Evolution of mechanical parameters of deep sandstone and its constitutive model under the condition of different stress paths

Using a multifunctional true triaxial fluid–solid coupling system, the mechanical properties of deep sandstone and its constitutive model were studied under the simulated depths 1000, 1500, and 2000 m and different stress paths. In stress path 1, σz is increased while unloading along the x-axis (σx decreased). Path 2 involves an increase in σz with two-sided unloading. Path 3 involves an increase in σz with one-sided unloading σx and σy true triaxial loading. In addition, the evolution law of the Poisson’s ratio μ and deformation modulus E of the deep sandstone under different stress paths and simulated depths were studied. The experimental results show that the following: (1) The μ–σz curves of the sandstone at different simulated depths and the same stress path are similar. As σz increases, the Poisson’s ratio first decreases and then increases to a maximum value, and subsequently, it abruptly decreases. On the other hand, the initial value of the deformation modulus E increases with an increase in the simulated depth. (2) Under the same simulated depth and different stress paths, the deformation trends of the E–σz curves of the deep sandstone are quite different. Under the stress paths 1 and 2, the deformation modulus E of the deep sandstone first increases and then sharply decreases as σz increases. On the other hand, it first decreases and then increases under the condition of stress path 3. (3) A constitutive model for the deep sandstone is proposed herein. The research results can provide some important references for deep rock mechanics.


Introduction
With the rapid development in the global economy, shallow mineral resources have remarkably depleted; thus, deep underground projects have received increasing attention. During the mining process of deep underground resources, in situ stress is an important force that causes mass destruction of the geological rocks. Meanwhile, the mechanical properties of rocks vary under different buried depths and stress paths. In this study, the evolution of the mechanical parameters and a constitutive model of a deep sandstone under the condition of different simulated depth and stress paths based on the reduction of the initial in situ stresses are studied, which can provide some guidance for deep underground engineering. At present, the research on the mechanical properties of coal and rocks under true triaxial forces has globally made certain progress. Feng et al. [1] proposed a three-dimensional strength criterion for hard rocks in combination with true triaxial test data on different types of hard rocks, aiming at the problem that the failure modes of hard rocks are quite different under true triaxial compression. Wu et al. [2] conducted directional hydraulic fracturing tests on a fine sandstone and obtained the hydraulic pressure-time and the fracturing volumestrain curves of the sandstone. They concluded that the crack growth in sandstone can be divided into two stages: the gentle wave shape and the fault-rock drop shape stages. Jia et al. [3] simulated the crack growth model of a fractured sandstone using PFC3D and established the relation that low medium principal stresses significantly affect the peak strength of sandstone. Li et al. [4] conducted true triaxial loading and unloading tests on sandstone with different stress paths using the autonomous true triaxial electro-hydraulic servo system. They revealed the effect of intermediate principal stress and established a three-dimensional mechanical model for the rock. Vachaparampil et al. [5] obtained the strength characteristic curves and test-fracture photographs of three kinds of shale by true triaxial compression tests. They concluded that medium principal stresses significantly affect the strength and failure characteristics of the three kinds of shale. Furthermore, Lu et al. [6] conducted true triaxial loading tests on raw coal in two different modes and thoroughly studied the strength and failure characteristics of raw coal under true triaxial ground pressure as well as the coupling effect of coal and gas outburst on the raw coal to establish the fracture mechanism of raw coal under composite dynamic disaster. Lu [7] conducted uniaxial, conventional triaxial, and true triaxial tests on the marble from the Jinping Hydropower Station. The failure modes of the marble under the three tests were compared and analyzed. Furthermore, the Mohr-Coulomb criterion was modified, and the corresponding model was established. Wu et al. [8] adopted a true triaxial loading test system to conduct laboratory tests on the deep-buried sandstone of the Jinping II hydropower station under different high-stress conditions. They obtained that the plate fracture failure phenomenon of rocks under different high-stress conditions can better represent the rule of the plate fracture failure phenomenon in the practice of the project. Yin et al [9][10][11] conducted true triaxial loading tests on a sandstone under different stress paths and at different loading and unloading rates. They investigated their effects on the failure mode, strength characteristics, deformation characteristics, and permeability evolution law of the sandstone. Du [12] conducted true triaxial compression tests on sandstone, marble, and granite and discovered that the lithology, σ2, and σ3 all affect the strength, fracture inclination, failure mode, and nonlinear mechanical behavior of rocks. In the course of studying the mechanical properties of coal and rocks under true triaxial loading conditions, most studies discovered the effects of principal stress on the strength, deformation, failure mode, permeability, and energy evolution of rocks. However, only few reports exist on the evolution law and constitutive model of the mechanical parameters of rocks under true triaxial loading conditions. As a result, this paper conducts true triaxial loading and unloading tests on deep sandstone with different initial stress paths and simulation depths. It further studies the evolution law of the mechanical parameters of the deep sandstone under the different stress paths of reducing the initial in situ stress and the structure model. The results can provide some theoretical basis for the construction of deep underground projects.

