Mechanism Analysis of Soil Disturbance in Sodic Saline–Alkali Soil Tillage Based on Mathematical Modeling and Discrete Element Simulation

Abstract

To elucidate the mechanism by which soil disturbance affects tillage performance during subsoiling remediation of northeastern primary sodic saline–alkali soil, this study established a mathematical prediction model linking subsoiler configuration parameters with draft force and soil porosity based on the soil dynamic equation and the fourth strength theory. Discrete element simulation and field experiments demonstrated the model’s accuracy in predicting draft force and soil looseness (error < 5%). Among three configuration patterns evaluated, the “W”-type arrangement was selected for further simulation testing and predictive analysis through parameter adjustment. The simulation results aligned with the prediction results. Particle flow analysis revealed a quadratic relationship between subsoiler spacing variation, draft force, and soil looseness. At the particle scale, soil particle movement patterns were found to govern macroscopic effects, where soil clogging and repeated disturbances emerged as primary drivers of nonlinear variations in draft force and soil porosity. Finally, field experiments and simulations were performed using the parameter combinations predicted by the mathematical model, confirming the accuracy of these parameters. Through a tripartite validation approach combining mathematical modeling, DEM simulation, and field trials, this study systematically elucidates the complete mechanism whereby subsoiler arrangement parameters influence the tillage performance of sodic saline–alkali soil via soil–tool interactions, providing theoretical foundations for optimizing subsoiling equipment design and reducing energy consumption in saline–alkali land cultivation.

1. Introduction

Saline–alkali soil refers to soil with excessive accumulation of soluble salts that inhibits or damages crop growth. China’s northeastern region serves as the main zone for sodic saline–alkali soil, which has a hard, compact texture and poor permeability and tends to form dense, water-resistant crusts after rainfall or irrigation, creating unfavorable conditions for crop growth [1,2,3]. At present, the improvement technology for saline–alkali land is mostly focused on traditional methods such as chemical leaching and gypsum improvement, but there are problems such as high cost and secondary pollution risk [4]. Subsoiling alleviates deep soil compaction by improving compaction status and enhancing soil aeration and water permeability, stimulating root development and boosting soil productivity [5,6]. This method has been demonstrated as a practical approach for saline–alkali soil reclamation [7].

The draft force during tillage operations mainly results from the dynamic interaction between soil properties and the subsoiling machine’s movement [8]. Analyzing these soil–tool interaction forces can optimize subsoiler geometric parameters or arrangements, reducing draft force and enhancing working efficiency [9,10,11,12]. Researchers employed analytical and experimental methods to investigate the interaction mechanisms between soil and subsoilers. They developed mathematical models characterizing soil–tool interaction to assess the tillage performance of various subsoiler designs across different soil types [13,14,15]. A model’s accuracy is notably affected by soil mechanical properties, implemented geometric parameters, and operational parameters [16,17].

Soil properties exhibit complex and dynamic variations, with soil dynamic effects occurring at the soil–subsoiler interface. Both field experiments and soil bin tests face limitations in controlling test conditions, making precise characterization of soil–tool interactions challenging [18,19]. The numerical method mainly simulates and analyzes the working process of a machine through the discrete element method and the finite element method. The finite element method is a numerical method to simulate the mechanical behavior of continuous media. It is mainly used to simulate the deformation and stress distribution of a machine, but it cannot accurately simulate discontinuous behavior such as soil fracture and particle separation [20]. The discrete element method (DEM) is a specialized numerical simulation approach for analyzing discrete media. It effectively captures the discrete nature of soil particles, providing more realistic simulations of particle displacement and force distributions during subsoiling operations while enabling visualization [21,22,23]. Developing accurate soil models is fundamental for advancing tillage dynamics and kinematics research. Researchers have constructed simulation models representing soil particles of diverse shapes and sizes, enabling an accurate representation of soil structure. In selecting contact models, soils with high moisture content or clay loam require consideration of the adhesion effect between soil and implements [24,25]. Regarding the wet cohesive characteristics of deep sodic saline–alkali soil, the Hertz–Mindlin (with JKR) model incorporating surface energy provides a more precise simulation of soil–subsoiler interactions during deep tillage operations.

Accurate draft force prediction and soil disturbance pattern analysis during subsoiling constitute fundamental scientific challenges for optimizing subsoiler design and minimizing energy consumption. This research combines mathematical modeling, DEM simulations, and field experiments to develop a predictive model based on soil micro-element–tillage performance. Through discrete element method simulation of the instantaneous velocity field in saline–alkali soil particles under subsoiler action, soil particles’ displacement and flow characteristics are comprehensively examined at macroscopic and microscopic scales. Investigating the soil disturbance mechanism during subsoiling and its influence on tillage performance helps overcome the limitations of conventional statistical models in mechanistic interpretation and predictive accuracy, thereby offering a theoretical basis and data support for designing sodic saline–alkali soil implements and reducing tillage resistance.

2. Materials and Methods

2.1. Structural Parameters of Subsoiler

The subsoiler comprises a frame, a hitching mechanism (Figure 1a), the broken-line subsoiler, and auxiliary subsoiler subsoiler (Figure 1b). Prior studies developed a mathematical analytical model for the broken-line subsoiler and validated its tillage performance during subsoiling operations [26]. The subsoiler geometric parameters are presented in

Table 1. Structural parameters of the subsoiler.

