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離散要素法を用いた粉粒体間付着力の重力依存性解析
Gravitational Dependency Analysis on Adhesive Forceof Granular Material using a Discrete Element Method
大村拓也, 濵﨑塁, 石上玄也,
Takuya OMURA, Rui HAMASAKI, and Genya ISHIGAMI
慶應義塾大学，Keio University

1.

Introduction

Solar rocky bodies, including Mars, the Moon, and asteroids, are covered with a fine granular material
called regolith. This powdery surface layer significantly impacts the risks and challenges of space missions
that utilize exploratory rovers and spacecraft. For example, NASA’s Spirit Rover could not continue further
exploration because its wheels were trapped in Martian regolith1). During the Apollo 12 lunar landing,scattered regolith exhaust from the landing engine damaged the Lunar Surveyor 32). The maximum velocityof the scattered regolith was estimated as being as much as 2000 m/s3). Therefore, understanding thecharacteristics of regolith, particularly its gravitational dependency, is not only an area of scientific curiosity
but also has practical significance. Clarifying these characteristics provides direct input into the design ofrovers and spacecrafts and enhances their reliability and effectiveness, which are essential attributes for
successful space missions.
Previous studies have utilized parabolic flights to analyze the static characteristics of the granularmaterial4,5). These showed that the angle of repose was constant, regardless of gravity. However, inherent
challenges of parabolic flights, such as their short duration and instability in experiments, are significant
drawbacks. Ozaki et al.6) conducted experiments at the International Space Station (ISS) to examine thecharacteristics of granular materials under stable artificial gravity conditions for long periods. Although thesestudies have yielded innovative insights, they have yet to be fully utilized in the numerical analysis of rovers
or spacecrafts. This is because they still need to clarify the gravitational dependency of the characteristic
parameters to represent granular materials, which is required for numerical analysis.
In response, our study utilized the ISS experimental results as the foundation for conducting simulations
using the discrete element method (DEM). We analyzed the gravitational dependency characteristics ofalumina beads by focusing on adhesive forces such as van der Waals and electrostatic forces between particles.
First, we determined the basic DEM parameters based on the soil test results conducted on Earth. Next, by
adjusting the adhesion parameters in our simulation to align with the ISS experimental results, we identified
the gravitational dependency of the adhesive force between the alumina beads. The purpose of this study is
to clarify the gravitational dependency of the DEM parameters to represent granular materials in numericalsimulations, in order to use them in numerical analysis of rovers or spacecrafts. The rest of this paper isorganized as follows: Section 2 describes the calibration method and results for the DEM parameters based on
the soil test results. Section 3 presents an overview of the ISS experiments and compares the experimental andsimulation results, focusing on the behavior of alumina beads under various artificial gravities in an hourglass
mission. In Section 4, the gravitational dependency of adhesive force is analyzed and discussed. Section 5
concludes the paper by encapsulating the key findings and suggesting future research directions.

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2.

DEM models and calibration

2.1. Physical models
The DEM is a numerical simulation method in which a granular material is modeled as an aggregate ofdiscrete particles. Solving for the motion of individual particles at each instant enables the analysis of theoverall behavior of granular material and the interactions between particles or between particles and target
objects (e.g., rover wheels). In this study, we used the commercial software Ansys Rocky7). For the force models
governing the interactions between particles, we adopted the Hertzian spring-dashpot model and the MindlinDeresiewicz model for the normal and tangential directions, respectively. These models accurately simulate
the dynamic responses of granular materials because they are both nonlinear spring-dash systems. In addition
to these fundamental force models, we introduced a rolling resistance model to express the actual shape of the
granular material and its resistance to rotation. This inclusion added a realistic factor to the simulation byaccounting for the intrinsic characteristics of real-world particles.
Particles are subject to attractive forces, such as van der Waals and electrostatic forces. These can becollectively modeled in the DEM as adhesive forces 𝐹𝐹. In this study, the constant model was used, defined by
the following equation:
𝑓𝑓 𝑔𝑔 min(𝑚𝑚1 , 𝑚𝑚2 ) ,𝐹𝐹 = � adh
0,𝑠𝑠 ≤ 𝛿𝛿adh,𝑠𝑠 > 𝛿𝛿adh(1)where 𝑓𝑓adh represents the adhesion coefficient relative to the weight, and 𝑔𝑔 is the gravitational acceleration.
𝑚𝑚1 and 𝑚𝑚2 are the masses of the particles in contact. 𝑠𝑠 is the contact overlap and 𝛿𝛿adh shows the distancethreshold at which the adhesive force acts between the particles. This modeling approach enables us tounderstand the impact of adhesive forces in a gravitational context, which is crucial in simulating the behaviorof regoliths, especially in low-gravity environments such as the Moon or Mars.
2.2. Calibration of DEM parameters
To conduct a valid simulation using DEM, quantitative evaluations and validations must be performedby comparing them with experimental results. In this study, we calibrated the DEM parameterscomprehensively in three steps: preconditioning, DEM parameter tuning, and validation.
2.2.1. Preconditioning
Based on particle density test data acquired by Ozaki et al.6), the particle density in our simulations wastreated as a fixed DEM parameter of 3.827 g/cm3. Three representative diameters were statistically chosen fromthe test results for the particle size distribution. Specifically, we approximated the measured mass-frequency
ratio shown in Figure 1(a) using a Gaussian function. We then chose the mean particle size of the Gaussian
fitting and the sizes corresponding to the mean ± one standard deviation as the representative diameters.
Figure 1 shows the mass frequencies and cumulative mass ratios for the experimentally obtained andstatistically selected representative particle diameters. The three representative diameters chosen in this study
were 222, 258, and 294 μm, with mass ratios of 27.4%, 45.2%, and 27.4%, respectively. The particle size wasscaled up by a factor of six during the tuning and validation of the DEM parameters for computational
efficiency. In addition, the distance threshold 𝛿𝛿adh for the adhesive force model was fixed at 10 nm.
2.2.2. DEM parameter tuning

