Discrete Meta-Modeling Method of Breakable Corn Kernels with Multi-Particle Sub-Area Combinations

Abstract

Simulation is an important technical tool in corn threshing operations, and the establishment of the corn kernel model is the core part of the simulation process. The existing modeling method is to treat the whole kernel as a rigid body, which cannot be crushed during the simulation process, and the calculation of the crushing rate needs to be considered through multiple criteria such as the contact force, the number of collisions, and so on. Aiming at the issue that kernel crushing during maize threshing cannot be accurately modeled in discrete element simulations, in this study, a sub-area crushing model was constructed; representative samples with 26%, 30% and 34% moisture content were selected from a double-season maturing region in China; based on the physical dimensions and biological structure of the maize kernel, three stress regions were defined; and mechanical property tests were conducted on each of the three stress regions using a texturometer as a way to determine the different crushing forces due to the heterogeneity of the maize structure. The correctness of the model was verified by stacking angle and mechanical property experiments. A discrete element model of corn kernels was established using the Bonding V2 method and sub-area modeling. Bonding parameters were calculated by combining stacking angle tests and mechanical property tests. The flattened corn kernel was used as a prototype, and the bonding parameters were determined through size and mechanical property tests. A 22-ball bonding model was developed using dimensional parameters, and the kernel density was recalculated. Results showed that the relative error between the stacking angle test and the measured mean value was 0.31%. The maximum deviation of axial compression simulation results from the measured mean value was 22.8 N, and the minimum deviation was 3.67 N. The errors between simulated and actual rupture forces at the three force areas were 5%, 10%, and 0.6%, respectively. The decreasing trend of the maximum rupture force for the three moisture levels in the simulation matched that of the actual rupture force. The discrete element model can accurately reflect the rupture force, energy relationship, and rupture process on both sides, top, and bottom of the grain, and it can solve the error problem caused by the contact between the threshing element and the grain line in the actual threshing process to achieve the design optimization of the threshing drum. The modeling method provided in this study can also be applied to breakable discrete element models for wheat and soybean, and it provides a reference for optimizing the design of subsequent threshing devices.

1. Introduction

Corn is the first largest food crop and the second largest cash crop, and it is important for ensuring national food security (National Bureau of Statistics, 2022) [1]. Mechanized harvesting of maize is an important way to achieve stable maize yields, and with the technological development of modern society, the development of threshing devices that ensure a low crushing rate even under high moisture content conditions has become a new development direction. The threshing device is the core component to achieve kernel harvesting [2]; the contact of the threshing element and corn kernel has become an urgent problem to be solved. The establishment of a corn kernel model with a high degree of reduction to simulate the process of corn kernel breaking is a prerequisite for the study of low-loss and high-efficiency corn threshing device under high water content.

Moisture content significantly affects the physical and mechanical properties of maize kernels during threshing, which in turn affects kernel breakage. Currently, research on the effect of moisture content on grain properties has taken several forms, one form being the relationship between moisture content and crushing force in grains by Guo et al., Chandio et al., (Guo et al., 2022; Li et al.; Chandio et al., 2021; Li et al., 2021; Kruszelnicka et al., 2022) [2,3,4,5,6]. In another form, Mousaviraad et al. and Kruszelnicka et al. explored the effect of grain moisture content on the stacking angle and found that seed moisture significantly affected the physical and mechanical properties of maize kernels (Mousaviraad et al., 2020; Kruszelnicka et al., 2022) [7,8]. In addition, researchers often use the discrete element method (DEM) to simulate the threshing process for designs’ evaluation (Wang et al. 2020; Geng et al., 2023; Dong et al., 2023; Shi et al., 2023; Fang et al., 2024) [9,10,11,12,13,14]. The main ideas of the existing discrete element studies work three ways. One way is to explore the optimal parameters to assist in the design of the device by modeling the grains; e.g., Wang et al. developed a hollow cylindrical elastic binder based on the DEM method for the rice plant model. The second way is to optimize the parameters of the existing device; e.g., Geng et al. used the DEM method to optimize the existing threshing drum in terms of rotational speed, seed force, etc. The third way is to directly establish the grain model and verify the correctness of the established model; e.g., Shi et al. established a straw model of wheat through the bonding model of DEM and optimized the parameters; Fang et al. simulated the interaction between seeds and rollers and verified the established discrete element model by stacking angle experiments.

