Features and Constitutive Model of Gypsum’s Uniaxial Creep Damage considering Acidization

School of Resource, Environment and Safety Engineering, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China School of Energy and Mining Engineering, China University of Mining and Technology (Beijing), Beijing 100083, China Work Safety Key Lab on Prevention and Control of Gas and Roof Disasters for Southern Coal Mines, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China Foreign Language School, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China


Introduction
Currently, about 44 percent of mines are exploited by the room-and-pillar method. French Lorraine iron ore accounts for 94% of the total production, while 58 mines in Lorraine are mined by the room-and-pillar method. In the United States, the method is used by 65 percent of metal mines and in Sweden, by over half of nonferrous metal mines [1,2]. During the mining, accidents are not uncommon, which threaten the engineering quality and the safety of miners and even bring substantial economic losses to the country [3][4][5][6]. The potential safety hazard of the method is mainly manifested as: the pillar is eroded and becomes sharp (see Figure 1), which leads to the overall instability of the house [7][8][9][10].
With the massive mining, shallow mineral resources are gradually exhausted, pushing the mining to the deep, where the underground environment becomes increasingly complex [11][12][13][14][15][16]. After excavation, due to the existence of groundwater, the pillar will be eroded, leading to the gradual decrease of its stability with time and the eventual failure [17][18][19]. In the process of being eroded to destruction, the influence of groundwater and mine pressure on pillar cannot be ignored [3,4,20]. Especially, the groundwater with partial acidity will seriously threaten the pillars [21][22][23]. Generally, the causes of the formation of groundwater are man-made and natural. Artificial acid groundwater is generally formed by the interaction of oxygen-containing water with surrounding rocks during mining [24,25]. The naturally formed acidic groundwater is due to the long-term evolution of water environment under natural conditions [26][27][28][29] and causes damage to the mechanical properties of rocks, which is an inescapable problem [12,[30][31][32][33].
Wawersik et al. [34] conducted tests on granite and sandstone, and the results showed that the time-varying effect would enhance with moisture content. Under uniaxial compression, the steady-state creep rate of dry specimens differs by about two orders of magnitude from that of saturated specimens. The tests by Dong et al. [35] showed that the intrusion of groundwater into rocks mainly produces two effects: one is that the friction and cohesion between the rock mineral grains decreases; the other is that it changes the mineral composition and microstructure, resulting in pores, caves, and cracks, which eventually leads to the softening of the rock and greatly reduce its strength. Taking tuff as the research object, Zhu et al. [36] carried out creep tests under dry and saturated state, respectively, and discussed the regularity of rock creep under water-bearing state. The result showed that, compared with the creep deformation, the water content slightly affects the instantaneous elastic deformation modulus, but greatly affect the ultimate creep deformation. By comparing the dry and saturated samples, the difference of their creep deformations was found to be 5-6 times. Huang et al. [37] conducted uniaxial compression creep tests on mudstone under different water-bearing conditions and found that with the increase of water content, the elastic modulus and uniaxial compressive strength greatly reduced, and the creep deformation and steady-state creep rate significantly increased. Li et al. [38] carried out creep tests on granite in air-dried and saturated states. The results showed that the long-term strength of the saturated granites was lower than the dried ones, while the creep rate and deformation are roughly larger. Liu et al. [39] used soft conglomerate to carry out single and biaxial creep tests to analyze the deformation characteristics of dry and saturated specimens. The instantaneous deformation modulus of the saturated conglomerate was found to be much lower than that of the dry state. Okubo et al. [40] carried out uniaxial compressive creep tests on tuff and andesite in dry and saturated state and studied their creep characteristics. The creep strain of rock in saturated state was found to be larger than that in dry, while the strength of creep failure is smaller. Xie et al. [41] performed chemical corrosion on porous limestone, and then carried out triaxial creep tests. The results showed that chemical reagent corrosion can increase the creep deformation and the permeability. Li et al. [42] executed