Research Article  Open Access
Yun Lin, Feng Gao, Keping Zhou, Rugao Gao, Hongquan Guo, "Mechanical Properties and Statistical Damage Constitutive Model of Rock under a Coupled ChemicalMechanical Condition", Geofluids, vol. 2019, Article ID 7349584, 17 pages, 2019. https://doi.org/10.1155/2019/7349584
Mechanical Properties and Statistical Damage Constitutive Model of Rock under a Coupled ChemicalMechanical Condition
Abstract
Chemical corrosion has a significant impact on the damage evolution behavior of rock. To investigate the mechanical damage evolution process of rock under a coupled chemicalmechanical (CM) condition, an improved statistical damage constitutive model was established using the DruckerPrager (DP) strength criterion and twoparameter Weibull distribution. The damage variable correction coefficient and chemical damage variable which was determined by porosity were also considered in the model. Moreover, a series of conventional triaxial compressive tests were carried out to investigate the mechanical properties of sandstone specimens under the effect of chemical corrosion. The relationship between rock mechanics properties and confining pressure was also explored to determine Weibull distribution parameters, including the shape parameter and scale parameter . Then, the reliability of the damage constitutive model was verified based on experimental data. The results of this study are as follows: (i) the porosity of sandstone increased and the mechanical properties degraded after chemical corrosion; (ii) the relationships among the compressive strength, the peak axial strain, and confining pressures were linear, while the relationships among the elastic modulus, the residual strength, and confining pressures were exponential functions; and (iii) the improved statistical damage constitutive model was in good agreement with the testing curves with . It is hoped that the study can provide an alternative method to analyze the damage constitutive behavior of rock under a coupled chemicalmechanical condition.
1. Introduction
The interaction between groundwater and rock has a significant influence on the deformation and strength of rock [1]. As known, underground water may lead to many hazards, for example, karst collapse of pillars [2]. In general, underground water is a complex aqueous solution with various ion components and pH values [3], and the stability of rock mass engineering, such as oil extraction, nuclear waste geological repository, energy underground reservoir, and carbon dioxide geological storage reservoir, is significantly affected by the corrosion of hydrochemical solutions [4–7]. Moreover, the influence of stress conditions on the mechanical behavior of rock is not negligible. Thus, it is of high importance to explore the mechanical behavior of rock under the coupled effects of hydrochemical environment and loading.
Due to the corrosion of chemical solutions, the pore structure would be changed and the physical and mechanical properties of rocks would deteriorate gradually with the increase in chemical damage. Many scholars have explored the effects of chemical corrosion on rock deformation and strength characteristics through theoretical and experimental methods [4, 8–18]. Ning et al. [16] studied the deterioration characteristics of mechanical properties of sandstone under acid corrosion and defined the chemical damage variable based on the change in the effective bearing area of sandstone specimens and then established the degradation equation of the strength and elasticity model. Feng et al. [9, 10] experimentally analyzed the evolution law of mechanical properties of rocks under the effect of chemical corrosion combined with neural network simulation and numerical analysis. Ding and Feng [19] studied the evolution laws of mechanical properties of limestone under the corrosion effect of different chemical solutions. Feucht and Logan [20], Dieterich and Conrad [21], and Logan and Blackwell [22] experimentally analyzed the deformation of rock immersed in chemical solutions and then explored the relations among compressive strength, elastic modulus, and chemical corrosion. Han et al. [12, 13] explored the effects of chemical solutions on the deterioration characteristics of fracture toughness and compressive strength of sandstone under different freezethaw cycle conditions. Ni et al. [23] investigated the relationship between deterioration of mechanical properties of rock and damage variable of granite under different chemical solutions and freezethaw cycles based on experimental data. These researches mainly focus on the influence of chemical corrosion on the mechanical effect and the chemical damage variables of rock, and the results are of great significance.
