Study on the Creep Characteristics and Fractional Order Model of Granite Tunnel Excavation Unloading in a High Seepage Pressure Environment

Abstract

The creep associated with unloading surrounding rock during the excavation of deep tunnels seriously affects the stability of the tunnel, and a high seepage pressure will aggravate the strength attenuation and structural deterioration of the surrounding rock. Based on the background of the excavation-induced unloading of the surrounding rock of a deeply buried granite tunnel with high seepage pressure, in this paper we carry out a triaxial unloading seepage creep test that considers the effects of both excavation disturbance and seepage pressure. We also analyze the mechanism of unloading and seepage pressure leading to sample failure and construct a fractional creep damage constitutive model that considers the unloading effect. The results include the following findings, firstly, seepage pressure will affect the creep deformation of rock for a long time, and the circumferential expansion of the granite creep process is more obvious than the axial expansion. Secondly, a high seepage pressure will reduce the rock bearing capacity. Under 0, 2 and 4 MPa seepage pressures, the long-term strength of the samples are 193.7 MPa, 177.5 MPa and 162.1 MPa, respectively. Thirdly, the rock damage factor increases with increasing seepage pressure, time and deviatoric stress. Finally, the rationality of a fractional-order model that considers the effect of unloading and seepage is verified by the test data. These research results may provide some reference for the stability analysis of surrounding rock during excavation in environments under high-stress and high-seepage-pressure.

1. Introduction

The creep characteristics of rock are important factors affecting the long-term stability of tunnel-surrounding rock, and the establishment of a constitutive model is another popular but challenging topic in the study of rock creep mechanics [1].

In recent years, scholars worldwide have conducted much research on the creep characteristics and constitutive models of rock masses and have accomplished many research achievements in theory and practice [2,3,4,5,6]. Some scholars have performed research on rock creep in fluid environments [7,8,9]; for instance, Yu et al. [10] studied the creep test of red sandstone specimens under different water soaking conditions. Their results show that the peak strength and elastic modulus of the red sandstone decrease with not only water content but also immersion time, which can be expressed by a negative exponential function. Deleruyelle et al. [11] presents an analytical approach for the post-closure behavior of a deep tunnel inside a rock mass considering both creep and hydromechanical coupling. Creep, supposedly slower than hydraulic flow, is described by a linear Norton-Hoff law in the particular case of an elastically incompressible rock mass (i.e., Poisson’s ratio $\nu$ = 0.5). These studies show that the creep of water causes continuous damage and deterioration of rocks. However, it has been found that the unloading effect from rock excavation disturbance then produces initial damage to rock creep [12,13,14,15], weakening its strength. Therefore, an increasing number of scholars have considered the rock damage effect when constructing rock creep models, unlike in the early Nishihara, Kelvin and Burgess models [16,17,18]. Cao et al. [19], by introducing time-hardening theory and damage theory, obtained analytical solutions of creep strengthening and weakening behaviors during the complete creep process. A unified creep model was established that can fully capture all the typical creep behavior stages for rock under multiple stress levels. Kang et al. [20], conducted experiments that took into account the viscoelastic–plastic characteristics and the damage effect, and a fractional nonlinear model was proposed to describe the creep behavior of coal. It is verified that the introduction of fractional parameters and the damage factor in the present model is essential to the accurate prediction of the full creep stage of coal. These models are more accurate and reasonable than traditional models for describing rock creep.

In practice, unloading deformation occurs in the excavation of tunnel-surrounding rock, accompanied by crack development or new cracks. With increasing tunnel depth, deep rock mass enters a complex environment of high stress and high seepage, which weakens the long-term strength of the surrounding rock; additionally, the deformation gradually increases with time, which may lead to the fracture and instability of the surrounding rock [21,22,23]. Therefore, it is of great practical value to study the creep mechanical characteristics of rock under the combined action of excavation disturbance and seepage water pressure for the stability analysis of tunnel-surrounding rock.

