# An Experimental Study on the Creep Characteristics of Sandstone in the Interval of Different Critical Stresses

Authors: Chao Yang; Xingchen Dong; Xuan Xu; Qiancheng Sun
Journal: Shock and Vibration
 (2021)

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## Abstract

Creep tests on brittle sandstone specimens were performed to investigate the time-dependent characteristics in the interval of different critical stresses. The results showed that failure will not occur when the loaded stressσ1 is less than the critical stress of dilation σcd, while all specimens were destroyed when σ1 is larger than σcd. In addition, the value of σcd was very close to the long-term strength obtained by the method of the isochronous stress-strain curve. Therefore, σcd can be regarded as the long-term strength of the sandstone specimens. When σ1 is larger than σcd, the time required for the failure of specimen tf decreases with the increase of σ1; the creep rate dε/dt increases with time t, and the specimen will be destroyed when it reaches a maximum value (dε/dt)max. Both relationships tf and σ1 and (dε/dt)max and σ1 can be described by the exponential function. Then, a nonlinear damage creep model considering the deformation damage and strength damage in the interval of different critical stresses was established, which can describe the whole creep process and predict the failure time of sandstone specimens.


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## Body

## 1. Introduction

Critical stresses (the crack closure stress levelσcc, crack initiation stress level σci, critical stress of dilation σcd, and peak strength σc) are important indexes for evaluating hard brittle rocks [1–3], which reflect the internal microfracture activity state of rocks under different stress levels. The failure process of rocks under different compression conditions can be divided into 4 stages, which are crack closure stage (σ1 < σcc), elastic region (σcc < σ1 < σci), stable crack growth stage (σci < σ1 < σcd), and unstable crack growth stage (σcd < σ1 < σc). Obviously, the creep characteristics of hard brittle rocks should also be different when under long-term loading in different critical stress intervals.As the critical point of time-dependent damage under the creep condition, the long-term strength of rocks has always been the focus of researchers [4–6]. Currently, there are mainly two methods to determine the long-term strength. (1) The steady-state creep rate [7]: carrying out creep tests at multiple stress levels by the “Chen method” and determining long-term strength as the maximum load with zero creep rate. (2) The stress-strain isochronal curves [8, 9]: treating the long-term strength as the yield strength and considering the time effect. Both methods mentioned above are based on a large number of creep tests, which makes the acquisition of long-term strength of rocks relatively complicated. If the results of uniaxial or triaxial tests can be used to establish the correlation with the long-term strength of rocks, the difficulty of the test will be greatly reduced. Some attempts have been made, such as Liu [10] who used the volume expansion method which determines the long-term strength by the volume expansion point and Ding et al. [11] who proposed a method to predict salt rock damage based on the stress corresponding to the damage initiation point as the long-term strength. However, there are few studies on the relationship between critical stresses and long-term strength of hard brittle rocks, as well as creep characteristics under different critical stress intervals.In this study, a series of laboratory tests, including uniaxial compression test, multistage loading creep test, and single-stage loading creep test, were carried out on sandstone specimens. The creep characteristics of specimens in the interval of different critical stresses were investigated. The conclusions can provide reference for the analysis of time-dependent behavior of hard brittle rocks.


## 2. Specimen Preparation and Testing

The sandstones were obtained from Chongqing in China, whose natural dry density was approximately 2.68  ×  103 kg/m3. Specimens made from sandstones had a columnar shape with a diameter of 50 mm and a height of 100 mm, which were in accordance with the standard recommended by the International Society for Rock Mechanics.All the tests were performed on a HYZW-500L Rock Mechanics Testing System at the Key Laboratory of Geological Hazards on Three Gorges Reservoir Area, Ministry of Education, China. The loading rate of all tests was maintained at constant 0.2 kN/s. Firstly, uniaxial compression tests were carried out to obtain the critical stresses of the specimens. The axial stressσ1 was loaded on specimens progressively from 0 MPa until specimens were destroyed. Then, multistage loading creep tests were carried out to obtain the long-term strength. σ1 was loaded progressively from 0 MPa till the first stage of loading, and then it was increased to the next stage when the deformation was stable. Finally, the single-stage creep tests were performed, in which σ1 was set based on the results of the first two tests. After reaching the design level with the same loading procedure mentioned above, σ1 was kept constant until specimens were destroyed or the deformation was stable without increase (in this case, the loading time was at least 200 hours). The load, time, and displacements were recorded in all the tests.


