Flexible DEM Model Development and Parameter Calibration for Rape Stem
Flexible DEM Model Development and Parameter Calibration for Rape Stem
Guan, Zhuohuai;Mu, Senlin;Li, Haitong;Jiang, Tao;Zhang, Min;Wu, Chongyou
2022-08-23 00:00:00
applied sciences Article Flexible DEM Model Development and Parameter Calibration for Rape Stem Zhuohuai Guan, Senlin Mu *, Haitong Li, Tao Jiang, Min Zhang and Chongyou Wu Nanjing Institute of Agricultural Mechanization, Ministry of Agriculture and Rural Affairs, Nanjing 210014, China * Correspondence: musenlin@caas.cn Abstract: The discrete element method (DEM) is an effective technical tool for simulating the dynamic behavior of granular materials in agricultural engineering. However, most of the agricultural materials, such as rape stems, are flexible bodies, and it is difficult to simulate their elastic-plastic characteristics such as bending and failure using the rigid discrete element model. In this research, a flexible DEM model for rape stems was developed and the related parameters were calibrated. The proposed model consists of sequentially arranged rigid units, which were bonded together by the Hertz–Mindlin bonding contact model in the EDEM. The range of the contact parameters of a rape stem was first determined by bench test. The rape stem repose angle test was carried out as an evaluation indicator for the calibration of the DEM contact parameters. The significant factors affecting the repose angle in the contact parameters were discovered using the Plackett–Burman simulation test, and the optimal combination of these parameters was determined based on the response surface simulation test. The rape stem repose angle simulation result was 26.55 with a relative error of 2.2% for the physical tests. The rape-stem flexible DEM model’s bonding parameters were calibrated based on a three-point bending physical test and a Box–Behnken simulation test. The test results show that a rape stem’s maximum damage force obtained from the constructed model was 16.76 N, and the relative error of the measured values from the physical tests was 3.5%. The flexible DEM model could demonstrate the deformation and fracture of rape stems under an external force and can be used for the simulation of harvesting processes such as cutting, Citation: Guan, Z.; Mu, S.; Li, H.; Jiang, T.; Zhang, M.; Wu, C. Flexible conveying, and threshing. DEM Model Development and Parameter Calibration for Rape Stem. Keywords: rape stem; discrete element method; flexible DEM model; parameter calibration Appl. Sci. 2022, 12, 8394. https:// doi.org/10.3390/app12178394 Academic Editor: 1. Introduction Roberto Romaniello 6 2 Rape is the most important oil crop in China, with about 7.2 10 hm planted year- Received: 3 August 2022 round, and it is an important source of high-quality vegetable oil, providing about 50% of Accepted: 19 August 2022 China’s high-quality edible oil [1]. Rape is mainly harvested with combine harvesters [2,3]. Published: 23 August 2022 The movement pattern of rape material in the harvester and the mechanical relationship Publisher’s Note: MDPI stays neutral with the working parts of the harvester are key to the operational performance of the with regard to jurisdictional claims in rape combine harvester, mainly including the feeding of the header, conveying, threshing, published maps and institutional affil- etc. [4,5]. Modeling and simulation of the above process can help optimize the performance iations. and cost-effectiveness of combine harvester devices used in rape cultivation [6,7]. The discrete element method (DEM) is a useful numerical technique for simulating discrete body interactions and has a strong potential for the simulation-based design and performance analysis of the crop-to-machine process. Copyright: © 2022 by the authors. For the simulative study of the rape-harvesting process, discrete element model Licensee MDPI, Basel, Switzerland. construction methods have been widely studied for rapeseed long stems, short stalks, This article is an open access article seeds, pod shells, light residues, and other materials [8–11]. However, the above models, in distributed under the terms and all the simulations mentioned above, were constructed by a multi-spherical method and conditions of the Creative Commons could not simulate the deformative behaviors of long stems since all the primitive elements Attribution (CC BY) license (https:// were mutually connected in a rigid way. The mechanical behaviors such as the bending creativecommons.org/licenses/by/ 4.0/). Appl. Sci. 2022, 12, 8394. https://doi.org/10.3390/app12178394 https://www.mdpi.com/journal/applsci Appl. Sci. 2022, 12, 8394 2 of 18 and breaking of rape stems during harvesting are difficult to describe when employing a rigid model. To address the problem of a simulative analysis of flexible bodies such as crop stems, various models have been proposed. Matthew Schramm constructed a flexible fiber wheat straw DEM model that includes a mega-particle comprising smaller particles and bonded by four beam shaped bonds. The related parameters were calibrated by the three-point bending test and uniaxial compression test of wheat straw. The regression meta-model was established between the particle contact Young’s modulus and the bond Young’s modulus and Poisson’s ratio to predict the Young’s modulus of flexible wheat straw [12]. Fanyi Liu developed a flexible wheat straw model based on the Hertz–Mindlin with bonding model using the discrete element method. A single-factor sensitivity analysis and calibration of the bonding parameters were performed using a three-point bending test. The small difference between the simulated and measured elastic modulus was less than 4.2% [13]. Tom Leblicq developed a segmented wheat stem for a DEM simulation with a number of realistic bending parameters. The effects of plastic deformation and damage could be incorporated into the model. Stem bending measurements showed that the model could predict the force–deformation relationship under different numbers of segments, stem lengths, support distances, and different degrees of plastic deformation [14]. Liao Yitao carried out a bolting-stage rape crop-stalk-chopping model and the parameters were calibrated. A number of spheres were bonded, replacing the whole stalk. The error between the simulation and the physical test value of the bending force until breaking was less than 4. 21% [15]. In this research, a flexible rape stem DEM model was developed based on rigid rape stem units and bonding keys. The basic range of rape stem interaction parameters was tested by the developed test bench. Repose angle physical and simulation experiments were carried out to determine the significant factors and optimal parameter combinations in the collision parameters. Based on the collision parameters, the Hertz–Mindlin with bonding model and its bonding parameters were investigated according to the three-point bending test. The bending behavior of a rape stem was simulated using the developed flexible DEM model and was experimentally verified. The rape-stem flexible DEM model could be used for the simulation of harvesting processes such as cutting, conveying, and threshing. 2. Materials and Methods 2.1. Rape’s Characteristic Material Parameters The characteristic material parameters for constructing a discrete element model of rape stems mainly include triaxial dimension, density, shear modulus, and Poisson’s ratio. According to the previous studies, the shear modulus and Poisson’s ratio of rape stems were determined as 1.1 10 Pa and 0.4, respectively [16–18]. Other characteristic material parameters were tested by field sampling. The rape variety in this study was Zheyou 51, collected in June 2022 at the lower reaches of Yangtze River Plains, located in Liyang, Jiangsu province, China (119.4837 E, 31.41538 N). The average planting density, biomass, 5 2 4 2 3 2 yield, and height were about 4.34 10 plant/hm , 1.25 10 kg/hm , 2.3 10 kg/hm , and 1457.0 mm, respectively. The rape stem was stout at the bottom and weak at the top of the plant. The diameter of the main stem was measured at 100.0 mm intervals starting from 50.0 mm above the ground. The average diameter of the stems was 7.1 mm, and the stem density was determined to be 486 kg/m by weighing the stem mass and calculating the volume. The material parameters required for the rape stem DEM simulation are shown in the Table 1. Appl. Sci. 2022, 12, 8394 3 of 18 Table 1. Simulation parameter. Parameter Value Poisson’s ratio 0.4 3 b Rape stem Density/kgm 486 7 a Shear modulus/Pa 1.1 10 Intrinsic parameter Poisson’s ratio 0.3 3 a Steel Density/kgm 10 a Shear modulus/Pa 7.9 10 Coefficient of restitution (A) 0.322~0.611 Rape stem–rape stem Coefficient of static friction (B) 0.381~0.752 Coefficient of rolling friction (C) 0.013~0.032 Contact parameter Coefficient of restitution (D) 0.298~0.623 Rape stem–steel Coefficient of static friction (E) 0.284~0.730 Appl. Sci. 2022, 12, x FOR PEER REVIEW Coefficient of rolling friction (F) 0.021~0.042 3 of 19 a b Note: indicates that the item is obtained from the literature; indicates that the item is determined by the physical test; indicates that the item is the parameter to be determined. 