Study on the creep constitutive model of layered rockconsidering anisotropic and damage factors after hightemperature exposure

Abstract

The failure of layered rock after high temperature exposure is a major concern in deep underground engineering projects. This paper proposes an improved Nishihara creep constitutive model that considers damage factors and the bedding angle, which overcomes the shortcomings of the deviation in the description of the conventional Nishihara model in the acceleration stage. The constitutive model is verified by the conventional triaxial creepiest. The theoretical curve has a high degree of fitting with the experimental curve. The experimental results show that a temperature of $400 \circ C$ has an obvious influence on the steady creep rate and the creep strain of layered sandstone, and $600 \circ C$ can be regarded as the temperature threshold for the long-term strength and change from anisotropic to isotropic of layered sandstone. The irreversible melting mixing phenomenon at the boundary of mineral particles with increasing temperature is the mechanism by which different treatment temperatures affect the anisotropy degree of layered rock.

Keywords

Layered rock after high temperature exposure constitutive model creep characteristics deviation coefficient

Introduction

In recent years, the rapid development of tunneling, mining and hydropower engineering has greatly promoted underground structure excavations to depth (Li et al., 2018), and therefore, high temperature has become one of the essential properties of deep layered sedimentary rocks such as sandstone. Due to this fact, the mechanical properties of layered sedimentary rocks in a natural state are no longer suitable. For the design and construction of underground spaces in high-temperature environments. The creep damage of deep layered sedimentary rocks is significantly influenced not only by the bedding surface but also by the high-temperature environment in the deep surrounding rock. Consequently, studying creep constitutive damage models of sedimentary rocks after high temperature exposure has always attracted extensive attention.

A large amount of literature has already focused on the fundamental physical parameters, deformation characteristics, mechanical properties, failure mechanism, and constitutive model of rocks under or after high temperature. For example, Ersoy et al. (2021), Hajpál (2002) and Zhao et al. (2009) studied the threshold value of sandstone porosity changes under real-time high temperature conditions. They all concluded that the porosity of sandstone increased slightly with increasing high temperature, and it decreased significantly when the temperature exceeded the threshold. The former, however, considers that the threshold for the decline in porosity of sandstone is $300 \circ C$ , while the latter considers that the threshold is $600 \circ C$ . Gautam et al. (2016) and Taheri et al. (2020) studied the rock mechanical behavior under different real-time high temperatures. Gasc et al. (2011), Rong et al. (2015), Shi et al. (2021), Sun et al. (2020) and Wu et al. (2005) studied the mechanism of the deterioration of the properties of rocks under different high-temperature effects and discussed other conditions with advanced micro experimental methods. Rybacki et al. (2017) and Yang and Hu (2018) studied the creep behavior of different rocks under high temperatures conditions. Some scholars have also studied the mechanical properties of rocks after exposure to high temperatures (Dwivedi et al., 2008; Heuze, 1983; Zha et al., 2021). The effect of temperature on the mechanical properties of granite have been investigated and it has been found that the mechanical properties of granite, such as the deformation modulus, Poisson’s ratio, tensile strength, and compressive strength, are different from those in the real-time high temperature process. For example, Tian et al. (2016) believed that the mechanical properties of sandstone will deteriorate to some extent after exposure to high temperatures. Overall, the mechanical properties of rocks will be affected by real-time high temperatures or after high temperature exposure. The higher the temperature is, the more pronounced the deterioration of the mechanical properties of rock. The difference is that the changes in the mechanical properties of rocks caused by different temperature treatment conditions are different. In fact, rocks do not completely exhibit isotropic characteristics but are anisotropic due to the existence of structural planes, especially in sedimentary rocks. The transverse isotropy in the sedimentary rock greatly affects the site selection and structural design of underground space projects. To describe the creep failure mechanism of sedimentary rock after high temperature conditions, it is necessary to establish the creep constitutive relationship of rock materials. Li et al. (2017) studied the creep behavior of salt rock based on micromechanics. Li et al. (2020), Li et al. (2019) and Yao and Fang (2020) proposed a constitutive model for the whole process of rock creep that consider initial damage and creep damage. Sagong et al. (2011) used numerical simulations and physical experiments to investigate the deformation characteristics of layered rock masses with different dip angles. The anisotropic creep law of brittle green schist was studied through the uniaxial compression creep test (Shen et al., 2020; Wu et al., 2014). It was concluded that the failure model of anisotropic rock is closely related to the inclination of the bedding plane, and the minimum compressive strength between the bedding plane and the maximum principal stress direction is usually between 30° and 60°. Xu et al. (2019) proposed a creep model that combines the Maxwell-Kelvin and nonlinear visco-plastic bodies to describe the whole creep process and the transversely isotropic characteristics of phyllite. Pellet et al. (2005), Shao et al. (2006) and Wu et al. (2020) proposed a constitutive model based on the visco-plastic constitutive law and damage theory and depicted the mechanical behavior of layered rock with time. Hou et al. (2019), Huang et al. (2021) and Shen et al. (2020) believed that the initial damage has a definite influence on the creep behavior of rock, and then a new nonlinear creep damage model was developed based on a multistep creep test. Song et al. (2019), Xu and Cui (2020) and Zhou et al. (2018) established new creep constitutive models based on fractional theory. Overall, most of the studies on creep constitutive equations considering damage factors were based on the average temperature, and few have considered the effect of after high temperature on the creep properties of layered rock.

