Effect of Salinity on the Steady-State and Dynamic Rheological Behavior of Illite Clay

Abstract

The rheological behavior of clay in a water–salt environment determines the long-term deformation and structural stability of building materials and geotechnical engineering. In this study, the effects of salinity on the rheological behavior and microstructure stability of the clay mineral illite were investigated through steady-state and dynamic rheological tests. The results reveal that specimens with different salinities exhibit shear thinning behavior during the steady-state rheological test. When the shear rate is higher than 0.5 s⁻¹, the flow curves are described well by the Herschel–Bulkley model. As the salinity increases from 0 to 1.8 mol/L, the yield stress varies from 1500 to 3500 Pa. With the increase in salinity, the consistency factor of the specimens increases, while the flow coefficient decreases. Under dynamic loading, high-salinity specimens exhibit higher modulus and yield stresses, thereby enhancing the stability of the microstructure. The viscoelastic–plastic constitutive model under dynamic loading has been established, which can effectively describe and calculate the long-term deformation of clay minerals. These research results provide reference and guidance for understanding the rheological behavior of clay.

1. Introduction

In southern China, soft clay is widely distributed and mainly consists of fine-grained clayey silt. The soil type contains a substantial proportion of clay minerals, with illite being the dominant component, accounting for over 50% [1,2,3]. These clay minerals primarily originate from the decomposition of various silicate minerals, including kaolin, illite, and montmorillonite, with particle sizes typically less than 2 μm [4,5,6]. The composition of these clay minerals reflects the mechanical properties and viscoelasticity of the soil from a microscopic perspective. Consequently, studying the viscoelasticity of the pure clay mineral lays the groundwork for a deeper understanding of the rheological properties of mixed soils, such as loess, silts, and other clayey soils [7,8,9].

Rheology, the discipline that investigates the deformation and flow of materials under external forces [10,11], is particularly relevant to clay, a frictional material characterized by cohesion and an internal friction angle [12,13]. The constitutive relationship of clay is not merely a simple stress–strain relationship but also incorporates a stress–strain–time relationship, resulting in time-dependent viscoelastic behavior under natural conditions [14,15,16,17]. Moreover, in estuarine and lacustrine environments, clay often presents high levels of salinity, which can significantly influence its rheological behavior and pose risks to the stability of various engineering structures, including slopes, excavation pits, and buildings. Therefore, understanding the rheological behavior of clay minerals and calculating their long-term deformation, particularly under dynamic loads such as seismic loads, wave loads, and vehicle traffic loads, holds substantial practical importance for clay-rich engineering projects. Furthermore, this knowledge provides a crucial reference for predicting long-term deformation in building foundations.

Currently, the majority of research on the rheological behavior of soil has been predominantly based on traditional rheological methodologies. These approaches typically investigate the rheological properties of soil through the lens of macroscopic soil mechanics, focusing on phenomena such as creep, stress relaxation, and long-term strength [18,19,20,21,22,23]. While these rheological parameters can reflect the rheological characteristics of soil, such methods commonly treat soil as a solid and analyze small strain behavior from the perspective of solid mechanics. However, in southern China or coastal areas, clay is often characterized by a high water content and substantial fluidity. In such cases, regarding soil as a fluid and investigating large deformation problems from the perspective of fluid mechanics has emerged as a crucial area of focus for researchers. In recent years, spurred by the progress of interdisciplinary research, rheometers have increasingly found applications in the rheological testing of soil. These instruments offer effective characterization of the rheological properties, viscoelasticity, and strength characteristics of soil, thereby providing new technical means for studying the rheological behavior of soil [17]. These fundamental studies demonstrate that rheological investigation methods can effectively and sensitively reflect the influence of various factors on the rheological properties and microstructure stability [24,25,26,27,28,29,30,31]. For instance, Zhang and Peng (2015) found that the interaction of clay mineral particles (particularly bentonite) in the flotation system increased the pulp viscosity, whereas poorly crystallized kaolinite had little impact on pulp viscosity [2]. Ni et al. (2020) reported that kaolin and illite exhibited shear thinning behavior, while montmorillonite behaved differently, showing higher shear stress. Moreover, they found that the viscoelasticity of mixed clay (kaolin, illite, and montmorillonite) was predominantly controlled by montmorillonite [4]. Wei et al. (2025) investigated the rheological behaviors of kaolin, montmorillonite, and three other mixed clays and analyzed their evolution mechanisms through SEM experiments [11].