2.2. Rape Stem Contact Parameters Test 2.2. Rape Stem Contact Parameters Test The construction of a discrete element model for rape stems requires the determina- The construction of a discrete element model for rape stems requires the determination tion of contact characteristic parameters between the stem–stem and stem–external mate- of contact characteristic parameters between the stem–stem and stem–external materials. rials. In this paper, bench tests were conducted to determine the basic range of rape stem In this paper, bench tests were conducted to determine the basic range of rape stem contact contact parameters, and then the optimal combination of the parameters was determined parameters, and then the optimal combination of the parameters was determined by by simulation tests. The contact parameters include coefficient of restitution, coefficient of simulation tests. The contact parameters include coefficient of restitution, coefficient of static friction, and coefficient of rolling friction. Q235 steel was selected as the contact ma- static friction, and coefficient of rolling friction. Q235 steel was selected as the contact terial since it is commonly used in agricultural machinery. material since it is commonly used in agricultural machinery. The test bench composed of Q235 is shown in Figure 1. The material box is a cube The test bench composed of Q235 is shown in Figure 1. The material box is a cube without the bottom plate. The material can fall freely after the baffle is pulled out. The without the bottom plate. The material can fall freely after the baffle is pulled out. The horizontal angle of the test plate can be freely adjusted for determining the coefficient of horizontal angle of the test plate can be freely adjusted for determining the coefficient restitution. The coefficient of static friction and coefficient of rolling friction can be deter- of restitution. The coefficient of static friction and coefficient of rolling friction can be mined by adjusting the angle of the base plate. The contact characteristic parameters of determined by adjusting the angle of the base plate. The contact characteristic parameters of the stem-Q235 can be directly determined by this device. When measuring the character- the stem-Q235 can be directly determined by this device. When measuring the characteristic istic collision parameters between stems–stems, the stems were bonded onto the test plate collision parameters between stems–stems, the stems were bonded onto the test plate and and the the baseb plate ase plate in advance. in advance. Figure 1. Rape stem contact parameters test bench. (1) Material box, (2) Baffle plate, (3) Test plate angle Figure 1. Rape stem contact parameters test bench. (1) Material box, (2) Baffle plate, (3) Test plate adjustment device, (4) Test plate, (5) Frame, (6) Base plate angle adjustment device, and (7) base plate. angle adjustment device, (4) Test plate, (5) Frame, (6) Base plate angle adjustment device, and (7) base plate. 2.2.1. Coefficient of Rest it ut ion The coefficient of restitution is an important mechanical parameter reflecting the col- lision characteristics during agricultural machine–rape stem interactions. The coefficient of restitution is defined as the ratio of the normal partial velocity of the center of mass of two objects before and after a collision [19], and is a parameter to measure the restitutive ability of an object after deformation. The test method for the rape stems’ coefficient of restitution is shown in Figure 2. Appl. Sci. 2022, 12, 8394 4 of 18 2.2.1. Coefficient of Restitution The coefficient of restitution is an important mechanical parameter reflecting the collision characteristics during agricultural machine–rape stem interactions. The coefficient of restitution is defined as the ratio of the normal partial velocity of the center of mass of two objects before and after a collision [19], and is a parameter to measure the restitutive Appl. Sci. 2022, 12, x FOR PEER REVIEW 4 of 19 ability of an object after deformation. The test method for the rape stems’ coefficient of restitution is shown in Figure 2. Figure Figure 2. 2. T Test est of of the the ra rape ste pe stems’ ms’ coef coeffic ficient ient of of restitution. restitution. h h is 1 is the the vertical vertica distance l distance betwe between e the n the baffle baffle plate and the center of the test plate, m; α is the angle between the test plate and the horizon, plate and the center of the test plate, m; a is the angle between the test plate and the horizon, ( ); h is (°); h2 is the vertical distance between the test plate and the base plate m; v1 is the normal fractional the vertical distance between the test plate and the base plate m; v is the normal fractional velocity of velocity of rape stem before the collision, m/s; v2 is the normal fractional velocity of rape stem after rape stem before the collision, m/s; v is the normal fractional velocity of rape stem after the collision, the collision, m/s; S is the horizontal distance between rape stem–bottom plate contact point and m/s; S is the horizontal distance between rape stem–bottom plate contact point and the center of the the center of the test plate, m. test plate, m. Before the test, the rape stem was sectioned into 20.0 mm, 30.0 mm, 40.0 mm, and Before the test, the rape stem was sectioned into 20.0 mm, 30.0 mm, 40.0 mm, and 50.0 mm lengths and placed in the material box. After the baffle plate was pulled, the rape 50.0 mm lengths and placed in the material box. After the baffle plate was pulled, the rape stem started to move in free-fall. The stem velocity when in contact with the test plate is stem started to move in free-fall. The stem velocity when in contact with the test plate is hg = t 1 2 h = gt 2 2 1 (1) (1) v = gt 1 1 vg = t 11 v = 2gh (2) 1vh = 2g1 (2) where h is the vertical distance between the baffle plate and the center of the test plate, m; where h1 is the vertical distance between the baffle plate and the center of the test plate, t is the free fall motion time, s; v is the normal fractional velocity of rape stem before the 1 1 m; t1 is the free fall motion time, s; v1 is the normal fractional velocity of rape stem before collision, m/s. the collision, m/s. The stem collides with the test plate and then falls on the bottom plate after an oblique The stem collides with the test plate and then falls on the bottom plate after an throwing motion: oblique throwing motion: 8 1 2 h = v t + gt 2 2y 2 2 2 1 S = v t 2x 2 hv =−+t gt 22 y2 2 (3) > v = v cos(90 2a) 2x 2 Sv = t v = v sin(90 2a) 2y 222 x (3) vv = cos() 90−α 2 22 x vv=sin() 90−α2 22 y v = 2c hS+αot2 () (4) sin 2α where h2 is the vertical distance between the test plate and the base plate, m; v2 is the normal fractional velocity of rape stem after the collision, m/s; t2 is the oblique throwing Appl. Sci. 2022, 12, 8394 5 of 18 v = q (4) 2(h +S cot 2a) sin 2a Appl. Sci. 2022, 12, x FOR PEER REVIEW 5 of 19 where h is the vertical distance between the test plate and the base plate, m; v is the normal 2 2 fractional velocity of rape stem after the collision, m/s; t is the oblique throwing motion time, s; v is the components of v in the vertical direction, m/s; v is the components of motion time, 2y s; v2y is the components 2 of v2 in the vertical direction, 2x m/s; v2x is the compo- v in the horizontal direction, m/s; a is the angle between the test plate and the horizon, ( ); nents of v2 in the horizontal direction, m/s; α is the angle between the test plate and the S is the horizontal distance between rape stem-bottom plate contact point and the center of horizon, (°); S is the horizontal distance between rape stem-bottom plate contact point and the test plate, m. the center of the test plate, m. According to Equations (2) and (4), the coefficient of restitution’s calculation formula is: According to Equations (2) and (4), the coefficient of restitution’s calculation formula is: v S c = = p (5) v S v 2 2 sin 2a h h + S cot 2a 1 ( ) 1 2 c == (5) 2sin 2α+ hh S cot 2α 1 () where c is the coefficient of restitution’s calculation. where cr is the coefficient of restitution’s calculation. In this research, h = 0.45 m, h = 0.45 m, and a = 45 . The variable that needs to be 1 2 In this research, h1 = 0.45 m, h2 = 0.45 m, and α = 45°. The variable that needs to be measured is S. Each length of rape stem was tested 30 times in duplicate. The coefficient measured is S. Each length of rape stem was tested 30 times in duplicate. The coefficient of restitution between rape stem–steel and rape stem–rape stem was [0.322, 0.611] and of restitution between rape stem–steel and rape stem–rape stem was [0.322, 0.611] and [0.298, 0.623], respectively. [0.298, 0.623], respectively. 