Study on the Mechanical Effect and Constitutive Model of Montmorillonite under the Action of Acid Rain: A Case Study on Montmorillonite-Quartz Remolded Soil

School of Geosciences and Info-Physics, Central South University, Changsha 410012, China Shanxi Earthquake Agency, Taiyuan 030000, China School of Earth Science and Engineering, Hebei University of Engineering, Handan 056038, China Institute of Mountain Hazards and Environment, Chinese Academy of Sciences, Chengdu 610041, China College of Water Conservancy, Yunnan Agricultural University, Kunming 650000, China


Introduction
Rainfall is the most common inducing factor of soil landslide, which is mainly reflected in the interaction between rainfall infiltration and soil, i.e., the water-soil interaction, among which the water-soil chemical action is important. Acid rain is a worldwide natural disaster with a strong corrosive effect. When acid rain infiltrates the soil, a strong water-soil chemical action will occur, which will change the composition and content of the soil and affect the physical and mechanical properties of the soil.
In response to the changes of soil physical and mechanical properties caused by acid rain, scholars have conducted relevant researches and made some progress. For example, Gu et al. [1] studied the shear strength of soil under the action of acid rain through the immersion experiment of remolded sandy silt and found that the cohesive force gradually decreased under the action of acid liquid and tended to be stable in the later period, while the internal friction angle slightly changed. e decrease in shear strength can induce landslide, and it was necessary to strengthen the research on the disaster mechanism of landslides caused by acid rain. Gratchev et al. [2] found that acid rain could erode CaCO 3 , the cement in soil, which resulted in a significant decrease in soil cohesion. Xiao et al. [3] found that compounds including iron, silicon, aluminum, and potassium in red clay could be eroded by acid rain and lost with the migration of groundwater, and this would increase the soil pore ratio. Bakhshipour et al. [4] studied the effect of acid rain on the physical and mechanical properties of residual soil and found that low-pH acid rain and high flux caused a decrease in soil strength and dry density, while the permeability coefficient, liquid limit, and optimal water content showed an increase. Gratchev et al. [5,6] conducted compression tests on the natural soil, which was immersed in sulfuric acid solution, and found that the compression coefficient of the soil under a long-term acidic environment significantly increased, and the increase of the compression coefficient was negatively correlated with pH. Zhang et al. [7] found that the shear strength of red clay in Yunnan showed a general downward trend under acid rain, and the reduction was positively correlated with the concentration of acid rain and contact time. e interactions between acid rain and red clay were summarized as follows: acid rain-red clay physical action, acid rain-red clay chemical action, and acid rain-red clay salt dissolution, replacement, crystallization, etc. Xia et al. [8] found that the strength of the river silt bubble mixed soil decreased after acid rain; the stronger the acid was, the more obvious the physical and mechanical properties changed. Li et al. [9] studied the disintegration mechanism of mudstone in acid rain area and found that the damage of mudstone mainly originated from the dissolution of specific minerals. With the decrease in pH, the chemical damage of mudstone increased.
Current research mainly focused on the soils with poor mechanical properties (such as residual soil) and special soils (such as red clay and river silt bubble mixed soil), etc., and the damage mechanisms were studied from the perspective of the dissolution of calcium cement, iron cement, and soluble salt cement under acid rain. In addition, these soils are often distributed in specific areas, and the cementing materials are also regional and poor representative.
Clay minerals are the widely presented secondary minerals in natural soils and greatly affect the mechanical properties of soils. However, few researchers studied the mechanical effects of clay minerals under acid rain. In this study, the montmorillonite-quartz sand remolded soil was taken as the research object; by the immersion experiment, the mechanical effect and constitutive model of montmorillonite under the action of acid rain were studied. is work has important theoretical and practical significance for understanding the mechanism of soil landslide under acid rain and will take valued assistance for the prediction and prevention of landslide disaster.

