant compte de la morphologie de la surface et des conditions aux limites. Néanmoins, les profils de contraintes avant et après le pic des interfaces roche-béton chargées sont d'une grande importance dans les environnements dynamiques sujets à des ruptures et à des éclatements de roche dépendant du temps. Le modèle complet de surface contrainte-déplacement, CSDS, a été récemment mis à jour par les auteurs pour représenter le comportement de cisaillement progressif aux interfaces roche-roche. Le modèle est utilisé dans cette étude en se concentrant spécifiquement sur le comportement de l'interaction roche-béton pour les problèmes d'instabilité des interfaces de joint. Tout d'abord, les facteurs clés influençant la déformabilité et la résistance au cisaillement aux interfaces roche-béton sont rappelés. Ensuite, l'application du modèle CSDS est validée pour des interfaces roche-béton non liées avec différentes rugosités de surface et conditions aux limites. Des conclusions et des suggestions sont fournies à la fin. 1 INTRODUCTION The shear behavior of contacts between rock (and rock-like) surfaces varies according to geological and mechanical properties of the interfaces. Quantifying this behavior is an important part of conventional stability analysis for rock excavations and large foundations. Notable complexity arises when the contact materials are different. Natural infill and intrusive formations (e.g. dykes) represent special conditions which influence the shear behavior of interfaces. Contacts between natural rock formations and man-made concrete structures is of great interest in foundation design and construction of large retaining infrastructures (e.g. hydroelectric dams). The present article addresses the nature and quantifiable behavior of rock-concrete interfaces under shear loading conditions. Several studies have been performed in the literature to investigate the shear behavior at the interface between concrete and rock with different morphologies and varied boundary conditions (e.g., Lam and Johnstone 1989;
... For P2 and P3, the mesh included both sides of the tunnel axis ( Figure 4). For the ground, the lining and the piles, the mesh is made of 10-node tetrahedral; the soil-pile interface is modelled using the so-called 'joint elements' of CESAR, which are zero-thickness elements derived from those proposed by Goodman et al. (1968). ...
... Since the jointed rock mass has obvious anisotropy and discontinuity, and the internal structure of surrounding rock is constantly developing and changing during tunnel or chamber excavation, it is difficult to reasonably predict the anisotropy and discontinuity of the jointed rock mass by using the existing rock mechanics theory, such as the elastic-plastic theory [8][9][10]. Currently, laboratory experiments [11,12] or numerical simulations [13,14] are often used for research, and most of these experiments are conducted using rock-like the subset network's correlation coefficients throughout the deforming process for the displacement field and the strain [25]. ...
For jointed rock mass with anisotropy and discontinuity, the structure of the surrounding rock is constantly developing and changing during tunnel excavation. It is difficult to reasonably predict localized deformation of jointed rock mass by using the existing rock mechanics theory. In this paper, the failure characteristic of pre-holed jointed rock mass with three joint angles is experimentally investigated by adopting the digital image correlation and acoustic emission methods. To avoid the influence of measurement error on Digital Image Correlation (DIC) from discontinuous deformation, parametric studies and an optimized algorithm are also included in DIC tests. Results indicate that the perpendicular-jointed condition (0° joints) is the most dangerous situation because of its comparatively lower strength and brittle failure mode with a shift energy release. For rocks with different jointed angles, localized deformation emerges after the material enters the plasticity. Significant localization occurs after the failure with cracks surrounding the center hole and pre-existing joints.
The shear and normal stiffness of fractures are important parameters governing deformation behaviour of sedimentary rocks. However, shear stiffness is challenging to measure and is often inadequately applied in geotechnical design. Shear stiffness is frequently estimated without testing, instead based on assumptions and represented by a single value, despite the knowledge it should be represented as a function to capture variability. Most published stiffness data for sedimentary rocks are for joints at low normal stresses, with limited data for bedding parallel fractures at normal stresses above 1 MPa. This paper addresses this gap by presenting 55 shear stiffness results for bedding parallel fractures, tested under applied effective normal stresses ranging from 1 to 15 MPa. Results show a strong correlation between applied normal stress and shear stiffness, with shear stiffness increasing as normal stress increases. The results are then used to determine the suitability of current shear stiffness estimation methods. The findings highlight the need for significant improvement in current shear stiffness guidelines and emphasise the need for guidelines to better account for system influence with data processing protocols incorporating actual intact data and using displacement for determining the seating influence cut-off. Additionally, the common practice of using normal-to-shear stiffness ratios for estimates should be discontinued, as it is unsupported by data. Furthermore, predictions demonstrate the impact shear stiffness has on depth of fracture, supporting the necessity of representing stiffness as a function rather than a single value.