2.2.2. Coefficient of Static Friction 2.2.2. Coefficient of Static Friction The coefficient of static friction is the ratio of the maximum static frictional force on the object to the normal pressure. The rape stem coefficient of the static friction test method The coefficient of static friction is the ratio of the maximum static frictional force on is shown in Figure 3. In the initial state, the rape stems to be tested were placed axially on the object to the normal pressure. The rape stem coefficient of the static friction test the horizontal test plate. The test plate was slowly lifted and stopped when the rape stem method is shown in Figure 3. In the initial state, the rape stems to be tested were placed slipped. The angle q displayed by the angle scale was recorded at this time, and the static axially on the horizontal test plate. The test plate was slowly lifted and stopped when the friction coefficient was calculated according to Equation (6). rape stem slipped. The angle θs displayed by the angle scale was recorded at this time, and the static friction coefficient was calculated according to Equation (6). m = tan q (6) s s μθ = tan ss (6) where m is the rape stem coefficient of static friction; q is the friction angle, ( ). s s where μs is the rape stem coefficient of static friction; θs is the friction angle, (°). Figure 3. Coefficient of static friction test method. θs is the friction angle, ( °). Figure 3. Coefficient of static friction test method. q is the friction angle, ( ). The rape stems of 20.0 mm, 30.0 mm, 40.0 mm, and 50.0 mm lengths were each rep- The rape stems of 20.0 mm, 30.0 mm, 40.0 mm, and 50.0 mm lengths were each licated 30 times. The coefficient of static friction between rape stem–steel and rape stem– replicated 30 times. The coefficient of static friction between rape stem–steel and rape rape stem was obtained as [0.284, 0.730] and [0.381, 0.752], respectively. stem–rape stem was obtained as [0.284, 0.730] and [0.381, 0.752], respectively. 2.2.3. Coefficient of Rolling Friction Rolling friction refers to an object on the surface of another object that possesses non- slip rolling properties or a tendency to roll, where the object in contact with the surface experiences deformation due to rolling hindrance. The rape stem’s coefficient of rolling Appl. Sci. 2022, 12, 8394 6 of 18 2.2.3. Coefficient of Rolling Friction Appl. Sci. 2022, 12, x FOR PEER REVIEW Rolling friction refers to an object on the surface of another object that possesses 6 of 19 non-slip rolling properties or a tendency to roll, where the object in contact with the surface experiences deformation due to rolling hindrance. The rape stem’s coefficient of rolling friction for the test method and the mechanical state of the rape stem are shown in Figure 4. friction for the test method and the mechanical state of the rape stem are shown in Figure The test plate was placed horizontally and the rape stems to be tested were placed radially 4. The test plate was placed horizontally and the rape stems to be tested were placed radi- on the test plate. The test plate was slowly lifted and stopped when the rape stem rolled on ally on the test plate. The test plate was slowly lifted and stopped when the rape stem the surface of the test plate. rolled on the surface of the test plate. Figure 4. Coefficient of rolling friction test method. θr is the angle between base plate and frame, Figure 4. Coefficient of rolling friction test method. q is the angle between base plate and frame, ( ); (°); Gs is rape stem gravity, N; Ff is rolling resistance, N; M1 is rolling torque, N·m; M2 is rolling G is rape stem gravity, N; F is rolling resistance, N; M is rolling torque, Nm; M is rolling resistance s f 1 2 resistance torque, N·m. torque, Nm. The stem tends to roll down the slope under the action of gravity. The rolling torque The stem tends to roll down the slope under the action of gravity. The rolling torque is: is: M MG = G = r sinr sinq θ (7) 1 s r 1 sr (7) wher where e rr is rape stem is rape stem radius, radius, mm; mm; M M1 is is r rol olling ling tor tor que, que N , N·m; m; G G is s irape s rapstem e stem gravi graty v,itN; y, N q ; isθr 1 s r the is the ang angle between le between base baplate se plate and fr and frame, ame, ( ).(°). The rape stem remains stationary under the action of the roll resistance. The rolling The rape stem remains stationary under the action of the roll resistance. The rolling r resistance tor esistance torque que is: is: M = f G cos q (8) 2 r s r Mf = G cosθ 2 rs r (8) where f is coefficient of rolling friction; M is the rolling resistance torque, Nm. r 2 where fr is coefficient of rolling friction; M2 is the rolling resistance torque, N·m. In the critical state: In the critical state: M = M 1 2 (9) G r sin q = f G cos q s r s r MM = (9) Gr sinθθ = f G cos The coefficient of rolling friction can be calculated as: srs r The coefficient of rolling friction can be calculated as: f = r tan q (10) r r fr = tanθ rr (10) The rape stems of 20.0 mm, 30.0 mm, 40.0 mm, and 50.0 mm lengths were each The rape stems of 20.0 mm, 30.0 mm, 40.0 mm, and 50.0 mm lengths were each repli- replicated 30 times. The coefficient of rolling friction between rape stem–steel and rape cated 30 times. The coefficient of rolling friction between rape stem–steel and rape stem– stem–rape stem was obtained as [0.013, 0.032] and [0.021, 0.042], respectively. rape stem was obtained as [0.013, 0.032] and [0.021, 0.042], respectively. 2.3. Rape Stem Contact Parameter Calibration 2.3.1. 2.3. RRape ape Ste Stem m CoRepose ntact Para Angle meter T C est alibration A cone will form when a discrete material is dumped in a plane, and the inner angle 2.3.1. Rape Stem Repose Angle Test made by the cone and the horizontal plane is the repose angle. The angle of repose is a A cone will form when a discrete material is dumped in a plane, and the inner angle made by the cone and the horizontal plane is the repose angle. The angle of repose is a macroscopic parameter characterizing the flow and friction of granular materials and is related to particle density, particle surface area, and particle friction coefficient. Therefore, Appl. Sci. 2022, 12, 8394 7 of 18 Appl. Sci. 2022, 12, x FOR PEER REVIEW macr oscopic parameter characterizing the flow and friction of granular materials and 7 of 19 is Appl. Sci. 2022, 12, x FOR PEER REVIEW 7 of 19 related to particle density, particle surface area, and particle friction coefficient. Therefore, the repose angle test is often used as the discrete elements parameter calibration for material [20,21]. The cylinder lifting method is a common means to obtain the repose the repose angle test is often used as the discrete elements parameter calibration for mate- the repose angle test is often used as the discrete elements parameter calibration for mate- angle [22]. rial [20,21]. The cylinder lifting method is a common means to obtain the repose angle [22]. rial [20,21]. The cylinder lifting method is a common means to obtain the repose angle [22]. Mixed rape stems with diameters of about 6.0 mm, 7.0 mm, and 8.0 mm were selected Mixed rape stems with diameters of about 6.0 mm, 7.0 mm, and 8.0 mm were selected Mixed rape stems with diameters of about 6.0 mm, 7.0 mm, and 8.0 mm were selected for the cylinder lifting repose angle test. Rape stems were prefabricated into 20.0 mm, for the cylinder lifting repose angle test. Rape stems were prefabricated into 20.0 mm, 30.0 for the cylinder lifting repose angle test. Rape stems were prefabricated into 20.0 mm, 30.0 30.0 mm, and 40.0 mm lengths, and mixed rape stems with 9 sizes of samples were used in mm, and 40.0 mm lengths, and mixed rape stems with 9 sizes of samples were used in the mm, and 40.0 mm lengths, and mixed rape stems with 9 sizes of samples were used in the the test. A steel cylinder (with a diameter of 120.0 mm and height of 100.0 mm) was fixed test. A steel cylinder (with a diameter of 120.0 mm and height of 100.0 mm) was fixed on test. A steel cylinder (with a diameter of 120.0 mm and height of 100.0 mm) was fixed on on the universal-material-testing machine and its bottom surface was in contact with the the universal-material-testing machine and its bottom surface was in contact with the test the universal-material-testing machine and its bottom surface was in contact with the test test bench. The rape stems were filled into the steel cylinder until it was full. The steel bench. The rape stems were filled into the steel cylinder until it was full. The steel cylinder bench. The rape stems were filled into the steel cylinder until it was full. The steel cylinder cylinder was lifted upward at a speed of 0.05 m/s so that the rape stems formed a granule was lifted upward at a speed of 0.05 m/s so that the rape stems formed a granule pile, as was lifted upward at a speed of 0.05 m/s so that the rape stems formed a granule pile, as pile, as shown in Figure 5a. After trial, the frontal image of the accumulated stems was shown in Figure 5a. After trial, the frontal image of the accumulated stems was captured shown in Figure 5a. After trial, the frontal image of the accumulated stems was captured captured and binarized. The boundary was extracted by the edge detection method and the and binarized. The boundary was extracted by the edge detection method and the linear and binarized. The boundary was extracted by the edge detection method and the linear linear fitting method was used to calculate the slope of the boundary to obtain the repose fitting method was used to calculate the slope of the boundary to obtain the repose angle. fitting method was used to calculate the slope of the boundary to obtain the repose angle. angle. The test was repeated 5 times and averaged. The test was repeated 5 times and averaged. The test was repeated 5 times and averaged. (a) (b) (a) (b) Figure 5. Rape stem repose angle test. (a) Physical test; (b) Simulation test. Figure Figure 5. 