In summary, we take the unloaded surrounding rock of a cavern excavation in a high-seepage area as the research object to study the creep characteristics resulting from the unloading of granite under the action of dynamic high seepage pressure. We analyze the influence of seepage pressure on instantaneous strain, determine the effective long-term strength of rock specimens, establish a fractional creep damage constitutive model according to the variation in specimen creep rate, compare the test and the model fitting curves, analyze the damage factors, and determine the rationality and applicability of the model to enhance the related engineering research on the long-term stability of deep excavation unloading processes conducted in surrounding rock masses.

2. Creep Sample Preparation and Testing Scheme

The samples were machined into a cylindrical shape with a diameter of 50 mm and a height of 100 mm according to the “Test Procedure for Water Conservancy and Hydropower Engineering” (SL/T 264-2020) implemented in China. During the machining process, it was ensured that the unevenness of the sample ends was within 0.05 mm; in addition, it was ensured that the periphery of the samples was smooth with no significant unevenness and that the overall length and diameter error was within 0.3 mm. The samples were screened using a wave velocity meter to reduce their dispersion. The loading scheme is detailed in

Table 1. Test loading scheme.

Confining Pressure/MPa	Seepage/MPa
	0	2	4	6
20 MPa	S-1	S-2	S-3	S-4

, and the specimen numbers are S-1, S-2, S-3 and S-4. Specimen 2 was chosen for the description of the experimental procedure; the confining pressure of specimen 2 was 20 MPa and the percolation pressure was 2 MPa.

(1)

Triaxial compression test: The hydrostatic pressure is loaded to 20 MPa in stages, the confining pressure is maintained at a constant value, and the axial pressure is loaded at a rate of 2 MPa/min until the specimen is damaged to obtain the triaxial peak compressive strength ${\sigma }_C$ of the specimen under the surrounding pressure of 20 MPa.

(2)

Triaxial unloading test: The hydrostatic pressure of the specimen is loaded to 20 MPa in stages, the confining pressure is maintained at a constant value, and the axial pressure is loaded at 2 MPa/min to 70% of its triaxial peak compressive strength ${\sigma }_C$; the axial pressure is maintained at a constant value and the confining pressure is unloaded at 1 MPa/min until the specimen is damaged in order to obtain the unloading failure confining pressure ${\sigma }_d$ under a 20 MPa confining pressure.

(3)

Triaxial creep test: A description of the creep test, which is performed in three stages based on the test stress path, is given below.

(a)

First stage (initial stress field stage): The stress state of the deep rock is simulated and the axial pressure and circumferential pressure are simultaneously loaded to 20 MPa according to the hydrostatic pressure loading mode.

(b)

Second stage (unloading test piece preparation stage): The constant confining pressure and axial pressure are increased to 70% of the triaxial compressive strength ${\sigma }_C$, and the unloading test piece is finished by reducing the pressure to 50% of the unloading failure confining pressure ${\sigma }_d$ after reaching the target value.

(c)

Third stage (creep stage): The axial pressure and circumferential pressure of the unloading specimen are simultaneously loaded to a magnitude of 50% of the unloading confining pressure and then reduced to 20–0.5 ${\sigma }_d$ MPa, the seepage pressure is loaded to 2 MPa, and the axial pressure is increased to 50% of the compressive strength ${\sigma }_C$ while maintaining the confining pressure. The stress path is shown in Figure 1a. After this process, 24 h is taken as the time period and the axial compression is increased to 5% of the triaxial compressive strength ${\sigma }_C$ and loaded in a graded manner until the sample is destroyed. The graded loading of axial compression is shown in Figure 1b.

3. Test Results

3.1. Creep Failure Curves

The creep test results from the unloading samples under different seepage pressures are shown in Figure 2; this figure shows the cumulative curves of the axial and circumferential strains with time under different stress levels.

