A Constitutive Model for Beach Sand Under Cyclic Loading and Moisture Content Coupling Effects with Application to Vehicle–Terrain Interaction

Abstract

Vehicle repeated passes over soft terrain alter the soil’s bearing and shear behavior, thereby affecting vehicle mobility and energy consumption. To address this issue, this study conducted cyclic compression and shear tests on beach sand with moisture contents of 5%, 15%, and 25%. A constitutive model incorporating the coupling effects of loading cycles (N) and moisture content (ω) was developed based on the Bekker and Janosi model framework. The model expresses compression parameters as functions of N and ω, and describes shear behavior through the strength evolution function k(N,ω) and deformation modulus function h(N,ω). Results show excellent agreement between the model predictions and experimental data (R² > 0.92). Furthermore, a vehicle–soil coupled dynamics model was established based on the proposed constitutive model, forming a comprehensive analytical framework that integrates soil meso-mechanics with full vehicle–terrain interaction. This work provides valuable theoretical and technical support for predicting vehicle trafficability on coastal soft soils and optimizing vehicle suspension systems.

1. Introduction

The evolution of soil mechanical properties under repeated loading is a classical problem in vehicle–terramechanics, directly influencing the driving performance and trafficability of off-road vehicles on soft terrain [1,2,3,4]. Under repeated wheel loading and cyclic shearing, the soil undergoes irreversible cumulative plastic deformation, degradation of bearing stiffness, and reorganization of its surface structure [5,6]. These changes affect the vehicle’s traction performance and ride quality. Such responses are particularly pronounced in low-bearing-capacity soils and under high moisture conditions [7], exhibiting strong strain accumulation effects and dynamic response characteristics [8,9].

Over the past few decades, triaxial tests and direct shear tests have been widely used to investigate the deformation characteristics of soils under loading, with key factors of study including load amplitude [10,11,12], confining pressure [13,14], and loading frequency [15,16,17]. For example, Aursudkij [10] observed that the cumulative axial deformation of soil increases with greater load amplitude. Majid [18] investigated the dynamic properties of sand from the Kutch region using cyclic triaxial tests, while Wang [19] studied the variation in stiffness of loess from the Gansu area under different overconsolidation ratios through triaxial testing. Their research indicated that as the overconsolidation ratio increases, soil stiffness enhances while its potential for deformation decreases. However, these traditional testing methods exhibit significant limitations in accurately simulating the specific scenario of repeated vehicle passes and capturing the resulting evolution of soil mechanical behavior. The loading mechanism of triaxial tests struggles to replicate the actual stress conditions experienced by subgrade soils under vehicle traffic, and the fixed shear plane in direct shear tests fails to represent the changing shear directions induced by repeated vehicle passages.

In the field of soft soil parameter identification, existing methods primarily fall into two categories: laboratory testing and road testing. For instance, Ray [20] utilized Bevameter to study the effects of plate size and shear rate on the bearing and shear characteristics of saturated sandy soil; Alberto [21] employed Bayesian methods to efficiently handle uncertainties in soil parameters for rover position prediction; in road testing, Luca [22] designed an optimization problem based on semi-empirical contact parameters to minimize the discrepancy between experimental and numerical traction performance, thereby estimating parameters such as settlement modulus, cohesion, internal friction angle, and multi-pass factors; James [23] extended the classical Bekker model to develop a reduced-order nonlinear terramechanics model as a substitute for the soil contact model to account for additional dynamic effects. While these methods have advanced the identification of soft soil parameters at different levels, none systematically consider the coupled effects of repeated loading cycles and moisture content on the evolution of soil parameters, which are critical characteristics governing soil behavior under repeated vehicle passes in moisture-varying environments such as coastal tidal flats.

To quantitatively describe the evolution of the aforementioned soil mechanical behavior, this study extends and modifies conventional models within the classical vehicle–terramechanics framework by incorporating the coupled effects of moisture content and cyclic loading frequency. The Bekker [24] pressure–sinkage model serves as a fundamental theory for describing the interaction between rigid bodies and deformable terrain, and is widely used in vehicle–terramechanics analysis. Its core formulation defines the relationship between the pressure per unit area p and the soil sinkage z:

$$P=\left({\displaystyle \frac{k_c}{b}}+k_{\phi }\right)Z^n=K{z}^n$$

(1)

In this equation, P denotes the pressure per unit area; Z represents the soil sinkage under load; b corresponds to the radius of the circular plate; n signifies the soil deformation exponent; k_c indicates the cohesive modulus; and k_φ refers to the frictional modulus. In classical models, n, k_c, and k_φ are typically treated as constants independent of soil state. However, in reality, these mechanical parameters undergo significant evolution under cyclic loading and variations in moisture content. Therefore, this study extends the aforementioned parameters into functions of the number of cyclic loading cycles N and the moisture content ω, so as to more accurately characterize soil response in complex environments.

The Janosi shear stress–displacement model is used to describe the development of shear stress with displacement in granular materials (e.g., sand). Its curve typically exhibits an asymptotic approach to the maximum shear stress without a distinct peak. The model expression is given as

$$\tau ={\tau }_{\max }(1-e^{-{\displaystyle \frac{j}{K}}})$$

(2)

In this equation, K represents the shear deformation modulus; τ denotes the shear stress; j indicates the terrain shear displacement; c stands for the soil cohesion; and φ refers to the soil internal friction angle.

The value of K can be regarded as a measure of the shear displacement required to reach the maximum shear stress, and it determines the shape of the shear curve. Specifically, K can be obtained by drawing a tangent to the shear curve at the origin and finding the horizontal distance from the intersection point of this tangent with the horizontal line representing the maximum shear stress τ_max to the vertical axis. The slope of the shear curve at the origin can be derived by differentiating τ with respect to j in Expression (2):

$${\left.{\displaystyle \frac{d\tau }{dj}}\right|}_{j=0}={\displaystyle \frac{{\tau }_{\max }}{K}}{\left.e^{-j/K}\right|}_{j=0}={\displaystyle \frac{{\tau }_{\max }}{K}}$$

(3)

Under cyclic shearing and variations in moisture content, the value of K undergoes significant changes. This study establishes a functional relationship between K and the number of cycles N, moisture content ω, and normal stress σ, thereby constructing a modified constitutive model capable of describing the evolution of soil shear behavior.