## 3. Experimental Results

### 3.1. Critical Stresses

Based on the uniaxial compression test, the crack volumetric strainεvc, the volumetric strain εv, and the elastic volumetric strain εve are calculated as follows [6]:(1)εvc=εv−εveεv=ε1+2ε3εve=1−2μσ1E,where ε1 is the axial strain, ε3 is the radial strain, σ1 is the stress level, E is the elastic modulus, and μ is Poisson’s ratio.Theσ1-ε1-ε3, εv-ε1, and εvc-ε1 curves of specimens are shown in Figure 1. According to the crack strain model method, the stress corresponding to the end of the concave section of the σ1-ε1 curve is point σcc, the stress corresponding to the beginning of the negative correlation between εvc and ε1 in the εvc−ε1 curve is point σci, and σcd is the value corresponding to the inflection point of the εv−ε1 curve, after which the whole specimen starts to expand. Therefore, the critical stresses and deformation parameters of the sandstone specimen were σcc ≈ 22.6 MPa, σci ≈ 50.3 MPa, σcd ≈ 72.6 MPa, σc ≈ 95.2 MPa, E ≈ 11.2 GPa, and μ ≈ 0.2.Figure 1
Uniaxial compression test curves and failure of the sandstone specimen.


### 3.2. Long-Term Strength of the Isochronal Curve Method

Based on the multistage loading creep tests, the time-dependent deformation curve underσ1 = 50, 60, 70, and 80 MPa is shown in Figure 2. The deformations were stable after 24 h at σ1 = 50, 60, and 70 MPa, while creep failure occurred after 17.4 h when σ1 = 80 MPa.Figure 2
Creep deformation curves of the multistage creep test and failure of the specimen.The isochronal curve composed of stress and strain values at loading timet of 2, 4, 8, 16, and 24 h is shown in Figure 3. The spacing between curves started to increase gradually from σ1 of 70 MPa, and the spacing with σ1 of 80 MPa between curves was significantly different from that of σ1 of 60 and 70 MPa. Therefore, the long-term strength of specimens was determined to be 70 MPa.Figure 3
Stress-strain isochronal curves.


### 3.3. Creep Rate of the Single-Stage Loading Creep Test

According to critical stresses of specimens and combined with results of the multistage loading creep test, the stress levelsσ1 of the single-stage loading creep tests were designed to be 50, 60, 72, 74, 75, 80, and 85 MPa. From the test curves and specimen failures as shown in Figure 4, specimens did not fail under σ1 of 50, 60, and 72 MPa, while the accelerated creep stage and failure occurred under σ1 of 74, 75, 80, and 85 MPa, and the creep failure time gradually decreased to 139.5, 108.7, 42.2, and 17.5 h, respectively. The long-term strength obtained by the stress-strain isochronal curves was about 70 MPa, and the stress of creep failure in the single-stage loading creep tests was between 72 and 74 MPa, which was extremely close to the critical stress of dilation (σcd ≈ 72.6 MPa) obtained in the uniaxial compression test. Therefore, σcd can be approximated as the long-term strength of specimens in this paper. It can be considered that when σ1 is less than σcd, there will be no accelerated creep stage and no creep failure to specimens; when σ1 is greater than σcd, the accelerated creep stage and failure of specimens will occur.Figure 4
Creep deformation curves of the single-stage creep test and failure of the specimens.As shown in Figure4, the creep curves only showed characteristics of decay creeping, and the creep rate (dε1/dt) finally approaches 0 in the crack closure stage, elastic region, and stable crack growth stage (σ1 < σcd).By analyzing the creep rate variation characteristics of four specimens at the accelerated creep stage shown in Figure5, the steady-state creep stage of specimens was relatively short, which decreased with an increase of stress levels (the durations under σ1 of 74, 75, 80, and 85 MPa were 10.1, 6.0, 5.0, and 2.8 h, respectively). Yu et al. [12] showed that the time-dependent deformation exhibited is unstable in the secondary creep stage, and the results of the tests are similar. After the steady-state creep stage, there was a stage of slowly increasing creep rate defined as stage IIIa (which is relatively evident in Figures 5(c) and 5(d)) in this paper. After stage IIIa, there was a stage in which the creep rate increases abruptly defined as stage IIIb; it was at the end of this stage that specimens failed. Due to the penetration and closure of some microfractures inside specimens under creep conditions, a steep increase and then a steep decrease of creep rate can be found in Figures 5(a) and 5(b). The failure modes of specimens were brittle failure, which is reflected in the creep curve as a sudden vertical ascent of the curve marked in Figure 4. The speed of the ascent was so fast that it presented a folded state rather than a smooth transition. Table 1 summarizes the creep rates and inclination angles of the marked area in Figure 4. There was a maximum creep rate (dε1/dt)max at which specimens failed. And there was a positive correlation between (dε1/dt)max and σ1; the larger σ1 was, the larger (dε1/dt)max was.Figure 5
Creep rate curves of specimens under different stress levels: (a)σ1 = 74 MPa; (b) σ1 = 75 MPa; (c) σ1 = 80 MPa; (d) σ1 = 85 MPa.
(a)(b)(c)(d)Table 1
Creep rate and curve inclination before failure.
σ1 (MPa)74758085(dε1/dt)max/h−10.63 × 10−53.67 × 10−510.00 × 10−522.00 × 10−5Inclination angles (°)15455085Creep curve before failure