Parameter	Value
Broken-line subsoiler penetration angle, θB (°)	11
Broken-line subsoiler fillet radius, R (cm)	41
Broken-line subsoiler straight section length, hB (cm)	30
Broken-line subsoiler thickness, WB (cm)	3
Broken-line subsoiler shank width, tB (cm)	18.5
Auxiliary subsoiler penetration angle, θA (°)	30
Auxiliary subsoiler thickness, WA (cm)	1.8
Auxiliary subsoiler shank width, tA (cm)	16
Auxiliary subsoiler tip length, LA (cm)	5
Auxiliary subsoiler tip width, JA (cm)	6
Auxiliary subsoiler wing width, IA (cm)	12

. A Cartesian coordinate system (Figure 1c) was established by defining the three-dimensional coordinates of the subsoiler, with the tip of the right-side broken-line subsoiler in the forward direction serving as the origin. The machine’s forward direction was designated as the positive x-axis, while the vertically upward direction was set as the positive z-axis. To implement various arrangement patterns of the subsoiler, the longitudinal spacing between the middle auxiliary subsoiler and the broken-line subsoiler was defined as X₁, whereas the spacing between the side auxiliary subsoiler and broken-line subsoiler was set as X₂ (with equal absolute values for X₁ and X₂). The lateral spacing between the auxiliary and broken-line subsoiler was designated as Y, and the auxiliary subsoiler penetration depth was Z. During the subsoiler tine’s operation, a soil wedge formed ahead of the implementation. We define the soil wedge as a geometric volume generated by rotating a plane about an axis [27], and we define L_B as the maximum distance between the broken-line subsoiler and the soil wedge and L_A as the corresponding distance for the auxiliary subsoiler (Figure 1d), as expressed in Equations (1) and (2).

In the diagram, L₁ is the horizontal distance from the bottom of the soil wedge to the handle of the subsoiling shovel in the rotating plane, in cm; L₂ is the horizontal distance from the bottom of the soil wedge to the front end of the soil wedge in the rotating plane, in cm; ε is the angle between the upper surface of the soil wedge and the bottom of the soil wedge, in °; w is the thickness of subsoiler, in cm; and L_f is the length of the shovel tip of the auxiliary subsoiler, in cm.

$$L_B=L_{1B}+L_{2B}+{\displaystyle \frac{w_B}{2\sin \epsilon }}$$

(1)

$$L_A=L_{1A}+L_{2A}+{\displaystyle \frac{I_A}{2\sin \epsilon }}$$

(2)

2.2. Development of the Mathematical Model

2.2.1. Construction of Soil Micro-Element Kinetic Equation

During the operation of the subsoiler, variations in the longitudinal spacing, lateral spacing, and auxiliary subsoiler penetration depth of the subsoiler may lead to soil overlap or intersection phenomena in the collaborative operation area between auxiliary subsoilers and the broken-line subsoiler, thereby influencing the draft force and soil looseness of the implement. To characterize the stress state of soil micro-elements at different positions during subsoiling, this study performed a force analysis on soil micro-elements within the disturbance zone of the broken-line subsoiler and auxiliary subsoiler based on the dynamic equation.

Without Interaction

When the subsoiler spacing satisfies Equation (3), the soil disturbance width area of the broken-line subsoiler and that of the auxiliary subsoiler do not overlap. The force model of soil elements is established separately for the soil in the S₁ region (in front of the broken-line subsoiler) and the soil elements in the S₂ region (in front of the auxiliary subsoiler).

$$\cos \epsilon L_B\le Y-\sin \epsilon L_A$$

(3)

When a soil element is only acted upon by the broken-line subsoiler, the positional relationship between the soil element and the broken-line subsoiler is shown in the S₁ region of Figure 2, and the triaxial stress formulas for the soil element in Region 1 are shown in Equations (4)–(6). When a soil element is only acted upon by the auxiliary subsoiler, the positional relationship between the soil element and the auxiliary subsoiler is shown in the S₂ region of Figure 2, and the formulas for the three stress components of the soil element are shown in Equations (7)–(9).

$$d{F}_{x{S}_1}=d_A\tan \epsilon \cdot \left({\displaystyle \frac{\rho \left(g+u\right){{\displaystyle \mathbf{\int }\mathbf{\int }}}_x^{L_B}{S}_{1\left(x,y,z\right)}dxdy}{\int S_{1\left(x,y,z\right)}dy}}\right)+d_{A}c$$

(4)

$$d{F}_{y{S}_1}=d_A\cdot \left({\displaystyle \frac{\rho \left(g+u\right)\left(\frac{Y}{\sqrt{X^2+Y^2}}\right){{\displaystyle \mathbf{\int }\mathbf{\int }}}_{-\sin \epsilon \left(L_B\right)}^y{S}_{1\left(x,y,z\right)}dxdy}{\int S_{1\left(x,y,z\right)}dx}}\right)$$

(5)

$$d{F}_{z{S}_1}={\displaystyle \frac{180}{\epsilon }}\cdot {\displaystyle \frac{\rho g\int \int S_{1\left(x,y,z\right)}-zdxdy}{{(L_B-\frac{z}{\tan \epsilon })}^2-{\left(\frac{w}{2\sin \epsilon }\right)}^2}}$$

(6)

$$d{F}_{x{S}_2}=d_A\tan \epsilon \cdot \left({\displaystyle \frac{\rho \left(g+u\right){{\displaystyle \mathbf{\int }\mathbf{\int }}}_x^{L_A}{S}_{2\left(x,y,z\right)}dxdy}{\int S_{2\left(x,y,z\right)}dy}}\right)+d_{A}c$$

(7)

$$d{F}_{y{S}_2}=d_A\cdot \left(\rho g{\displaystyle \frac{{{\displaystyle \mathbf{\int }\mathbf{\int }}}_{Y-\sin \epsilon \left(L_A\right)}^y\left(\frac{\left(y-Y\right)}{\sqrt{{\left(x-X_2\right)}^2+{\left(y-Y\right)}^2}}\right).S_{2\left(x,y,z\right)}dxdy}{\int S_{2\left(x,y,z\right)}dx}}\right)$$

(8)

$$d{F}_{z{S}_2}={\displaystyle \frac{180}{\epsilon }}\cdot {\displaystyle \frac{\rho g\int \int S_{2\left(x,y,z\right)}-zdxdy}{{\left(L_A\right)}^2-{\left(\frac{\tan \epsilon {\mathsf{\omega }}_1}{2\sin \epsilon }\right)}^2}}$$

(9)

where ρ is the soil density in g/cm³, g is the gravitational acceleration in m/s², d_A is the lateral surface area of the soil element, and u is the velocity of the wedge-shaped soil block relative to the control volume bottom in m/s.