To more realistically simulate the repetitive vertical loading and horizontal shear forces exerted by vehicle tires on the ground, this study selected coastal tidal flat sand from Zhangzhou, Fujian Province, as the research subject. By controlling the moisture content (5%, 15%, 25%), systematic repetitive settlement tests and repetitive shear tests were conducted. Building upon the Bekker and Janosi model framework, a constitutive model for settlement and shear was further developed to account for the coupled effects of repeated loading cycles (N) and moisture content (ω). This addresses the limitations of traditional triaxial and direct shear tests in characterizing vehicle repeated passage conditions. Moreover, the constructed model was applied to simulate vehicle–soil coupling dynamics in soft soils. A quarter-vehicle suspension model incorporating terrain subsidence characteristics was developed to systematically analyze vehicle ride comfort (spring mass acceleration) and control stability (tire deflection) under varying moisture contents and load cycles. This research provides a scientific foundation and data support for evaluating coastal soft ground vehicle passability and optimizing travel strategy designs for adaptive engineering.

2. Materials and Methods

2.1. Physical Parameters of Tested Soil

Coastal beach sand is a typical superficial sedimentary medium [25], characterized by its loose structure and high porosity [26]. Its mechanical properties are highly susceptible to external loading and hydrological conditions. Under natural states, it typically exhibits low shear strength and bearing capacity [27], and is prone to mechanical responses such as sinkage, localized shear failure, and particle reorganization under load [28]. Macroscopically, coastal beach sand displays significant structural instability and pronounced discrete characteristics in its mechanical behavior [29]. For this study, sand collected from the coastal area of Zhangzhou, Fujian Province (as shown in Figure 1), was used and subsequently prepared into test samples with moisture contents of 5%, 15%, and 25%, respectively.

The sand preparation process is illustrated in Figure 2 and mainly includes the following steps. First, sand samples are collected from Zhangzhou, Fujian Province, and sieved through a 16-mesh (1.18 mm) sieve to remove larger particles such as stones. Subsequently, the sieved sand is placed in an oven at (105 ± 5) °C and dried to constant weight, serving as the dried sample (defined as the reference sample with a moisture content of ω ≈ 0%). This dried sample forms the basis for preparing specimens with varying moisture contents. Based on the target moisture content (ω = 5%, 15%, 25%), the required amounts of dried sand and water are measured by mass. The required water content was determined by Formula (4). The measured water was evenly sprayed onto the sand using a sprayer, and the mixture was placed in a sealed container and stirred thoroughly for at least 10 min to ensure uniform distribution of moisture in the soil. The prepared wet soil samples are then sealed and placed in a constant-temperature environment for curing for over 24 h, allowing the moisture to fully migrate and achieve a uniform equilibrium state among the soil particles.

$$m_{\omega }={\displaystyle \frac{\omega }{1-\omega }}\times m_s$$

(4)

where m_ω is the mass of water; m_s is the mass of dried sand.

To minimize experimental errors introduced by inherent soil heterogeneity, a detailed particle size analysis was conducted on the sandy soil used in the tests. Particle size analysis conducted using a laser diffraction granulometer on the three soil samples revealed that the particle composition is predominantly fine sand. The coefficient of uniformity (Cu) is shown in

Table 1. Gradation index of beach sand.

Beach Sand	d10 (mm)	d30 (mm)	d50 (mm)	d60 (mm)	D90 (mm)	Cu
Sample-1	0.01960	0.11712	0.18067	0.22041	0.44239	1.1245
Sample-2	0.01987	0.12316	0.19017	0.23147	0.45314	1.1649
Sample-3	0.02450	0.13122	0.19845	0.23975	0.04593	0.9978

. According to the internal stability criterion proposed by Istomina, soils are considered well-graded if their coefficient of uniformity (Cu) is greater than or equal to 5. Therefore, the sand used in this study can be classified as uniform. The cumulative percentage distribution of the soil is presented in Figure 3, with Figure 3a specifically illustrating the percentage frequency distribution by particle size.

2.2. Repeated Pressure-Sinkage Test Program

Building upon known influences of loading rate on soil mechanical behavior, this experiment focuses on investigating the effects of repeated loading on soil deformation and strength characteristics under varying moisture contents. Water content alters the mechanical response of beach sand under repeated vehicle loading [27], particularly in intertidal zones where water level fluctuations cause the soil to experience conditions ranging from dry to saturated [30], substantially complicating the soil’s mechanical behavior during vehicle passage. The test was conducted with low (5%), medium (15%), and high (25%) moisture content levels. A high-precision electronic universal testing machine was used to apply vertical cyclic loading at a compression rate of 200 mm/min. Based on the development trend of soil sinkage deformation observed in preliminary tests, the number of non-recovered compression loading cycles per group was set to five. In non-recovered compression loading, on the same soil sample, a series of consecutive loading cycles is performed using a loading plate. After each cycle reaches the target load value, the load is fully unloaded. During this process, the soil undergoes irreversible deformation, i.e., plastic deformation. Subsequently, the loading plate is raised until it no longer contacts the soil surface, and the next loading cycle commences. This simulates the cumulative plastic deformation of the soil when subjected to continuous vehicle passage. Repeated compression tests were performed using both 150 mm and 100 mm plates to systematically analyze the cumulative plastic deformation and evolution of mechanical response in the soil under the coupled effects of moisture content and cyclic loading. The equipment used for the repeated compression tests is illustrated in Figure 4: Figure 4a shows the overall setup of the electronic universal testing machine, and Figure 4b and Figure 4c show the circular plates with diameters of 150 mm and 100 mm, respectively.

2.3. Repeated Shear Test Program

The repeated shear tests utilized a disc with protruding lugs as the shearing tool, which was rotated on the soil surface under a specified normal load. The shearing speed was set to 200°/min with a rotation angle of 90°. A schematic diagram of the disc is shown in Figure 5c, and its dimensional parameters are listed in

Table 2. Shear disk dimensional parameters.

Location	Parameters
Inner diameter (mm)	100
Outer diameter (mm)	200
Thickness (mm)	6
Spike height (mm)	4.2
Spike width (mm)	6
Number of spines: pieces	12

. By adjusting the mass on the disc to achieve different normal stresses (6.1 kPa, 10.3 kPa, and 14.4 kPa), the shear mechanical properties of the soil under varying normal stress conditions were systematically investigated. The tests were conducted using a Bevameter device (shown in Figure 5). Each set of tests involved five non-recovered shear loading cycles. A linear electric actuator and a servo motor provided precise rotational drive and displacement control. The system primarily consists of a testing unit, a data acquisition system, and a frame. Through shear tests under different normal stress conditions, the testing platform enables systematic observation of the deformation and strength evolution of loose granular media, such as beach sand, under cyclic loading.

