Creep strain analysis and an improved creep model of granite based on the ratio of deviatoric stress-peak strength under different confining pressures

Creep refers to the deformation of rock with time under long-term applied stress, which occurs in most underground engineering. The creep behavior of granite in Shuangjiangkou underground powerhouse in Western Sichuan Province, China, was studied by creep tests. Based on test results, a new parameter DPR, the ratio of deviatoric stress to peak strength, is proposed. DPR is found to be a key parameter to describe creep parameters such as instantaneous elastic modulus, creep elastic modulus, and viscosity coefficient of rock under different confining pressures. Creep tests show that instantaneous elastic modulus increases with the increase of DPR. Creep elastic modulus increases when DPR changes from 0.54 to 0.7004, but decreases when DPR is from 0.7004 to 0.88, indicating fractures in rock closes first and then new fractures are generated. The viscosity coefficient of the rock increases first and then decreases with the increase of DPR, and when DPR = 0.7171, viscosity coefficient is maximum, indicating the time for the rock to reach stability is the longest in creep tests. By introducing DPR and confining pressure into the creep model, which interconnects creep parameters in a unified expression, an improved generalized Kelvin creep model is proposed which can accurately describe the primary and the secondary creep behavior of granite under given deviatoric stresses and confining pressures.


Introduction
The time-dependent (creep) behavior of rock refers to the continuous deformation under constant applied stress (Brantut et al. 2013;Shao et al. 2003). The deformation does not occur instantaneously but is a time-dependent delayed deformation (De-shun et al. 2013;Yang et al. 2014). The creep behavior is understandable in soft rocks (Hunsche and Cristescu 1997), but hard rocks under high stress can also exhibit obvious creep behavior. The time-dependent displacement of stone underneath a force that really is smaller than stone's relatively brief toughness is known as creep. When pressures are reduced, creep stress is rarely entirely restored, hence it is mostly a permanent deformation. Excessive deformation caused through creep can affect the design function of rock organization as well as enhance the cost of repair (Wang et al. 2019). Therefore, creep characteristics are a significant mechanical basis to the long-term stability of rock engineering (Bourgeois et al. 2002), and the effect of creep on the stability of geotechnical engineering should be fully considered. This requires the establishment of a model that can predict the creep process of rock under different stress states and determine the creep converge time of rock.
A number of studies have been reported on creep characteristics of rock by analyzing creep strain or deformation. Wang et al. (2019) and Goodman (1989) indicated that the creep strain of a rock mass under different applied stresses 1 3 109 Page 2 of 11 can be divided into three stages: primary creep (attenuated creep), secondary creep (steady creep), tertiary creep (accelerated creep), as shown in Fig. 1. However, if the strain rate of rock in the second creep stage becomes zero, then the rock will not pass in the tertiary creep step and its strain will not increase. Zhang et al. (2015) indicated that only when the deviatoric stress is increased to or exceeds a certain value, the rock will enter the tertiary creep stage and its strain rate increases rapidly. An element of a program's strain that is made up of asymmetrical primary stress. By removing the mean stress of each major stress, three deformations stresses are produced. The amount of body deformation is controlled by deviator stress forces. Lajtai and Schmidtke (1986) and Li et al. (Cheng et al. 2007) considered the certain value to be the long-term strength of the rock. Therefore, whether the creep strain rate of rock decreases to zero depends on whether the deviatoric pressure is greater as compared to the long-term strength or not. Zhang et al. (Liang-quan et al. 2010), Wu et al. (Qingzhao et al. 2012) andCui et al. (Dongsheng et al. 2016) also indicated that when the deviatoric stress is less than the long-term strength, the steady-state creep strain rate should be zero, and the rock will not enter tertiary creep. Zhao et al. (Cui Xihai and Zhiliang 2006) have shown that the creep strain ε v of rock consists of viscoelastic strain (reversible deformation) ε ve and visco-plastic strain (irreversible deformation) ε vp , and the visco-plastic strain ε vp is the main reason for time-dependent delayed failure in rock engineering, as shown in Fig. 2. The existence of a yield stress is the major viscosity distinction between such viscoplastic and viscoelastic materials. A viscoplastic substance does have a compressive strength below which this will not break, but a viscoelastic material will compress no matter how much force is applied. If the strain rate is zero, only visco-elastic strain ε ve exists in the creep process and the rock finally reaches a stable state. Therefore, it is of great significance to study the reversible deformation in the creep process for the long-term stability of rock engineering. Viscoelastic activity combines viscous and elastic activity, with an immediate strain energy accompanied by a viscosity, time-dependent stress as a response to applied pressure.