A reasonable constitutive model is a useful method to investigate the damage evolution behavior of materials [24–26]. In recent years, many damage constitutive models have been established based on various strength theories or criterions, such as rock strain strength theory [27–29], DruckerPrager (DP) criterion [30–33], MohrCoulomb (MC) criterion [34–37], and HoekBrown (HB) criterion [38, 39]. However, the strength of microelements, which is affected by many factors, cannot be calculated accurately using the strain strength theory [39]. Meanwhile, the MC criterion may cause great error at a high stress level, and the form of HB criterion is too complex because it was developed from experimental results [40]. Though the DP criterion is conservative in calculating the intensity of the microelement, it is still widely used due to its simple form. In the present work, the DP criterion is used to describe the strength of microelements. So far, studies on the damage constitutive model of rocks treated with chemical corrosion are limited [41, 42]. Moreover, these studies mainly focused on the constitutive equations of rock under the chemical corrosion and uniaxial compression coupling. As known, the mechanical behaviors of rock under triaxial compression are different, but the chemicalmechanical (CM) coupling damage constitutive model of rock has not been well studied, especially the mechanical behavior in the postpeak stage of rock treated with chemical corrosion. So the results of published literatures are far from fully understanding the characteristics of deformation and destruction of rock affected by coupling chemical corrosion and load. All of these address a need to establish a chemicalmechanical coupling damage constitutive model to explore the mechanical response of rock.
As known, the model parameters in the constitutive models are highly affected by some mechanical properties, such as peak strength, peak axial strain, and elastic modulus [36, 37, 39]. In general, the values of these mechanical properties are obtained by conventional mechanical tests. Obviously, it is inconvenient to calculate the model parameters based on the data of conventional mechanical tests. The reason lies in the fact that the values of mechanical properties will change in response to various confining pressure values [36, 43, 44]. This also addresses a need to develop a more efficient method to obtain mechanics parameters of rocks. Recently, some methods for quantitative estimation of rock mechanical parameters have been explored. For example, Li et al. [36] established the relationship between peak stress, peak strain, and confining pressure by MC criterion and data fitting. Then they calculated the values of peak stress and peak strain of rock under different confining pressures and verified the applicability of these relationships by the constitutive model established. Yu et al. [45] explored the relationship between peak stress, peak strain, and elastic modulus and confining pressure of rock at different temperatures by data fitting. However, the mechanical behavior of rock after chemical corrosion is different, and further efforts are still needed to determine the model parameters for rock treated with chemical corrosion.
In this work, considering the chemical damage variable and damage variable correction coefficient, an improved statistical damage constitutive model of sandstone after chemical corrosion was established based on twoparameter Weibull distribution and DP strength criterion. To verify the reliability of the constitutive model, a series of conventional triaxial compression tests of sandstone treated with chemical corrosion were carried out, and the relationships between mechanical properties of sandstone and confining pressures were explored. The Weibull parameters of the improved constitutive model, including the shape parameter and scale parameter , were calculated based on these relationships, and then the theoretical curves were simulated and compared with the testing curves. The results showed that the improved model can well reflect the damage evolution behavior of sandstone treated with chemical corrosion.
2. Statistical Damage Constitutive Model
2.1. Damage Constitutive Model of Sandstone without Chemical Corrosion
In terms of damaged materials, it is usually difficult to analyze the pattern and damage mechanism of each microelement defect to determine the effective bearing area of the material. In order to measure the damage of materials through intermediate variables, Lemaitre [46] proposed an equivalent strain hypothesis. The hypothesis can be described as the following [47]: the strain generated by the effective stress on the intact materials is equivalent to the strain generated by the nominal stress on the damaged materials. The strain equivalence has been widely used in the establishment of the constitutive model of rocklike materials. Based on the assumption, the effective stress of rock can be represented by the nominal stress, that is, where is the nominal stress, is the effective stress, and is the material damage variable under load, which measures the degree of material damage. When , the material is in a nondestructive state. When , the material completely destroys, and corresponds to different degrees of damage.