By comparing the data analyses of the axial and circumferential strains in Figure 2, we can see that the axial strain change trend unclear, but that the circumferential strain first decreases and then increases with the increase in seepage pressure of the unloading sample. Under a seepage pressure of 0 MPa, the S-1 specimen yields when it is loaded to the sixth stress, which lasts for 120.39 h, with a maximum axial strain of 1.04 × 10⁻² and a maximum circumferential strain of −1.93 × 10⁻². Under a seepage pressure of 2 MPa, the S-2 specimen yields and breaks during the fifth level of stress loading, which lasts for 96.31 h, with a maximum axial strain of 0.93 × 10⁻² and a maximum circumferential strain of −1.34 × 10⁻². Under a seepage pressure of 4 MPa, the S-3 specimen yields and breaks during the fifth level of stress loading, which lasts for 81.68 h, with a maximum axial strain of 0.98 × 10⁻² and a maximum circumferential strain of −2.10 × 10⁻². However, under a seepage pressure of 6 MPa, the S-4 sample yields and fails when it is loaded to its fourth axial stress level, which lasts for 72.79 h. The axial strain reaches a maximum value of 1.01 × 10⁻² and the circumferential strain reaches a maximum value of −2.67 × 10⁻². From the analysis of the change in axial strain, the damage to the specimen is mainly brittle damage, because the difference in deformation of the axial strain to the damage is not large, but under different conditions, with increases in the axial stress and circumferential tensile force, the internal damage gradually accumulates until the specimen is suddenly damaged. On the other hand, the results show that the seepage pressure exhibits a considerable effect on the failure of the specimen and that the axial compressive strength of the specimen decreases with increasing seepage pressure.

The creep curve in Figure 2 shows that the variation range of each level of strain before failure is smaller than that in the accelerated creep stage, and the main strain before failure corresponds to the attenuation creep stage; that is, the sudden increase in stress leads to the instantaneous elastic and plastic deformation of the rock. With the slow change in the internal structure of the rock, the internal stress of the specimen in the steady creep stage is adjusted to achieve a new dynamic balance, and the deformation is relatively stable and lasting, resulting in viscoelastic and viscoplastic deformation. At the initial stages of loading, the internal structure of the sample is less damaged. With the increase in the loading level, the strain curve becomes steeper, indicating that the increase in the loading level increases the internal structural damage to the sample, and that the macroscopic strain of the sample is in a state of accelerated change.

3.2. Graded Strain Curve Analysis

To further analyze the changes in the specimens at all levels, according to the creep test data, the results shown in Figure 2 were processed by the Boltzmann linear superposition principle, the graded loading creep curve was obtained, and the graded loading creep characteristic curve of the unloaded specimen was drawn, as shown Figure 3.

Figure 3 shows that the S-2 sample creeps to 0.866 × 10⁻² in axial strain and −1.148 × 10⁻² circumferential strain before loading the fifth stress level of axial stress at 2 MPa of seepage pressure, and after loading the fifth level of axial stress, the axial and circumferential strains increased to 0.983 × 10⁻² and −1.610 × 10⁻², respectively, and damage occurred. The deformations in the axial and circumferential directions are 11.90% and 28.70% of the total strain, respectively. The S-4 sample creeps to 0.787 × 10⁻² in axial strain and −1.292 × 10⁻² circumferential strain before loading the fourth stress level of axial stress at 6 MPa of seepage pressure, and after loading the fourth level of axial stress, the axial and circumferential strains increased to 1.002 × 10⁻² and −2.671 × 10⁻², respectively, and damage occurred. The deformations in the axial and circumferential directions are 21.46% and 51.63% of the total strain, respectively. The change degree of the circumferential strain is significantly greater than that of the axial strain, which shows that circumferential expansion is more obvious than axial expansion in the creep process of unloading specimens under high seepage pressure.

From the graded loading results of the specimens, it can be seen that, on one hand, the relative ductility of the unloaded specimens is lower in the high-seepage-pressure environment compared with that in the low-seepage-pressure environment, and brittle damage is more likely to occur. On the other hand, in the specimen under high seepage pressure, the internal cracks are subjected to a higher seepage volume and to tangential stress during the development of damage, so that the fracture tensile stress is more significant and prompts the continuous development of the internal crack tip, which in turn provides a seepage channel for the fluid. The internal specimen will be subjected to higher seepage volume and tangential stress, further aggravating the deterioration of the specimen, so that the specimen is more prone to damage under high seepage pressure.

