Calibration and Experimental Validation of Discrete Element Parameters of Fritillariae Thunbergii Bulbus

Abstract

The development of slicing equipment for Fritillariae Thunbergii Bulbus (FTB) has been constrained by the absence of precise and reliable simulation model parameters, which has hindered the optimization of structural design through simulation techniques. Taking FTB as the research object, this study aims to resolve this issue by conducting the calibration and experimental validation of the discrete element parameters for FTB. Both intrinsic and contact parameters were obtained through physical experiments, on the basis of which a discrete element model for FTB was established by using the Hertz–Mindlin with bonding model. To validate the calibrated bonding parameters of this model, the maximum shear force was selected as the evaluation index. Significant influencing factors were identified and analyzed through a single-factor test, a two-level factorial test, and the steepest ascent method. Response surface methodology was then applied for experimental design and parameter optimization. Finally, shear and compression tests were conducted to verify the accuracy of calibrated parameters. The results show that the mechanical properties of FTB are significantly affected by the normal stiffness per unit area, the tangential stiffness per unit area, and the bonding radius, with optimal values of 1.438 × 10⁸ N·m⁻³, 0.447 × 10⁸ N·m⁻³, and 1.362 mm, respectively. The relative errors in the shear and compression tests were all within 5.18%. The maximum error between the simulated and measured maximum shear force under three different types of blades was less than 5.11%. The percentages of the average shear force of the oblique blade were reduced by 52.23% and 29.55% compared with the flat and arc blades, respectively, while the force variation trends for FTB remained consistent. These findings confirm the reliability of the simulation parameters and establish a theoretical basis for optimizing the structural design of slicing equipment for FTB.

1. Introduction

Fritillariae Thunbergii Bulbus (FTB), also known as “Xiang Bei” in Chinese, is a renowned authentic medicinal herb from Zhejiang Province, China. FTB has a long cultivation history, and holds significant economic value and promising market potential. In recent years, its cultivation area has been steadily increased. Slicing plays a fundamental role in the initial processing of FTB. Therefore, an in-depth investigation of the material property parameters of FTB is vital for optimizing the design of slicing equipment, which provides essential guidance for equipment development and process optimization [1,2,3,4].

Advancements in the discrete element method (DEM) and EDEM simulation software have facilitated new approaches for analyzing the contact characteristics between agricultural products and machine components, leading to their widespread application in the field of agricultural engineering [5,6,7]. Many scholars have conducted DEM modeling and simulation research on a variety of agricultural products, including fruits, vegetables [8,9], soils [10,11,12,13], stems [14,15,16,17], and seeds [18,19,20,21]. Focusing on apples, Jianhong Zhang et al. [22] established a simulation model in EDEM software and calibrated the contact parameters between apples and contact materials by combining bench tests with simulation experiments. The angle of repose tests, steepest ascent method, and central composite design were utilized to further calibrate the contact parameters between apples and obtain the fundamental physical properties of three apple varieties (Red Fuji, Hua Niu, and Golden Delicious). The results of this study offer theoretical support for DEM-based research on the material characteristics of apples. Guozhong Zhang et al. [23,24] carried out virtual calibration of bonding parameters by employing DEM to reveal the mechanical forces variation during water chestnut shearing by different types of blades. Accurate simulation models and reliable parameter calibration methods for mechanical operations (e.g., peeling and segmenting) on water chestnut were developed, which offer theoretical support for equipment design. 9determine key contact parameters, including the coefficient of restitution, static friction, and rolling friction. Additionally, bonding parameters were determined by the Hertz–Mindlin with bonding model combined with shear and compression tests of Panax notoginseng stems, including bonding radius, normal stiffness, critical normal stress, tangential stiffness, and critical tangential stress. The accuracy of the calibrated parameters was verified by comparing simulation outcomes with experimental results from the angle of repose test and shear tests, confirming the credibility of the model. Mengchen Wu et al. [25] simulated the accumulation process of peanut seed particles in EDEM software, and validated the fidelity of both the constructed model and its simulation parameters by bench tests. Xirui Zhang et al. [26] employed the Hertz–Mindlin with bonding contact model to develop a bonded-particle DEM model for banana stalks. The calibrated parameters enabled an in-depth analysis of how banana stalk fibers fracture under crushing. In conclusion, previous studies show that material properties vary significantly across different agricultural products. However, research on the parameter calibration and simulation modeling of FTB remains limited.

The present study focuses on FTB. The intrinsic and contact parameters were obtained through physical experiments, and external morphological features of FTB were acquired via 3D scanning techniques. The Hertz–Mindlin with bonding contact model was employed to develop a simulation model for FTB. The maximum shear force of FTB was used as the evaluation index for virtual parameter calibration. The accuracy of the model was validated by shear and compression tests, and the reliability of calibrated parameters was further examined by using different types of shear blades.