Qian and Yin [48] believed that the residual strength of rock has high influence on the stability of geotechnical systems. Based on experimental analysis, Xu et al. [39] found that the residual strength would become the main factor affecting the softening mechanical behavior in the postpeak stage of the stressstrain curves after macrocracks appear, and the damage variables should be improved in the postpeak stage. Gao et al. [32] improved the damage variable of rock by introducing a damage variable correction coefficient which was defined as the ratio of the residual strength and the peak strength and verified the reliability of the method using experimental results. In this study, the damage variable correction coefficient is also applied, . Then, equation (1) can be written as
Then, the strain can be written as equation (3) according to Hooke’s law [49]. where , and are three principal stresses; is the strain which corresponds to the stress of ; is the elastic modulus; and is Poisson’s ratio.
Combing equations (2) and (3), the improved statistical damage constitutive model can be expressed as
Under triaxial compression, the rock can be divided into a number of microelements with different defects and the damage of rock microelement is random. Then, we define as the number of failed microelements and is the total number of microelements. The ratio of to could be seen as the mechanical damage variable.
The fracture of rock is caused by the failure of the weakest link according to the theory of fracture mechanics [50]. The Weibull distribution [51] was also proposed based on the weakest link theory and has applied to many groups of problems, for example, the size effect on failure in solids. Assuming that the strength level of the microelements of materials satisfies the Weibull distribution function, Krajcinovic and Silva [52] first proposed to calculate the damage variable of materials based on the Weibull distribution. Then, many scholars applied the distribution to establish various statistical constitutive models of rocks [27, 36, 42, 53–55].
On the other hand, the expression of the Weibull distribution can be divided into two kinds, i.e., threeparameter Weibull distribution and twoparameter Weibull distribution. The expression of the threeparameter Weibull distribution is [32, 55] where is the microelement strength distribution function, is the parameter which indicates the strength level of the microelement, is the shape parameter denoting the degree of homogeneity, is the scale parameter associated with the strength of microelements, and is the position parameter which represents the damage evolution threshold.
In most cases, the threshold is assumed to be zero [56]. Then, it becomes the twoparameter Weibull distribution and could be expressed as
Quinn (Quinn and Quinn 2010) believed that it is very risky to assume that a finite threshold strength exists without careful screening or nondestructive evaluation, and the twoparameter Weibull function is the most commonly used for simplicity. Deng et al. [56] found that the twoparameter Weibull function is sufficient suitable for describing the statistical strength variation of materials. Moreover, the twoparameter Weibull distribution is the most widely accepted function in the treatment of rock properties, and the results of a large number of studies indicate that it is capable of describing the statistical strength of rock microelements [30, 36, 53, 54, 57, 58]. Therefore, it is appropriate to introduce the twoparameter Weibull distribution to calculate the damage variable of rocks.
Then, the damage variable can be obtained based in equations (5) and (7).
Combining equation (8) and equation (4), the improved statistical damage constitutive model is
2.2. Damage Constitutive Model of Sandstone under a Coupled CM Condition
Under the action of chemical corrosion and loading, the rocks show different damage characteristics. The chemicalmechanical coupling damage can be obtained by the generalized damage variables derived from the equivalent strain principle, and the total damage variable can be expressed as [39] where is the total damage variable and is the chemical damage variable.
Thus, the damage variable in equation (4) should be replaced by the total damage variable in equation (10). Then the damage constitutive model of sandstones after chemical corrosion is as follows:
Combining equations (8) and (11), the improved damage constitutive model of rock treated with chemical corrosion can be expressed as
2.3. Chemical Damage Variable
According to equation (12), the chemical damage variable has a significant influence on the mechanical evolution behavior of rock. Thus, the determination of the chemical damage variable should be identified firstly to establish the damage constitutive model of rock. Mechanical properties have been widely used to calculate the damage variable [28, 59], such as maximum strain, elastic modulus, and residual strain. However, it is not easy to obtain the value of mechanical property which depends on a large number of mechanical tests. Therefore, other simple, fast, and convenient methods have been explored by many scholars, such as the chemical damage variable based on CT number [60–62] and rock density [63–65]. Note that CT number, which can be expressed by the change rate of the attenuation coefficient of water, is often used to describe the attenuation coefficient of rock materials relative to the attenuation coefficient of water [60]. The change in porosity has also been widely used to define the damage variable of rock [12, 16, 23, 66]. Due to that, the porosity can be obtained by nuclear magnetic resonance (NMR) technique easily; the chemical damage variable characterized in terms of porosity is also used in this study, and it can be expressed as [67] where is the porosity of sandstone treated with chemical corrosion, and is the initial porosity.