Given the above background, both the effect of high temperature and the surface structure of rock obviously affect the failure properties of layered rock and the creep damage evolution. Researchers have realized the importance of the two aspects in rock, but few have considered the creep constitutive model of layered rock after exposure to a high temperature condition. The main contribution of this study is to investigate a constitutive creep model that considers thermal damage and the bedding angle in terms of damage. First, based on previous research, an elastic-plastic element that considers the damage factors and bedding angle is proposed, and a creep damage constitutive model after high temperature exposure is constructed. Next, we analyze the creep characteristics, creep rate, and long-term strength characteristics of layered rock after different treatment temperatures. Then, the microstructure characteristics of layered rock after the different treatment temperature are analyzed by thin section tests, and the influence of different treatment temperatures on the anisotropy degree of layered rock is explained from the microscopic perspective. Next, the proposed creep constitutive model is verified by the creep test results of layered rock after different processing temperatures. Finally, a relevant indicator basis for underground space engineering design and construction is provided, and the study of layered rock creep characteristics after exposure to high temperatures is enriched. The relevant indicators were proposed to provide a systematic methodology to quantify the creep constitutive model in layered rock after a high temperature exposure.

Establishment of the constitutive model

The classic Nishihara model consists of a Hooke’s body, viscoelastic body, and viscoplastic body in series. This model has been widely used to characterize the viscoelastic and viscoplastic deformation characteristics of rock creep. However, the classic Nishihara model can neither simulate the accelerated creep stage well nor consider the effects of thermal damage caused by the natural cooling of the rock after high temperatures and bedding angles. In the present study, we will construct an elastic-plastic damage element that considers the thermal damage after exposure to high temperatures and the bedding angle. The elastoplastic element will replace Hooke’s body in the classical Nishihara model. The parameters of the improved Nishihara model’s constitutive model will be identified to optimize the Nishihara model’s simulation effect on the whole creep process.

Derivation of damage variables considering bedding angle and thermal damage

It is assumed that the angle between the maximum principal stress $σ_{1}$ and the bedding surface is $β$ , Adobe Illustrator was used to draw Figure 1.

The strain of layered rock is only a function of stress under normal temperature. However, the strain of layered rock after high temperature exposure is not only a function of stress but also a function of temperature. It as assumed that the layered rock has undergone high temperature damage ( $D_{T}$ ) before it begins to be compressed if the room temperature is $20 \circ C$ . Liu et al. (2001) pointed out that the thermal damage of rock after cooling is instantaneous. Therefore, the thermal damage variable ( $D_{T}$ ) is defined as the rate of change of the instantaneous elastic modulus as follows:

D_{T} = 1 - E_{(T, β)} / E_{(20 \circ C, 0 \circ)}

(1)

where

D_{T}

denotes the thermal damage of layered rock caused by high temperature.

E_{(T, β)}

denotes the instantaneous elastic modulus after high treatment temperature T and bedding angle β.

E_{(20 \circ C, 0 \circ)}

denotes the instantaneous elastic modulus at room temperature (

20 \circ C

) and a bedding angle of 0°. It is generally believed that the conventional triaxial creep straintime curve result of layered rock can be divided into three stages: the attenuation stage, stabilization stage, and accelerated creep stage. The axial strain rate of the layered rock decreases in the first two stages but increases in the last stage. In the present study, the damage variable of layered demonstration creep compression after exposure to high temperatures is defined as

D_{k (T, β)}

. Under a low stress level, the rock is compact, and the original crack is closed, resulting in a reduction in the damage variable

D_{k (T, β)}

. Compared with the initial stage of creep pressure, the rock forms a denser continuum when the microcracks in the layered rock are basically healed. The damage variable at this stage reaches a minimum value

D_{k (T, β), \min}

. The initiation and propagation of microcracks gradually appear with increasingstress level, and thereby, the value of the damage variable

D_{k (T, β)}

increases. The relationship between the damage value

D_{k (T, β)}

of layered rock and time is shown in Figure 2. The damage is

D_{T}

if the initial time

t

is equal to 0.

The damage variable reaches $D_{k (T, β), \min}$ until the rheology progresses to the end time of the creep stable stage $t_{s}$ . Damage variables are derived based on Kachanov’s creep damage evolution equation:

D_{k (T, β)}^{'} = {[σ / λ (1 - D_{k (T, β)})]}^{γ}

(2)

where

λ

and

γ

are model parameters related to materials.

σ

represents the stress exerted on the materials.

D_{k (T, β)}

is the damage variable of layered rocks during creep compression, and

D_{k (T, β)}^{'}

is the first derivative of the damage variable of layered rock during creep compression.