5. Rape Rape stem stem repose repose ang angle le test test.. ( (a a) ) P Physical hysical t test; est; ( (b b) ) Simulation test. Simulation test. 2.3.2. 2.3.2. R Rape ape St Stem em Repose Repose Angle Angle Sim Simulation ulation Model Model 2.3.2. Rape Stem Repose Angle Simulation Model The rape s The rape stems’ tems’ repose angle w repose angle was as fur further ther studied studied by DEM. The by DEM. The rape rape stem discrete stem discrete The rape stems’ repose angle was further studied by DEM. The rape stem discrete element model was built in the DEM software EDEM 2018 (DEM Solutions Limited, Ed- element model was built in the DEM software EDEM 2018 (DEM Solutions Limited, Edin- element model was built in the DEM software EDEM 2018 (DEM Solutions Limited, Edin- inburgh, UK). Each rape stem was approximated as a cylinder and filled with spherical burgh, UK). Each rape stem was approximated as a cylinder and filled with spherical par- burgh, UK). Each rape stem was approximated as a cylinder and filled with spherical par- particles as shown in Figure 6. To match the repose angle physical test, rape stem models ticles as shown in Figure 6. To match the repose angle physical test, rape stem models with ticles as shown in Figure 6. To match the repose angle physical test, rape stem models with with lengths of 20.0 mm, 30.0 mm, and 40.0 mm and diameters of 6.0 mm, 7.0 m, and lengths of 20.0 mm, 30.0 mm, and 40.0 mm and diameters of 6.0 mm, 7.0 m, and 8.0 mm lengths of 20.0 mm, 30.0 mm, and 40.0 mm and diameters of 6.0 mm, 7.0 m, and 8.0 mm 8.0 mm were established. The steel cylinder ’s size and lifting speed and rape stem’s quality were established. The steel cylinder’s size and lifting speed and rape stem’s quality in the were established. The steel cylinder’s size and lifting speed and rape stem’s quality in the in the simulation were consistent with the physical test, as shown in Figure 5b. simulation were consistent with the physical test, as shown in Figure 5b. simulation were consistent with the physical test, as shown in Figure 5b. 20.0 mm 30.0 mm 40.0 mm 20.0 mm 30.0 mm 40.0 mm 20.0 mm 30.0 mm 40.0 mm 20.0 mm 30.0 mm 40.0 mm 20.0 mm 30.0 mm 40.0 mm 20.0 mm 30.0 mm 40.0 mm (a) (b) (c) (a) (b) (c) Figure 6. Rape stem repose angle test. (a) Diameter = 6.0 mm; (b) Diameter = 7.0 mm; (c) Diameter Figure 6. Rape stem repose angle test. (a) Diameter = 6.0 mm; (b) Diameter = 7.0 mm; (c) Diameter Figure 6. Rape stem repose angle test. (a) Diameter = 6.0 mm; (b) Diameter = 7.0 mm; = 4.0 mm. = 4.0 mm. (c) Diameter = 4.0 mm. The Hertz–Mindlin (no slip) model was chosen as the particle model and particle con- The Hertz–Mindlin (no slip) model was chosen as the particle model and particle con- tact model in the EDEM software for rape stems. The simulation time step, environmental tact model in the EDEM software for rape stems. The simulation time step, environmental −6 −2 gravitational acceleration, and preservation interval were 0.5 × 10 s, 9.8 m·s , and 0.01 s, −6 −2 gravitational acceleration, and preservation interval were 0.5 × 10 s, 9.8 m·s , and 0.01 s, Appl. Sci. 2022, 12, 8394 8 of 18 The Hertz–Mindlin (no slip) model was chosen as the particle model and particle contact model in the EDEM software for rape stems. The simulation time step, environmen- 6 2 tal gravitational acceleration, and preservation interval were 0.5 10 s, 9.8 ms , and 0.01 s, respectively. The simulation time was 2.0 s to ensure that the rape stems remained stationary and so the repose angle no longer changed. The related contact-mechanical characteristic parameters between the rape stem and the steel are shown in Table 1. 2.3.3. Rape Stem Contact Parameter Calibration Method Due to the large number of stem contact parameters, the significant parameters af- fecting the repose angle needed to be screened first. The Plackett–Burman method is a two-level experimental design method whereby the significance of the factors is deter- mined by comparing the difference between the two levels of each factor with the overall difference. In this research, based on the basic parameter range determined by the physical repose angle test, the key simulation model parameters affecting the repose angle were investigated by the Plackett–Burman test. The test was conducted in 14 groups, with 2 groups of central levels for model’s validation. To reduce the influence of random factors such as material generation position and angle on the test results, each group of tests was repeated 5 times and averaged. Contact parameters that significantly affected the repose Appl. Sci. 2022, 12, x FOR PEER REVIEW 9 of 19 angle were determined by ANOVA and t-test. Response surface tests were conducted with these significance parameters as variables to determine the optimal combination of parameters for the rape stem contact parameters. unit contained 24 particles. Secondly, several rigid units were arranged sequentially along 2.4. Calibration of Flexible Discrete Element Parameter for Rape Stem their axes through the EDEM software particle factory API. Finally, the rigid units were 2.4.1. Flexible Discrete Element Model bonded together (Figure 7c) by the Hertz–Mindlin with bonding contact model to replace The position of each spherical unit in the rigid stem model is fixed, so it cannot simulate the initial rape stem rigid discrete element model (Figure 7b). The rigid units are bonded the deformation behaviors of the stem such as bending and twisting. To address the problem to each other and able to withstand a certain amount of normal and tangential displace- wherein the rape stem is a flexible body and the rigid model cannot accurately express its ment. The bond will fracture when reaching the critical normal and tangential stresses. mechanical characteristics, the rape stem flexible discrete element model (Figure 7) was Thus, the developed rape stem flexible model can simulate a variety of mechanical behav- built based on EDEM API and Hertz–Mindlin with Bonding contact model. iors such as tensile, bending, torsion, and fracture behaviors. (b) (a) (c) Figure 7. Rape stem flexible discrete element model. (a) Rigid discrete unit; (b) Rigid discrete ele- Figure 7. Rape stem flexible discrete element model. (a) Rigid discrete unit; (b) Rigid discrete element ment model; (c) Flexible discrete element model. model; (c) Flexible discrete element model. 2.4.2. Three-Point Bending Test Firstly, a rigid unit for the rape stem flexible model was built through spherical particle The three-point bending test could quantify and comprehend the elastoplastic flex- filling as shown in Figure 7a. The rape stem was approximated as a hollow structure since ural behav the internal ior of crop stalk [2 spongy structure3 of ]. In t the h rape is research, stem is loose the ma and ximum dama of low strength. ge force of t According he rato pe stems was obtained by the three-point bending test, which was used to calibrate the bond- the result of the rape stems’ structure test, the overall diameter of the rigid discrete unit and ing parameters of the flexible discrete element model. the internal hollow core diameter were determined to be 7.0 mm and 3.0 mm, respectively. Spherical The test dev particles ice is shown with a diameter in Figuof re 8a. Th 2.0 mm e loading were used device was th for filling,e electron and each ic un rigid iversal unit testin contained g mach 24 ine (utm6503, Suns, Shen particles. Secondly, several zhrigid en, Chin units a). T wer he test mater e arranged sequentially ials were selec along ted from their axes through the EDEM software particle factory API. Finally, the rigid units were bonded rape stems with an average diameter of 7.0 mm (with 95% confidence interval lower and upper limit together (Figur s of 6.7 mm a e 7c) by the nd 7 Hertz–Mindlin .4 mm, respect with ively bonding ). The moistur contact e model content of to replace the ra the pe stems initial rape stem rigid discrete element model (Figure 7b). The rigid units are bonded to each was measured to be 51.3%. The distance between the support points on both sides of the other and able to withstand a certain amount of normal and tangential displacement. The sample was 60.0 mm, and the loading speed was 1 mm/s. The support cylinder and loading bond will fracture when reaching the critical normal and tangential stresses. Thus, the cylinder radii were 4.0 mm. The rape stem was extruded and deformed under the load until broken, and the maximum bending-damage force was recorded. The test was re- peated 