# Review of Discrete Element Method Simulations of Soil Tillage and Furrow Opening

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## Abstract

**:**

## 1. Introduction

^{2}N

_{γ}+ cdN

_{c}+ c

_{a}dN

_{a}+ qdN

_{q})w

^{−3}), d is working depth (m), c is soil cohesion (Pa), c

_{a}is soil–metal adhesion (Pa), q is surcharge stress (Pa), w is tool width (m), and N

_{γ}, N

_{c}, N

_{a}, and N

_{q}are dimensionless N factors that are dependent on gravity, cohesion, adhesion, and surcharge, respectively.

## 2. Modelling Agricultural Soils with DEM

#### 2.1. DEM Contact Models

^{s}) and damping (F

^{d}) forces. Some commonly used contact models in DEM simulations are listed in Table 1 and reviewed below. These contact models have been implemented in commercially available software such as Bulk Flow Analyst™, Chute Analyst™, Chute Maven

^{®}, DEMpack™, Altair

^{®}EDEM™, ELFEN, GROMOS-96, ITASCA PFC (2D & 3D), LiGGGHTS

^{®}, MIMES, PASSAGE/DEM, Rocky, SimPARTIX

^{®}, StarCCM+, UDEC, 3DEC, and YADE [25,26]. Table 1 also shows the various DEM software that have been used in tillage and furrow opening research and the types of soil the contact models have been used to model.

_{i}and radii r

_{i}, located at x

_{i}in contact, taking F as contact force, g as acceleration due to gravity, I

_{i}as the moment of inertia of a particle, ω

_{i}as the angular velocity of a particle, and T

_{i}as torque due to the tangential component of the contact force:

#### 2.1.1. Linear Spring Contact Model

_{n}is the normal stiffness, while d

_{n}is the damping coefficient. Considering an imaginary rod of radius $r=\left({r}_{1}+{r}_{2}\right)/2$ between the centres of the two particles and Young’s Modulus E:

_{t}, ${\delta}_{t}$, d

_{t}, and ${\dot{\delta}}_{t}$ are tangential components of stiffness, overlap, damping coefficient, and relative velocity.

#### 2.1.2. Hertz–Mindlin Contact Model

_{n}:

_{t}:

#### 2.1.3. Hysteretic Spring Contract Model

_{n}:

_{1}and k

_{2}are the loading and the unloading stiffnesses, respectively, and e is the coefficient of restitution of the particles, and they are related as $e=\sqrt{{k}_{1}/{k}_{2}}$.

_{t}:

_{k}is the stiffness factor equal to the tangential stiffness ratio to normal loading stiffness.

#### 2.1.4. Accounting for Cohesion with DEM Contact Models

#### Linear Cohesion Model

_{ca}(Equation (21)) is the cohesive or adhesive force, then the normal contact force is modified, as shown in Equation (22).

_{c}is the contact radius between particles and can be determined using Equation (23). Equation (21) is called the constant cohesion model because the cohesive stress $\widehat{c}$ is a constant. The constant cohesion model makes the model particles too sticky [52]. A modification has therefore been proposed, depending on the degree of compression between two adjacent particles. If compression between two adjacent particles is given by Equation (24), then the cohesive stress increases with time t according to Equation (25).

#### Parallel Bond Model

#### Johnson–Kendall–Roberts Cohesion

^{2}).

**Table 2.**Relative errors in DEM-predicted soil–tool reaction forces, travel speeds, and operating depths reported in the various literature across soil types.

Reference | Relative Error in DEM Prediction (%) Relative to Measured Data | Travel Speed (km h^{−1}) | Operating Depth (mm) | Tillage Tools | Soil Texture | Dry Bulk Density (kg m^{−3}) | Soil Water Content (%, w/w) | Cohesive Strength (kPa) | Contact Model | |
---|---|---|---|---|---|---|---|---|---|---|

Draught | Vertical Force | |||||||||

Sadek et al. [58] | n/a | n/a | n/a | n/a | n/a | Sandy soil | 990 1280 1360 1500 | 0.02 13 21.5 | 1.23-32.70 | PBM |