2. Materials and Methods

2.1. Experimental Materials

FTB was used as the experimental material in this study, which is sourced from Pingtian Township, Taizhou City, Zhejiang Province, China. FTB is generally oblate-spherical in shape, featuring a central depression and consisting of two lobes, between which there are typically 2 to 3 buds. As shown in Figure 1. A total of 200 medium-sized commercial FTB samples were chosen at random to ensure the generality and variability of experimental samples. The dimensions of samples were measured using a vernier caliper, yielding an average major axis diameter (a) of 37.60 mm, a minor axis diameter (b) of 35.06 mm, and an average height (h) of 24.37 mm. In addition, a total of 20 samples were tested to obtain the average moisture content using the drying method and the average density using the water displacement method, yielding values of 68.7% and 1.04 g/cm³, respectively [27].

2.2. Intrinsic Parameters

In this study, mechanical testing of FTB was conducted using a TMS-Pro texture analyzer (The Food Technology Corporation TMS-Pro texture analyzer from the Fairfax, VA, USA), incorporating both compression and shear tests.

Poisson’s ratio and elastic modulus are the primary intrinsic parameters of a material. As shown in Figure 2, a texture analyzer equipped with a flat probe was utilized to measure the intrinsic parameters of FTB, and its loading rate was set to 25 mm/min [9].

Considering the anisotropic nature of FTB, whole bulbs were used for the compression tests, which were repeated 20 times to ensure reliability.

Poisson’s ratio refers to the ratio of transverse contraction (or expansion) strain to longitudinal extension strain under uniaxial tension or compression, and functions as an elastic constant that quantifies the lateral deformation of a material. The calculation formula is shown in Equation (1):

$$\mu =\left|{\displaystyle \frac{{\delta }_1}{{\delta }_2}}\right|={\displaystyle \frac{W_1-W_2}{L_1-L_2}}$$

(1)

In Equation (1), δ₁ is the transverse deformation (mm); δ₂ is the axial deformation (mm); W₁ is the initial transverse dimension before compression (mm); W₂ is the transverse dimension after compression (mm); L₁ is the initial axial dimension before compression (mm); and L₂ is the axial dimension after compression (mm).

The Poisson’s ratio (μ) of FTB was obtained as 0.373 through 20 repetitions of mechanical tests.

The elastic modulus E serves as an indicator of a material’s capacity to withstand elastic deformation. The elastic modulus of FTB was determined to be 7.93 MPa based on texture analyzer measurements conducted across 20 experimental trials. The calculation formula is provided in Equation (2):

$$E={\displaystyle \frac{F\cdot L}{S\cdot \Delta L}}\sqrt{b^2-4ac}$$

(2)

In Equation (2), F is the maximum bearing capacity of FTB within the elastic deformation range (N); L is the original length of the FTB sample (mm); S is the cross-sectional area of the FTB sample (mm²); and ΔL denotes the variation in the length of the FTB sample before and after compression (mm).

2.3. Contact Parameters

2.3.1. The Coefficient of Friction

The coefficient of static friction refers to the ratio of the maximum static friction force to the normal force between two surfaces, providing a reliable indication of the frictional properties between a material and the contacting solid surface. Since slicing and other preprocessing equipment used for FTB after harvest are primarily made of stainless steel plates and industrial rubber, this study employed the inclined plane method to evaluate the coefficient of static friction [28] between FTB and these two contact surfaces. A friction testing apparatus [19,29] equipped with a high-speed camera (The manufacturer of the Phantom high-speed camera is Vision Research, Inc., which is located in Wayne, NJ, USA.)and a protractor(The manufacturer of the protractor is Ming Sen Company, which is located in Ningbo, Zhejiang, China.) was used to measure the critical angle, as shown in Figure 3. Ten randomly selected FTB samples were used in the test. At the initial stage of the experiment, the stainless steel plate and industrial rubber were placed on the apparatus, and the FTB sample was placed in a stationary position on the contact surface. The angle of the device was then gradually increased at a uniform speed until the sample showed signs of sliding. The critical angle at the onset of slipping was recorded using a high-speed camera.

$$f={\displaystyle \frac{mg\mathrm{s}\mathrm{i}\mathrm{n}\alpha }{mg\mathrm{c}\mathrm{o}\mathrm{s}\alpha }}=\mathrm{t}\mathrm{a}\mathrm{n}\alpha$$

(3)

Each FTB sample was tested three times, during which the angle formed between the plate surface and the horizontal surface was recorded. Based on Equation (3), the static friction coefficients between FTB and the stainless steel plate (f₁), as well as between FTB and industrial rubber (f₂), were determined through a total of 30 tests, yielding values of 0.63 and 0.66, respectively.