3.3. Instantaneous Strain and Creep Characteristics

To compare the differences in the creep characteristics of granite under four seepage pressures and analyze their changing trends, the instantaneous and creep strain increments of each unloading sample at all levels of loading were statistically processed, as shown in

Table 2. Instantaneous and creep strain of specimens under different osmotic pressures.

Confining Pressure	Seepage Pressure	Deviatoric Stress	Instantaneous Strain		Creep Strain
			Axial	Circumferential	Axial	Circumferential
20 MPa	0 MPa	144.45	0.581	−0.264	0.036	−0.060
		162.33	0.040	−0.028	0.016	−0.070
		178.22	0.044	−0.038	0.018	−0.101
		194.00	0.044	−0.057	0.034	−0.172
		210.12	0.051	−0.092	0.053	−0.263
		225.73	0.056	−0.140	-	-
	2 MPa	145.59	0.59	−0.310	0.025	−0.069
		162.00	0.043	−0.035	0.020	−0.091
		178.03	0.045	−0.043	0.032	−0.167
		194.11	0.048	−0.079	0.058	−0.374
		210.10	0.068	−0.159	-	-
	4 MPa	146.81	0.609	−0.322	0.028	−0.074
		162.77	0.047	−0.036	0.018	−0.102
		178.55	0.041	−0.050	0.034	−0.199
		194.24	0.046	−0.075	0.154	−1.251
	6 MPa	145.45	0.574	−0.368	0.031	−0.132
		162.40	0.045	−0.044	0.018	−0.158
		178.10	0.045	−0.058	0.074	−0.533
		193.72	0.125	−0.808	-	-

.

As seen from Table 2, under the creep conditions of unloading samples with different seepage pressures, the axial transient strain and circumferential transient strain at level 1 are both greater than the transient strain increased by stress loading at all levels prior to damage. This phenomenon occurs because the deviatoric stress of the unloading rock instantly increases during the initial stages of axial force under creep loading, resulting in the large deformation of the samples. When the axial stress is loaded before failure, the change in the instantaneous strain is relatively small, and the elastic modulus of the rock undergoes no obvious change at this time; however, the creep of each stage of the specimen shows an increasing trend with the increase in seepage pressure, which indicates that the pore water pressure does not completely penetrate the rock during the initial stages of stress and cause the structure to change. However, with an increase of test time, the grain structure in the rock changes slowly due to the long-term load and pore water pressure, which indicates that the seepage pressure has a long-term influence on the creep deformation of the rock. In addition, Table 2 shows that the axial strain and circumferential strain decrease from the first to the second loading level, increase slowly before the granite is destroyed and then increase sharply when the sample is destroyed at the last loading level. The larger the seepage pressure is, the more obvious is the increase in the creep for each loading level tested.

From the comprehensive analysis in Figure 3, it can be seen that, under different seepage pressures, the axial and circumferential instantaneous strains of the first loading level are much larger than those of the loading level before failure, resulting in obvious axial and circumferential strains. To facilitate the observation of the changes in the instantaneous strain of several stages before failure, the axial and circumferential instantaneous strain data under pressures of 0 MPa and 2 MPa (Table 2) were selected for fitting, and the first instantaneous strain of each sample under axial compression is omitted. The fitting curve of the instantaneous strain is shown in Figure 4. In Figure 4a,b, the axial and circumferential instantaneous strains increase linearly with increasing loading levels until the specimen is damaged. The slopes of the fitting equations of the axial instantaneous strain under seepage pressures of 0 MPa and 2 MPa are 0.0039 and 0.0074 respectively, and the circumferential and axial strains under seepage pressure are larger than those without seepage pressure. Considering the instantaneous strain, the seepage pressure increases the internal damage to the unloaded specimen, increases the deformation of the specimen, and promotes the development and expansion of cracks until the specimen forms a penetrating failure surface.

3.4. Long-Term Strength

In this paper, creep tests were used to determine the long-term creep strength of granite based on the loading creep curve by the isochronous stress–strain curve cluster method [24,25]. According to the creep test results, the isochronous stress–strain curves of the unloaded samples under three seepage pressures were obtained, as shown in Figure 5.

