Experimental Study on the Mechanical Properties of Deep Granite under Gradient-Confining Pressure

: In deep layers, the complex geological environment is characterized by high temperature and high stress which causes marked differences in the mechanical properties of granite compared to those of granite in shallow areas. To investigate the mechanical properties of deep granite, this paper utilizes conventional triaxial compression tests to determine the various mechanical properties and failure modes of deep granite under different confining pressures. The findings indicate that the elastic modulus, Poisson’s ratio, and peak strength of deep granite display greater dispersion than those of shallow granite under the influence of confining pressure. Based on the stress-strain curve, various characteristic stresses of deep granite under different confining pressures are calculated, and a distinct exponential function relationship exists between the characteristic stresses and the confining pressure. Furthermore, the finite element numerical simulation software Abaqus is employed to simulate the conventional triaxial compression of granite under different confining pressures, thereby revealing the stress and deformation evolution process of granite during the compression process. This research unveils the mechanical properties of deep granite under gradient-confining pressure, which can offer crucial theoretical evidence and data to support engineering applications in relevant fields.


Introduction
The deformation properties of rocks are extremely important in geological engineering and geology research [1][2][3].In particular, under diverse confining pressure conditions, the mechanical behavior and deformation mechanisms of rocks may undergo substantial alterations [4,5].For underground engineering, especially mining engineering, development towards greater depths is an inescapable trend [6].Various countries have different definitions for depth, and most countries consider 600 to 1200 m underground as the demarcation line between deep and shallow depths [7,8].Granite, as a crucial component of the deep earth, possesses unique geological features and has a complex formation process [9][10][11].
In-depth research is of tremendous significance for unveiling the evolutionary laws of the inner earth and ensuring the safety and stability of underground engineering [12,13].There are apparent differences between deep granite and shallow granite in many aspects.Deep granite exhibits some distinctive characteristics.It is formed in the relatively deep part of the crust and is subjected to a high-pressure and high-temperature environment.This results in its crystal size generally being small, with fine and tightly packed grains.The rock structure is compact, the rock strength is high, the rock has outstanding resistance to weathering, and the rock can maintain stability under complex geological conditions [14].The confining pressure exerts multiple influences on the mechanical properties of rocks.It can notably enhance the strength of rocks, enabling them to endure pressure and shear force more effectively [15].Moreover, it can modify the deformation characteristics of rocks, transitioning from brittle to ductile and increasing their toughness to better withstand deformation and destruction.Additionally, the confining pressure affects the fracture mode and permeability of rocks, and variations in confining pressure may lead to changes in the fracture mode and a reduction in permeability [16,17].
In previous studies, a multitude of scholars have conducted extensive and in-depth discussions regarding the properties of granite through a substantial number of tests and theoretical analyses, and these studies have revealed the substantial influence of confining pressure on the strength of the rocks, the deformation modulus, and the failure mode.These research findings have provided a solid foundation for a better understanding the failure mechanism of deep granite.Zhu et al. investigated the mechanical behavior of granite after heat treatment via conventional triaxial compression tests, and the results indicated that, with increasing temperature, the brittleness of the granite specimen decreased, while the ductility and plasticity increased [18].Yin et al. investigated the thermodynamic properties of granite under triaxial stress and discovered that the elastic modulus of coarse-grained granite exhibited an initial slow increase followed by a sharp decrease with increasing temperature [19].Sun et al. carried out cooling and heating cycles on granite at diverse temperatures and reported that, at a given heat treatment temperature, the thermal diffusivity of the granite decreased as the number of cycles increased [20].Nara et al.'s research revealed the presence of anisotropy in the long-term strength of granite [21].Zhao et al. examined the mechanical properties of jointed granite under high-temperature conditions and analyzed the failure patterns of jointed rock masses in conjunction with increases in temperature, decreases in temperature, and variations in joint type.Research has indicated that high-temperature conditions modify the mechanical properties of granite, particularly when the temperature is higher than 400 • C, at which point the peak stress sharply decreases [22].Hu et al. quantitatively investigated the damage to granite through mineral analysis, uniaxial tensile tests, uniaxial compression tests, and acoustic wave tests [23].Yang et al. conducted uniaxial compression tests on granite specimens under high confining pressure cyclic loads.Through nuclear magnetic resonance (NMR) and acoustic emission (AE) analyses, the porosity, secondary uniaxial peak strength, and dynamic fracture characteristics were obtained throughout the pre-damage test [24].Peng et al. investigated the type I fracture characteristics of granite specimens under diverse temperatures and burial depths via the semicircular bending (SCB) test.The test results demonstrated that there were significant differences in the mineral composition and crack characteristics of granite at different burial depths.The effects of the high-temperature notably weakened the influence of burial depth on the fracture characteristics of granite [25].Through triaxial compression creep tests under the action of temperature and disturbance loads, Li et al. reported that the triaxial compression creep curve of granite under the action of temperature and disturbance loads encompasses attenuation, stability, acceleration, and disturbance phases.Alongside the increase in the loading rate, both the shear strength and the residual strength decreased [26].Luo et al., based on the nanoindentation test, explored the microcrack evolution and crack characteristics of different minerals within granite.The results indicated that intracrystalline cracks play a crucial role in the failure process of brittle rocks and largely dominate the macroscopic properties of the materials [27].
In summary, despite the numerous existing studies on deep granite, the changes in the mechanical properties and characteristic stresses under the effects of different confining pressures are still not well understood.Moreover, the stress and deformation evolution processes during triaxial compression under high confining pressure and low confining pressure have yet to be investigated thoroughly.This paper will attempt to address these issues.