shear creep tests on weak structural planes of sandstone under different moisture content. The result showed that the effect of moisture content on the shear creep deformation of weak structural plane is very significant. With the increase of water content, its deformation amount increases gradually, and the creep strength decreases. Moreover, the time reaching to a stable creep state increases continuously. Based on the engineering background of Wuhan Yuejiang Tunnel, Li et al. [43] carried out dry and saturated shear creep tests, respectively, and found that water can accelerate the creep strain rate and reduce the strength value of creep failure. Therefore, in engineering practice, the effect of water on sandstone creep cannot be ignored. Wang et al. [44] carried out triaxial creep tests on silty mudstone with dry and saturated state, respectively, to analyze the influence of water on its creep deformation and long-term strength. The result showed that the instantaneous strain, creep strain, and total strain of silty mudstone in dry state are all smaller than those in saturated, and water can lead to a significant change in its creep mechanical properties. Brzesowsky et al. [45] investigated the creep effects of the hydrochemical environment and stress state coupling effect on sandstone under compression. Most scholars study the creep of rock in the hydrochemical environment by hydration treatment before testing. Few researches perform the two tests simultaneously.
Constitutive model is the key and difficult point of rock creep [46]. Many scholars have established creep constitutive models of rock materials from different ways and achieved fruitful results, including empirical model, component combination model, and nonlinear theoretical model [47][48][49].
Empirical models are the simplest, which refers to the establishment of stress-strain-time function relationships based on test data by using mathematical methods. They can be obtained from different rock materials with different experimental conditions. At present, common empirical model equations are power type, exponential type, logarithmic type, and the combination of the three. The theory mainly includes aging, flow, reinforcement theory, and elastic continuation [46,50]. Xu et al. [51] took granite as the research object and carried out creep tests and obtained the empirical formula of the negative exponential type by summing up. Jiang et al. [52] executed uniaxial creep tests on sandstone. By using the creep test data and substituting power function, the power function type empirical model was obtained. Singh et al. [53] and Mesri et al. [54] summed up the power function constitutive relation between strain and time by the consolidation creep test of clay. Li et al. [55] performed uniaxial creep tests on marble and obtained the empirical formula of the creep by fitting the test curve. Lu et al. [56] carried out three-axis consolidation undrained   which made it difficult to reflect the creep mechanism and characteristics inside the rock, so empirical models are rarely used at present. Component combination models idealize the rock medium into basic elements in series or parallel to reflect the properties in the process of creep, such as elasticity, viscosity, and plasticity [58]. The combination model is characterized by its simple concept, which can reflect the creep characteristics of various rocks by changing the size, quantity, and combination mode of various mechanical parameters of the basic elements. Therefore, it is widely used. Currently Great achievements have been made in creep tests and constitutive models of rocks under the influence of the hydrochemical environment [76]. However, most of the related studies are conducted on the acidification of rocks before the mechanical properties are tested, which is not quite consistent with the actual engineering [77]. The mine pressure and the immersion of groundwater continuously influence on the pillars. At present, the creep constitutive model of rock is mainly applied to the analysis of rock mass itself, but rarely to the external environmental factors such as acid, alkali, temperature, and humidity [78,79]. Based on the abovementioned engineering 3 Geofluids practice, the hydrogeological survey of the Gypsum mine in Lilin County, Hunan province, was carried out. Through investigation, weak acid groundwater was found in this mine area, where the pillar is damaged by its erosion for a long time. Therefore, this paper intends to take the gypsum rock in the mine area as the research object, simulate the groundwater erosion environment, conduct immersion on the gypsum rock, study the uniaxial compression creep mechanical characteristics and constitutive model of the gypsum rock under the condition of acid erosion, and thus provide certain guiding significance for the retaining and protection of pillars.