Many creep constitutive models have been developed based on different assumptions. Visco-elastic strain (reversible deformation) is one of the assumptions considered in traditional models. Fahimifar et al. (Haridas et al. 1995) proposed a new formulation to determine the wall displacement and convergence of tunnel based on the visco-elastic body, and compared the formulation with numerical analysis. It found that the results of the numerical analysis were consistent with the results of the proposed solution. Zhao et al. (Xihai and Zhiliang 2006) analyzed the visco-elastic-plastic strain characteristics of rock based on the creep test results and proposed a creep model which can describe the loading and unloading creep behavior precisely and the full stages of creep. Zhao et al. (2017) conducted a true triaxial compression test and proposed a creep model which can describe the creep behavior of hard rock under different 3D stress states. The model is composed of a parallel combination of Hooke component and damper component to describe visco-elastic behavior, and a nonlinear visco-plastic body to demonstrate irreversible creep behavior. A solid displaying Hookean behavior is characterized by immediate linear relationships among stress and pressure elements, as well as elasticity.
Most creep model can designate the association betwixt the strain as well as the time of rock, and the factors in these creep models are identified by creep tests and data fitting. Zhang et al. (2015) and Zhang et al. (Fahimifar et al. 2010) utilized Burgers model to fit the strain-time relationship of rocks, in addition to summed up the relationship between axial stresses and parameters of the Burgers model. that is widely used to explain a creep recovery assessment. The model is comprised of springs and comfortable getting that are placed as a series and in parallel arrangement of distinct components. Cong et al. (Zhao et al. 2018) proposed an improved Burgers model, and indicated the relationship between deviatoric stresses with initial elastic modulus and strain rate. Mansouri et al. (Zhang et al. 2012) found that the axial stress has a linear relationship with the initial elastic strain, the initial elastic strain rate and the creep strain. However, the proposed creep models need different parameters for different stress state but these parameters are not related in a unified expression. In addition, confining pressures are not explicitly considered in the constitutive equations of these creep models, so the rock strain cannot be directly expressed by the creep models beneath dissimilar constrictive anxiety besides axial pressures.
In fact, confining pressure is an important factor affecting rock characteristics in creep tests. In this paper, a new parameter DPR, the ratio of deviatoric stress to peak strength, is proposed. The highest value of tensile force or the highest amount of the proportion of shear stress over efficient mean or stress distribution is the peak strength. The moisture content, gradient, bulk density, soil properties, thermal conductivity, and the typical effective pressure exerted on the fault plane all influence the advice to ensure a compressed soil sample. Lei et al. show that an infinite strain must be supplied to the Kelvin system in order for it to experience immediate stress of continuous because the dash-pot will not respond instantly to limited stress. The springs in the kelvin version will not be able to compress their lengths back to their initial form quickly after relieving the stress. Rather, the springs will exert forces on the damper, resulting in compression creeping due to external pressure. After a certain amount of time, all creeping stress would be restored. This is known as viscoelastic contraction, and it occurs in actual viscoelastic materials. The rapid strain as well as visco-elastic strain of rocks in the creep test were analyzed by DPR, and an improved generalized Kelvin creep model is proposed which can accurately describe the relationship between rock strain and time under different confining pressures and deviatoric stresses. To measure elevated temperatures displacement of material, a variety of creep treatments are performed. Tensile creeping, compression creep, bending creep, indented creep, and other types of creep are common examples.

Rock samples and test equipment
Rock samples for tests were granite, taken from the central guide tunnel of the chief besides auxiliary power stations of the Shuangjiangkou hydropower station in western Sichuan Province, China. The specified samples are gray-white in color besides are mostly collected of muscovite, plagioclase, quartz, and feldspar. As shown in Fig. 3, these rock samples were handled into standard cylinders ϕ50 × 100 mm samples in accordance with the experimental specifications endorsed through the International Association Rock Mechanics (ISRM). The International Society for Rock Mechanics (ISRM) is a non-profit organization committed to the research and scientific progress of rock engineering, as well as related subjects like geologist, geophysical, soil mechanics, mining engineering, petrochemical engineering, and construction management.
The MTS815 rock triaxial test machine at Sichuan University's College of Water Resources as well as Hydropower was used for the testing, as shown in Fig. 4. The America MTS corporation's triaxial control flexible monitoring equipment for rocks and concrete features three distinct closedloop mechanical transmission capabilities for axial pressure, confinement stress, and hydraulic conductivity. The major technical parameters of MTS815 are listed in Table 1. The load in the test is applied by a force sensor, and the deformation of those samples is measured by extensometers. The precision of all measured parameters is 0.5%. The measurement information is collected by computers automatically to eliminate manual error.