Chen et al. [7] | 4 to 31 | n/a | 3.19 (average) | 100 | Sweep tine | Coarse sand Loamy sand Sandy loam | 1410 1330 1410 | 8.98 14.84 18.2 | 15.7 25.2 36 | PBM |

Obermayr et al. [52] | n/a | n/a | 2.16–4.5 | 10–200 | Bulldozer blade | n/a | 1900 | n/a | 11.16 | LSCM + cohesion |

Tamas et al. [30] | 4 to 12 | n/a | 1.8–8.64 | 200 | Sweep tine | Sandy soil | 1850 | 6.33 | 11.86 | PBM |

Bravo et al. [18] | 9, 24 | n/a | - | 150–500 | Para-plough and mouldboard plough | Clay (Vertosol) | 1000 1200 1400 | 8 18 20 35 | 25–125 | LSCM + cohesion |

Li et al. [56] | 3 to 15 | n/a | 3.6 | 180–260 | Subsoiler | n/a | n/a | 19 | n/a | PBM |

Mak and Chen [61] | n/a | n/a | 2.2–6.59 | 50–200 | Sweep tine | Loamy sand | 1320 | 11.3 | 13.9 | PBM |

Obermayr et al. [72] | n/a | n/a | 100–200 | Straight-vertical blade and bulldozer blade | Sand | 1520 1980 1870 | 10 15 | 6–22.5 | LSCM + cohesion | |

Ucgul et al. [38] | ≤11.6 | ≤15.2 | 5–12.5 | 70 | Sweep tine | Sandy loam | 1750 | 8 | 6 | HSCM |

Ucgul et al. [53] | n/a | n/a | 4–12 | 75 | Sweep tine | Sandy loam | 1320 1780 1880 | 1 15 13 | 3 15 22 | HSCM + LCM |

Kotrocz et al. [60] | n/a | n/a | n/a | 50–150 | Cone penetrometer | Loamy sand | 1632 | 15.8 | 6.61–8.66 | PBM |

Li et al. [70] | 2.99 | 3–18 | Claw | Sandy loam | 1300 | n/a | 17.5 | PBM | ||

Murray [69] | 1.86 | 50.7 | 8 | 38 | Disc and hoe openers | Clayey lacustrine | 1560 | 19.6 | n/a | PBM |

Hang et al. [32] | n/a | n/a | 3 | 300 | Subsoiler | Loamy clay | 1346 | 12.5 | 11.8 | PBM |

Milkevych et al. [62] | n/a | n/a | 3.2 | 100 | Sweep tine | Coarse sand Loamy sand | 1410 1330 | 9 14.8 | 15.8 25.1 | PBM |

Tekeste et al. [55] | 9, 12 | -59, -49 | 0.79–9.65 | 102 | Sweep tine | Loam | 1307 | 8.99 | 33 | PBM |

Tong et al. [73] | <10 | <10 | 7.2 | 300–450 | Subsoiler (straight shank-sweep tine, curved shank-chisel tine, curved shank-sweep tine, bentleg-chisel tine) | n/a | 1230–1420 | n/a | n/a | not stated |

Kim et al. [44] | 5.16 to 9.9 | n/a | 7.64–7.9 | 5–200 | Mouldboard plough | Loam | 1496–1904 | 24.5–34.02 | n/a | EEPA |

Aikins et al. [41] | 5 to 31 | 8, 20 and greater | 8 | 100 | Bentleg and narrow point openers | Clay (Vertosol) | 1504 | 23.7 | 46.4 | HSCM + LCM |

Wang et al. [74] | 15.08 | n/a | 3 | 300 | Winged subsoiler | Sandy loam | 1404–1833 | n/a | n/a | PBM |

Sadek et al. [75] | ≤20.2 | n/a | 4–16 | 127 | Disc | Sandy loam | 1700 | 16.32 | n/a | PBM |

Saunders et al. [76] | n/a | n/a | 4.5–10 | 25–100 | Plough skimmers | Sandy loam | 1523.8 | 8.3 | n/a | HSCM + LCM |

Ma et al. [77] | 2.88 to 5.97 | n/a | 1.08–2.16 | 120 | Scraper | Sandy loam | 1389 | 10 | n/a | not stated |

Hoseinian et al. [35] | 2 | 2.5 | 0.9 | 150 | Dual sideway-share | Sandy clay loam | 1565 | 11.5 | 15.4 | PBM |

#### Edinburgh Elasto-Plastic Adhesion Model

_{n}-δ) curve” for the EEPA contact model. A full description of the EEPA contact model can be found in Morrissey et al. [78]. The EEPA contact model has been used recently for modelling the interaction between tillage tools and agricultural soils [44,45,46].