Experimental Materials.
e purity of the montmorillonite sample in this study is approximately 97%, the average particle size is approximately 21.31 μm, the measured swelling degree is 7.50 ml/g, and the specific surface area is 55.18 m 2 /g. According to the XRD analysis, the chemical formula of the montmorillonite mineral in this study is Ca 0.2 (Al,Mg) 2 Si 4 O 10 (OH) 2 ·4H 2 O (unit cell parameter: a � 0.5169 nm, b � 0.5169 nm, c � 1.502 nm, and density (m) � 2.30 g/cm 3 ) [10]. In addition, the montmorillonite sample also contains a small amount of quartz, tridymite, and muscovite.
As is well known, the main energy source in China is coal, so acid rain is mainly the sulfuric acid type. However, in recent years, with the rapid increasing number of cars, the demand for oil has increased year by year, and acid rain gradually changed to the nitric acid type with an increasing trend of acidity. e pH of acid rain in some areas is even less than 3, and the minimum is approximately 2.8 [11]. In this study, the HNO 3 solution with pH � 3 was used instead of acid rain to immerse montmorillonite and quartz sand remolded soil, and the mechanical effect of montmorillonite under acid rain was studied by the quick shear test.

Immersion Experiment.
Previous studies [12] showed that the density of cohesive soil was 1.8∼2.0 g/cm 3 , and the density of sand soil was 1.6∼2.0 g/cm 3 , so the density of remolded soil in this study is set to 2.0 g/cm 3 . In addition, the montmorillonite content in the sliding surface soil is much higher than that in the surrounding soil, and the content is approximately 10% [13,14]. erefore, the content of montmorillonite in this study was set at 10%. e preparation process of remolded soil is as follows: firstly, we fully mixed the montmorillonite-quartz sand mixture, which was weighed according to the mixing ratio, added a certain amount of pH � 3 HNO 3 solution, and thoroughly mixed again; after the controlled moisture content uniformly reached 17.65%, 120 g wet soil sample (including 91.8 g quartz sand, 10.2 g montmorillonite, and 18 g pH � 3 HNO 3 solution) was compacted in a 60 cm 3 cutting ring (Φ61.8 × 20 mm). Four prepared remolded soil samples were stacked together with the cutting ring in a 2 L large-diameter beaker. To prevent the soil samples from being dispersed, the soil samples were separated by permeable stones. en, 1.8 L pH � 3 HNO 3 solution was injected into the beaker, which was immediately sealed with plastic wrap and placed in the incubator at constant temperature (15°C, 20°C, and 25°C) for the reaction. After the reaction reached the set time (1 d, 5 d, 10 d, and 15 d), pH and electrical conductivity (EC) of the immersion solution were measured by using the multiparameter water quality detector (HQ30d), and then the quick shear tests were immediately performed to obtain the cohesion (c) and internal friction angle (φ).