To evaluate the coefficient of rolling friction between FTB and the stainless steel plate, a combined approach involving the inclined plane rolling method and the law of conservation of energy was used. The friction apparatus was fixed at an angle of 25° (θ), allowing the FTB sample to roll down the inclined surface of the stainless steel plate with an initial velocity of zero. As illustrated in Figure 4, the sample eventually came to rest at a certain point along the plate due to the effect of friction. The coefficient of rolling friction (μ) can be calculated through the law of conservation of energy, with the distance M traveled by the sample on the inclined stainless steel plate being 150 mm, and the final position P representing the point where the sample remains stationary on the plate. Given the irregular shape of FTB, which cannot be simplified to a simple spherical model, the rolling trajectory on the horizontal plane tends to deviate from a straight line, making it challenging to measure the accurate path. Consequently, a large number of experiments were conducted, and 10 sets of data with reasonably straight rolling paths were selected for analysis [30]. The calculation is based on Equation (4):

$$mgM\sin \theta ={\mu }_{f}mg(M\cos \theta +P_0)$$

(4)

In Equation (4), θ is the fixed inclination of the friction apparatus (°); μ_f is the coefficient of rolling friction between the FTB and the stainless steel plate; M is the rolling distance that FTB slides on the friction apparatus (mm); P₀ is the distance that FTB slides on the stainless steel plate before coming to rest (mm).

After conducting several repeated experiments, the coefficient of rolling friction for the interaction between FTB and the stainless steel plate (μ_f) was measured to be 0.14.

2.3.2. The Coefficient of Restitution

The coefficient of restitution indicates the capacity of one material to restore its original form after impact. Considering that frequent collisions occur between FTB and harvesting equipment, as well as among bulbs themselves, during pre-treatment processes, the coefficient of restitution was measured by a high-speed camera and coordinate paper as reference tools [31,32,33].

The free-fall method was employed to measure the coefficient of restitution between FTB and a stainless steel plate [34,35,36]. The free-fall motion was captured by a high-speed camera to collect trajectory data, as shown in Figure 5. Each FTB sample was released from a fixed height H₁ and allowed to fall freely onto a stainless steel plate. The rebound height H₂ was subsequently recorded by the high-speed camera. Because only the gravitational force acts on the sample during this process, the effect of both dynamic and static friction coefficients between the sample and the surface can be neglected. The coefficient of restitution e₁ between FTB and stainless steel is calculated by Equation (5):

$$e_1=\sqrt{{\displaystyle \frac{H_2}{H_1}}}$$

(5)

The experiment was conducted 20 times, from which the coefficient of restitution between FTB and the stainless steel plate was determined as e₁ = 0.606.

To evaluate the coefficient of restitution between FTB samples themselves, the suspension method was used, as illustrated in Figure 6. In this setup, sample B was suspended vertically, while sample A was raised to a height, H₁, and then released to collide with sample B. The extent of oscillation following the collision reflects the restitution behavior between the two samples. The motion was captured by a high-speed camera, and the swing heights of sample A (H₂) and sample B (H₃) were recorded by using coordinate paper as a reference. The coefficient of restitution between two samples, denoted as e₂, was calculated by Equation (6):

$$e_2={\displaystyle \frac{\sqrt{H_3}-\sqrt{H_2}}{\sqrt{H_1}}}$$

(6)

The experiment involved 20 FTB samples, and each measurement was repeated 10 times. Based on the result, the coefficient of restitution (e₂) between FTB samples themselves was calculated to be 0.366. The inherent parameters of FTB are shown in

Table 1. The intrinsic parameters of FTB.

Intrinsic Parameters	Values
Poisson’s ratio (μ)	0.373
Elastic modulus/Mpa	7.93
Static friction coefficients between FTB and the stainless steel plate f1	0.63
Static friction coefficients between FTB and industrial rubber f2	0.66
Coefficient of rolling friction for the interaction between FTB and the stainless steel plate μf	0.14
Coefficient of restitution between FTB and the stainless steel plate e1	0.606
Coefficient of restitution between FTB themselves, e2	0.366

.

2.4. Construction of Discrete Element Model of FTB

2.4.1. FTB Model

Due to the irregular external shape of FTB, it is challenging to accurately reproduce its external morphological features by traditional modeling methods. To obtain a more realistic three-dimensional representation and improve the fidelity of simulation experiments, a sample—whose length, width, and height closely approximate the average dimensions—was selected for modeling, as shown in Figure 7. A K500 automatic turntable 3D scanner (The manufacturer of the K500 automatic turntable 3D scanner is Nankai 3D Technology Co., Ltd., which is located in Shenzhen, China) was employed to perform a full 3D scan of the selected sample. The K500 scanner offers a reconstruction accuracy of less than 0.001 mm and can acquire up to 1 × 10⁶ precise laser points per second, allowing for the generation of high-density 3D point cloud data. The scan data were processed using Techlego 3D scanning software (Version: V3.2.1, 64-bit) to remove outliers and fill any missing regions, resulting in an editable 3D model file. The finalized model was exported in IGS format and then imported into the EDEM software (Version: 2025.1, BETA, Developed by CAE Systems). The particle generation function in EDEM was used to fill the scanned geometry with particles; the number of filling particles generated was 2532, thereby constructing the discrete element model of FTB, as presented in Figure 7b [37].