Laboratory Tests
2.1. Preparation of Rock Specimens. All the specimens are taken from the gypsum mine in Lilin County, Changde, Hunan Province. To reduce the dispersion of rock specimens, we sample them from the same rock block by drilling and coring (see Figure 2(a)-2(c)). To be specific, we make their ends and sides flat with the cutting machine and grind them with sandpaper to ensure the integrity and smoothness. Finally, we obtain the 50 × 100 mm standard cylinder specimens with smooth surfaces. We prepare hydrochloric acid solutions with pH values of 5, 6, and 7 in advance, respectively (see Figure 3). Then, the processed specimens are put into the prepared acidic solutions (see Figure 2(d)).

Experiment Content and Test Method.
To observe the mechanical properties of gypsum before and after the acidification treatment, the scanning electron microscope (SEM) observations (see Figure 4(a)) and uniaxial compression creep tests (see Figure 4(b)) are performed.

SEM Observation.
To observe the changes in the gypsum rock microstructure before and after acid corrosion, we perform SEM tests on four kinds of specimens. One is the original gypsum specimen in dry state, and the others are the specimens after the saturation with pH values of 7, 6, and 5 for 49 days, respectively. The sample surfaces are scanned with a magnification coefficient of 400-450 (see Figure 4(a)).

Uniaxial Compression Creep
Test. There are many loading methods for creep tests, among which one-stage loading, multistage monotonic loading, and multistage cyclic loading are commonly used [80] (see Figures 5(a)-5(c)).
The uniaxial compression creep tests are carried out on RMT-150C rock mechanics testing machine in the Hunan University of Science and Technology (see Figure 4(b)). Considering the dispersion of the samples and the addition of acid solutions, we combine the multistage monotonic loading (see Figure 5(b)) with the multistage cyclic loading (see Figure 5(c)) in this test and call them the multistage monotonic cyclic loading mode (see Figure 5(d)). The stress in uniaxial compression creep tests is divided into three levels: 50%, 60%, and 70% of the uniaxial compressive strength. Since the previously measured average uniaxial compressive strength of gypsum rock is 64 MPa, the three stress levels are set to 32 MPa, 38.4 MPa, and 44.8 MPa, respectively. Each stress level is maintained for 48 h. After the first monotonic loading stage, we unload the stress to 0.1 kN for 24 h and observe the unloading curve. Then, we add the hydrochloric acid solution with pH values of 5, 6, and 7 into the molds, which are divided into the first, the second, and the third group. Next, we carry out the second stage loading on the specimens and observe the change of the stress-strain curves. Note that the stress level and the holding time are the same as in the first stage.

Laboratory Results and Discussion
3.1. Chemical Damage Analysis. The main component of gypsum is CaSO 4 . However, there exist other oxides in gypsum rocks, such as SiO 2 and CaO. As the reaction goes on, the contents of Ca, K, and Na in the gypsum rock decrease, while the proportion of Si and Mg increases. It is because the reaction of hydrochloric acid with CaSO 4 and other substances forms solutions or gas. The nonhydrophilic minerals or the minerals that cannot react with hydrochloric acid will be exposed on the rock surface. Since the acidity affects the reaction efficiency, the proportions of elements are different under the four conditions (see Figure 6). In the acidic solutions, the oxidized mineral of gypsum reacts with H + as follows:   . The rock specimen saturated in the distilled water (pH = 7) has no obvious change in size, but it changes in color (see Figure 7(b)). Compared with the specimen saturated in the acid solution with pH = 6 (see Figure 7(c)), the one in the solution with pH = 5 has more powdery particles falling off its surface and has fine cracks (see Figure 7(d)). Therefore, the acid solution with the lower pH value causes stronger chemical damage to the gypsum.
Microscopically, the rock specimen shows a clear layered structure and lamellar crystalline morphology before immersion. It has good homogeneity, close internal structure, and small interlayer distance. The microfracture and micropore are of small size and relatively scattered distribution. There is almost no large pore in the specimens (see Figure 8(a)). It means that the gypsum rocks have good macroscopic mechanical properties before erosion.
After being saturated in the neutral water, the structure of the specimen does not change significantly, but the color became lighter (see Figure 8(b)). After being saturated in the acid solution, the original lamellar structure or lamellar crystal morphology changes to sponge or floc shape, the structural porosity increases, and the interlayer boundary becomes fuzzy (see Figure 8(c)). The numbers of microcracks and micropores increase, and some independent small-sized micropores are connected to form large-scale "gullies" (see Figure 8(d)). Besides, the solution with the lower pH value causes more serious damage to the internal microstructure.    7 Geofluids the solution [81]. This phenomenon also reflects that more serious damage will occur to the microstructure of gypsum with stronger acidity. Note that the quality of the specimens saturated in the distilled water also decreases slightly. Thus, there exist water-soluble minerals in the gypsum.