3. Results and Discussion

3.1. Analysis of Repeated Pressure-Sinkage Results

3.1.1. Analysis of Soil Pressure-Sinkage Characteristics

Figure 6 shows the load–sinkage relationship curves of soil samples with different moisture contents under repeated loading (N = 1~5) using a 150 mm circular plate. During the first loading cycle (N = 1), the soil with low moisture content (5%) exhibited significant sinkage exceeding 10 mm under a relatively small load (approximately 450 N), indicating a loose structure and weak interparticle cohesion. Initial loading readily induced shear failure and rapid compaction, reflecting poor overall stability and low bearing capacity. In contrast, the curves for soils with higher moisture content (15% and 25%) showed steeper slopes during the first loading, requiring higher loads to achieve the same sinkage depth. This effect was particularly pronounced at 25% moisture content, suggesting that moderate water content promotes particle rearrangement and the formation of a denser, more stable structure, thereby enhancing initial bearing capacity [31].

The repeated loading process revealed significant compaction effects and evolutionary patterns. During the repeated loading phase (N = 2~5), the load–sinkage curves of the soil with 5% moisture content remained relatively flat, indicating that the soil skeleton remained weak after the initial structural failure, and continued to undergo substantial plastic deformation under loading [32]. In contrast, the soils with 15% and 25% moisture content exhibited notable strengthening effects. At the beginning of the second loading cycle, the stiffness of the curves increased markedly—the load required to achieve the same sinkage depth rose, while the cumulative plastic deformation under the same load level decreased progressively. In particular, in the N = 3~5 curves, the slope increased sharply, and the curves shifted toward higher load levels. As the number of loading cycles increased, the load–sinkage curves of the soils with 15% and 25% moisture content became nearly identical, indicating the formation of a stable and dense load-bearing structure and a significant enhancement in mechanical response. This demonstrates that near the optimal moisture content, repeated loading effectively promotes particle rearrangement and densification, substantially improving the soil’s stability and bearing capacity.

The simplified model of the plate sinkage process under repeated loading, as shown in Figure 7, demonstrates that the mechanical response and deformation evolution can be categorized into three progressive stages with increasing number of loading cycles (N): an initial compression and particle rearrangement phase (N = 1); followed by a densification and stable skeleton formation phase (N = 2); and finally a shakedown and stabilized elastoplastic state phase (N = 3, 4, 5). The figure illustrates the corresponding evolution of both macroscopic mechanical behavior and meso-scale structure throughout these stages, highlighting characteristic nonlinearity and cumulative plastic deformation.

During the first loading stage (N = 1), as illustrated by the initial state of the beach sand under compression, the soil skeleton was highly loose with high porosity, and interparticle contacts were primarily point-to-point with weak cohesion. Under external load, intense particle displacement occurred, large pores collapsed rapidly, and significant irreversible plastic deformation was observed, resulting in the maximum sinkage depth H₁. The mechanical behavior in this stage was dominated by global shear failure and initial compaction, with energy dissipation mainly through friction. Upon entering the second loading stage (N = 2), the soil structure had already undergone considerable alteration from the first cycle: porosity decreased, interparticle contact points increased, and a force chain network began to develop. As shown in the corresponding figure, particles rearranged into a more stable configuration, greatly reducing the remaining compressible pores and enhancing interparticle frictional resistance. Consequently, even under the same external load, the additional plastic sinkage H₂ was much smaller than H₁ (H₂ << H₁). This stage represents a critical period for the construction of a load-bearing skeleton, during which the soil transitions from a “loose state” to a “dense state” with a notable increase in stiffness.

In the third and subsequent loading stages (N = 3, 4, 5), the soil deformation entered a shakedown state. As indicated by the gradually stabilizing morphology of the “Beach Sand” from N = 3 to N = 5, the spatial distribution of particles became steady, and the force chains were optimized and homogenized. Macroscopically, the incremental plastic deformation per cycle became minimal and consistent, with the sinkage depths following the sequence H₃ > H₄ > H₅, reflecting a relatively stable elastoplastic state. At this point, the applied load was primarily borne by the stabilized skeleton, and the soil exhibited significant elastic recovery and cyclic stability.

A systematic analysis was conducted on the pressure-sinkage characteristics of the soil under three moisture content conditions (5%, 15%, and 25%) and two plate sizes (D = 150 mm and 100 mm). The load–sinkage curves in Figure 8 clearly demonstrate that, under the same moisture content, the plate size influences the mechanical response of the soil. When achieving the same sinkage depth, the load required for the larger plate (D = 150 mm) is substantially greater than that for the smaller plate (D = 100 mm). For example, as shown in the figure, at 5% moisture content, the load corresponding to a sinkage depth of 10 mm is higher for the 150 mm plate than for the 100 mm plate. Conversely, under the same applied load, the sinkage depth generated by the larger plate is much smaller than that of the smaller plate. The curves show that the load–sinkage curve of the larger plate consistently lies above that of the smaller plate and exhibits a steeper slope, indicating higher stiffness and bearing capacity.

This phenomenon can be explained by Boussinesq’s elastic theory: when a load is transferred into the soil through a plate, stress is distributed and dissipates with depth. Due to its larger contact area, the large plate (D = 150 mm) induces stress to greater depths and tends to cause general shear failure. The deeper influence mobilizes a stronger soil mass to jointly resist the load [33], resulting in smaller deformation per unit pressure and a curve that exhibits higher stiffness and ultimate bearing capacity. In contrast, the small plate (D = 100 mm) influences a shallower zone with significant stress concentration, making it more prone to local shear or punching failure [34]. Under repeated loading, the shallow soil beneath the small plate is more rapidly compacted to failure [35,36]. Furthermore, as the number of loading cycles increases (from N = 1 to N = 5), all curves show a noticeable strengthening effect, which is particularly pronounced under high moisture content and large plate conditions. The curve for N = 5 is shifted upward and exhibits a steeper slope compared to N = 1, indicating that the soil gradually densifies and its structure stabilizes under repeated loading. The coupling of this size effect and the compaction effect collectively influences the long-term mechanical behavior of the soil under load.

