Discrete Element Method-Based Analysis of Tire-Soil Mechanics for Electric Vehicle Traction on Unstructured Sandy Terrains

Abstract

In order to tackle the issues of poor mobility and unstable traction of electric vehicles on sandy landscapes, this research develops a high-accuracy numerical model for wheel–sand interaction relying on the Discrete Element Method (DEM). An innovative parameter calibration procedure is proposed herein, which optimizes the sand contact parameters. This reduces the error between the simulated and measured angles of repose to merely 1.2% and substantially improves the model’s reliability. The model was then used to systematically compare the performance of a 205/55 R16 slick tire with a treaded tire on sand. Simulations demonstrate that at a 30% slip ratio, the treaded tire exhibited significantly higher traction and greater sinkage than the slick tire. This indicates that tread patterns enhance traction mechanically by increasing the contact area and promoting shear deformation of the sand. The trends of traction with slip ratio and the corresponding sand flow patterns showed excellent agreement with experimental observations, which validated the simulation approach. This research provides an efficient and accurate tool for evaluating tire-sand interaction, providing critical support for the design and control of electric vehicles on complex terrains.

1. Introduction

Driven by the global demand for sustainable transport and extreme environment exploration, electric vehicles (EVs) are increasingly used beyond urban roads—on unstructured sandy terrains such as deserts, beaches, and even planetary surfaces. As the only interface between the vehicle and the ground, tires play a critical role in determining the mobility, traction efficiency, and energy consumption of EVs on loose soils like sand. EVs differ from traditional internal combustion engine vehicles primarily in their torque characteristics and energy efficiency requirements. During startup on sandy terrain, EVs need instantaneous traction to avoid slipping. Additionally, they rely on tire optimization to reduce energy consumption. Therefore, a deeper understanding of tire-sand contact mechanics is essential for improving the performance of electric vehicles and planetary rovers. However, tire–sand interaction involves large deformations, granular flow, and discontinuous media. This complexity makes it difficult for traditional continuum mechanics theories to accurately describe the underlying meso-scale mechanisms. The discrete element method can be used to analyze the sand particle motion [1] by assigning properties to the particles and obtaining the corresponding motion laws [2]. The angle of repose, as a macroscopic representation of particle properties, can be used as an intuitive basis for simulating whether the particle properties are correct or not [3]. The accuracy of the results through simulation alone is yet to be examined. The combination of simulation and experiment has become an increasingly sophisticated method [4,5,6,7,8,9] for particle model construction and validation.

Currently, the Discrete Element Method (DEM) has been used to study wheel-terrain interaction. Jasoliya et al. [5] summarized numerical modeling methods for soil-tire interaction, although their review did not address the specific requirements of electric vehicles on unstructured sandy terrain—such as traction performance under low-speed, high-torque conditions. Sedara et al. [2] improved tillage machine design using the Discrete Element Method, but their research did not explore how tire tread patterns regulate the shear behavior of sand particles at a micro level. Although Alkhalifa et al. [10] experimentally analyzed the influence of vertical load on tire-soil contact, their study did not employ the DEM to quantify the movement of soil particles. Dong et al. [8] used the DEM to design tillage wheels for soft terrain, while also analyzing the mechanism by which wheel teeth disturb sand particles. Similarly, Patidar et al. [6] employed the DEM to simulate the interaction between a rotary tiller’s rotor and soil, and further optimized the rotor structure. As for tread patterns, Pacejka [11] utilized the FEM to analyze the contact pressure distribution of treads on the ground, though this work did not cover the particle scale. Meanwhile, Jelinek et al. [12] applied the DEM to study the traction performance of rigid wheels on sand, but failed to quantify the impact of tread patterns.

Addressing the aforementioned gaps, this study integrates the advantages of the aforementioned methods and develops a wheel–sand DEM model suitable for electric vehicles. First, we calibrated key sand contact parameters, including coefficients of restitution, static friction, and rolling friction, by combining physical angle of repose tests with numerical simulations. The steepest ascent method and Box–Behnken response surface methodology were used to optimize these parameters, ensuring high physical accuracy. Using this calibrated model, we created detailed numerical models of a 205/55 R16 radial tire in both slick and treaded versions. We then systematically compared their traction performance, sinkage behavior, and sand flow patterns under different slip ratios. The simulation results were validated using experimental data from published soil bin tests [13]. This study improves our understanding of how tread patterns enhance performance on sandy terrain. It also offers a powerful simulation tool and theoretical support for selecting tires, designing treads, and optimizing electric vehicles on loose, deformable surfaces.

2. Wheel and Soil Contact Modeling

2.1. Wheel–Soil Interaction Modeling

The basic model of wheel–sand interaction is shown in Figure 1 [14]. Here, $\theta$ is the wheel-loading contact angle and represents the circumferential angle formed by the tire in contact with the ground. This angle is divided into the angle of incidence and the angle of departure. The angle of incidence ${\theta }_1$ and the angle of departure ${\theta }_2$ are the circumcentric angles formed by the front edge of the tire and the rear edge of the tire, respectively, in contact with the ground. The contact angle of the wheel and soil ranges between the angle of incidence and the angle of departure. Positive stress $\sigma \left(\theta \right)$ on the tire contact surface represents the force of the ground on the tire and is directed towards the center of the circle. The shear stress $\tau \left(\theta \right)$ represents the force of the ground on the tire in the direction of the tire surface. The ground force on the tire can be obtained by integrating the positive and shear stresses. When $\sigma \left(\theta \right)$ reaches its maximum value, at this point, the wheel-loading contact angle is defined as the maximum stress angle recorded as ${\theta }_m$.