Let be the elastic modulus of rock without chemical corrosion, then the elastic modulus and improved damage constitutive model of rock treated with chemical corrosion can be expressed as
2.4. Determination of and
A reasonable measurement of the strength level of microelements in rock is the key to establishing the damage evolution equation based on the theory of statistical damage. Li and Tang [29, 68, 69] preliminarily established a statistical damage evolution model of rocks using axial strain to measure the strength level of microelements. As known, the strength level of the rock microelement is not directly determined by the axial strain, but directly related to the stress state of the rock microelement. Therefore, it is unreasonable to adopt the axial strain to measure the strength level of the rock microelement. Some strength criterions have also been used to measure the strength of the microelement, where the DruckerPrager yield criterion [70] is the most commonly used [30, 33, 71, 72]. In this study, the DP criterion is proposed to measure the strength level of the rock microelement.
The DP criterion is a generalized Mises yield criterion, which takes into account the effect of intermediate principal stress and hydrostatic pressure. It has a simple form and is widely used in geotechnical engineering. The expression is [73] where and are material parameters, and are the first invariant of the stress tensor and the second invariant of the stress deviator, respectively, and the parameters can be expressed as where , , and are effective stresses and is the internal friction angle.
Combining equations (2), (4), (15), and (16), the strength level of microelement can be calculated easily as where and .
Under the condition of conventional triaxial compression testing,, then the strength of microelement can be calculated as
Substituting equation (15) into equation (9), and then differentiating equation (9), the following expression can be obtained:
According to the characteristics of the curve of stressstrain of rocks, equation (9) satisfies the following geometrical conditions of the peak point on the curve: where is the peak stress of rock and is the strain corresponding to the peak stress, which is called the peak strain in this study.
Then, we have where is the strength of the microelement when the axial stress reaches the peak value.
Solving equation (21), the parameter and can be calculated as
Then the improved constitutive model of rock under chemicalloading conditions was established by combining equations (14) and (22). To analyze the reliability and feasibility of the improved model, the damage constitutive model without the damage variable correction coefficient can be expressed as equation (23). In the model, the parameters and can also be determined based on the above method, which is not described in this study.
3. Tests of Porosity and Triaxial Mechanical Properties
3.1. Materials and Experimental Methods
To analyze the change in porosity and the mechanical properties of rock treated with chemical corrosion and verify the reliability of the improved constitutive model established in this work, a series of conventional triaxial compression tests were conducted. The rock used in this study is finegrained sandstone collected from Hunan Province, China. The selected rock samples belong to the same block to ensure the reliability of the test results. According to the standards of ISRM suggested [74, 75], these sandstones are cylinders with diameter of and height of . In addition, the specimens with larger discreteness were excluded by comparing the quality and longitudinal wave velocity (LWV) of specimens. The mean density and LWV of specimens retained are 2.41 g/cm^{3} and 2310 m/s, respectively. The mineral components of the specimens were detected by Xray detection, and the results are listed in Figure 1. It can be found that the mineral composition of sandstone is mainly quartz, feldspar, mica, calcite, etc.
Since the chemical corrosion of rock mass engineering is a longterm and slow process [4, 11, 76], chemical solutions with large concentration were selected to observe the effect of chemical solutions on rock in a laboratory environment. In the tests, three chemical solutions were prepared, i.e., an acidic solution with 0.01 mol/L H_{2}SO_{4} (), distilled water (), and an alkaline solution with 0.01 mol/L NaOH (). Specimens were divided into four sets, i.e., A, B, C, and D. Specimens of groups A, B, and C were immersed in H_{2}SO_{4} solution, NaOH solution, and distilled water, and specimens of group D were in the natural state and used as references.