Specimen Preparation
The granite specimen for the triaxial test in this paper was obtained from the main shaft at the −1465 m level of the Shaling Gold Mine in Laizhou city, Shandong Province, China.Figure 1 presents a picture of the deep granite specimen under a binocular transmitted-reflected polarized light microscope.The specimen rock exhibits an obvious schistose structure, and the dark-colored minerals have directional arrangement.Under the microscope, the rock assumes a porphyritic structure, with the phenocrysts being orthoclase (approximately 65%), with a grain size of approximately 0.5 mm, and a matrix primarily consisting of quartz (approximately 30%), with a finer grain size (<0.1 µm), and the dark-colored minerals mainly being biotite (~5%).

Specimen Preparation
The granite specimen for the triaxial test in this paper was obtained from the main shaft at the −1465 m level of the Shaling Gold Mine in Laizhou city, Shandong Province, China.Figure 1 presents a picture of the deep granite specimen under a binocular transmitted-reflected polarized light microscope.The specimen rock exhibits an obvious schistose structure, and the dark-colored minerals have directional arrangement.Under the microscope, the rock assumes a porphyritic structure, with the phenocrysts being orthoclase (approximately 65%), with a grain size of approximately 0.5 mm, and a matrix primarily consisting of quartz (approximately 30%), with a finer grain size (<0.1 µm), and the dark-colored minerals mainly being biotite (~5%).Before carrying out the triaxial compression test, the process of preparing the granite specimen was carried out as follows: First, the specimen collection was carried out to select a granite rock block with good integrity, no fissures, and uniform texture to ensure that it could meet the requirements of the test.In line with the recommendations of the International Society for Rock Mechanics and Engineering (ISRM), the rock specimen was processed into a standard component of Φ50 × 100 mm.A rock column with a diameter of 50 mm was drilled onto the surface of the rock block, and then it was cut into a cylinder with a height of 100 mm using a cutting machine.The surface of the cut specimen was treated to remove its sharp edges and uneven parts by grinding.Upon the completion of the process, the dimensions of the samples need to be measured precisely to ensure that their side lengths strictly adhered to the standards proposed by the International Society for Rock Mechanics and Engineering (ISRM).After that, each sample was marked and recorded, as shown in Figure 2. In this test, a total of 12 standard rock specimens were fabricated.Throughout the entire preparation process, it was necessary to strictly control the quality of each step to ensure that the prepared granite specimen meets the requirements of the triaxial compression test to obtain accurate and reliable test results.Before carrying out the triaxial compression test, the process of preparing the granite specimen was carried out as follows: First, the specimen collection was carried out to select a granite rock block with good integrity, no fissures, and uniform texture to ensure that it could meet the requirements of the test.In line with the recommendations of the International Society for Rock Mechanics and Engineering (ISRM), the rock specimen was processed into a standard component of Φ50 × 100 mm.A rock column with a diameter of 50 mm was drilled onto the surface of the rock block, and then it was cut into a cylinder with a height of 100 mm using a cutting machine.The surface of the cut specimen was treated to remove its sharp edges and uneven parts by grinding.Upon the completion of the process, the dimensions of the samples need to be measured precisely to ensure that their side lengths strictly adhered to the standards proposed by the International Society for Rock Mechanics and Engineering (ISRM).After that, each sample was marked and recorded, as shown in Figure 2. In this test, a total of 12 standard rock specimens were fabricated.Throughout the entire preparation process, it was necessary to strictly control the quality of each step to ensure that the prepared granite specimen meets the requirements of the triaxial compression test to obtain accurate and reliable test results.

