Development of a DEM-Based Flexible Plant Model for Mature Peanut Plants

Abstract

Accurate discrete element method (DEM) modelling of mature peanut plants is essential for simulating peanut harvesting, pod detachment, and harvest-loss formation. However, existing peanut DEM models are usually simplified as isolated pods, rigid cylindrical particles, or partial stem–pod structures, which limits their ability to represent the flexible deformation of vines and pod stalks and the fracture behaviors at the pod–pod stalk junction. In this study, a DEM-based flexible plant model was developed for mature peanut plants. The geometric dimensions, contact parameters, and mechanical properties of peanut pods, pod stalks, and stems were measured through physical experiments. The Hertz–Mindlin model was used for non-bonded contacts, whereas the Hertz–Mindlin with Bonding model was adopted to represent the flexible connections among plant organs and the fracture behaviors of the pod–pod stalk junction. The main DEM parameters were calibrated using Plackett–Burman screening, steepest ascent experiments, and central composite design. The results showed that the tangential stiffness per unit area and tangential critical stress at the pod–pod stalk junction were the dominant factors affecting pod detachment force. The optimized parameter combination was a tangential stiffness per unit area of 4.738 × 10⁵ N/m³ and a tangential critical stress of 9.350 × 10⁵ Pa, corresponding to a simulated tensile force of 6.73 N. Model validation was performed by comparing peanut harvesting simulations with field trials. The relative error of pod loss rate between simulation and field measurement was less than 7.55%, and the t-test result indicated no significant difference between the two datasets (p > 0.05). These results demonstrate that the proposed flexible peanut plant model can effectively characterize pod–pod stalk separation and can provide a reliable DEM modelling basis for peanut harvesting process analysis and equipment optimization.

1. Introduction

Peanuts, as a globally significant oilseed and cash crop, represent China’s most internationally competitive high-quality oilseed crop. China ranks first worldwide in peanut harvest area, accounting for approximately 40% of global production, and stands as one of the few Chinese agricultural products with strong international competitiveness in export earnings. According to data from the 2025 China Statistical Yearbook [1], China’s peanut cultivation area and output have shown steady growth. Peanut production has shown an upward trend from 2014 to 2024. Annual output increased from 15.901 million tons in 2014 to 19.613 million tons in 2024. This production growth has further heightened the industry’s urgent demand for mechanization and intelligent solutions.

With the continuous advancement of computer technology, the Discrete Element Method (DEM) has been widely applied in agricultural material modelling [2,3,4,5]. Sun Kai researched the DEM modelling method for wheat plants. The Hertz–Mindlin contact model was employed to characterize inter-particle forces. A parallel bonding model was applied to simulate the connection characteristics between plant organs and constituent spheres. A flexible DEM model of wheat plants was constructed using a geometry-mechanics coupling strategy [6]. Shi Ruijie established a multi-level bonding system comprising root stems, middle stems, neck stems, lateral branches, and capsules based on the Hertz–Mindlin with bonding contact model, thereby constructing a DEM model of flax plants [7]. Wang Qirui developed a modelling and simulation method for rice threshing using DEM, establishing a discrete element model of rice plants with hollow cylindrical elastic keys to simulate the complex threshing process [8]. The Lensert team developed a segmented stalk model on the DEMeter++ platform. Rigid cylindrical elements connected via spherical joints incorporate six-degree-of-freedom spring-damper systems at joints to simulate bending characteristics [9]. Scholar further introduced a torque spring model to optimize stem bending dynamics [10]. Xu et al. developed a model of rice plants in their natural bending state. They determined the parameters for the rice plant model and validated the accuracy of these parameters through comparative experiments [11]. Similarly, other researchers have used the discrete element method to develop flexible body models for rapeseed [12], notoginseng [13], maize [14], safflower [15], citrus fruit stalks [16], tobacco leaves [17], and white radish roots [18]. However, in theory, the continuous bending of crops can be modeled as long as there are sufficient nodes. Yet this approach leads to a dramatic increase in computational costs. Therefore, given the practical limitations of discrete element method simulations, the existing literature indicates that there are relatively many modeling methods for the stems of crops such as wheat and rice [19,20,21,22,23]. However, detailed modeling of peanut plants remains an unexplored area of research. This work provides new insights for establishing a discrete element model of peanut plants.

The overall structure of peanut plants is relatively complex. Wang Bing investigated the operational mechanism of a semi-feed harvester. By simplifying the peanut plant structure, a single-seed peanut model was constructed [24]. Su Lingbin employed a discrete element particle bonding method to construct a flexible stem structure. Pods were directly simulated as rigid cylindrical particles. Through stem–pod coupling modelling, a discrete element model of peanut stems and pods was developed [25]. Zhang Jinbiao analyzed the interaction mechanism between peanut vines and the crushing mechanism, establishing a discrete element model of peanut stems and vines [26]. Currently, discrete element modeling of peanut plants remains at a simplified stage, and existing research suffers from the following shortcomings: First, some studies simplify the plant into a single-pod model, ignoring the connections between the pod, the pod stalk, and the vine, and are thus unable to dynamically simulate the pod-picking process. Second, some studies simplify the pod into a rigid cylindrical particle, making it difficult to simulate the pod–pod stalk separation process under impact loads. These simplified models have significant limitations in terms of structural fidelity, accuracy of mechanical response, and the ability to simulate fracture processes, making it difficult to accurately characterize the separation behavior between pods and pod stalk during peanut harvesting.

However, previous studies on peanuts have only achieved modelling of individual components like stems or pods. A whole-plant model incorporating stems, pods, leaves, and other parts has yet to be constructed. First, the complexity of the plant’s structure and its connections poses a fundamental challenge to modeling. Peanut plants form pods below ground level, and their underground parts—including the taproot, multi-tiered lateral roots, and pods attached to the root system—exhibit a complex spatial structure. Second, there is a lack of data on the mechanical properties of the weak mechanical nodes at peanut pod detachment points and on the calibration of adhesion parameters. Furthermore, peanut plants consist of both rigid pods and flexible components such as vines, stem, pod stalk, and root systems; a single contact model struggles to accurately describe the mechanical behavior of both types of materials simultaneously. This significantly increases the modeling workload and complexity.

To address these limitations, this study aimed to develop and validate a DEM-based flexible model for mature peanut plants that can represent both the flexible deformation of plant organs and the fracture behaviors at the pod–pod stalk junction. The specific objectives were: (1) to measure the geometric, contact, and mechanical parameters of mature peanut plants during the harvest period; (2) to construct a flexible whole-plant DEM model by combining multi-sphere geometric representation with the Hertz–Mindlin and Hertz–Mindlin with Bonding contact models; (3) to identify and optimize the key bonding parameters governing pod detachment using Plackett–Burman, steepest ascent, and central composite experiments; and (4) to validate the reliability of the calibrated model through comparative harvesting simulations and field trials. The proposed model is expected to provide a reproducible modelling framework for peanut harvesting simulations and to support the optimization of peanut harvesting equipment.