The rheological constitutive model of soil can effectively characterize its internal structure and mechanical properties for the prediction of long-term soil deformation. In recent years, a substantial number of constitutive models have been based on rheometer research methods. However, most of these models are applicable only under steady-state conditions. For viscoelastic materials, based on the relationship curve between viscosity and shear rate, constitutive models can be mainly classified into the power law model (1923), the Carreau model (1968), the Ellis rheological model (1965), the Bingham Plastic model (1922), the Herschel–Bulkley model (1926), the Kelvin–Voigt model (1892), and the Maxwell model (1867) [32]. Based on these models, scholars have conducted numerous research studies. For example, Hotta et al. (2024) conducted steady-state rheological tests on submarine debris. The experimental results were consistent with the Bingham model [33]. They also established a mathematical relationship between the model parameters and the fluidity index. Liang et al. (2023) conducted steady-state and dynamic rheological tests on Ili loess with different moisture contents and soluble salt concentrations. Their findings indicated that the rheological behavior of Ili loess conforms to the Bingham model under low shear rates [32]. Wei et al. (2024) further revealed the steady-state and dynamic rheological behavior of five clayey materials and proposed a dynamic yield stress model [11]. These results effectively described the yield stress and the liquid limit under dynamic loading conditions. In the aforementioned studies, various researchers have established and refined different types of constitutive models to describe the relationship between shear rate and shear stress in soils. These findings provide theoretical support and technical guidance for disaster prediction and prevention in geotechnical engineering, as well as for the calculation of long-term deformation.

Although numerous researchers have focused on the rheological properties of clay or clay minerals [34,35,36], only a limited number have taken into account the influence of salinity on rheological behavior. Moreover, the existing rheological models are mainly applicable to steady-state conditions; few researchers have focused on rheological constitutive models for soils under dynamic loading, especially long-term deformation calculations.

Consequently, to further reveal the impact of salinity on the rheological behavior and structural stability of clay minerals (specifically illite), this study conducted steady-state and dynamic rheological tests with four salinity levels and under two water content conditions. Through steady-state shear tests and strain amplitude sweep tests, the rheological property parameters were obtained. Based on the results of the rheological tests, a viscoelastic–plastic constitutive model for clay under dynamic loads was established, and a mathematical model for calculating long-term deformations was obtained. The research findings provide valuable references for engineering construction and disaster prediction in clay-rich regions.

2. Materials and Methods

2.1. Materials

The pure mineral illite used as the test material was purchased from a materials manufacturer in Xiangxi, Hunan Province, China. According to the data provided by the manufacturer, the contents of illite and quartz were found to be 99.5% and 0.5%, respectively, and the results were verified by an XRD test (Figure 1a). The detection and analysis were performed using an X-ray diffractometer (D8 ADVANCE, Bruker, Mannheim, Germany, Figure 1b). The scanning range was set from 5° to 60° (2θ), with a step size of 0.02° and a counting time of 0.5 s per step.

The grain size distribution of illite is shown in Figure 2 and contains 91.4% clay-sized particles (<0.002 mm). The particle size analysis was conducted using a laser diffraction particle size analyzer (LS 13302). Meanwhile, a cone penetration method with deionized water was used to determine the liquid limit and plastic limit, and the corresponding results are 29.7% and 18.1%, respectively. The soluble salt used in the study was anhydrous sodium sulfate (Na₂SO₄). The sodium sulfate is in powder form, with a Na₂SO₄ content of ≥ 99.0% and a pH value ranging from 5 to 8.

2.2. Specimen Preparation

To prepare the soil specimens, the illite was first dried at 105 °C. The anhydrous sodium sulfate was then added to deionized water to dissolve and obtain different target saline solutions of 0, 0.35, 1.0, and 1.8 mol/L. The classification boundaries of salinity are determined based on the grading criteria of saline soils. Then, the saline solution was added to the dry illite and mixed evenly, to obtain the determined salinity and water content specimens. For ease of specimen preparation, the initial water content was 21.5% (w_L). Finally, specimens with different salinities were produced by the static pressure method using self-made molds (Figure 3). Before the experiments, the target water contents of the specimens were 37.2% (1.25 w_L) and 44.6% (1.5 w_L), which were obtained by the titration method. After mixing, illite samples with different salinities were prepared and placed in a sealed tank for 24 h to ensure the uniformity of water and salt. The target dry density was 1.45 g/cm³ and the size of the specimen was 25 mm in diameter and 4 mm in height. Five specimens were prepared for each group of the repeated tests.

2.3. Rheological Test

Rheological tests were carried out using an MCR 302 rheometer (Anton Paar, Austria). A plate–plate sensor (pp25) was used in the steady shear test (SST) and amplitude sweep test (AST).

The main aims of the steady shear test are to measure the yield stress τ₀ and viscosity η of the specimens. The yield stress refers to the critical stress and is used to determine whether plastic failure has occurred, along with the minimum stress required to initiate a flow in the static state. Viscosity is the viscous resistance that impedes the relative motion within a material and reflects the morphological changes in the soil during shear conditions [37]. As shown in Figure 4a, continuous torque in the same direction is applied to the upper plate, and the relationship between torsional shear stress and shear rate is measured to determine the yield stress and viscosity of the soil specimen. h represents the distance between the upper plate and the lower plate, m; φ is the deflection angle, °; s is the displacement of the upper plate. The expression is shown in Equation (1):

$$\tau ={\tau }_y+\eta \dot{\gamma }$$

(1)

where τ is the shear stress, Pa; τ_y is the yield stress, Pa; η is the viscosity, Pa·s; $\dot{\gamma }$ is the shear rate, 1/s.