Based on the initial conditions, the modified form of equation (2) can be obtained

\{\begin{array}{l} [{(1 - D_{T})}^{γ + 1} - {(1 - D_{k (T, β)})}^{γ + 1}] / (γ + 1) = σ^{γ} t / λ^{γ}, t = 0 \\ [{(1 - D_{T})}^{γ + 1} - {(1 - D_{k (T, β), \min})}^{γ + 1}] / (γ + 1) = σ^{γ} t / λ^{γ}, t = t_{s} \end{array}

(3)

The damage variable $D_{k (T, β)} = 1$ , when rheology progresses to the end time of the accelerated creep stage $t_{1}$ . The evolution equations of the rheology damage variables in the attenuation, stabilization, and accelerated creep stages can be obtained by equation (4) from the two equations in the joint equation of equation (3):

D_{k (T, β)} = \{\begin{array}{l} 1 - (1 - D_{T}) {\{1 - t [1 - {((1 - D_{k (T, β), \min}) / (1 - D_{T}))}^{γ + 1}] / t_{s}\}}^{1 / γ + 1}, t < t_{s} \\ 1 - {[σ^{γ} (γ + 1) (t_{1} - t) / λ^{γ}]}^{1 / γ + 1}, t_{1} \geq t > t_{s} \end{array}

(4)

Construction of damaged rheological element

In the creep process of layered rock after high temperature exposure, especially in the accelerated creep stage, the creep characteristics are affected by the high temperature and closely related to the damage and fracture caused by long-term stress. Therefore, the microscale crack evolution and damage accumulation should be considered. In Section 2.1, the damage variables $D_{T}$ and $D_{k (T, β)}$ are defined to represent thermal damage caused by natural cooling of layered rock after high temperature and mechanical property deterioration caused by long-term stress, respectively. The transversely isotropic damage rheological element after high temperature exposure is constructed in combination with the influence law of transversely isotropic creep characteristics, Adobe Illustrator was used to draw Figure 3.

The cross-sectional area of the above elements is assumed to be $A$ . The damaged areacaused by natural cooling after high temperature and long-term stress is $A_{1}$ , which is notstressed. The undamaged area is $A_{2}$ which is stressed. The equilibrium relationship can be expressed as

σ_{0} A = σ_{0}^{'} A_{2}

(5)

where

σ_{0}

denotes the total stress acting on the cross-sectional area

A

of the element, and

σ_{0}^{'}

denotes the effective stress acting on the undamaged area

A_{2}

. The damage area ratio isregarded as the damage of the components

D_{k (T, β)} = 1 - A_{2} / A

(6)

where

D_{k (T, β)}

is the amount of damage of the components. It is assumed that the undamaged part of the element satisfies Hooke’s law:

σ_{0}^{'} = E_{0} ε^{'}

(7)

where

E_{0}

is the elastic modulus of the element and

ε^{'}

represents the strain caused by the effective stress of the undamaged part. According to the deformation coordination principle, the total strain of the element caused by the total stress should be equal to the strain caused by the effective stress of the undamaged part.

Equation (6) is substituted into equation (5), and the combined results are linked with equations (5) and (7)

ε = σ_{0} / E_{0} (1 - D_{k (T, β)})

(8)

Equation (8) is calculated based on the differential principle. Under the condition of three-dimensional compression, the expression of the stain rate is obtained by combining equation (3)

\begin{array}{l} \dot{ε} = Δ ε / Δ t = σ_{0} Δ D_{k (T, β)} / E_{0} Δ t (1 - D_{k (T, β)}) (1 - D_{k (T, β)} - Δ D_{k (T, β)}) \\ = σ_{0} |D_{k (T, β)}| / E_{0} {(1 - D_{k (T, β)})}^{2} \\ = - σ_{0}^{γ + 1} / λ^{γ} E_{0} {(1 - D_{k (T, β)})}^{γ + 2} \end{array}

(9)

where

Δ t

is a small time step,

Δ ε

is a small strain value, and

Δ D_{k (T, β)}

is a small damage value. At the accelerated creep stage,

\dot{ε}

> 0 go to zero.

By integrating the piecewise functions in equation (9), the strain expressions of attenuation and steady creep and accelerated creep can be obtained as follows:

ε = \{\begin{array}{l} ε_{0} + σ_{0} / E_{0} (1 - D_{T}) - σ_{0} {({(1 - D_{T})}^{γ + 1} + t σ^{γ} (γ + 1) / λ^{γ})}^{- 1 / γ + 1} / E_{0}, & t < t_{s} \\ ε_{1} - σ_{0} / E_{0} (1 - D_{k (T, β), \min}) + σ_{0} {((t_{s} - t) σ^{γ} (γ + 1) / λ^{γ})}^{1 / γ + 1} / E_{0}, & t \geq t_{s} \end{array}

(10)

where

ε

represents the total strain value of the element caused by the total stress at anytime;

ε_{0}

represents the initial strain value in the steady creep stage; and

ε_{1}

represents the final strain value in the steady creep stage.

The creep equation of the elastic-plastic element after high temperature exposure has the function of expressing three-stage creep strain. Based on the reduction in the number of parameters in the model, the improved Nishihara model’s constitutive model is straightforward. Only a single element of the constitutive model will be discussed. Therefore, equation (10) is the constitutive model.

Parameter determination of the constitutive model

Analytical methods and numerical methods are usually used for the identification of the parameters of the constitutive model. An analytical approach can be directly analyzed by experiments and data curves. This test’s rheology curves show evident stages of instantaneous deformation, viscoelastic deformation, and viscoplastic deformation. Three rules are followed. First, the elastic and elastoplastic parameters are determined by instantaneous deformation. Second, viscoelastic parameters are determined by viscoelastic deformation or stable creep deformation. Third, viscoplastic deformation or unstable creep deformation is used to determine viscoplastic parameters.