Constitutive Model of Cementing Force between
Montmorillonite Layers. Researchers who comprehensively study the interaction between layers of montmorillonite minerals are Larid DA and Jia J. Larid [15] classified and studied the interaction between crystal layers from the energy perspective. Considering the deficiency of the Larid model, Jia [16] revised it from the force perspective and divided the forces between layers into two categories: attractive forces and repulsive forces. In this study, Jingchao Jia's model [16] was used to analyze the forces between montmorillonite layers. e specific model and parameter values are as follows: (1) Van der Waals force [16]: 2 Advances in Civil Engineering where F v is the Van der Waals force, H is the Hamaker constant with the value of 2.2 × 10 −20 J, x is the thickness of the crystal layer, and λ is the distance between crystal layers. (2) Electrostatic attraction of cations between layers [16]: where F ai is the electrostatic attraction of cations between layers to the crystal layers on both sides; ω is the surface charge density of the crystal layer (C/m 2 ); ε 0 is the vacuum permittivity, which takes a value of 8.854 × 10 −12 C 2 /Nm 2 ; and ε r is the relative permittivity between layers, which is a function of the distance between crystal layers: where ε b is the relative permittivity of the bulk phase with a value of 78.15; ε c is the relative permittivity of the crystal layer with a value of 6; and C 1 and C 2 are the empirical parameters with values of 1.46 × 10 −9 m and 9.0 × 10 −10 m, respectively.
where e is the unit charge quantity, which takes the value of 1.602 × 10 −19 C; v is the ion valence; the ion between the montmorillonite layer groups in this study is calcium ion, so v is taken as 2; n is the cation concentration between crystal layers; N A is the Avogadro constant with the value of 6.02 × 10 23 mol −1 ; T is the absolute temperature; and k is Boltzmann constant with the value of 1.38 × 10 −23 J/K. (3) Cation hydration force [16]: where F hi is the cation hydration force; r i is the effective radius of cation, and it is taken as 0.1 nm for calcium ion; and δ s is always related to the radius of the cation and usually taken as 0.054 nm.
(4) Hydration force of the crystal layer surface [16]: where F hs is the hydration force of the crystal layer surface and r s is half of the thickness of the crystal layer. (5) Electrostatic repulsion force between layer groups [16]: where F rs is the electrostatic repulsion force between layer groups. (6) Born Repulsion Force [16]: where F b is the Born repulsion force and L is a constant with a value of 1.4627 × 10 −124 Jm 10 . (7) Osmotic pressure [16]: where F o is the osmotic pressure and n and n 0 are the cation concentration inside and outside the crystal layer, respectively.
us, cementing force F between the montmorillonite layers is the combined force of the above seven types of forces:

Constitutive Cohesion Model of Saturated Montmorillonite-Quartz Sand Remolded Soil.
In this study, montmorillonite is the only one cement in the remolded soil. e key to obtaining the theoretical value of cohesive force is to find the total cementation strength generated by the cementing forces between the montmorillonite layers on the shear plane. e unit of cementing force between montmorillonite layers is N/m 2 . In order to calculate cohesion, the interaction area of montmorillonite (effective contact area of montmorillonite) S E on the shear plane must be obtained. en the cohesive force of saturated montmorillonite-quartz sand remolded soil can be calculated as follows:

Advances in Civil Engineering
where c is the cohesive force, S E is the effective contact area of montmorillonite on the shear plane, and S is the shear area.
(1) Method to obtain the spacing between montmorillonite layers: the above models of all seven-component force calculation formulas of cementing force F showed that the spacing λ between montmorillonite layers was the key to obtain F. In this study, the saturated soil sample contains high swelling montmorillonite, and λ is affected by the volume of montmorillonite bound water.
If the swelling volume of montmorillonite is greater than or equal to the pore volume of soil, the following relationship can be obtained [17,18]: where V w is the volume of water in the remolded soil, V v is the pore volume of soil, M is the mass of montmorillonite sample in soil, S s is the specific surface area of montmorillonite sample, and d is half of the spacing between montmorillonite layers.
To verify whether montmorillonite fills the soil pores, the concept of net swelling degree Z is introduced in this study, which is the volume increment of montmorillonite per unit mass after free expansion in water: where Z is the net swelling degree, M m is the mass of montmorillonite, V mw is the total volume of montmorillonite after saturated, and V m is the volume of montmorillonite in the dry state. e pore volume V v in the soil can be obtained as follows: where W s is the mass of soil particles and c w is the bulk density of water.
If Z · M m ≥ V v , the montmorillonite completely fills the pores of soil sample and satisfies the requirements of the model. If Z · M m < V v , the following relationship works: In this study, specific gravity G s is tested as 2.65, so pore volume V v in the remolded soil can be calculated as 21.45 ml.
e swelling degree of the montmorillonite sample is approximately 7.50 ml/g, net swelling degree Z is 7.30 ml/g, and the volume increment of the montmorillonite sample with a purity of 97% of 10.2 g is 72.23 ml, which is much larger than the pore volume of 21.45 ml. us, montmorillonite in the remolded soil swells and fills the pores of the soil, and the water in the soil is all located between montmorillonite crystal layers, which satisfies the requirements of the model.
(2) Method to obtain the effective contact area of montmorillonite on the shear plane: in this study, the interaction area of montmorillonite on the shear plane is defined as the effective contact area, and it can be obtained as follows: (1) Calculation of the montmorillonite area on shear plane: assuming that montmorillonite is evenly distributed in the entire remolded soil, the following relation can be obtained: where S m is the montmorillonite area on the shear plane, h is the height of the remolded soil sample, and ρ is the density of montmorillonite. (2) Calculation of the effective contact area: the contact probability with montmorillonite on the shear plane is the ratio of the montmorillonite area S m on the shear plane to the shear plane area S. en, the effective montmorillonite contact area S E on the shear plane can be expressed as follows: (3) Cohesion model of saturated montmorillonitequartz sand remolded soil:

Method to Establish the Chemical Reaction Rate Equation.
Previous studies [3,6,7] showed that acid rain could react with cementing substances in soil, which affected the mechanical effect of soil particles. In this study, only one cement-montmorillonite is in the remolded soil, while quartz sand has strong acid resistance, so only montmorillonite will react with the HNO 3 solution.
During the experimental process of this study, the contents of montmorillonite and HNO 3 were constantly changing, and these changes followed the chemical reaction kinetics. Only by establishing the corresponding chemical reaction rate equation, we can provide parameters for the cohesive force model and realize the simulation calculation of the cohesive force. e chemical reaction in this process is as follows: In this study, the following method [19] was used to establish the reaction rate equation: where c is the reaction rate of montmorillonite and HNO 3  ]/dt can be measured from the experiment, k, α, and β are the three desired parameters, and there must be three equations to solve.
us, these parameters can be obtained by arranging a set of three tests, but the three tests must have a certain concentration relationship [19] (Table 1).
By substituting the data in Table 1 into (20), one ternary equation set can be obtained: e results are as follows: By substituting the solved k, α, and β into (20), the reaction rate equation at T°C can be obtained. e montmorillonite chemical formula Ca 0.2 (Al, Mg) 2 Si 4 O 10 (OH) 2 · 4H 2 O in this study is the chemical formula after the isomorphous replacement and cation exchange. Its initial chemical formula is Al 2 Si 4 O 10 (OH) 2 · 4H 2 O, and the chemical reaction equation is as follows: In addition, the chemical reaction rate in (20) is the formation rate of Al 3+ , and the consumption rate of montmorillonite under acid rain can be calculated based on the chemical reaction formula by using the generated amount of Al 3+ . e principle is as follows [20].
For any chemical reaction, e chemical reaction rate J can be written as For a sealed reaction vessel with a certain volume, it can be characterized by unit volume reaction rate r:

Variation of the Immersion Liquid Parameters and
Cohesive Force. Under certain temperature conditions (15°C, 20°C, and 25°C), by immersing montmorillonitequartz sand remolded clay in the pH � 3HNO 3 solution for a certain time (1 d, 5 d, 10 d, and 15 d), and combining the changes of the immersion liquid parameters and cohesive force, the influence of acid rain on the cohesive force was studied. e results are shown in Figure 1. At all temperatures, the pH of the soaking solution increased, and the electrical conductivity decreased, which indicated that montmorillonite reacted with the strong acid HNO 3 , consumed H + , and produced the corresponding nitrate. Under the same concentration conditions, because the electrical conductivity of soluble salt was usually weaker than that of the corresponding strong acid, the electrical conductivity of the immersion liquid generally showed a decline. After 1 day, the pH at 15°C, 20°C, and 25°C was 3.45, 3.51, and 3.78, respectively; after 15 days, the pH at 15°C, 20°C, and 25°C was 4.84, 5.51, and 6.57, respectively; this indicated that, after the same immersion time, the pH increased with the increase of temperature. e decrease of electrical conductivity also showed the same trend. Hence, temperature has a great promoting effect on the chemical reaction between HNO 3 and montmorillonite. In addition, the cohesive force showed the characteristic of first decreasing, subsequently increasing, and then decreasing again Advances in Civil Engineering in general. At 15°C, the cohesive force of soil first decreased and subsequently increased, while at 20°C and 25°C, the cohesive force first decreased, subsequently increased, and then decreased. At the higher temperature, the change stage of cohesion was more complete, which also reflects the promoting effect of temperature.
To further clarify the acid rain effect on the cohesive force, the time points at 20°C were added, and the results are shown in Figure 2. With the extension of time, pH gradually increased, and the electrical conductivity decreased. e cohesive force generally first decreased, subsequently increased, and subsequently decreased. ese changes were completely consistent with the previously summarized laws.