Uniaxial Creep Mechanics and Deformation
Characteristics of Gypsum. Through three groups of uniaxial compression creep tests under acid corrosions, we obtain data of gypsum in terms of different pH values. Considering the discreteness of the samples, we choose the specimen with the ideal result from each test group and make analysis. Table 1 shows the creep damage of gypsum rocks, and Figure 10 illustrates the creep curve of the specimens under different acid conditions.
When the three groups of specimens loaded with three levels of stress in the first loading process, their strain increments have nearly no difference. When the axial stress is unloaded to the preload value, there exist residual deformations in the specimens, which are almost the same (see Figure 10). It means that the dispersion of the three groups is small, and the test results are reliable. During the second loading process, the specimen soaked in the solution with pH = 5 is damaged under the first stress level, whereas the specimen soaked in the solution with pH = 6 and the one  8 Geofluids soaked in distilled water both failed under the second stress level. The difference is that the former specimen is damaged earlier than the latter, which confirms that the stronger acidity causes greater creep damage. According to the test data, we draw the creep curves of specimens under different stress levels (see Figure 11) and analyze the test results in detail.
In the first loading process, the rocks have instantaneous strain when they are under axial stress. For the three groups of specimens in the dry state, when the stress is 32 MPa, their instantaneous axial strains are 1.10%, 1.00%, and 1.09%. As time goes on, the increase of the strains becomes more slowly, and the final strains are stable at 1.11%, 1.13%, and 1.10%, respectively. When the stress increases to 38.4 MPa, the instantaneous strain increments are 0.11%, 0.10%, and 0.13%, respectively. The strains increase more slowly with time, and the final strain increments are stable at 0.13%, 0.12%, and 0.15%, respectively. When the stress increases to 44.8 MPa, the instantaneous strain increments are 0.10%, 0.09%, and 0.09%, respectively. The strain increases slowly with time, and the final strain increments are stable at 0.13%, 0.11%, and 0.10%. With axial stress loading, the strains increase slowly with time, while the growth rate decreases. When the growth rate is zero, the specimen goes into a stable creep state. When the stress increases from the first level to the second level, the strain increment of specimens is greater, compared to the case that the stress increases from the second level to the third level. This indicates that the specimens are fully compacted under the second stress level instead of the first.
In the unloading process, the strains of the three groups of specimens all drop sharply to about 0.29% and then go into the stable stage. The deformations of the specimens are not fully recovered. Residual deformations still exist. It indicates that the specimens suffer plastic deformation (see Figure 11(a)).
In the second loading process, we add different solutions into the molds. The rock specimen soaked in the pH = 5 solution shows the most obvious change. After being loaded with the first stress level, the rock sample is damaged after 11.3 h. At the beginning of loading, the instantaneous strain appears and grows from 0.29% to 1.19%. It can be seen from Figure 11(d) that the first creep stage is very short. The specimen soon enters the steady-state creep stage, during which the strain rate is almost unchanged and keeps at about 1:03 × 10 −4 /h. After about 11 h, the specimen goes into the accelerated creep stage. At this time, the strain begins to accelerate, the strain rate increases, the creep curve becomes concave, and the specimen is damaged quickly. As for the specimen soaked in the solution with pH = 6 and the one soaked in the neutral distilled water, their initial creep stages are very short under the first stress level. They enter the steady-state creep stage quickly and do not have the accelerated creep stage. Under the second stress level, these two groups of specimens experience the initial stage, steady stage, and accelerated creep stage. Their strain rates in the steady creep stage are 1:46 × 10 −5 /h and 1:16 × 10 −5 /h, respectively. Besides, the specimen saturated in the acid solution is destroyed much earlier than the one soaked in neutral distilled water (see Figures 11(b) and 11(c)).
We compare the strain rates of rock specimens in the steady-state creep stage before entering the accelerated creep stage. As shown in Figure 12, the saturation with the smaller pH value produces greater strain rate. With the decrease of the pH value, the strain rate increases exponentially. It shows that hydrochloric acid affects the creep characteristics of gypsum, which can accelerate the creep. To be specific, the stronger acidity causes greater influence on the creep damage.