2. Materials and Methods

To ensure the reliability of parameter calibration during the peanut pod separation process, the “Huayu” series varieties, predominantly cultivated in northern China, were selected. The sample data were collected in October 2025. Sampling was primarily conducted in the major peanut-producing areas of the Yellow River Basin. The peanut plant samples were collected from Dongying City (37°37′ N, 118°39′ E) and Zhucheng City (35.99° N, 119.42° E) in Shandong Province, and Xinxiang City (35.18° N, 113.52° E) in Henan Province. At each location, a standard plot was selected and planted with the “Huayu” series of peanuts. Planting density, fertilization regimens, and field management practices were kept consistent across all plots. During the peanut maturity stage, 10 healthy plants were randomly selected from each plot, for a total of 30 plants, for subsequent experiments. The data presented below represent the statistical averages of samples from the three locations, reflecting the average conditions in the major peanut-producing regions of the Yellow River Basin.

2.1. Measurement of Primary Dimensions of Peanut Plants

Mature peanut plants were divided into aboveground vine sections and belowground pod systems. The vine section mainly consisted of the main stem, lateral stems, and leaves, whereas the pod system consisted of roots, pod stalks, and peanut pods. To reduce computational cost while retaining the mechanical characteristics directly related to pod detachment, the peanut plant was simplified according to the load-bearing hierarchy and the relevance of each organ to pod–stalk separation.

The main stem, primary lateral stems, pod stalks, peanut pods, and pod–stalk junctions were retained because these components constitute the main force-transmission path during peanut harvesting. Leaves and small lateral branches were not regarded as primary load-bearing components during pod picking. Their influence on pod detachment was mainly reflected in secondary collision and frictional disturbance rather than in the fracture behavior at the pod–stalk junction. Therefore, these components were simplified by reducing their geometric representation and by incorporating their secondary contact effects into the global contact parameters of the vine system. This treatment reduced the number of DEM particles and bonding elements while maintaining the main mechanical pathway from external harvesting load to pod–stalk junction fracture.

For the subterranean pod system, a finer geometric description was adopted because the pod, pod stalk, and pod–stalk junction directly determine pod detachment behavior. In contrast, the aboveground vine section was subjected to coarse calibration because it mainly serves as a flexible support and load-transfer structure in the present model. The contact parameters of the primary components, including peanut pods, pod stalks, and steel harvesting components, were systematically determined through physical tests and DEM calibration.

Dimensions of peanut stems, pods, and pod stalks were collected using measuring tapes (3 m HY66-5019 Huiyuan Precision Engineering) and vernier calipers (0.02 mm Shanghai Shengong Measuring Tools Co., Ltd., Shanghai, China). To standardize the geometric description of each peanut pod, a local coordinate system O–XYZ was established for each individual pod. The pod geometric center was defined as the origin O. The distance between the two lateral sutures was defined as the pod width W, and the width direction was set as the X-axis. The pod thickness H was set along the Y-axis. The distance from the pod–stalk junction to the pod tip was defined as the pod length L, and the length direction was set as the Z-axis, as shown in Figure 1. Each pod therefore has its own local coordinate system, which moves and rotates with the pod. In subsequent DEM simulations, the local coordinates of each pod were transformed into the global coordinate system using rotation matrices according to the initial spatial orientation of the pod.

For peanut pods, W denotes pod width. For stems and pod stalks, D denotes the equivalent diameter measured perpendicular to the axial direction. The geometric dimensions of peanut plants were obtained by measuring the main stem, pod stalk, and peanut pods. The measurements were statistically analyzed and averaged. The results are presented in

Table 1. Geometric characteristic dimensions of peanut plants.

Peanut Plant Structure	Average Value
	L/(mm)	W or D/(mm)	H/(mm)
Peanut pods	34.27	18.29	15.76
Stem	763.81	6.74	/
Pod stalk	108.26	1.93	/

Note: H is the pod thickness, W is the pod width, D is the equivalent diameter of the stem or pod stalk, and L is the length of the corresponding component.

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2.2. Determination of Contact Parameters

During peanut plant modelling, the Hertz–Mindlin contact model was used to describe non-bonded contacts among peanut pods, pod stalks, and steel components. In this model, contact behavior includes both normal and tangential components. The normal contact component is mainly characterized by the elastic modulus, Poisson’s ratio, equivalent contact radius, normal overlap, and restitution coefficient, which together determine the normal elastic response and damping behavior during collision. The tangential contact component is mainly characterized by the static friction coefficient, rolling friction coefficient, and tangential contact stiffness and damping, which determine sliding resistance, rolling resistance, and tangential energy dissipation.

Therefore, the contact parameter measurements in this study were designed according to the physical meaning of each DEM parameter. The inclined-plane test was used to determine the static friction coefficient, which characterizes tangential sliding resistance. The inclined impact test was used to determine the collision restitution coefficient, which characterizes the normal rebound and energy dissipation behavior during collision. The rolling test was used to determine the rolling friction coefficient, which characterizes resistance to rotational motion. Three main contact pairs were considered according to the main contact events occurring during peanut harvesting: peanut pod–peanut pod, peanut pod–steel, and pod stalk–steel.

2.2.1. Determination of Static Friction Coefficient

The static friction coefficient between contacting materials was determined using an inclined-plane test based on Coulomb’s friction law. As shown in Figure 2. Three types of contact pairs were considered according to the main contact events occurring during peanut harvesting: peanut pod–peanut pod, peanut pod–steel, and pod stalk–steel. For the peanut pod–peanut pod contact pair, two pod-arrangement conditions were prepared. In the aligned arrangement, peanut pods were fixed on the lower plate with their longitudinal axes approximately parallel to one another. In the random arrangement, peanut pods were randomly distributed on the lower plate to better reproduce the disordered pod orientation during mechanical harvesting. The random-orientation result was used as the input value for the DEM model because the pod posture in actual harvesting is generally irregular and dynamically changing.

During the test, the tested sample was placed on the prepared contact plate, and the inclination angle of the plate was gradually increased at a uniform rate. The critical angle θ at which the sample started to slide was recorded. Each contact pair was tested 10 times, and the average value was used as the measured static friction coefficient. The static friction coefficient was calculated as follows:

$$\gamma =\tan \theta$$

(1)

where γ is the static friction coefficient between the tested material and the contact material, θ is the inclination angle of the plane.

To evaluate the effect of pod arrangement on the measured static friction coefficient, two peanut pod–peanut pod contact conditions were compared: aligned arrangement and random arrangement. In the aligned arrangement, the longitudinal axes of peanut pods were approximately parallel, which represented an idealized and ordered contact condition. In the random arrangement, the peanut pods were placed with irregular orientations, which was closer to the actual contact state during mechanical harvesting.

The measured static friction coefficients under the aligned and random arrangements were 0.62 and 0.71, respectively. Compared with the aligned arrangement, the random arrangement produced a higher static friction coefficient and greater fluctuation in the critical sliding angle. This difference was mainly caused by the irregular surface morphology of peanut pods and the local interlocking between adjacent pods under random orientation. Since peanut pods are generally disordered and continuously changing in posture during harvesting, the random-arrangement friction coefficient was selected as the DEM input parameter for the peanut pod–peanut pod contact pair.