Simplifying equation (10), we obtain the constitutive creep equation for layered rock after exposure to high temperatures:

ε = \{\begin{array}{l} a_{1} - a_{2} {(1 - a_{3} t)}^{a 4} \\ b_{1} + b_{2} {(b_{3} (t_{s} - t) + 1)}^{b 4} \end{array}

(11)

Where the simplified parameters $a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}$ and $b_{4}$ are represented as:

a_{1} = ε_{0} + σ_{0} / E_{0} (1 - D_{T})

(12)

a_{2} = a_{1} - ε_{0}

(13)

a_{3} = (γ + 1) σ^{γ} / λ^{γ} {(1 - D_{T})}^{γ + 1}

(14)

a_{4} = - 1 / γ + 1

(15)

b_{1} = ε_{1} - σ_{0} / E_{0} (1 - D_{k (T, β), \min})

(16)

b_{2} = ε_{1} - b_{1}

(17)

b_{3} = (γ + 1) σ^{γ} / λ^{γ} {(1 - D_{k (T, β), \min})}^{γ + 1}

(18)

b_{4} \neq a_{4}

(19)

Based on the above formula, there are four independent position parameters in the model: $E_{0}$ , $D_{k (T, β), \min}$ , $λ^{γ}$ , $γ$ .

Triaxial creep test

Test equipment and scheme

In this test, the rock samples were layered structured sandstone with different bedding angles. The cylindrical size of the rock samples is approximately Φ 50 mm × 100 mm (diameter × height) according to Rock Test Regulations for Water Resources and Hydropower Engineering (SL264-2007). The bedding angles $β$ were 0°, 30°, 45°, 60°, and 90°, respectively. The article employs an orthogonal method to study samples at different temperatures and angles. By controlling the sole variable, two sets of samples at angles of 0° and 90° are selected to analyze the effect of temperature on the creep characteristics of sandstone at temperatures of 20°C, 200°C, 400°C, 600°C, and 800°C, respectively. Additionally, samples at temperatures of 20°C, 400°C, and 800°C are selected to analyze the influence of different bedding angles (0°, 30°, 45°, 60°, and 90°) on creep characteristics. Therefore, a total of 19 rock samples were prepared. The layered sandstone samples with different bedding angles obtained after high temperature are shown in Figure 4.

The prepared samples were kept in an environment with a constant temperature and humidity. The test heating device adopted aKXX-10-12A box resistance furnace, which can heat up to $1200 \circ C$ . In the present study,the rock samples were heated from $20 \circ C$ (room temperature) to $200 \circ C$ , $400 \circ C$ , $600 \circ C$ and $800 \circ C$ . The YSJ-01-00 creep testing machine was selected and can accurately recordthe axial load, confining pressure, axial displacement, time, and other related data in the experiment. At the beginning of the sandstone heating test, the rock samples were thoroughly heated at a rate of $20 \circ C$ /min and maintained for 4 hours after reaching the specified temperature. Then, they were naturally cooled to $20 \circ C$ in the hearth and placed in a container with desiccant. XRD (X-ray powder diffraction) of naturally cooled layered sandstone samples after high temperature exposure was carried out by a DX-2700 diffractometer. The test results show that the mineral components of these layered sandstones were mainly quartz and potash feldspar without clay minerals. Of them, Quartz accounted for 93% ∼ 95%. The confining pressure, the initial loading grade, and the loading modes of the conventional triaxial creep test were 5 MPa, 10.1 MPa, and 10.1 MPa, respectively, based on the results of the average temperature conventional triaxial compressive strength test of samples of the same batch. The loading process adopts load control at a rate of 0.1 kN/s, and the data collection is carried out according to time control (120 s) and displacement control (0.02 mm). In the creep test, the holding period for each load level of the rock sample is designed to be 3 days. When the deformation rate under each load level is less than 0.0004 mm/h, the next load level is applied. The strain values of the rock under lower axial load levels quickly stabilize over time, so the actual consolidation time under the first two load levels is approximately 2 days, while the actual consolidation time under the remaining load levels is approximately 3 days.

Experimental results

Creep characteristic analysis

The creep curves of layered sandstone after different high temperature are obtained when the bedding angle $β$ is 0° and 90°, respectively, OriginLab 2020 was used to draw Figure 5.

The results of the rest of the layered sandstone are not listed here due to page limitations. It can be observed that (1) the axial strain of the layered sandstone after different high temperatures in both cases increases with time. (2) The amount of initial creep of the specimens exhibits a small difference if the treatment temperature is within the range of $20 \sim 600 \circ C$ . After $800 \circ C$ , the amount of initial creep of the specimens increases significantly compared with those of other temperature conditions. (3) The total amount of creep and ductility of the samples increase with increasing treatment temperature. The total amount of creep of the sample increases obviously when the treatment temperature reaches $600 \circ C$ , and the creep test ductility increases significantly again when the treatment temperature reaches $800 \circ C$ . (4) In the accelerated creep stage, the failure time of the sample under the final load is shorter than the time under other high temperatures when the processing temperatures are $200 \circ C$ and $400 \circ C$ , respectively. In addition, the phenomenon that the creep duration of the specimens increases significantly is observed at $600 \circ C$ and $800 \circ C$ . (5) Based on the consideration of the number of load grades, the number of load grades at average temperatureis the smallest, and the load grade at $600 \circ C$ is the greatest. Creep still shows prominent three-stage creep characteristics after exposure to high temperatures.