5 times and averaged, resulting in a mean maximum damage force of 16.20 N and a coefficient of variation of 4.94%. (a) (b) Figure 8. Rape stem three-point bending test: (a) Physics test; (b) Simulation test. 2.4.3. Bonding Parameter Calibration In the Hertz–Mindlin with Bonding contact model, the key parameters include nor- mal stiffness per unit area, shear stiffness per unit area, critical normal stress, critical shear stress, and bonded radius. Referring to the vine materials and existing studies on Appl. Sci. 2022, 12, x FOR PEER REVIEW 9 of 19 unit contained 24 particles. Secondly, several rigid units were arranged sequentially along their axes through the EDEM software particle factory API. Finally, the rigid units were bonded together (Figure 7c) by the Hertz–Mindlin with bonding contact model to replace the initial rape stem rigid discrete element model (Figure 7b). The rigid units are bonded to each other and able to withstand a certain amount of normal and tangential displace- ment. The bond will fracture when reaching the critical normal and tangential stresses. Thus, the developed rape stem flexible model can simulate a variety of mechanical behav- iors such as tensile, bending, torsion, and fracture behaviors. (b) Appl. Sci. 2022, 12, 8394 9 of 18 (a) (c) Figure 7. Rape stem flexible discrete element model. (a) Rigid discrete unit; (b) Rigid discrete ele- developed rape stem flexible model can simulate a variety of mechanical behaviors such as ment model; (c) Flexible discrete element model. tensile, bending, torsion, and fracture behaviors. 2.4.2. 2.4.2. Thr Thr ee-Point ee-Point Bend Bending ing Test Test The three-point bending test could quantify and comprehend the elastoplastic flex- The three-point bending test could quantify and comprehend the elastoplastic flexural behavior ural behav ofior of crop stalk [2 crop stalk [23]. In 3 this ]. In t resear his research, ch, the maximum the maximum dama damage force ge of force of t the rapehstems e rape was stemobtained s was obta by ined the bthr y the ee-point three-p bending oint bending test, t which est, which w was used as us to ed calibrate to calibra the te th bonding e bond- parameters ing paramete ofrs of th the flexible e flexib discr le discre ete element te element mode model. l. The test device is shown in Figure 8a. The loading device was the electronic universal The test device is shown in Figure 8a. The loading device was the electronic universal testing testing mach machine ine (utm6503, Suns, Shen (utm6503, Suns, Shenzhen, zhen, Chin China). a). T The he test mater test materials ials were se were selected lected from from rape stems with an average diameter of 7.0 mm (with 95% confidence interval lower and rape stems with an average diameter of 7.0 mm (with 95% confidence interval lower and upper limits of 6.7 mm and 7.4 mm, respectively). The moisture content of the rape stems upper limits of 6.7 mm and 7.4 mm, respectively). The moisture content of the rape stems was measured to be 51.3%. The distance between the support points on both sides of was measured to be 51.3%. The distance between the support points on both sides of the the sample was 60.0 mm, and the loading speed was 1 mm/s. The support cylinder and sample was 60.0 mm, and the loading speed was 1 mm/s. The support cylinder and loading loading cylinder radii were 4.0 mm. The rape stem was extruded and deformed under the cylinder radii were 4.0 mm. The rape stem was extruded and deformed under the load load until broken, and the maximum bending-damage force was recorded. The test was until broken, and the maximum bending-damage force was recorded. The test was re- repeated 5 times and averaged, resulting in a mean maximum damage force of 16.20 N and peated 5 times and averaged, resulting in a mean maximum damage force of 16.20 N and a coefficient of variation of 4.94%. a coefficient of variation of 4.94%. (a) (b) Figure 8. Rape stem three-point bending test: (a) Physics test; (b) Simulation test. Figure 8. Rape stem three-point bending test: (a) Physics test; (b) Simulation test. 2.4.3. 2.4.3. Bondin Bonding g Parameter Parameter C Calibration alibration In the He In the Hertz–Mindlin rtz–Mindlin w with ith Bonding Bonding contact contacmodel, t modethe l, th key e key p parameters arameters include includ normal e nor- stif mal fness stiffne per ssunit per un area, it are shear a, shear stiffness stiffnes per unit s per ar un ea, itcritical area, cri normal tical no str rm ess, al s critical tress, cri shear ticalstr sh ess, ear and stress, an bonded d b radius. onded r Referring adius. Referrin to the vine g to materials the vine mate and existing rials an studies d existing studie on agricultural s on material simulation parameters, the critical normal stress and critical shear stress are not significant in the discrete element model of crop stems and can be taken as the values of 45.0 Mpa and 7.0 Mpa, respectively [24,25]. The stiffness per unit area, shear stiffness per unit area, and bonded radius were determined by the Box–Behnken response surface test. The established discrete element model for rape stems’ flexibility was applied to conduct a three-point bending simulation test. The maximum damage force of the three- point bending test was used as the response value to determine the optimal combination of bonding parameters. The simulation test is shown in Figure 8b. The simulation fixed time step was 5.0 10 s and other conditions were consistent with the physical test. The bonding parameter range is shown in Table 2. The effect of bonding parameters and their interaction on stem bending damage was analyzed by ANOVA. The relationship between the bonding parameters and the rape stems’ maximum bending-damage force was modeled to solve the optimal combination of parameters. Appl. Sci. 2022, 12, x FOR PEER REVIEW 10 of 19 agricultural material simulation parameters, the critical normal stress and critical shear stress are not significant in the discrete element model of crop stems and can be taken as the values of 45.0 Mpa and 7.0 Mpa, respectively [24,25]. The stiffness per unit area, shear stiffness per unit area, and bonded radius were determined by the Box–Behnken response surface test. The established discrete element model for rape stems’ flexibility was applied to con- duct a three-point bending simulation test. The maximum damage force of the three-point Appl. Sci. 2022, 12, 8394 10 of 18 bending test was used as the response value to determine the optimal combination of bonding parameters. The simulation test is shown in Figure 8b. The simulation fixed time −7 step was 5.0 × 10 s and other conditions were consistent with the physical test. The bond- ing parameter range is shown in Table 2. The effect of bonding parameters and their inter- Table 2. Bonding parameter. action on stem bending damage was analyzed by ANOVA. The relationship between the bonding parameters and the rape stems’ maximum bending-damage force was modeled Level to solve the optimal combination of parameters. Factor 1 0 1 Table 2. Bonding parameter. 3 9 9 10 Stiffness per unit area (G)/Nm 1.0 10 5.5 10 1.0 10 Level 3 8 8 9 Shear Stiffness per unit area (H)/Nm 1.0 10 5.5 10 1.0 10 Factor −1 0 1 Bonded radius (I)/mm 1.0 1.5 2.0 −3 9 9 10 Stiffness per unit area (G)/N·m 1.0 × 10 5.5 × 10 1.0 × 10 −3 8 8 9 Shear Stiffness per unit area (H)/N·m 1.0 × 10 5.5 × 10 1.0 × 10 3. Results and Discussion Bonded radius (I)/mm 1.0 1.5 2.0 3.1. Contact Parameter 3.1.1. Repose Angle Test 3. Results and Discussion The results of the physical and simulation tests of the rape stems’ repose angle are 3.1. Contact Parameter shown in Figure 9a. The result of the binarization of the stem-piling image is shown in 3.1.1. Repose Angle Test Figure 9b. The discrete rape stems were tapered and stacked on a flat surface, but some of The results of the physical and simulation tests of the rape stems’ repose angle are the stems protruded from the tapered boundary. The boundary of the stacked stems was shown in Figure 9a. The result of the binarization of the stem-piling image is shown in Figure 9b. The discrete rape stems were tapered and stacked on a flat surface, but some of further extracted and linearly fitted as shown in Figure 9c. The rape stem repose angle was the stems protruded from the tapered boundary. The boundary of the stacked stems was obtained as shown in Figure 9d, and the mean value of the rape stem repose angles was further extracted and linearly fitted as shown in Figure 9c. The rape stem repose angle was 27.15 with a coefficient of variation of 1.27% from five replicate trials. The same algorithm obtained as shown in Figure 9d, and the mean value of the rape stem repose angles was was applied to the simulation results. The simulation results indicate that the established 27.15° with a coefficient of variation of 1.27% from five replicate trials. The same algorithm was applied to the simulation results. The simulation results indicate that the established discrete element model can simulate the piling process of the rape stems, and that the DEM discrete element model can simulate the piling process of the rape stems, and that the DEM contact parameter can be calibrated by the repose angle test. contact parameter can be calibrated by the repose angle test. (a) Appl. Sci. 2022, 12, x FOR PEER REVIEW 11 of 19 (b) (c) (d) Figure 9. Rape stem repose angle test. (a) Image of rape stem pile; (b) Binarized image; (c) Bound- Figure 9. Rape stem repose angle test. (a) Image of rape stem pile; (b) Binarized image; (c) Boundary ary Fitting; (d) Test result. Fitting; (d) Test result. 