Congestion Effect

Congestion will occur when the soil at the same position is simultaneously disturbed by the broken-line subsoiler and the auxiliary subsoiler. The zone where soil congestion occurs is defined as S₃ (Figure 3). The force acting on the soil element in the x-direction is shown in Equation (10), while the force acting on the soil element in the y-direction is divided into two components:${dF}_{{ys}_3}$ and ${dF}_{{-ys}_3}$, with the y-axis as illustrated in Equations (11) and (12). The force acting in the z-direction is presented in Equation (13).

$$d{F}_{x{S}_3}=d_A\tan \epsilon \cdot {\displaystyle \frac{\rho \left(g+u\right)\left({{\displaystyle \mathbf{\int }\mathbf{\int }}}_X^{X+L_A}{S}_{2\left(x,y,z\right)}dxdy+{{\displaystyle \mathbf{\int }\mathbf{\int }}}_x^{L_B}{S}_{1\left(x,y,z\right)}dxdy\right)}{\int S_{1\left(x,y,z\right)}dx+\int S_{2\left(x,y,z\right)}dx}}$$

(10)

$$d{F}_{y{S}_3}=\mu d_A\cdot \left({\displaystyle \frac{\rho g{{\displaystyle \mathbf{\int }\mathbf{\int }}}_y^{\sin \epsilon \left(L_B\right)}\left(\frac{y}{\sqrt{x^2+y^2}}\right).S_{1\left(x,y,z\right)}dxdy}{\int S_{3\left(x,y\right)}dx}}\right)$$

(11)

$$d{F}_{-y{S}_3}=\mu d_A\cdot \left(\rho g{\displaystyle \frac{{{\displaystyle \mathbf{\int }\mathbf{\int }}}_{Y-\sin \epsilon \left(L_A\right)}^y\left(\frac{\left(y-Y\right)}{\sqrt{{\left(x-X_2\right)}^2+{\left(y-Y\right)}^2}}\right).S_{2\left(x,y,z\right)}dxdy}{\int S_{2\left(x,y,z\right)}dx}}\right)$$

(12)

$$d{F}_{z{S}_3}={\displaystyle \frac{\rho g\int \int S_{3\left(x,y,z\right)}-zdx.\int S_{3\left(x,y,z\right)}dy}{\int S_{3\left(x,y,z\right)}dz}}$$

(13)

Repetitive Perturbation Effect

Soil repeated disturbance should be calculated under two scenarios. When the soil element is first acted upon by the broken-line subsoiler and then by the auxiliary subsoiler, the zone of repeated disturbance is defined as S₄, whose position is shown in Figure 4a. The three stress components of the soil element are provided in Equations (14)–(16). When the soil element is first acted upon by the auxiliary subsoiler and then by the broken-line subsoiler, the position of the soil element is shown in Figure 4b, and Equations (17)–(19) give the three stress components of the soil element, where $\dot{u}$ is the acceleration of the soil element in the x-direction and c is the soil cohesion.

$$d{F}_{x4}=d_A\tan \epsilon \cdot \left({\displaystyle \frac{\rho \left(g+\dot{u}\right){{\displaystyle \mathbf{\int }\mathbf{\int }}}_x^{X+L_A}{S}_{4\left(x,y,z\right)}dxdy}{\int S_{4\left(x,y,z\right)}dy}}\right)$$

(14)

$$d{F}_{y{S}_4}=d_A\rho g.{\displaystyle \frac{{{\displaystyle \mathbf{\int }\mathbf{\int }}}_{Y-\sin \epsilon \left(L_A\right)}^y\frac{\left(y-Y\right)}{\sqrt{{\left(x-X_2\right)}^2+{\left(y-Y\right)}^2}}.S_{4\left(x,y,z\right)}dxdy}{\int S_{4\left(x,y,z\right)}dx}}$$

(15)

$$d{F}_{z{S}_4}=d_A{\displaystyle \frac{180}{\epsilon }}\cdot {\displaystyle \frac{\rho g\int \int S_{4\left(x,y,z\right)}-zdxdy}{\int S_{4\left(x,y,z\right)}dz}}-c{d}_A$$

(16)

$$d{F}_{x{S}_5}=d_A\tan \epsilon \cdot \left({\displaystyle \frac{\rho \left(g+\dot{u}\right){{\displaystyle \mathbf{\int }\mathbf{\int }}}_x^{L_B}{S}_{5\left(x,y,z\right)}dxdy}{\int S_{5\left(x,y,z\right)}dy}}\right)$$

(17)

$$d{F}_{yS5}=d_A\rho g.{\displaystyle \frac{{{\displaystyle \mathbf{\int }\mathbf{\int }}}_y^{\sin \epsilon \left(L_B\right)}\frac{y}{\sqrt{x^2+y^2}}.S_{5\left(x,y,z\right)}dxdy}{\int S_{5\left(x,y,z\right)}dx}}-c{d}_A$$

(18)

$$d{F}_{zS5}=d_A{\displaystyle \frac{180}{\epsilon }}.{\displaystyle \frac{\rho g\int \int S_{5\left(x,y,z\right)}-zdxdy}{\int S_{5\left(x,y,z\right)}dz}}-c{d}_A$$

(19)

2.2.2. Establishment of Prediction Model of Subsoiler Arrangement–Tillage Performance

Establishment of Draft Force Prediction Model

The total draft force of the subsoiler is the combined force of the auxiliary subsoiler and broken-line subsoiler. The subsoiling resistance comprises cutting force, frictional force, and inertial force. Based on the established Cartesian coordinate system, we separately analyzed the auxiliary subsoiler’s and broken-line subsoiler’s working force at different positions. The subsoiling resistance, F_S1, of the broken-line subsoiler consists of the cutting force, F_c1; the frictional force, F_f1; and the inertial force, F_i1, with F_f1 being affected by the auxiliary longitudinal spacing of subsoilers X₁ and X₂. The left-side force acting on F_f1L and the right-side force acting on F_f1R are calculated independently, as shown in Equations (20)–(23).