The whole creep curves of the axial strain of layered sandstone specimens at different bedding angles change with time when the treatment temperatures are $20 \circ C$ and $400 \circ C$ , respectively. OriginLab 2020 was used to draw Figure 6.

The creep curves with a bedding angle of 60° at the two temperatures are the shortest among the different bedding angles, while the creep curves of 90° are the longest. This indicates that the layered sandstone with a bedding angle of 60° or 90° has the lowest or highest failure load level among the different bedding angles. With increasing time, the creep curve with $β = 0 \circ$ is at the bottom of the whole curve cluster, while that of $β = 60 \circ$ is always at the top, and those of other groups of angles are staggered between them. The results show that the ductility of the layered rock specimens is the worst when $β = 60 \circ$ after different temperature treatments, and the bedding angle significantly influences the creep ability of layered rock samples.

Steady creep rate analysis

In the conventional triaxial creep test of rock, the creep rate of rock will maintain a nonzero constant growth and eventually accelerate creep failure when the load applied to layered rock reaches or exceeds the long-term strength. Based on the last load data and creep process curve, the stable creep rate of layered sandstone after different treatment temperatures is statistically analyzed and obtained, OriginLab 2020 was used to draw Figure 7.

Figure 7(a) shows that the steady creep rates of samples with different bedding angles show a slow upward trend when the treatment temperature is lower than $400 \circ C$ . Compared withthe steady creep rates at $20 \circ C$ , the steady creep rates increase by 324.44% and 25% whenthe treatment temperature reaches $400 \circ C$ . If the treatment temperature is more significantthan that at $400 \circ C$ , the steady creep rate of samples with different bedding angles shows a rapid downward trend. The steady creep rates decrease by 75.71% and 87.08%, respectively.

When the treatment temperature is higher than $600 \circ C$ , the steady creep rate does not change, and the creep rates of samples with different bedding angles are similar, which indicates thatthe anisotropy degree of rock decreases and tends to be isotropic under the effect of high temperature. Figure 7(b) shows that the steady creep rate changes in a nearly M shape with increasing bedding angle. It reached the minimum at $β = 0 \circ$ and the maximum at $β = 60 \circ$ .

Long-term strength characteristics

The isochronal curve method is used to analyze the long-term strength of samples (Shen and Chen, 2011). Under the same stress level, the stress and strain values at different times are used to draw the isochronous stress-strain curves carved at 0.01 h, 1 h, 5 h, 10 h, 20 h,40 h, 80 h, and 100 h after different treatment temperatures. The inflection point value of the curve cluster is selected as the long-term strength value. The long-term strength curves of layered sandstone at different bedding angles and after different temperatures are obtained, OriginLab 2020 was used to draw Figure 8.

Figure 8(a) shows that the long-term intensity of samples with different bedding angles continues to increase and reaches a maximum at $600 \circ C$ . The long-term strength begins todecrease when the temperature exceeds $600 \circ C$ .

Figure 8(b) shows that the long-term strength of layered sandstone decreases first and then increases with increasing bedding angle. When $β = 0 \circ$ and $β = 30 \circ$ , the change in long-term strength is small. The long-term strength decreases and reaches the lowest value at $β = 60 \circ$ , which corresponds to the steady creep rate law in Figure 6. The long-term strength increases with a further increase in bedding angles.

Verification of the constitutive model and parameter identification

The conventional triaxial creep test of layered sandstone under different experiment conditions is substituted, and the unknown parameters are solved by the MATLAB least square method to verify the reliability and rationality of the constitutive model established in this paper. Considering that there are many test data, the data corresponding to the load of the previous stage and the load of the last stage load the long-term strength of the sample are selected for fitting. Conventional triaxial creep test data of four representative samples were selected: different treatment temperatures of $200 \circ C$ , $400 \circ C$ , $800 \circ C$ at the bedding angle $β = 90 \circ$ , and $β = 0 \circ$ after the $400 \circ C$ temperature treatment.

According to the constitutive equation of equation (10), the main model parameters are the initial elastic modulus of an elastic-plastic element $E_{0}$ , the minimum damage value $D_{k (T, β), \min}$ , and the model-related material parameters $λ^{γ}$ and $γ$ . The combination of different model parameters is expressed by $a_{i}$ and $b_{i}$ (where i = 1,2,3,4).

Equation (10) is used for the calculation to simplify the constitutive model, and equation (11) is used for the simplified expression of the parameters. The results of the calculation parameters are shown in Tables 1 to 4, and the comparison of the calculation results and test results of the samples at different treatment temperatures and different bedding angles is shown in Figure 9, which is drawed by OriginLab 2020:

Table 1.

Model parameters of the samples with a treated temperature at $200 \circ C$ and a bedding angle of 90°.