Interfaces widely present in nature and civil engineering are the most fundamental elements causing discontinuous deformation and failure. The Goodman joint element is incorporated into the vast majority of commercial and proprietary finite element programs due to its simplicity. However, the numerical properties of the Goodman element are not ideal. A lot of effort has gone into enhancing its numerical properties, only to make some empirical suggestions for adjusting mesh and computation parameters, yet there are still many examples that fail to reach convergence or incur drastic numerical oscillations of contact stress. Considering that the Goodman element is implemented within the framework of the displacement method, and poorlyposed problems in the displacement method are usually well‐posed problems in the force method, this study proposes a force method version of the Goodman element, abbreviated as FMVGE. The primal unknowns in FMVGE are the contact stresses on the interface rather than nodal deformations. The core problem in FMVGE represents the contact conditions in the form of a quasi‐variational inequality (QVI). By employing process iteration rather than state iteration used in the displacement method, at the same time, FMVGE precisely satisfies the contact conditions, with convergence being theoretically guaranteed. Analysis of classic examples and engineering cases indicates that the numerical properties of FMVGE are significantly superior to the widely adopted master–slave block method. Appendices A and B provide the proof of the convergence of the FMVGE solution and the core Matlab code for solving the QVI, respectively.
A comprehensive review on Geo-interface modeling with material point method (MPM) was published in the Journal of Rock Mechanics and Geotechnical Engineering.
RESUMO - A análise de estabilidade de taludes é de fundamental importância em projetos de infraestrutura, não importa seu tamanho. A falha de um talude tem um grande impacto econômico e ambiental e pode inviabilizar o projeto, causando danos irreparáveis ao meio ambiente e à vida. Portanto, a análise de estabilidade de taludes tem como objetivo avaliar a possibilidade de um talude sofrer movimentos de massa e propor soluções de estabilização. Esta pesquisa focou na obtenção dos parâmetros de resistência dos materiais encontrados no talude de escavação do vertedouro da Barragem de Misicuni aplicando o método de análise reversa e usando dados de instrumentação deformacional. A modelagem de parâmetros pode dar suporte a projetos futuros em termos de comportamento mecânico e segurança para a Barragem de Misicuni, além de projetos de engenharia subsequentes em massas de rochas sedimentares deformadas por influência tectônica. A Barragem Misicuni faz parte do Projeto Múltiplo Misicuni localizado no departamento de Cochabamba, parte central da Bolívia. Este projeto consiste em usar água das bacias dos rios Misicuni, Viscachas e Putucuni. Para realizar este artigo, três investigações de núcleo de perfuração foram usadas ao longo da seção de análise, bem como dados de leitura de dois inclinômetros instalados na escavação do talude de bancada, levantamento topográfico e 39 testes de compressão uniaxial em amostras coletadas dos núcleos de perfuração. Inclinômetros são instrumentos comumente utilizados para monitorar taludes de solo ou rocha, e auxiliam na determinação da superfície de falha e estimativa da velocidade de movimentação. O uso de dados de inclinômetros e modelagem geológica possibilitou analisar e obter parâmetros de tensão e deformação ocorridos durante as fases de escavação, por meio da aplicação da análise do Método dos Elementos Finitos (MEF), que permitiu comparar os resultados previstos e observados durante a execução e conclusão do local. Portanto, este artigo apresenta os parâmetros de resistência obtidos por meio de retroanálise usando dados de deformação obtidos de instrumentação de campo em um contexto de geologia estrutural complexa. Palavras-chave: Maciço rochoso. Estabilidade de taludes. Instrumentação. MEF. ABSTRACT - Slope stability analysis is of fundamental importance in infrastructure projects no matter its size. The failure of a slope has a major economic and environmental impact and can turn the project unfeasible, causing irreparable damage to the environment and life. Therefore, slope stability analysis aims to evaluate the possibility of a slope suffering mass movements and to propose stabilization solutions. This research focused on obtaining the resistance parameters of the materials found in the excavation slope of the Misicuni Dam spillway applying the back analysis method and using deformational instrumentation data. The parameter modeling can support future projects in terms of mechanical behavior and safety for the Misicuni Dam in addition to subsequent engineering projects in sedimentary rock masses deformed by tectonic influence. The Misicuni Dam is part of the Misicuni Multiple Project located in the Cochabamba department, central part of Bolivia. This project consists of using water from the Misicuni, Viscachas and Putucuni river basins. To carry out this paper, three drill core investigation were used along the analysis section, as well as read out data from two inclinometers installed on the bench slope excavation, topographic survey and 39 uniaxial compression tests on samples collected from the drill cores. Inclinometers are instruments commonly used to monitor soil or rock slopes, and help to determine the failure surface and estimate the speed of movement. The use of inclinometer data and geological modeling made it possible to analyze and obtain stress and strain parameters that occurred during the excavation phases, by applying Finite Element Method (FEM) analysis, which allowed to compare the predicted and observed results during the execution and completion of the site. Therefore, this paper presents the resistance parameters obtained through back analysis using deformation data obtained from field instrumentation in a context of complex structural geology. Keywords: Rock Mass. Slope Stability. Instrumentation. FEM.