Specimen Preparation
The granite specimen for the triaxial test in this paper was obtained from the main shaft at the −1465 m level of the Shaling Gold Mine in Laizhou city, Shandong Province, China.Figure 1 presents a picture of the deep granite specimen under a binocular transmitted-reflected polarized light microscope.The specimen rock exhibits an obvious schistose structure, and the dark-colored minerals have directional arrangement.Under the microscope, the rock assumes a porphyritic structure, with the phenocrysts being orthoclase (approximately 65%), with a grain size of approximately 0.5 mm, and a matrix primarily consisting of quartz (approximately 30%), with a finer grain size (<0.1 µm), and the dark-colored minerals mainly being biotite (~5%).Before carrying out the triaxial compression test, the process of preparing the granite specimen was carried out as follows: First, the specimen collection was carried out to select a granite rock block with good integrity, no fissures, and uniform texture to ensure that it could meet the requirements of the test.In line with the recommendations of the International Society for Rock Mechanics and Engineering (ISRM), the rock specimen was processed into a standard component of Φ50 × 100 mm.A rock column with a diameter of 50 mm was drilled onto the surface of the rock block, and then it was cut into a cylinder with a height of 100 mm using a cutting machine.The surface of the cut specimen was treated to remove its sharp edges and uneven parts by grinding.Upon the completion of the process, the dimensions of the samples need to be measured precisely to ensure that their side lengths strictly adhered to the standards proposed by the International Society for Rock Mechanics and Engineering (ISRM).After that, each sample was marked and recorded, as shown in Figure 2. In this test, a total of 12 standard rock specimens were fabricated.Throughout the entire preparation process, it was necessary to strictly control the quality of each step to ensure that the prepared granite specimen meets the requirements of the triaxial compression test to obtain accurate and reliable test results.