$a_{1} / 10^{- 3}$	$a_{2} / 10^{- 3}$	$a_{3} / h^{- 1}$	$a_{4}$	$b_{1} / 10^{- 3}$	$b_{2} / 10^{- 3}$	$b_{3} / h^{- 1}$	$b_{4}$
1.726	4.414	1.237	−0.25	–	–	–	–
1.896	−0.414	−152.2	1.32	2.854	2.359	2.290	−0.320

Table 2.

Model parameters of the samples with a treated temperature at $400 \circ C$ and a bedding angle of 0°.

$a_{1} / 10^{- 3}$	$a_{2} / 10^{- 3}$	$a_{3} / h^{- 1}$	$a_{4}$	$b_{1} / 10^{- 3}$	$b_{2} / 10^{- 3}$	$b_{3} / h^{- 1}$	$b_{4}$
1.442	3.451	1.613	−0.257	–	–	–	–
1.052	−0.594	−145.2	2.530	2.456	2.565	1.540	−0.530

Table 3.

Model parameters of the samples with a treated temperature at $400 \circ C$ and a bedding angle of 90°.

$a_{1} / 10^{- 3}$	$a_{2} / 10^{- 3}$	$a_{3} / h^{- 1}$	$a_{4}$	$b_{1} / 10^{- 3}$	$b_{2} / 10^{- 3}$	$b_{3} / h^{- 1}$	$b_{4}$
1.649	3.911	1.452	−0.338	–	–	–	–
2.483	−0.315	−125.100	3.417	3.757	2.343	1.890	−0.417

Table 4.

Model parameters of the samples with a treated temperature at $800 \circ C$ and a bedding angle of 90°.

$a_{1} / 10^{- 3}$	$a_{2} / 10^{- 3}$	$a_{3} / h^{- 1}$	$a_{4}$	$b_{1} / 10^{- 3}$	$b_{2} / 10^{- 3}$	$b_{3} / h^{- 1}$	$b_{4}$
1.617	3.512	1.971	−0.327	–	–	–	–
4.233	−0.485	−179.400	4.246	3.061	2.691	1.780	−0.246

According to the comparative analysis of creep test curves and theoretical curves of representative samples in Figure 9, the fitting degree is high, which overcomes the disadvantage of the low description of the acceleration deformation stage obtained by the traditional Nishihara model. The whole creep process can be accurately described, which shows that the proposed model is reasonable and reliable.

Discussion

Discussion on the parameter sensitivity of the constitutive model

To understand the influence of the parameters in the creep constitutive model on the creeps train law, the sensitivity analysis of the simplified parameters $a_{1} \sim a_{4}$ , $b_{3}$ and $b_{4}$ in equation (11) will be performed by following the parameter sensitivity analysis method (Li et al., 2019). Essentially, the constitutive model established in Section 2 is a staged model. Based on the staged model, the sensitivity analysis can be divided into two parts: attenuation and stablecreep and accelerated creep. In the stage of attenuation and steady creep, two sets of relevant data of the samples are mainly analyzed, including (1) $β = 90 °$ , T = $200 \circ C$ , $a_{1} = 1.896$ , $a_{2} = - 0.414$ , $a_{3} = - 152.2$ , $a_{4} = 1.32$ , and a load of 152.79 MPa; (2) $β = 90 °$ , T = $800 \circ C$ , $a_{1} = 4.233$ , $a_{2} = - 0.485$ , $a_{3} = - 179.4$ , $a_{4} = 2.46$ , and a load of 50.93 MPa. In Case (1), each of $a_{1}$ and $a_{3}$ is discussed through the parameter sensitivity analysis method, and other parameters remain unchanged. The parameter discussion for Case (2) is similar to that of Case (1), except that only $a_{2}$ and $a_{4}$ change. In the stage of accelerated creep, only one setoff data is analyzed due to lack of data points, that is, (3) $β = 90 °$ , $b_{1} = 3.061$ , $b_{2} = 2.691$ , $b_{3} = 1.78$ , $b_{4} = - 0.246$ , and T = $800 \circ C$ . The relation curves of the axial strain oflayered sandstones with time when only the values of the parameters $a_{1} \sim a_{4}$ , $b_{3}$ are given, OriginLab 2020 was used to draw Figure 10.

Figure 10(a) shows the creep curves of the axial strain of layered sandstone samples with time when the values of $a_{1}$ are 1.735, 1.700, and 1.800. Figure 10(b) shows the creep curves of the axial strain of layered sandstone samples with time when the values of $a_{2}$ are 399, 300, and 500. The changes in parameters a₁ and a₂ directly affect the amount of instantaneous creep and total creep but not the trend and line type of creep. Under the same stress level,the axial strains of the instantaneous creep and total creep increase and decrease correspondingly with the increase in $a_{1}$ and $a_{2}$ , respectively. It can be observed from Figure 10(c) that a₃ has a noticeable influence on the creep rate change rate, the time entering stable creep, and the total amount of creep. A smaller value of $a_{3}$ will shorten the time required to enter the stable creep stage and reduce the rate of creep change, resulting in a larger total amount of creep. Similarly, the larger the value of $a_{4}$ is, the shorter the duration of attenuation creep and the larger the total creep variable, as shown in Figure 10(d). Changing the value of $b_{3}$ has a little effect on the stable creep rate but has a significant control effect on the total creep variable, as shown in Figure 10(e). In other words, the amount of total creep increases with the increase in $b_{3}$ . However, both the steady creep rate and the total amount of creep are verydependent on the value of $b_{4}$ , as shown in Figure 10(f). The larger the value of $b_{4}$ is, the larger the creep rate of the total amount of creep.