2.2.2. Determination of the Collision Recovery Coefficient

Traditional methods for determining impact recovery coefficients typically rely on free-fall impact tests. However, these methods are susceptible to errors caused by the drop height. When the drop height is too great, air resistance can cause the material’s trajectory to deviate. When the drop height is too small, the material fails to achieve sufficient bounce height. Therefore, a new impact drop test rig has been developed based on the principle of inclined-plane impact.

Construct an impact drop test rig based on the inclined plane collision principle, as shown in Figure 3. Measure the velocity of the peanut pod before collision and the resultant value of velocity components along the three coordinate axes after collision. Use Equation (2) to determine the collision recovery coefficient.

$$e={\displaystyle \frac{v_2}{v_1}}$$

(2)

where e is the collision recovery coefficient between materials, v₂ is the instantaneous rebound velocity of peanut pods after collision, v₁ is the instantaneous velocity of the peanut pod during its descent prior to collision.

The principle of the improved impact test bench is shown in Figure 3. The test bench consisted of an impact cylinder, a collision platform, a test sample, a reflective surface, and a high-speed camera. During the test, the peanut pod was released from the impact cylinder and then collided with the material plate mounted on the collision platform. The collision platform could be kept horizontal or adjusted to a specified inclination angle, so that the impact behavior under different collision angles could be measured.

Compared with the conventional free-fall rebound method, in which the restitution coefficient is usually calculated from the drop height and rebound height under vertical impact, the present test bench was designed to obtain the velocity components of the irregular peanut pod before and after collision. This design is more suitable for peanut pods because their irregular shape often causes oblique rebound, rotation, and non-planar motion after impact. Therefore, using only the rebound height may introduce errors in the restitution coefficient.

The reflective surface in Figure 3 was used as an optical mirror rather than as a collision surface. It allowed the high-speed camera to record the side-view projection of the peanut pod trajectory while the reference wall provided the front-view projection. By combining the coordinates obtained from the reference wall and the reflective surface, the three-dimensional position of the peanut pod center could be reconstructed. The velocity components along the X-, Y-, and Z-directions after collision were then calculated from two consecutive frames. The resultant rebound velocity was used to calculate the collision restitution coefficient.

Perform a kinematic analysis of the falling, collision, and rebound process of a peanut pod. The peanut pod undergoes uniformly accelerated motion in the vertical direction prior to colliding with the material plate. The material plate is positioned horizontally or at a certain angle of inclination. When the collision angle is 0°, establish a horizontal Cartesian coordinate system XYZ, as shown in Figure 4a. The spatial reference wall coordinates before collision are (Ya, Za). The coordinates on the reflecting mirror surface are (Xa, Za). At the moment of collision, the spatial reference wall coordinates are (Y₀, Z₀). The coordinates on the reflecting mirror surface are (X₀, Z₀).

Before collision, the peanut pod moved approximately with uniform acceleration in the vertical direction. The average velocity during the falling interval was first calculated from the displacement and time. Then, according to the uniformly accelerated motion equation, the instantaneous velocity immediately before collision was obtained. The average velocity during this period is expressed as:

$${\overline{v}}_a={\displaystyle \frac{z_a- z_0}{t_a}}$$

(3)

The instantaneous velocity before collision can be expressed as:

$$v_1={\displaystyle \frac{Z_a-Z_0}{t_a}}+{\displaystyle \frac{1}{2}}g{t}_a$$

(4)

Perform a kinematic analysis of the peanut pod’s descent, impact, and rebound process. Obtain the component velocities along the three spatial axes. The equation is expressed as:

$$v_2=\sqrt{v_x^2+v_y^2+v_z^2}$$

(5)

The collision recovery coefficient equation between peanut pods and materials is simplified as follows:

$$e={\displaystyle \frac{v_2}{v_1}}={\displaystyle \frac{2t_a\sqrt{v_x^2+v_y^2{+}_z^2}}{2Z_a-2Z_0+g{t}_a^2}}$$

(6)

where v_x is the component velocity of the pod along the X-direction after collision, v_y is the component velocity of the pod along the Y-direction after collision, v_z is the component velocity of the pod along the Z-direction after collision, g is the acceleration due to gravity, and t_a is the acceleration time.

When the collision material plate is inclined at an angle α, the spatial coordinate system XYZ is transformed into the inclined spatial coordinate system X′Y′Z′, as shown in Figure 4b. The relationship equations for the spatial coordinate systems are:

$$\left\{\begin{array}{l}X\prime =X\\ Y\prime =Y\cos \alpha +Z\sin \alpha \\ Z\prime =Z\cos \alpha -Y\sin \alpha \end{array}\right.$$

(7)

The contact between peanut pods and the inclined material plate can be decomposed into a normal collision recovery coefficient and a tangential collision recovery coefficient. The following equations express these:

$$\left\{\begin{array}{l}{e}_n={\displaystyle \frac{\left|v_{z\prime }\right|}{v_{1z^{\prime }}}}={\displaystyle \frac{2t_a\left|v_z\sin \alpha -v_y\cos \alpha \right|}{\left(2z_a-2z_0+g{t}_a^2\right)\sin \alpha }}\\ e_t={\displaystyle \frac{\sqrt{v_{x^{\prime }}^2+v_{y^{\prime }}^2}}{\left|v_{1y^{\prime }}\right|}}={\displaystyle \frac{2t_a\sqrt{v_x^2+{(v_z\cos \alpha +v_y\sin \alpha )}^2}}{\left(2z_a-2z_0+g{t}_a^2\right)\cos \alpha }}\end{array}\right.$$

(8)

where $v_x^{\prime }$ is the component velocity of the pod along direction x′ after collision, $v_y^{\prime }$ is the component velocity of the pod along direction y′ after collision, $v_z^{\prime }$ is the component velocity of the pod along direction z′ after collision.

Through kinematic analysis of the falling, collision, and rebound process of peanut pods, it was found that when the collision angle remains constant, the collision recovery coefficient between peanut pods and the material plate depends on the displacement change of the peanut pods before and after collision.

Decompose the motion of the peanut pod after collision rebound into uniform horizontal motion and uniformly decelerated vertical motion. Use a high-speed camera to record the coordinate points at the instant the peanut pod rebounds. The reference wall coordinates are (Y₀, Z₀). The mirror surface coordinates are (X₀, Z₀). When rebounding to a certain height, the motion time is denoted as t_n. The reference wall coordinates are (Y_n, Z_n). The mirror surface coordinates are (X_n, Z_n). Then the instantaneous horizontal velocity is expressed as:

$$v_x={\displaystyle \frac{x_n-x_0}{t_n}}$$

(9)

$$v_y={\displaystyle \frac{Y_n-Y_0}{t_n}}$$

(10)

Vertical uniform deceleration motion can be expressed as:

$$v_z={\displaystyle \frac{Z_n-Z_0}{t_n}}+{\displaystyle \frac{1}{2}}g{t}_n$$

(11)

Substitute Equations (9)–(11) into Equation (6). Rearrange to obtain the collision recovery coefficient equation:

$$e={\displaystyle \frac{\sqrt[2t_a]{\frac{{\left(x_n-x_0\right)}^2+{\left(Y_n-Y_0\right)}^2+{\left(Z_n-Z_0\right)}^2}{t_n^2}+\left(Z_n-Z_0\right)g+\frac{1}{4}{g}^2{t}_n^2}}{2Z_a-2Z_0+g{t}_a^2}}$$

(12)

Using a high-speed camera, capture the coordinate values of the center point of the peanut pod between two frames. Substituting these values into Equation (12) allows calculation of the collision recovery coefficient between the peanut pod and the material.