The discrete element model consists of multiple particles, and the force between these particles is determined by the contact model. Among the available models, the Hertz–Mindlin with bonding contact model was adopted to construct the discrete element model for its capacity to form the bonding interactions between particles, enabling the simulation of biomechanical properties of FTB through the breakage of these bonds. After the model was imported into GOM Inspect software (Version: 2018.0.1, 64-bit, Developed by ZEISS Group), sharp edges and noise points were refined to generate a high-quality 3D model. This model was then imported into EDEM, where particles were generated and 14,444 bonding links were added, as illustrated in Figure 7c, resulting in the bonded-particle model of FTB. In the bonding model, cohesive forces were introduced between adjacent elements, causing the individual units to behave as a unified structure under loading. When external forces are applied, internal stress is generated within the bonding network. If the stress exceeds a critical threshold, some bond links between units are broken, eliminating their cohesive effect and leading to the separation of individual elements [38].

2.4.2. Shear and Compression Models

Models of the cutting blade, compression plate, and base platform were constructed in SolidWorks (Version: 2023 SP2, Developed by Dassault Systemes), with all dimensions matching those used in the physical experiments. After being exported in STEP format, the models were imported into EDEM software. The loading speed for both the blade and the compression plate was set to 400 mm/min, with motion applied in the vertically downward direction. Shear and compression simulations were then conducted by the discrete element method to replicate the physical test conditions.

2.5. Parameter Calibration Experimental Design

2.5.1. Shear Test

Shear tests were conducted on FTB using a TMS-Pro texture analyzer to obtain practical reference values indicative of bonding parameters. Fresh FTB samples were used as the experimental materials. Considering its unique structure and anisotropic properties, each sample was cut in half and placed on the support platform of the texture analyzer with the inner surface facing downward. A custom-designed blade was used to perform the shearing operation, as presented in Figure 8.

During the test, the blade fixed to the terminal position of the texture analyzer was operated at a speed of 400 mm/min. Due to the brittle nature of FTB, the maximum force typically occurred at the moment of structural rupture, although the position at which this peak appeared varied unpredictably along the shear zone. Therefore, the maximum shear force served as the evaluation index. A total of 25 repeated experiments were conducted, yielding an average measured maximum shear force of 37.6 N. This value was subsequently used as the physical reference for the parameter calibration in the simulation experiments.

2.5.2. Single-Factor Test Design

Based on materials with similar mechanical properties and the results of multiple preliminary tests [23,37], the parameter ranges for the discrete element simulation of FTB were defined, as presented in

Table 2. Simulation parameters of discrete elements of FTB.

Parameters	Values
Particle contact radius x1/mm	1.2~2.0
Normal stiffness per unit area x2/(N·m−3)	1.0 × 107~9.0 × 108
Tangential stiffness per unit area x3/(N·m−3)	1.0 × 107~9.0 × 108
Critical normal stress x4/Pa	1.0 × 106~90.0 × 106
Critical tangential stress x5/Pa	1.0 × 106~90.0 × 106
Bonding radius x6/mm	1.2~2.0

. A single-factor test was applied to examine the effect of each parameter on the maximum shear force of FTB [23,39].

The single-factor test was designed in the EDEM discrete element simulation software, with maximum shear force as the evaluation index. Parameters x₁ to x₆ were each tested at five levels, as detailed in

Table 3. Levels of single-factor tests.

Levels	x1	x2	x3	x4	x5	x6
1	1.2	1.0 × 107	1.0 × 107	1.0 × 106	1.0 × 106	1.2
2	1.4	5.0 × 107	5.0 × 107	5.0 × 106	5.0 × 106	1.4
3	1.6	1.0 × 108	1.0 × 108	10.0 × 106	10.0 × 106	1.6
4	1.8	5.0 × 108	5.0 × 108	50.0 × 106	50.0 × 106	1.8
5	2.0	9.0 × 108	9.0 × 108	90.0 × 106	90.0 × 106	2.0

Note: x₁ is the particle contact radius, x₂ is the normal stiffness per unit area, x₃ is the tangential stiffness per unit area, x₄ is the critical normal stress, x₅ is the critical tangential stress, and x₆ is the bonding radius.

. During the simulations, the median value among the five levels, namely the third level, was set as a constant.

2.5.3. Two-Level Factorial Level Test

A two-level factorial test was conducted to identify the parameters that significantly affect the maximum shear force. Simulation experiments were conducted via the EDEM software, while the Design-Expert software (Version: 13.0.1.8, Developed by Stat-Ease) [40] was utilized to design a total of 16 experimental runs for statistical significance analysis [41,42,43,44,45].