Testing Equipment
The hard rock triaxial testing machine of the Key Laboratory of Deep Metal Mine Safety Mining of the Ministry of Education, Northeastern University, was used in the test.This testing machine can complete single-axis compression, triaxial compression, and rheological tests under complex paths.The maximum axial force output of this system is 2000 kN, and the maximum confining pressure is 100 MPa.In terms of test control, it can achieve axial deformation control, circumferential deformation control, and load control.The stress control rate range of the system is 0.1 kN/s to 5 kN/s, and the deformation control rate range is 0.003 mm/min to 0.02 mm/min.This hard rock triaxial testing machine consists of a high-stiffness load frame, a pressure chamber, an axial pressure loading system, a confining pressure loading system, a data acquisition system, and a servo control system.The overall machine stiffness is 9 GN/m, allowing for the acquisition of the curve of the complete stress-strain failure process of hard rock.The measurement of axial and circumferential deformations employs built-in differential variable pressure displacement sensors (LVDTs).The deformation measurement system consisted of 2 vertical LVDTs and 1 radial LVDT.This system is designed with a new type of low-friction self-stable ball head, ensuring the consistency of the measurement results of the vertical LVDTs on the left and right sides.A schematic diagram and an actual photograph of the testing equipment are shown in Figure 3.
The hard rock triaxial testing machine of the Key Laboratory of Deep Metal Mine Safety Mining of the Ministry of Education, Northeastern University, was used in the test.This testing machine can complete single-axis compression, triaxial compression, and rheological tests under complex paths.The maximum axial force output of this system is 2000 kN, and the maximum confining pressure is 100 MPa.In terms of test control, it can achieve axial deformation control, circumferential deformation control, and load control.The stress control rate range of the system is 0.1 kN/s to 5 kN/s, and the deformation control rate range is 0.003 mm/min to 0.02 mm/min.This hard rock triaxial testing machine consists of a high-stiffness load frame, a pressure chamber, an axial pressure loading system, a confining pressure loading system, a data acquisition system, and a servo control system.The overall machine stiffness is 9 GN/m, allowing for the acquisition of the curve of the complete stress-strain failure process of hard rock.The measurement of axial and circumferential deformations employs built-in differential variable pressure displacement sensors (LVDTs).The deformation measurement system consisted of 2 vertical LVDTs and 1 radial LVDT.This system is designed with a new type of low-friction self-stable ball head, ensuring the consistency of the measurement results of the vertical LVDTs on the left and right sides.A schematic diagram and an actual photograph of the testing equipment are shown in Figure 3.
The loading steps of the triaxial compression test are as follows: The confining pressure is applied to the predetermined value at a loading rate of 0.5 MPa/s, and the confining pressure is kept constant throughout the test.Using the axial stress control method, the axial pressure is applied at a loading rate of 0.5 kN/s.When the specimen shows dilatancy, it is switched to circumferential deformation control, and the loading rate is set at 0.015 mm/min until the specimen is damaged.
The loading steps of the triaxial compression test are as follows: The confining pressure is applied to the predetermined value at a loading rate of 0.5 MPa/s, and the confining pressure is kept constant throughout the test.Using the axial stress control method, the axial pressure is applied at a loading rate of 0.5 kN/s.When the specimen shows dilatancy, it is switched to circumferential deformation control, and the loading rate is set at 0.015 mm/min until the specimen is damaged.

Rock Mechanical Properties and Failure Morphology
Based on the test data, the various mechanical properties of deep granite under a gradient confining pressure are presented in Table 1, and the relationship curves of the elastic modulus, Poisson's ratio, and peak strength of deep granite under different confining pressures are depicted in Figure 4.As shown in Figure 4, with increasing confining pressure, the trends in the elastic modulus, Poisson's ratio, and peak strength of the deep granite are basically consistent, and there is a relatively large dispersion.This presents a substantial distinction from the outcomes of traditional triaxial tests on shallow granite.In traditional triaxial compression tests on granite subjected to gradient-confining pressure gradients, as the confining pressure increases, the elastic modulus tends to increase.This occurs as the increase in confining pressure leads to the closure of cracks within the rock, reducing the defects and discontinuities within, thereby enhancing the overall integrity and stiffness of the rock.Simultaneously, under the influence of confining pressure, the contact between the particles Appl.Sci.2024, 14, 4598 6 of 16 within the rock becomes closer, enhancing the bearing capacity of the rock.For deep granite situated in a high-stress environment, the rock is typically more compact, with a stronger binding between particles and a larger elastic modulus, while the sensitivity of the elastic modulus to the influence of confining pressure is not high.
Along with the increase in confining pressure, the inhibitory effect on the radial strain of the rock also correspondingly intensifies, resulting in a possible decrease in the Poisson's ratio.However, due to the greater stiffness and more compact internal structure of the deep granite, this leads to a reduction in the inhibitory effect of confining pressure on the radial strain, consequently exhibiting a significant degree of dispersion.
In most cases, as the confining pressure increases, the peak strength of the rock likewise increases.This is attributed to the fact that the confining pressure can suppress the expansion of cracks within the rock, facilitating a closer arrangement between the particles, resulting in a better overall integrity of the rock and an enhanced ability to withstand the load.However, considering that the internal structure of deep granite itself is highly compact and possesses a greater strength, coupled with the more complex geological conditions in deep rock compared to shallow rock, significant differences in the composition of the rock are observed.Consequently, there is also a degree of dispersion in the peak strength.Nevertheless, the dispersion is smaller than that of the elastic modulus and Poisson's ratio, indicating that the increase in confining pressure has a more substantial impact on the increase in the strength of the rock.
Figure 5 depicts the failure modes of the deep granite under diverse confining pressures.From the figure, it can be observed that at low confining pressures (ranging from 8 to 16 MPa), the rock exhibits multiple crack expansion and tensile failure.The failure surface of the tensile failure is approximately parallel to the axial loading direction, and in some instances, local peeling occurs within the rock.With increasing confining pressure, the tensile failure of the rock decreases, and the shear failure mode assumes a dominant position, with the failure surface also tending to become smoother.This is a result of the increase in confining pressure making it more difficult for cracks within the rock to expand, enhancing the overall integrity of the rock and increasing the overall strength and stiffness, enabling the sliding between the rock particles to become more stable, and thereby altering the failure mode.