Changes of Internal Friction Angle and Shear Strength under Acid Rain.
To clarify the acid rain effect, this study also analyzed the internal friction angle and the shear strength by combining with the cohesion and vertical pressure of soil landslide. Figure 3 shows that, under acidic conditions, when the cohesive force changed, the internal friction angle remained Advances in Civil Engineering basically unchanged, and the overall variation was smaller than that of the cohesive force. Based on the above changes in cohesion and internal friction angle, combined with the lower vertical pressure provided by the soil landslide body (the thickness of the soil landslide body was mainly 0-5 m and 5-10 m [21] and the vertical pressure was only approximately 0.1-0.2 MPa), according to Coulomb's law [22], it was not difficult to infer that the change in soil shear strength under acid rain was consistent with the change in cohesive force.

Mechanism of Montmorillonite Mechanical Effect under Acid Rain.
To clarify the mechanism of montmorillonite mechanical effect under acid rain, X-ray diffraction analysis was performed on the montmorillonite samples before and after immersing in pH � 3HNO 3 , and the XRD spectrums were overlaid as shown in Figure 4.
After montmorillonite being immersed in the acid solution, the characteristic peak intensity of the montmorillonite mineral at the diffraction angle of 5.71°s howed a large drop from 1990 to 1204, and the diffraction angle before and after soaking changed from 5.71°to 5.79°. In addition, the characteristic peak intensity of montmorillonite at approximately 29°decreased from 495 to 201, and the diffraction angle before and after immersion also changed from 29.47°to 29.06°. e intensity and the characteristic diffraction angle of other components in the montmorillonite sample did not significantly change, and no new characteristic peaks appeared. is indicated that, after the interaction with HNO 3 , the montmorillonite had a large loss, and no new solid material was generated.
Considering the changes in pH and conductivity, it could be believed that HNO 3 would greatly erode montmorillonite and cause the decrease of cohesive force. e increase of cohesive force in the middle of immersion indicates that there must be a substance that provides greater cohesion, but the XRD results show that no new substance was generated. In this study, the phenomenon of increased cohesion in the middle stage of acid immersion appeared more than once. e reason was not clear and required further study.
In terms of internal friction angle, previous studies [23 -26] showed that the main factors that affected the internal friction angle were the density, particle size, particle shape, mineral composition, etc. e X-ray diffraction analysis result showed that, under acidic conditions, only montmorillonite was eroded in the remolded soil, quartz sand had no loss, and no new substance was generated. Since the remolded soil in this study was mainly composed of quartz sand, the content of montmorillonite was low, and the acid solution had limited effect on the above factors, so the internal friction angle basically remained unchanged.