## 3.1. Critical Stresses

Based on the uniaxial compression test, the crack volumetric strainεvc, the volumetric strain εv, and the elastic volumetric strain εve are calculated as follows [6]:(1)εvc=εv−εveεv=ε1+2ε3εve=1−2μσ1E,where ε1 is the axial strain, ε3 is the radial strain, σ1 is the stress level, E is the elastic modulus, and μ is Poisson’s ratio.Theσ1-ε1-ε3, εv-ε1, and εvc-ε1 curves of specimens are shown in Figure 1. According to the crack strain model method, the stress corresponding to the end of the concave section of the σ1-ε1 curve is point σcc, the stress corresponding to the beginning of the negative correlation between εvc and ε1 in the εvc−ε1 curve is point σci, and σcd is the value corresponding to the inflection point of the εv−ε1 curve, after which the whole specimen starts to expand. Therefore, the critical stresses and deformation parameters of the sandstone specimen were σcc ≈ 22.6 MPa, σci ≈ 50.3 MPa, σcd ≈ 72.6 MPa, σc ≈ 95.2 MPa, E ≈ 11.2 GPa, and μ ≈ 0.2.Figure 1
Uniaxial compression test curves and failure of the sandstone specimen.


## 3.2. Long-Term Strength of the Isochronal Curve Method

Based on the multistage loading creep tests, the time-dependent deformation curve underσ1 = 50, 60, 70, and 80 MPa is shown in Figure 2. The deformations were stable after 24 h at σ1 = 50, 60, and 70 MPa, while creep failure occurred after 17.4 h when σ1 = 80 MPa.Figure 2
Creep deformation curves of the multistage creep test and failure of the specimen.The isochronal curve composed of stress and strain values at loading timet of 2, 4, 8, 16, and 24 h is shown in Figure 3. The spacing between curves started to increase gradually from σ1 of 70 MPa, and the spacing with σ1 of 80 MPa between curves was significantly different from that of σ1 of 60 and 70 MPa. Therefore, the long-term strength of specimens was determined to be 70 MPa.Figure 3
Stress-strain isochronal curves.