Because peanut pods are irregular biological particles, post-collision rotational motion may contribute to energy dissipation [27]. Therefore, the rotation of the pod after collision was also evaluated using high-speed images. The rotational kinetic energy was estimated as:

$$E_r={\displaystyle \frac{1}{2}}(I_x{\omega }_x^2+I_y{\omega }_y^2+I_z{\omega }_z^2)$$

(13)

and the translational kinetic energy was calculated as:

$$E_t={\displaystyle \frac{1}{2}}m(v_x^2+v_y^2+v_z^2)$$

(14)

The proportion of rotational energy was then expressed as:

$${\eta }_r={\displaystyle \frac{E_r}{E_t+E_r}}\times 100\%$$

(15)

where E_r is the rotational kinetic energy, E_t is the translational kinetic energy, I_x, I_y, and I_z are the moments of inertia of the peanut pod around the three principal axes, ω_x, ω_y, and ω_z are the angular velocities, m is the pod mass, and v_x, v_y, and v_z are the rebound velocity components. This treatment improves the reliability of the restitution coefficient used in the DEM model.

The peanut pod was approximated as an equivalent triaxial ellipsoid for estimating the moments of inertia. The semi-axes of the equivalent ellipsoid were defined as a = L/2, b = W/2, and c = H/2, where L, W, and H are the length, width, and thickness of the pod, respectively. Based on the high-speed image analysis, the average rotational kinetic energy was Er = 0.895 × 10⁻⁵ J, and the average rotational energy ratio was ν_r = 15.84%. This result indicates that post-collision rotation accounted for 1/5 proportion of the total kinetic energy. Therefore, considering rotational motion improved the reliability of the restitution coefficient used in the DEM model.

2.2.3. Determination of Rolling Friction Coefficient

The determination of the coefficient of rolling friction utilizes the law of conservation of energy. The change in a material’s kinetic energy is equal to the work done on it by the net external force. The experimental principle is shown in Figure 5.

The rolling friction coefficient was determined based on the energy balance of a cylindrical sample rolling along an inclined plane and then along a horizontal plane. Since peanut pod stalks have irregular shapes that make stable rolling difficult, pod stalk samples were processed into short cylindrical specimens while retaining their inherent surface characteristics as much as possible. During the test, the cylindrical sample rolled down the inclined plane and continued moving along the horizontal plane until it stopped.

At the initial and final states, the translational and rotational velocities of the sample were both zero. Therefore, the gravitational potential energy lost during rolling along the inclined plane was assumed to be dissipated by rolling resistance along both the inclined and horizontal sections. The energy balance can be expressed as:

$$\left\{\begin{array}{l}\Delta E=E_p-W_s=0\\ E_p=mgh=mgx\sin \alpha \\ W_s=F_{f}s=mgf(x\cos \alpha +l)\end{array}\right\}$$

(16)

where ΔE represents the change in kinetic energy; Ep represents the gravitational potential energy of the material; Ws represents the work done by the frictional force; m represents the mass of the material; g represents the acceleration due to gravity; Ff represents the frictional force; f represents the coefficient of rolling friction; x represents the distance the material rolls down the inclined plane; l represents the distance the material rolls along the horizontal plane; and α represents the angle of inclination of the plane.

By substituting the gravitational potential energy and the work done by rolling resistance into the energy balance equation and cancelling m and g, the rolling friction coefficient can be obtained as:

$$f={\displaystyle \frac{x\sin \alpha }{x\cos \alpha +l}}$$

(17)

This derivation clarifies that the rolling friction coefficient was calculated from the measured rolling distance along the inclined plane, the horizontal stopping distance, and the inclination angle of the plane.

The results of the physical tests described in Section 2.2.1, Section 2.2.2 and Section 2.2.3 are summarized in

Table 2. Measured contact parameters of peanut plant components.

Parameters	Value
	Coefficient of Static Friction	Coefficient of Rolling Friction	Collision Recovery Coefficient
Peanut pods—Peanut pods	0.62	0.47	0.34
Peanut pods—Steel	0.54	0.42	0.29
Pod stalk—Steel	0.71	0.58	0.21

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2.3. Determination of Mechanical Parameters

Taking peanut pods as an example, compression testing was performed using a texture analyzer. As shown in Figure 6a. The compression diagram is shown in Figure 6c. Peanut pods are irregular hollow shell-like biological structures rather than solid regular bodies. Therefore, the mechanical parameters obtained from compression tests should be interpreted as apparent or equivalent parameters for DEM parameterization, rather than as intrinsic material constants of the pod shell.

During the compression test, the peanut pod was placed between two parallel compression plates. The flat indenter moved along the transverse direction of the pod at a rate of 10 mm/min. To obtain the apparent deformation characteristics, a loading–unloading compression procedure was conducted within the small-deformation stage before macroscopic rupture of the pod. The force–displacement curve was recorded continuously. The closed curve shown in Figure 6b represents the loading–unloading hysteresis loop. The loading branch corresponds to the compression process, whereas the unloading branch corresponds to partial elastic recovery of the pod after unloading. The area enclosed by the loop reflects the energy dissipated by viscoelastic deformation, shell-wall friction, local irreversible deformation, and micro-damage of the hollow pod structure.

The apparent Poisson’s ratio was calculated from the ratio between the longitudinal strain and transverse strain in the small-deformation stage:

$${\epsilon }_i={\displaystyle \frac{\left|{e_i}^{\prime }\right|}{\left|e_i\right|}}={\displaystyle \frac{L_{i1}-L_{i2}}{W_{i1}-W_{i2}}}$$

(18)

where ε_i is the apparent Poisson’s ratio of the specimen. $e_i^{\prime }$ is the longitudinal deformation of the specimen. $e_i$ is the transverse deformation of the specimen. L_i1 is the longitudinal length of the specimen before loading. L_i2 is the longitudinal length of the specimen after loading. W_i1 is the transverse length of the specimen before loading. W_i2 is the transverse length of the specimen after loading. i is the number of specimens.

The apparent compressive elastic modulus of the peanut pod was estimated according to the classical Hertz contact theory for local contact between curved biological bodies and rigid compression plates. Considering the irregular hollow structure of peanut pods, the calculated modulus was regarded as an apparent compressive modulus reflecting the overall contact stiffness of the pod under compression:

$$E_1={\displaystyle \frac{0.338F(1-{\mu }^2)}{D^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}{\left[K_1{\left({\displaystyle \frac{1}{R_1}}+{\displaystyle \frac{1}{R_1^{\prime }}}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}+K_2{\left({\displaystyle \frac{1}{R_2}}+{\displaystyle \frac{1}{R_2^{\prime }}}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right]}^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$$

(19)

where E₁ is the apparent compressive elastic modulus of the peanut pod, F is the compression force; D is the deformation of the peanut pod under compression, μ is Poisson’s ratio; R₁ and $R_1^{\prime }$ are the minimum and maximum curvature radii of the peanut pod at the contact point with the upper compression plate, R₂ and $R_2^{\prime }$ are the minimum and maximum curvature radii of the peanut pod at the contact point with the lower compression plate, and K₁ and K₂ are the Hertz contact coefficients determined according to the calculated value of cos θ.