The supporting force is the soil’s vertical force on the tire. It comes from accumulated vertical components of normal stress in the contact angle range. The maximum stress angle contributes most, as normal stress is highest there. Affected by soil parameters and sinkage, it increases with the load. The drawbar pull is the horizontal driving force. It forms from accumulated horizontal components of shear stress in the contact angle range. The shear stress in the entry and exit zones work together. The slip ratio affects it by changing the exit angle. When the slip ratio is low, the drawbar pull increases with it. Beyond a certain value, the drawbar pull stabilizes. The resistance moment hinders tire rolling. It includes the compression resistance moment from normal stress and the slip resistance moment from shear stress. It is most significant near the maximum stress angle, where both stresses reach their peaks. It correlates positively with sinkage, and treaded tires offset part of its increase by adjusting the shear stress acting angle. Sinkage is a function of the contact angle, with a parabolic distribution in the contact area. It is largest at the maximum stress angle. Its magnitude relates to load and soil properties. Greater load or softer soil leads to more significant sinkage. Soil rebound height is the soil’s elastic recovery height in the tire’s exit zone. It relates to soil elastic parameters and exit sinkage. Usually, harder soil or greater exit sinkage leads to higher rebound height. Treaded tires have slightly lower rebound height than slick tires, as they disturb the soil more with shear. In addition, it reduces the subsequent tires’ secondary sinkage.

On this basis, the study of ground mechanics will also make some corrections in combination with the tire’s pivoting effect in the actual driving process. It is usually assumed that there is a certain linear relationship between the departure angle and the slip rate and cut-in angle, as shown in Equation (1).

$${\theta }_m={\theta }_1\left(a+bi\right), {\theta }_2=c{\theta }_{1}i$$

(1)

$$i=1-{\displaystyle \frac{v}{\omega r}}$$

(2)

where $i$ represents the slip rate of the tire, which is mainly related to the radius of the tire, the current travel speed and the angular velocity. The a, b and c in Equation (1) are fitting coefficients. They are usually taken empirically to simplify the calculation volume. In order to facilitate the calculation and improve the calculation efficiency, the above three fitting coefficients are taken as reasonable values, where a is taken as 0.5 and b and c are taken as 0.

2.2. Positive Stress Distribution Model

In the theory of ground mechanics, the traction force provided by the tire to the road surface comes mainly from the positive and shear stresses generated by the pressure of the load on the road surface. Only by accurately analyzing the stress–strain relationship generated by the soil under tire loading can the positive and shear stresses in the part of the tire in contact with the pavement be analyzed. In order to calculate the positive stress, the classical positive stress–strain equation [15] proposed by Bekker was chosen:

$$\sigma =\left({\displaystyle \frac{k_c}{b}}+k_{\varphi }\right)z^n$$

(3)

where $\sigma$ and $z$ are both functions of the wheel–loam contact angle, which is integrated over the wheel–loam contact angle during the operation. Here, $\sigma$ is the positive stress (Pa), $k_c$ is the soil cohesion modulus (N/mⁿ⁺¹), $k_{\varphi }$ is the soil friction modulus (N/mⁿ⁺²), $b$ is the tire width (m), $z$ is the tire sinkage (m), and $n$ is the soil deformation index.

The positive stress distribution in the contact part of the wheel loading is shown in Figure 2 [16].

When ${\theta }_m\le \theta \le {\theta }_1$, the positive stress provided by the pavement can satisfy Bekker’s positive stress-sinking equation, so the positive stress in this part can be calculated simply by Equation (3) [16], in which the required tire sinking $z_1$ can be obtained by the simple geometric relationship in Figure 1, and the expression of Equation (4). However, for the part of ${\theta }_2\le \theta \le {\theta }_m$, the positive stress provided by the road surface cannot be calculated according to Equation (3), because, through the analysis of test results, Wong found that the stress–strain in this part does not satisfy Bekker’s positive stress–strain formula. Instead, it is symmetric with the distribution of positive stress in part ${\theta }_m\le \theta \le {\theta }_1$. As shown in Figure 2, the positive stress distribution in part ${\theta }_2\le \theta \le {\theta }_m$ is shown on the left, that in part ${\theta }_m\le \theta \le {\theta }_1$ is shown on the right, and both of them are approximately symmetrical with respect to the stress line at the maximum stress angle [16]. Through the simple similarity relationship, the relation between the contact angle of the tire’s leading edge and trailing edge can be obtained for ${\sigma }_1={\sigma }_2$. This lays the foundation for further calculations.

$$z_1=r\left(\cos \theta -\cos {\theta }_1\right)$$

(4)

When ${\theta }_2\le \theta \le {\theta }_m$, substituting the expression for the tire contact angle into Equation (4), it is possible to obtain the amount of tire subsidence along the back edge of the contact under the same stress $z_2$.

$$z_2=r\left\{\cos \left[{\theta }_1-\left({\displaystyle \frac{\theta -{\theta }_1}{{\theta }_m-{\theta }_2}}\right)\left({\theta }_1-{\theta }_m\right)\right]-\cos {\theta }_1\right\}$$

(5)