Stress-Strain Curves and Characteristic Stresses
The rock stress-strain curve is an important tool used to show the relationship between the stress and strain acting on the rock during the loading process.It presents the mechanical properties and deformation laws of the rock in the form of a graph.Through

Stress-Strain Curves and Characteristic Stresses
The rock stress-strain curve is an important tool used to show the relationship between the stress and strain acting on the rock during the loading process.It presents the mechanical properties and deformation laws of the rock in the form of a graph.Through this curve, we can gain an in-depth understanding of the behavior and response of the rock under different stresses.The deformation of the rock mainly includes five stages, as shown in Figure 6.
the stress corresponding to the point where this stage begins is the yield stress σ f .In this stage, the rock fractures continue to develop until they are damaged.
The post rock fracture stage (after point D): The post rock fracture stage refers to the state in which the rock is in a state after fracture has already occurred.In this stage, the physical properties and mechanical behavior of the rock will change significantly.After fracture, the integrity and continuity of the rock are damaged, and the bearing capacity is greatly reduced.Cracks further expand and interconnect with each other.At the same time, the permeability of the rock increases, making it easier for water and other substances to penetrate it.This stage holds great significance for geological engineering and geotechnical engineering because it directly affects the stability and safety of the engineering structure.In practical engineering, it is necessary to conduct a detailed assessment and analysis of the post rock fracture stage to take appropriate measures to ensure the safety and stability of the project, such as using reinforcement measures or conducting monitoring.Figure 7 represents the stress-strain relationship curve of deep granite under a confining pressure gradient, which shows the interrelationship between axial strain, radial strain, and volumetric strain.The rock fracture compaction stage (the OA segment): The sedimentary or crystallization defects when the rock is formed, crustal movement, temperature changes, weathering, the action of water, mineral expansion, seismic activities, and heterogeneity can all cause small fractures in the rock.It has a significant impact on the mechanical properties and permeability of the rock.The volume of the rock in this stage decreases with increasing load, which is due to the gradual closure of the fractures inside the rock after being compressed and the formation of nonlinear deformation.The stress corresponding to the end of this stage is the closure stress σ cc .However, for hard rocks, because of their hard and dense texture, this stage is often not very obvious.
The rock elastic stage (AB segment): At this stage, the response of the rock to the external force is linear.After unloading, the rock will return to its size and shape at the initial stage of this stage.
The stable development stage of rock fractures (BC segment): After the elastic stage ends, the rock begins to exhibit initial fractures, and the stress corresponding to the initial crack is the initiation stress σ ci , which is the critical point for the rock to transition from a stable state capable of bearing the load to a state of producing cracks.The initiation stress is affected by various factors, such as the properties of the rock itself, the loading conditions, and the external environment.By studying the initiation stress, the mechanical properties of the rock can be further understood.In practical applications, the initiation stress has a important significance, and it provides a key indicator for assessing the stability of the rock and predicting the risk of rock fracture.After the rock is subjected to axial loading, it is axially manifested as a decrease in volume, while radially, it is manifested as an increase in volume.However, the initial stage of axial strain is often greater than that of radial strain, so the rock macroscopically experiences a decrease in volume.However, with increasing load, when the volume of the rock is compressed to the minimum, the volume of the rock will begin to slowly increase, and the stress corresponding to this point is the dilatancy stress σ cd .
The unstable development stage of rock fractures (CD segment): After entering this stage, the rock begins to transform from elastic deformation to plastic deformation, and the stress corresponding to the point where this stage begins is the yield stress σ f .In this stage, the rock fractures continue to develop until they are damaged.