The parameters in equation (11) correspond to the following physical symbols in the creep model: the $a_{1}$ and $a_{2}$ reactions for $E_{0}$ , $λ^{γ}$ from $a_{3}$ by a separate reaction, the a₃ and a₄ reactions of $γ$ , and the joint influence of $b_{3}$ reactions $λ^{γ}$ , $γ$ , and $D_{k (T, β), \min}$ , $b_{3}$ reactions of $γ$ .

In summary, the total amount of creep is affected by all parameters. In the attenuation and steady creep stage, the changes in the elastic modulus of the elastic-plastic element $E_{0}$ directly affect the instantaneous amount of creep. The model parameters related to materials $λ^{γ}$ and $γ$ affect the span of entering the steady creep time. The model parameters related to material g control the span of attenuation creep time and affect the steady creep rate. In the accelerated creep stage, the values of the minimum damage variables of layered rocks during creep compression $D_{k (T, β), \min}$ and the model parameters related to materials $λ^{γ}$ and $γ$ have a noticeable influence on the total amount of creep.

Discussion on the influence of high temperature on rock anisotropy

Niandou H. et al. (1997) proposed the following formula to describe the degree of anisotropy of rock:

k_{T} = E_{0 \circ} / E_{90 \circ}

(20)

where

k_{T}

denotes an anisotropic parameter of the rock elastic modulus.

E_{0^{\circ}}

and

E_{90^{\circ}}

denote the elastic modulus of rock when the bedding angle is

β = 0 °

and

β = 90 °

, respectively.In the present study, a relative index isotropic deviation coefficient (IDC) is proposed to describe the anisotropy of layered rocks as follows:

IDC = |E_{90 \circ} - E_{0 \circ}| / E_{90 \circ} \times 100 %

(21)

Among them, 0% < IDC < 100%. IDC indicates the intensity of anisotropy. The greater the value is, the stronger the anisotropy. The ratio of stress to strain at half of the peak load is the conventional creep elastic modulus based on the test data of Section 3.2. The creep elastic modulus is statistically summarized after different treatment temperatures with $β = 0 °$ and $β = 90 °$ . The creep elastic modulus is listed in Table 5.

Table 5.

Table of the values of the creep elastic modulus.

Treatment temperature/ $\circ C$	Bedding angle/°	Elastic modulus/GPa
20	0	6.48
20	90	5.67
200	0	7.14
200	90	7.26
400	0	8.49
400	90	8.10
600	0	6.58
600	90	5.71
800	0	3.92
800	90	3.87

The anisotropy parameter $k_{T}$ and the IDC can be calculated according to equations (20) and (21) and are listed in Table 6. The relationship between the anisotropy parameter $k_{T}$ , and isotropic deviation coefficient IDC and temperature is shown in Figure 11.

Table 6.

Anisotropy parameter $k_{T}$ and the IDC results of layered rocks after high temperature exposure.

Treatment temperature/ $\circ C$	20	200	400	600	800
$k_{T}$	1.2	0.98	1.05	1.15	1.01
IDC/%	12.50	1.68	4.59	13.22	1.28

When the treatment temperatures are $200 \circ C$ , $400 \circ C$ , and $800 \circ C$ , the anisotropy parameter $k_{T}$ differs from 1 by 0.02, 0.05, and 0.01, respectively, and the IDC values are 1.68%, 4.59%, and 1.28%, respectively. After $20 \circ C$ and $600 \circ C$ , the anisotropy parameter differs from 1 by 0.2 and 0.15, respectively, and the IDC values are 12.50% and 13.22%, respectively. Itis found that the changes in $k_{T}$ and IDC with temperature are in accordance with the power law relationship obtained by nonlinear fitting. The anisotropy of layered rocks weakens and gradually tends to be isotropic with increasing treatment temperature. A lower treatment temperature will make the anisotropic strength reduction faster. When the treatment temperature exceeds $600 \circ C$ , the anisotropic degree tends to be flat. As illustrated by the results in Figure 7(a), when the treatment temperature of the rock samples exceeds 600°C, the steady creep rate remains unchanged, and the creep rates of samples with different bedding angles become similar. This indicates a reduction in the degree of anisotropy of the rock, tending towards isotropy under the influence of high temperature. Therefore, the treatment temperature at $600 \circ C$ can be regarded as the temperature threshold for layered rock to change from anisotropic to isotropic. OriginLab 2020 was used to draw Figure 11, it shows that the IDC can reflect the anisotropy degree of layered rock more intuitively and more accurately reflect the evolution trend of anisotropy degrees with temperature change than the anisotropy degree parameter $k_{T}$ . The IDC only needs to compare the percentage values under different treatment temperature conditions to determine the degree of anisotropy.