Figure 9 illustrates the relationship between plate sinkage depth and the number of loading cycles in the repeated pressure-sinkage tests, clearly demonstrating the significant influence of moisture content on the mechanical behavior of the beach sand. As the moisture content of the sand increased from 5% to 25%, the cumulative sinkage decreased noticeably. A comparison between Figure 9a (150 mm plate) and Figure 9b (100 mm plate) shows that the sinkage under 5% moisture content was substantially greater than that under 15% and 25% moisture content. The dry sand (5%) exhibited a loose structure with negligible cohesion between particles, leading to high compressibility during the initial loading stage and rapid development of sinkage. Although its ultimate bearing capacity was provided by interparticle friction, the overall higher position of its curve indicates large deformation and poor resistance to sinkage.

As the number of loading cycles increased, the sinkage under all moisture conditions exhibited a gradually decreasing trend, indicating a compaction effect induced by repeated loading. Among them, the soils with higher moisture content (15% and 25%) showed faster convergence of sinkage and lower final stabilized values. Particularly at 2 5% moisture content, the sinkage was the smallest, and the curve was the steepest, suggesting that moderate moisture enhances the formation of a more stable skeletal structure through capillary-induced apparent cohesion, resulting in a hardened bearing layer near the surface. These results demonstrate that the moisture content of beach sand is a critical factor influencing vehicle trafficability. Sand with optimal moisture content provides higher bearing strength and reduced sinkage, improving vehicle driving performance, while overly dry sand leads to increased sinkage due to its loose structure, thereby impairing vehicle mobility.

3.1.2. Analysis of Soil Pressure-Sinkage Parameters

Figure 10 and Figure 11 illustrate the variation of the deformation exponent (n), cohesive modulus (k_c), and frictional modulus (k_φ) of the beach sand with the number of repeated loading cycles (N) under different moisture contents (5%, 15%, 25%). As shown in Figure 10a–c, the soil deformation exponent n exhibits an increasing trend with additional loading cycles, and higher moisture content corresponds to greater values of n. At 5% moisture content, the most significant increase in nn occurs between the first and second loading cycles, reaching 105.36%, which is consistent with the results in Figure 8a,b. Under low moisture conditions, the large interparticle voids allow intense particle displacement under load, while at higher moisture levels, water facilitates particle rearrangement and structural adjustment. Figure 11a–c further reveal the strengthening effect of repeated loading on the soil strength parameters. Both the cohesive modulus k_c and the frictional modulus k_φ show continuous growth with increasing N, indicating that repeated compaction enhances the overall pressure–sinkage capacity of the soil. In summary, under repeated loading, moisture content is a critical factor influencing the evolution of the mechanical properties of beach sand. The presence of moderate moisture (e.g., 15%, 25%) more effectively promotes particle reorganization and structural densification, thereby markedly improving the cohesive strength and overall bearing capacity of the soil.

3.2. Analysis of Repeated Shear Test Results

3.2.1. Analysis of Soil Shear Characteristics

The angular displacement–torque curves under different moisture contents and loading rates are shown in Figure 12. The entire shear process can be broadly divided into an initial stage and a stable stage. Based on the curve morphology, in the initial shear stage, the torque increases rapidly with the shear angle until reaching a peak value, at which shear failure occurs. Thereafter, the torque stabilizes, entering the steady-state phase [37]. Experimental results indicate that with an increasing number of repeated shear cycles, the shear strength of the soil tends to increase, which can be attributed to particle rearrangement and enhanced structural compactness under shearing [38]. Under the 5% moisture content condition, the shear torque shows a significant increasing trend with higher normal stresses (6.1 kPa, 10.3 kPa, and 14.4 kPa) applied by the disc. The increase in normal stress substantially enhances the frictional resistance and interlocking effects between soil particles, and the shear torque represents the moment required to overcome this resistance. The shear strength of the soil is generally described by the Mohr–Coulomb criterion. When the moisture content remains constant, the cohesion c and the internal friction angle φ can be considered approximately constant, and the shear strength τ increases linearly with the normal stress σ.

$$\tau =c+\sigma \tan \phi$$

(5)

From a microscopic perspective, under low normal stress conditions (6.1 kPa), the shear behavior of the soil primarily involves particle rearrangement in a loose state. The shear strength in this case mainly originates from limited initial interlocking and sliding friction. When shear is applied, energy is predominantly consumed to overcome these relatively weak mechanical constraints, allowing particles to slide with relative ease. In contrast, under high normal stress conditions, the soil response shifts to a compressive shear mode. The significant increase in shear torque [39] directly reflects the substantial energy required to overcome the greatly enhanced internal friction and interlocking resistance activated by the external load.

Under low moisture content (5%), soil particles primarily rely on mechanical interlocking and dry friction. The cyclic shearing process forces particle rearrangement and strengthens the interlocking structure, resulting in “compaction hardening.” Conversely, at medium to high moisture levels (15% and 25%), moisture forms capillary bridges and lubricating films. Cyclic shearing disrupts the temporarily stabilized structure created by capillary forces among particles and promotes particle alignment along the shearing direction, forming a smooth shear plane. Additionally, under high moisture content and high normal stress, pore water pressure may develop, reducing effective stress. These mechanisms collectively lead to a decrease in soil shear strength with increasing loading cycles, manifesting as “shear softening.”

Under moisture contents of 15% and 25%, repeated shear loading leads to a reduction in shear strength. At a normal stress of 6.1 kPa, the maximum decrease in shear strength reaches approximately 33%. The primary reason is that repeated shearing no longer promotes soil densification but instead causes structural disruption, particle reorientation, and the formation of lubricating water films [40,41], resulting in a “softening” tendency. The shear failure patterns of soils at different moisture contents are shown in Figure 13.

In the dry state (approximately 0–5% moisture content), the interactions between sand particles are dominated by mechanical interlocking and direct grain-to-grain contact. Shear strength primarily derives from interparticle friction and interlocking effects. At this stage, the soil exhibits low plasticity, and shear marks are less distinct. At low moisture content (approximately 10–15%), water does not fully saturate the interparticle voids. The limited water exists mainly in the form of capillary water and strongly adsorbed bound water, forming an extremely thin film on particle surfaces. In this state, capillary forces provide additional “apparent cohesion,” enhancing the soil’s plasticity. Particles are reoriented along the shear direction, forming a smoother shear plane, which results in relatively clear shear traces. At 25% moisture content, capillary water is maximized. Water acts as a lubricant, facilitating particle sliding and rearrangement into a denser configuration under external forces, while capillary forces restrict excessive particle displacement. However, when a higher normal load is applied—such as a disc bearing mass corresponding to 14.4 kPa—the entrapped pore water cannot dissipate rapidly, leading to a sharp increase in pore water pressure. This mechanism results in the most pronounced shear marks among all conditions.