According to previous literatures [59, 77–84], the porosity could be tested by nuclear magnetic resonance (NMR). In this study, the AniMR150 NMR imaging system of Central South University (China) was used to measure the porosity of sandstone treated with different chemical corrosions. Since the porosity increments of specimens from all groups are very small (almost 0) after 50 days of immersion, then the porosity with immersion time of 50 days is selected as the final porosity of sandstone specimens treated with chemical corrosion.
The conventional triaxial compression tests with various confining pressures (0, 5, 10, 15, 20, 25, and 30 MPa) were conducted on the MTS 815 material testing system with a maximum loading capability of 2600 kN. The experiment used the axial stress control, and the loading rate was a constant of 0.1 kN/s. During testing, the load and displacement were recorded directly by the system. In this study, the triaxial compressive strength was the peak load when the specimen failed, and the elastic modulus was the slope of the approximate line segment in the stressstrain curve. Note that there are three samples in each group, and the results are the average values of each group.
3.2. Experimental Results
Table 1 displays the average results of porosity and mechanical parameters of sandstone sample immersion in different chemical solutions. Note that the results of sandstone under the confining pressure of 30 MPa were used to verify the feasibility of the improved constitutive model, and the experimental data was not discussed in this section.
 
Note: : cohesion of rocks. 
From Table 1, it can be seen that the porosity increases when sandstone specimens immersed in chemical solutions, and the porosity change of specimens immersed in H_{2}SO_{4} solution is the largest, followed by that of specimens immersed in NaOH solution and distilled water, respectively. Thus, the order of the chemical damage variable of sandstone specimens treated with different chemical corrosion is . In terms of mechanical properties of sandstone in the same chemical environment, the peak strength, peak axial strain, elastic modulus, and residual strength increase with the increase in confining pressures. Under the condition of the same confining pressure, the peak strength, elastic modulus, and residual strength decrease with the chemical damage variable increasing, while the peak axial strain shows no apparent trend. Meanwhile, it can also be found that the internal friction angle and the cohesion decrease with the chemical damage variable increasing, while the evolution law of Poisson’s ratio is opposite. Overall, we can draw that the mechanical properties deteriorate under the condition of chemical corrosion, and the ascending order of the effect of chemical solution on mechanical properties is distilled water, NaOH solution, and H_{2}SO_{4} solution, respectively.
3.3. Relationships between Mechanical Properties and Confining Pressures
From equation (22), it is obvious that the mechanical properties, such as , , , and , contribute to the determination of and . However, the values of the four mechanical parameters will change in response to various confining pressures, and they must be remeasured by a series of mechanical tests. This seems to be a rather awkward situation for the operation of the above method. Therefore, further efforts are still needed for the estimation of ,, , and .
According to the previous studies [36, 39, 85], the four mechanical properties have a certain relationship with confining pressures. Many studies suggested that it is a feasible method to obtain the mechanical parameters of rocks under different confining pressures by experimental fitting with small amounts of data. For example, Alam et al. (Alam et al. 2014) explored the relationship among triaxial strength, tangent modulus, and confining pressure using four points (different confining pressures). Li et al. [36] discussed the relationship among stress, peak axial strain, and confining pressure through six points and then verified the statistical damage constitutive model established in the literature. By fitting five sets of data, the relationship among stress, elastic modulus, and confining pressure was studied by Xu and Karakus [86]. Wang et al. [47] investigated the relationship among strain, elastic modulus, and confining pressure using six confining pressures. Golshani et al. (Golshani et al. 2006) established the relationship between stress and confining pressure using six points. Based on six sets of data, Yoshinaka et al. (Yoshinaka et al. 2008) established the relationship between Young’s modulus and confining pressure. Therefore, the method of experimental fitting is also used to determine the values of the four mechanical properties under different confining pressures in this study.