The post rock fracture stage (after point D): The post rock fracture stage refers to the state in which the rock is in a state after fracture has already occurred.In this stage, the physical properties and mechanical behavior of the rock will change significantly.After fracture, the integrity and continuity of the rock are damaged, and the bearing capacity is greatly reduced.Cracks further expand and interconnect with each other.At the same time, the permeability of the rock increases, making it easier for water and other substances to penetrate it.This stage holds great significance for geological engineering and geotechnical engineering because it directly affects the stability and safety of the engineering structure.In practical engineering, it is necessary to conduct a detailed assessment and analysis of the post rock fracture stage to take appropriate measures to ensure the safety and stability of the project, such as using reinforcement measures or conducting monitoring.
Figure 7 represents the stress-strain relationship curve of deep granite under a confining pressure gradient, which shows the interrelationship between axial strain, radial strain, and volumetric strain.
The volumetric strain of the rock can be considered to be composed of the crack volumetric strain and the elastic volumetric strain.The indirect method is ordinarily employed to compute the crack volumetric strain, which is derived by deducting the elastic volumetric strain from the total volumetric strain of the rock.Under the conventional triaxial stress state where σ 2 = σ 3 , the volumetric strain of the rock mass can be expressed via Equation (1): In the equation, ε V is the volumetric strain, ε 1 is the axial strain, and ε 3 is the radial strain.
According to the generalized Hooke's law, combined with the conventional triaxial stress state, the elastic strain of the rock is [28]: In the equation, σ 1 is the maximum principal stress, σ 3 is the minimum principal stress, ε e 1 is the axial elastic strain of the rock, ε e 3 is the radial elastic strain of the rock, µ is Poisson's ratio, and E is the elastic modulus.
By combining Equations ( 1) and ( 2), the elastic volumetric strain of the rock can be derived as: where ε e V is the elastic volumetric strain of the rock.Then, the volumetric strain of the crack can be expressed as: Substituting the mechanical properties in Table 1 into Equation ( 4) enables f the crack volumetric strain of the rock under different stress states at each confining pressure to be derived.Via the stress-strain curve, four characteristic stress values of the rock can be Appl.Sci.2024, 14, 4598 9 of 16 acquired: the closure stress σ cc , the initiation stress σ ci , the dilatancy stress σ cd , and the yield stress σ f .The starting and ending points of the horizontal section of the crack volumetric strain-axial strain curve correspond to σ cc and σ ci , respectively, the turning point of the rock volumetric strain pertains to σ cd , and the point where the rock volumetric strain is zero relates to σ f .(g) (h) The volumetric strain of the rock can be considered to be composed of the crack volumetric strain and the elastic volumetric strain.The indirect method is ordinarily employed to compute the crack volumetric strain, which is derived by deducting the elastic volumetric strain from the total volumetric strain of the rock.Under the conventional triaxial stress state where  =  , the volumetric strain of the rock mass can be expressed via Equation (1): Based on the aforementioned calculation process, the characteristic stresses of the deep granite under the gradient confining pressure are obtained, as presented in Table 2.After eliminating the points with greater dispersion, a diagram of the relationship between the gradient confining pressure and the characteristic stress is constructed, as depicted in Figure 8, and an exponential function is utilized to fit the relationship between the confining pressure and the characteristic stress to obtain a fitting function such as Equation (5).
,  = 0.5267 Figure 8 and Equation (5) show that the relationship between the characteristic stresses of the deep granite and the confining pressure essentially conforms to an expo-   (5) show that the relationship between the characteristic stresses of the deep granite and the confining pressure essentially conforms to an exponential function.With increasing confining pressure, the values of each characteristic stress also increase accordingly, and the rate of increase becomes increasingly faster.Nevertheless, dispersion still occurs, which is attributed to the heterogeneity of the deep granite.The reasons for analyzing the variation in the characteristic stress resulting from the increase in the confining pressure mainly include the following aspects:

•
Restrict the fracture of the rock: the increase in the confining pressure enables the rock to receive more lateral constraints, reducing the potential for crack expansion and fracture within the rock.• The increase in the integrity of the rock contributes to enhancing the overall strength of the rock.• Alter the internal stress state of the rock: the stress distribution within the rock becomes more uniform, thereby enhancing the bearing capacity of the rock.