3.2.2. Analysis of Soil Shear Parameters

Figure 14 illustrates the variation trend of the shear deformation modulus K under different moisture content conditions. Analysis indicates that with an increasing number of shear cycles, the soil with 25% moisture content exhibits strengthening behavior, characterized by a stable or rising shear modulus, particularly under loads of 6.1 kPa and 10.3 kPa. In contrast, samples with 5% and 15% moisture content generally display softening behavior, where the shear modulus decreases sharply after the first shear cycle and subsequently fluctuates within a certain range. This suggests that the soil structure under dry or medium moisture conditions undergoes damage after the initial shear, leading to a reduction in shear resistance. Under high load conditions (14.4 kPa), the shear moduli of all soil types converge to a relatively low and narrow range (approximately 0.4–0.9), indicating that high load suppresses the influence of soil structural differences, resulting in more similar shear behavior across varying soil states. Starting from the second shear cycle, the shear modulus values enter a period of relatively stable fluctuation, demonstrating that the soil structure reaches a new and relatively stable critical state after the initial shear-induced damage.

4. Establishment of Soil Constitutive Models

4.1. Development of the Pressure-Sinkage Constitutive Model

The classical Bekker pressure-sinkage model serves as a fundamental theory for describing the interaction between rigid bodies and deformable terrain. Its parameters n, k_c, k_φ are typically treated as constants. However, experimental results from this study (as shown in Figure 10 and Figure 11) demonstrate that under the coupled effects of repeated loading and varying moisture content, the mechanical behavior of soil exhibits significant evolutionary characteristics.

To quantify the influence of the number of repeated loading cycles N and moisture content ω, this section systematically extends the classical Bekker model by developing a constitutive model that captures the evolution of soil mechanical behavior under coupled conditions. By expressing the model parameters as functions of N and ω, a more universal load–sinkage relationship is derived. This enhanced model provides a precise theoretical predictive tool for engineering applications involving repeated loading and variable environmental conditions, such as off-road vehicle mobility.

Based on the experimental results presented in Section 3.1.2, the parameters n, k_c, and k_φ exhibit regular variations with the number of loading cycles N and moisture content ω. The deformation index n increases monotonically with N, following a logarithmic functional trend. Meanwhile, ω influences both the initial values and growth rates of n, k_c, and k_φ. Therefore, the evolution functions of n, k_c, and k_φ are established as follows:

$$n(N,w)=n_0(w)+\alpha (w)·\ln (N)$$

(6)

where n₀(ω) represents the initial deformation index as a function of moisture content ω, and α(w) denotes the growth coefficient of the deformation index, which also depends on ω.

$$k_c(N,w)=k{c}_0(w)·e^{{\lambda }_c(w)·N}$$

(7)

where k_c0(w) denotes the initial cohesive modulus as a function of moisture content ω, and λ_c(w) represents the growth coefficient of the cohesive modulus, which is also dependent on ω.

$$k_{\phi }(N,w)=k_{\phi 0}(w)·e^{{\lambda }_{\phi }(w)·N}$$

(8)

where k_φ0(w) denotes the initial frictional modulus as a function of moisture content ω, and λφ(w) represents the growth coefficient of the frictional modulus, which is also dependent on ω.

Extended Bekker Model Formulation:

$$p(N,w)=\left[{\displaystyle \frac{k_c(N,w)}{b}}+k_{\phi }(N,w)\right]·z^{n(N,w)}$$

(9)

And by substituting the above functions into the extended model, the complete expression is obtained as follows:

$$p(N,w)=\left[{\displaystyle \frac{k{c}_0(w)·e^{{\lambda }_c(w)·N}}{b}}+k_{\phi 0}(w)·e^{{\lambda }_{\phi }(w)·N}\right]·z^{n_0(w)+\alpha (w)·\ln (N)}$$

(10)

where P(N,ω) denotes the soil pressure under the combined conditions of the number of repeated loading cycles N and the moisture content ω.

The parameter values under different operating conditions are shown in

Table 3. Bekker model parameter fitting results.

n	ω	n0(ω)	α(w)	R2
	5%	1.12	1.20	0.98
	15%	1.59	0.45	0.96
	25%	2.14	0.18	0.92
kc	ω	kc0(w)	λc(w)	R2
	5%	1.8	1.8	0.99
	15%	50.1	1.2	0.98
	25%	80.4	1.6	0.97
kφ	ω	kφ0(w)	λφ(w)	R2
	5%	36	1.8	0.99
	15%	1000	1.2	0.98
	25%	1600	1.6	0.97

.

Based on a systematic modification of the Bekker model parameters considering the number of repeated loading cycles N and the moisture content ω, an extended constitutive model has been developed to capture the evolution of soil mechanical behavior under cyclic loading. A comparison between the fitted model curves and the experimental load-sinkage curves under different moisture contents shows that they are highly consistent across all loading stages (R² > 0.92). The model accurately reflects the trends of soil stiffness hardening and sinkage development and physically rationalizes the evolutionary behavior of soil structure under the coupled effects of moisture content and repeated compaction.

4.2. Development of the Shear Constitutive Model

The aforementioned experimental results indicate that moisture content and the number of loading cycles exhibit a significant coupled influence on the shear mechanical behavior of coastal beach sand. To accurately predict the traction performance and soil stability of vehicles during repeated passes under complex hydrological conditions such as those in coastal tidal flats, it is necessary to develop a shear constitutive model that simultaneously accounts for the effects of moisture content ω and the number of shear cycles N. Based on the classical Janosi model framework, a modified shear constitutive model suitable for beach sand is constructed by incorporating quantitative functions of ω and N. The model parameters were calibrated and validated using experimental data.