3.3.1. Peak Strength and Peak Axial Strain
The triaxial strength of rock under a confining pressure could be obtained by a certain strength criterion. The mostly widely used strength criterion is the MohrCoulomb strength criterion, and the criterion has been successfully used in many studies [36, 87–90] to obtain the peak axial stress of rocks. The MC criterion is also adopted in this study, and then the peak axial stress of sandstone specimens can be expressed as
From equation (24), the value of the peak axial stress of sandstone under different confining pressures could be obtained based on the experimental data in Table 1.
The peak axial strain has a strong linear relationship (shown in equation (25)) with confining pressures [13, 36, 86]. Based on the experimental data in Table 1, the relationship between the peak axial strain and confining pressures of sandstone specimens treated with different chemical corrosion are regressed in Figure 2. where and are constants. It is obvious that is the peak axial strain of rock under uniaxial compression when the confining pressure is zero, then the parameter is the strain of rock under uniaxial compression, and represents the strain increasing rate with the increasing of confining pressure.
(a)
(b)
(c)
(d)
To evaluate the performance of the fitting curves, the root mean square error (RMSE) and coefficient of determination () were introduced [30, 91]. measures how well the association is between changes in two variables, and RMSE is used to find the sample standard deviation of the error between regressed and actual values. An ideal regression curve has and . where is the number of data points, and, , and are the fitting value, the actual value, and the average value of the stress at the data point.
The relationships between the peak axial strain and confining pressure are expressed in Table 2. It can be found that the linear relationship fits well the experimental data with and . Then, the peak axial strain of sandstone specimens under different confining pressures can be calculated based on the fitting results.

3.3.2. Elastic Modulus
Figure 3 gives the elastic modulus of sandstone specimens with different confining pressures in the natural state. We can see that the increasing rate of elastic modulus decreases gradually with the confining pressure increasing, indicating that the relationship between elastic modulus and confining pressures could be fitted by an exponential equation shown in equation (27), which has been proposed and verified by Pourhosseini and Shabanimashcool [40] and Yang et al. [85]. where , , and are constants which can be obtained by exponential fitting.
When the confining pressure is zero, then is the elastic modulus of rock under the uniaxial compressive state. So the parameter is a fitting coefficient defining the increasing rate of elastic modulus with the increasing of confining pressures.
Using the method, the regression result of sandstone specimens under the natural state is shown in Figure 3 and the fitting equation is listed in equation (28). It can be seen that the exponential equation can fit the experimental data well with and . where represents the elastic modulus of sandstone specimens under natural state.
As mentioned in Sections 2.2 and 2.3, the elastic modulus of sandstone specimens under the effect of loading and chemical corrosion could be described as
Combining equation (29) and the porosity of sandstone specimens under different states in Table 1, the elastic modulus of sandstone specimens immersed in different chemical solutions are calculated and listed in Table 3. It is obvious that the error between the actual and predicted elastic modulus is less than 11%. Thus, it is feasible to calculate the elastic modulus of sandstone specimens under a coupled chemicalmechanical condition by using equations (28) and (29).

3.3.3. Residual Strength
Under the condition of different chemical corrosions, the testing results of the average residual strength with various confining pressures and the fitting curves are shown in Figure 4. The regression results listed in Table 4 reveal that the relationship between residual strength and confining pressures can be expressed by exponential equations (, ), which is similar to the results of Xu et al. [39].
(a)
(b)
(c)
(d)

4. Verification of the Improved Damage Constitutive Model
In this study, the experiment results of conventional triaxial compression tests of sandstone treated with different chemical corrosions under a confining pressure of 30 MPa were used to verify the reliability of the improved damage constitutive model. The mechanical parameters of sandstone under the confining pressure of 30 MPa, which are used to simulate the mechanical behavior based on the improved damage constitutive model, can be calculated according to the relationships between mechanical properties and confining pressures in Section 3.3. The damage variable correction coefficient can be calculated based on the definition in Section 2.1. Then, the model parameters are obtained and listed in Table 5.