Dynamic rheological tests are mainly used to study the rheological behavior of soil under dynamic loads and reveal its viscoelastic–plastic properties. For dynamic rheological tests, the main output relationship is between the shear strain γ and shear stress τ. An amplitude sweep test in oscillatory mode is carried out by applying oscillatory torque to the upper plate, generating dynamic torque in the soil sample between the upper and lower plates (as shown in Figure 4b, where the numbers 1, 2, 3, 4, 5 and 6 are the motion process of the upper plate). Using the rheometer calculation software, the initial storage modulus G₀′, the storage modulus G′, and initial loss modulus G₀″, the loss modulus G″ can be obtained [24,25], as well as the end of the linear viscoelastic range (LVR, including the linear viscoelastic strain γ_LVR, the linear viscoelastic stress τ_LVR, and the linear viscoelastic stress G_LVR) and the yield point (YP, including the yield strain γ_YP for the yield modulus G_YP). Finally, the loss factor tanδ = G″/G′ (0° < δ < 90°) and the integral I_z (the area between tanδ = 1 and the actual tanδ up to YP) were calculated. The representative results of the AST are shown in Figure 5, wherein all parameters are plotted.

For all the tests, the control contact mode was the normal force, and the value was 3 N. The temperature was controlled at 20 °C and the waiting time before each test was 1 min. The specific parameters of SST and AST are shown in

Table 1. The test parameters of SST and AST.

Test Type	Parameter	Value
SST	Distance between plates/mm	4
	Shear rate/s−1	10−4~10
	Measuring points	30
	Measuring time/min	10
AST	Distance between plates/mm	4
	Shear strain/%	10−4~102
	Frequency/Hz	1
	Measuring points	30
	Measuring time/min	15

.

3. Results and Discussion

3.1. Steady-State Rheological Behavior of Illite

3.1.1. Influence of Salinity on Yield Stress

The flow curves between shear stress and shear rate are shown in Figure 6 for illite at four salinities, prepared with water contents of 1.25 w_L and 1.5 w_L, respectively. The flow curves can be divided into three stages: (1) at a smaller shear rate (10⁻⁴~5 × 10⁻⁴ s⁻¹), the shear stress increases linearly; (2) at a shear rate between 5 × 10⁻⁴ and 0.5 s⁻¹, the shear stress decreases; (3) when the shear rate is higher than 0.5 s⁻¹, the shear stress increases nonlinearly. This is due to the interaction between clay mineral particles, which increases with the increase in shear rate. In addition, shear stress with a salinity of 1.8 mol/L is significantly higher than the other salinities. The main reason is that the different salinity levels in the water will cause different surface charges on the particles, thus resulting in different interaction forces between the particles [17]. Similarly, Zhang et al. (2015) found that it is more difficult for illite to collide and form flocs in fresh water [38]. However, as the salinity of the solution increases, the attraction between particles intensifies, and the Zeta potential of the particles decreases (as shown in Figure 7). It indicates that as the salinity increases, individual particles form small flocs, and simultaneously, small flocs collide and bond with each other to form larger flocs. Their research findings further confirm the reliability of our experimental results that illite exhibits greater shear stress as the salinity increases. In addition, compared to Figure 6a,b, the shear stress of specimens for 1.25 w_L is higher than that for 1.5 w_L. The shear stress varies from 1000 to 4000 Pa when the share rate is between 10⁻⁴ and 10² s⁻¹. This is because as the water content increases, the bound water film on the particle surface thickens, reducing the adhesion force at the soil particle–water–air interface and weakening the cohesion between particles [39].

When the shear rate is higher than 0.5 s⁻¹, the Herschel–Bulkley model (H-B model) [40] is applied to characterize the flow curve. The H-B model contains three parameters, which can be used to describe the rheological characteristics of materials showing yield stress and exhibiting nonlinear shear thinning or shear thickening behavior at high shear rates [40]. The model expression can be expressed as in Equation (2):

$$\tau =\left\{\begin{array}{ll}{\tau }_y+K{\dot{\gamma }}^n\hfill & \tau >{\tau }_y\hfill \\ \tau \hfill & \tau \le {\tau }_y\hfill \end{array}\right.$$

(2)

where τ is the shear stress, Pa; τ_y is the yield stress, Pa; $\dot{\gamma }$ is the shear rate, 1/s; K is the consistency factor, Pa·sⁿ; n is the rheology index. Under shear thinning conditions in soil, the value of n varies from 0 to 1. The higher the value of the rheology index n, the better the flow properties of specimens [41,42].

The fitting results with the H-B model are shown in Figure 8a,b, and the corresponding fitting parameters, such as the consistency factor K, the rheology index n, and the coefficient determination R², are listed in

Table 2. The Herschel–Bulkley model, fitting the results of specimens with different salinities and water contents of 1.25 w_L and 1.5 w_L.