## 3.3. Creep Rate of the Single-Stage Loading Creep Test

According to critical stresses of specimens and combined with results of the multistage loading creep test, the stress levelsσ1 of the single-stage loading creep tests were designed to be 50, 60, 72, 74, 75, 80, and 85 MPa. From the test curves and specimen failures as shown in Figure 4, specimens did not fail under σ1 of 50, 60, and 72 MPa, while the accelerated creep stage and failure occurred under σ1 of 74, 75, 80, and 85 MPa, and the creep failure time gradually decreased to 139.5, 108.7, 42.2, and 17.5 h, respectively. The long-term strength obtained by the stress-strain isochronal curves was about 70 MPa, and the stress of creep failure in the single-stage loading creep tests was between 72 and 74 MPa, which was extremely close to the critical stress of dilation (σcd ≈ 72.6 MPa) obtained in the uniaxial compression test. Therefore, σcd can be approximated as the long-term strength of specimens in this paper. It can be considered that when σ1 is less than σcd, there will be no accelerated creep stage and no creep failure to specimens; when σ1 is greater than σcd, the accelerated creep stage and failure of specimens will occur.Figure 4
Creep deformation curves of the single-stage creep test and failure of the specimens.As shown in Figure4, the creep curves only showed characteristics of decay creeping, and the creep rate (dε1/dt) finally approaches 0 in the crack closure stage, elastic region, and stable crack growth stage (σ1 < σcd).By analyzing the creep rate variation characteristics of four specimens at the accelerated creep stage shown in Figure5, the steady-state creep stage of specimens was relatively short, which decreased with an increase of stress levels (the durations under σ1 of 74, 75, 80, and 85 MPa were 10.1, 6.0, 5.0, and 2.8 h, respectively). Yu et al. [12] showed that the time-dependent deformation exhibited is unstable in the secondary creep stage, and the results of the tests are similar. After the steady-state creep stage, there was a stage of slowly increasing creep rate defined as stage IIIa (which is relatively evident in Figures 5(c) and 5(d)) in this paper. After stage IIIa, there was a stage in which the creep rate increases abruptly defined as stage IIIb; it was at the end of this stage that specimens failed. Due to the penetration and closure of some microfractures inside specimens under creep conditions, a steep increase and then a steep decrease of creep rate can be found in Figures 5(a) and 5(b). The failure modes of specimens were brittle failure, which is reflected in the creep curve as a sudden vertical ascent of the curve marked in Figure 4. The speed of the ascent was so fast that it presented a folded state rather than a smooth transition. Table 1 summarizes the creep rates and inclination angles of the marked area in Figure 4. There was a maximum creep rate (dε1/dt)max at which specimens failed. And there was a positive correlation between (dε1/dt)max and σ1; the larger σ1 was, the larger (dε1/dt)max was.Figure 5
Creep rate curves of specimens under different stress levels: (a)σ1 = 74 MPa; (b) σ1 = 75 MPa; (c) σ1 = 80 MPa; (d) σ1 = 85 MPa.
(a)(b)(c)(d)Table 1
Creep rate and curve inclination before failure.
σ1 (MPa)74758085(dε1/dt)max/h−10.63 × 10−53.67 × 10−510.00 × 10−522.00 × 10−5Inclination angles (°)15455085Creep curve before failure