Experimental methods
The fundamental parameters of the granite samples were attained through a conservative triaxial density test, listed in Table 2. According to the result of conventional triaxial compression tests, creep tests under different confining pressures were carried out. Cheng et al. discussed the cylindrical specimen material which is added to circumferential pressures constraining pressures and regulated increase in axial stresses or axial separations in a traditional triaxial test. The cylindrical specimen sample is typically a specified dimension. A thin rubber barrier surrounds the experiment laterally. The processing of the sample is determined by the type of soil. The creep test of four samples was carried out by Chen's loading method, i.e., multi-step loading method (Cong and Xinli 2017). Using a small measurement, multiple-step loading test can be used to determine a breakdown boundary for various confining levels of stress. The test consists of a number of consolidating and shear phases, with axial load being halted or reverse until full fail for each stage. The goal of the creep test is to determine the pace where the secondary or synchronism creep develops. The gradient of the system increases as the tension or temperature rises, indicating that the quantity of distortion at a given point in time results in an increase. The creep test program for every granite sample was given below as: (1) Constraining compression was useful to each granite sample to a predetermined value at 0.1 MPa/s, and remained stable.
(2) The axial stress was applied to each granite sample to a predetermined value at 30kN/min. 55 ~ 60% of the peak strength in the conventional triaxial compression test is taken as the first stress level of the creep test under the same confining pressure, maintaining the axial stress until the strain rate was stable.
(3) The stress was enhanced through 10% of the peak strength as the next stress level. (4) If the strain rate at the fourth stress level remains stable, the test was finished. Table 3 shows the fundamental factors of granite samples besides every stress stage structure of triaxial creep tests.

Analysis of creep tests
A novel dimensionless factor DPR, the proportion of deviatoric stress towards peak strength, was developed in this work, which is used to analyze creep tests. DPR can be expressed as: where σ 1 − σ 3 is the deviatoric stress; σ p is the peak strength.
(1)  since the conventional triaxial compression tests, σ p can be attained which is illustrated in Table 2; σ p is related to confining pressures and can be expressed as: where k 1 and b 1 are parameters; in this creep test, k 1 = 8.8387 and b 1 = 111.34.
It is found that the new parameter DPR is a key variable to determine the creep behavior of granite instantaneous as well as creep strain under different confining pressures. Based on DPR, the traditional creep model is improved. Compared to the previous creep model which needs different parameters for different stress states, the developed technique can conveniently describe the deformation of rock with time in a unified expression beneath to given deviatoric stresses besides constraining weights.

Strain of creep tests
According to the test program, four creep tests under different confining pressures were carried out. Figure 5 shows the strain curvatures for granite samples along with the time period beneath to dissimilar constraining burdens by using the Boltzmann superposition principle. The Boltzmann superposition definition assumes that the current response of a substance is established by superimposing the object's reaction to its whole load experience. (Mansouri and Ajallocian 2018) Here, if the creep strain rate (h −1 ) is < 3.0 × 10 -5 , the creep rate is considered to be "0". In Fig. 5, the rate of creep strain is almost close to '0' with the time period in the secondary creep of all creep tests beneath dissimilar constraining burdens. It is indicated that the last distortion of each level in all creep tests is stable and the strain is no longer increased.