$$F_{c1}={{\displaystyle \int }}_{{\displaystyle \frac{w}{2sin\epsilon }}}^{L_B}d{F}_{x{S}_1}+{{\displaystyle \int }}_{\min \left(L_B,Y+sin\epsilon \left(L_1+L_2\right)\right)}^{L_B}d{F}_{x{S}_3}+{{\displaystyle \int }}_{\min \left(L_B,Y+{\displaystyle \frac{w}{2\sin \epsilon }}\right)}^{L_B}d{F}_{x{S}_4}$$

(20)

$$\begin{array}{l}{F}_{f1L}=\mu {{\displaystyle \int }}_{-sin\epsilon \left(L_B\right)}^0{A}_{S1}d{F}_{y{S}_1}+\mu {{\displaystyle \int }}_{\min \left\{{\displaystyle \frac{1}{2}}\left[sin\epsilon \left(L_1+L_2\right)+\left(Y-sin\epsilon \left(L_A\right)\right)\right],sin\epsilon \left(X_1+{\displaystyle \frac{w}{2sin\epsilon }}\right)\right\}}^{sin\epsilon \left(L_B\right)}{A}_{S1}d{F}_{y{S}_3}\\ +\mu {{\displaystyle \int }}_{\min \left\{{\displaystyle \frac{1}{2}}\left[\sin \epsilon \left(L_1+L_2\right)+\left(Y-\sin \epsilon \left(L_A\right)\right)\right],sin\epsilon \left(X_1+{\displaystyle \frac{w}{2sin\epsilon }}\right)\right\}}^{\sin \epsilon \left(L_1+L_2\right)}{A}_{S1}d{F}_{y{S}_4}\end{array}$$

(21)

$$\begin{array}{l}{F}_{f1R}=\mu {{\displaystyle \int }}_0^{sin\epsilon \left(L_B\right)}{A}_{S_1}d{F}_{y{S}_1}+\mu {{\displaystyle \int }}_{\min \left\{{\displaystyle \frac{1}{2}}\left[sin\epsilon \left(L_1+L_2\right)+\left(Y-sin\epsilon \left(L_A\right)\right)\right],sin\epsilon \left(X_1+{\displaystyle \frac{w}{2sin\epsilon }}\right)\right\}}^{sin\epsilon \left(L_B\right)}{A}_{S1}d{F}_{y{S}_3}\\ +\mu {{\displaystyle \int }}_{\max \left\{{\displaystyle \frac{1}{2}}\left[\sin \epsilon \left(L_1+L_2\right)+\left(Y-\sin \epsilon \left(L_A\right)\right)\right],sin\epsilon \left(X_2+{\displaystyle \frac{w}{2sin\epsilon }}\right)\right\}}^{\sin \epsilon \left(L_1+L_2\right)}{A}_{S_1}d{F}_{y{S}_4}\end{array}$$

(22)

$$F_{i1}={\displaystyle \frac{1}{tan\frac{\theta }{2}}}{{\displaystyle \int }}_0^{L_A}d{F}_{z{S}_1}+{\displaystyle \frac{1}{tan\frac{\theta }{2}}}{{\displaystyle \int }}_{\min (L_A-sin{X}_{1)}}^{60}d{F}_{z{S}_4}$$

(23)

The subsoiling force, F_S2, of the central auxiliary subsoiler comprises the cutting force, F_c2; the friction force, F_f2; and the inertial force, F_i2. The calculation formula is shown in Equations (24)–(26).

$$F_{c2}={{\displaystyle \int }}_{X_1-L_A}^{X_1+L_A}d{F}_{x{S}_2}+{{\displaystyle \int }}_{\min \left(X_1+L_A,Y+sin\epsilon \left(L_1+L_2\right)\right)}^{X_1+L_A}d{F}_{x{S}_3}+{{\displaystyle \int }}_{\min \left(X_1,Y+{\displaystyle \frac{w}{2\sin \epsilon }}\right)}^{X_1+L_A}d{F}_{x{S}_4}$$

(24)

$$\begin{array}{l}{F}_{f2}=2\mu {{\displaystyle \int }}_{{\displaystyle \frac{1}{2}}\left[sin\epsilon \left(L_1+L_2\right)+\left(Y-sin\epsilon \left(L_A\right)\right)\right]}^Y{A}_{S_2}d{F}_{y2}\\ +2\mu {{\displaystyle \int }}_{\min \left\{Y-sin\epsilon \left(L_A\right),{\displaystyle \frac{1}{2}}\left[sin\epsilon \left(L_1+L_2\right)+\left(Y-sin\epsilon \left(L_A\right)\right)\right]\right\}}^{{\displaystyle \frac{1}{2}}\left[sin\epsilon \left(L_1+L_2\right)+\left(Y-sin\epsilon \left(L_A\right)\right)\right]}{A}_{S_2}d{F}_{-y3}\\ +2\mu {{\displaystyle \int }}_{\max \left\{Y-sin\epsilon \left(L_A\right),{\displaystyle \frac{1}{2}}\left[sin\epsilon \left(L_1+L_2\right)+\left(Y+sin\epsilon \left(L_A\right)\right)\right]\right\}}^{{\displaystyle \frac{1}{2}}\left[sin\epsilon \left(L_1+L_2\right)+\left(Y-sin\epsilon \left(L_A\right)\right)\right]}{A}_{S_1}d{F}_{y{S}_4}\end{array}$$