#### 2.2. Particle Size and Shape

## 3. Calibration Techniques for Determining DEM Input Parameters

#### 3.1. Angle of Repose Test

#### 3.2. Inclined Plane Test

_{s}), while the ball bearing is used in the determination of the soil–tool coefficient of rolling friction (µ

_{r}). If the mass of the block is m

_{s}, the mass of the ball is m

_{r}, and the angles at which sliding and rolling occur are Ψ

_{s}and Ψ

_{r}, respectively, then the coefficients are calculated according to Equations (28) and (29) [29,97].

#### 3.3. Direct Shear Test

_{a}) is applied while an increasing horizontal (shearing) force (F

_{b}) is applied to the movable half till a certain amount of displacement occurs [98]. At that point, the horizontal force would have reached a maximum value and remain constant or slightly increase or decrease afterward [30]. The experiment is repeated several times with different normal force values.

_{b}= cA + F

_{a}tan ϕ

#### 3.4. Triaxial Compression Test

_{1}) and radial (σ

_{3}) stresses as indicated in Figure 12 until soil failure is achieved [98]. The radial stress (σ

_{3}) is first applied around the specimen to a set level via the confining water pressure. An axial strain is then mechanically applied at a controlled rate which generates a corresponding additional deviator stress (q) logged over time and combining with σ

_{3}to form a resultant axial stress σ

_{1}. The above steps are repeated several times under increasing radial stress. The plots of the deviator stress (q = σ

_{1}– σ

_{3}) against axial strain identify each deviator stress value at failure and a simple process—for instance using the Mohr circle method—is then used to quantify soil cohesion and internal friction angle [101].

#### 3.5. In Situ Approaches

## 4. Prediction of Soil Failure, Loosening, and Disturbance Parameters

#### 4.1. Soil Failure and Loosening

^{TM}), respectively, to measure the degree of particles loosening in DEM. In the work of Tamas et al. [30], for instance, it was found that the DEM modelled soil porosity and soil-break-up resulting from loosening by sweeps increased with both increasing speed and rake angle, which agrees with experimental results.

_{g}being grid volume and V

_{p}the total volume of particles whose centroids are located within the grid, voidage can be calculated according to Equation (31).

- Minimum particle displacement caused directly by an opener occurs with particles just adjacent to the bottom part of the opener (for wide tines) or particles aligning the walls of the slot below critical depth (for narrow tines).
- To establish a sharp contrast between displaced and undisturbed particles, particle locations immediately after particle loosening (i.e., before the particle settle) has to be used.

#### 4.2. Soil Movement and Disturbance Parameters

## 5. Prediction of Tillage Forces

^{−1}with a sweep tine. Bo et al. [31] observed a similar trend between draught force measured for four subsoilers in the soil bin and that obtained through DEM simulation. A winged subsoiler among the four had the highest draught force in both soil bin tests (up to 50% more) and simulations (up to 55% more). With all four subsoilers, DEM predicted draught force with relative errors below 4%.

^{−1}and 300 mm, respectively, had an RE of 9.71%, whereas the non-winged subsoiler obtained an RE of 15.08% when the draught force was compared with the experimental result. Chen et al. [7] observed about 4–31% RE between the draught of experimental and DEM results. A good correlation was obtained between the measured and predicted draught forces (r = 0.95), whereas a more limited correlation was observed for vertical force (r = 0.71). With blunt (R90B) and chamfered (C2S) narrow openers with a 90° rake angle, a blunt opener with a 45° rake angle, and a bentleg opener, Aikins et al. [41] predicted draught force with REs of 20%, 22%, 31%, and 5%, respectively. Vertical force was also predicted with 8% and 20% relative error for R90B and C2S, respectively, but poorly for the two other openers.