Water Content %	Salinity mol/L	K Pa·sn	n	R2
1.25 wL	0	417.4	0.56	0.997
	0.35	651.5	0.39	0.998
	1.0	752.1	0.25	0.998
	1.8	959.6	0.19	0.994
1.5 wL	0	223.4	0.72	0.996
	0.35	346.2	0.55	0.991
	1.0	633.6	0.28	0.991
	1.8	810.9	0.14	0.989

. As is evident, a very good fit is obtained (R² > 0.980). From Table 2, it can be observed that with the increase in salinity, the consistency factor K increases, while the rheology index n decreases. The corresponding values for a water content of 1.5 w_L are smaller than for 1.25 w_L. In particular, the value of the rheology index is less than 0.15 for specimens prepared with 1.8 mol/L with a water content of 1.5 w_L, while the value reaches 0.56 and 0.72 for specimens prepared with 0 mol/L when the water content values are 1.25 w_L and 1.5 w_L. This suggests that specimens with a salinity of 0 mol/L exhibit the strongest shear thinning behavior and an increase in water content increases the degree of shear thinning behavior.

Yield stress is a crucial characteristic parameter of non-Newtonian fluids and is directly related to illite initiation flow [37]. In recent years, different methods have been proposed to define and obtain the yield stress. In this study, the H-B model is used to estimate yield stress, and we only focus on studying the effect of salinity on the yield stress of illite. The yield stress curves of the specimens at different salinities are shown in Figure 9. It is worth noting that the yield stress of specimens with different water contents all increase linearly with increasing salinity. Specimens with lower water content exhibit higher yield stress, with a higher position of the curve. Specifically, the yield stresses for specimens with different salinities and water contents range from 1500 to 3500 Pa. The yield stress increases by 43.7% and 50.5% when the salinity increases from 0 to 1.8 mol/L under water contents of 1.25 w_L and 1.5 w_L, respectively.

3.1.2. Influence of Salinity on Viscosity

Figure 10 shows the relationship between the viscosity and shear rate of illite with different salinities. The viscosity values with different salinities decrease linearly with the shear rate, and the values vary from 10⁴ Pa·s to 10⁻¹ Pa·s, suggesting that illite exhibits shear thinning behavior. The relationship between the viscosity and shear rate of all sets of experimental data can be uniformly expressed by Equation (3):

$$\eta ={10}^A\times {\dot{\gamma }}^{B-1}$$

(3)

where η is the viscosity, Pa·s; A and B are the fitting parameters. The fitting results of A, B, and the coefficient determination R² are listed in

Table 3. The fitting results for A and B for Equation (3) with the experimental data in Figure 8.

Water Content %	Salinity mol/L	A	B	R2
1.25 wL	0	0.386	0.028	0.998
	0.35	0.435	0.030	0.999
	1.0	0.516	0.035	0.997
	1.8	0.525	0.030	0.997
1.5 wL	0	0.323	0.048	0.998
	0.35	0.359	0.040	0.998
	1.0	0.444	0.030	0.997
	1.8	0.524	0.043	0.996

. As the salinity increases, the parameter of A increases, while the parameter of B remains unchanged.

By analyzing the fitting parameters of Equation (3), it is evident that fitting parameter A has a linear relationship with salinity, while fitting parameter B almost remains unchanged (B = 0.035), as shown in Figure 11. The viscosity of illite while considering salinity is obtained by normalization, which uses r, $\dot{\gamma },$ and the average of fitting parameter B (Equation (4)). Thus, Equation (4) is able to reveal the viscosity when under the coupling effects of salinity and shear rate.

$$\eta ={10}^{cr+d}\times {\dot{\gamma }}^{0.035}$$

(4)

Here, c and d are the fitting parameters; c = 0.407 and d = 0.077 (w = 1.25 w_L), otherwise, c = 0.322 and d = 0.112 (w = 1.5 w_L).

3.2. Dynamic Rheological Behavior of Illite

3.2.1. Storage Modulus and Loss Modulus

Figure 12 shows the storage modulus G′ and loss modulus G″ of illite at different salinities with water contents of 1.25 w_L and 1.5 w_L, respectively. The curves are similar in shape for all specimens and can be divided into three phases: (1) elastic phase; (2) transgression phase; (3) structural collapse and viscous deformation phase. The specimens with high salinity show a higher storage modulus and loss modulus, and, for each condition, the storage modulus is relatively higher than the loss modulus. This phenomenon can also be explained by the results of the steady-state rheological tests (Figure 7). This behavior is of great significance in foundation reinforcement treatments and river dredging projects. For example, in the treatment of soft clay roadbeds in coastal areas, the shear strength of the foundation can be effectively increased by regulating the salinity of the groundwater level. In river dredging projects, the strength and mobility of clay can be reduced by decreasing the salinity of the clay. Furthermore, in this study, the illite specimens with a higher water content of 1.5 w_L exhibit lower storage modulus and loss modulus. In particular, the initial storage modulus G₀′ varies from 1850 to 4790 kPa as salinity increases from 0 to 1.8 mol/L, and the initial loss modulus G₀″ varies from 1050 to 2420 kPa with a water content of 1.25 w_L. The specimens with a water content of 1.5 w_L have a lower initial storage modulus and loss modulus, with an initial storage modulus of between 1280 and 2530 kPa and an initial loss modulus of between 630 and 1210 kPa, respectively. Generally, both the storage modulus and loss modulus obviously decreased with the increasing shear strain, which indicates that illite shows shear thinning behavior. This result is consistent with the literature results of Hyun et al. in 2002 [43] and Ni et al. in 2020 [4].