(25)

$$\begin{array}{l}{F}_{i_2}={\displaystyle \frac{1}{tan\frac{\theta }{2}}}{{\displaystyle \int }}_0^{\min \left\{60,{\displaystyle \frac{60-Z}{\cos 15°}}+s+\sin \left(X_1+{\displaystyle \frac{\tan \epsilon {\mathsf{\omega }}_1}{2\sin \epsilon }}\right)\right\}}d{F}_{z{S}_1}\\ +{\displaystyle \frac{2}{tan\frac{\theta }{2}}}{{\displaystyle \int }}_{\min \left\{60,{\displaystyle \frac{60-Z}{\cos 15°}}+s+\sin \left(X_1+{\displaystyle \frac{\tan \epsilon {\mathsf{\omega }}_1}{2\sin \epsilon }}\right)\right\}}^{60}d{F}_{z{S}_4}\end{array}$$

(26)

The subsoiling force, F_S3, of the side auxiliary subsoiler includes the cutting force, F_c3; the frictional force, F_f3; and the inertial force, F_i3, with corresponding equations provided in Equations (27)–(29). The mathematical expression for the total draft force, F, of the subsoiler is shown in Equation (30).

$$F_{c3}=2{{\displaystyle \int }}_{X_2}^{X_2+L_A}d{F}_{x{S}_2}+{{\displaystyle \int }}_{\min \left(X_1+L_A,Y+sin\epsilon \left(L_1+L_2\right)\right)}^{X_2+L_A}d{F}_{x{S}_3}+{{\displaystyle \int }}_{\min \left(X_1,Y+{\displaystyle \frac{w}{2\sin \epsilon }}\right)}^{X_2+L_A}d{F}_{x{S}_4}$$

(27)

$$\begin{array}{l}{F}_{f3}=2\mu {{\displaystyle \int }}_{{\displaystyle \frac{1}{2}}\left[sin\epsilon \left(L_1+L_2\right)+\left(Y-sin\epsilon \left(L_A\right)\right)\right]}^Y{A}_{S_2}d{F}_{y2}\\ +\mu {{\displaystyle \int }}_{\min \left\{Y-sin\epsilon \left(L_A\right),{\displaystyle \frac{1}{2}}\left[sin\epsilon \left(L_1+L_2\right)+\left(Y-sin\epsilon \left(L_A\right)\right)\right]\right\}}^{{\displaystyle \frac{1}{2}}\left[sin\epsilon \left(L_1+L_2\right)+\left(Y-sin\epsilon \left(L_A\right)\right)\right]}{A}_{S_2}d{F}_{-y3}\\ +\mu {{\displaystyle \int }}_{\max \left\{Y-sin\epsilon \left(L_A\right),{\displaystyle \frac{1}{2}}\left[sin\epsilon \left(L_1+L_2\right)+\left(Y+sin\epsilon \left(L_A\right)\right)\right]\right\}}^{{\displaystyle \frac{1}{2}}\left[sin\epsilon \left(L_1+L_2\right)+\left(Y-sin\epsilon \left(L_A\right)\right)\right]}{A}_{S_1}d{F}_{y{S}_4}\end{array}$$

(28)

$$\begin{array}{l}{F}_{i_3}={\displaystyle \frac{1}{tan\frac{\theta }{2}}}{{\displaystyle \int }}_0^{\min \left\{60,{\displaystyle \frac{60-Z}{\cos 15°}}+s+\sin \left(X_2+{\displaystyle \frac{\tan \epsilon {\mathsf{\omega }}_1}{2\sin \epsilon }}\right)\right\}}d{F}_{z{S}_1}\\ +{\displaystyle \frac{1}{tan\frac{\theta }{2}}}{{\displaystyle \int }}_{\min \left\{60,{\displaystyle \frac{60-Z}{\cos 15°}}+s+\sin \left(X_2+{\displaystyle \frac{\tan \epsilon {\mathsf{\omega }}_1}{2\sin \epsilon }}\right)\right\}}^{60}d{F}_{z{S}_4}\end{array}$$

(29)

$$F=2\left(F_{C1}+F_{f1L}+F_{f1R}+F_{i1}\right)+F_{C2}+F_{f2}+F_{i2}+2\left(F_{c3}+F_{f3}+F_{i3}\right)$$

(30)

Establishment of Soil Disturbance Area Prediction Model

According to the fourth strength theory, the force acting on the state of the soil element is verified as expressed in Equation (31). The three-dimensional coordinates of soil elements under triaxial force acting on conditions meeting the allowable stress criteria are determined (Figure 5b), and the response nephogram of overall soil disturbance elements during stable tool operation is generated. The response nephogram is projected onto the yz-plane (Figure 5b), with edge point coordinates extracted to plot the soil disturbance profile during multi-subsoiler cooperative operation, where σ₁ represents dF_x, σ₂ denotes dF_y, σ₃ corresponds to dF_z, and [σ] indicates the allowable normal stress of sodic saline–alkali soil.

$$\sqrt{{\displaystyle \frac{1}{2}}\left[{\left({\mathsf{\sigma }}_1-{\mathsf{\sigma }}_2\right)}^2+{\left({\mathsf{\sigma }}_2-{\mathsf{\sigma }}_3\right)}^2+{\left({\mathsf{\sigma }}_2-{\mathsf{\sigma }}_3\right)}^2\right]}=\left[\mathsf{\sigma }\right]$$

(31)

Establishment of Soil Bulkiness Prediction Model

Soil looseness is a comprehensive index comprising the theoretical soil disturbance area and the surface-heaved soil area (Figure 6), as expressed by Equation (32), where P_O denotes soil looseness (%), A_e represents the cross-sectional area between the soil surface after subsoiling and the original surface (mm²), and A_O indicates the cross-sectional area from the surface to the theoretical subsoiling trench bottom (mm²).

$$P_O={\displaystyle \frac{A_e}{A_O}}$$

(32)

The surface-heaved soil area, A_e, depends on the displacement of disturbed particles. When a soil element’s velocity reaches zero, with its z-coordinate exceeding 60, it is counted as surface heaving. Thus, A_e is mathematically defined by Equation (33), where ρ_s means the bulk density of soil particles after subsoiling (g/cm³).