2.5.4. Steepest Ascent Method

Based on the significant influencing factors identified through the two-level factorial test, the steepest ascent method was implemented to determine their optimal value range. During the test, non-significant influencing factors were held constant at their level 3 setting, as previously determined in the single-factor test.

2.5.5. Box-Behnken Design Experiment

To identify the optimal simulation parameters, the response surface methodology was applied to identify the optimal simulation parameters, following the principles of the Box-Behnken design (BBD). The significant influencing factors identified through the two-level factorial test were selected as the influencing variables. Their value ranges were refined through the steepest ascent method. Subsequently, a five-level BBD test was conducted, where the response variable was defined as the relative error (E) between the measured maximum shear force and the simulation result. The formula used to calculate relative error is shown in Equation (7):

$$E={\displaystyle \frac{|F_1-F_2|}{F_1}}\times 100\%$$

(7)

In Equation (7), E is the relative error; F₁ is the measured maximum shear force (N); F₂ is the simulated maximum shear force (N).

3. Results and Analysis

3.1. Analysis of the Single-Factor Test Results

Simulation tests were performed in the EDEM discrete element software with maximum shear force serving as the evaluation index. The outcomes of the single-factor test are shown in Figure 9. The simulation outcomes for varying values of particle contact radius (x₁), critical normal stress (x₄), and critical tangential stress (x₅) exhibited minimal deviation from the experimentally measured maximum shear force, indicating that these parameters have a minor effect on shear performance. In contrast, as the normal stiffness per unit area (x₂) increased, the maximum shear force initially increased and then decreased. When x₂ ranged between 0.5 × 10⁸ and 1.0 × 10⁸, the simulation results closely matched the measured values. Similarly, the simulated shear force approached the experimental results when the tangential stiffness per unit area (x₃) ranged from 0.5 × 10⁸ to 1.0 × 10⁸ and when the bonding radius (x₆) was between 1.4 mm and 1.6 mm. These results indicate that factors x₂, x₃, and x₆ significantly affect the maximum shear force, as variations in their values caused notable changes in the simulation outcomes.

3.2. Analysis of the Two-Level Factorial Test Results

Through the two-level factorial test, significant influencing factors on the maximum shear force were identified. Then, simulation experiments were conducted in the EDEM software, the results of which are shown in

Table 4. Design and results of two-level factorial tests.

Serial Numbers	x1/mm	x2/(×108 N·m−3)	x3/(×108 N·m−3)	x4/(×106 Pa)	x5/(×106 Pa)	x6/mm	Maximum Shear Force/N
1	1.6	1	1	5	5	1.4	43.1
2	1.4	0.5	1	5	5	1.4	29.4
3	1.4	1	0.5	5	5	1.4	29.8
4	1.6	1	0.5	10	10	1.6	40
5	1.4	1	0.5	10	10	1.6	40
6	1.4	1	1	10	10	1.6	64.8
7	1.6	1	1	10	10	1.6	64.8
8	1.6	0.5	0.5	10	10	1.6	27.1
9	1.6	0.5	0.5	5	5	1.4	22.9
10	1.6	1	0.5	5	5	1.4	29.8
11	1.6	0.5	1	10	10	1.6	37.9
12	1.4	0.5	0.5	10	10	1.6	27.1
13	1.6	0.5	1	5	5	1.4	29.4
14	1.4	1	1	5	5	1.4	43.1
15	1.4	0.5	0.5	5	5	1.4	22.9
16	1.4	0.5	1	10	10	1.6	37.9

Note: x₁ is the particle contact radius, x₂ is the normal stiffness per unit area, x₃ is the tangential stiffness per unit area, x₄ is the critical normal stress, x₅ is the critical tangential stress, and x₆ is the bonding radius.

. A significance analysis was then performed in the Design-Expert software, with the results presented in

Table 5. Significance analysis of the two-level factorial test results.

Source	Effects	Degrees of Freedom	Mean Square	F	p
Model	169,000	6	28,141.88	19.49	<0.0001 **
A-x1	539.24	1	539.24	0.373	0.5563
B-x2	136,000	1	136,000	93.9	<0.0001 **
C-x3	19,185.07	1	19,185.07	13.29	0.0054 **
D-x4	3754.61	1	3754.61	2.6	0.1413
E-x5	9.14	1	9.14	0.006	0.9383
F-x6	9003.42	1	9003.42	6.23	0.034 *
Residual	12,996.85	9	1444.09

Note: ** indicates extremely significant (p < 0.01), * indicates significant (0.01 ≤ p < 0.05). x₁ is the particle contact radius, x₂ is the normal stiffness per unit area, x₃ is the tangential stiffness per unit area, x₄ is the critical normal stress, x₅ is the critical tangential stress, and x₆ is the bonding radius.

.