Uniaxial Creep Failure Patterns of Gypsum under Acid
Corrosion. In different external environments, the rocks present different failure patterns [82]. As for the three groups of gypsums under acid corrosion, their uniaxial creep failure patterns are roughly the same (see Figure 13). All of them suffer splitting failure modes and end damages with different degrees. Especially, the specimen saturated in the acid solution with pH = 5 is most clearly destroyed. Since some minerals in specimens can react with hydrochloric acid and change into solution and gas, the end of specimens becomes sharper after the reaction. Moreover, with the axial stress, the rock specimens are finally destroyed. It can be seen from the damage degrees that the solution with the smaller pH value has more obvious effect on the creep characteristics of gypsum.

The Creep Constitutive Model of Gypsum under Acid
Corrosion. Under the external loads, rocks show complex mechanical characteristics, such as elasticity, plasticity, and creep. In the study of creep characteristics, the most important parts are the creep model construction and application [83]. To describe different creep characteristics of rocks, researchers usually take the methods of empirical formulas and differential equations. Based on the experimental data of the uniaxial compression creep of gypsum obtained in the previous sections, we introduce a nonlinear element and combine it with basic elements to establish a new constitutive model. The model can describe the accelerated creep stage of gypsum under the acid solution.  [84,85]. Its mechanic model and dynamics are shown in Figure 14.
The constitutive equation of the Kelvin model is The constitutive equation of the Maxwell model is According to the series relation, there is According to equations (2), (3), and (4), we can get: By differentiating and simplifying equation (5), we get the constitutive equation of the Burgers model as The creep equation of the Kevin model is The creep equation of the Maxwell model is According to the superposition principle, the creep equation of the Burgers model can be obtained by superimposing the creep equations of the Kelvin and Maxwell models as According to equation (9), when t = 0, there is  It means that the burgers model has instantaneous elastic deformation. Besides, when t = 0, no other component except the spring k 2 has deformation. As time goes on, the strain increases, and the viscous element flows at a constant velocity.
At the moment of t 1 , the unloading process begins. Its curve is shown in Figure 14(b). During the unloading process, the instantaneous spring back happens on the spring k 2 . With the increase of time, the deformation of k 2 continues to recover until the deformation of spring k 2 restores. At this time, the deformation is When t 1 is large enough, the deformation recovery of the elastic after effect is Finally, the remaining deformation of the model is The above analysis shows that this model has the characteristics of instantaneous deformation, deceleration creep, and constant velocity creep.
According to the above test results, under the condition of acid corrosion and low stress, the gypsum rocks have instantaneous deformation and suffer the deceleration and stable creep stages. Under the condition of high stress, the rocks also have instantaneous deformation and suffer the deceleration creep, stable creep, and accelerated creep stages. Although the Burgers model can well reflect the properties of instantaneous deformation, deceleration creep, and stable creep, it cannot represent the accelerated creep stage. Therefore, based on the Burgers model, this paper constructs a new model to describe the characteristics of gypsum rocks in different creep stages under acid corrosion.
Under the corrosion of acid, the mechanical properties of gypsum become weaker. At the same time, the external stress makes the internal structure of rock dislocated [86]. The creep failure of the specimen finally occurs under the combined action of force and acid. Since there exists acidification in the whole creep process, the parameters that reflect the creep characteristics of gypsum rocks must be related to the pH value. The main creep parameters related to the pH value are the elastic modulus and viscosity coefficient of the gypsum rock. In the accelerated creep stage, the presence of acid can promote the accelerated creep process and rock failure. Xu et al. [87] proposed a new nonlinear viscoplastic body (NVPB) model by paralleling a nonlinear viscous element with a plastic element. NVPB can well reflect the accelerated creep stage of rock. Its mechanical model and creep curve are shown in Figure 15. In this paper, we will use this model to describe the accelerated creep stage of gypsum rocks under acid corrosion.
The creep equation of the model is