The development of an accurate discrete elemental model of maize seeds is crucial for studying high-moisture threshing mechanisms (Zhou et al., 2020; Wang et al., 2016) [15,16]. Zhou, et al. analyzed the shape and size of maize seeds and constructed seed models with different accuracies using a multi-sphere filling method. Typically, 8–25 spheres were used to represent malleable seeds, and the model accuracy increased with the number of filled spheres. Wang, X. et al. classified seed shapes and modeled wedge-shaped seeds by sub-sphere filling. The highest model accuracy was achieved when 14–16 spheres were used. Significant progress has been made in modeling maize seeds, which can be divided into two main approaches. The first is an unbreakable maize kernel model developed using a multi-sphere filling approach (Coetzee et al., 2009) [17]. The maize kernel is reduced to two rigidly connected spheres with diameters ranging from 5 to 12 mm. Xu et al. reduced soybean seeds to an ellipsoid of average size and produced principal dimensions via a normal distribution (Xu et al., 2018) [18]; Lu et al. modeled wheat seeds with an ellipsoid and two types of polyspheres (Lu et al., 2023) [19]. These models were tested under shear and compression conditions with a maximum error of 26% in the final test. These models need to be subjected to a full crushing analysis, including metrics such as contact cohesion, collision frequency, and normal force. The second method is a model based on 3D scanning (Li et al., 2022; Zhang et al., 2024) [20,21]. Li et al. used 3D scanning methods to obtain the 3D dimensions of maize kernels and constructed a polyhedral kernel model by the cross-section method. The contact parameters were optimized by the inclined drop method and the funnel method, combined with response surface analysis. The final errors of these two methods were 4.38% and 6.98%, respectively, confirming the accuracy of the model. However, Zhang pointed out that although these geometrically precise seeds provided an accurate representation of size, their unbreakable nature in DEM simulations led to the introduction of systematic errors in quantifying the breakage rate. The analysis suggests that current models of unbreakable maize kernels prevent the direct quantification of fragmentation rates in simulations. This limitation highlights the need to develop breakable discrete metamodels to assess kernel damage mechanisms. Existing modeling approaches focus on the overall force distribution within the model. However, their uniform and unbreakable nature requires experimental results to determine whether the impact force is in the breakage range or not, depending on the number of contacts and the total contact force. As a result, these models are unable to accurately simulate seed crushing under real conditions, leading to significant systematic errors in statistical analyses and experimental evaluations. To address these issues, a maize kernel model capable of directly simulating kernel breakage was developed to better assess damage mechanisms. In the current unbreakable kernel model, the rate of kernel breakage during simulation is usually calculated from metrics such as the number of contacts and contact force. However, as a rigid model, it cannot simulate the actual crushing effect of the kernel. Therefore, there is a need to develop a breakable discrete element model to evaluate the damage mechanism of maize kernels.

In this study, a breakable discrete element model considering structural heterogeneity is proposed for mature maize kernels. Unlike traditional methods, in this paper, based on the structural heterogeneity of maize kernels, the force regions of the kernels were divided, and the mechanical properties of each region were tested separately, identifying the critical crushing forces in specific force regions and obtaining a more accurate crushing force for the kernels. Core crushing was simulated using the EDEM (v2018) Bond V2 model (Han et al., 2024) [22], which creates a multi-particle bonded mass, as different particles need to be bonded during model building. Key parameters were calibrated by experimental–computational integration, and the validity of the model was verified by stacking angle tests.

2. Materials and Methods

2.1. Measurement of Physical Parameters of Maize Kernels

Dimensional characterization was performed on fifty randomly selected maize kernels, measuring five parameters, W1 (width at the point of maximum curvature change), W2 (maximum width), T1 (maximum thickness), H2 (height excluding the germ), and H1 (total height), as shown in Figure 1. The detailed measurements are presented in

Table 1. Maize kernel size measurements.

Measuring Position	Gauge/mm
W1	5.42
W2	9.30
T1	4.31
H1	11.88
H2	9.54

; the seed size distribution basically conformed to a normal distribution, as shown in Figure 1.

The Huanghuaihai region is characterized by biannual crop maturation. Corn is typically harvested with a moisture content ranging from 25% to 35%. Due to the need for immediate wheat sowing after the corn harvest, there is insufficient time to dry the corn kernels to lower moisture levels. Therefore, threshing operations must be conducted under high-moisture conditions. Due to the different times for the planting and harvesting of maize in different regions, the moisture content will deviate, combined with the local maize planting spacing and planting pattern; in order to be able to expand the scope of this paper’s research, the selection of maize kernels with a moisture content of 26%, 30% and 34% was made.

In accordance with the GB/T 10362-2008 standard for corn moisture’s determination, intact kernels were randomly sampled from the cob and analyzed using an LDS-1G computerized moisture meter (Beijing YINO Technologies, Beijing, China), as shown in Figure 2c. The measured moisture content was 26%, 30%, and 34%, with a measurement tolerance of ±0.5%.

2.2. Seed Force Area Division

In DEM simulations of threshing processes, contact forces between threshing components and kernels must reach critical thresholds to detach kernels from the cob while preserving their integrity. Excessive rotational speeds can cause mechanical forces that rupture kernels, increasing grain breakage rates. Therefore, threshing forces must be optimized to achieve complete detachment without compromising the kernels’ integrity.

Current DEM simulations estimate kernel breakage rates using contact parameters such as collision frequency and normal forces. However, corn kernels have heterogeneous material properties, with distinct hardness gradients in the pericarp, endosperm, and germ. Traditional quasi-static compression tests with universal testing machines (UTMs) measure bulk crushing forces by recording the peak stress during whole-kernel compression, but they fail to accurately capture localized failure thresholds.

This method measures the maximum crushing force, but during threshing and simulation, grains rarely achieve full surface contact with the threshing element. Instead, the contact is often limited to line contact. Due to this line contact phenomenon and the material heterogeneity of grains, localized contact forces may fail to reach region-specific failure thresholds or exceed bulk compressive strength thresholds, leading to inaccurate breakage classifications. Grain rupture is misclassified when applied stresses fall below localized critical values or exceed bulk material limits. Additionally, the crushing rates vary depending on the impact location, introducing a certain degree of error.