Constitutive Model of the Cohesive Force of Saturated
Remolded Soil under Acid Rain. e previous study showed that, under the action of acid rain, the cohesion of the montmorillonite-quartz sand remolded soil greatly changed, while the internal friction angle was little changed. erefore, this study mainly focused on the cohesion model of saturated montmorillonite-quartz sand remolded soil under acid rain.
According to the foregoing analysis, acid rain can erode montmorillonite, which leads to changes of cohesive force. By establishing the chemical reaction rate equation of montmorillonite-HNO 3 , the mass change of montmorillonite at any time can be obtained.
In addition, to make the model closer to reality, it is necessary to revise the cohesive force model of saturated montmorillonite-quartz sand remolded soil. e revisions are mainly reflected in two aspects: (1) Assuming that the soil particles in the saturated remolded soil were distributed evenly, when the montmorillonite particles are eroded by HNO 3 , the remaining montmorillonite cannot continue to expand crossing the quartz sand due to the quartz sand barrier. e dissolution volume existed in the form of pores, which is consistent with the scanning electron microscopy results [27]. erefore, the spacing between montmorillonite layers would not increase during the reaction.
(2) e effective contact area is the area where montmorillonite layers contact each other on the shear plane. Once the remolded soil is prepared, the contact relationship between soil particles is determined. If the montmorillonite on either side is eroded by acid, the other side loses contact object, so the effective contact area will be reduced. For the two sides of the shear plane, the actual reduced effective contact area is twice that of one side.
Based on the above considerations, the cohesive force of saturated montmorillonite-quartz sand remolded soil under acid rain can be expressed as where c ′ is the cohesive force of saturated montmorillonitequartz sand remolded soil under acid rain, S E ′ is the effective montmorillonite contact area on the shear plane after reaction time t, and c is the reaction rate of montmorillonite at T°C. Since the mechanism of the increase of cohesive force in the middle stage was not clear and the total change trend of the cohesive force was reduction, the increased cohesion was not considered in this model.

Reaction Rate Equation.
According to the abovementioned method, a certain amount of montmorillonite was immersed in 600 ml of HNO 3 solution with different concentrations, and the temperature was constant at 20°C. Once the reaction reached the set time, the Al 3+ concentration in the immersion solution was immediately measured by using an ultraviolet-visible spectrophotometer (Shimazu Co., UV-2600). e specific experimental data are shown in Table 2, and the calculation results of the reaction order and rate constant are shown in Table 3.
en, the chemical reaction rate of montmorillonite and HNO 3 can be expressed as follows:  Figure 5. Figure 5 shows that the error between the theoretical calculated value and the measured value is small, and the theoretical calculated value changed from a rapid decline in the early stage to a slow decrease in the later stage and was consistent with the change of the measured value (except for the middle period of acid immersion). is change trend was also consistent with the convergence characteristics of the closed chemical reaction [20].
In the preparation process of remolded soil, despite repeated stirring, it was still difficult to achieve a theoretically uniform distribution of montmorillonite in the remolded soil, and it was inevitable that montmorillonite will be concentrated in the local position of soil sample. is increased the cement-montmorillonite contact area to a certain extent, making the measured value of cohesive force of montmorillonite-quartz sand remolded soil (22.31-26.18 kPa) larger than the theoretical value of cohesion (21.74-23.15 kPa), but the overall error of the two was small, and the change trend was consistent. All of these showed that the calculation model of soil cohesive force under acid rain was successful.

Conclusions
(1) Montmorillonite reacted with the acid solution. pH of the immersion solution increased gradually, and the electrical conductivity (EC) generally showed a decrease. e cohesive force c first decreased, subsequently increased, and then decreased again. e internal friction angle φ basically remained unchanged. e change of the shear strength under acidic conditions was consistent with the change of cohesive force. Temperature greatly promoted the chemical reaction between HNO 3 and montmorillonite.
(2) XRD comparison results before and after immersion indicated that the acid solution could dissolve montmorillonite, and no new substance was generated after the immersion, which mainly caused the decrease of soil cohesion. (3) Combined with the change mechanism of cohesion, the cohesive force constitutive model of saturated montmorillonite-quartz sand remolded soil under acid rain was established. ere was little error   between the calculated value and the measured value, and the change trend of the theoretical calculated values was consistent with the one of the measured values, which indicates that the model established in this study was successful.

Data Availability
e data used to support the findings of the study are available from the corresponding author upon request.

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