Geofluids
When n ≠ 1, the relationship between time and strain is nonlinear. When n > 1, the strain rate increases with time. When n < 1, the strain rate decreases with time. When the model is used to describe the accelerated creep stage of gypsum rock, n must be greater than "1". In this equation, η is the viscosity coefficient. Since this model is used to describe the creep characteristics of the gypsum rock under acid corrosion conditions, η must be related to the pH value and the stress σ. That is, η = η (pH, σ). As we know, the Burgers model can well describe the creep mechanical properties of gypsum rocks. Therefore, by connecting the NVBP model with the Burgers model in series, we obtain a nonlinear creep model that can fully reflect the accelerated creep stage of gypsum rocks under acid corrosion. The elastic modulus and viscosity coefficient in the Burgers model are related to the pH value of the acid solution, so the specific structure of the nonlinear creep model is shown as Figure 16.
According to the creep equation (9)     According to equation (15), when σ ≤ σ s ðpHÞ, the model degenerates to the Burgers model. Then, it can describe the properties of the gypsum rock in terms of deceleration creep and stable creep. When σ > σ s ðpHÞ, the model is a nonlinear creep model that can describe the accelerated creep stage.
3.3.2. Verification of the Creep Constitutive Model. By substituting the identified parameters into the above model, we can get the constitutive model curves. Figure 17 shows its comparisons with the test curves.
According to Figure 17, the established constitutive model curve and creep test curve have good coincidence under various conditions. It means that the creep process of gypsum rock under acid corrosion can be well described by connecting the Burgers model with the NVPB model proposed by Xu et al. [79]. Besides, the square of the parameter identification correlation coefficient approaches to "1", which also verifies the correctness and rationality of the new model.

Conclusions
In light of the above work, the main conclusions of this paper are as follows: (1) After acidic saturation, the original lamellar structures and crystal forms were spongy or flocculent. The sample structure loosened, and the boundary between layers became fuzzy. Meanwhile, the number of microcracks and micropores increased, which weakened the macromechanical properties of the gypsum (2) The gypsum specimens with pH = 5 hydrochloric acid were damaged at the first stress level, while that with pH = 6 and pH = 7 were destroyed at the second stress level. The difference is that the failure time of the former is earlier than that of the latter, which indicates that the stronger acidity causes greater corrosion on the creep of the samples (3) The failure modes of the three groups were basically the same, with cleavage and end damage of different degrees. The damage to the end of gypsum sample with pH = 5 hydrochloric acid is the most obvious (4) A new nonlinear creep constitutive model was established by using the Burgers model and NVPB model in series, which agrees well with the creep test results and provides guidance for practical calculation

Data Availability
The numerical data used to support the findings of this study have not been made available because the nature of this research and participants of this study did not agree for their data to be shared publicly.

Conflicts of Interest
The authors wish to confirm that there are no known conflicts of interest associated with this publication, and there has been no significant financial support for this work that could have influenced its outcome.