Following the established protocol, the kernel length excluding the germ (H2) was divided into three biomechanical stress zones based on force distribution patterns. Force areas 2 and 3 were positioned at the upper and lower thirds of H2, respectively. i.e., below the blue and green dotted lines in Figure 3. Similarly, the total kernel width (W2) was divided into lateral thirds, with symmetrical force application areas located at the left and right third-width positions; the symmetrical force application points were located outside the red dotted lines on the left and right, respectively. These areas, designated as FP1, FP2, and FP3 (Figure 3), correspond to the vertices of an internal tangent quadrilateral geometry. Quasi-static compression tests were performed at each force application area to determine the critical failure thresholds of individual kernels under localized loading. In order to ensure that the platen corresponded to the force area of the seed, when placing the seed position, the seed size was measured and placed in strict accordance with the size of the force.

2.3. Seed Mechanical Tests

Based on the references (Hommood, et al., 2020; Shi, et al., 2020; Han D, et al., 2025) [23,24,25], uniaxial compression tests were conducted at the three designated force areas using a YINO ENS-DVU texture analyzer (Beijing YINO Technologies, Beijing, China), as illustrated in Figure 4a. The instrument consists of a piston rod, base, pressure plate, control knob, and peripheral device. As shown in Figure 4b–d, seeds were positioned on the edge of the pressure plate according to the locations of the three force areas. This ensured that the edge of the pressure plate applied force directly to the corresponding areas. The tests aimed to determine the maximum crushing force of the grains under varying force area conditions. Each test was repeated three times, with results recorded sequentially and averaged to obtain the final value. The experimental data are shown in

Table 2. Maximum crushing force at different force areas for seeds with different moisture content.

Moisture Content/%	Critical Crushing Force 1/N	Critical Crushing Force 2/N	Critical Crushing Force 3/N
26	333.4	226.9	550.9
30	205.90	147.30	237.88
34	172.70	134.02	233.62

, and the ruptured seeds in the experiment are shown in Figure 5.

2.4. Definition of Spherical Particle Coordinates

Maize kernels are not homogeneous due to their composition, so the distribution of moisture within the kernel is also not homogeneous, which can negatively affect the modeling and parameter calculation of the kernel. Before determining the coordinates of spherical grains, the following assumptions were made: 1. The internal moisture of the seed grain is uniformly distributed and remains unchanged under external forces. 2. External environmental factors, such as temperature and humidity, are assumed to have no influence on the seed grain during model construction.

Building upon prior specifications, kernel geometry was constrained by a maximum width W₂ and thickness T₁. A 2D coordinate system originated from the kernel’s apex particle center, with constituent spherical particles mapped via discrete coordinates (i, j), where i and j represent the horizontal row and vertical column indices, respectively. That is, the center of the ball of each particle is regarded as a mass point; the coordinates have nothing to do with the distance between the centers of the balls of each particle but only represent the position of the center of the ball in relation to the whole corn kernel; e.g., the coordinates of the topmost particle are (0, 0), those of the leftmost particle of the second row are (−1, −1) and those of the middle particles of the second row are (0, −1), and so on, and the coordinates of each particle are as shown in

Table 3. Coordinates of individual maize kernel constituent spherical particles.

Particle Position	R/mm	Particle Position	R/mm
(0, 0)	0.825	(2, −3)	1.00
(−1, −1)	1.04	(−2, −4)	1.08
(0, −1)	0.825	(−1, −4)	1.08
(1, −1)	1.04	(0, −4)	0.85
(−1, −2)	1.00	(1, −4)	1.08
(0, −2)	0.85	(2, −4)	1.08
(1, −2)	1.00	(−2, −5)	1.08
(−2, −3)	1.00	(−1, −5)	1.08
(−1, −3)	1.00	(0, −5)	1.08
(0, −3)	0.85	(1, −5)	1.08
(1, −3)	1.00	(2, −5)	1.08

.

In the process of building up individual maize kernels, to ensure simulated crushing conditions of the maize kernels during threshing and that the crushing site is at the same location as the previously described force area, a gap was left between the different fracture sites of the same kernel to allow for the use of a bond to connect the individual spherical particles later. According to the location of the force area, a gap was left between the 3rd row of particles and the 4th row of particles; and a gap was left between (±1, j) (0 < j ≤ 5) and (0, j) (0 < j ≤ 5) particles.

When establishing the corn kernel model, it should be ensured that the size of each force area is the same as the size of the broken kernel, so rows 4–5 at the bottom of force area 1 are modeled by the superposition of two spheres, and according to the shape and structure of the corn kernel, which presents a four-pronged shape with a narrow upper part and a wide lower part, rows 2 and 3 of force area 1 are modeled with a single sphere connection. To simulate the irregular boundary when the grain breaks, and at the same time to ensure that the bond key can be generated smoothly when establishing the discrete element model of the grain, the radius of the spherical particles at (1, 3) and (−1, 3) is set to 0.9 mm when establishing the grain model, and the same is done for force areas 2 and 3. A single corn kernel consists of 22 spherical particles of different sizes, whose coordinates and contact radii are shown in Table 3, and the distribution of the individual spherical particles and breaking areas that make up the kernel is shown in Figure 6a,b.

To ensure variability in bond parameters between the force areas of the maize kernel, the kernel model was divided into seven distinct regions, labeled area 1 through to area 7. Adjacent regions are connected by bonds, with each bond type assigned unique parameters. The division of these regions is illustrated in Figure 6c.