As shown in Table 5, the p-value of this model is less than 0.0001, indicating that it is highly statistically significant and, thus, both reliable and well-fitted. The coefficient of determination (R² = 0.9285) is close to 1, which suggests a strong correlation between the model and the data. The adjusted R_adj² = 0.8809 further confirms the precision and reliability of the two-level factorial test results. Specifically, the p-values for factors x₂ and x₃, which are both less than 0.001, indicate that these factors have a highly significant effect. x₆ had a p-value less than 0.1, which suggests a statistically significant effect. In contrast, x₁, x₄, and x₅ were not significant influencing factors influencing shear force. As illustrated in Figure 10, the Pareto chart confirms the significance of x₂, x₃, and x₆, as their effects exceed the critical t-value threshold, further supporting their classification as significant influencing factors.

3.3. Analysis of the Steepest Ascent Method Results

Based on the results of the two-level factorial test, the steepest ascent method was conducted to further determine the optimal range of the significant influencing factors. The non-significant influencing factors were held constant. Based on the defined ranges of the influencing factors, the initial levels for factors x₂ and x₃ were established at 0.3 × 10⁸ N·m⁻³, with increments of 0.8 × 10⁸ N·m⁻³. For factor x₆, the starting value was set to 1.3 mm, and the factor levels were determined using a gradient increment of 0.08 mm. The design and results of the steepest ascent method are presented in

Table 6. Schemes and results of the steepest ascent method.

Serial Numbers	x2/(×108 N·m−3)	x3/(×108 N·m−3)	x6/mm	Shear Force/N
1	0.3	0.3	1.3	14.3
2	1.1	1.1	1.38	37
3	1.9	1.9	1.46	75
4	2.7	2.7	1.54	106.8
5	3.5	3.5	1.62	132.5
6	4.3	4.3	1.7	141.2

Note: x₂ is the normal stiffness per unit area, x₃ is the tangential stiffness per unit area, and x₆ is the bonding radius.

. As the levels of the significant influencing factors increased, the shear force also gradually increased. Since the simulated shear force in Group 2 (37 N) closely approximates the experimentally measured value, the factor settings from Groups 1 to 3 were selected for the subsequent Box-Behnken Design.

3.4. Response Surface Experimental Design and Result Analysis

To determine the optimal parameter combination of discrete element simulation for FTB, a Box-Behnken experiment was employed with the maximum shear force serving as the response variable. The detailed experimental design and corresponding results are shown in

Table 7. Shear test, Box-Behnken experimental design, and results.

Serial Numbers	x2	x3	x6	Maximum Shear Force/N	Relative Error E/%
1	0.3	1.1	1.46	26.6	27.80
2	1.1	0.3	1.3	22.6	38.65
3	0.3	1.9	1.38	26.6	10.24
4	1.1	1.1	1.38	37	0.434
5	1.1	1.1	1.38	42.6	15.64
6	1.1	1.1	1.38	45.7	24.05
7	1.9	1.1	1.46	63.6	72.64
8	1.9	0.3	1.38	36.9	0.163
9	1.1	1.9	1.46	58.3	58.25
10	1.9	1.9	1.38	80.8	119.3
11	0.3	1.1	1.3	15.96	56.68
12	1.1	1.1	1.38	37.8	2.606
13	1.1	1.9	1.3	51.3	39.25
14	0.3	0.3	1.38	19.69	46.55
15	1.1	1.1	1.38	38.5	4.506
16	1.9	1.1	1.3	39.1	6.135
17	1.1	0.3	1.46	33	10.42

Note: x₂ is the normal stiffness per unit area, x₃ is the tangential stiffness per unit area, and x₆ is the bonding radius.

. Based on the results of the steepest ascent method, a regression equation was established with the maximum shear force (Y) as the response, as well as the significant influencing factors x₂, x₃, and x₆ as the independent variables.

Y = −583.67 − 62.65 × 10⁸x₂ + 9.08 × 10⁸x₃ + 835.5x₆ + 14.45 × 10¹⁶x₂x₃ + 54.14 × 10⁸x₂x₆ − 13.28 × 10⁸x₃x₆ − 3.37
× 10¹⁶x₂² + 4.42 × 10¹⁶x₃² − 289.26x₆²

(8)

Significance analysis of the Box-Behnken experimental data was conducted in the Design-Expert software, with the outcomes presented in

Table 8. Analysis of response surface variance.

Source	Effect	Degrees of Freedom	Mean Square	F	p
Model	4339.0	9	482.11	30.32	<0.0001 **
A-x2	2163.18	1	2163.18	136.03	<0.0001 **
B-x3	1373.14	1	1373.14	86.35	<0.0001 **
C-x6	345.06	1	345.06	21.70	0.0023 **
AB-x2x3	342.07	1	342.07	21.51	0.0024 **
AC-x2x6	48.02	1	48.02	3.02	0.1258
BC-x3x6	2.89	1	2.89	0.1817	0.6827
A2-x22	19.53	1	19.53	1.23	0.3044
B2-x32	33.75	1	33.75	2.12	0.1885
C2-x62	14.43	1	14.43	0.9074	0.3725
Residual	111.32	7	15.90
Lack of fit	56.49	3	18.83	1.37	0.3714
Pure error	54.83	4	13.71

Note: ** indicates extremely significant (p < 0.01), x₂ is the normal stiffness per unit area, x₃ is the tangential stiffness per unit area, and x₆ is the bonding radius.