Materials and method
The test employed the multifunctional true triaxial fluid-solid coupling system developed by the Chongqing University, which comprises a frame rack, true triaxial pressure chamber, loading system, internal sealed seepage system, control and data measurement devices, acquisition system, and acoustic emission monitoring system. The maximum pressure obtainable in the three directions are 6000, 6000, and 4000 KN. A two-way rigidity, always rigid and flexible loading method, can be performed in the true triaxial loading under different stress paths for the mechanical properties of deep sandstone. The test instrument is shown in Figure 1.

Extraction of sandstone
A typical sandstone was collected from the Huafeng Coal Mine of Shandong Xinwen Group, as shown in Figure 2. The surface of the sandstone was devoid of macro cracks and flaws, which made it suitable for the test. The sandstone was cut into cubic specimens of 100 mm × 100 mm × 100 mm using a cutter. The physical and mechanical properties of the deep sandstone are listed in Table 1. To avoid the damage of the instrument, the specimens were completely wrapped with thin plastic films after cutting and grinding, and they were then placed in the multifunctional true triaxial fluid-solid coupling system.

Initial in situ stress determination at different simulated depths
From the results of the initial in situ stress measurements, as shown in Figure 3, the axial stresses of the sandstone sample at simulated depths of 1000, 1500, and 2000 m were calculated using the following empirical formula:  Figure 3. In situ stress measurement results in various countries [15] The horizontal stress was calculated using the statistical formula developed by O.
where σz is the axial stress, σy the maximum horizontal stress, and σx the minimum horizontal stress. To facilitate the experiment, the calculated value of the initial in situ stress state was reduced. The initial in situ stress states of the deep sandstone at simulated depths of 1000, 1500, and 2000 m are tabulated in Table 2.  Table 2 shows the simulated depths and initial in situ stress levels of the deep sandstone. The true triaxial test stress paths 1, 2, and 3 on the deep sandstone can be divided into two steps. The first step is the initial in situ stress reduction stage. In this stage, the deep sandstone is synchronously subjected to three-directional stresses. This stage is completed when the three-directional stresses at the different simulated depths of 1000, 1500, and 2000 m reach the preset values. For example, the three-directional stress preset values for the deep sandstone at the simulated depth of 1000 m are σz = 27 MPa, σy = 51 MPa, and σx = 33 Mpa.