Finally, the two different expressions for tire sinkage in parts ${\theta }_m\le \theta \le {\theta }_1$ and ${\theta }_2\le \theta \le {\theta }_m$ are substituted into Bekker’s positive stress–sinkage Equation (3) to obtain the final positive stress distribution prediction equation. Here, ${\sigma }_1$ represents the positive stress provided by pavement ${\theta }_m\le \theta \le {\theta }_1$ and ${\sigma }_2$ represents the positive stress provided by pavement ${\theta }_2\le \theta \le {\theta }_m$.

$${\sigma }_1=\left({\displaystyle \frac{k_c}{b}}+k_{\varphi }\right)r^n{\left(\cos \theta -\cos {\theta }_1\right)}^n$$

(6)

$${\sigma }_2=\left({\displaystyle \frac{k_c}{b}}+k_{\varphi }\right)r^n\left\{\cos \left[{\theta }_1-\left({\displaystyle \frac{\theta -{\theta }_1}{{\theta }_m-{\theta }_2}}\right)\left({\theta }_1-{\theta }_m\right)\right]-\cos {\theta }_1\right\}$$

(7)

2.3. Modeling of Shear Stress Distribution

Janosi, Bekker’s assistant, proposed a shear stress–strain formula based on the positive stress-sinking formula, borrowing some of the theory from soil mechanics [17]:

$$\tau =\left(c+\sigma \tan \phi \right)\left(1-e^{{\displaystyle \frac{j}{k}}}\right)$$

(8)

where $\tau$ is the shear stress (Pa), $c$ is the soil cohesion (Pa), $\sigma$ is the positive stress (Pa), $\phi$ is the soil internal friction angle (rad), $j$ is the shear strain, and $K$ is the soil shear deformation modulus (m).

Here, $\tau$, $\sigma$ and $j$ are functions of the wheel-loading contact angle $\theta$, by taking a partial derivative of $j$ in Equation (8):

$${\displaystyle \frac{\partial \tau }{\partial j}}={\displaystyle \frac{{\tau }_{\max }}{K}}{e}^{-\frac{j}{K}}$$

(9)

The variation curve of $\tau$ about shear strain $j$ can be obtained as shown in Figure 3. From the curve, it can be found that when $j=0$, its slope is the ratio between ${\tau }_{\max }$ and $K$. Therefore, a common method of solving $K$ is the track plate test, through which the relationship between shear strain $j$ and shear stress $\tau$ [17] is obtained, and the value of $K$ is obtained from the curve.

The shear displacement of the pavement soil is solved for by the velocity of the tire relative to the pavement. From this, the shear stress distribution relationship on the contact surface of the tire–pavement interaction is obtained by means of the shear stress–strain equation. The decomposition of soil shear velocity in wheel–soil interaction is shown in Figure 4, and the radial velocity of the soil is neglected under reasonable assumptions in the actual calculation.

By means of the principle of decomposition and superposition of velocities, it is possible to express the tangential velocity $v_{\tau }$ at any point of the wheel-loading contact surface:

$$v_{\tau }=\omega r-v\cos \theta =\omega r+\omega r\left(i-1\right)\cos \theta$$

(10)

Integrating Equation (10) over time $t$ gives an expression for the soil shear strain:

$$j={\displaystyle {\int }_0^t{v}_{\tau }dt={\displaystyle {\int }_0^t\left[\omega r+\omega r\left(i-1\right)\cos \theta \right]}d\frac{\theta }{\omega }}$$

(11)

After obtaining the shear strain of the soil, it was substituted into Janosi’s shear stress–strain Equation (8) to obtain an expression for the shear stress:

$$\tau =\left[c+\sigma \tan \varphi \right]\left\{1-\exp \left\{-{\displaystyle \frac{r}{K}}\left[\left({\theta }_1-\theta \right)+\left(i-1\right)\left(\sin {\theta }_1-\sin \theta \right)\right]\right\}\right\}$$

(12)

2.4. Theoretical Modeling of Traction Characteristics in Linear Conditions

In theoretical models of ground mechanics, the theory of equivalent radius is often used to address the effect of tire tread on traction performance. This equates the tire tread to an additional radius of the tire in a certain proportion, thus improving the traction performance of the tire. Here, $r_o$ represents the original radius of the tire, $r$ represents the equivalent radius, and $k$ is the equivalence coefficient. That is:

$$r=r_0+k{b}_L$$

(13)

Based on the Rankine soil pressure theory, the equivalent coefficients were derived and calculated, and the expression formulae were obtained [18].

$$k=1-{\displaystyle \frac{3\left(r_o+b_L\right)}{8\left(1+\cot \frac{\alpha }{2}\cot X_c\right)b_L}}$$

(14)

where $\alpha$ is the angle between the tire treads (rad), $b_L$ is the thickness of the tire tread (m), and $X_c$ is the horizontal angle of the logarithmic helix of soil shear in the Tyshaki theory (rad).

The formulae for the positive and shear stresses were obtained and corrected according to the formula for the equivalent radius of the tire. The expressions for the load $W$, the hook traction $F_{DP}$, and the torque $M$ of the tire are obtained by integrating the positive and shear stresses in polar coordinates over the contact angle of the wheel yoke in different directions.

