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

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 stress σ 1 is less than the critical stress of dilation σ cd , while all specimens were destroyed when σ 1 is larger than σ cd . In addition, the value of σ cd was very close to the long-term strength obtained by the method of the isochronous stress-strain curve. Therefore, σ cd can be regarded as the long-term strength of the sandstone specimens. When σ 1 is larger than σ cd , the time required for the failure of specimen t f decreases with the increase of σ 1 ; the creep rate d ε /d t increases with time t , and the specimen will be destroyed when it reaches a maximum value (d ε / d t ) max . Both relationships t f and σ 1 and (d ε /d t ) max and σ 1 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.


Introduction
Critical stresses (the crack closure stress level σ cc , crack initiation stress level σ ci , critical stress of dilation σ cd , and peak strength σ c ) are important indexes for evaluating hard brittle rocks [1][2][3], which reflect the internal microfracture activity state of rocks under different stress levels. e failure process of rocks under different compression conditions can be divided into 4 stages, which are crack closure stage (σ 1 < σ cc ), elastic region (σ cc < σ 1 < σ ci ), stable crack growth stage (σ ci < σ 1 < σ cd ), and unstable crack growth stage (σ cd < σ 1 < σ c ). 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][5][6]. Currently, there are mainly two methods to determine the long-term strength. (1) e 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) e 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. e creep characteristics of specimens in the interval of different critical stresses were investigated. e conclusions can provide reference for the analysis of time-dependent behavior of hard brittle rocks.

Specimen Preparation and Testing
e sandstones were obtained from Chongqing in China, whose natural dry density was approximately 2.68 × 10 3 kg/ m 3 . 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 ree Gorges Reservoir Area, Ministry of Education, China. e 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. e axial stress σ 1 was loaded on specimens progressively from 0 MPa until specimens were destroyed. en, multistage loading creep tests were carried out to obtain the long-term strength. σ 1 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 σ 1 was set based on the results of the first two tests. After reaching the design level with the same loading procedure mentioned above, σ 1 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). e load, time, and displacements were recorded in all the tests.

Critical Stresses.
Based on the uniaxial compression test, the crack volumetric strain ε c v , the volumetric strain ε v , and the elastic volumetric strain ε e v are calculated as follows [6]: where ε 1 is the axial strain, ε 3 is the radial strain, σ 1 is the stress level, E is the elastic modulus, and μ is Poisson's ratio. e σ 1 -ε 1 -ε 3 , ε v -ε 1 , and ε c v -ε 1 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 σ 1 -ε 1 curve is point σ cc , the stress corresponding to the beginning of the negative correlation between ε c v and ε 1 in the ε c v − ε 1 curve is point σ ci , and σ cd is the value corresponding to the inflection point of the εv − ε 1 curve, after which the whole specimen starts to expand. erefore, the critical stresses and deformation parameters of the sandstone specimen were σ cc ≈ 22.6 MPa, σ ci ≈ 50.3 MPa, σ cd ≈ 72.6 MPa, σ c ≈ 95.2 MPa, E ≈ 11.2 GPa, and μ ≈ 0.2.

Long-Term Strength of the Isochronal Curve Method.
Based on the multistage loading creep tests, the time-dependent deformation curve under σ 1 � 50, 60, 70, and 80 MPa is shown in Figure 2. e deformations were stable after 24 h at σ 1 � 50, 60, and 70 MPa, while creep failure occurred after 17.4 h when σ 1 � 80 MPa. e isochronal curve composed of stress and strain values at loading time t of 2, 4, 8, 16, and 24 h is shown in Figure 3. e spacing between curves started to increase gradually from σ 1 of 70 MPa, and the spacing with σ 1 of 80 MPa between curves was significantly different from that of σ 1 of 60 and 70 MPa. erefore, the long-term strength of specimens was determined to be 70 MPa.