Waterflood-induced fracture (WIF) is often found in tight reservoirs with the water injection operation, which can significantly exacerbate reservoir heterogeneity and result in unidirectional advancement of injected water. Accurate characterization of dynamic propagation behavior of WIFs is crucial during development plan design, reservoir numerical simulation, and stimulation measure selection. However, currently-used reservoir simulation software tends to overlook dynamic propagation behavior of WIFs, and simplify the WIFs into a time-independent fracture with a specified direction and fixed length. In response to this issue, we established a damage-based finite element model for WIF, considering the stress sensitivity effect of matrix and the interaction effect of natural fractures and matrix.
A coupled hydro-mechanical-damage (HMD) model is established. We define strain-related damage variables to facilitate the calculation of fracture and matrix evolution in a unified form. The presence of filler content in natural fractures under initial conditions is regarded as a filled joint element with a certain thickness. The stiffness of fractures is derived with reference to the Goodman joint, which is used to calculate the normal/shear displacement of natural fractures. The porosity and permeability are related to stress and strain, and dynamically change during the simulation process. The coupling model is solved using a finite-element numerical simulator to obtain the deformation and pressure change of the reservoir during the water injection process. Finally, a case study of China's AS Oilfield is conducted using the proposed method to discuss the pressure response characteristics, mechanical characteristics of natural fractures and WIF extension trajectories, under two working schemes of single well injection-stewing-production and one injection well and two production wells.
The results show that the fracture characteristics during water injection period can be summarized in three forms: generation of WIFs, activation of natural fractures, and communication of natural fractures. WIFs appeared first in the injection well, extended along the direction of the maximum horizontal principal stress, and appear to be locally deflected when the natural fractures are around. With formation pressure increasing, the natural fractures near the injection well gradually open up, while the distant ones appear to close. When WIFs communicate natural fractures, the width of which is significantly increased. The results also show that the WIFs show a better effect of enhancing the water supply capacity of the reservoir, which helps to replenish the formation pressure. The closer the leading edge of the WIFs is to the production well, the better the effect of replenishment of energy is. The established model furnishes a visual representation and offers a quantitative analysis of the fracture evolution process, presenting an analytical idea for time-dependent WIF research.
Persistency, as a key geometric parameter of joints, significantly affects shear strength parameters of jointed rock mass. A good understanding of how persistency affects shear behavior of joint is therefore crucial for better evaluation of stability of rock slope. To investigate the failure and micro-cracking behavior of non-persistent rock joint under direct shear, a novel Voronoi generation algorithm is first used to establish an improved grain-based model (GBM) of granite which considers the shape of feldspar. The calibrated model is then used to simulate the direct shear test of numerical models possessing different joint persistency under various normal stresses. The results reveal that the developed micro-cracks generally increase rapidly when the shear strain reaches to a value approximately 50 % of the peak shear strain and the grain boundary tensile micro-crack is dominant among these initiated micro-cracks. Micro-cracks generally initiate at the ends of rock bridge, and then gradually propagate to the central of rock bridge, forming en-echelon fractures. The failure mode of numerical model is closely related to the generated en-echelon fractures. An increase in both joint persistency and normal stress can lead to a shear failure. The finding in this study provides an important basis for understanding the mechanical behavior and failure mechanism of jointed rock mass.