According to schematic diagram 6 (d) and (e), the maximum and minimum curvature radii are measured, and the following calculation is obtained.

$$\cos \theta ={\displaystyle \frac{R-R^{\prime }}{R+R^{\prime }}}$$

(20)

where R is the minimum radius of curvature of the peanut pod at the contact point, R′ is the maximum radius of curvature of the peanut pod at the contact point.

The peanut pod was approximated as a curved biological body in contact with the compression plate. The minimum curvature radius R and maximum curvature radius R′ at the contact region were calculated from the measured pod thickness H and length L. The curvature radii were expressed as:

$$R={\displaystyle \frac{H}{2}}$$

(21)

$$R^{\prime }={\displaystyle \frac{H^2+{(\frac{L}{2})}^2}{2H}}$$

(22)

where L is the length of the peanut pod, H is the thickness of the peanut pod.

Additionally, tensile tests were conducted on the pod–pod stalk junction using a texture analyzer. Peanut pod–pod stalk junction specimens were clamped at both ends of the texture analyzer, ensuring no slippage between the grips and the specimen during stretching. After testing, the elastic modulus of each sample was calculated using Equation (23). Each test was repeated 20 times, and the results were averaged.

$$E_2={\displaystyle \frac{F_2\times L}{A\times ∆L}}$$

(23)

where F₂ is the tensile force, L is the effective length of the specimen, ΔL is the deformation of the specimen, A is the cross-sectional area of the specimen, and E₂ is the elastic modulus.

The mechanical parameters of the main parts of peanut plants were measured, with the results shown in

Table 3. Mechanical parameters of peanut plant components.

Component or Connection Region	Value
	Apparent Poisson’s Ratio	Apparent Elastic Modulus/MPa	Maximum Tensile Force/N
Peanut pod	0.372	96	/
Pod–stalk junction	0.247	118	12.74
Pod stalk	/	/	10.21

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2.4. DEM Modelling of Mature Peanut Plants

2.4.1. Contact and Bonding Model Implementation

In the DEM model, the Hertz–Mindlin contact model was used for non-bonded contacts among peanut pods, pod stalks, and steel components. This contact model describes the normal and tangential contact responses during collision, sliding, and rolling. However, the flexible deformation of stems and pod stalks and the fracture behavior at the pod–stalk junction cannot be represented by non-bonded contact alone. Therefore, the Hertz–Mindlin with Bonding model was introduced to describe the mechanical connection among adjacent particles within flexible plant organs and at the pod–stalk junction.

In the bonding model, adjacent particles are connected by a virtual bond with a given bonding radius, stiffness, and strength. The bond can transmit both forces and moments, thereby allowing the particle assembly to show bending, tensile, and shear resistance. During harvesting simulation, external impact and pulling loads are transferred through the stem, pod stalk, and pod–stalk junction. Therefore, it is necessary to calculate the stress state of each bond to determine whether local failure occurs. For this reason, the maximum normal and tangential stresses in the bond were calculated as:

$$\begin{array}{l}{\sigma }_{\max }={\displaystyle \frac{F_n}{A}}+{\displaystyle \frac{2M_t{R}_b}{J}}\\ {\tau }_{\max }={\displaystyle \frac{F_t}{A}}+{\displaystyle \frac{2M_n{R}_b}{J}}\end{array}$$

(24)

where σ_max is the maximum normal stress; τ_max is the maximum tangential stress; F_n and F_t are the normal and tangential forces; M_n and M_t are the normal and tangential moments; A is the bonding cross-sectional area; J is the polar moment of inertia of the bond section; and R_b is the bonding radius. The bond failure criterion is expressed as:

$$\begin{array}{l}{\sigma }_{\max }\ge {\sigma }_c\\ {\tau }_{\max }\ge {\tau }_c\end{array}$$

(25)

where σ_c is the normal critical stress and τ_c is the tangential critical stress. Once this criterion is satisfied, the bond is deleted, and separation between the peanut pod and pod stalk occurs.

The initial values of the bonding stiffness and critical stress were estimated according to the measured mechanical properties of the pod–stalk junction and then further calibrated through the Plackett–Burman screening test, steepest ascent test, and central composite design. Therefore, the bonding parameters used in the final DEM model were not only derived from theoretical estimation but also optimized according to the measured pod detachment force.

$$\left\{\begin{array}{l}{K}_n={\displaystyle \frac{4}{3}}{({\displaystyle \frac{1-{\epsilon }_a^2}{E_a}}+{\displaystyle \frac{1-{\epsilon }_b^2}{E_b}})}^{-1}{({\displaystyle \frac{r_a+r_b}{r_a{r}_b}})}^{-{\displaystyle \frac{1}{2}}}\\ K_s=({\displaystyle \frac{1}{2}}\sim {\displaystyle \frac{2}{3}})K_n\end{array}\right\}$$

(26)

where K_n is the normal stiffness coefficient. K_s is the tangential stiffness coefficient. ε_a and ε_b are the Poisson’s ratios of the particles at both ends of the connecting key. E_a and E_b are the elastic moduli of the particles at both ends of the connecting key. r_a and r_b are the radii of the particles at both ends of the connecting key.

Compressive strength is expressed as:

$$\sigma ={\displaystyle \frac{F}{A_1}}={\displaystyle \frac{F}{\pi R^2}}$$

(27)

where б is the critical normal stress. F is the critical pressure of the material. A is the compression area. R is the compression surface radius.

The critical tangential stress can be expressed as:

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

(28)

where τ is the critical tangential stress. φ is the internal friction angle of the material. c is the cohesive force of the material particles.

The normal stiffness coefficient, tangential stiffness coefficient, normal critical stress, and tangential critical stress per unit area of the peanut pod–pod stalk junction system were calculated. The results are shown in

Table 4. Bonding parameters of peanut plant discrete element model.

Adhesive Components	Parameters	Value
Peanut pods—Pod stalk	Normal stiffness per unit area/105 (N/m3)	5
	Tangential stiffness per unit area/105 (N/m3)	5
	Normal critical stress/105 Pa	1
	Tangential critical stress/105 Pa	1
	Bonding radius x9/(mm)	1.1

. The particle bonding radius is typically 1.2 to 2 times the particle radius. The particle radius was set based on the geometric characteristics of peanut pods, pod stalks, and stems. The radius of the bond between the pod and the pedicel is set to 1.1 mm.