. The model was found to be significant, while the non-significant lack-of-fit term indicates that no other unaccounted factors influenced the response variable. The coefficient of determination (R²) was 0.9750, and the adjusted coefficient of determination (R_adj²) was 0.9428, with both values approaching 1. The difference between the adjusted (R_pre²) and R_adj² coefficients was less than 0.2, indicating that the model fits well and reliably reflects the relationship between the maximum shear force and factors x₂, x₃, and x₆. The p-values for x₂ and x₃ were both less than 0.001, indicating their statistically significant effect on the maximum shear force. The p-values for x₆ and the interaction term x₂x₃ were both less than 0.01, suggesting a substantial influence. The remaining terms exhibited p-values above 0.05, indicating no statistically significant effect on the maximum shear force.

The response surfaces of normal stiffness (x₂) per unit area, tangential stiffness (x₃) per unit area, and bonding radius (x₆) with respect to the maximum shear force are obtained based on the regression equation, as, respectively, presented in Figure 11a–c. The response surfaces of normal stiffness (x₂) per unit area, tangential stiffness (x₃) per unit area, and bonding radius (x₆) in relation to the relative error (e) are shown in Figure 11d–f. From Figure 11a,d, it can be observed that when the bonding radius (x₆) remains constant, the maximum shear force increases with the normal stiffness (x₂) and tangential stiffness (x₃) per unit area, while the relative error initially decreases and then increases. When the tangential stiffness (x₃) per unit area is held constant, both the normal stiffness (x₂) per unit area and the bonding radius (x₆) increase gradually, leading to an increase in the maximum shear force, with the relative error initially decreasing and then increasing. Similarly, when the normal stiffness (x₂) per unit area remains constant, an increase in both tangential stiffness (x₃) per unit area and bonding radius (x₆) contributes to an upward trend in the maximum shear force, with the relative error first decreasing and then increasing. It is noteworthy that the maximum shear force and relative error show more significant changes with variations in normal stiffness (x₂) and tangential stiffness (x₃) per unit area.

4. Validation Experiment

The validation experiments comprised shear and compression tests under the optimal parameter combination, as well as shear tests using different types of blades. These tests were conducted to validate the fidelity and reliability of the discrete element model for FTB and its calibrated parameters, and to investigate the mechanical properties of FTB under shear using various blade configurations. These findings offer theoretical guidance and simulation-based support for the subsequent development of slicing equipment and cutting tools for FTB.

4.1. Validation Experiment of Optimal Parameter Combination

4.1.1. Shear Verification Experiment

The optimal parameter combination was acquired by analyzing the regression model developed from the Box-Behnken simulation experiments for FTB shearing, with the normal stiffness (x₂) per unit area of 1.438 × 10⁸ N·m⁻³, the tangential stiffness (x₃) per unit area of 0.447 × 10⁸ N·m⁻³, and the bonding radius (x₆) of 1.362 mm. The remaining non-significant influencing factors were set to constant values obtained from the single-factor test. Discrete element model simulations were conducted in EDEM software under these parameter combinations to validate the model.

As shown in Figure 12, the curve of shear force-time for FTB was obtained through a uniaxial shear test by a TMS-Pro texture analyzer. The experimental curve was then compared with the simulation curve, which is shown in Figure 13, revealing that both exhibited a similar trend. The findings indicate that the discrete element model for FTB can effectively capture the temporal variation in force during the shearing process. The experimentally obtained maximum shear force was 36.84 N, while the simulated value was 37.6 N. A relative error of 2.06% suggests a strong agreement between the experimental and simulation results.

4.1.2. Compression Verification Experiment

Considering the anisotropic characteristics of FTB, a compression validation experiment was conducted under the previously calibrated parameters to validate the fidelity of the discrete element model.

As shown in Figure 14, the mean pressure-time curve was obtained through a uniaxial compression test using a TMS-Pro texture analyzer. This experimental curve was compared with the simulation curve, which is shown in Figure 15, and both exhibited a consistent trend. This demonstrates that the established discrete element model for FTB accurately captures the variation in force over time during the compression process. The rupture force measured in the physical experiment was 40.61 N, while the simulation produced a rupture force of 40.28 N, resulting in a relative error of 0.82%. This close agreement between the simulation and experimental results confirms the accuracy of the established model.