To ensure that fractures at each force area align with the measured size of the maize kernel, the model was constructed using spherical particles with small radii. However, this approach affects the thickness of the kernel model. To compensate for this limitation, a granular unit for the maize kernel was introduced, as shown in Figure 7a. This granular unit features a three-layer arrangement, where three spherical particles are connected to form strips with overlapping sections between each granule. Each strip particle is treated as a single structural unit representing the corn seed grain. The final maize kernel model is shown in Figure 7b,c.

2.5. Calculation of Seed Density

After constructing the seed model based on the Martingale rule, its density must be recalculated. This adjustment is necessary because the spherical particles forming the seed model are connected by bonds, which create gaps to accommodate bond formation. These gaps result in changes to the seed’s overall density. So seed density should be recalculated.

As shown in Figure 8, the seed model consists of 22 spherical particles. The total seed volume can be calculated in the EDEM pre-processing module by summing the individual volumes of the spheres that form the seed. However, certain particles overlap, specifically pairs such as (0, 0) and (0, −1), (±1, −1) and (±1,−2), (±1, −3) and (±2, −3), (±1, −4) and (±2, −4), and (±1, −5) and (±2, −5). These overlapping regions, referred to as intersection areas 1–5, must be subtracted from the total volume to accurately determine the seed grain volume.

The overlapping volume between particles primarily takes the form of a spherical crown. This can be simplified by calculating the volume of the spherical crown using the formula for the overlapping volume of spheres with equal radii.

$$V={\displaystyle \frac{\pi (2r-d{)}^2}{12d}}(d^2+4dr)$$

(1)

In the formula, V represents the overlap volume of spherical particles (mm³), r represents the radius of spherical particles (mm), and d represents the spherical distance (mm).

By substituting the radius of the spherical particles and the distance between their centers, the overlapping volume for spherical particles of an equal radius is calculated. The results are presented in

Table 4. The volume of overlapping spherical particles of equal radius in a single seed layer.

r1/mm	r2/mm	Spherical Distance d/mm	Overlap Volume V/mm3
0.75	0.75	0.75	0.552
1.08	1.08	0.92	2.11
1.08	1.08	1.07	1.68

.

According to the formula for the overlapping volume of spheres of unequal radii,

$$V={\displaystyle \frac{\pi (r_1+r_2-d{)}^2}{12d}}[d^2+2d(r_1+r_2)-3{(r_1-r_2)}^2]$$

(2)

In the formula, V represents the overlap volume of spherical particles (mm³), r₁, and r₂ represent the radius of overlapping spherical particles (mm), and d represents the spherical center distance between two spherical particles (mm).

Substituting the radius of the spherical particles with the sphere center distance gives the overlapping volume of spherical particles of unequal radii, and the results are shown in

Table 5. The volume of overlapping spherical particles of unequal radius in a single seed layer.

r1/mm	r2/mm	Spherical Distanced d/mm	Overlap Volume V/mm3
1.08	0.90	0.78	1.740
1.00	1.04	1.54	0.370
1.00	0.90	0.10	0.381

.

After calculating the overlapping volume of each spherical particle, the total volume of each particle composing the seed grain was measured using EDEM software. First, individual spherical particles were defined in the Bulk Material section during the pre-processing stage of EDEM and classified by radius. Seven radii were established: 0.75 mm, 0.825 mm, 0.85 mm, 0.9 mm, 1 mm, 1.04 mm, and 1.08 mm. The corresponding volumes of these spherical particles, measured under the Properties section of the Bulk Material module, are presented in

Table 6. Volume and number of each spherical particle.

r/mm	V/mm3	Number of Particles
0.75	1.77 × 109	1
0.825	2.35 × 109	1
0.85	2.57 × 109	2
0.90	3.05 × 109	2
1.00	4.19 × 109	4
1.04	4.71 × 109	2
1.08	5.27 × 109	10

The volume of each spherical particle is named V_i (0 < i ≤ 7) and is obtained by summing the volumes as follows:

$$V_Z={{\displaystyle \sum V}}_i-{{\displaystyle \sum V}}_C$$

(3)

In the formula, V_z represents monolayer seed volume (mm³), V_i represents the volume of each spherical particle (mm³), and V_c represents the overlap volume of spherical particles (mm³).

Based on the given equation, the volume of a single seed grain monolayer is calculated to be 72.712 mm³. Since the seeds are arranged in three layers, the overlapping volume of the spherical particles within a single particle unit must also be considered. As the spherical particles in the particle unit have identical radii, the overlap volume can be determined using the formula for spheres of equal radius. The calculated overlap volumes between particle units are presented in

Table 7. Overlap volume of spherical particles in the particle unit.

r1/mm	r2/mm	Spherical Distanced d/mm	Overlap Volume V/mm3
0.75	0.75	0.75	0.552
0.75	0.75	1.00	0.262
0.825	0.825	1.00	0.476
0.85	0.85	1.00	0.564
0.90	0.90	1.00	0.770
1.00	1.00	1.00	1.308
1.04	1.04	1.00	1.576
1.08	1.08	1.00	1.874

.

Considering the overlapping volume V_c1 of spherical particles in the granule unit in the seed volume, the total volume of seeds should be

$$V_{Z_1}=({\displaystyle \sum V_i}-{\displaystyle \sum V_c}-{\displaystyle \sum V_{c1}})\times 3$$

(4)

In the formula, V_z1 represents the total volume of individual seeds (mm³), V_i represents the overlap volume of individual spherical particles (mm³), V_c represents the overlap volume of spherical particles of monolayer seeds (mm³), and V_c1 represents the overlap volume between particulate units (mm³).