Experimental results indicate that moisture content ω influences the shear strength parameters of the soil, namely cohesion c and the internal friction angle φ, while the number of repeated loading cycles N alters the soil structure, thereby affecting its ultimate shear strength τ_max and shear deformation characteristics K. Therefore, by expressing c, φ, τ_max and K as functions of ω and N, the modified constitutive model can be formulated as follows:

$$\tau (N,\omega )={\tau }_{\max }(N,\omega )(1-e^{(-{\displaystyle \frac{j}{K(N,\omega )}})})$$

(11)

$${\tau }_{\max }(N,\omega )=\left[c(\omega )+\sigma \tan \phi (\omega )\right]k(N,\omega )$$

(12)

$$K(N,\omega )=K_0\left(\omega \right)h(N,\omega )$$

(13)

where

c(ω) and φ(ω) are functions of moisture content representing cohesion and the internal friction angle, respectively;

k(N,ω) is the strength evolution function with repeated loading, characterizing the strengthening or softening effects induced by repeated shearing;

h(N,ω) is the deformation characteristics evolution function with repeated loading, describing the variation in the shear deformation modulus with the number of loading cycles;

K₀(ω) denotes the initial shear deformation modulus at the first loading cycle (N = 1).

By performing linear regression fits to the Mohr–Coulomb strength envelope based on the peak shear strengths under different normal stresses, the corresponding values of c and φ for various moisture contents were obtained, as listed in

Table 4. Shear strength parameters at different moisture contents.

ω (%)	c (kpa)	φ (°)	tanφ
5	8.52	30	0.57
15	15.86	24	0.45
25	26.44	22	0.40

. The quadratic polynomial functions for c(ω) and tan tanφ(ω) can then be fitted as follows:

$$c\left(\omega \right)=0.058{\omega }^2-0.426\omega +9.784$$

(14)

$$tan\phi \left(\omega \right)=-0.0005{\omega }^2-0.0115\omega +0.626$$

(15)

The function k(N,ω) is defined as the ratio of the peak shear strength at the N-th shearing cycle to that at the first cycle, serving to quantify the effect of repeated loading:

$$k\left(N,\omega \right)={\displaystyle \frac{{\tau }_{\max ,N}}{{\tau }_{\max ,1}}}$$

(16)

By calculating the values of k under different ω and N, the evolutionary patterns shown in Figure 15 reveal that when ω = 5%, k(N,ω) > 1 under both low normal stress (σ = 6.1 kPa) and high normal stress (σ = 14.4 kPa); when ω = 15%, k(N,ω) < 1 under all normal stress conditions, indicating that repeated shearing induces “softening” of the soil at this moisture content; and when ω = 25%, k(N,ω) > 1 at σ = 10.3 kPa, while k(N,ω) < 1 under both low normal stress (σ = 6.1 kPa) and high normal stress (σ = 14.4 kPa). Based on these observations, a parameterized model is established, and the parameter values are shown in

Table 5. Parameter values (α₁–α₆).

	α1	α2	α3	α4	α5	α6
P(ω,σ)	0.002	−0.001	−0.005	0.032	0.018	−0.125
Q(ω,σ)	−0.0003	0.0001	0.0008	−0.008	−0.004	0.035
R(ω,σ)	−0.002	0.001	0.005	−0.032	−0.018	1.125

.

$$k\left(N,\omega ,\sigma \right)=P\left(\omega ,\sigma \right)e^{Q\left(\omega ,\sigma \right)(N-1)}+R\left(\omega ,\sigma \right)$$

(17)

where P, Q, and R are quadratic functions of ω and σ:

$$P\left(\omega ,\sigma \right)={\alpha }_1{\omega }^2+{\alpha }_2{\sigma }^2+{\alpha }_3\omega \sigma +{\alpha }_4\omega +{\alpha }_5\sigma +{\alpha }_6$$

(18)

$$Q\left(\omega ,\sigma \right)={\alpha }_1{\omega }^2+{\alpha }_2{\sigma }^2+{\alpha }_3\omega \sigma +{\alpha }_4\omega +{\alpha }_5\sigma +{\alpha }_6$$

(19)

$$R\left(\omega ,\sigma \right)={\alpha }_1{\omega }^2+{\alpha }_2{\sigma }^2+{\alpha }_3\omega \sigma +{\alpha }_4\omega +{\alpha }_5\sigma +{\alpha }_6$$

(20)

Based on experimental data under varying moisture contents and normal stress conditions, an in-depth analysis was conducted on the evolution pattern of the shear deformation modulus K. The ratio of the shear deformation modulus at the N-th cycle to that at the first cycle is defined as follows:

$$h\left(N,\omega \right)={\displaystyle \frac{K_N}{K_1}}$$

(21)

Across all test conditions, the value of K exhibits a noticeable declining trend from the first to the fifth loading cycle, despite some fluctuations during the process. This indicates that repeated shearing causes irreversible damage or reorganization of the soil structure. Given that the evolution of K is simultaneously influenced by N, ω, and σ, it is modeled using a function of the following form:

$$h\left(N,\omega ,\sigma \right)={\displaystyle \frac{K_N}{K_1}}=\left[A\left(\omega ,\sigma \right)e^{B\left(\omega ,\sigma \right)N}+C\left(\omega ,\sigma \right)\right]$$

(22)

where the fitted results for the parameters A(ω,σ), B(ω,σ), and C(ω,σ) are determined using the equations below; the parameter values are shown in

Table 6. Parameter values (β₁–β₆).

	β1	β2	β3	β4	β5	β6
P(ω,σ)	0.023	−0.005	−0.042	0.352	0.128	−2.145
Q(ω,σ)	−0.001	0.003	0.002	−0.018	−0.007	0.105
R(ω,σ)	−0.022	0.005	0.040	−0.334	−0.121	2.040

.

$$A(\omega ,\sigma )={\beta }_1{\omega }^2+{\beta }_2{\sigma }^2+{\beta }_3\omega \sigma +{\beta }_4\omega +{\beta }_5\sigma +{\beta }_6$$

(23)

$$B(\omega ,\sigma )={\beta }_1{\omega }^2+{\beta }_2{\sigma }^2+{\beta }_3\omega \sigma +{\beta }_4\omega +{\beta }_5\sigma +{\beta }_6$$

(24)

$$C(\omega ,\sigma )={\beta }_1{\omega }^2+{\beta }_2{\sigma }^2+{\beta }_3\omega \sigma +{\beta }_4\omega +{\beta }_5\sigma +{\beta }_6$$

(25)

Based on the Janosi model framework, a modified shear constitutive model suitable for beach sand was developed by incorporating functional relationships with moisture content ω and the number of shear cycles N. This model utilizes the functions k(N,ω) and h(N,ω) to describe strength evolution and variations in the deformation modulus under repeated shearing, respectively, effectively capturing the “hardening” or “softening” behavior of the soil under different working conditions. Comparisons between the model-predicted curves and experimental shear curves under various moisture contents and normal stresses show that the model outputs align well with the test data in most scenarios. Although slight deviations occur under high moisture content and high stress conditions, the model overall demonstrates strong engineering applicability and predictive accuracy.