$$A_e=\underset{{\displaystyle \frac{\sqrt[2]{\Delta x^2+\Delta y^2+\Delta z^2}}{\Delta t}}\to 0}{\lim }{{\displaystyle \int }}_{60}^{\underset{0\le x\le 1}{\max }x{e}^{-x^2}}{\rho }_{s}dz$$

(33)

The theoretical soil disturbance area, A_O, is calculated by multiplying the theoretical tillage depth by the furrow width, D_A, with the mathematical expression of D_A shown in Equation (34).

$$D_A=4Y+2sin\epsilon \left(L_A\right)$$

(34)

2.3. Establishment of the Discrete Element Simulation Model

The simulation was performed using Altair EDEM 2022 software. Following prior studies, the Hertz–Mindlin with JKR contact model was adopted. The water contents and cohesion of different soil depths were different, and different surface energy parameters were set to characterize the viscoplastic differences caused by the water content gradient of saline–alkali soil [26]. The soil particle contact parameters are presented in

Table 2. Simulation parameters.

Parameters	Value (mm)
	0–50	50–200	50–200	200–800
Density of soil particles (kg/m3)	2.05	2.1	2.1	2.3
Poisson’s ratio of soil	0.28	0.36	0.36	0.36
Shear modulus of soil (Pa)	100	300	300	400
Particle radius (mm)	10	10	12	10
Coefficient of restitution, soil–soil	0.4	0.6	0.6	0.6
Coefficient of static friction, soil–soil	0.4	0.5	0.5	0.55
Coefficient of rolling friction, soil–soil	0.2	0.4	0.4	0.5
Coefficient of restitution, soil–steel	0.5	0.5	0.5	0.5
Coefficient of static friction, soil–steel	0.45	0.5	0.5	0.55
Coefficient of rolling friction, soil–steel	0.08	0.1	0.1	0.1
JKR surface energy	1.2	3.7	3.7	5.9
Density of steel (kg/m3)	7865
Poisson’s ratio of steel	0.3
Shear modulus of steel (1010)	7.9

.

The simulation soil bin measures 2880 mm (length) × 2700 mm (width) × 800 mm (depth) (Figure 7). The top layer comprises the natric horizon, A_z, representing surface soil at a 0–50 mm depth, formed by 10 mm particles (pink). Below lies the impermeable layer, E (50–200 mm depth), comprising a blend of 10 mm and 12 mm particles (blue and yellow). The basal subsoil layer, C_y, extends from a depth of 200 to 800 mm and contains 10 mm particles (green). Considering soil heterogeneity and the irregular shapes of soil particles, the multi-sphere clump model was adopted to construct the soil particle model. In the simulation, the subsoiler’s penetration speed was maintained at 0.15 m/s, while its forward speed remained at 0.83 m/s. The implement ceased descending upon reaching the auxiliary subsoiler penetration depth of 30 cm, and it continued advancing until the subsoiler tip made contact with the soil bin, whereupon the simulation was terminated (Figure 7).

2.4. Field Test

The experimental site was in Tongquan Village, Liangjiazi Town, Da’an City, Baicheng, Jilin Province (longitude: 123°88′, latitude: 45°47′). The area was classified as severely sodic saline–alkali soil with an average soil pH of 10.42 and a total salt content of 22.83 g/kg. The uncultivated land preserved the pristine geomorphic features of sodic saline–alkali soil (Figure 8).

The experiment was conducted on 24 September 2024, with an average daytime temperature of about 22 °C, a humidity of about 63%, and sunny weather. The Dongfeng 2104 tractor (Dongfeng Agricultural Machinery Group Co., Ltd., Changzhou, China) provided the power (210 horsepower), while the Case 1404 tractor (Case New Holland Industrial Machinery Co., Ltd., Harbin, China), equipped with a subsoiler, performed the implement’s lifting operations. At a constant tractor speed of 4 km/h, the D. R920WL wireless tension meter system (Descent Sensor System Engineering Co., Ltd., Shenzhen, China) measured the overall soil draft force in the saline–alkali soil (Figure 9); the force measurement range is 10⁵ N, and the accuracy is 10 N. After five experimental repetitions, data were extracted from the recorder for subsequent analysis.

3. Results

3.1. Comparison of Test, Simulation, and Prediction Results

The fixed vertical spacing, X, of subsoilers was set at 35 cm; the horizontal spacing, Y, of subsoilers was set at 35 cm; and the subsoiling depth, Z, of auxiliary subsoilers was set at 30 cm. Field experiments were performed by varying the relative positions between the broken-line subsoiler and the auxiliary subsoiler, testing three arrangement modes of subsoiler configurations (“W”, “V”, and “∧”) with different spacing combinations. The draft force and soil looseness under the three arrangement patterns in the field test were compared with data extracted from the EDEM simulation post-processing module and the predicted value of the mathematical model, as illustrated in Figure 10. For tillage resistance, the simulation test results were smaller than in the field test because the subsoiling plot in the field test was compacted by the traction tractor and the no-load tractor, resulting in an increase in tillage resistance; the predicted value of the mathematical model was greater than the field test and simulation results because the shape of the soil particles affected the interaction between the soil and the subsoiler. The mathematical model divided the soil into soil micro-elements, whose shapes were different from the actual soil particle shapes, so that there was an error between the prediction value and tillage resistance.

For all three arrangement patterns, the simulated and mathematically predicted draft force and soil looseness values demonstrated excellent consistency with field measurements, showing less than 5% error. These results confirm the reliability of the simulation analysis and the mathematical modeling approach. Figure 11 compares the draft force curve and soil looseness performance for three subsoiler configurations, transitioning from initial soil penetration at 0 s to stabilized operation. According to Figure 11, the order of draft force in the three arrangement modes is V > Λ > W. The draft force increases with the increase in the depth of the subsoiler. At the time of 2 s, the broken-line and the auxiliary subsoiler reached a stable working state. At this time, the draft force of the W-type arrangement subsoiler was small, and the soil looseness was the largest.