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 levels σ 1 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 σ 1 of 50, 60, and 72 MPa, while the accelerated creep stage and failure occurred under σ 1 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. e long-term strength obtained by the stress-strain isochronal curves was about 70 MPa, and the stress of creep failure in the singlestage loading creep tests was between 72 and 74 MPa, which was extremely close to the critical stress of dilation (σ cd ≈ 72.6 MPa) obtained in the uniaxial compression test. erefore, σ cd can be approximated as the long-term strength of specimens in this paper. It can be considered that  when σ 1 is less than σ cd , there will be no accelerated creep stage and no creep failure to specimens; when σ 1 is greater than σ cd , the accelerated creep stage and failure of specimens will occur. As shown in Figure 4, the creep curves only showed characteristics of decay creeping, and the creep rate (dε 1 /dt) finally approaches 0 in the crack closure stage, elastic region, and stable crack growth stage (σ 1 < σ cd ).
By analyzing the creep rate variation characteristics of four specimens at the accelerated creep stage shown in Figure 5, the steady-state creep stage of specimens was relatively short, which decreased with an increase of stress levels (the durations under σ 1 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 III a (which is relatively evident in Figures 5(c) and 5(d)) in this paper. After stage III a , there was a stage in which the creep rate increases abruptly defined as stage III b ; 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). e 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. e 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. ere was a maximum creep rate (dε 1 /dt) max at which specimens failed. And there was a positive correlation between (dε 1 /dt) max and σ 1 ; the larger σ 1 was, the larger (dε 1 /dt) max was.

Damage Variable.
Damage variable D 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][14][15][16][17]. Combined with the test results, it was considered in this paper that when σ 1 < σ cd , the specimens will not fail, and there is no damage to specimens; when σ 1 > σ cd , the specimens will finally fail, and the damage exists in specimens. e 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) e damage is also affected by σ 1 . e greater σ 1 is, the shorter the time for specimens to reach the fully damaged state.

Shock and Vibration 3
In terms of the strength of specimens, creep failure occurred in specimens under σ 1 of 74, 75, 80, and 85 MPa, which was reduced by 23%, 21%, 16%, and 10% compared with σ c .
In terms of the deformation of specimens, the viscosity coefficient η at each time in the creep process can be expressed as When σ 1 is constant, the creep rate dε 1 /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 where σ 1 is the long-term load applied on specimens and σ f (t) is the stress corresponding to the creep failure time t.
When σ f (t) � σ c , specimens are instantaneously damaged, namely, t � 0. When σ f (t) � σ cd , 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.
It can be seen from Figure 6 that the larger σ 1 was, the shorter the time required for the creep failure of specimen t f was. is relationship can be described as 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 D 1 (σ, t) of specimens can be described as e deformation damage variable D 2 (σ, t) of specimens proposed in this paper is where η(σ, t) is the viscosity coefficient function with creep time t, η 0 is the viscosity coefficient at the steady-state creep stage, and t w is the time required to reach stage III a . Creep failure of specimens occurred when D 1 (σ, t) � 1, while D 2 (σ, t) is not strictly equal to 1, and η(σ, t) corresponded to the maximum creep rate (dε 1 /dt) max as described in Table 1. Figure 7 shows the relationship between tw and σ 1 and the variation of η(σ, t) with time. e larger σ 1 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 where β is the correlation coefficient of damage degree, which is related to σ 1 . As σ 1 increases, β increases gradually. e fitting curve combined with η 0 , β, and σ 1 is shown in Figure 8.
It can be observed that, with the increase of σ 1 , η 0 , and β, different variation laws were shown: η 0 decreased with σ 1 , while β increased with σ 1 . By combining equations (6) and (7), the expression of deformation time-dependent damage variable D 1 (σ, t) can be described as

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 which σ cd is the stress threshold that controls the damaged viscous element, as shown in Figure 9. According to equation (7), the nonlinear expression of the constitutive equation of the damaged viscous element can be obtained as Taking ε 1 � 0 when t � 0, the creep equation of the damaged viscous element can be written as   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 (σ 1 < σ cd ), the damage creep model is a generalized Kelvin model; then, the equation is (2) When in the unstable crack growth stage (σ 1 < σ cd ), the damage creep model is made up of generalized Kelvin model in series with a damaged viscous element, and the equation is   Shock and Vibration Table 2 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.

Conclusions
(1) Creep tests were carried out, and the creep deformation curves of sandstone in the interval of different critical stresses were acquired. e 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 timedependent damage of hard brittle rocks were proposed. e 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) e 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. ey 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.
Data Availability e datasets generated and/or analysed during the current study are available from the first author upon reasonable request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.