At a low axial stress level, the rock is in the stage of attenuation creep without obvious deformation, and the isochronous curve is almost linear. With increasing stress levels, the sample begins to creep in a steady state, and the slope of the stress-strain isochronous curve begins to increase. With increasing test time, the bending degree of the isochronous curve increases. Finally, the rock enters the accelerated creep stage, the strain increases rapidly, and the rock is destroyed. Figure 5a shows that the curve cluster changes from dense to sparse, and that there is an obvious divergence starting point, which indicates the transformation of the sample from the viscoelastic deformation stage to the viscoplastic deformation stage. A straight line perpendicular to the horizontal axis is drawn through the divergence starting point, and the stress corresponding to the intersection of the straight line and the horizontal axis is the long-term strength of the rock. Similarly, by analyzing Figure 5b,c, it can be seen that the long-term strengths of the rock under the 2 MPa and 4 MPa seepage pressures are 8.4% and 16.3% lower than that under the 0 MPa seepage pressure, respectively, which indicates that the existence of seepage pressure has a great influence on the long-term strength of the rock. With the increase in seepage pressure, the long-term strength of the rock decreases significantly. Under the action of water pressure, the cementation between the particles in the rock sample is weakened, and the water pressure reduces the effective stress at the fracture surfaces and the damage surfaces of the rock. The unloading effect also weakens the cementation between the particles in the rock sample, which makes the cohesion decrease, and the decrease in cohesion is expressed in the decrease in the unloading strength of the rock sample. Therefore, the attenuation of the long-term strength of the rock mass should be considered in deep tunnel engineering projects in high-seepage-pressure environments.

3.5. Creep Rate

To establish a reasonable creep model, it is necessary to further explore the change in the creep rate of the specimen, and then select suitable components to describe the creep stages. Therefore, the creep rate calculation method proposed by Lyu et al. [26] for an incremental loading creep test was adopted.

$$\left\{\begin{array}{c}\begin{array}{c}\Delta \epsilon =\Delta {\epsilon }_1+\Delta {\epsilon }_2+\cdots +\Delta {\epsilon }_{n-1}\hfill \\ \Delta {\epsilon }_1={\epsilon }_2-{\epsilon }_1,\Delta {\epsilon }_2={\epsilon }_3-{\epsilon }_2,\cdots ,\Delta {\epsilon }_{n-1}={\epsilon }_n-{\epsilon }_{n-1}\hfill \\ \Delta \epsilon =({\epsilon }_2+{\epsilon }_3+\cdots +{\epsilon }_n)-({\epsilon }_1+{\epsilon }_2+\cdots +{\epsilon }_{n-1})={\epsilon }_n-{\epsilon }_1\hfill \end{array}\hfill \\ v_i=\frac{\Delta \epsilon }{\Delta t_i}\hfill \end{array}\right.$$

(1)

In Equation (1), $\epsilon$ is the creep strain; $\mathrm{∆}{t}_i$ is the creep time; $n$ is the number of creep test data; and $v_i$ is the strain rate.

In the curve of the rock strain-time–creep rate, we can see that, according to the creep rates, rock creep can be divided into three stages: the decay creep stage, steady creep stage and accelerated creep stage. Since the creep curve clusters in Figure 3 are similar under different seepage pressures, only attenuation creep and steady-state creep occur before failure. Therefore, the whole creep curve at the last stress level was taken as the research object; that is, the sample contained three creep stages for analysis, and the creep rate is calculated by Equation (1) and plotted. As shown in Figure 6, we only analyzed the fifth creep rate curve under a seepage pressure of 4 MPa.

Under the last axial compression loading stage, the creep deformation of the specimen shows an obvious nonlinear accelerated creep stage, which lasts for a short time and corresponds to a large creep deformation, indicating that the specimen was quickly damaged by compression at this stage. Figure 6 shows that the steady-state creep stage lasts longer than the decay creep stage and the acceleration creep stage. The creep rate of granite decreases rapidly from 0.295 × 10⁻² mm/min. This decrease is because the internal structure of the specimen is not obviously changed in the early stages of the increasing axial compression level, and the whole specimen is in a state of relatively balanced stress distribution: the decay creep stage. Then, the creep rate remains constant at a rate of 0.025 × 10⁻² mm/min for a long time as the specimen continues to creep. In the steady-state creep stage, the internal damage accumulates continuously and causes the crack to develop and expand gradually. Finally, the creep rate increases sharply to 1.794 × 10⁻² mm/min, and the specimen cracks. In the accelerated creep stage, the internal micro-element structure is destroyed, and the crack develops and rapidly expands to form a crack surface; thus, the bearing capacity of the specimen decreases sharply, and the specimen is destroyed.