3.1.2. Significant Factor for Repose Angle The Plackett–Burman simulation test results for the rape stems’ contact parameters are shown in Table 3. The test results were subjected to ANOVA to obtain the effect of each parameter, as shown in Table 4. Table 3. Results of Plackett–Burman simulation test of rape stems’ contact paramaters. Test No. A B C D E F y1 1 0.611 0.752 0.032 0.298 0.284 0.021 21.29 2 0.611 0.381 0.013 0.298 0.73 0.021 25.58 3 0.4665 0.5665 0.0225 0.4605 0.507 0.0315 27.18 4 0.611 0.752 0.013 0.298 0.284 0.042 24.89 5 0.611 0.381 0.032 0.623 0.73 0.021 30.33 6 0.611 0.752 0.013 0.623 0.73 0.042 33.60 7 0.4665 0.5665 0.0225 0.4605 0.507 0.0315 26.95 8 0.322 0.381 0.013 0.623 0.284 0.042 24.45 9 0.322 0.752 0.032 0.623 0.284 0.021 20.78 10 0.322 0.752 0.032 0.298 0.73 0.042 25.17 11 0.322 0.381 0.032 0.298 0.73 0.042 34.36 12 0.322 0.752 0.013 0.623 0.73 0.021 26.11 13 0.611 0.381 0.032 0.623 0.284 0.042 25.26 14 0.322 0.381 0.013 0.298 0.284 0.021 17.69 Note: A is rape stem–rape stem coefficient of restitution; B is rape stem–rape stem coefficient of static friction; C is rape stem–rape stem coefficient of rolling friction; D is rape stem–steel coeffi- cient of restitution; E is rape stem–steel coefficient of static friction; F is rape stem–steel coefficient of rolling friction; y1 is repose angle, (°). Table 4. ANOVA of Plackett–Burman test. Factor Standardized Effects Sum of Squares Contribution/% Significance Ranking A 2.065 12.793 4.305 3 B −0.972 2.832 0.953 5 C 0.813 1.985 0.668 6 D 1.925 11.117 3.741 4 E 6.800 138.720 46.687 1 Appl. Sci. 2022, 12, 8394 11 of 18 3.1.2. Significant Factor for Repose Angle The Plackett–Burman simulation test results for the rape stems’ contact parameters are shown in Table 3. The test results were subjected to ANOVA to obtain the effect of each parameter, as shown in Table 4. Table 3. Results of Plackett–Burman simulation test of rape stems’ contact paramaters. Test No. A B C D E F y 1 0.611 0.752 0.032 0.298 0.284 0.021 21.29 2 0.611 0.381 0.013 0.298 0.73 0.021 25.58 3 0.4665 0.5665 0.0225 0.4605 0.507 0.0315 27.18 4 0.611 0.752 0.013 0.298 0.284 0.042 24.89 5 0.611 0.381 0.032 0.623 0.73 0.021 30.33 6 0.611 0.752 0.013 0.623 0.73 0.042 33.60 7 0.4665 0.5665 0.0225 0.4605 0.507 0.0315 26.95 8 0.322 0.381 0.013 0.623 0.284 0.042 24.45 9 0.322 0.752 0.032 0.623 0.284 0.021 20.78 10 0.322 0.752 0.032 0.298 0.73 0.042 25.17 11 0.322 0.381 0.032 0.298 0.73 0.042 34.36 12 0.322 0.752 0.013 0.623 0.73 0.021 26.11 13 0.611 0.381 0.032 0.623 0.284 0.042 25.26 14 0.322 0.381 0.013 0.298 0.284 0.021 17.69 Note: A is rape stem–rape stem coefficient of restitution; B is rape stem–rape stem coefficient of static friction; C is rape stem–rape stem coefficient of rolling friction; D is rape stem–steel coefficient of restitution; E is rape stem–steel coefficient of static friction; F is rape stem–steel coefficient of rolling friction; y is repose angle, ( ). Table 4. ANOVA of Plackett–Burman test. Factor Standardized Effects Sum of Squares Contribution/% Significance Ranking A 2.065 12.793 4.305 3 B 0.972 2.832 0.953 5 C 0.813 1.985 0.668 6 D 1.925 11.117 3.741 4 E 6.800 138.720 46.687 1 F 4.328 56.203 18.916 2 According to Table 4, the significance ranking of the repose angles’ effect on the contact parameters of the rape stems was E, F, A, D, B, and C. t-test and Pareto charts (Figure 10) were created to determine the parameters that needed to be further discussed. As shown in the Pareto charts, the t-value of the coefficient of static friction (E), the rape stem–steel coefficient of the rolling friction contact plasticity ratio (F), the rape stem–rape stem coefficient of restitution (A), and the rape stem–steel coefficient of restitution (D) were all larger than 1, which shows they are the significance fartors for the repose angle. The rape stem–rape stem coefficient of static friction (B) and rape stem–rape stem coefficient of rolling friction (C) had almost no effect on the repose angle. Therefore, E, F, A, and D were the test factors for the further response surface test. For B and C, the intermediate values were taken as 0.5665 and 0.0225, respectively. Appl. Sci. 2022, 12, x FOR PEER REVIEW 12 of 19 F 4.328 56.203 18.916 2 According to Table 4, the significance ranking of the repose angles’ effect on the con- Appl. Sci. 2022, 12, 8394 12 of 18 tact parameters of the rape stems was E, F, A, D, B, and C. t-test and Pareto charts (Figure 10) were created to determine the parameters that needed to be further discussed. Figure 10. Pareto charts. Figure 10. Pareto charts. 3.1.3. Contact Parameter Calibration As shown in the Pareto charts, the t-value of the coefficient of static friction (E), the The rape stems’ contact parameter response surface simulation test results are shown rape stem–steel coefficient of the rolling friction contact plasticity ratio (F), the rape stem– in Table 5. The ANOVA of the test results is shown in Table 6. rape stem coefficient of restitution (A), and the rape stem–steel coefficient of restitution (D) were all larger than 1, which shows they are the significance fartors for the repose angle. Table 5. Results of rape stems’ contact-parameter-response surface simulation test. The rape stem–rape stem coefficient of static friction (B) and rape stem–rape stem coeffi- cient of rolling friction (C) had almost no effect on the repose angle. Therefore, E, F, A, and Test No. A D E F y D were the test factors for the further response surface test. For B and C, the intermediate 1 0.322 0.298 0.507 0.0315 22.88 values were taken as 0.5665 and 0.0225, respectively. 2 0.611 0.298 0.507 0.0315 23.88 3 0.322 0.623 0.507 0.0315 22.77 4 0.611 3.1.3. Contact Par0.623 ameter Calibration 0.507 0.0315 25.91 5 0.4665 0.4605 0.284 0.021 17.98 The rape stems’ contact parameter response surface simulation test results are shown 6 0.4665 0.4605 0.73 0.021 24.72 in Table 5. The ANOVA of the test results is shown in Table 6. 7 0.4665 0.4605 0.284 0.042 23.88 8 0.4665 0.4605 0.73 0.042 31.40 Table 5. Results of rape stems’ contact-parameter-response surface simulation test. 9 0.322 0.4605 0.507 0.021 22.35 10 0.611 0.4605 0.507 0.021 26.12 Test No. A D E F y1 11 0.322 0.4605 0.507 0.042 27.40 1 0.322 0.298 0.507 0.0315 22.88 12 0.611 0.4605 0.507 0.042 27.35 13 0.4665 0.298 0.284 0.0315 18.58 2 0.611 0.298 0.507 0.0315 23.88 14 0.4665 0.623 0.284 0.0315 17.98 3 0.322 0.623 0.507 0.0315 22.77 15 0.4665 0.298 0.73 0.0315 23.79 4 0.611 0.623 0.507 0.0315 25.91 16 0.4665 0.623 0.73 0.0315 31.40 5 0.4665 0.4605 0.284 0.021 17.98 17 0.322 0.4605 0.284 0.0315 20.41 18 0.611 0.4605 0.284 0.0315 19.08 6 0.4665 0.4605 0.73 0.021 24.72 19 0.322 0.4605 0.73 0.0315 23.87 7 0.4665 0.4605 0.284 0.042 23.88 20 0.611 0.4605 0.73 0.0315 29.72 8 0.4665 0.4605 0.73 0.042 31.40 21 0.4665 0.298 0.507 0.021 27.25 22 0.4665 0.623 0.507 0.021 21.82 23 0.4665 0.298 0.507 0.042 26.19 24 0.4665 0.623 0.507 0.042 29.29 25 0.4665 0.4605 0.507 0.0315 27.18 26 0.4665 0.4605 0.507 0.0315 24.72 27 0.4665 0.4605 0.507 0.0315 27.01 Appl. Sci. 2022, 12, 8394 13 of 18 Table 6. ANOVA for contact-parameter-response surface test’s results. Coefficient Estimate Source Sum of Squares df Mean Square F-Value p-Value (Coded Factors) Model n 349.62 14 24.97 11.66 <0.0001 ** A 1.03 12.8 1 12.8 5.97 0.0309 * D 0.5515 3.65 1 3.65 1.70 0.2163 E 3.92 183.94 1 183.94 85.85 <0.0001 ** F 2.11 53.2 1 53.20 24.83 0.0003 ** AD 0.5345 1.14 1 1.14 0.53 0.4792 AE 1.80 12.89 1 12.89 6.02 0.0304 * AF 0.9551 3.65 1 3.65 1.70 0.2164 DE 2.05 16.84 1 16.84 7.86 0.0159 * DF 2.13 18.19 1 18.19 8.49 0.013 * EF 0.1959 0.1535 1 0.154 0.072 0.7935 A 1.10 6.47 1 6.47 3.02 0.1077 D 1.10 6.47 1 6.47 3.02 0.1078 E 2.22 26.23 1 26.23 12.24 0.0044 ** F 0.6495 2.25 1 2.25 1.05 0.3257 Intercept 26.30 n n n n n Residual n 25.71 12 2.14 n n Lack of Fit n 21.94 10 2.19 1.16 0.5479 Pure Error n 3.77 2 1.89 n n Cor Total n 375.33 26 n n Note: ** means highly significant (p < 0.01), * means significant (0.01 p < 0.05). According to the quadratic multi-variate fitting regression ANOVA results of the coefficient of variation y , the regression model’s p-values are less than 0.01, which indicates that the model is highly significant. The lack-of-fit F-value is more than 0.05, which implies the model is not significant relative to the pure error. It indicates that the model can correctly reflect the relationship among y and A, D, E, and F as well as predict test results. Among them, A, E, F, and E , are highly significant model terms and AE, DE, and DF are significant model terms. After excluding the insignificant factors, the quadratic