Instantaneous strain and Instantaneous elastic modulus
Elastic modulus is below the proportional limit, the ratio of stress towards the associated strain. It is a measurement of a material's rigidity else stiffness. The higher the modulus, the stiffer the material is, then the lower the elastic strain caused through a given load. The modelled material will flex to a certain strain with constant tension, which is the instantaneous elastic fraction of the strain. It will then expand more and logarithmically approximate steady-state stress, which would be the delayed elasticity part of the strain. The total strain ε in creep tests can be divided into the instantaneous strain ε m and the creep strain ε v , which can be written as (Xihai and Zhiliang 2006): The immediate strain ε m alludes towards the total strain of each rock when the loading stress reaches the predetermined value in the test, that is, the total strain when t = 0 in Fig. 5, and the creep strain ε v refers to the strain increment of rock with time after axial stress. The value of ε, ε m , ε v and converge time t are listed in Table 4.
The instantaneous elastic modulus E M can be calculated by: Substituting Eq. (1) in Eq. (4), E M can be expressed as: under different confining pressures and DPR are shown in Fig. 6. The instantaneous elastic modulus of rock increases with the increase of axial stresses, and shows a linear relationship which can be expressed as: where E 0 is the instantaneous elastic modulus at a certain confining pressure but without deviatoric stress; E p is the instantaneous elastic modulus at peak strength σ p .
When DPR = 1 and 0, E p and E 0 can be obtained by the fitting expressions in Fig. 6, and the value of E p and E 0 under different confining pressures are shown in Table 5. The confining pressure has a linear relationship with E p and E 0 . When DPR = 1, 100% of the peak strength as well as confining pressures are applied to the rock, then E p can be expressed as: Similarly, when DPR = 0, only the confining pressure is applied to the rock, then E 0 can be obtained by the confining pressure:   When σ 3 = 0, the instantaneous elastic modulus of the rock under the peak strength is 17.87 GPa at DPR = 1; the instantaneous elastic modulus of the rock without applied stress is 11.76 GPa at DPR = 0. Therefore, Substituting Eq. (6) into Eq. (5), the instantaneous strain can be expressed as follows:

Creep strain and creep elastic modulus
Creep strain is one of the most concerned parameters in the creep test. Under the applied stress state, the creep strain increases with time, and the creep strain rate also changes with time. Hence, it is significant and necessary to study the creep strain in the creep test.
In this creep test, the concluding creep strain rate of the secondary creep of each level is 0, indicating that the creep strain is a visco-elastic strain ε ve (Xihai and Zhiliang 2006), as shown in Table 4. The visco-elastic strain ε ve in creep increases with time and finally reaches maximum.
The Kelvin model can well describe the visco-elastic strain in the creep with time, as shown in Fig. 7. The Kelvin model can be expressed as follows: where, ve (t)-visco-elastic strain with time;E K and η K -Kelvin's creep elastic modulus besides viscosity coefficient, correspondingly.
It can be seen from Eq. (10) that when t = 0, ve (t) = 0, and when t → ∞, ve (t) = 1 − 3 ∕E K . Therefore, E K can be defined by the final creep strain. The final creep strain can be represented by the deformation of the elastic element in Fig. 7. When t → ∞, by Eq. (1) and (10), E K can be obtained as: The relationship between E K and DPR of rock under different confining pressures is shown in Fig. 8. The which E K,max is the maximum E K at a given confining pressure.
When DPR < 0.7004, with the increase of deviatoric stress, the internal fracture of the rock gradually close and the rock stiffness increase. When DPR = 0.7004, E K reaches the maximum, indicating that the closing process of internal fracture of the rock is finished and the stiffness of the rock reaches the maximum. When DPR > 0.7004, a new fracture begins to form in the rock owing to the increase of deviatoric stress in addition to the rock stiffness decrease. Therefore, with the enhancement of DPR, the creep damage of rock can be divided into the fracture closure stage and fracture propagation stage. E K can be expressed by the closure degree of the internal fracture in the rock after the rock creep is stable, which can be expressed as: where D cr (σ) is the closure degree of the internal fracture in the rock. Figure 9 shows the association betwixt D cr (σ) and DPR under different confining pressures. Under different confining pressures, D cr (σ) corresponding to the same DPR is similar. Therefore, the relationship among D cr (σ) and DPR underneath dissimilar confining pressures can be fitted.
(12) E K,max = 117.29 3 + 545.06 When DPR = 0.7004, D cr (σ) = 1, which indicates that when the deviatoric stress is 70.04% of the peak strength, the internal fracture of rock can be closed. Table 4 shows the creep strain and converge time t c in each level of the creep test. The value of η K can be calculated by substituting the value of creep strain and converge time obtained from the creep test into Eq. (10) and (11), expressing as: As shown in Fig. 10, under different confining pressures, η K at the same DPR are approximately the same. The quadratic function can well describe the relationship between η K and DPR, expressed as: where l, m, n are said to be as factors and its values based on the proposed research is illustrated in Fig. 10.
When DPR = 0.7171, η K reaches the maximum, that is, η K,max = 927.83. Combined with E K analysis, it is found that E K and η K have the same trend of change, and reach the maximum when DPR = 0.70-0.72, indicating that the larger E K is, the smaller the creep strain rate of rock is, leading to a longer time for rocks to reach a stable state.