3.2. The Effect of Arrangement on Tillage Performance

3.2.1. The Change in Draft Force and Looseness with the Arrangement Mode

The W-type arrangement configuration was selected for subsequent study of subsoiler spacing and auxiliary subsoiler penetration depth through a comprehensive analysis of draft force and soil looseness during tillage operations. Utilizing the previously established predictive models, we computed soil looseness and draft force values across varying subsoiler spacing and auxiliary subsoiler penetration depth conditions. These computational results were compared with discrete element simulation data extracted via the post-processing module. The variation in draft force and soil looseness with longitudinal spacing, X; transverse spacing, Y; and auxiliary subsoiler penetration depth, Z, is shown in Figure 12.

Figure 12a demonstrates the correlation between draft force, soil looseness, and longitudinal spacing, X. The simulation results exhibit excellent agreement with the model’s predictive curve trend, confirming strong correlation. The mathematical model prediction results show that at X = 40 cm, the phase difference in soil disturbance between the auxiliary subsoiler and broken-line subsoiler enables particles to reach their peak height, where both the z-axis velocity component and the inertial force reach their minimum values. For X > 40 cm, particle descent forces the broken-line subsoiler to expend additional energy lifting soil, resulting in a 1.67% increase in force. The draft force reaches its minimum at a 40 cm longitudinal spacing depth, while soil looseness attains its maximum value. Figure 12b demonstrates that the subsoiler’s draft force exhibits an upward trend with increasing lateral spacing, Y, showing more pronounced variations within the 45–55 cm range. Soil looseness displays significant numerical fluctuations as Y increases, peaking at 45 cm spacing. Furthermore, the standard deviation of soil porosity grows with larger Y values, suggesting that increased Y leads to greater instability in furrow width. Figure 12c presents the curves of draft force and soil looseness versus auxiliary subsoiler penetration depth, Z. The draft force exhibits a quadratic growth pattern with increasing Z, demonstrating a more pronounced rate of change within the 30–40 cm range. Soil looseness initially increases, then decreases with depth. With the increasing subsoiling depth, Z, of auxiliary subsoilers, soil looseness rises before declining. Excessive subsoiling depth leads to lateral soil failure and reduced disturbance efficiency of the subsoiler, consequently yielding lower soil looseness.

3.2.2. Change of Soil Disturbance Area

The soil profile is obtained by the post-processing module of EDEM 2022 software, and the target contour points are extracted and fitted by MATLAB R2020a software to obtain the soil disturbance contour curve of the simulation process. Figure 13 illustrates the variation trend of total soil disturbance area with respect to lateral spacing, Y, and auxiliary subsoiler penetration depth, Z. The X arrangement pattern shows minimal influence on the total disturbed area due to the repeated disturbance phenomenon induced by longitudinal spacing adjustments, consequently exhibiting less impact on soil looseness and disturbance area than parameters Y and Z.

It can be seen from Figure 13 that the predicted total soil disturbance area is consistent with the simulation. While increased Y values show limited variation in soil surface heave height, they cause a significant expansion in furrow width. This may be because the increase in lateral spacing reduced the repeated disturbance of soil but increased the total area of soil disturbance (Figure 13a). When Z increases, soil surface heave height shows marked elevation (Figure 13b), primarily because the wing fails to achieve optimal operational status at shallow auxiliary subsoiler penetration depths. This condition causes soil lifted by the subsoiler tip to encounter wing obstruction during its upward movement, reducing soil looseness. When the depth of the auxiliary subsoiler is increased to 40 cm, the soil area that bulges out of the surface is affected by the boundary effect, thereby reducing the soil looseness.

3.3. Verification Test

With the objectives of maximizing soil looseness and minimizing draft force, the optimization of the established mathematical prediction model determined the optimal operating parameters for the subsoiler in a W-type arrangement pattern configuration: a longitudinal spacing (X) of 42.6 cm, a lateral spacing (Y) of 39.7 cm, and an auxiliary subsoiler penetration depth (Z) of 27.8 cm. The draft force predicted by the model was 30503N, and the soil looseness was 11.77% (Figure 14). The parameters were rounded to X = 43 cm, Y = 40 cm, and Z = 28 cm for the field test. Field validation tests and simulation experiments demonstrated less than 5% deviation between measured, simulated data, and prediction results. These results further confirm the reliability of the developed mathematical prediction model.

4. Discussion

The kinematic properties of soil particles govern the macroscopic flow behavior of particulate flow. Figure 15 presents instantaneous velocity vector diagrams of soil particles surrounding the polyline and auxiliary subsoiler under a W-type arrangement configuration at various time intervals. Upon soil penetration, the subsoiler tip applies compression forces while the tine body shears through the soil, inducing upward and forward movement tendencies in the preceding soil particles. This creates an obliquely ascending soil particle flow. With increasing subsoiler penetration depth, the quantity of disturbed particles ahead escalates, resulting in heightened subsoiling resistance. Notably, during the 1.2 s–1.6 s timeframe, the soil particle flow under the auxiliary subsoiler (Figure 15a) and the broken-line subsoiler (Figure 15b) is in a disorderly state. This phenomenon arises as the subsoiler tine’s penetration depth increases during forward motion, subjecting the underlying soil to compression while simultaneously dragging it forward due to frictional resistance.

Meanwhile, the soil behind the tine develops excess internal stress, generating an upward movement tendency. The rear soil particles exhibit free-fall behavior, creating a turbulent particle flow that moves obliquely downward toward the tine. This results in a disordered soil arrangement behind the tine, a characteristic particularly evident with polyline-shaped subsoilers [8,10]. At the time of 2.0, the subsoiler’s depth into the soil reaches a stable level. At this time, the soil particles behind the subsoiler are subjected to gravity to form a stable downward soil particle flow. The soil particles below are no longer squeezed and gradually separated from the bottom of the subsoiler. This phenomenon can be observed more clearly at 2.4 s. At this time, the resistance of the subsoiler is mainly caused by the force and reaction force given by the soil particle flow in front, and the particle motion state is stable. At this time, the draft force remains stable.