5. Soil–Vehicle Coupled Dynamics Simulation Analysis

The constitutive model established in this paper, which accounts for the coupled effects of repeated loading cycles $N$ and moisture content $\omega$, not only provides a theoretical basis for rapid assessment of soil mechanical properties but, more importantly, offers profound guidance for the design and performance optimization of off-road vehicle suspension systems. As the critical component connecting the vehicle to the ground, the performance of the suspension system is directly constrained by the mechanical characteristics of the ground. By more accurately describing the dynamic evolution of ground with varying moisture content during repeated vehicle passage, the proposed constitutive model offers new perspectives and quantitative tools for dynamic response analysis, parameter optimization, and intelligent control of suspension systems.

Unlike conventional solid surfaces, soft soil induces significant tire sinkage, substantially altering vehicle dynamics. Figure 16 presents the quarter tire-deformable terrain interaction suspension model, which incorporates ground sinkage characteristics beyond traditional suspension models to more accurately reflect force transmission under real-world conditions. By accounting for nonlinear tire-terrain interactions and sinkage parameters, the model simulates how varying soil properties (e.g., hardness, moisture content) affect the suspension system. Furthermore, the model captures dynamic response changes induced by road sinkage and considers energy dissipation effects from terrain deformation. The quarter tire-deformable terrain interaction suspension model is expressed in Formula (27).

$$\left\{\begin{array}{l}{m}_s·{\ddot{x}}_s+c_s({\dot{x}}_s-{\dot{x}}_u)+k_s(x_s-x_u)=0\hfill \\ m_u{\ddot{x}}_u-c_s({\dot{x}}_s-{\dot{x}}_u)-k_s(x_s-x_u)+k_t(x_u-(x_r+z_0))=0\hfill \end{array}\right.$$

(26)

The tire-terrain interaction model is derived from Bekker’s simplified tire-terrain interaction model, as shown in Figure 17. This model treats tire-terrain interaction as the rigid contact between the tire and deformable terrain. Since the behavior of pneumatic tires on soft road surfaces is similar to that of rigid wheels, the mechanical analysis of rigid wheels on soft road surfaces is of great significance. Bekker assumes that the reaction forces at all points within the contact patch are purely radial and equal to the normal pressure measured beneath a flat plate at the same depth in pressure–sinkage tests. Thus, $\sigma r\mathit{cos}\theta d\theta =pdx$, allowing the vertical load $W$ acting on the tire to be expressed by Formula (28).

$$W=b{\displaystyle {\int }_0^{{\theta }_0}\sigma r\cos \theta \mathrm{d}\theta }=-b{\displaystyle {\int }_0^{z_0}p\mathrm{d}x}=-b{\displaystyle {\int }_0^{z_0}\left(\frac{k_c}{b}+k_{\varphi }\right)z^n\mathrm{d}x}$$

(27)

Based on the geometric structure depicted in the figure and assumptions, the soil sinkage under the tire vertical load $W$ can be calculated as shown in Formula (29).

$$z_0={\left[\frac{3·W}{b(3-n)·\left((k_c/b)+k_{\varphi }\right)·\sqrt{D}}\right]}^{2/(2n+1)}$$

(28)

Here, $b$ is the tire width, $D$ is the tire diameter, b is the smaller dimension of the contact patch, and $n$, $k_c$, and $k_{\phi }$ are physical parameters in the soil pressure–sinkage model. The vertical load $W$ on the tire is determined by the quarter-vehicle model:

$$W=k_{\mathrm{t}}(x_{\mathrm{u}}-(x_{\mathrm{r}}+z_0))$$

(29)

By solving Equations (27), (29) and (30) simultaneously, the three unknowns $x_s$, $x_u$, $z_0$ can be determined at each time instant in the model simulation through integration twice. This section conducts simulation experiments using the established vehicle suspension model, whose parameters are listed in

Table 7. Quarter-vehicle suspension model parameters.

Parameter Name	Symbol	Numerical Value	Unit
Sprung mass	$m_s$	400	Kg
Unsprung mass	$m_u$	40	Kg
Suspension stiffness	$k_s$	20,000	N/m
Tire stiffness	$k_t$	200,000	N/m

.

For the selected random road as the simulation condition, the time-domain model of pavement unevenness can be expressed by the following equation:

$${\dot{x}}_r\left(t\right)=-2\pi f_0{x}_r\left(t\right)+2\pi w\left(t\right)\sqrt{{\Phi }_0\left(n_0\right)u}$$

(30)

Assuming a vehicle traveling at 90 km/h on a Class C random road, with $u=25 \mathrm{m}/\mathrm{s}$, ${\Phi }_0\left(n_0\right)=256\times {10}^{-6}$ m³ and a lower cutoff frequency $f_0=0.1\mathrm{H}\mathrm{z}$. Figure 18 shows the time-domain simulation results, while Figure 19 displays the power spectral density for this road, ABCD represent non-typical random paths. The figures demonstrate that the power spectral density obtained using the filtered white noise method aligns well with that of the standardized road, meeting simulation requirements.

Soil–vehicle coupled dynamic simulation experiments were conducted with the repeated load cycles $N=1$ and the moisture content $\omega =5\%$. Figure 20 and Figure 21 present simulation results comparing scenarios with and without consideration of deformable terrain. It is evident that incorporating deformable terrain impacts two key indicators of vehicle ride comfort and handling stability: sprung mass acceleration and tire deflection. This underscores the necessity of considering deformable terrain when conducting dynamic response analysis, parameter optimization, and intelligent control of vehicle suspension systems.

To investigate the dynamic response of vehicle suspension systems under the same moisture content but varying numbers of repeated load cycles, multiple soil–vehicle coupled dynamic simulation experiments were conducted with the moisture content $\omega =25\%$ and the repeated load cycles $N=1~5$. The simulation comparison results for the same moisture content but different repeated load cycles are shown in Figure 22, Figure 23, Figure 24 and Figure 25. Figure 22 and Figure 23 present time-domain comparison curves of sprung mass acceleration and tire deflection, while Figure 24 and Figure 25 display bar charts of the relative root mean square (RMS) values for sprung mass acceleration and tire deflection. The simulation results indicate that both the sprung mass acceleration and tire deflection gradually increase with the number of repeated load cycles. This aligns with the earlier conclusion: as the number of loading cycles increases (from $N=1$ to $N=5$), the soil exhibits a pronounced strengthening effect. The soil gradually consolidates, its structure stabilizes, and stiffer soil yields greater sprung mass acceleration and tire deflection. This result also demonstrates that the number of soil loading cycles impacts vehicle suspension systems. Therefore, investigating soil mechanical properties under repeated loading cycles holds considerable value for suspension system research, particularly for multi-axle wheeled or tracked vehicles.