In summary, in engineering practice, we should reduce the damage to rock masses caused by excavation unloading and seepage pressure. First, we should perform the necessary support work during the excavation and subsequent construction. Second, we should reduce the influence of the high-seepage-pressure environment on the excavation unloading of the surrounding rock and focus on monitoring the surrounding rock’s adjacent face. When the normal displacement and strain of the free face increase significantly, we should focus on these phenomena.

4. Establishment of the Creep Damage Model

4.1. Fractional Calculus Theory

Fractional calculus has obvious advantages in describing the steady-state creep and accelerated creep stages of rocks. Unlike integer calculus, fractional calculus has a good global correlation degree, and the experimental results are in high agreement with theoretical simulation results [27,28,29,30]. Fractional calculus was developed from traditional integral calculus theory, and it can describe any derivative or integral. Fractional calculus is defined in the form of an integral. According to different time domains, there are three types of fractional calculus: Caputo, Grunwald–Letnikov and Riemann–Liouville. We adopt the calculus operator defined by Riemann–Liouville.

In the fractional order, the elastic and plastic elements are exactly the same as the mechanical elements in the integer order; however, the viscous dashpot in the integer order is replaced by viscoelastic Abel elements in the fractional order. A schematic diagram is shown in Figure 7a. The stress–strain relationship of the Abel dashpot [31] is as follows:

$$\sigma \left(t\right)=\eta \frac{d^{\gamma }\epsilon \left(t\right)}{d{t}^{\gamma }}$$

(2)

where $\eta$ is the viscosity coefficient of the Abel dashpot, and $\gamma$ is the order of the Abel dashpot. To study the creep constitutive model of geotechnical materials, $\sigma \left(t\right)$ is considered to be a constant value. At this time, the creep constitutive equation of the Abel dashpot is obtained by using the Riemann–Liouville fractional differential operator theory as follows:

$$\epsilon \left(t\right)=\frac{{\sigma }_0}{\eta }\frac{t^{\gamma }}{\mathsf{\Gamma }\left(1+\mathsf{\gamma }\right)}\left(0\le \mathsf{\gamma }\le 1\right)$$

(3)

As seen from the above formula, when the stress level remains constant, the strain of the material between the fluid and the solid increases slowly; the strain does not show a linear increase like a Newtonian fluid, nor does it remain the same as in a linear elastic body. After the instantaneous deformation of the rock, the creep process enters the attenuation creep stage. At this stage, the creep deformation of the rock increases, and the creep rate decreases. Over time, the creep deformation of the rock increases at a fixed slope. The Abel dashpot simulates the material properties of an ideal elastomer and a Newtonian fluid.

4.2. Establishment of the Fractional Damage Model

According to the curve change of the accelerated creep failure stage, an exponential viscous element was selected, and was connected in parallel with the plastic element through stress triggering to form a new nonlinear viscoplastic element; this element is used to describe the accelerated creep failure stage of the sample, as shown in Figure 7b.

The creep equation of the nonlinear viscoplastic element is as follows:

$$\epsilon =\frac{\sigma -{\sigma }_s}{{\eta }_2}\exp {\left(t-t_F\right)}^{\lambda }$$

(4)

In Equation (4), ${\eta }_2$ is the viscosity coefficient of the nonlinear viscoplastic element, ${\sigma }_s$ is the stress threshold of the specimen in the accelerated failure stage, $t_F$ is the time corresponding to ${\sigma }_s$, and $\lambda$ is a parameter describing the time change of the accelerated failure stage.