Specific test design
In the true triaxial loading process, the detailed steps of the stress paths 1, 2, and 3 are as follows: Path 2: This path is similar to path 1. The difference is that path 2 involves two-sided unloading σx at an unloading rate of 2 KN/s. Path 3: To unload the rate 2 KN/s onesided unloading σy, the corresponding side with displacement control to keep the displacement constant, while unloading rate 2 KN/s onesided unloading σx, the corresponding side with displacement control to keep the displacement constant, and then in the displacement control way to load the slug σz, rate 0.003 mm/s, until the σx unload to 0. The corresponding diagrams of the stress paths are shown in Figure 4.   Table 3 lists the strengths of the deep sandstone under the same simulated depth and different stress paths. Figure

Evolution analysis of rock deformation parameters under different stress paths and the same simulation depth
Poisson's ratio and deformation modulus E are the two most important parameters in the study of the evolution law of rock deformation. The Poisson's ratio of a rock is the ratio of the radial strain to the axial strain when subjected to a load. It is expressed as follows: On the other hand, the deformation modulus E of a rock is the ratio of the applied stress to the strain when subjected to a load: In rock mechanics, μ and E are usually constant in the elastic region, and in this stage, E is known as the elastic modulus. In the plastic region, both μ and E vary with the applied stress. Under three-dimensional stresses, μ and E of a rock cannot be calculated using the formula for the single-axis stresses. Therefore, to obtain μ and E of a rock under the triaxial stress, we employed the formula adapted from the literature [16]. We assumed that the rock obeys Hook's law under the three-direction stress state, and the stress-strain relationship of the deep sandstone is as follows: where εz is the axial strain, εy is the maximum horizontal strain, and εx is the minimum horizontal strain. Through simple mathematical transformation, the expressions for E and μ under true triaxial stress are obtained as follows: Under the same simulated depth, the stress paths of deep sandstone vary, which results in deep sandstones having different stress state under true triaxial loading and unloading. This difference in the stress state leads to a difference in the evolution law of the mechanical parameters of the sandstone. According to Eq. (5), the E-σz and μ-σz curves of deep sandstone at the depth of 1500 m under different stress paths were obtained. In addition, the effect of the stress paths on the mechanical parameters of the deep sandstone was analyzed. As shown in Figure 7, the E-σz and μ-σz curves of the deep sandstone at 1500 m simulated depth under different stress paths are divided into two stages: the AB and BC stages. The AB stage is the pre-peak stage, whereas the BC stage is the post-peak stage. Figure 7 shows that with axial stress loading, the E of the deep sandstone under the same depth and different stress paths  11 greatly varies. Under the conditions of 1500 m of simulated depth and stress path 1, stage AB shows a slight increase in E as the axial stress increases. When the axial stress reaches point B (peak stress), E also reaches its peak value. This is because, after the initial in situ stress reduction process of deep sandstone and with the increase in the axial and maximum horizontal unloading stresses, the stiffness of deep sandstones increases, leading to the increase in the deformation modulus. In the BC stage, due to the strain loading, the rear axle that decreases the stress in deep sandstones ruptures; thus, E rapidly reduces. The formation of the inflection point (the turn-back phenomenon) is due to the outburst of the deep sandstone. Macroscopic cracks in the rock quickly reduce the bearing capacity and the stiffness of the rock, resulting in a decrease in the deformation modulus of the rock. Herein, the variation in E for the deep sandstone at the simulated depth of 1500 m and under stress path 2 is also established. At the simulated depth of 1500 m and under stress path 3, as the axial stress increased in stage AB, E first rapidly decreased and then slowly increased. As shown in Figure 7, with the axial stress loading, the evolutions of μ under the same simulated depth and different stress paths slightly differ. In stage AB, under the conditions of paths 1, 2, and 3 and the simulated depth of 1500 m, the Poisson's ratio first decreased with the axial stress loading and then rapidly increased to a maximum value after which it slowly decreased. In the BC stage, under the conditions of paths 1, 2, and 3 at the depth of 1500 m, the axial stress decreased when the strength of the deep sandstone reached its peak value and the Poisson's ratio rapidly decreased. This indicates that the stress path has little influence on the evolution of the Poisson's ratio of deep sandstones.   12 sandstones varies. Therefore, studying the evolution of the deformation parameters of the deep sandstone at different simulated depths is necessary. Figure 8 shows the μ-σz and E-σz curves under different simulated depths and stress path 1. It is divided into two stages. The AB stage is the pre-peak stage, and the BC stage is the post-peak stage. Figure 8 shows that with the axial stress loading, there is little difference in the evolution of the deformation modulus of deep sandstone under the same stress path and different simulated depths. The deformation modulus E increases with an increase in the simulated depth. For example, under the simulated depth of 1000 m in the AB stage, E slowly increased with an increase in the axial stress. However, in the BC stage, it rapidly decreased with a decrease in the axial stress, and the inflection-point phenomenon was observed. The evolutionary trends of the deformation modulus of deep sandstone under path 1 at depths of 1500 and 2000 m are similar. It can be observed that with axial loading, the evolution of the Poisson's ratio of deep sandstone under different simulation depths and the same stress path differs. In the AB stage, at the simulated depth of 1000 m, with an increase in the axial stress, the Poisson ratio first rapidly decreased first and then slowly decreased. At 1500 m simulated depth, as the axial stress increased, the Poisson ratio first decreased and then rapidly increased, whereas at the depth of 2000 MPa, it first rapidly decreased and subsequently slowly increased. In the BC stage, the Poisson's ratio evolution of deep sandstone at the simulated depths of 1000, 1500, and 2000 m show that after the rupture of the deep sandstone, with the axial loading and lateral unloading, the change in the axial strain was much less than that of the lateral strain; hence, the Poisson's ratio rapidly decreased.