2.4.2. Time-Step Determination and Solver Settings

The simulation time step was determined according to the Rayleigh time-step criterion. To ensure numerical stability, the actual time step was set as a fixed proportion of the Rayleigh time step rather than as an independent empirical value. In this study, the calculated Rayleigh time step was T_R = 7.6575 × 10⁻⁶ s. The simulation time step was set to 6.5% of the Rayleigh time step:

$$\Delta t=0.065T_R$$

(29)

Thus, the time step used in all simulations was 4.98 × 10⁻⁷ s. Particle generation time was 0.0042 s, and bond generation time was 0.008 s. The total simulated physical time was 1 s. The simulations were conducted using EDEM (2024). The gravity acceleration was set to 9.81 m/s². The solver used 12th Gen Intel^® Core™ i7-12700, and the data output interval was 0.01 s. The same solver settings were used for all calibration and validation simulations to ensure consistency.

2.4.3. Peanut Plant Model Morphology

The main stem and branches of peanut plants are characterized based on actual growth patterns. Key features include peanut pods and pod stalks distributed around the plant’s root system. This forms the primary structure of the peanut plant model. The coordinate values for the main structural components of the peanut plant model are shown in Figure 7.

Particle generation time is 0.0042 s. Bond generation time is 0.008 s. The total number of adhesive bonds between peanut pods and pod stalks is 191. The total number of pod-to-pod adhesive bonds is 201. The total number of stem-to-leaf adhesive bonds is 150. The average computational time for a simulation with a physical time of 1 s was 16.384 h. This configuration forms a flexible peanut plant body, as shown in Figure 8.

2.5. Experimental Design

Because the flexible peanut plant DEM model contains multiple intrinsic, contact, and bonding parameters, a stepwise parameter calibration strategy was adopted. First, Plackett–Burman design was used to screen the factors that significantly affected pod–pod stalk tensile force. Second, the steepest ascent method was applied to approach the optimal parameter region. Finally, central composite design was used to establish a response surface model and determine the optimal combination of significant bonding parameters.

The coded values in

Table 5. Plackett–Burman test factors and codes.

Parameter	Encoding
	−1	1
Static friction coefficient (pod stalk–steel) x1	0.61	0.82
Rolling friction coefficient (pod stalk–steel) x2	0.45	0.76
Collision recovery coefficient (pod stalk–steel) x3	0.16	0.33
Static friction coefficient (peanut pods–steel) x4	0.42	0.68
Rolling friction coefficient (peanut pods–steel) x5	0.32	0.54
Collision recovery coefficient (peanut pods–steel) x6	0.18	0.39
Tangential stiffness per unit area (peanut pods–pod stalk) x7/(105 N/m3)	3.00	7.00
Tangential critical stress (peanut pods–pod stalk) x8/(105 Pa)	6.00	14.00
Bonding radius (peanut pods–pod stalk) x9/(mm)	1.00	1.20

were determined based on physical measurements, preliminary simulations, and the theoretically calculated bonding parameter range. For the contact parameters, including static friction coefficient, rolling friction coefficient, and restitution coefficient, the low and high levels were selected according to the measured range and preliminary DEM sensitivity tests. For the bonding parameters at the pod–pod stalk junction, the initial range was determined from the calculated stiffness and critical stress values, and then adjusted through preliminary tensile simulations to ensure that the simulated tensile force covered the measured tensile force range.

The coded value of each factor was calculated as:

$$x_i={\displaystyle \frac{X_i-X_0}{\Delta X_i}}$$

(30)

where x_i is the coded value of the i-th factor, X_i is the actual value, X₀ is the central value, and ΔX_i is the step size. This coding method makes the factors dimensionless and comparable in the statistical analysis.

3. Results and Discussion

3.1. Plackett–Burman Design of Experiments

Plackett–Burman experimental design is primarily used to efficiently identify a small number of key parameters that significantly influence the response value from a large set of candidate parameters. In this study, the significant parameters identified by the PB method will serve as input parameters for subsequent steepest ascent experiments and response surface design. Consequently, the interactions not resolved during the PB phase will be explicitly modeled and estimated during the response surface optimization phase. This standard workflow of “PB screening + steepest ascent + response surface optimization” has been widely adopted and validated in the field of discrete-element parameter calibration.

This result is consistent with previous DEM parameter-calibration studies on agricultural biological materials, in which a screening experiment is commonly used to identify the dominant parameters before response surface optimization is performed [2,3,4,5,18]. However, most previous studies focused mainly on the calibration of contact parameters for seeds, stems, roots, or soil–plant interfaces. In contrast, the response variable in the present study was the pod–pod stalk tensile force, which is directly associated with the fracture behavior of the weak biological junction during peanut harvesting. Therefore, the calibrated parameters were not only contact parameters but also bonding parameters governing local detachment failure.

Based on the separation characteristics between peanut pods and pod stalks during peanut harvesting, a discrete element model for peanut pods–pod stalks was experimentally investigated. The intrinsic parameters and contact parameters of peanut pods and pod stalks were employed as experimental factors, with tensile strength serving as the test metric. The Plackett–Burman contact parameter experimental design and results are presented in

Table 6. Plackett–Burman experimental design and results.

Order	X1	X2	X3	X4	X5	X6	X7	X8	X9	F/(N)
1	0.61	0.45	0.16	0.42	0.32	0.18	3.00	6.00	1.00	8.05
2	0.82	0.76	0.16	0.68	0.54	0.39	3.00	6.00	1.00	8.29
3	0.82	0.45	0.16	0.42	0.54	0.18	7.00	14.00	1.00	6.89
4	0.82	0.45	0.33	0.68	0.32	0.39	7.00	14.00	1.00	6.17
5	0.61	0.45	0.16	0.68	0.32	0.39	7.00	6.00	1.20	10.69
6	0.715	0.605	0.245	0.55	0.43	0.285	5.00	10.00	1.10	6.91
7	0.61	0.76	0.33	0.42	0.54	0.39	7.00	6.00	1.00	12.16
8	0.82	0.76	0.33	0.42	0.32	0.18	7.00	6.00	1.20	11.79
9	0.715	0.605	0.245	0.55	0.43	0.285	5.00	10.00	1.10	6.23
10	0.82	0.45	0.33	0.68	0.54	0.18	3.00	6.00	1.20	7.87
11	0.82	0.76	0.16	0.42	0.32	0.39	3.00	14.00	1.20	6.17
12	0.61	0.76	0.33	0.68	0.32	0.18	3.00	14.00	1.00	5.31
13	0.715	0.605	0.245	0.55	0.43	0.285	5.00	10.00	1.10	6.11
14	0.61	0.76	0.16	0.68	0.54	0.18	7.00	14.00	1.20	7.39
15	0.61	0.45	0.33	0.42	0.54	0.39	3.00	14.00	1.20	6.72

.

Perform analysis of variance and significance testing on the Plackett–Burman experimental results in Table 6. The results are shown in

Table 7. Significance analysis of the results of the Plackett–Burman test.

Source	Sum of Squares	df	Mean Square	F-Value	p-Value	Significance
Model	55.67	9	6.19	35.08	0.0019	**
X1	0.8216	1	0.8216	4.66	0.0970
X2	1.86	1	1.86	10.53	0.0315	*
X3	0.5376	1	0.5376	3.05	0.1557
X4	3.06	1	3.06	17.36	0.0141	*
X5	0.1083	1	0.1083	0.6142	0.4770
X6	0.7008	1	0.7008	3.97	0.1170
X7	13.40	1	13.40	75.98	0.0010	**
X8	34.00	1	34.00	192.84	0.0002	**
X9	1.18	1	1.18	6.68	0.0610
Residual	0.7053	4	0.1763	/	/
Lack of Fit	0.3331	2	0.1665	0.8947	0.5278
Pure Error	0.3723	2	0.1861	/	/
Cor Total	63.37	14	/	/	/

Note: * indicates significant effect (0.01 < p < 0.05), ** indicates extremely significant effect (p ≤ 0.01).