The statistical damage constitutive model established in equation (23) is also simulated to compare the fitting performance of the improved model established in this study. The fitting results of sandstone specimens simulated by the two damage constitutive models and the stressstrain curves of the testing results are shown in Figure 5. Figure 6 shows the coefficient of determination between the two models and experimental data.
(a)
(b)
(c)
(d)
(a)
(b)
From Figure 5, it is obvious that the fitting curves of the improved damage constitutive model proposed in this study are highly similar to the experimental results, and the improved model can well reflect the damage evolution behavior of sandstone treated with chemical corrosion. Compared with the simulation results of the improved model in this study, the fitting results of the model in equation (23) agree well with the testing curves in the prepeak stage, but the postpeak curves are greatly deviated from the testing curves, which may be caused by the reason that the damage variable correction coefficient is not considered in the model of equation (23).
In Figure 6, the coefficients between the improved model and experimental data are greater 0.98, indicating that the model is feasible and reliable to describe the mechanical behavior of rocks under a coupled chemicalmechanical condition. Both of the two damage constitutive models have good fitting performance with the coefficients greater than 0.9. It is worth noting that the coefficients of the improved model are larger than that of the model in equation (23), while the values of RMSE of the improved model are smaller, which reveals that the fitting performance of the improved model proposed in this work is better. Figure 6 further reveals that chemical corrosion has influence on the fitting performance of the improved model, and the ascending order of the effect of different chemical corrosions is H_{2}SO_{4}, NaOH, and distilled water, respectively. This may be caused by the error between the chemical damage variable calculated in this study and the actual chemical damage of rocks. The former is calculated based on the change of porosity and an approximation of the actual chemical damage variable, and the error increases with the increasing of chemical corrosion. Thus, the effect of H_{2}SO_{4} solution and NaOH solution is larger.
Moreover, the improved curves have an increasing trend in the postpeak stage, which is not in accordance with the testing results. This may be due to that Lemaitre’s strainequivalent principle does not take into account that the partial compressive and shear stresses can continue to be propagated after the failure of the microelements of the rock [39]. Therefore, in order to establish a more reasonable statistic constitutive model for rock under the effect of chemical corrosion and loading, more factors should be considered in the future, such as the residual strain and postpeak elastic modulus.
5. Discussion
Figure 7 shows the relationships among , , and the chemical damage variable of sandstone specimens. The regression results reveal that the relationships can be described by exponential equations shown in equation (30). Figure 7 further shows that and are negatively correlated to the chemical damage variable, and both of them decrease with the chemical damage variable increasing. Based on the results of previous studies [39], represents the brittleness of rocks and is the average macroscopic strength of rocks. As known, both of the brittleness and macroscopic strength of rocks reduce as the chemical damage increases, which is consistent with the evolution law of the two parameters with the chemical damage variable. Thus, the parameter can reflect the brittleness of sandstone, and can be used to describe the rock average strength. Moreover, we can obtain the deterioration characteristics of average strength and brittleness of rocks using the chemical damage variable. It may be a meaningful method to investigate the damage evolution behavior and establish the damage constitutive model of rock based on the relationships between , , and chemical damage variable, and this will be the further study.
In this study, the method of experimental fitting is used to determine the values of , , , and . The result showed that the method is convenient and feasible to obtain the Weibull distribution parameters ( and ). Though it is only tested on sandstones under a coupled chemicalmechanical condition, it is expected to be suitable for other types of rocks in various environments. This claim needs verifications in future research.
The twoparameter Weibull distribution is the most widely used distribution function, and the results in this paper and other literatures [27, 42, 53, 54, 57, 58] also indicate that the distribution is capable of describing the strength level of the microelements of rocklike materials, while the threeparameter Weibull distribution is considered to be the most valuable statistical distribution and has been applied to establish constitutive models of rocks under a coupled thermalmechanical condition [32, 86]. Therefore, it is necessary and meaningful to investigate the constitutive relationship of corroded rocks based on the threeparameter Weibull distribution, which will be carried out in the next work.