•
Suppressing the brittle failure of the rock: this approach enables the rock to exhibit more plastic properties and reduces its tendency towards brittle failure.• Augment the stability of the rock: Enhance the stability of the rock under the condition of being under force.• The microstructure of the rock may change due to changes in the microstructure within the rock, thereby influencing its mechanical properties.

Model Establishment
The finite element numerical simulation software Abaqus 2021 was utilized to simulate the aforementioned triaxial compression test.The dimensions of the model and the material properties were precisely determined based on the results of the above tests.Initially, two analysis steps for applying the confining pressure and load were established for the model.To simulate the axial deformation at the bottom of the rock specimen by the testing machine while simultaneously eliminating the influence of the end effect, a reference point was added at the center of the bottom of the model, and a coupling constraint between the bottom surface and the reference point was constructed, with boundary restrictions of axial displacement and the rotation angle applied to this constraint while allowing it to have a radial deformation capability.Confining pressures of 8 MPa, 9 MPa, 10 MPa, 16 MPa, 18 MPa, 20 MPa, 24 MPa, 27 MPa, 30 MPa, 32 MPa, 36 MPa, and 40 MPa were applied to the model, and an axial load with a loading rate of 0.5 kN/s was applied.Eventually, a uniform hexagonal body was employed to divide the mesh for the model, and the numerical model shown in Figure 9 was established.

•
The microstructure of the rock may change due to changes in the microstructure within the rock, thereby influencing its mechanical properties.

Model Establishment
The finite element numerical simulation software Abaqus 2021 was utilized to simulate the aforementioned triaxial compression test.The dimensions of the model and the material properties were precisely determined based on the results of the above tests.Initially, two analysis steps for applying the confining pressure and load were established for the model.To simulate the axial deformation at the bottom of the rock specimen by the testing machine while simultaneously eliminating the influence of the end effect, a reference point was added at the center of the bottom of the model, and a coupling constraint between the bottom surface and the reference point was constructed, with boundary restrictions of axial displacement and the rotation angle applied to this constraint while allowing it to have a radial deformation capability.Confining pressures of 8 MPa, 9 MPa, 10 MPa, 16 MPa, 18 MPa, 20 MPa, 24 MPa, 27 MPa, 30 MPa, 32 MPa, 36 MPa, and 40 MPa were applied to the model, and an axial load with a loading rate of 0.5 kN/s was applied.Eventually, a uniform hexagonal body was employed to divide the mesh for the model, and the numerical model shown in Figure 9 was established.

Analysis of Simulation Outcomes
In line with the Mohr-Coulomb strength criterion, the failure of the rock is shear failure resulting from shear stress.Thus, the study of the shear stress distribution within the rock is highly important for explaining the failure of the rock.Figure 10 presents the shear stress nephogram of the established deep granite numerical model under gradient confining pressure.It can be observed from the figure that, under the action of the axial load, there is an obvious shear stress concentration within the rock, and it exhibits a certain