3.2.2. Viscoelasticity Parameters

Figure 13 shows the viscoelastic parameters of illite at different salinities with water contents of 1.25 w_L and 1.5 w_L, including the end strain of the linear viscoelastic range γ_LVR, the yield strain γ_YP, the integral z, and the loss factor tanδ. In general, salinity has different effects on the illite’s viscoelasticity parameters.

As a parameter to measure recoverable elasticity [25], the value of the linear viscoelastic range γ_LVR is between 0.030% and 0.035%, and the specimens of the water content of 1.5 w_L exhibit lower values (Figure 13a). Generally, γ_LVR tends to increase with an increase in salinity, indicating that the increase in salinity can lead to a wider range of LVR and improve the elasticity of the illite. As pointed out by Holthusen et al. (2019; 2020) [25,26] and Pértile et al. (2018) [44], the viscoelastic parameter γ_LVR reflects the presence and magnitude of electrostatic interparticle forces in illite and thus has an elastic property. As the salinity increases, the thickness of the double electron layer decreases and the attraction between the particles increases, thereby leading to an increase in γ_LVR. Meanwhile, the effect of water content on γ_LVR cannot be ignored. The surface of the particle-binding water film thickens as the water content increases, lowering the adhesion of the soil particle–water–air interface and the elasticity.

The relationship between yield strain γ_YP and salinity at water contents of 1.25 w_L and 1.5 w_L is almost linear (Figure 13b). As the salinity increases, the yield strain gradually increases, and the yield strain is between 1.5% and 1.8%, which demonstrates that the plastic failure of specimens with higher salinity requires wider shear strain. Also, the specimens with a water content of 1.5 w_L exhibit a lower yield strain than those of 1.25 w_L. At a higher water content, there is more pore water solution between the soil particles, resulting in less resistance to sliding between the particles and a lower angle of internal friction; thus, the specimen reaches the yield point earlier. This is consistent with the observations in the study by Pértile et al. (2016) [45].

The integral I_z can also be used to describe the stability of soil microstructure, and it is closely related to the yield strain γ_YP [25,44,45]. It can be observed from Figure 13c that the relationship between the integral I_z and salinity is the same as that of yield strain in Figure 13b. The integral I_z is positively correlated with salinity, and specimens with a higher water content exhibit a lower value. Similar to the scenario for yield strain, the increase in salinity increases the percentage of elastic deformation and the stiffness of the specimen.

To further investigate the effect of salinity on the rheological properties and the stability of the soil microstructure, the loss factor tanδ is used to help us quantify and classify the decay process of the modulus [46,47,48]. As shown in Figure 14, with the increase in shear strain, the loss factor tanδ of all specimens increases continuously when the shear strain is between 10⁻²% and 10²%. However, the loss factor tanδ remains almost unchanged when the shear strain is between 10⁻⁴% and 10⁻²%, indicating that the microstructure of the soil has not changed or has changed slightly. When the shear strain is between 10⁻²% and 10⁰%, the loss factor tanδ starts to increase, but the trend is not obvious. At this point, the soil microstructure changes and the stability decreases. Finally, as the shear strain continues to increase, the loss factor tanδ increases rapidly, leading to structural failure.

The salinity has an obvious influence on the loss factor tanδ. It can be observed that the specimens with low salinity have a larger loss factor tanδ, suggesting that the microstructure of the specimens with low salinity is more unstable. The loss factor tanδ of the low salinity specimens first reaches 1, indicating that the low salinity specimens are destroyed first, which is similar to the result of yield strain. The specimens with higher water content exhibit a higher loss factor tanδ, which may also explain the reason why specimens with a higher water content have a faster transition from elasticity to viscosity and the lowest structural stability.

3.2.3. Shear Resistance Parameters

Figure 15 shows the shear strength parameters of illite at different salinities with water contents of 1.25 w_L and 1.5 w_L, respectively. Generally, all three shear resistance parameters are positively correlated with salinity. The linear viscoelastic stress τ_LVR has a linear relationship with salinity at both water contents of 1.25 w_L and 1.5 w_L (Figure 15a). At a small shear strain (i.e., 10⁻⁴~10⁻²%), the values of linear viscoelastic stress are small and range from 8 to 21 Pa at a water content of 37.2%. A similar effect occurs at a water content of 44.6%, but the corresponding values are lower (from 3 to 10 Pa). The yield stress τ_YP exhibits a similar variation to linear viscoelastic stress, but it shows the exponential regression of salinity. Compared to linear viscoelastic stress, the yield stress has greater shear strain and stress values, and the values are about 300 times higher than linear viscoelastic stress (Figure 15a,b). Note that for yield stress at a water content of 1.5 w_L, the trend of value with increasing salinity is significantly less than the water content of 1.25 w_L. Similar to linear viscoelastic stress and yield stress, the storage modulus and loss modulus at the yield point increase with increasing salinity (Figure 15c). It also can be observed that the effect of water content on the shear strength parameter increases with the increase in salinity.