^{−1}and a depth of 70 mm. Prediction of the effect of rake angle on soil forces followed a similar trend (r values of 0.98 and 0.97) and had an RE of 11.6% and 15.2% for draught and vertical forces, respectively. Ucgul et al. [111] again obtained an accurate prediction of draught and vertical forces of a sweep tillage tool at varying speeds and geometry with r values ranging from 0.84 to 0.92. Murray [69] estimated an average RE of 1.86% for draught and 50.7% for vertical force with a flat single disc opener. For rotary tillers, Zhang et al. [109] reported a 12% RE in power consumption, while Du et al. [36] predicted increasing torque with tillage depth (150 to 180 mm) and travel speed (about 2 to 3 km h

^{−1}).

## 6. Soils Modelled in DEM Simulations

^{−3}) and relatively dry (soil water content of about 18%). Aikins et al. [41] also modelled a Vertosol with cohesive strength of 46.4 kPa.

## 7. Conclusions

- Even though the Hertz–Mindlin contact model (HMCM) has been used in most DEM studies of tillage and furrow opening, it consistently fails to predict vertical soil force accurately. The Hysteretic Spring contact model (HSCM) can more accurately predict soil forces and particle movement.
- Angle of repose, inclined plane, direct shear, triaxial compression, and some in situ tests (grouser shear, plate sinkage, and cone penetration tests) have been used to measure and calibrate DEM input parameters. The angle of repose test has been used mainly for cohesionless soils due to the poor flowability of cohesive soils. However, using results from reproducible phases of the angle of repose experiment, successful calibrations for cohesive soils have been achieved.
- Unlike other numerical models, DEM is able to closely predict not only soil forces, but it is also capable of modelling soil failure mechanisms, soil loosening, and soil particle movement. Soil rupture and crack propagation, critical depth, three-dimensional particle movement within the soil profile and lateral particle movement on top of the soil have all been predicted in DEM.
- Using voidage or porosity grids to determine loosened furrow cross-sectional profiles has been found to be superior to using particle velocity and displacement profiles. However, some researchers have successfully used a particle displacement approach to determine accurate furrow profiles with a more objective criteria for defining loosened furrow boundary.
- Close predictions of draught and vertical forces (≤20%) have been obtained with DEM. These predictions can be improved by using smaller particles of a near-real shape. However, this must be balanced with computation time requirements.

- The Edinburgh elasto-plastic adhesion model (EEPA) has been successfully used to model consolidated or cohesive powders. This contact model is recommended to be studied more extensively for cohesive soils, although some researchers have used it.
- Due to pore water pressure within wet and soft soils, coupling DEM and CFD is likely to produce more accurate simulations. This idea can be explored in future research.
- A comprehensive analysis of soil disturbance parameters has been successfully done using voidage grids in EDEM
^{®}DEM software. Replication of this approach in other DEM software is recommended. - The criteria introduced by Aikins et al. [41] for defining particle displacement threshold for DEM furrow profile identification need further investigation with particles of smaller radii than the 5 mm used in the study. This approach can provide greater details on the three-dimensional soil translocation process.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

γ | Surface energy (J/m^{2}) |

${\mathit{\delta}}_{n}$ | Linear overlap |

${\dot{\mathit{\delta}}}_{\mathit{n}}$ | Normal component of relative velocity |

${\dot{\mathit{\delta}}}_{\mathit{t}}$ | Tangential component of relative velocity |

$\widehat{\mathit{c}}$ | Cohesive stress |

${\dot{\mathit{\delta}}}_{\mathit{n}}$ | Linear relative velocity |

${\mathit{\delta}}_{\mathit{t}}$ | Tangential component of overlap |

${\mathit{\mu}}_{\mathit{t}}$ | Friction coefficient |

$\mathit{\varnothing}$ | Internal friction angle between the particles (Degree) |