To investigate the dynamic response of vehicle suspension systems in soils with varying moisture contents, multiple soil-vehicle coupled dynamic simulation experiments were conducted with the repeated loading cycles $N=2$, and the moisture contents $\omega =5\%$, $\omega =15\%$, and $\omega =25\%$. The simulation comparison results for the same number of repeated loading cycles under different moisture contents are shown in Figure 26 and Figure 27. Among these, Figure 26 and Figure 27 present time-domain comparison curves of sprung mass acceleration and tire deflection. Figure 28 and Figure 29 display the relative root mean square (RMS) values of the sprung mass acceleration and tire deflection. The simulation results indicate that the sprung mass acceleration and tire deflection under soil with a moisture content $\omega =5\%$ are greater than those under soils with moisture contents of $\omega =15\%$ and $\omega =25\%$. Meanwhile, the sprung mass acceleration and tire deflection under soils with moisture contents of $\omega =15\%$ and $\omega =25\%$ are relatively similar. This aligns with the earlier conclusion: the load–sinkage curve for soil with 5% moisture content is relatively flat, indicating that the soil skeleton remains weak after initial structural failure, leading to substantial plastic deformation during loading. In contrast, soils with 15% and 25% moisture content exhibit a pronounced strengthening effect. Starting from the second loading cycle, the stiffness of the curve increases—the load required to achieve the same sinkage increases substantially, while the cumulative plastic deformation at the same load level gradually decreases. This result also demonstrates that the soil moisture content has a significant impact on vehicle suspension systems. Therefore, studying the mechanical properties of soil at different moisture contents holds considerable value for research on vehicle suspension systems.

6. Conclusions

This study investigates the evolution of the mechanical properties of coastal beach sand from Zhangzhou, Fujian, under repeated loading through laboratory tests. The focus is placed on analyzing the effects of key factors such as moisture content, number of loading cycles, plate size, and normal stress. Furthermore, based on the Bekker and Janosi model frameworks, pressure–sinkage and shear constitutive models incorporating the coupling effects of repeated loading cycles and moisture content were developed, and we conducted simulation, validation, and application analysis of the aforementioned mechanical evolution patterns through the development of a vehicle–soil coupled dynamics model, quantifying the impact of soil state on vehicle dynamic response. The main conclusions are as follows:

(1)

Moisture content is a key factor governing the pressure–sinkage response and its evolution. Under low moisture content (5%), the soil structure is loose, exhibiting significant plastic deformation during the initial loading cycle, with continued notable accumulation of deformation in subsequent cycles and consistently low bearing capacity. In contrast, under medium to high moisture content (15%~25%), moisture enhances interparticle capillary forces, resulting in high initial bearing capacity. As the number of loading cycles increases (N = 2~5), the soil demonstrates a “strengthening effect”: cumulative sinkage decreases progressively, while bearing capacity and stiffness improve, leading to a denser and more stable structure.

(2)

Plate size influences stress propagation and failure modes. A smaller plate (100 mm) tends to cause stress concentration and punching failure, resulting in larger sinkage. In contrast, a larger plate (150 mm) promotes deeper stress diffusion, triggering general shear failure and engaging deeper soil layers in bearing the load. Under the same applied load, the larger plate induces smaller sinkage and exhibits higher bearing efficiency.

(3)

The soil exhibits either “hardening” or “softening” under repeated shearing, depending on moisture content and normal stress. Under dry conditions (5%), repeated shearing promotes particle densification, leading to increasing torque and shear strength, manifesting as “hardening.” In contrast, under high moisture content (15% and 25%), repeated shearing causes structural degradation: water lubrication and particle reorientation form slip surfaces, while under higher loads (14.4 kPa), a sharp rise in pore water pressure reduces effective stress. These mechanisms collectively result in a decrease in shear strength with increasing cycles, with a maximum reduction of 33%, representing a “softening” effect.

(4)

By establishing a vehicle–soil coupled dynamics model that integrates constitutive relations, this study reveals the quantitative influence of soil state on vehicle performance. Simulation results demonstrate that both soil moisture content and the number of loading cycles (N) collectively determine the equivalent surface stiffness: higher moisture content or more loading cycles lead to greater soil stiffness, which in turn causes a significant increase in vehicle sprung mass acceleration and tire dynamic deflection. This finding confirms that the evolution of soil mechanical state is a critical factor affecting vehicle ride comfort and handling stability, providing a direct theoretical basis and simulation tool for the design of suspension systems and the optimization of trafficability for vehicles operating on soft terrain.

This study elucidates the evolution of mechanical behavior in coastal beach sand under the coupled effects of moisture content and repeated loading, revealing the dynamic process from deformation accumulation to structural stability and its underlying meso-mechanisms. The constitutive models developed provide a theoretical basis and data support for evaluating and optimizing vehicle trafficability on coastal soft ground.

However, this study has certain limitations. First, it was conducted only at three fixed moisture content levels, thus failing to continuously simulate the dynamic variation of soil moisture in natural environments, especially leaving the mechanical behavior under fully saturated conditions unclear. Second, the limited number of loading cycles and normal stress levels in the tests was insufficient to fully cover extreme scenarios of repeated vehicle passes or long-term cumulative deformation effects. Furthermore, although the Discrete Element Method (DEM) simulations revealed meso-scale mechanical mechanisms, the particle models used were relatively idealized and did not fully account for complex factors such as real particle shapes, fragmentation, and interparticle liquid bridge forces. Due to the highly complex internal structural response of soil under the coupled effects of moisture variation and repeated loading, the current models still exhibit deviations in accurately predicting actual soil mechanical behavior and have not yet established a constitutive, rigorous relationship that precisely reflects multi-factor coupling effects. Nonetheless, this study advances the theoretical understanding of the response mechanisms of loose granular media under cyclic loading and provides critical data support and a scientific basis for evaluating vehicle trafficability in special terrains such as coastal tidal flats and deserts.