Improves kelvin creep model considering DPR
Conventional creep models may be used to mimic granite's time-dependent behavior. Traditional creep testing has two major weaknesses: it takes a very long time to get findings and produces results with a lot of variability. As a result, new testing methods, including the stress relaxation approach, have also been developed. With an ultimate strain rate of 0, the generalised Kelvin model may accurately represent primary and secondary creep, as illustrated in Fig. 11. The generalized Kelvin model is composed of a Hooke body and a Kelvin model. Hooke's body has no storage pressure at any one time and is solely affected by displacement at the very same moment. The Hooke law asserts that the movement or magnitude of distortion is directly proportional to the displacement load applied for relatively modest deflections of an item. The model is expressed by theoretical parts built up of synergistic and antagonistic parts coupled in series or in parallel to study the behavior of viscoelastic materials. The modified Kelvin model's thermal conductivity is made up of an elasticity spring that represents immediate rigidity and Kelvin-Voigt branches that are coupled in series. The equation for the generalized Kelvin model can be expressed as follow: Based on the analysis in Sect. 3.2-3.3, an improved model, which can accurately describe the time-dependent strain of rock, is obtained by substituting Eqs. (1), (6), (13) and (15) into Eq. (16): As shown in Fig. 12, comparing the rock strain curve obtained by Eq. (17) with the creep test, it is found that the improved model for the generalized Kelvin model can effectively describe the time-dependent behavior of the granite as well as the primary and secondary creep behaviors with a final strain rate of 0 under a given deviatoric stress besides constraining weight. Figure 13 shows the relationship of creep strain with time besides DPR underneath dissimilar confining pressures. When DPR is about 0.7, the creep strain shall be kept to a bare minimum under different confining pressure. (17)

Discussion
To grow a time-dependent creep model which can efficiently describe the creep distortion of granite, improvements were created through developing DPR besides constraining weight into the generalized Kelvin model dependent on the analysis of instantaneous strain as well as visco-elastic strain.
The tests illustrate that immediate elastic modulus enhances the growth of DPR underneath dissimilar confining compressions. Based on the analysis of the visco-elastic creep strain, it is found that the creep elastic modulus related to the final creep strain E K reaches the maximum when DPR = 0.7004. The viscosity coefficient η K associated with the creep converge time of rock reaches a maximum when DPR = 0.7171. Through data analysis of different rock tests in many previous studies (Zhang et al. 2015;Fahimifar et al. 2010;Zhao et al. 2018;Liu 1994;Landel and Nielsen 1993), it is found that under different confining pressures, when E K reaches the maximum, DPR may vary from 0.45-0.76; and when η K reaches the maximum, DPR is between 0.42-0.80. In this paper, the maximum of E K and η K reach maximum when DPR = 0.70-0.72, showing the same trend of change with other researchers' work. But for the diversity of rock types, the DPR corresponding to maximum E K and η K may change in a rather large range (Lei et al. 2015;Wenbo et al. 2019;Shuguang et al. 2019;Ateya, B.g,, et al. 2007). The generalized Kelvin model considering DPR can effectively predict the strain-time curve, but parameters in the model are different beneath dissimilar stresses. The association betwixt creep parameters as well as DPR is set up, so the improved generalized Kelvin typical can define the strain-time relationship underneath dissimilar confining pressures in addition to deviatoric stresses.

Conclusions
Through creep tests on the rock samples from the Shuangjiangkou hydroelectric power station, the creep-strain time curves under different stresses are studied as well as the influence of stress on the creep deformation of the rock samples is obtained. Here, a novel DPR factor was proposed, which is the relationship between deviator stress and maximum force.
The creep tests with DPR = 0.54-0.88 is analyzed, and the following conclusions are drawn: (1) Under different confining pressures, with the increase of DPR, the instantaneous elasticity modulus of rock increases linearly. (2) The creep converge strain of rock can be reflected through Kelvin's elastic modulus E K which increases firstly and then decreases with the increase of DPR.
When DPR = 0.7004, E K reaches the maximum. With the increase of DPR, the creep damage of rock can be divided into the fracture closure stage and fracture propagation stage.
(3) The Kelvin viscosity coefficient η K can reflect the relationship between creep strain rate and converge time. η K increases first and then decreases with the increase of DPR, and the value of η K is approximately equal under When DPR = 0.7171, η K reaches the maximum, that is, the time for the rock to reach creep stability is the longest. (4) DPR and confining pressure determine rock creep behavior and parameters. An improved generalized Kelvin creep model is provided, which can successfully explain the primary creep behavior of granite as well as the secondary creep behavior under specified deviatoric stresses and confining pressures, by including DPR and confining pressure into the creep model.