However, considering the initial damage to the sample during the unloading process and the sustained damage effects of seepage pressure during the creep of the sample, a rock damage variable $D$ was introduced. The damage factor $D$ is a function of axial stress $\sigma$, osmotic water pressure $P$, unloading magnitude $U$ and creep time $t$ [32] and is calculated as follows:

$$D=f\left(\sigma ,P,U,t\right)$$

(5)

The fractional nonlinear creep model of granite under seepage pressure constructed in this paper consists of three parts connected in series, as shown in Figure 8. The damaged elastic element was selected to represent the instantaneous deformation of granite at the initial stage of loading. Fractional damage viscoplastic elements were selected to characterize the decay creep stage and steady creep stage of the sample. The nonlinear element triggered by stress represents the accelerated creep stage of the sample.

• When $\sigma +\mathrm{∆}\sigma <{\sigma }_s$, $t<t_F$, only the damaged elastic element and the fractional damaged viscoplastic element participate in the deformation in the model, and the nonlinear creep equation is obtained as follows:

  $$\epsilon =\frac{\sigma }{E\left(1-D\right)}+\frac{\sigma }{{\eta }_1\left(1-D\right)}\frac{t^{\gamma }}{\mathsf{\Gamma }\left(1+\gamma \right)},\left(0<\gamma <1\right)$$

  (6)
• When $\sigma +\mathrm{∆}\sigma \ge {\sigma }_s$, $t>t_F$, the disturbance element in the model functions correctly, and the equation is as follows:

  $$\epsilon =\frac{\sigma }{E\left(1-D\right)}+\frac{\sigma }{{\eta }_1\left(1-D\right)}\frac{t^{\gamma }}{\mathsf{\Gamma }\left(1+\gamma \right)}+\frac{\sigma -{\sigma }_s}{{\eta }_2}exp{\left(t-t_F\right)}^{\lambda }$$

  (7)

4.3. Model Verification and Parameter Identification

To verify the applicability of the creep damage model under the seepage of unloading samples, according to the unloading creep test results and creep curve analyses, we choose the Levenberg-Marquardt and general global optimization methods to calculate the solution. The creep results of unloading granite specimens with different seepage pressures were selected for fitting analysis. The fitting results of the unloading creep damage model of selected granite specimens are shown in Figure 9, and the parameter inversion results are shown in

Table 3. Parameter fitting results of unloading creep damage model of samples.

Seepage Pressure/MPa	$\mathit{\sigma }$	$\mathit{E}$	${\mathit{\eta }}_1$	$\mathit{\gamma }$	${\mathit{\eta }}_2$	$\mathit{\lambda }$	$\mathit{D}$	${\mathit{R}}^2$
0	146	26,274	5.63 × 105	0.1434	8.98 × 107	1.3990	0.0228	0.9832
	162	25,688	7.44 × 106	0.3627			0.0482	0.9619
	178	26,722	3.44 × 106	0.3265			0.0719	0.9759
	194	26,998	2.30 × 106	0.3483			0.0796	0.9744
	210	26,928	1.80 × 106	0.4832			0.0927	0.9853
	226	26,009	3.6 × 105	0.9889			0.1134	0.8926
2	146	25,875	6.18 × 105	0.1010	9.11 × 107	4.1414	0.0370	0.9794
	162	25,356	6.69 × 106	0.4662			0.0491	0.9753
	178	26,452	2.77 × 106	0.3846			0.0830	0.9840
	194	26,369	1.44 × 106	0.4125			0.0882	0.9930
	210	27,654	4.11 × 106	0.9764			0.1517	0.8722
4	146	26,552	5.61 × 105	0.0878	9.02 × 106	0.7977	0.0681	0.9656
	162	26,044	2.25 × 106	0.2506			0.0761	0.9588
	178	26,495	3.10 × 106	0.4525			0.0964	0.9832
	194	27,241	1.13 × 106	0.5317			0.1274	0.9855

.