Moreover, the aim of this work is to provide an alternative method to describe the damage evolution process of rock under a coupled chemicalmechanical condition. However, it should be noted that it is still far from fully understanding the damage evolution characteristics of rocks. For example, the cumulative damage develops rapidly and the bearing capacity decay rapidly in the postpeak stage of the brittle materials [39, 55], then the model in this study would not be suitable.
6. Conclusions
In this work, an improved statistical damage constitutive model under a coupled chemicalmechanical condition is established and the following conclusions can be drawn: (1)An improved statistical damage constitutive model by combining Weibull distribution, chemical damage variable, and correction coefficient is proposed, which considers the chemical effects and residual strength of sandstone in postpeak stage(2)The chemical damage variable is defined in terms of porosity which can be obtained easily. The elastic modulus of sandstone treated with chemical corrosion can be calculated based on the chemical damage variable and initial elastic modulus of rock in the natural state. The fitting performance of the model would decrease with the chemical damage variable increasing(3)The relationships between mechanical parameters and confining pressures are determined. The physical meaning of parameters in the damage constitutive model is discussed based on the correlations between the parameters and chemical damage variable, which enhanced the adoptability of the model(4)Based on testing results, the improved statistical damage constitutive model is verified. The fitting performance of the improved model in this study is better with , indicating that the improved model is feasible to describe the damage evolution behavior of sandstone treated with different chemical corrosions. Though the proposed model is only tested on sandstones, the model is expected to be applicable to other types of rocks when the dominant damage mechanism due to chemical corrosion is the same
Symbols
:  Nominal stress (MPa) 
:  Effective stress (MPa) 
:  Material damage variable under load 
:  Damage variable correction coefficient 
,:  Three principal stresses (MPa) 
:  Strain correspond to the stress of 
:  Elastic modulus (GPa) 
:  Poisson’s ratio 
:  Number of failed microelements 
:  Total number of microelements. 
:  Microelement strength distribution function 
, :  Shape parameter and scale parameter of Weibull distribution 
:  Parameter which indicates the strength level of the microelement 
:  Total damage variable 
:  Chemical damage variable 
:  Initial porosity of specimen, porosity of specimen after days of immersion in chemical solutions (%) 
:  Elastic modulus of rock without chemical corrosion (GPa) 
, k_{0:}  Material parameters 
, :  First invariant of the stress tensor and the second invariant of the stress deviator 
,,:  Effective stresses of the three principal stresses (MPa) 
:  Internal friction angle (°) 
:  Peak stress of rock (MPa) 
:  Strain corresponding to the peak stress 
:  Strength of the microelement at peak point 
:  Residual strength (MPa) 
c:  Cohesion (MPa) 
:  Elastic modulus of sandstone specimens under natural state (GPa) 
RMSE:  Root mean square error 
R^{2}:  Coefficient of determination 
n:  Number of data points 
, ,:  Actual value, fitting value, and average value of the stress at the data point 
Data Availability
Most of the data generated or analysed during this study are included in this manuscript, and all of the data are available from the corresponding author on reasonable request.
Conflicts of Interest
The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Authors’ Contributions
The following are the authors’ contributions in this study: Yun Lin—methodology, formal analysis, writing of original draft preparation, and review and editing; Feng Gao—formula derivation and paper revision; Keping Zhou—resources and supervision; Rugao Gao—mechanical testing, review, and editing; and Hongquan Guo—mechanical testing and paper revision.
Acknowledgments
The research presented in this paper was jointly supported by the National Natural Science Foundation of China (Grant No. 51474252 and No. 51774323), the Project of the State Key Laboratory of Safety and Health for Metal Mines of China (Grant No. 2016JSKSSYS02), and the Fundamental Research Funds Project of Central South University (Grant No. 2016zzts095). The first author would like to thank the Chinese Scholarship Council for the financial support to the joint PhD studies at the University of Adelaide.
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