In practical calculations, the vertical load $W$ of the tire is generally a known condition, which is solved by the first equation in Equation (15), and then substituted into the next two equations to obtain the tire’s hooked traction as well as torque.

$$\begin{array}{c}W=rb\left[{\displaystyle {\int }_{{\theta }_m}^{{\theta }_1}{\tau }_1\sin \theta d\theta +{\displaystyle {\int }_{{\theta }_2}^{{\theta }_m}{\tau }_2\sin \theta d\theta }}\right]+\\ rb\left[{\displaystyle {\int }_{{\theta }_m}^{{\theta }_1}{\sigma }_1\left(\theta \right)\cos \theta d\theta +{\displaystyle {\int }_{{\theta }_2}^{{\theta }_m}{\sigma }_2\left(\theta \right)\cos \theta d\theta }}\right]\\ F_{DP}=rb\left[{\displaystyle {\int }_{{\theta }_m}^{{\theta }_1}{\tau }_1\cos \theta d\theta +{\displaystyle {\int }_{{\theta }_2}^{{\theta }_m}{\tau }_2\cos \theta d\theta }}\right]\\ -rb\left[{\displaystyle {\int }_{{\theta }_m}^{{\theta }_1}{\sigma }_1\left(\theta \right)\sin \theta d\theta +{\displaystyle {\int }_{{\theta }_2}^{{\theta }_m}{\sigma }_2\left(\theta \right)\sin \theta d\theta }}\right]\\ M=r^{2}b\left[{\displaystyle {\int }_{{\theta }_m}^{{\theta }_1}{\tau }_{1}d\theta +{\displaystyle {\int }_{{\theta }_2}^{{\theta }_m}{\tau }_{2}d\theta }}\right]\end{array}$$

(15)

3. Pavement Modeling

3.1. Simulation Principle

This study focuses on dry sand (moisture content < 1%). The reason is that arid areas are typical off-road scenarios for electric vehicles. However, it should be noted that the moisture content of sand significantly affects the adhesion between particles. The sand used in the experiment is dry sand. Inter-particle adhesion was less than 1 Pa and negligible. Therefore, the Hertz–Mindlin–Deresiewicz no-slip contact model [19] was chosen as the basis of establishment. The model is shown in Figure 5, in which the tangential force model is based on the Mindlin–Deresiewicz theory [20] and the normal force model is the Hertzian contact theory [21].

There are elastic forces $F_s$, damping forces $F_d$, and rolling friction ${\tau }_i$ between the two particles. Among them, the elastic force $F_s$ and damping force $F_d$ can be further divided into normal elastic force $F_s^n$, tangential elastic force $F_s^{\tau }$, normal damping force $F_d^n$, and tangential damping force $F_d^{\tau }$. $K_n$, $K_{\tau }$, $C_n$, $C_r$, and $\mu$ are the normal spring stiffness coefficient, spring tangential stiffness coefficient, normal damping coefficient, tangential damping coefficient, and coefficient of static friction, respectively [22].

$$F_s^n={\displaystyle \frac{4}{3}}{E}^{\ast }\sqrt{R^{\ast }}{\delta }_n^{\frac{3}{2}}$$

(16)

where

$${\displaystyle \frac{1}{E^{\ast }}}={\displaystyle \frac{1-{\upsilon }_i^2}{E_i}}+{\displaystyle \frac{1-{\upsilon }_j^2}{E_j}}$$

(17)

$${\displaystyle \frac{1}{R^{\ast }}}={\displaystyle \frac{1}{R_i}}+{\displaystyle \frac{1}{R_j}}$$

(18)

where $E^{\ast }$ is the equivalent Young’s modulus; $R^{\ast }$ is the equivalent radius; ${\delta }_n$ is the normal overlap; $E_i$, $E_j$ are the elastic modulus of contact particle A and particle B, respectively; ${\upsilon }_i$, ${\upsilon }_j$ are Poisson’s ratios; and $R_i$, $R_j$ are the radii of the contact spheres.

$$F_d^n=-2\sqrt{{\displaystyle \frac{5}{6}}}\beta \sqrt{S_n{m}^{\ast }}{\upsilon }_n^{\overline{rel}}$$

(19)

$$m^{\ast }={\left({\displaystyle \frac{1}{m_1}}+{\displaystyle \frac{1}{m_j}}\right)}^{-1}$$

(20)

where $F_d^n$ is the normal damping, $m^{\ast }$ is the equivalent mass, and ${\upsilon }_n^{\overline{rel}}$ is the normal component of the relative velocity.

$$\beta ={\displaystyle \frac{\ln e}{\sqrt{{\ln }^{2}e+{\pi }^2}}}$$

(21)

$$S_n=2E^{\ast }\sqrt{R^{\ast }{S}_n}$$

(22)

where $\beta$ is the damping ratio, $S_n$ is the normal stiffness, and $e$ is the coefficient of restitution.

$$F_s^{\tau }=-S_{\tau }{\delta }_{\tau }$$

(23)

where

$$S_{\tau }=8G^{\ast }\sqrt{R^{\ast }{\delta }_n}$$

(24)

where $F_s^{\tau }$ is the tangential force, $S_{\tau }$ is the tangential stiffness, ${\delta }_{\tau }$ is the tangential overlap, and $G^{\ast }$ is the equivalent shear modulus.

$$F_d^{\tau }=-2\sqrt{{\displaystyle \frac{5}{6}}}\beta \sqrt{S_{\tau }{m}^{\ast }}{\upsilon }_{\tau }^{\overline{rel}}$$

(25)

where $F_d^{\tau }$ is the tangential damping force and ${\upsilon }_{\tau }^{\overline{rel}}$ is the tangential component of the relative velocity.