Analysis of Simulation Outcomes
In line with the Mohr-Coulomb strength criterion, the failure of the rock is shear failure resulting from shear stress.Thus, the study of the shear stress distribution within the rock is highly important for explaining the failure of the rock.Figure 10   To probe the impact of the confining pressure on the axial deformation of the rock specimen, the axial displacement nephograms of the numerical model under the action of the axial load at both low and high confining pressures were separately derived, with a total of four stages selected from the beginning to the end, as depicted in Figure 11.As seen from the figure, upon being subjected to the axial load, distinct axial deformation emerged within the numerical model, and the amount of deformation decreased from the end of the model towards the middle, with the overall deformation presenting a layered distribution.With increasing axial load, this distribution trend became increasingly evident.The axial deformation of the model under high confining pressure was also somewhat restrained, and the overall deformation amount was smaller than that under low To probe the impact of the confining pressure on the axial deformation of the rock specimen, the axial displacement nephograms of the numerical model under the action of the axial load at both low and high confining pressures were separately derived, with a total of four stages selected from the beginning to the end, as depicted in Figure 11.As seen from the figure, upon being subjected to the axial load, distinct axial deformation emerged within the numerical model, and the amount of deformation decreased from the end of the model towards the middle, with the overall deformation presenting a layered distribution.With increasing axial load, this distribution trend became increasingly evident.The axial deformation of the model under high confining pressure was also somewhat restrained, and the overall deformation amount was smaller than that under low confining pressure, while the layered distribution effect of the deformation is also weaker than that under low confining pressure.

Conclusions
Using conventional triaxial compression tests and the finite element software Abaqus, this paper studied the mechanical properties of deep granite under gradient confining pressure, and the following conclusions were drawn:

•
Owing to its peculiar and complex geological environment, under gradient-confining pressure, the elastic modulus, Poisson's ratio, and peak strength of the deep granite present a greater degree of dispersion than those of the shallow granite.• The characteristic stress of the deep granite exhibits an evident exponential function distribution under the gradient-confining pressure, which indicates that the greater the confining pressure is, the greater the characteristic stress and the faster the growth rate.• The results of the numerical simulation indicate that in the conventional triaxial compression test, there is a marked shear stress concentration in the radial direction of the specimen, and with increasing confining pressure, the range of the shear stress concentration correspondingly expands.The axial deformation of the specimen decreases from both ends to the middle, there is a distinct layered distribution, and the layered distribution effect becomes weaker with increasing confining pressure.

Figure 1 .
Figure 1.Observation of deep granite under polarized light microscope.

Figure 2 .
Figure 2. Preparation process of the rock specimens.

Figure 1 .
Figure 1.Observation of deep granite under polarized light microscope.

Figure 1 .
Figure 1.Observation of deep granite under polarized light microscope.

Figure 2 .
Figure 2. Preparation process of the rock specimens.Figure 2. Preparation process of the rock specimens.

Figure 2 .
Figure 2. Preparation process of the rock specimens.Figure 2. Preparation process of the rock specimens.

17 Figure 5 .
Figure 5. Failure forms of deep granite under gradient confining pressure.

Figure 5 .
Figure 5. Failure forms of deep granite under gradient confining pressure.

Figure 6 .
Figure 6.Typical stress-strain curve of the rock.

Figure 6 .
Figure 6.Typical stress-strain curve of the rock.

Figure 8
Figure8and Equation(5) show that the relationship between the characteristic stresses of the deep granite and the confining pressure essentially conforms to an exponential

Figure 9 .
Figure 9. Numerical model of the rock specimens.

Figure 9 .
Figure 9. Numerical model of the rock specimens.
presents the shear stress nephogram of the established deep granite numerical model under gradient confining pressure.It can be observed from the figure that, under the action of the axial load, there is an obvious shear stress concentration within the rock, and it exhibits a certain symmetric distribution along the transverse central axis.With increasing confining pressure, the position of the shear stress concentration notably expands.Under the same numerical model, the stress concentration positions under each confining pressure are basically consistent, and they are more in line with the position of the rock failure surface in the real physical test results.Appl.Sci.2024, 14, x FOR PEER REVIEW 14 of 17

Figure 10 .
Figure 10.Cloud diagram of the shear stress distribution of the numerical model under gradient confining pressure.

Figure 10 .
Figure 10.Cloud diagram of the shear stress distribution of the numerical model under gradient confining pressure.

Table 1 .
The results of triaxial compression tests on deep granite under gradient-confining pressure.Based on the test data, the various mechanical properties of deep granite under a gradient confining pressure are presented in Table1, and the relationship curves of the elastic modulus, Poisson's ratio, and peak strength of deep granite under different confining pressures are depicted in Figure4.

Table 1 .
The results of triaxial compression tests on deep granite under gradient-confining pressure.

Table 2 .
Statistical results of the characteristic stresses of deep granite under gradient confining pressure.