It is widely recognized that high CFRDs, especially those with thinner slabs, exhibit significant three-dimensional effects. Three-dimensional finite element analysis offers a more realistic depiction of the stress distribution of the dam. However, there is a lack of studies that utilize stochastic dynamics and probabilistic analyses to investigate the seismic safety of three-dimensional CFRDs and establish corresponding safety evaluation criteria.
During shield tunneling through existing steel reinforced concrete structures, superstructure deformation is an important parameter that reflects the disturbance degree of engineering construction to existing structure. Precisely predicting structural deformation can help engineers adjust shield machine operational parameters and ensure the success of the project. There has been no attempt to study the feasibility and applicability of machine learning for predicting structural deformation when shield machine cut through existing structure. To address this problem, this paper proposes a novel hybrid model (DSGCN-TCN), combining dynamic spatial graph convolutional network (DSGCN) and temporal convolutional network (TCN), to predict structural deformation. First, dynamic adjacency matrix is constructed based on correlation coefficient and attention mechanism to describe the dynamic change of irregular graph structure. Then dynamic adjacency matrices and feature matrices as the input of the GCN model to extract the dynamic spatial feature of structural deformation data. Followed by TCN and attention layer to capture the temporal correlation of structural deformation data. Finally, the prediction performance of the proposed method is verified using measured data from practical engineering. The experiment results show that compared with the selected baseline models and sub-models, the proposed model can predict the structural deformation more accurately.
Piled raft foundation is a sustainable foundation approach in the current era due to the development of immense infrastructure. Initially, this foundation was primarily used for the construction of high-rise buildings in weaker soil bases, however, the application of piled raft systems is now extending to offshore and marine structures. This paper mainly concentrates on the review of available literature in the domain of the piled raft foundation and presents a critical review of the evolution of the piled raft from the past to till date. This state-of-the-art review starts from the early-age research and analytical works on the piled raft foundation. Various approaches suggested by several researchers are described here. The review also contains the experimental research conducted on piled rafts including the numerical analysis carried out on the existing case studies. The applications of different numerical software for the modelling of piled rafts are also described elaborately and the research related to the parametric analysis is accumulated together. The load sharing nature, interaction behaviour, and the influence of the loading system are reviewed separately depending upon the contribution of the research papers to the particular domain. There exist several design processes and selection criteria for the piled raft foundation and this is mainly due to the inadequate perception regarding the nature of the piled raft foundation during its overall applications. This literature review would be helpful for other researchers to acquire a clear idea of the behaviour of piled rafts and also would be useful to identify the future prospects of the piled raft foundation.
Considering different sources of internal heterogeneity is crucial for studying the mechanical response and fracture process of brittle rocks. This approach enables the analysis of the relationship between phenomena occurring at the particle scale and the emerging macroscopic nonlinear behavior. Given the promising results of the Continuum Voronoi Block Model (CVBM) in representing rock behavior across different scales, the integration of material and contact heterogeneity into the model is presented in this paper. By integrating these sources of heterogeneity, the CVBM can better align with the Grain-Based Models (GBMs), which have proven effective in simulating brittle rock behavior. As a case study, the documented UCS and BTS tests on Äspö diorite samples were simulated. Sensitivity analyses were conducted for calibration parameters, mesh density, and Voronoi block size. Furthermore, the effects of spatial distribution and mineral structure on the overall model response were evaluated, comparing the results with the natural dispersion observed in laboratory tests. Notably, the characteristic and peak stresses exhibited higher sensitivity to the change in mineral arrangement. Finally, the capabilities of the heterogeneous CVBM to capture the average behavior and fracture process of brittle rocks are analyzed.
The Continuum Voronoi Block Model (CVBM), a pseudo-discontinuum modeling technique based on the Finite Element Method, was employed to investigate the impact of discontinuities on spalling phenomena around excavations in rocks under high-stress conditions. The CVBM’s ability to produce numerical results consistent with spalling was demonstrated through a case study of the Mine-by tunnel. The results show that the model can explicitly capture the formation of macro-fractures parallel to excavation walls, intact rock slabs, and V-shaped notches. The results of this case study support the application of CVBM for parametric analyses to investigate the role of discontinuities in spalling failure, integrating discrete fracture network (DFN) into the model. The influence of DFN parameters such as dip angle, spacing, persistence, position, and mechanical properties is also evaluated. It was found that discontinuities promote stress relief due to shear, thereby altering spalling damage around the excavation in highly stressed rock. This finding highlights the crucial role of discontinuities in influencing the behavior of excavations under such conditions.
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