µ_{r} | Coefficient of rolling friction |

µ_{s} | Coefficient of static friction |

A | Cross-sectional area of the shear box |

a | JKR contact radius |

c | Soil cohesion (Pa) |

c_{a} | Soil–metal adhesion (Pa) |

d | Working depth (m) |

d_{n} | Damping coefficient |

d_{t} | Tangential component of damping coefficient |

e | Coefficient of restitution of the particles |

E | Young’s modulus |

E_{eq} | Equivalent Young’s modulus |

F | Contact force |

F_{a} | Normal force in direct shear test |

F_{b} | Horizontal (shearing) force in direct shear test |

F_{ca} | Cohesive or adhesive force |

F^{d} | Damping force |

F_{n} | Normal contact force |

F^{s} | Spring force |

F_{t} | Tangential component of the contact force |

g | Acceleration due to gravity |

G_{eq} | Equivalent shear modulus |

I_{i} | Moment of inertia of a particle |

k_{1} | Loading stiffnesses |

k_{2} | Unloading stiffnesses |

k_{n} | Normal stiffness |

k_{t} | Tangential component of stiffness |

m_{eq} | Equivalent particle mass |

m_{i} | Mass of spherical particle |

m_{r} | Mass of the ball used in inclined plane test |

m_{s} | Mass of block used in inclined plane test |

N | N factor. Suffixes: γ = gravitational, c = cohesive, a = adhesive, q = surcharge |

P | Soil cutting force (N) |

q | Surcharge stress (Pa) |

q | Deviator stress in triaxial compression test |

r_{c} | Contact radius |

r_{eq} | Equivalent particle radius |

r_{i}, | Radius of spherical particle |

T_{i} | Torque due to the tangential component of the contact force |

V_{g} | Voidage grid volume |

V_{p} | Total volume of particles with centroids within voidage grid |

w | Tool width (m) |

x_{i} | Location of spherical particle |

γ | Specific weight of soil (N m^{−3}) |

ε_{a} | Axial strain in triaxial compression test |

σ_{1} | Axial stress in triaxial compression test |

σ_{3} | Radial stress in triaxial compression test |

Ψ | Inclined plane tilt angle. Subscripts: s = sliding, r = rolling |

ω_{i} | Angular velocity of a particle |

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**Figure 1.**Discrete element method simulation of a narrow tine furrow opener operating in a moist sandy-loam soil. Modified from Barr et al. [21].

**Figure 2.**A schematic diagram of the normal (

**left**) and tangential (

**right**) components of the linear spring contact model.

**Figure 3.**Hysteretic spring force –displacement relationship used in the Hertz–Mindlin contact model (HMCM). Redrawn from Ucgul et al. [29]. The dashed line contrasts the basic linear elastic model relationship.

**Figure 4.**Hysteretic spring force-displacement relationship used in the hysteretic spring contact model (HSCM). Redrawn from Ucgul et al. [29].

**Figure 6.**Edinburgh elasto-plastic adhesion model (EEPA) normal contact force-displacement relationship (

**a**) linear and (

**b**) non-linear [50].

**Figure 7.**Different associations of spherical DEM particles used to more realistically account for the effects of soil particle shape on bulk behaviour.

**Figure 9.**Repeatable dome-like soil pile obtained in the angle of repose test for a loose cohesive soil by Aikins et al. [41]: (

**a**) laboratory experiment and (

**b**) optimized DEM simulation.

**Figure 13.**Instruments for in situ measurements of soil mechanical properties: (

**a**) sampled undisturbed soil specimen, (

**b**) soil–metal static friction, (

**c**) soil–metal rolling friction, (

**d**) direct shear test, (

**e**) shear modulus, and (

**f**) Young’s modulus. Retrieved from Kim et al. [44].

**Figure 14.**(

**a**) Motorised cone penetrometer for in situ measurements, and (

**b**) DEM cone penetration simulation used by Aikins et al. [41].

**Figure 15.**Furrow profiles created using voidage grid bins in EDEM 2.7

^{TM}. Adapted from Barr [49].

**Figure 17.**Comparison of furrow profiles for 16 mm wide narrow openers at different rake angles (35°, 53°, 72°, and 90°) obtained from soil bin experiment and DEM simulations. Retrieved and modified from Barr [49].