For discrete element simulations, rolling friction between particles is important and needs to be addressed by applying a moment to the contact surface.

$${\tau }_i=-{\mu }_r{F}_s^n{R}_i{\omega }_i$$

(26)

where ${\mu }_r$ is the coefficient of rolling friction, $R_i$ is the distance from the point of contact to the center of mass, and ${\omega }_i$ is the unit angular velocity vector of the object at the point of contact.

3.2. Particle Parameters

This study temporarily uses spherical particles to simplify calculations. However, a note is necessary. The irregular shapes of real sand particles affect inter-particle interlocking. According to a study by Yang et al. [3], the multi-sphere cluster particle model can reduce the prediction error of sand shear strength by 15%. Thus, in subsequent studies, super-quadric particles will be used to reproduce the real shape of sand particles so as to further improve the model accuracy. In order to reduce the number of generated particles to shorten the simulation time, the radius of the particles is enlarged appropriately and taken as 2 mm.

The intrinsic parameters of granular materials include material density, Young’s modulus, and Poisson’s ratio. The above parameters are taken empirically and the intrinsic parameters of sandy soil are shown in

Table 1. Sandy soil intrinsic parameter values.

Particle Radius (mm)	Material Density (kg/m3)	Young’s Modulus (Pa)	Poisson’s Ratio
2	2650	2 × 108	0.21

.

The contact parameters of granular materials in the Hertz–Mindlin–Deresiewicz no-slip contact model mainly include the coefficient of recovery, static friction coefficient, and rolling friction coefficient. Due to the small radius of the sand sample particles, the material properties are strongly influenced by these parameters. These parameters usually have no empirical values to refer to and need to be calibrated using parametric inversion methods.

In order to make the density of simulated sand samples consistent with the density of real sand samples, the number of generated particles was calculated according to the following formula:

$$N={\displaystyle \frac{\pi r^{2}h\rho }{m}}$$

(27)

where $N$ is the number of particles generated, $r$ is the radius of the specimen, $h$ is the height of the specimen, $\rho$ is the dry density, and $m$ is the mass of the particles.

3.3. Contact Parameter Calibration

When EDEM simulation operations are performed, there exist some fine-scale parameters that have a very strong influence on the experimental results. The main methods to determine these parameters are the precision chemical experiment method, elasticity theory calculation method, and parameter calibration method. The first two methods are not widely used because they require complex equipment or the theory is not rigorous enough, so the commonly used method is the parameter calibration method. The core concept of the parameter calibration method is to continuously adjust the initial parameters until the simulation outcomes are consistent with the experimental results; after this adjustment, the parameters can be applied in subsequent simulations. In this paper, the idea of using the parameter calibration method is to compare the simulation test of the stacking angle with the actual test by continuously adjusting the three initial parameters: the coefficient of restitution, coefficient of static friction and coefficient of rolling friction. Then, the particle contact parameters are further adjusted to make the numerical simulation calculations consistent with the test results.

3.3.1. Angle of Repose Test

Commonly used methods include the funnel method, cylinder method, split plate method, and so on. In the present study, the cylinder method is employed to conduct the angle of repose test, and the specific experimental procedure is detailed as follows: (1) A cylinder with an inner diameter of 50 mm and a height of 250 mm is filled with sand up to a height of 150 mm. (2) Slowly lift up the cylinder, and the sand and soil will naturally accumulate and form a slope angle. The foot of the slope is the accumulation angle. (3) Take a vertical picture of the foot of the slope after it has stabilized. (4) Import the picture into image-processing software to measure the angle of accumulation. Figure 6 shows the process of the stacking angle test.

The stacking test was repeated four times, and the average value was taken to be recorded as the angle of repose of the test sand grains, which turned out to be 31.579°.

In the stacking angle simulation test, a cylinder with an inner diameter of 50 mm and a height of 250 mm is first modeled. Then the particles are generated inside the circular pass with reference to the intrinsic parameters of sand particles. After the particle generation is finished, an upward velocity of 0.5 m/s is applied to the cylinder, and the particles are automatically stacked after losing the constraint.

3.3.2. Path of Steepest Ascent Method

The intrinsic parameters and simulation scales of the sandy soil were entered into the EDEM. The range values and recommended values of the simulation parameters were obtained from the GEMM database as: the sand-sand coefficient of restitution, 0.15~0.75, 0.6; the sand–sand coefficient of static friction, 0.44~2.5, 0.85; and the sand–sand coefficient of rolling friction, 0.05~0.3, 0.1. The parameters were further narrowed down by the steepest climb test.

Table 2. Steepest climb test design options and results.

No.	Parameter A	Parameter B	Parameter C	Repose Angle/(°)	Relative Error/%
1	0.15	0.44	0.05	9.41	70.2
2	0.27	0.852	0.1	12.05	61.84
3	0.39	1.264	0.15	17.45	44.74
4	0.51	1.676	0.2	23.17	26.63
5	0.63	2.088	0.25	29.85	5.5
6	0.75	2.5	0.3	34.1	7.98

shows the design scheme and results of the steepest climb test, in which parameters A, B, and C represent the coefficient of restitution, coefficient of static friction and coefficient of rolling friction, respectively. The results show that the angle of repose derived from the simulation test increases gradually with the increase in the values of A, B, and C. The relative error between the simulation and the actual measured angle of repose shows a tendency of decreasing and then increasing. The relative error in the angle of repose reaches its minimum at test level No. 5. It can be seen that the optimum interval of the test variables is near the test level No. 5. Therefore, the No. 5 level was selected as the center point and set as the medium level. The No. 4 and No. 6 levels were selected as the low and high levels, respectively, for the Box–Behnkens test.