## 4. Nonlinear Damage Creep Model

### 4.1. Damage Variable

Damage variableD is an important mechanical parameter for evaluating material properties and establishing the material damage model. When D  =  0, there is no damage to the material; when D  =  1, the material is fully damaged and loses its strength [13–17]. Combined with the test results, it was considered in this paper that when σ1 < σcd, the specimens will not fail, and there is no damage to specimens; when σ1 > σcd, the specimens will finally fail, and the damage exists in specimens. The damage has the following characteristics: (1) the damage is a time-dependent damage which increases gradually with time and finally reaches complete damage, namely, D  =  1. (2) The damage is also affected by σ1. The greater σ1 is, the shorter the time for specimens to reach the fully damaged state.In terms of the strength of specimens, creep failure occurred in specimens underσ1 of 74, 75, 80, and 85 MPa, which was reduced by 23%, 21%, 16%, and 10% compared with σc.In terms of the deformation of specimens, the viscosity coefficientη at each time in the creep process can be expressed as(2)dε1dt=σ1η.Whenσ1 is constant, the creep rate dε1/dt is inversely proportional to η. As shown in Figure 5, the creep rate of the specimen was constant in stage II and began to increase as it entered stage III, which indicates that η began to decay with increasing creep time. Therefore, it can be assumed that no time dependence was produced before stage III, while it started to appear after stage III. When σ1 was 74, 75, 80, and 85 MPa, η of creep failure was lost about 92%, 90%, 99%, and 95%, respectively.It was obvious that the damage of the specimen in strength was very different from that in deformation, and the same damage variable cannot be used to describe both. Therefore, two types of damage variables, named strength damage variable and deformation damage variable, were proposed from the analysis above.The strength damage variableD1 (σ, t) can be calculated as(3)D1σ,t=σc−σftσc−σ1,where σ1 is the long-term load applied on specimens and σf(t) is the stress corresponding to the creep failure time t.Whenσf(t) = σc, specimens are instantaneously damaged, namely, t = 0. When σf(t) = σcd, the specimen will be permanently undamaged, namely, t = ∞. Combined with the stress and time corresponding to the creep failure of specimens in the single-stage loading creep test, the fitting curve is presented in Figure 6.Figure 6
Relationship between load and time in the creep failure of specimens.It can be seen from Figure6 that the larger σ1 was, the shorter the time required for the creep failure of specimen tf was. This relationship can be described as(4)σftf=σcd+σc−σcdexp−αtf,where α is the correlation coefficient of damage degree, which can be fitted according to the test data. For specimens in this paper, α = −0.025. By combining equations (3) and (4), the expression of strength time-dependent damage variable D1(σ, t) of specimens can be described as(5)D1σ,t=σc−σcd1−exp−αtσc−σ1.The deformation damage variableD2(σ, t) of specimens proposed in this paper is(6)D2σ,t=0,t≤tw,1−ησ,t/η0,t>tw,where η(σ, t) is the viscosity coefficient function with creep time t, η0 is the viscosity coefficient at the steady-state creep stage, and tw is the time required to reach stage IIIa.Creep failure of specimens occurred whenD1 (σ, t) = 1, while D2 (σ, t) is not strictly equal to 1, and η(σ, t) corresponded to the maximum creep rate (dε1/dt)max as described in Table 1.Figure7 shows the relationship between tw and σ1 and the variation of η(σ, t) with time. The larger σ1 is, the shorter the time experienced before entering the steady-state creep stage is and the small η in the steady-state creep stage is. Moreover, η(σ, t) decreases gradually with time after stage II; this relationship can also be described as(7)ησ,t=η0,t≤tw,η0exp−βt,t>tw,where β is the correlation coefficient of damage degree, which is related to σ1. As σ1 increases, β increases gradually. The fitting curve combined with η0, β, and σ1 is shown in Figure 8.Figure 7
Relationship betweentw and σ1 and η(σ, t) and t: (a) tw-σ1; (b) η(σ, t)-t.
(a)(b)Figure 8
The relation graph of viscosity coefficientη0-σ1 and β-σ1 in stage II of the specimen: (a) η0-σ1; (b) β-σ1.
(a)(b)It can be observed that, with the increase ofσ1, η0, and β, different variation laws were shown: η0 decreased with σ1, while β increased with σ1. By combining equations (6) and (7), the expression of deformation time-dependent damage variable D1(σ, t) can be described as(8)D2σ,t=0,t≤tw,1−exp−βt,t>tw.