Eventually, the total volume V_z1 of individual kernels can be calculated to be 218.136 mm³, and according to the density equation, the density ρ of the established maize kernels can be calculated to be 1375 kg/m³, which matches with the density of the actual maize kernels (1100–1400 kg/m³).

2.6. Calculation of Seed Bonding Parameters

Prior to the generation of bonds, the normal stiffness per unit area, shear stiffness per unit area, normal strength, and shear strength of the seeds must be calculated to establish the bond’s relevant parameters.

To calculate the bond-related parameters, the force area of the modeled seed must first be defined and determined. As shown in Figure 9a, force area 1 corresponds to the area of the red trapezium formed when force area 1 is crushed under pressure. The dimensions of this trapezium—its length, width, and height—are approximated as follows: the bottom length is one-third of W₁, the lower bottom length is one-third of W₂, and the height is equal to H₂. These dimensions are derived based on the seed’s overall size, where each is one-third of the corresponding seed dimension. As shown in Figure 9b, the area of force area 2, when crushed under pressure, is referred to as stress area 2. Based on the pressure dimensions described earlier, stress area 2 is simplified to the area of the blue trapezoid. The dimensions of the blue trapezoid are approximated as follows: the height of the upper base is W₁, the height of the lower base is one-third of the seed grain’s width at one-third of the height of H₂, and the trapezoid’s overall height is one-third of H₂. Similarly, as shown in Figure 9c, the area of force area 3, when crushed under pressure, is referred to as force area 3. This area is simplified to the green trapezoid for ease of calculation. The upper base of the green trapezoid is defined by approximating the arc-shaped bottom edge of the seed grain as a horizontal straight line in space. This produces two tangent points located on both the arc-shaped bottom edge and the horizontal line. The horizontal distance between these tangent points defines the length of the upper base of the green trapezoid, while the length of the lower base is equal to W₂.

Substituting the maize kernel size parameters measured in 2.1, 1/3W₁ is used as the upper bottom length of the trapezoid of stress area 1, 1/3W₂ is used as the lower bottom length of the trapezoid of stress area 1, and H₁ is used as the trapezoid height of the trapezoid of stress area 1.

At the stage of elastic deformation of the seeds (deformation < 5%), the elastic deformation of the seeds is 0.046. The shear stiffness per unit area of the constructed seed model was calculated based on the strain formula, the equation for the modulus of elasticity, the equation for the normal stiffness, and the Poisson’s ratio of 0.35:

$$\left\{\begin{array}{c}\epsilon ={\displaystyle \frac{\Delta L}{L_0}}\\ \sigma ={\displaystyle \frac{F}{A}}\\ E={\displaystyle \frac{\sigma }{\epsilon }}\\ K_n={\displaystyle \frac{EA}{L_0}}\end{array}\right.$$

(5)

According to the relationship between tangential stiffness per unit area and the normal stiffness equation,

$$K_s^{\prime }={\displaystyle \frac{K_n^{\prime }}{2(1+\mu )}}$$

(6)

In the formula, K_s^′ represents tangential stiffness per unit area (N/m²), K_n^′ represents normal stiffness per unit area, and μ represents the Poisson’s ratio.

The compressive strength of the seeds was calculated according to the compressive strength formula,

$${\sigma }_n={\displaystyle \frac{F}{A}}$$

(7)

The shear strength of the seeds was calculated and according to the shear strength formula,

$$\tau =k{\sigma }_n$$

(8)

In the formula, τ represents seed shear strength (Pa), and k represents the scale factor, usually 0.5–0.7; here the average of the two is taken (Pa).

The other two regions of maize kernels were modeled in the same way, and the three bond parameters were finally calculated as shown in

Table 8. Force areas 1–3 seed bonding parameters.

Force Area 1
Parameter Category	Calculation Result
Normal stiffness per unit area	389.67 MPa
Tangential stiffness per unit area	114.32 MPa
Compressive strength	17.89 MPa
Shear strength	10.73 MPa
Force Area 2
Parameter Category	Calculation Result
Normal stiffness per unit area	218.16 MPa
Tangential stiffness per unit area	80.8 GPa
Compressive strength	10.04 MPa
Shear strength	6.02 MPa
Force Area 3
Parameter Category	Calculation Result
Normal stiffness per unit area	550.87 MPa
Tangential stiffness per unit area	204.03 MPa
Compressive strength	25.34 MPa
Shear strength	15.2 MPa

.

2.6.1. Simulation of Mechanical Tests

Simulation of mechanical property tests is a common method used in research (Liu, et al., 2022, Ramaj, et al., 2024) [26,27]. To validate the accuracy of the simulation model, mechanical property tests were conducted in EDEM. The three-dimensional plastron model was constructed using SolidWorks (v2020), and a seed grain was generated through the particle factory, as shown in Figure 10. Since moisture content significantly influences the mechanical properties of seed grains, bonding parameters were set based on a moisture content of 26%. When the generated seed grain was placed flat on the base of the texture meter, the control platen applied force at three designated areas, as illustrated in Figure 11. Existing studies indicate that the critical crushing force of seed grains is independent of the loading rate; therefore, the loading rate of the indenter was increased to 0.01 m/s to reduce the simulation time. The experiment was terminated when the seed grain collapsed, as individual spherical particles composing the seed grain could not be compressed in EDEM.