. The tensile strength coding equation is obtained as follows:

$$F=8.13-0.2617x_1+0.3933x_2+0.2117x_3-0.505x_4+0.095x_5+0.2417x_6+1.06x_7-1.68x_8+0.3133x_9$$

(31)

As shown in Table 7, The analysis of variance indicated that the Plackett–Burman regression model was statistically significant (p < 0.01), and the coefficient of determination R² was 0.9875, indicating that the model could explain most of the variation in the simulated tensile force. The lack-of-fit term was not significant (p > 0.05), suggesting that the regression equation was adequate for screening the key factors affecting pod–pod stalk separation. Among the nine candidate parameters, the tangential stiffness per unit area X₇ and tangential critical stress X₈ showed the strongest effects on the tensile force. This result is mechanically reasonable because X₇ determines the elastic resistance of the bonded pod–pod stalk junction before fracture, whereas X₈ defines the tangential stress threshold for bond failure. In contrast, friction and restitution parameters mainly affect external contact and rebound behaviors and therefore have relatively weaker influence on the tensile fracture response of the pod–pod stalk junction. Based on the screening results, X₇ and X₈ were selected for subsequent steepest ascent and response surface optimization.

Analysis of variance was performed to evaluate the significance of each factor in the Plackett–Burman design. In Table 7, df denotes the degrees of freedom, which represents the number of independent pieces of information used to estimate each source of variation. The mean square was calculated by dividing the sum of squares by the corresponding df. The F-value was calculated as the ratio of the mean square of each factor to the residual mean square. A larger F-value indicates that the factor has a stronger influence on the response variable relative to random error. The p-value was used to determine the statistical significance of each factor.

Based on the aforementioned test results, X₇ and X₈ were selected as experimental factors. The relative error between the simulated tensile force and the actual tensile force was used as the response value. The steepest incline test was conducted. The test plan and results are shown in

Table 8. Steepest climbing test plan and results.

Order	X7/(105 N/m3)	X8/(105 Pa)	F/(N)	Relative Error/(%)
1	3.00	6.00	5.17	27.99%
2	4.00	8.00	5.95	17.13%
3	5.00	10.00	6.87	4.31%
4	6.00	12.00	7.91	10.17%
5	7.00	14.00	8.53	18.80%

.

As shown in Table 8, the measured peanut pods–pod stalk tensile force gradually increases with the progressive increase of experimental factors X₇ and X₈. The relative error compared to the actual tensile force first decreases and then increases. The relative error was minimal in the third experimental group, at 4.31%. Therefore, the parameters from the third experimental group were selected as the central point, with the parameters from the second and fourth groups serving as the low and high levels, respectively, for the subsequent experiments.

The dominant influence of tangential stiffness per unit area and tangential critical stress is also in agreement with previous flexible crop DEM models, where the stiffness and strength of bonding elements determine the bending, tensile, and fracture responses of plant organs [6,7,8,9,10,11,16,17]. For flexible plant materials, external contact parameters such as friction and restitution mainly influence sliding, rebound, and energy dissipation during collision, whereas bonding stiffness and critical stress directly control the load-transfer capacity and failure threshold of connected biological tissues. Therefore, the stronger effects of X₇ and X₈ observed in this study indicate that pod detachment is governed primarily by the mechanical resistance and fracture threshold of the pod–pod stalk junction rather than by the external collision behavior of peanut pods.

3.2. Central-Composite Experiment

To investigate the effects of tangential stiffness per unit area X₇ and tangential critical stress X₈ on tensile force, a Central Composite design was employed. The factor codes for the experiment are shown in

Table 9. Central—Composite test factors and coding.

Order	Experimental Factors
	X7/(105 N/m3)	X8/(105 Pa)
−1.414	4.00	8.00
−1	4.293	8.586
0	5.00	10.00
1	5.707	11.414
1.414	6.00	12.00

. The experimental plan and results are presented in

Table 10. Central—Composite test scheme and results.

Order	X7/(105 N/m3)	X8/(105 Pa)	F/(N)
1	4.293	8.586	6.59
2	4.293	11.414	6.97
3	5.00	12.00	7.84
4	6.00	10.00	7.21
5	5.00	10.00	6.79
6	5.00	8.00	6.63
7	5.00	10.00	6.84
8	5.00	10.00	6.91
9	5.00	10.00	7.08
10	5.707	8.586	7.01
11	4.00	10.00	6.59
12	5.00	10.00	7.17
13	5.707	11.414	7.91

.

A significance analysis was conducted on the test result data in Table 10, with the results shown in

Table 11. Central—Composite test significance analysis.

Source	Sum of Squares	df	Mean Square	F-Value	p-Value	Significance
Model	1.98	5	0.3961	17.09	0.0008	**
X7	0.6254	1	0.6254	26.99	0.0013	**
X8	1.12	1	1.12	48.26	0.0002	**
X7×8	0.0676	1	0.0676	2.92	0.1314
X72	0.0018	1	0.0018	0.0757	0.7912
X82	0.1599	1	0.1599	6.90	0.0341	*
Residual	0.1622	7	0.0232	/	/
Lack of Fit	0.0579	3	0.0193	0.7408	0.5807
Pure Error	0.1043	4	0.0261	/	/
Cor Total	2.14	12	/	/	/

Note: * indicates significant effect (0.01 < p < 0.05), ** indicates extremely significant effect (p ≤ 0.01).

.

The significance of the quadratic regression model obtained from the central composite design was evaluated using analysis of variance. In Table 11, df represents the degrees of freedom for the model, each model term, residual error, lack of fit, and pure error. The F-value is the ratio of the mean square of a given source of variation to the residual mean square, and it was used to test whether the corresponding model term had a statistically significant effect on the tensile force. The p-value represents the probability that the observed effect was caused by random error. Therefore, model terms with smaller p-values and larger F-values were considered to have more significant effects on the response.

The analysis of variance showed that the quadratic regression model was highly significant, indicating that the model could effectively describe the relationship between the bonding parameters and the tensile force. The lack-of-fit term was not significant, demonstrating that the regression model had good fitting accuracy within the selected factor range. The linear terms X₇ and X₈ were both extremely significant. This indicates that the tensile force was mainly controlled by the tangential stiffness per unit area and tangential critical stress at the pod–pod stalk junction. The interaction term X₇X₈ was not significant, suggesting that the two parameters mainly affected the tensile force independently within the tested range.

The fitted regression equation was:

$$F=6.96+0.2796X_7+0.3739X_8+0.13X_7{X}_8-0.0159X_7^2+0.1516X_8^2$$

(32)

Based on the regression equation, the contour plot (a) and response surface diagram (b) showing the effect of factor interactions on tensile force are obtained. As shown in Figure 9, tensile force increases with rising tangential stiffness per unit area and tangential critical stress.