regression model (actual factor) of the repose angle y is: y = 40.03 21.00 A + 4.38 E 375.02 F + 55.71 AE + 56.62 DE + 1249.82 DF 38.34 E (11) The effect of the interaction factors on the repose angle is shown in Figure 11. When D and F are at the center level (0.4605 and 0.0315), y increases with the increase of A and E. The response surface’s curve changes faster along the E direction, and the effect of the rape stem–steel coefficient of static friction on the repose angle is more significant than that of the rape stem–rape stem coefficient of restitution. When A and F are at the center level (0.4665 and 0.507), y increases with the increase of D and E. The response surface’s curve changes faster along the F direction, and the effect of the rape stem–steel coefficient of static friction on the repose angle is more significant than that of the rape stem-steel coefficient of restitution. When A and E are at the center level (0.4665 and 0.0315), y increases with the increase of D and F. The response surface curve changes faster along the F direction, and the effect of the rape stem–steel coefficient of rolling friction on the repose angle is more significant than that of the rape stem–steel coefficient of restitution. Appl. Sci. 2022, 12, x FOR PEER REVIEW 14 of 19 Cor Total \ 375.33 26 \ Note: ** means highly significant (p < 0.01), * means significant (0.01 ≤ p < 0.05). According to the quadratic multi-variate fitting regression ANOVA results of the co- efficient of variation y1, the regression model’s p-values are less than 0.01, which indicates that the model is highly significant. The lack-of-fit F-value is more than 0.05, which implies the model is not significant relative to the pure error. It indicates that the model can cor- rectly reflect the relationship among y1 and A, D, E, and F as well as predict test results. Among them, A, E, F, and E², are highly significant model terms and AE, DE, and DF are significant model terms. After excluding the insignificant factors, the quadratic regression model (actual factor) of the repose angle y1 is: y1 = 40.03 − 21.00 A + 4.38 E − 375.02 F + 55.71 AE + 56.62 DE + 1249.82 DF − (11) 38.34 E² The effect of the interaction factors on the repose angle is shown in Figure 11. When D and F are at the center level (0.4605 and 0.0315), y1 increases with the increase of A and E. The response surface’s curve changes faster along the E direction, and the effect of the rape stem–steel coefficient of static friction on the repose angle is more significant than that of the rape stem–rape stem coefficient of restitution. When A and F are at the center level (0.4665 and 0.507), y1 increases with the increase of D and E. The response surface’s curve changes faster along the F direction, and the effect of the rape stem–steel coefficient of static friction on the repose angle is more significant than that of the rape stem-steel coef- ficient of restitution. When A and E are at the center level (0.4665 and 0.0315), y1 increases with the increase of D and F. The response surface curve changes faster along the F direc- Appl. Sci. 2022, 12, 8394 14 of 18 tion, and the effect of the rape stem–steel coefficient of rolling friction on the repose angle is more significant than that of the rape stem–steel coefficient of restitution. (a) (b) (c) Figure 11. The effect of interaction factors on repose angle: (a) y1 = f (A 0.4605, E, 0.0315); (b) y1 = f Figure 11. The effect of interaction factors on repose angle: (a) y = f (A 0.4605, E, 0.0315); (0.4665, D, E, 0.507); (c) y1 = f (0.4665, D, 0.0315, F). (b) y = f (0.4665, D, E, 0.507); (c) y = f (0.4665, D, 0.0315, F). 1 1 Using the physical test measurement of the stacking angle of 27.15° as the optimiza- Using the physical test measurement of the stacking angle of 27.15 as the optimization tion target, the combination of the contact parameters’ optimization model is: target, the combination of the contact parameters’ optimization model is: y =27.15° 0.y32= 2≤≤ 27.15 A 0.611 > 1 > 8 0.322 A 0.611 < > 0.298≤≤ D 0.623 (12) s.t. 0.298 D 0.623 (12) 0.284≤≤ E 0.730 > s.t. > > 0.284 E 0.730 > > 0.021≤≤ F 0.042 : : 0.021 F 0.042 Based on Equations (11) and (12), the solution for the optimal combination of param- eters was solved as A = 0.514, D = 0.463, E = 0.568, and F = 0.029. The rape stem repose angle simulation result was 26.55 through this combination of parameters. The relative error to the measured values from physical tests (27.15 ) was 2.2%. 3.2. Flexible DEM Model 3.2.1. Bonding Parameter Box–Behnken Simulation Test The result of the bonding parameters’ Box–Behnken simulation is shown in Table 7. A quadratic multivariate fitting of the experimental results was carried out by applying Design-Expert 12 software, and an ANOVA and a regression coefficient significance test were carried out for the regression model. The result is shown in Table 8. Table 7. Bonding parameter Box–Behnken simulation result. 3 3 Test No. G/Nm H/Nm I/mm y /N 9 9 1 2.0 54.80 5.50 10 1.00 10 9 9 2 5.50 10 1.00 10 1.0 16.16 10 8 3 1.00 10 1.00 10 1.5 35.90 9 8 4 1.00 10 1.00 10 1.5 8.72 9 8 5 5.50 10 5.50 10 1.5 27.60 9 8 6 1.00 10 5.50 10 1.0 4.91 9 8 7 5.50 10 5.50 10 1.5 27.90 10 8 8 1.00 10 5.50 10 2.0 86.50 9 8 9 2.0 47.80 5.50 10 1.00 10 10 8 10 1.00 10 5.50 10 1.0 19.26 9 8 11 5.50 10 5.50 10 1.5 25.30 10 9 12 1.00 10 1.00 10 1.5 41.80 9 8 13 5.50 10 1.00 10 1.0 14.51 9 8 14 1.00 10 5.50 10 2.0 31.20 9 9 15 1.00 10 1.00 10 1.5 12.43 3 3 Note: G is normal stiffness per unit area, Nm ; H is shear stiffness per unit area, Nm ; I is bonded disk radius, mm; y is maximum damage force, N. 2 Appl. Sci. 2022, 12, 8394 15 of 18 Table 8. ANOVA for bonding parameter Box–Behnken simulation result. Appl. Sci. 2022, 12, x FOR PEER REVIEW 16 of 19 Coefficient Estimate Sum of Squares df Mean Square F-Value p-Value Source (Coded Factors) Model n 6184.30 9 687.14 40.93 0.0004 ** G 15.77 1990.80 1 1990.80 118.59 0.0001 ** H 2.28 41.68 1 41.68 2.48 0.1759 Lack of Fit \ 79.89 3 26.63 13.16 0.0714 I 20.68 3422.13 1 3422.13 203.86 <0.0001 ** GH 0.5475 1.20 1 1.20 0.0714 0.7999 Pure Error 4.05 2 2.02 \ \ \ GI 10.24 419.23 1 419.23 24.97 0.0041 ** HI 1.34 7.16 1 7.16 0.4263 0.5426 Cor Total \ 6268.24 14 \ \ \ G 0.0354 0.0046 1 0.0046 0.0003 0.9874 2.19 17.63 1 17.63 1.05 0.3524 Note: ** means highly significant (p < 0.01), * means significant (0.01 ≤ p < 0.05). I 8.57 271.15 1 271.15 16.15 0.0101 * Intercept 26.93 n n n n n Residual n 83.93 5 16.79 According to Table 8, the regression model’s p-value is less than 0.01, which indicates Lack of Fit n 79.89 3 26.63 13.16 0.0714 that the model is very significant. The lack of fit F-value is more than 0.5, which implies Pure Error n 4.05 2 2.02 n n Cor Total n 6268.24 14 n n n that the model is not significant relative to the pure error. It indicates that the model can Note: ** means highly significant (p < 0.01), * means significant (0.01 p < 0.05). correctly reflect the relationship among y2 and G, H, and I as well as predict the test results. According to Table 8, the regression model’s p-value is less than 0.01, which indicates Among them, G, I, and GI are highly significant model terms and I is a significant model that the model is very significant. The lack of fit F-value is more than 0.5, which implies that the model is not significant relative to the pure error. It indicates that the model can term. After excluding the insignificant factors, the quadratic regression model (actual fac- correctly reflect the relationship among y and G, H, and I as well as predict the test results. tor) of the maximum damage force y2 is: Among them, G, I, and GI are highly significant model terms and I is a significant model term. After excluding the insignificant factors, the quadratic regression model (actual −9 −9 y2 = 57.64 − 3.32 × 10 G − 88.40I + 4.55 × 10 GI + 34.91I² (13) factor) of the maximum damage force y is: 9 9 2 y = 57.64 3.32 10 G 88.40I + 4.55 10 GI + 34.91I (13) The effect of the interaction factors on the maximum damage force is shown in Figure 12. When H is at the center level (5.50 × 10 ), y2 increases with the increase of G and I. The The effect of the interaction factors on the maximum damage force is shown in response sur Figure 12. When fHace is atcthe urve center chlevel anges faster along (5.50 10 ), y increases the with I dire thection increase , and of the effect of the bonded G and I. The response surface curve changes faster along the I direction, and the effect of disk radius on the maximum damage force is more significant than that of normal stiffness the bonded disk radius on the maximum damage force is more significant than that of per unit area. normal stiffness per unit area. Figure 12. The effect of interaction factors on maximum damage force. y = f (G, 5.50 10 , I). 