2.6.2. Seed Bonding

Seed bonding (bond generation) was carried out in the Meta-Particle section of the EDEM pre-treatment plate. Prior to generating bond keys, particle models were created in the Bulk Material section. A total of seven particle models were developed based on their radii, as detailed in

Table 9. Coordinates and contact radius of each spherical particle.

r/mm	X	Y	Z	Contact Radius/mm
0.75	0	0	0	1
0.825	0	0	0	1
0.85	0	0	0	1.2
0.9	0	0	0	1.2
1.00	0	0	0	1.2
1.04	0	0	0	1.2
1.08	0	0	0	1.2

.

In the Physics section of the EDEM pre-processing board, we selected the Bonding V2 model and input the contact parameters for the bulk material. Next, we imported the coordinates of the kernel model into the cell below the Meta-Particle. To ensure a seamless connection with the mandrel, the overall coordinates of the kernel model were re-planned. A two-dimensional coordinate system was established based on the corn cob cross-section, as shown in Figure 12. The connecting line was defined using the center of an adjacent kernel and the mandrel center. The kernel spacing angle was determined by the angle between two connecting lines. For the variety “Zheng dan 958,” the number of kernels in a single circle of corn is 16, resulting in a neighboring kernel spacing angle of 22.5°. This value was calculated using trigonometric formulas.

$$\begin{array}{l}x=R\sin \theta \\ y=R\cos \theta \end{array}$$

(9)

In the formulas, x represents the seed’s horizontal coordinate, y represents the seed’s longitudinal coordinates, R represents the distance between the mandrel and the center of the seed(mm), and $\theta$ represents the seed spacing angle (°).

The final calculation results are shown in

Table 10. Coordinates of each spherical particle of the seed model.

X	Y	Z	R/mm
0	17.825	−1	0.825
0	18.655	−1	0.75
0	20.335	−1	1.08
0	22.335	−1	0.85
0	24.435	−1	0.85
0	26.285	−1	1.08
1.86	18.865	−1	1.04
−1.86	18.865	−1	1.04
2.21	20.365	−1	1
−2.21	20.365	−1	1
2.61	22.335	−1	1
−2.61	22.335	−1	1
1.85	22.335	−1	0.9
−1.85	22.335	−1	0.9
2.95	24.435	−1	1.08
−2.95	24.435	−1	1.08
2.03	24.435	−1	1.08
−2.03	24.435	−1	1.08
3.33	26.285	−1	1.08
−3.33	26.285	−1	1.08
2.26	26.285	−1	1.08
−2.26	26.285	−1	1.08

.

From the above, the process of building a discrete meta-model of breakable seeds is shown in Figure 13.

2.7. Stacking Angle Test

Stacking angle experiments are a common method for analyzing the mechanical properties of grains and verifying the accuracy of discrete element models (Li, et al., 2025, Chen, et al., 2020) [6,28]. To verify the accuracy of the established seed model, a seed stacking angle test was conducted using a custom-built measuring instrument, as shown in Figure 12a. The instrument consists of an outer frame, motor, traction wire, cylinder, and base, with the cylinder and base made of organic glass. A total of 260 g of seeds was weighed and placed in the cylinder, which was lifted at a constant speed of 0.05 m/s. As the cylinder rose, the seeds scattered freely onto the base, forming a stacking angle, as illustrated in Figure 12b. The experiment was repeated three times, and the average value was recorded as the final result.

The acquired stacking angle images were processed in MATLAB (v2023) through binarization, edge extraction, and linear fitting. Simulation tests of the stacking angle were conducted in EDEM, as illustrated in Figure 14c. A three-dimensional model of the stacking angle tester was created using SolidWorks. Based on reference (Cui T et al., 2023) [29], the collision recovery coefficient between seeds was set to 0.086, the static friction coefficient to 0.484, and the kinetic friction coefficient to 0.103. The test step size was set to 6.34 × 10⁻⁷ s. The seed stacking process, as shown in Figure 14d,e, was simulated until the seeds became completely stationary.

2.8. Traditional Model Stacking Angle Experiment

The stacking angle test mainly measures the coefficient of friction of the kernel, the flow characteristics and the friction characteristics between the particles, which are affected by the roughness; the grain size distribution; and the water content of the kernel. While in the actual threshing process, the cob is often threshed by rubbing, i.e., the friction and impact effects of the maize kernel and the threshing element; in this paper, the friction effects can be measured by the stacking angle, and the impact effects can be measured by the mechanical properties test. According to reference (Wang Y, et al., 2020) [30], the discrete element model was built by modeling the seeds with a traditional 10-ball filling, as shown in Figure 15c.

The model was applied to stacking angle experiments, ensuring that parameters such as dynamic and static friction factors, contact coefficients, time steps, and mesh sizes matched the experimental settings described in the paper. The experiment was conducted three times, and the average value was used as the final result. The simulation process is illustrated in Figure 15a.

The 10-ball filling model underwent edge detection and binarization steps using MATLAB. The resulting binarized image of the model is shown in Figure 15b.

The stacking angle of the 10-ball filling model was calculated, with an average value of 26.71°. The experiment was conducted three times, and the experimental stacking angles are presented in Figure 16.

3. Results and Discussion

3.1. Mechanical Test Analysis

As can be seen from Figure 17, in the case of the same moisture content, the different force areas directly affect the critical force of seed crushing, and there is a significant difference between the various force areas. Among them, the crushing force of force area 3 is the largest, and the crushing force of force area 2 is the smallest, and the difference in the crushing critical force between different force areas ranges from 13.4% to 58.8% of its own crushing critical force, and it is most significant in the case of the grain with a 26% moisture content, and the difference in the average value of the critical crushing force of force area 2 and force area 3 is 58.8%.