When the subsoiler tines are arranged in a W-shaped configuration with a 35 cm lateral spacing and a 30 cm working depth, the draft force initially decreases, then increases with greater longitudinal spacing (Figure 12a); the soil looseness follows an inverse trend. This behavior likely stems from the repeated disturbance between subsoilers in this spatial arrangement. With a 30 cm longitudinal spacing, soil particles first experience cutting and lifting forces from the auxiliary subsoiler, gaining initial momentum. However, before their motion trajectory peaks, the particles enter the influence zone of the broken-line subsoiler. The dual constraints from both subsoiler types create localized particle congestion, resulting in soil compression (Figure 16a, zone 1). Therefore, the draft force of the subsoiler is larger and the soil looseness is smaller. When the longitudinal spacing is 40 cm, the soil begins to be cut and lifted by the auxiliary subsoiler, initiating movement. At peak elevation, soil particles experience secondary disturbance from the broken-line subsoiler tine, preventing lateral escape. Consequently, the broken-line subsoiler encounters minimal inertial resistance from the soil, resulting in lower subsoiler draft force and increased soil looseness (Figure 16b, zone 2). With a longitudinal spacing of the subsoiler of 50 cm, soil disturbed by the auxiliary subsoiler has already passed its apex when entering the broken-line subsoiler’s working zone. The downward vertical velocity component necessitates additional lifting force from the broken-line subsoiler, increasing draft force. Furthermore, having fully disengaged from the auxiliary subsoiler’s influence, most soil particles escape laterally past the broken-line subsoiler (Figure 16c, zone 3). This makes the soil bulge out of the surface area to a minimum, decreasing soil looseness.

When the subsoilers are arranged in a W-shaped pattern with a longitudinal spacing of 40 cm and a working depth of 30 cm, the subsoiling resistance increases with greater lateral spacing between the subsoilers (Figure 12b). This occurs because wider lateral spacing results in more soil being disturbed by the subsoilers (Figure 16d–f), consequently increasing the draft force. Soil looseness rises but then declines as the subsoilers’ lateral spacing increases. This trend is attributed to the expanding surface area of soil upheaval with greater spacing, but the expansion demonstrates an apparent diminishing marginal effect (Figure 16f, zone 6). While the furrow width shows a linear increase with wider subsoiler spacing, the soil upheaval surface area growth rate gradually decreases. Soil looseness is directly proportional to the surface area of soil upheaval and inversely proportional to furrow width, resulting in an initial increase followed by a decrease in soil looseness as the lateral spacing between subsoilers widens.

With a W-shaped subsoiler arrangement and a 40 cm longitudinal spacing and a 45 cm lateral spacing, the draft force increases sharply with a greater penetration depth of the auxiliary subsoilers, where the soil looseness increases first and then decreases with the increase in the penetration depth of the auxiliary subsoilers (Figure 12c). The effect of too shallow a penetration depth of the auxiliary subsoiling shovel wing on soil disturbance is not apparent, resulting in a small surface area of soil bulging (Figure 16g, zone 7), which leads to the lowest soil looseness. When the subsoiling depth of the auxiliary subsoiler is 30 cm, the depth of the auxiliary subsoiler is moderate, the disturbance effect on the soil is good (Figure 16h, zone 8), the width of the soil is moderate, and the soil looseness is greater. As the subsoiling depth of the auxiliary subsoiler exceeds the critical threshold, it induces lateral soil failure with reduced upward soil particle displacement and increased frictional resistance on the implement. Conversely, an increased subsoiling depth of the auxiliary subsoiler causes soil congestion between the subsoilers (Figure 16i, zone 9), substantially elevating draft force while diminishing soil looseness.

5. Conclusions

In this study, aiming at the problem that the mechanism of soil disturbance affected by the subsoiling shovel tillage process in sodic saline–alkali soil is unknown, a soil micro-elemen–tillage performance prediction model was constructed based on soil dynamics. By analyzing the variation law of soil particle instantaneous velocity and particle trajectory under different subsoiling shovel arrangements and spacings, the tillage performance of the subsoiler under different subsoiling shovel arrangements and spacings can be predicted. The draft force and soil looseness values of “W”, “V”, and “∧” subsoilers were obtained through field experiments. The accuracy of the mathematical model and the simulation model was verified by comparing the discrete element simulation results and the predicted values of the mathematical model. The results showed that the “W”-type arrangement performed better in tillage than the other two. The mathematical model and discrete element simulation were used to predict and analyze the subsoiler’s draft force and soil looseness under different spacings of the “W”-type arrangement. The results demonstrate that soil looseness increases and decreases with greater spacing intervals. The draft force follows a quadratic function relationship with longitudinal spacing, lateral spacing, and auxiliary subsoiler penetration depth, showing the increase in resistance as auxiliary subsoiler penetration depth deepens. The congestion effect of soil particles and repeated disturbances constitutes the primary cause behind this nonlinear variation in draft force. Optimization of arrangement parameters using the established mathematical prediction model yields optimal tillage performance at a longitudinal spacing (X) of 42.6 cm, a lateral spacing (Y) of 39.7 cm, and an auxiliary subsoiler penetration depth (Z) of 27.8 cm. Simulation and field tests were carried out, and the error was within 5%. The model’s prediction results carry substantial theoretical and practical significance. This study elucidates how subsoiler structure influences tillage performance by examining the correlation between subsoiler configuration and soil particle displacement patterns, thereby contributing to soil management optimization. The model provides a theoretical framework and empirical data for designing related tillage implements. In practical agricultural operations, draft force reduction can be achieved by considering multiple factors, including soil texture, subsoiling depth, and implement geometry, underscoring the model’s field applicability. For soil containing roots, it is necessary to consider the effect of root–soil composite strength on tillage resistance. The mathematical model for this problem needs to be further calibrated to adapt to more types of soil parameters.