2 8 Figure 12. The effect of interaction factors on maximum damage force. y2 = f (G, 5.50 × 10 , I). Using the physical test measurement of the maximum damage force, 16.20 N, as the optimization target, the combination of the bonding parameters’ optimization model is: y =16.2 N 9 10 1.0×× 10 ≤≤ G 1.0 10 (14) s.t. 1.0 ×10≤≤ H 1.0 ×10 1.0mm≤≤ I 2.0 mm Based on Equations (13) and (14), the solution for the optimal combination of param- 9 −3 8 −3 eters was solved as G = 4.4 × 10 N m , H = 7.73 × 10 N m , and I = 1.26 mm. 3.2.2. Test Verification The developed rape-stem flexibility discrete element model and its parameter combi- nations were used for the three-point bending verification test. The comparison of the dis- placement-force curve of the simulation with the physical test is shown in Figure 13. The load–displacement curve obtained from the simulation is consistent with the trend of the physical test curves. The load states of the rape stems at each stage were also consistent between the simulation and physical tests. The simulation result of the rape stems’ maxi- mum damage force was 16.76 N, and the relative error of the measured values from the Appl. Sci. 2022, 12, 8394 16 of 18 Using the physical test measurement of the maximum damage force, 16.20 N, as the optimization target, the combination of the bonding parameters’ optimization model is: y = 16.2 N 9 10 1.0 10 G 1.0 10 (14) 8 9 > s.t. 1.0 10 H 1.0 10 1.0 mm I 2.0 mm Based on Equations (13) and (14), the solution for the optimal combination of parame- 9 3 8 3 ters was solved as G = 4.4 10 N m , H = 7.73 10 N m , and I = 1.26 mm. 3.2.2. Test Verification The developed rape-stem flexibility discrete element model and its parameter com- binations were used for the three-point bending verification test. The comparison of the displacement-force curve of the simulation with the physical test is shown in Figure 13. The load–displacement curve obtained from the simulation is consistent with the trend of the Appl. Sci. 2022, 12, x FOR PEER REVIEW 17 of 19 physical test curves. The load states of the rape stems at each stage were also consistent be- tween the simulation and physical tests. The simulation result of the rape stems’ maximum damage force was 16.76 N, and the relative error of the measured values from the physical p tests hysic was al tes 3.5%. ts was The 3.5%. Th verification e veriftest icatishows on test sh that ow the s th calibrated at the calibr flexible ated flexible discrete el- discrete element model’s bonding parameters are accurate for rape stems and can be used for simulations of ement model’s bonding parameters are accurate for rape stems and can be used for simu- deformation and fracturing under the action of an external force. lations of deformation and fracturing under the action of an external force. Figure 13. Displacement–force variation curve of rape stems under three-point bending. Figure 13. Displacement–force variation curve of rape stems under three-point bending. 3.2.3. Process of Rape Stem Bending and Rupture 3.2.3. Process of Rape Stem Bending and Rupture According to Figure 13, the rape stem-bending process can be divided into three stages, According to Figure 13, the rape stem-bending process can be divided into three namely, compression, bending, and rupture. In the stage of compression, the central axis of stages, namely, compression, bending, and rupture. In the stage of compression, the cen- the stem remains stable under the support of the mechanical tissue of the rape stem, and tral axis of the stem remains stable under the support of the mechanical tissue of the rape only a small local compression deformation occurs between the tool and the supporter. stem, and only a small local compression deformation occurs between the tool and the There were some differences in the elastic moduli of the rape stem samples, so the actual supporter. There were some differences in the elastic moduli of the rape stem samples, so bending tests had different compression deformation rates, but the bending load always the actual bending tests had different compression deformation rates, but the bending load increased rapidly with the increase in the tool displacement. The compression deformation always increased rapidly with the increase in the tool displacement. The compression de- rate of the simulation is relatively small but still within the realistic range. In the stage formation rate of the simulation is relatively small but still within the realistic range. In the of bending, the stem is further deformed by compression and the central axis of the stem stage of bending, the stem is further deformed by compression and the central axis of the is bent, resulting in plastic deformation. The loads extracted from both the physical tests stem is bent, resulting in plastic deformation. The loads extracted from both the physical and simulations showed a fluctuating lifting trend under the axial tension of the stem. In tests and simulations showed a fluctuating lifting trend under the axial tension of the stem. the stage of rupture, the load exceeds the bending strength of the rape stem and fractures In the stage of rupture, the load exceeds the bending strength of the rape stem and fractures occur in the radial direction of the stem away from the tool. The stem’s cross-sectional morphology is destroyed, and the load after the stem’s rupture falls rapidly until the stem is completely ruptured. 4. Conclusions In this research, a flexible DEM model for rape stems was developed. The proposed model consists of sequentially arranged rigid units that were bonded together by the Hertz–Mindlin with Bonding contact model in the EDEM. The flexible DEM model could demonstrate the deformation and fracture of the rape stems under external force. The contact parameter ranges of the rape stems were determined by a bench test and calibrated based on the repose angle. The significant factors for the repose angle were static friction, the rape stem–steel coefficient of rolling friction’s contact plasticity ratio, the rape stem–rape stem coefficient of restitution, and the rape stem–steel coefficient of restitution, and the optimal combination of the parameters was 0.568, 0.029, 0.514, and 0.463, respectively. The repose angle’s simulation result relative error to the physical tests was 2.2%. The rape stems’ flexible DEM model’s bonding parameters were calibrated based on the three-point bending physical test and the Box–Behnken simulation test. The optimal Appl. Sci. 2022, 12, 8394 17 of 18 occur in the radial direction of the stem away from the tool. The stem’s cross-sectional morphology is destroyed, and the load after the stem’s rupture falls rapidly until the stem is completely ruptured. 4. Conclusions In this research, a flexible DEM model for rape stems was developed. The proposed model consists of sequentially arranged rigid units that were bonded together by the Hertz–Mindlin with Bonding contact model in the EDEM. The flexible DEM model could demonstrate the deformation and fracture of the rape stems under external force. The contact parameter ranges of the rape stems were determined by a bench test and calibrated based on the repose angle. The significant factors for the repose angle were static friction, the rape stem–steel coefficient of rolling friction’s contact plasticity ratio, the rape stem–rape stem coefficient of restitution, and the rape stem–steel coefficient of restitution, and the optimal combination of the parameters was 0.568, 0.029, 0.514, and 0.463, respectively. The repose angle’s simulation result relative error to the physical tests was 2.2%. The rape stems’ flexible DEM model’s bonding parameters were calibrated based on the three-point bending physical test and the Box–Behnken simulation test. The 9 3 optimal combination of bonding parameters was obtained as G = 4.40 10 N/m , 8 3 H = 7.73 10 N m , and I = 1.26 mm. The test results show that the rape stems’ maximum damage force obtained from the constructed model was 16.76 N, and the relative error of the measured values from the physical tests was 3.5%. The load–displacement curve obtained from the simulation is consistent with the trend of the physical test curves. The load states of the rape stems at each stage were also consistent between the simulation and physical tests. The developed rape-stem flexible DEM model can be used for the simulation of harvesting processes such as cutting, conveying, and threshing. Author Contributions: Conceptualization, Z.G. and S.M.; methodology, Z.G.; software, T.J.; valida- tion, H.L.; formal analysis, Z.G.; investigation, Z.G.; resources, Z.G.; data curation, Z.G.; writing— original draft preparation, Z.G.; writing—review and editing, S.M.; visualization, C.W.; supervision, M.Z. and C.W. project administration, S.M.; funding acquisition, Z.G. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Key Research and Development Program of China (2021YFD200050201), Basic Scientific Research Professional Expenses of Chinese Academy of Agricul- tural Sciences (S202102-01, S202203), and China Agriculture Research System of MOF and MARA (CARS-12). Data Availability Statement: The data presented in this study are available on request from the authors. Acknowledgments: The authors thank the editor and anonymous reviewers for providing helpful suggestions for improving the quality of this manuscript. 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