**Table 1.**Discrete element method contact models, their advantages and disadvantages, types of soil modelled, and software used.

Contact Model | Advantages | Disadvantages | Types of Soil Modelled | References | Software Used by Researchers |
---|---|---|---|---|---|

Linear spring contact model | - Simple to use.
| - Does not account for nonlinearity in loading and unloading cycles and plastic deformation of soil.
| Sandy | Tanaka et al. [27], Asaf et al. [10], Shmulevich et al. [6], Ono et al. [28] | PFC2D, EDEM |

Linear spring contact model with cohesion | - Allows users to consider cohesion in the linear spring contact model.
| - Only considers the cohesion through the normal direction.
| Vertosol | Bravo et al. [18] | DEMeter++ |

Hertz–Mindlin contact model | - Simple to use.
- Although this model was designed for fine, dry particles, it can be used to model wet particles as well.
| - Inaccurate prediction for vertical tillage force.
| Sandy | Ucgul et al. [29] | EDEM |

Parallel bond model (PBM) or Hertz–Mindlin contact model with cohesion | - Allows users to model cohesion.
| - Excessive forces cause the bonds to be broken irrationally.
| Coarse sand, loamy, sandy loam, loessal, clay, sandy clay loam, loamy clay | Tamas et al. [30], Chen et al. [7], Bo et al. [31], Hang et al. [32], Cheng et al. [33], Yang et al. [34], Hoseinian et al. [35] | EDEM PFC3D |

Hertz–Mindlin contact model with Johnson–Kendall–Roberts (JKR) | - Enables the modelling of strongly adhesive bonds such as exist in dry powders or wet materials.
| - -
| Clay, silty clay loam | Cheng et al. [33], Du et al. [36], Zhai et al. [37] | EDEM |

Hysteric spring contact model | - Accounts for plastic deformation during loading and unloading of soil.
- Suitable for both cohesive cohesionless soils.
| - It requires a large number of input parameters, making its setup and calibration complex.
| Sandy | Ucgul et al. [38], Ucgul et al. [29] | EDEM |

Hysteric spring contact model with linear cohesion contact model | - Allow users to consider cohesion in the hysteretic spring contact model.
| - Only considers the cohesion through the normal direction.
| Sandy loam, clay (Vertosol) | Barr et al. [21], Barr et al. [39], Makange et al. [40], Aikins et al. [41], Awuah et al. [42], Wang et al. [43] | EDEM |

Edinburgh elasto-plastic adhesion model | - It is versatile since it can be used as a linear or non-linear Hertzian spring model.
- It also allows tensile forces to develop and a non-linear force-displacement behaviour in compression.
| - It requires a lot of input parameters.
| Clay, clay loam, sandy loam, loam, sandy | Kim et al. [44], Wu et al. [45], Zhao et al. [46], Sun et al. [47] | EDEM, PFC3D, LiGGGHTS |

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**MDPI and ACS Style**

Aikins, K.A.; Ucgul, M.; Barr, J.B.; Awuah, E.; Antille, D.L.; Jensen, T.A.; Desbiolles, J.M.A.
Review of Discrete Element Method Simulations of Soil Tillage and Furrow Opening. *Agriculture* **2023**, *13*, 541.
https://doi.org/10.3390/agriculture13030541

**AMA Style**

Aikins KA, Ucgul M, Barr JB, Awuah E, Antille DL, Jensen TA, Desbiolles JMA.
Review of Discrete Element Method Simulations of Soil Tillage and Furrow Opening. *Agriculture*. 2023; 13(3):541.
https://doi.org/10.3390/agriculture13030541

**Chicago/Turabian Style**

Aikins, Kojo Atta, Mustafa Ucgul, James B. Barr, Emmanuel Awuah, Diogenes L. Antille, Troy A. Jensen, and Jacky M. A. Desbiolles.
2023. "Review of Discrete Element Method Simulations of Soil Tillage and Furrow Opening" *Agriculture* 13, no. 3: 541.
https://doi.org/10.3390/agriculture13030541