As Mohr-Coulomb pointed out, the increase in shear strength is related to vertical pressure, cohesion, and the angle of internal friction [49]. In our study, all specimens are subjected to a vertical pressure of 3 N (the contact stress set in the test), and the vertical pressure is kept constant during the test. Using the soil rheology method, this aspect effect has been confirmed by Holthusen et al. (2017) [24]. Therefore, the main contribution to the increase of shear strength is made by cohesion and the angle of internal friction. As the salinity in illite increases, the flocculation of the particles and the contact area between the particles also increase, leading to an increase in cohesion and internal friction angle. This result is consistent with the recorded effect of salinity on kaolinite [45] and subtropical Oxisols [26].

In addition, when the salt concentration of the pore solution increases, the effect of osmotic suction should be considered, especially for soils containing clay minerals (i.e., illite kaolinite and montmorillonite). In our study, when the concentration of sodium sulfate in illite increases, the cations in the pore solution and the negatively charged soil particles are attracted to each other, causing the osmotic suction to increase. This is consistent with the salt effects that have been studied by Markgraf and Horn (2009) [30]. Meanwhile, the pore water content increases with the increase in water content, thereby making the angle of internal friction smaller and decreasing the relative sliding resistance between the particles, leading to lower shear strength.

As analyzed above, in a steady state, the increasing salinity causes the yield stress and viscosity to increase, and the ability to flow is reduced. In a dynamic state, both the viscoelastic and shear strength parameters increase with increasing salinity, resulting in an improvement in the microstructural stability of illite. As a result, when estuarine clay minerals are deposited, the flow ability of the clay minerals decreases as salinity increases. In addition, the microstructural stability of illite increases with the increase in salinity, which is detrimental to the transport of estuarine sediment. However, it is beneficial in terms of the rheological properties and structural stability of clay building materials and clay foundations.

4. Viscoelastic-Plastic Rheological Constitutive Model

4.1. Model Analysis

From the steady-state tests of the clay mineral illite, it is evident that illite samples with different salinities exhibit a certain level of yield stress and viscosity. When the shear stress is less than the yield stress, the specimen does not undergo plastic deformation or flow or exhibit stress behavior. When the shearing external force is greater than the yield stress, the specimen begins to undergo plastic deformation (irreversible) and flow. However, due to the viscosity of illite, the specimen exhibits viscosity during the plastic flow process. Therefore, in a steady state, the specimen mainly shows visco-plastic characteristics. However, most real-life engineering projects are in special environments, such as the rapid (repeated cyclic) vibration of vehicles, earthquakes, wave scouring, and a series of dynamic shearing behaviors. It can be seen from the dynamic test results that the elasticity of the specimen manifests under these special environmental conditions. Therefore, under the action of shearing with external forces, the specimen exhibits both elasticity and viscosity before yielding, meaning that it is referred to as a viscoelastic body. After yielding, plastic behavior occurs, thus classifying illite as a viscoelastic-plastic body under the action of dynamic loads. It should be noted here that visco-plastic deformation itself is a nonlinear deformation (relative to viscoelasticity). This linear visco-plasticity does not mean that the deformation is itself linear; rather, it refers to a linear relationship between the overstress (the difference between the total stress and the yield stress, σ-σ_y) of the specimen itself and of the visco-plastic deformation.

Therefore, from the above analysis, it can be seen that under the action of dynamic loads (with a frequency of 1 Hz), before yielding, no flow (plastic) deformation occurs, and the specimen exhibits solid viscoelastic characteristics. An elastic element and a viscous element are connected in parallel, according to the Voigt model (Kelvin model), which can be used to describe the viscoelastic solid (Figure 16). After yielding, the specimen begins to undergo plastic deformation and exhibits viscoelastic-plastic characteristics. In the modified model, a slider and a dashpot are connected in parallel and then in series, with the original model used to describe the viscoelastic-plastic characteristics of illite under the action of dynamic loads.

4.2. Model Establishment

As shown in Figure 16, before the specimen yields, the model consists of an elastic element (a spring) and a viscous element (the dashpot) connected in parallel. Due to the elastic element and the viscous element being in parallel, they share the same shear strain, denoted as γ_₁. The shear stresses are τ_₁ (elastic) and τ_₂ (viscous), respectively. Therefore, the total dynamic shear stress τ is the sum of the stresses borne by the elastic and viscous elements.

Therefore, the viscoelastic part before yielding can be represented by Equations (5)–(7):

$${\tau }_1=E{\gamma }_1$$

(5)

$${\tau }_2={\eta }_1{{\dot{\gamma }}_1}^n$$

(6)

$$\tau ={\tau }_1+{\tau }_2=E{\gamma }_1+{\eta }_1{{\dot{\gamma }}_1}^n$$

(7)

where E is the elastic modulus, in kPa; ${\dot{\gamma }}_1$ is the shear rate before yielding; η₁ is the viscosity before yielding, in Pa·s.