3.3.3. Box–Behnkens Test

The Box–Behnkens experimental design scheme and results are shown in

Table 3. Box–Behnken experimental design scheme and results.

No.	Parameter A	Parameter B	Parameter C	Repose Angle/(°)
1	−1 (0.51)	−1 (1.676)	0 (0.25)	26.81
2	1 (0.75)	−1	0	25.41
3	−1	1 (2.5)	0	22.43
4	1	1	0	26.84
5	−1	0 (2.088)	−1 (0.2)	21.82
6	1	0	−1	20.38
7	−1	0	1 (0.3)	27.02
8	1	0	1	34.17
9	0 (0.63)	−1	−1	20.46
10	0	1	−1	20.23
11	0	−1	1	31.15
12	0	1	1	31.78
13	0	0	0	28.37
14	0	0	0	28.63
15	0	0	0	29.13
16	0	0	0	28.74
17	0	0	0	29.77

. Based on the experimental results, a second-order regression model for the angle of repose of sand particles with three independent variables was developed using Design-Expert 13 software. The quadratic polynomial equation is

$$\begin{array}{l}\theta =28.93+1.09A-0.3188B+5.15C+1.45AB+2.15AC\\ +0.215BC-1.81A^2-1.75B^2-1.27C^2\end{array}$$

(28)

The analysis of variance (ANOVA) was performed on the experimental model in Table 3, and the results were obtained as shown in

Table 4. Box–Behnken test quadratic polynomial model ANOVA.

Error Source	Mean Square	Freedom	Quadratic Sum	p Value
Model	31.91	9	287.15	<0.0001
A	9.50	1	9.50	0.0075
B	0.8128	1	0.8128	0.3132
C	212.49	1	212.49	<0.0001
AB	8.44	1	8.44	0.0100
AC	18.45	1	18.45	0.0013
BC	0.1849	1	0.1849	0.6203
A2	13.74	1	13.74	0.0029
B2	12.88	1	12.88	0.0035
C2	6.83	1	6.83	0.0161
Residual error	0.6885	7	4.812
Lack of fit	1.21	3	3.63	0.1036
Pure error	0.2963	4	1.19
Total		16	291.97
R2 = 0.9835; R2adj = 09623; CV = 3.11%; Adeq Precision = 22.9452

, which showed that: the p-value of the model of the equation was less than 0.0001, which showed extreme significance (p < 0.01), indicating that the model showed extreme significance between the dependent variable (angle of repose) and all the independent variables; the misfitting term was not significant (p = 0.1036 > 0.05), which showed that, although the analysis of the results was performed by using this model, the simulation of this equation was credible, although there was a certain probability of error; the coefficient of determination R² = 0.9835 and the corrected coefficient of determination R²_adj = 0.9623, both of which are close to 1, indicate that the fitted equation is meaningful and its reliability is high; the coefficient of variation CV = 3.11%, and the precision Adeq Precision reaches 22.9452, indicating that the model has a good degree of credibility and precision.

3.3.4. Optimal Parameter Validation

The optimization function of Design-Expert was used to optimize the regression model with the measured angle of repose of 31.579°. The resulting optimized solutions are: the coefficient of recovery, 0.638; the coefficient of static friction, 1.89; and the coefficient of rolling friction, 0.283.

The comparison between the simulated and measured results of the angle of repose under the optimized solution is shown in Figure 7. There is no obvious difference between the shapes of the sand pile obtained from the simulation and the measured test, and the angles of repose obtained from four repetitions of the simulation test are 32.6°, 30.9°, 32.9°, and 31.4°, with an average value of 31.95° and a standard deviation of 0.8261°. The relative error in the measured angle of repose was 1.2%. The results show that it is effective to optimize the sand simulation parameters by the response surface method.

3.4. Road Surface Simulation Model

The pavement model for the simulation of tire traction characteristics was built based on the contact parameters obtained from previous stacking angle experiments and simulations. The parameters of the sandy pavement simulation model are shown in

Table 5. Pavement Simulation Model Parameters.

Intrinsic Parameter				Contact Parameter
Particle Radius (mm)	Material Density (kg/m3)	Young’s Modulus (Pa)	Poisson’s Ratio	Coefficient of Restitution	Coefficient of Static Friction	Coefficient of Kinetic Friction
2	2650	2 × 108	0.21	0.638	1.89	0.283

.

The pavement model was constructed using the parameters from the table, with dimensions of 5 m in length, 0.4 m in width, and 0.15 m in thickness. The granular box material is steel, and the material parameters are shown in

Table 6. Material parameters of pellet box and tire.

Material Type	Material Density (kg/m3)	Young’s Modulus (Pa)	Poisson’s Ratio
Steel	7800	7 × 1010	0.3
Caoutchouc	1800	2 × 106	0.49

. The coefficient of restitution, static friction factor, and rolling friction factor of the sand and steel are 0.6, 0.5, and 0.05 [23], respectively. The generated pavement is shown in Figure 8. The discrete element sand particles reach a steady state under self-weight.