### 4.2. A New Damage Creep Model with Different Critical Stress Intervals

Based on the creep characteristics and two damage variables of specimens, a new 5-element nonlinear damage creep model was established, in whichσcd is the stress threshold that controls the damaged viscous element, as shown in Figure 9.Figure 9
Damage creep models with different critical stress intervals.According to equation (7), the nonlinear expression of the constitutive equation of the damaged viscous element can be obtained as(9)dε1dt=σ1η0expβt.Takingε1 = 0 when t = 0, the creep equation of the damaged viscous element can be written as(10)ε1=σ1η0βexpβt−1.Substituting equation (7) into the generalized Kelvin model, the equation of the nonlinear damage creep model can be determined as follows.(1)
When in the crack closure stage, elastic region, and stable crack growth stage (σ1 < σcd), the damage creep model is a generalized Kelvin model; then, the equation is(11)εσ,t=σ1E0+σ1E11−exp−E1η0t.(2)
When in the unstable crack growth stage (σ1 < σcd), the damage creep model is made up of generalized Kelvin model in series with a damaged viscous element, and the equation is(12)εσ,t=σ1E0+σ1E11−exp−E1η1t+σ1η0t,t≤tw,σ1E0+σ1E11−exp−E1η1t+σ1η0βexpβt−1,t>tw.Table2 shows the fitting values of parameters, and Figure 10 shows the comparison of creep fitting curves and test curves. It can be observed that the damage creep model can better describe all three stages of the creep process and can accurately predict the creep failure time of specimens. Due to the influence of specimen discreteness, the fitting effect under 75 MPa is poor.Figure 10
The scatter plot of one-step loading creep test and its fitting curve.Table 2
The parameters of the damage creep model for specimens.
σ1 (MPa)E0 (GPa)E1 (GPa)η1⋅GPa⋅hη0⋅GPa⋅hβ509.3160500——609.93170600——7211.35195600——7411.03300900800000.018847511.11300900550000.029788010.45300600250000.074008510.3930060098000.23350


## 4.1. Damage Variable

Damage variableD is an important mechanical parameter for evaluating material properties and establishing the material damage model. When D  =  0, there is no damage to the material; when D  =  1, the material is fully damaged and loses its strength [13–17]. Combined with the test results, it was considered in this paper that when σ1 < σcd, the specimens will not fail, and there is no damage to specimens; when σ1 > σcd, the specimens will finally fail, and the damage exists in specimens. The damage has the following characteristics: (1) the damage is a time-dependent damage which increases gradually with time and finally reaches complete damage, namely, D  =  1. (2) The damage is also affected by σ1. The greater σ1 is, the shorter the time for specimens to reach the fully damaged state.In terms of the strength of specimens, creep failure occurred in specimens underσ1 of 74, 75, 80, and 85 MPa, which was reduced by 23%, 21%, 16%, and 10% compared with σc.In terms of the deformation of specimens, the viscosity coefficientη at each time in the creep process can be expressed as(2)dε1dt=σ1η.Whenσ1 is constant, the creep rate dε1/dt is inversely proportional to η. As shown in Figure 5, the creep rate of the specimen was constant in stage II and began to increase as it entered stage III, which indicates that η began to decay with increasing creep time. Therefore, it can be assumed that no time dependence was produced before stage III, while it started to appear after stage III. When σ1 was 74, 75, 80, and 85 MPa, η of creep failure was lost about 92%, 90%, 99%, and 95%, respectively.It was obvious that the damage of the specimen in strength was very different from that in deformation, and the same damage variable cannot be used to describe both. Therefore, two types of damage variables, named strength damage variable and deformation damage variable, were proposed from the analysis above.The strength damage variableD1 (σ, t) can be calculated as(3)D1σ,t=σc−σftσc−σ1,where σ1 is the long-term load applied on specimens and σf(t) is the stress corresponding to the creep failure time t.Whenσf(t) = σc, specimens are instantaneously damaged, namely, t = 0. When σf(t) = σcd, the specimen will be permanently undamaged, namely, t = ∞. Combined with the stress and time corresponding to the creep failure of specimens in the single-stage loading creep test, the fitting curve is presented in Figure 6.Figure 6
Relationship between load and time in the creep failure of specimens.It can be seen from Figure6 that the larger σ1 was, the shorter the time required for the creep failure of specimen tf was. This relationship can be described as(4)σftf=σcd+σc−σcdexp−αtf,where α is the correlation coefficient of damage degree, which can be fitted according to the test data. For specimens in this paper, α = −0.025. By combining equations (3) and (4), the expression of strength time-dependent damage variable D1(σ, t) of specimens can be described as(5)D1σ,t=σc−σcd1−exp−αtσc−σ1.The deformation damage variableD2(σ, t) of specimens proposed in this paper is(6)D2σ,t=0,t≤tw,1−ησ,t/η0,t>tw,where η(σ, t) is the viscosity coefficient function with creep time t, η0 is the viscosity coefficient at the steady-state creep stage, and tw is the time required to reach stage IIIa.Creep failure of specimens occurred whenD1 (σ, t) = 1, while D2 (σ, t) is not strictly equal to 1, and η(σ, t) corresponded to the maximum creep rate (dε1/dt)max as described in Table 1.Figure7 shows the relationship between tw and σ1 and the variation of η(σ, t) with time. The larger σ1 is, the shorter the time experienced before entering the steady-state creep stage is and the small η in the steady-state creep stage is. Moreover, η(σ, t) decreases gradually with time after stage II; this relationship can also be described as(7)ησ,t=η0,t≤tw,η0exp−βt,t>tw,where β is the correlation coefficient of damage degree, which is related to σ1. As σ1 increases, β increases gradually. The fitting curve combined with η0, β, and σ1 is shown in Figure 8.Figure 7
Relationship betweentw and σ1 and η(σ, t) and t: (a) tw-σ1; (b) η(σ, t)-t.
(a)(b)Figure 8
The relation graph of viscosity coefficientη0-σ1 and β-σ1 in stage II of the specimen: (a) η0-σ1; (b) β-σ1.
(a)(b)It can be observed that, with the increase ofσ1, η0, and β, different variation laws were shown: η0 decreased with σ1, while β increased with σ1. By combining equations (6) and (7), the expression of deformation time-dependent damage variable D1(σ, t) can be described as(8)D2σ,t=0,t≤tw,1−exp−βt,t>tw.