By transforming Equation (7), the expression for the shear rate is obtained as:

$${\dot{\gamma }}_1={\left({\displaystyle \frac{\tau -E{\gamma }_1}{{\eta }_1}}\right)}^{{\displaystyle \frac{1}{n}}}$$

(8)

By differentiating both sides of Equation (8), the first derivative of the shear rate, i.e., the acceleration, is obtained:

$${\ddot{\gamma }}_1={\displaystyle \frac{1}{n}}{\left({\displaystyle \frac{\tau -E{\gamma }_1}{{\eta }_1}}\right)}^{{\displaystyle \frac{1-n}{n}}}\left({\displaystyle \frac{\tau ’-E{\dot{\gamma }}_1}{{\eta }_1}}\right)$$

(9)

where ${\ddot{\gamma }}_1$ is the acceleration before yielding.

For the viscous part after yielding, the shear stress can be expressed as:

$$\tau -{\tau }_y={\eta }_2{{\dot{\gamma }}_2}^n$$

(10)

where τ_y is the yield stress, in Pa; ${\dot{\gamma }}_2$ is the shear rate after yielding; η₂ is the viscosity after yielding, in Pa·s.

By transforming Equation (10), the expression for the shear rate of the viscous part is obtained as:

$${\dot{\gamma }}_2={\left({\displaystyle \frac{\tau -{\tau }_y}{{\eta }_2}}\right)}^{{\displaystyle \frac{1}{n}}}$$

(11)

By differentiating both sides of Equation (11), the acceleration is obtained:

$${\ddot{\gamma }}_2={\displaystyle \frac{1}{n}}{\left({\displaystyle \frac{\tau -{\tau }_y}{{\eta }_2}}\right)}^{{\displaystyle \frac{1-n}{n}}}{\displaystyle \frac{\dot{\tau }}{{\eta }_2}}$$

(12)

where ${\ddot{\gamma }}_2$ is the shear strain acceleration after yielding.

The total shear rate is the sum of the shear rates before and after yielding, as shown in Equation (13).

$$\dot{\gamma }={\dot{\gamma }}_1+{\dot{\gamma }}_2$$

(13)

Substituting Equation (10) into Equation (13) and combining it with Equation (8), we have:

$$\left({\displaystyle \frac{\tau -E{\gamma }_1}{{\eta }_1}}\right)={{\dot{\gamma }}_1}^n={\left(\dot{\gamma }-{\dot{\gamma }}_2\right)}^n={\left[\dot{\gamma }-{\left({\displaystyle \frac{\tau -{\tau }_y}{{\eta }_2}}\right)}^{{\displaystyle \frac{1}{n}}}\right]}^n$$

(14)

The total shear strain acceleration is the sum of the shear strain accelerations before and after yielding, as shown in Equation (15).

$$\ddot{\gamma }={\ddot{\gamma }}_1+{\ddot{\gamma }}_2$$

(15)

If we substitute Equations (9) and (11) into Equation (15), we have:

$$\ddot{\gamma }={\displaystyle \frac{1}{n}}{\left({\displaystyle \frac{\tau -E{\gamma }_1}{{\eta }_1}}\right)}^{{\displaystyle \frac{1-n}{n}}}\left({\displaystyle \frac{\dot{\tau }-E{\dot{\gamma }}_1}{{\eta }_1}}\right)+{\displaystyle \frac{1}{n}}{\left({\displaystyle \frac{\tau -{\tau }_y}{{\eta }_2}}\right)}^{{\displaystyle \frac{1-n}{n}}}\left({\displaystyle \frac{\dot{\tau }}{{\eta }_2}}\right)$$

(16)

By substituting Equation (14) into Equation (16) and after further simplification, the viscoelastic–plastic rheological equation of illite under dynamic loading is obtained, as shown in Equation (17).

$$\ddot{\gamma }={\displaystyle \frac{1}{n}}{\left[\dot{\gamma }-{\left({\displaystyle \frac{\tau -{\tau }_y}{{\eta }_2}}\right)}^{{\displaystyle \frac{1}{n}}}\right]}^{1-n}\left[{\displaystyle \frac{\dot{\tau }}{{\eta }_1}}-{\displaystyle \frac{E}{{\eta }_1}}{\left[\dot{\gamma }-{\left({\displaystyle \frac{\tau -{\tau }_y}{{\eta }_2}}\right)}^{{\displaystyle \frac{1}{n}}}\right]}^n\right]+{\displaystyle \frac{1}{n}}{\left({\displaystyle \frac{\tau -{\tau }_y}{{\eta }_2}}\right)}^{{\displaystyle \frac{1-n}{n}}}\left({\displaystyle \frac{\dot{\tau }}{{\eta }_2}}\right)$$

(17)

To simplify the model, the rheological model is analyzed when n = 1. The variable n represents the flow index of the specimen under dynamic loading, ranging between 0 and 1. When n = 1, this indicates that the specimen exhibits the best flow performance.