Under the condition of different water content, it still transpires that the crushing force of force area 3 is the largest, and the crushing force of force area 2 is the smallest. The crushing critical force at different force areas of individual grains decreases with the increase in water content, but the effect of force areas on the crushing critical force of grains decreases relative to the increase in the water content of grains, but the difference in the maximum crushing force of each force area is still significant, and the maximum difference between different force areas is 42.6%. Using the traditional method of measuring the critical force of grain crushing, (i.e., the whole grain is extruded until it is crushed, as shown in

Table 11. Conventional measurement of critical force test results for seed breakage.

Moisture Content/%	Critical Crushing Force/N
26%	260
30%	408
34%	328

), the difference is large. Therefore, when the threshing element is in contact with the kernel, the difference in force areas will result in a different maximum crushing force on the kernel.

As can be seen from Table 11, the crushing critical force measured by the traditional measurement method is within the interval between the maximum and minimum crushing critical force of the force area division method with the same moisture content, and the crushing critical force obtained by the traditional measurement method will cause a large error in EDEM if it is utilized, and the method of force area division is very useful, for the force area division method is more nuanced in measuring the crushing critical force of the seeds and avoids a large error in the simulation process.

Response surface tests were conducted using Design Experts 13 with ANOVA and linear fitting with the

Table 12. Simulation experiment program levels.

Test Level	Experimental Factors
Encodings	Pressure Area A	Moisture Content B
−1	1	26%
0	2	30%
1	3	34%

test factors:

Using the Box-Behnken plate in Design Experts, the crushing rate test program and results for different parameter combinations were obtained, as in

Table 13. Simulation experiment scheme and results.

Serial Number	A	B	Breaking Force F/N
1	0	1	134.02
2	0	0	147.3
3	−1	1	172.7
4	0	0	147.3
5	−1	−1	333.4
6	1	−1	550.9
7	−1	0	205.9
8	0	0	147.3
9	1	0	237.88
10	1	1	233.62
11	0	0	147.3
12	0	−1	226.9
13	1	0	237.88
14	0	0	147.3
15	0	−1	226.9
16	0	1	134.02
17	−1	0	205.9

:

The data in Table 13 were analyzed using ANOVA, revealing p-values of 0.0187 and 0.0002 for force area and moisture content, respectively, as the

Table 14. ANOVA of simulation mechanics experiment. Generation of seed force area–water content response surfaces.

Source	Sum of Squares	df	Mean Squares	F-Value	p-Value
Model	1.453 × 105	5	29,051.38	15.05	0.0001
A—Broke Area	14,653.01	1	14,653.01	7.59	0.0187
B—Moisture Content	55,068.85	1	55,068.85	28.53	0.0002
AB	6129.32	1	6129.32	3.18	0.1024
A2	47,991.56	1	47,991.56	24.86	0.0004
B2	17,939.71	1	17,939.71	9.29	0.0111
Residual	21,234.19	11	1930.38
Lack of Fit	21,234.19	3	7078.06
Pure Error	0	8	0
Cor Total	1.665 × 105	16

. The p-value of the overall model was 0.0001, indicating that both force area and moisture content had extremely significant effects on the critical crushing force of grains. The regression model effectively explained the corresponding variables. Moisture content, with a highly significant p-value of 0.0002, was identified as the dominant factor influencing the crushing force of grains. Force area, with a significant p-value of 0.0187, demonstrated a substantial impact on the crushing force. However, the AB value was not significant, suggesting no direct relationship between breaking force and moisture content. This indicates that changes in kernel moisture content do not affect the delineation of force areas, allowing the division of force areas to be applied across a wider range of kernel moisture content. These findings suggest that the sub-regional mechanical property test based on force area division is broadly applicable to maize kernels. Additionally, the model effectively predicts trends in the critical crushing force.

The response surface (Figure 18) shows that when the force area remains constant, the critical crushing force of seeds initially decreases and then increases as the water content rises. Conversely, when the water content remains constant, the critical crushing force increases with changes in the force area. At the same time, the rupture force was slightly changed by changing the order of the moisture content and the force area, while it can be learnt from the response surface that a change in the force area caused the rupture force of the seeds to change drastically, and a change in the moisture content caused the rupture force of the seeds to change relatively gently.

3.2. The Results of Simulation of Mechanical Tests

In the EDEM post-processing module, select “Compressive Force” under the Particle section and export the data to obtain the extrusion force on the seeds during the experiment. The detailed data are presented in

Table 15. Simulation of mechanical test results.

Number	Strains 1/N	Strains 2/N	Strains 3/N
1	325.4	241.2	684.3
2	374.8	268	478.3
3	350.5	239.9	479.1
Average Value	350.23	249.7	547.23
Standard Deviation	24.70	15.86	118.70

.

From Table 15, it can be seen that there is a significant change in the pressure exerted on seed crushing when the moisture content is the same and the force areas are different, which is the same as the conclusions of the previous mechanical property tests in the sub-region when the platen is pressed at force area 1; there is a difference of 16.83 N between the actual mechanical property experiments and the simulation results when the platen is pressed at force area 2; there is a difference of 22.8 N between the actual mechanical property experiments and the simulation results when the platen is pressed at the force area 3. The difference between the actual mechanical characteristic experiment and the simulation result is 3.67 N; according to the above conclusions, it can be considered that the correctness of the bond parameter is proven.