## 4.2. A New Damage Creep Model with Different Critical Stress Intervals

Based on the creep characteristics and two damage variables of specimens, a new 5-element nonlinear damage creep model was established, in whichσcd is the stress threshold that controls the damaged viscous element, as shown in Figure 9.Figure 9
Damage creep models with different critical stress intervals.According to equation (7), the nonlinear expression of the constitutive equation of the damaged viscous element can be obtained as(9)dε1dt=σ1η0expβt.Takingε1 = 0 when t = 0, the creep equation of the damaged viscous element can be written as(10)ε1=σ1η0βexpβt−1.Substituting equation (7) into the generalized Kelvin model, the equation of the nonlinear damage creep model can be determined as follows.(1)
When in the crack closure stage, elastic region, and stable crack growth stage (σ1 < σcd), the damage creep model is a generalized Kelvin model; then, the equation is(11)εσ,t=σ1E0+σ1E11−exp−E1η0t.(2)
When in the unstable crack growth stage (σ1 < σcd), the damage creep model is made up of generalized Kelvin model in series with a damaged viscous element, and the equation is(12)εσ,t=σ1E0+σ1E11−exp−E1η1t+σ1η0t,t≤tw,σ1E0+σ1E11−exp−E1η1t+σ1η0βexpβt−1,t>tw.Table2 shows the fitting values of parameters, and Figure 10 shows the comparison of creep fitting curves and test curves. It can be observed that the damage creep model can better describe all three stages of the creep process and can accurately predict the creep failure time of specimens. Due to the influence of specimen discreteness, the fitting effect under 75 MPa is poor.Figure 10
The scatter plot of one-step loading creep test and its fitting curve.Table 2
The parameters of the damage creep model for specimens.
σ1 (MPa)E0 (GPa)E1 (GPa)η1⋅GPa⋅hη0⋅GPa⋅hβ509.3160500——609.93170600——7211.35195600——7411.03300900800000.018847511.11300900550000.029788010.45300600250000.074008510.3930060098000.23350


## 5. Conclusions

(1)
Creep tests were carried out, and the creep deformation curves of sandstone in the interval of different critical stresses were acquired. The test results showed that the long-term strength is very close to the critical stress of dilation.(2)
Two new damage variables to describe the time-dependent damage of hard brittle rocks were proposed. The creep failure time of specimens was controlled by the strength damage variable, and the deformation of specimens was controlled by the deformation damage variable.(3)
The nonlinear damage creep models with different critical stress intervals can well match the test data and predict the creep failure time and deformation of specimens in the creep process. They can also describe the phenomenon that the creep failure mode of hard brittle rocks was brittle failure.(4)
Further research will be carried out in the numerical calculation of the damage model established in this paper, so as to realize the application in practical engineering.



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