4.2. Shear Validation Experiment with Different Types of Blades

To validate the reliability of the FTB discrete element model under different types of blades, a shear validation experiment was conducted to compare the effects of three types of distinct blades (Figure 16) on the shear force of FTB. Shear force-time curves for each blade type were obtained using a TMS-Pro texture analyzer, and these experimental results were compared with the corresponding simulation curves, which are shown in Figure 17. The trends in both sets of curves were generally consistent. The simulated average shear force for the flat blade was 37.6 N, with a relative error of 2.06% compared to the experimental value. For the arc blade, the simulated average shear force was 32.5 N, with a relative error of 2.14%. The oblique blade produced a simulated average shear force of 24.7 N, with a relative error of 5.11%. Overall, the simulation results exhibited a high degree of consistency with the experimental data, as depicted in

Table 9. Shear test and simulation results of different types of blades on FTB.

Blade Types	Simulated Values/N	Measured Values/N	Relative Error/%
Flat blade	37.6	36.84	2.06
Arc blade	32.5	31.82	2.14
Oblique blade	24.7	26.03	5.11

. Both the flat and arc blades generated higher average shear forces compared to the oblique blade. The oblique blade, by contrast, applied shear more gradually and produced lower average shear forces. Specifically, the percentages of the average shear force with the oblique blade were reduced by 52.23% and 29.55% compared to the flat and arc blades, respectively. In addition, across a series of replicated experiments, the observed differences manifested with a high degree of consistency. It is highly improbable that these differences are merely accidental occurrences stemming from experimental errors. This strongly indicates that the impact of blade shape on shear force is statistically significant.

5. Conclusions

The present study adopted the Hertz–Mindlin with bonding model to calibrate the discrete element parameters of FTB through a combined approach involving physical experiments and virtual simulation-based calibration tests. The optimal parameter combination was determined and subsequently validated through experimental trials. The main conclusions are as follows:

(1) Experimental measurements yielded average dimensions for FTB as follows: major axis diameter (a) of 37.60 mm, minor axis diameter (b) of 35.06 mm, and height (h) of 24.37 mm. The moisture content measured by the drying method was 68.7%, and the density measured by the water displacement method was 1.04 g/cm³. The Poisson’s ratio of FTB was 0.373. The elastic modulus of FTB, measured by a texture analyzer, was 7.93 MPa. The coefficient of friction between FTB and the stainless steel plate was 0.63, while the coefficient of friction between FTB and industrial rubber was 0.66. The coefficient of rolling friction between the FTB and stainless steel plate was 0.14. The coefficient of restitution (e₁) of FTB with a stainless steel plate was 0.606, and the coefficient of restitution (e₂) between FTB was 0.366.

(2) Shear tests were conducted on FTB using a TMS-Pro texture analyzer, yielding an experimentally measured maximum shear force of 37.6 N. This value served as the physical reference for the simulation-based parameter calibration. Utilizing EDEM software and the Hertz–Mindlin with bonding model, a bonding model of FTB was constructed through particle replacement. The discrete element parameters of FTB were calibrated by targeting the maximum shear force obtained from the shear tests. Single-factor tests and two-level factorial tests were performed to investigate the influence of diverse parameters on the maximum shear force. Subsequent significance analysis using Design-Expert software yielded a quadratic regression equation describing the relationship between the significant influencing factors and the maximum shear force, as well as the optimal parameter combination of significant influencing factors. The simulation results showed a strong agreement with the experimental results, further confirming the reliability of the parameters used for constructing and calibrating the FTB simulation model, effectively mitigating the influence of anisotropy on the study. The applicability of the discrete element model for materials with high moisture content has been improved.

(3) Three different types of blades were designed and subjected to both shear experiments and simulations. A comparative analysis of the results showed that the simulated shear forces for the three types of blades closely matched the experimental values, with the maximum error not exceeding 5.11%, indicating a high level of reliability of the simulation outcomes. The shear processes using the flat and arc blades were observed to be unstable. Both blades exerted higher average shear forces on FTB compared to the oblique blade. In contrast, the oblique blade exhibited a smoother shear process and lower average shear force. Specifically, the percentages of the average shear force of the oblique blade were reduced by 52.23% and 29.55% compared to the flat and arc blades, respectively. Therefore, the oblique blade is recommended as the preferred option for subsequent slicing operations of FTB. These findings provide valuable guidance for optimizing mechanized peeling and slicing equipment for FTB, particularly in the development of key components.

To further enhance the operational performance of FTB slicing equipment, future research should focus on optimizing the design of the slicing mechanism to improve its response accuracy and stability. In addition, future studies should delve deeper into the areas related to FTB harvesting and damage assessment, with an emphasis on integrating mechanical engineering with intelligent agricultural technologies. Simultaneously, it is crucial to extend the research on FTB to other crops and mechanical components. This expansion will not only promote the comprehensive development of agricultural mechanization but also provide more efficient solutions for advancing agricultural mechanization.