3.3. Comparison of Stacking Angle Test

According to the stacking angle error calculation formula,

$$Y_1={\displaystyle \frac{\beta -{\beta }_1}{\beta }}\times 100\%$$

(10)

In the formula, β represents test stacking angle measurements (°); β₁ represent the stacking angle simulation test value (°).

From Figure 19 and Equation (10), we cropped the stacked corner images obtained from the three experiments into six images, numbered 1–6, as the

Table 16. Experimental data on stacking angle.

Test Number/°	Experimental Angle/°	Simulation Angle/°
1	22.75	18.94
2	22.32	26.10
3	20.62	16.29
4	20.22	22.50
5	24.93	22.72
6	22.48	27.16
Average Value	22.22	22.29

. However, the error of stacking angle often comes from human error when the image is taken; in order to avoid the error caused by human factors, we use MATLAB to perform the image binarization, edge extraction and other operations, to facilitate the subsequent processing; it can be seen that the simulation results are close to the experimental results in the three stacking angle experiments and there is no large deviation from the experimental results. It can be concluded that Y is 0.31%; due to the small error, it can be assumed that the correctness of the modeled contact parameters is built into the EDEM.

3.4. Comparison with Conventional Seed Models

In a comparison of the experimental values and the simulation of the seed stacking angle of the 10-ball filling model, it can be seen that the experimental angle is 24.81°, and the simulation angle is 26.71°; according to the stacking angle error formula (15), it is calculated that the simulation of the 10-ball filling model of the seed and the actual stacking angle results in an error of 7.65%, as shown in

Table 17. Comparison between traditional 10-ball model and crushable model.

Test Number	Experimental Angle (°) (10-Ball Model)	Simulation Angle (°) (10-Ball Model)	Experimental Angle (°) (Breakable Model)	Simulation Angle (°) (Breakable Model)
1	24.71	24.71	22.75	18.94
2	25.62	25.62	22.32	26.10
3	26.18	31.92	20.62	16.29
4	23.04	25.70	20.22	22.50
5	27.34	28.61	24.93	22.72
6	21.96	23.71	22.48	27.16
Average Value	24.81	26.71	22.22	22.29

.

From the above, it can be seen that the error accuracy of the 10-ball filling model is 7.65%, while the error accuracy of the breakable seed model is 0.31%, and the precision was improved over previous studies.

In conclusion, the corn kernel model developed in this paper can accurately reflect the mechanical properties of actual corn kernels and can simulate the breaking of actual kernels. In the actual threshing process, the contact between the corn and the threshing element is often more complex. In order to simulate complex threshing situations, an overall model of the crushable corn kernel, as well as the kernel, can be established in future research. However, this requires further exploration of the crushable cores and the way the kernels are connected to the cores, to promote discrete element simulation technology in the field of agricultural engineering from an idealized model to the direction of the high-fidelity modeling of real operating scenarios and to provide more accurate simulation tools for the digital design of agricultural equipment. The maize kernel model proposed in this study can be used to optimize the interaction between the kernel and the threshing element during the simulation process to simulate the deformation and fragmentation of the kernel. The optimum moisture content for harvesting maize can also be determined by investigating the breaking force of kernels with a different moisture content; and comparing a moisture content of 26%, 30% and 34%, the lowest moisture content should be selected, which has the highest breaking force for the kernels and can withstand greater impacts from the threshing elements. However, it still has some limitations. For example, kernels can only be broken into fixed shapes, and each kernel has the same broken area. In addition, the connection between the kernel and the kernel shaft is also an important issue, which will directly affect the threshing effect of the harvester. In future research, the kernel model will be further improved to explore the strength of the connection between the kernel and the mandrel and the method of connection in order to build a model of crushable corn kernels. In addition, this modeling and model parameter calibration approach will be applied to the simulation of other crops (e.g., wheat and rice) with threshing elements. Finally, the model can be optimized by exploring the influence of dynamic moisture effects on the establishment of a discrete elemental model for maize kernels and the connection between the kernel and the kernel shaft and the mechanical interaction between the two during threshing.

4. Conclusions

(1)

Based on the differences in the mechanical properties of maize kernel size and kernel structure, a discrete elemental model of maize kernel bonding was established by using the force area division and sub-area mechanical property test method, combined with the bonding contact model.

(2)

The rupture force at each stress area of maize was determined by mechanical property tests, and the bonding parameters of the maize kernel model were verified by combining it with the experimental design method, and the relative error between the simulated kernel rupture force and the average value of the measured kernel rupture force was 0.6–10%. Corn kernel bonding discrete element model force areas and kernel mechanical properties have a significant effect (p < 0.05), consistent with the actual corn kernel structure and mechanical properties.

(3)

The physical properties of the seeds were analyzed using the stacking test and discrete element simulation, and the test was repeated three times, resulting in a final stacking angle of 22.22° for the stacking angle test and 22.29° for the stacking angle under the discrete element simulation, with an error of 0.31%, which is extremely accurate and can be considered as a correct model of the seeds.

(4)

The modeling and simulation methods presented in this paper can be applied to the optimization of threshing drums, which can crush maize kernels, to analyze more accurately the energy and force relationships between the kernels and the threshing elements, thus comparing them with real-life tests.