Constitutive analysis of rock mass
The constitutive model of the rock mass represents an essential attribute of the rock mass. The rock-mass structure model can be divided into elastic plastic and nonlinear elastic models. The elastic model is less than the nonlinear elastic model needs to determine the parameters, so it is widely used in rock mechanics. At present, most research on the structure models of a rock mass is based on uniaxial and triaxial stresses. However, the rock mass in the geological environment exhibits the true triaxial stress state. Therefore, the structure model of rock under the minimum horizontal stress under axial loading and single surface unloading is deduced herein by analyzing the true triaxial test results of deep sandstone under different stress paths.  Based on the previous analysis of the rock strength guidelines, the following assumptions are made with regards to the structural model of the deep sandstone: (1) Stages OA1 and A1A involve the initial stress reduction process, and stage AB is the secondary pressure stage; all belong to the elastic stage. In this stage, the stress-strain relationship of deep sandstones obeys Hooke's law. (2) In the BC stage, the strength criterion of the deep sandstone applies to the Mogi-Coulomb strength criterion.

(3)
In the CD stage, the residual strength of the deep sandstone conforms to the revised Hoek-Brown strength criterion.

(4)
In the BC stage, the stress-strain line for deep sandstones is a curve, whereas other stages show linear relationships. Figure 9 shows the stress-strain curve of the deep sandstone of the entire process under the simulated depth of 1500 m and stress path 1. The OA1 and A1A stages represent the initial in situ stress reduction stages, which are considered to be in the elastic region. This is because of the small load applied to the deep sandstone. The OB stage is also an elastic stage, the BC stage is the plastic stage, the CD stage is the strain -softening stage, and the DE stage is the ideal plastic stage.
(1) Elastic stage (OB) In the elastic stage of rocks, with an increase in the three-direction stress, the strain can be determined using Hooke's law: H H where [Ce] is the softness matrix expressed as follows: where μe is the Poisson's ratio in the elastic stage and Ee is the elasticity modulus.
(2) Plastic yield stage (BC) At the BC stage, the stress-strain constitutive relation is obtained by linear fitting as follows: =36.6718+177.7434 V H (8) where σ is the directional stress and ε the directional strain in the BC stage.
(3) Strain softening stage CD From the analysis of the strength criterion, the deep sandstone conforms to the Mogi-Coulomb strength criterion at the maximum load strength and the residual strength conforms to the revised Hoek-Brown strength guideline. The peak yield function fF can be expressed as On the other hand, the residual yield function fc can be expressed as