In the comparative analysis of the test curve and the fitted curve of the model in Figure 9, it can be seen that the fitted curve shows elastic deformation at the initial stage under low axial compression. There are then two stages of decay creep and steady creep, and the granite enters the accelerated creep stage under the last axial compression level. The changing trend of the curve accurately describes the creep process of granite specimens, which indicates the applicability and rationality of the model established in this paper. Because of the heterogeneity of rock, there is some difference between the test data and the fitting curve, but the overall fitting degree is relatively high, and the correlation fitting coefficient is approximately 0.97. The curve fitting degree of the accelerated creep stage is slightly lower than those of the first two creep stages. This phenomenon occurs because, when the rock enters the accelerated creep stage, the internal damage is more serious, and the strain changes sharply. Once the last level of loading stress reaches the target value, the creep time shortens, the sample mainly shows obvious instantaneous elastic–plastic strain, and the damage finishes quickly in a short time, occurring in the form of brittle failure. However, overall, the fitted curve is in good agreement with the test curve.

According to the identification results of the creep parameters of the unloading samples in Table 3, the regularity of the variations among the axial creep parameters is obvious and can be analyzed as follows.

• The viscosity variable ${\eta }_1$ first increases and then decreases with the decay creep, steady creep and accelerated creep of the rock mass, which shows that the viscosity variable exhibits damage and deterioration effects.
• The larger the seepage pressure is, the larger the initial damage factor $D$, which indicates that the seepage pressure causes damage to the excavation unloading; the larger the seepage pressure is, the greater the rock damage.
• With increasing deviatoric stress, the damage factor $D$ values of the elastic elements and viscoplastic elements tend to increase. The increases in the damage factors indicate that the damage to the rock increases with time and deviatoric stress.

5. Discussion

In this paper, we considered the effects of both unloaded excavation and seepage pressure on granite, creep damage experiments were carried out on unloaded rocks under seepage pressure, with the results showing that granite failure damage is preceded by a significant expansion phenomenon, and that high seepage pressure has a significant reduction in long-term strength of the rock, and the results can be used to guide the deformation monitoring of granite tunnels after excavation in a high seepage pressure environment. Based on the fractional-order theoretical derivation, a new creep damage intrinsic model that can describe the nonlinear creep properties of granite is proposed, and the results of the study can have a better prediction of the long-term creep behavior of unloaded granite under high stress and high seepage pressure.

The variation pattern of damage factor and the correlation with the influencing factors of damage factor still need to be further investigated, and the damage caused by unloading magnitude and seepage pressure on the rock needs to continue to be tested for quantitative analysis. We will conduct microscopic analyses of the changes of rock grain structure under multiple stresses in the next study.

6. Conclusions

In this paper, the excavation and unloading of deep surrounding rocks in a high-seepage-pressure environment was taken as an example, and a creep test of unloading samples under different seepage pressures was conducted. The main findings are as follows:

• With the increase in each axial compression level, the axial strain and circumferential strain of the specimen show an increasing trend; with the increase in the axial compression loading level, the circumferential strain gradually exceeds the axial strain, which indicates that the circumferential expansion of granite is more obvious than the axial expansion.
• With the increase in the seepage pressure, the instantaneous strain of each stage of the specimen changes minimally while the creep deformation shows an increasing trend. This finding indicates that the instantaneous deformation of the specimen is minimally affected by the seepage pressure, but that the seepage pressure affects the rock creep deformation for a long time. The creep failure process of the specimen under high seepage pressures is obviously faster than that under low seepage pressures.
• The long-term strengths of granite under 0, 2 and 4 MPa seepage pressures are 193.7 MPa, 177.5 MPa and 162.1 MPa, respectively. The long-term strengths of specimens under 2 MPa and 4 MPa seepage pressure decreases are 8.4% and 16.3% smaller than that under 0 MPa seepage pressure, respectively, indicating that rocks are more prone to yield failure under a high seepage pressure. Furthermore, the long-term strength attenuation effect should be considered in engineering design.
• The analysis of the creep constitutive model shows that the larger the seepage pressure is, the larger the initial damage factor $D$, which indicates that the seepage pressure damages the rock before it undergoes excavation unloading; the larger the seepage pressure is, the greater the rock damage. With increasing deviatoric stress, the damage factor increases, which indicates that the damage to rock increases with time and deviatoric stress.