Optimize using the Design Expert 13 software module. Investigate the optimal parameter combinations through simulation and establish the following constraints:

$$\left\{\left.\begin{array}{c}min\left(X_7,X_8\right)\\ s.t.\left\{\left.\begin{array}{c}4\le X_7\le 6\\ 8\le X_8\le 12\end{array}\right\}\right.\end{array}\right\}\right.$$

(33)

Through optimization analysis, the tangential stiffness per unit area between the peanut pod and pod stalk was determined to be 4.738 × 10⁵ N/m³, with a Tangential critical stress of 9.350 × 10⁵ Pa. At this point, the force required to separate the peanut pod from the stalk is 6.73 N.

Compared with previous peanut DEM studies, the present model provides a more detailed representation of the pod–pod stalk separation process. Earlier peanut models usually simplified the plant as a single-pod structure, a rigid pod particle, or a partial stem–pod system [24,25,26]. Such simplifications are useful for reducing computational cost, but they have limited ability to describe the progressive load transfer from the vine and pod stalk to the pod–stalk junction and the subsequent fracture of the weak connection. In the present study, the optimized tangential stiffness and tangential critical stress enabled the bonding model to reproduce the measured pod detachment force. This indicates that the proposed model can capture not only pod motion but also the local mechanical failure process responsible for harvest-induced pod separation.

In the discrete element bonding model, the tangential stiffness per unit area reflects the ability of the peanut pod–pod stalk junction to resist elastic deformation. The tangential critical stress simulates the stress threshold at which the peanut pod–pod stalk junction fractures during the harvesting process. From a botanical perspective, the cellular arrangement at this node undergoes significant changes, intercellular spaces widen, and the vascular bundle is discontinuous at this point, with no continuous vessels passing through the region. This node thus becomes the weak point where the peanut pod detaches. The cross-sectional area inferred by calculating the relationship between the breaking force and the critical stress is biologically plausible.

3.3. Validation Test

To validate the accuracy and reliability of the peanut pod and pod stalk contact parameters and adhesion parameters in peanut plants, comparative experiments were conducted between simulation and field conditions. The comparative trial was conducted on 18 October 2025, at a peanut trial field in Dongying City, Shandong Province, China (37°37′ N, 118°39′ E). The reliability of the model was verified by statistically analyzing the loss rate error during peanut harvesting, as shown in Figure 10.

For the field trials, samples were taken from mature peanuts ready for harvest. One-meter-long sections of peanut ridges were selected, and the trials were repeated four times. The peanut harvesting simulation was also conducted four times. For each trial, results were calculated by taking the average. The statistical results of the validation tests are shown in

Table 12. Deviation between physical and simulation test results.

Test	Loss Rate During Peanut Harvesting/%
Simulation Testing (y1)	3.75	4.02	4.27	3.18
Field Trials (y2)	4.01	3.83	4.04	3.42
Absolute value of relative error	6.93	4.73	5.39	7.55

.

To determine the extent of deviation between the simulation results and the field trial measurements, the absolute values of the relative errors were calculated. The relative error in the peanut pods drop rate between simulation and field trial results was less than 7.55%. The above statistical results confirm that the simulation test results are consistent with those of the field trials. To avoid mistaking random errors for actual experimental effects, a t-test was conducted to further validate the model’s validity. First, the weighted average of the two sample means was calculated using the following equation:

$$s_e^2={\displaystyle \frac{S{S}_1+S{S}_2}{{\nu }_1+{\nu }_2}}={\displaystyle \frac{{\displaystyle \sum {(y_1-{\overline{y}}_1)}^2+{\displaystyle \sum {(y_2-{\overline{y}}_2)}^2}}}{(n_1-1)+(n_2-1)}}$$

(34)

Thus, the standard error of the difference between the means of the two samples is:

$$s_{{\overline{y}}_1-{\overline{y}}_2}=\sqrt{{\displaystyle \frac{s_e^2}{n_1}}+{\displaystyle \frac{s_e^2}{n_2}}}$$

(35)

Further calculations yielded the following results:

$$\left|t\right|=\left|{\displaystyle \frac{{\overline{y}}_1-{\overline{y}}_2}{s_{{\overline{y}}_1-{\overline{y}}_2}}}\right|<t_{0.05}$$

(36)

Based on the above analysis, p > 0.05, confirming that there is no statistically significant difference between the two groups. This validates the accuracy of the discrete element parameters in the flexible peanut plant model established in this paper.

The validation results also indicate that the flexible bonding representation of the pod–pod stalk junction is effective for simulating harvest-induced pod separation. Compared with simplified rigid models, the proposed model can describe not only the translational motion of peanut pods but also the local fracture process at the weak biological connection between pod and pod stalk. This provides a useful DEM modelling basis for analyzing peanut harvesting mechanisms and optimizing the structural and operating parameters of peanut harvesters.

Field validation is an essential step for evaluating the reliability of DEM models used in agricultural machinery simulations. Previous studies on crop harvesting and plant-material interaction have also combined DEM simulations with bench or field tests to verify model accuracy [12,18,23,26]. In this study, the relative error of pod loss rate between the simulation and field test was less than 7.55%, and the t-test showed no significant difference between the two datasets. These results suggest that the calibrated flexible peanut plant model can reasonably reproduce the pod detachment and loss formation behavior under harvesting conditions. Nevertheless, the remaining error may be related to biological variability among peanut plants, spatial differences in pod distribution, moisture-content variation, and simplifications in the representation of root–soil interaction. These factors should be further considered when the model is extended to different varieties, soil conditions, and harvesting devices.

4. Conclusions

In this study, a DEM-based flexible model of mature peanut plants was developed and validated for peanut harvesting simulations. The main conclusions are as follows:

(1) The contact and mechanical parameters of peanut pods, pod stalks, and stems were measured through physical experiments. The Plackett–Burman screening results showed that the tangential stiffness per unit area and tangential critical stress at the pod–pod stalk junction were the dominant parameters affecting pod detachment force. This indicates that pod separation during harvesting is mainly governed by the elastic resistance and fracture threshold of the weak biological connection between pod and pod stalk.

(2) The optimized bonding parameter combination was obtained through steepest ascent and central composite experiments. The optimal tangential stiffness per unit area was 4.738 × 10⁵ N/m³, and the optimal tangential critical stress was 9.350 × 10⁵ Pa. Under this parameter combination, the simulated pod–pod stalk tensile force was 6.73 N, which was close to the measured tensile force and satisfied the calibration requirement.

(3) Field validation showed that the relative error of pod loss rate between DEM simulation and field trial was less than 7.55%, and the t-test indicated no significant difference between the simulated and measured results (p > 0.05). These results confirm that the proposed flexible peanut plant model can reasonably reproduce pod detachment behaviors during harvesting.

The developed model can be used to support the mechanism analysis of peanut harvesters and the optimization of harvesting equipment. Future work will focus on extending the model to different peanut varieties, moisture contents, and soil conditions, as well as improving the description of rotational collision behaviors and root–soil interaction during the complete harvesting process.