Therefore, by further simplification of Equation (17), the differential equation of the dynamic rheological model can be obtained, as shown in Equation (18).

$${\eta }_1{\eta }_2\ddot{\gamma }+E{\eta }_2\dot{\gamma }=\left({\eta }_1+{\eta }_2\right)\dot{\tau }+E\left(\tau -{\tau }_y\right)$$

(18)

If we further transform Equation (18):

$$\ddot{\gamma }\left(t\right)+{\displaystyle \frac{E}{{\eta }_1}}\dot{\gamma }(t)-{\displaystyle \frac{{\eta }_1+{\eta }_2}{{\eta }_1{\eta }_2}}\dot{\tau }(t)={\displaystyle \frac{E\left(\tau -{\tau }_y\right)}{{\eta }_1{\eta }_2}}$$

(19)

From further analysis of Equation (18), it can be seen that the form of the rheological model under dynamic loading is similar to Newton’s second law. In structural dynamics, the state of a system under external forces is described by displacement, velocity, and acceleration. However, soil is a frictional material with damping characteristics. Therefore, the left side of the equation includes the mass acceleration term and the damping term, while the right side represents the time-dependent external force.

4.3. Model Solving

If we let $2\chi ={\displaystyle \frac{E}{{\eta }_1}}$, $K_{\gamma }=-{\displaystyle \frac{{\eta }_1+{\eta }_2}{{\eta }_1{\eta }_2}}\dot{\tau }(t)$, $F\sin \omega t={\displaystyle \frac{E\left(\tau -{\tau }_y\right)}{{\eta }_1{\eta }_2}}$, Equation (19) can be further transformed into Equation (20):

$$\ddot{\gamma }\left(t\right)+2\chi \dot{\gamma }\left(t\right)+K_{\gamma }=F\sin \omega t$$

(20)

The general solution of Equation (20) is the general solution of the following second-order homogeneous linear differential equation with constant coefficients:

$$\ddot{\gamma }\left(t\right)+2\chi \dot{\gamma }\left(t\right)+K_{\gamma }=0$$

(21)

Since the dynamic shear test involves forced vibration, the characteristic equation has two conjugate complex roots, as shown in Equations (22) and (23).

$$\left.\begin{array}{l}{r}_1=-\chi +i\sqrt{{\chi }^2-K_r}\\ r_2=-\chi -i\sqrt{{\chi }^2-K_r}\end{array}\right\}$$

(22)

$$\gamma \left(t\right)=A_1{e}^{\left(-\chi +i\sqrt{{\chi }^2-K_r}\right)t}+B_1{e}^{\left(-\chi -i\sqrt{{\chi }^2-K_r}\right)t}$$

(23)

With the initial condition $\gamma \left(0\right)=0 \dot{\gamma }\left(0\right)=0$, after solving for A₁ and B₁, the relevant solution can be obtained, as shown in Equation (24).

$$\gamma \left(t\right)={\displaystyle \frac{F\sin \left(\omega t-\phi \right)}{\sqrt{{\left(K_{\gamma }-{\omega }^2\right)}^2+4{\chi }^2{\omega }^2}}}$$

(24)

By substituting χ, K_γ, and F into Equation (24), the final long-term deformation model can be obtained, as shown in Equation (25):

$$\gamma \left(t\right)={\displaystyle \frac{E\left(\tau -{\tau }_y\right)\sin \left(\omega t-\phi \right)}{{\eta }_1{\eta }_2\sin \omega t\sqrt{{\left(\frac{{\eta }_1+{\eta }_2}{{\eta }_1{\eta }_2}\dot{\tau }(t)+{\omega }^2\right)}^2+{\left(\frac{E}{{\eta }_1}\right)}^2{\omega }^2}}}$$

(25)

where ω is the vibration frequency, in s⁻¹. The relationship between angular frequency and frequency satisfies ω = 2πf; δ is the phase angle, °.

From Equation (25), the long-term deformation in dynamic tests is mainly related to the elastic modulus E, elastic viscosity η₁, plastic viscosity η₂, and vibration frequency ω. The numerical calculation of long-term foundation deformation, slope stability analysis, river dredging, disaster prevention, and control provides both a theoretical basis and technical support.

5. Conclusions

Rheological methods can reflect the rheological properties and structural stability of soils at the micro-scale. In the case of the pure clay mineral illite, the effect of salinity on its rheological properties and structural stability was mainly investigated by a series of steady and dynamic rheological tests, and the intrinsic rheological mechanism was revealed. In addition, a viscoelastic-plastic constitutive model for clay under dynamic loads has been established. The main conclusions are as follows:

(1)

The H-B model fits better for the flow curves of illite. The yield stress of specimens increases linearly with salinity, and specimens with lower water content exhibit higher yield stress. The yield stress increases about 50.0% when the salinity increases from 0 to 1.8 mol/L.

(2)

The consistency factor decreases with reducing salinity, while the rheology index increases. The viscosity of samples with different salinities decreases linearly with the shear rate, and the illite shows shear thinning behavior.

(3)

The specimens with higher salinity exhibit a higher storage modulus and loss modulus, which indicates greater elastic behavior. In the meantime, the increase in salinity causes an increase in the viscoelastic parameter and the shear strength parameter in different trends and, thus, improves the microstructural stability of illite.

(4)

Based on the viscoelastic-plastic constitutive model under dynamic loads, the long-term deformation of clay minerals is calculated, providing a theoretical basis for the calculation of settlement by engineering foundations in clay areas and with river dredging. The reliability of the long-term deformation predictions will be further verified in future research.