The mechanism of the change in anisotropy with different treatment temperatures can be observed using slice images of layered rock, as shown in Figure 12. The main components of this batch of rock samples are quartz and potash feldspar. The difference in the thermal elastic modulus and thermal expansion coefficient between quartz and potash feldspar is the fundamental reason for the anisotropy of rocks (Homand and Houpert, 1989). With increasing treatment temperature, different minerals demonstrate irreversible deformation and expansion to varying degrees, and the sericitization of feldspar is gradually apparent. Another apparent phenomenon is that both ends of cracks are almost distributed at the unsmooth points of the crystal geometric boundary shape. The irregular geometric shape of the crystal leads to stress concentration under the action of thermal stress, thereby resulting in crystal cracking. This phenomenon was founded by (Liang et al., 2006).

Based on the above process development, the internal stress of the crystals exceeds the cohesion and strength of the minerals, and intragranular cracks appear and gradually evolve into trans granular cracks. The number of trans granular cracks increases with increasing treatment temperature. When the treatment temperature reaches $600 \sim 800 \circ C$ , ironimpregnation along the microcracks becomes increasingly apparent, explaining how layeredsandstone samples change from yellow-brown to reddish-brown with increasing treatment temperature.

In summary, the irreversible melting mixing phenomenon at the boundary of mineral particles caused by high temperature is the essential reason for the weakening of the anisotropy of layered rock. The treatment temperature has a significant influence on the strength of mineral crystals, and the reaction is the decrease in elastic modulus and long-term strengthen macroscopic mechanical properties, which is mutually confirmed with the test data of long-term strength characteristics in Subsection 3.2.3 and the creep elastic modulus after the high temperature in Table 5. In addition, Figure 12(c) shows that the melting convergence phenomenon of the mineral boundary appears with increasing treatment temperature. The number of intergranular fractures is reduced compared with those of Figure 12(a) and (b), and the boundaries between mineral crystals are blurred, which indicates that the increase in treatment temperature causes the mineral crystals to melt to a certain extent, and the mineral crystals near the intergranular fractures produce new fusion. After natural cooling, a local mixture of mineral components is formed in this part of the region, which weakens the anisotropic strength of the local mixture and tends to be isotropic; then, the macroscopic anisotropy of layered rock is affected.

Conclusions

This research examines the creep characteristics of layered sandstone after exposure to high temperatures based on physical experiments. The key findings are summarized below:

An improved Nishihara creeps’ constitutive model for layered rock is proposed based on the consideration of various damage factors and bedding angles. The types of damage include thermal damage caused by natural cooling after high temperature and creep damage caused by conventional triaxial compression, and the constitutive creep model is verified by the conventional triaxial creep compression test. The theoretical curve has a high degree of fitting with the experimental curve, which overcomes the shortcomings of the description deviation of the conventional Nishihara model in the acceleration stage. Parameter sensitivity analysis shows that in the attenuation and steady creep stage, the changes in the elastic modulus of the elastic-plastic element $E_{0}$ directly affect the instantaneous amount of creep. The model parameters related to materials $λ^{γ}$ and $γ$ affect the time of entering steady creep. The model parameters related to material g control the span of attenuation creep time and affect the steady creep rate. In the accelerated creep stage, the values of the minimum damage variables of layered rocks during creep compression $D_{k (T, β), \min}$ and the model parameters related to materials $λ^{γ}$ and $γ$ have a noticeable influence on the total amount of creep.

The results of the creep test after high temperature exposure show that the long-term strength first decreases and then increases and the steady creep rate changes in an M shape with increasing bedding angle. The treatment temperature of $400 \circ C$ has an obvious influence on the steady creep rate and the creep strain of layered sandstone. The treatment temperature at $600 \circ C$ mainly affects the long-term strength of layered sandstone. The temperature of $600 \circ C$ can be regarded as the temperature threshold for layered rock to change from anisotropic to isotropic.

The isotropic deviation coefficient (IDC) is proposed to represent the anisotropy of layered rock. The IDC has the advantages of simple calculation and accurately reflects the evolutionary trend of anisotropy degrees with temperature change. The relationship between the IDC and temperature conforms to the power law.

The mechanism by which different treatment temperatures affect the anisotropy degree of layered rock is revealed from the micro perspective by thin section tests. The irreversible melting mixing phenomenon at the boundary of mineral particles caused by high temperature exposure is the essential reason for the weakening of the anisotropy of layered rock. Both the strength of mineral crystals and the number of intergranular fractures decrease with increasing treatment temperature, and the melting convergence phenomenon of mineral boundaries appears with increasing treatment temperature.

Highlights

An improved Nishihara creeps constitutive model for layered rock is proposed based on the consideration of various damage factors and bedding angles.

The isotropic deviation coefficient (IDC) is proposed to represent the anisotropy of layered rock.

The mechanism by which different treatment temperatures affect the anisotropy degree of layered rock is revealed from the micro perspective by thin section tests.

Footnotes

CRediT authorship contribution statement

Lubo Meng: Conceptualization, Writing-original draft, Methodology. Shan Zhang: Data curation. Haoyu Li: Software, Validation. Tianyi Liu: Experimental program, implementation, and operation. Tianbin Li: Writing-review editing, Supervision, Funding acquisition.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the National Natural Science Foundation of China (No. 42130719), State Key Laboratory of Geohazard Prevention and Geoenvironment Protection Independent Research Project (SKLGP2022Z003), and the Sichuan Science and Technology Planning Project (No. 2022NSFSC0411).

ORCID iD

Shan Zhang

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