The stabilization of the sand assembly under self-weight compaction was determined by monitoring the temporal evolution of its total kinetic energy. As shown in Figure 9, the total kinetic energy rises sharply initially, then decreases gradually, and finally levels off. After 1 s, the kinetic energy stabilizes and approaches zero, indicating that the sand assembly has reached an equilibrium state and the self-weight compaction process is complete.

4. Tire Modeling

The subject of this paper is a 205/55 R16 radial tire on a soft sandy road surface. The deformation of the tire is negligible compared to the sinking of the road surface because the road surface is soft and the tire pressure is usually large. Therefore, this paper assumes that the test wheel is a rigid uniform tire model.

The process of building a 205/55/R16 radial tire model is as follows: create a half-2D symmetric cross-section tire model in SOLIDWORKS as shown in Figure 10a; rotate the half-2D symmetric cross-section model by 360° around the rolling axis to obtain a half-3D symmetric cross-section model of the tire as shown in Figure 10b; and map the symmetry plane to a complete full-3D tire model, as shown in Figure 10c.

The model created is shown in Figure 11.

The tire material parameters are shown in Table 6.

5. Numerical Simulation

In order to verify the feasibility of the DEM-based method in studying the wheel–sand contact mechanics, a simulation model of tire driving in sand was developed based on the indoor soil-tank experiment [13], as shown in Figure 12.

5.1. Simulation Setup

The sand model is fixed with displacement constraints in the x, y, and z directions. The tire is only allowed to translate in the x-direction, sink in the z-direction, and rotate around the y-axis. The simulation time step is 20% of the total simulation time. The simulation process is as follows: (1) the discrete element cell is balanced under self-weight; (2) the tire is balanced under self-weight and external load (1000 N); and (3) the tire center of mass is loaded with a horizontal velocity of 5 m/s and the corresponding angular velocity to analyze the traveling behavior of the tire under different slip rates.

5.2. Simulation Variable Design

The simulation setup contains two types of tires. They are glossy tires and treaded tires. First, we applied a horizontal velocity to the tire. Then, we applied different angular velocities on this basis. This forms operating conditions with a slip ratio from 0% to 60%. Considering the randomness of sand particle movement in discrete element simulations, each set of tests is repeated three times. The average value is obtained through multiple repetitions to reduce random errors.

5.3. Data Acquisition

The horizontal force between the tire and sand particles is collected in real time via the built-in force monitoring module of EDEM 2021.2 software. The sampling frequency is once every 0.01 s. Data from the transition phase is excluded, and the arithmetic mean of the stable phase is taken as the traction result for this set of simulations.

Sinkage is obtained by monitoring the displacement change in the tire’s center of mass along the z-axis. The z-coordinate of the center of mass when the tire just contacts the sand bed is taken as the reference. The difference in the z-coordinate during the stable phase is the final sinkage.

6. Simulation Results and Discussion

6.1. Simulation Results and Analysis of Smooth Tires

Figure 13 shows the cloud diagram of the motion trajectory of the glossy tire at a 30% slip rate. The simulation results show clear rut formation under the tire. The soil particles are compressed and sink, while the material on both sides is extruded sideways and upwards, forming bulges.

The travel speed was varied to observe tire behavior at different slip ratios. The resulting relationship between drawbar pull and slip ratio was then compared with experimental data from the literature [20], as shown in Figure 14.

The traction force in Figure 14 is taken as the mean value of the tire after stabilizing the movement in the sand. From the figure, it can be observed that there is a discrepancy between the simulation results and the experimental values, and this difference arises from variations in tire models and applied loads. Nevertheless, the overall trend remains consistent: when the slip rate is below 30%, the traction force rises noticeably as the slip rate increases; once the slip rate exceeds 30%, the traction force tends to stabilize.

Figure 15a shows the vector diagram of sand flow of tire at a 30% slip rate condition. It is observable that the sand flow can be categorized into two distinct zones. The front zone flows in a clockwise direction and the rear zone flows in an anti-clockwise direction. This phenomenon aligns with the experimental results [11] presented in Figure 15b.

6.2. Comparison of Simulation Results of Different Tread Patterns

To further verify the validity of the Discrete Element Method in simulating and evaluating wheel–sand contact mechanics on the basis of the original glossy tire, the sand-driving performance of a uniform specification tire with a grooved tread pattern was analyzed. The patterned tire is shown in Figure 16. Figure 17 shows the simulation results of the motion trajectory of the grooved tread tire at a 30% slip rate. Figure 18 shows the trend of road surface normal reaction force with time for two tread structure tires at the same slip rate. It shows that the vertical reaction force on both tire types exhibits significant initial fluctuations before stabilizing around 1000 N during travel on sand.

Figure 19 shows the variation in traction force with time for both tires. As can be seen from the figure, when the tire contacts the sand movement, the traction force first shows large fluctuations, then decreases rapidly and enters a relatively stable state. The traction force of the treaded tire is about 116 N, which is larger than that of the glossy tire, 73 N. The grooves on the tire tread expand the contact area between the tire and the sand, leading to a rise in the tire’s driving force, and this observation aligns with established engineering principles.

Figure 20 shows the variation in sinkage with time for both tires. As shown in the figure, the tire sinkage increases rapidly upon initial contact with the sand and soon stabilizes. The treaded tire exhibits a final sinkage of approximately 21 mm, which is greater than that of the smooth tire (18 mm). This difference is attributed to the tangential action of the tread patterns, which displace sand particles laterally and result in greater sinkage.

These results demonstrate the effectiveness of the discrete element method for evaluating the mechanics of wheel–sand contact.

7. Discussion

7.1. Result Interpretation and Hypothesis Verification

This study’s core hypothesis is that a finely calibrated DEM model can reliably simulate the complex mechanical interactions between wheels and sand. This calibration was validated using key quantitative metrics. The optimized sand contact parameters are: the coefficient of restitution (0.638), the coefficient of static friction (1.89), and the coefficient of rolling friction (0.283). With these parameters, the average simulated angle of repose was 31.95° (standard deviation: 0.8261°), showing only a 1.2% relative error compared to the measured angle of repose (31.579°) from physical tests. Additionally, the quadratic regression model for the angle of repose showed a coefficient of determination $R^2=0.9835$ and adjusted $R_{adj}^2=0.9623$. This confirms the model’s statistical reliability. The simulation results strongly support this assumption. Firstly, the trend of the relationship between traction force and slip ratio for the slick tire was consistent with experimental results, indicating that the model captures the essential characteristic of how shear strength in sand changes with slip. Secondly, phenomena observed in the simulation, including soil compaction at the leading edge, uplift on both sides, and the formation of two counter-rotating flow zones, closely align with classically observed sand movement patterns in experiments. This validates, from a particle kinematics perspective, the physical processes behind rut formation and traction generation at the macroscopic level. For the treaded tire, its higher traction and greater sinkage visually confirm the conventional theory that treads improve traction by penetrating the soil and increasing the shear area. This study successfully visualizes and quantifies this process at the particle scale through DEM simulation.

7.2. Significance of Studying

The significance of this study extends beyond simple tire performance comparison. It establishes a powerful numerical platform for investigating interactions between vehicles and highly deformable terrain. Unlike traditional physical soil bin tests, which are costly and difficult to reproduce, the DEM simulation approach allows convenient adjustment of numerous variables such as particle shape, size distribution, soil moisture, complex tread patterns, and multi-wheel interactions. This capability enables systematic investigation of how various factors affect vehicle mobility. Compared with wheeled mobile robots and forestry machinery, EVs have moderate load and torque characteristics that require tires to balance between traction and sinkage, and this study provides a solution to this problem. These advantages are particularly valuable for electric off-road vehicles and autonomous vehicles, which are undergoing rapid development and require accurate dynamics models to achieve the optimal torque distribution and trajectory planning on unfamiliar terrain. The methodology validated in this study can generate training or validation data for such high-fidelity models.

7.3. Limitations and Future Research Directions

This study achieved positive results, but some limitations remain. These point to clear directions for future work. First, the model used spherical particles. The current spherical particle assumption ignores the effect of sand particle shape on dilatancy. This leads to a smaller particle embedding depth of tread patterns. A correction via shape-parameterized modeling will be needed in subsequent work. It would also help to study how shape affects macro-scale mechanical behavior. Second, this research only used dry sand conditions. For cohesive soils, a cohesion module must be added. DEM-FEM coupling may also be needed to handle plastic flow. For mixed terrains, the model can be adapted by adjusting particle gradation. However, this often leads to underestimated terrain stiffness. Gradation effects are neglected here. For wet sand, a capillary force model can be introduced. But the model cannot be adapted to frozen soil. It lacks thermal mechanical coupling. For loose rocky terrains, super-quadric particles are required to simulate irregular rocks. This, however, brings the issue of increased computational costs. In the future, it is necessary to establish sand models with different moisture contents, to introduce modified contact models, and to verify the impact of the moisture content on the traction efficiency of electric vehicles as well. Finally, the findings should be built into full vehicle dynamics models. They should also be tested on real electric off-road vehicles. The subsequent work will focus on establishing a vehicle testing platform centered on this 205/55 R16 tire. This step is the key to applying micro-scale mechanical insights to real engineering applications.

8. Conclusions

This study uses the Discrete Element Method to simulate the interaction between wheels and sand with high precision. It can reliably reproduce the key aspects of wheel–sand interaction. A key contribution is the development of a systematic parameter calibration method that combines the steepest ascent test and the Box–Behnken design. This method accurately determines microscopic sand contact parameters, providing a reliable basis for the simulation model. Using this model, the study quantitatively measures how tread patterns improve traction on sand. It shows that treads enhance performance by increasing the contact area and changing how the sand flows.

The results confirm that the DEM-based approach effectively reproduces important phenomena, such as tire sinkage, sand movement, and the relationship between traction and slip. These simulations, while exhibiting numerical discrepancies from the physical experiments, demonstrate excellent consistency with the core experimental trends, including the increasing-then-stabilizing pattern of traction with slip ratio and the characteristic dual-zone sand flow. This supports the use of the Discrete Element Method as an effective and reliable way to study vehicle performance on soft terrain.

The refined simulation framework developed in this work and the specific conclusions obtained can be directly applied to the structural design and optimization of tires for special vehicles, such as electric vehicles, lunar rovers, and construction machinery. Additionally, they provide essential input parameters and mechanistic insights for developing advanced vehicle-terrain dynamics models and traction control strategies. This study offers clear practical value for engineering applications and theoretical significance.