A direct shear test was conducted to tune the DEM parameters. The direct shear test enabled theacquisition of two characteristics: the peak strength and residual strength. The peak strength depends on theinitial packing density of the granular material before the test begins8). However, Afzali-Nejad et al. reported
that the residual strength was constant regardless of the initial packing9). Therefore, we tuned the DEM
parameters based on the three correlation diagrams shown in Figure 2. Figure 2(a) and (b) correspond to thepeak strength. The black squares represent the experimental results, and the gray shaded areas represent the
95% confidence intervals estimated from the experimental results. Figure 2(c) illustrates the residual strength,
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and the gray-shaded area represents the 95% confidence ellipse calculated from the experimental results. The
goal of this tuning was to ensure that the simulation results fell within the gray-shaded areas. A trial-and-error
approach was employed to tune the DEM parameters. The circles in Figure 2 represent the simulation results.
For both the loose- and dense-packing states, DEM parameters that fit within the 95% confidence intervals andellipses in all three correlation diagrams could be found.
2.2.3. Validation
A minimum density test was conducted to evaluate the calibrated parameters. The mean minimum bulk
density obtained in the previous study6) was 2.116 g/cm3. However, the simulation results obtained using thecalibrated parameters yielded 2.152 g/cm3. Consequently, the error rate was less than 2%, indicating that valid
results were obtained.
The DEM parameters obtained through the above process are summarized in Table 1.

(a)

(b)

Figure 1. Representative particle diameters of alumina beads. (a) shows mass-frequency ratio, and (b) shows
cumulative mass ratio.

(a)

(b)

(c)

Figure 2. Experimental and simulated results in the direct shear test. (a) and (b) show the peak strength. (c)shows the residual strength.
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Table 1. Calibrated discrete element method parameters for alumina beads.

Density [g/cm3]
Diameter [μm]3.827222, 258, 294

Mass ratio [%]27.4, 45.2, 27.4

Young’s modulus [MPa]20

Poisson’s ratio [-]0.3

Rolling resistance [-]0.13
Particle-particle

Particle-solid boundary

Static friction coefficient [-]

0.34

0.33

Dynamic friction coefficient [-]

0.34

0.33

Restitution coefficient [-]

0.38

0.3

Adhesion coefficient [-]Distance threshold [nm]

0.015
10

0
0

(a)

Capturing start(b)

Drop end

Flipping

Falling particles

Figure 3. Hourglass experiments overview. Artificial gravity is generated by controlling the centrifuge’s
rotation velocity, and the particle flow is captured by a camera, as shown in (a). (b) shows a snapshot as
particles fall.

3.

Hourglass experiments and simulation results

3.1. Hourglass experiments overview
The hourglass mission was conducted on the ISS. During the mission, a centrifuge installed in theJapanese Experimental Module of the ISS was used to generate stable artificial gravity. This innovative setup
ensures high-quality, long-term, and stable artificial gravity conditions. By controlling the centrifuge's angularvelocity 𝜔𝜔 as shown in Figure 3(a), the mission produced arbitrary levels of artificial gravity. In this specificcampaign, gravity levels were achieved from 0.063 G to 2.0 G. Moreover, an onboard camera capable ofcapturing at 25 frames per second was installed on the front of the hourglass. Figure 3(b) shows an example
snapshot captured using this camera. Therefore, the alumina beads can be observed and analyzed as they fallwithin the hourglass, which is maintained in a vacuum under various artificial gravity conditions. This study
used experimental results in six artificial gravities of 0.063 G, 0.17 G, 0.38 G, 0.50 G, 0.75 G, and 1.0 G tocompare with simulation results.
3.2. Comparison of results
We consider the findings derived from the hourglass mission, focusing primarily on the angle of repose(AoR) and mass flow rate. These results are compared with the simulation results obtained using the DEM
parameters calibrated in Section 2.2.
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3.2.1. AoR results
The AoR was estimated from the heights of the alumina beads at the front and back of the hourglass, as
shown in Figure 4(a). However, in the simulation, these angles were calculated through a linear approximationof the slope as viewed from the side, as shown in Figure 4(b). Figure 5 shows the results of the AoR measured
in the experiments and simulations. Figure 5 confirms that AoR is an almost constant value regardless ofgravity, which is the same as in previous research4,5). Moreover, even without adjusting the adhesive force, theexperimental and simulation results were almost identical. Therefore, we suggest that granular materials
behave similarly, regardless of gravity, with respect to static characteristics such as AoR.
3.2.2. Mass flow rate results
The mass flow rate 𝑣𝑣m is calculated as:
𝑣𝑣m =

𝑚𝑚
,
𝑡𝑡e − 𝑡𝑡0(2)

where 𝑚𝑚 is the mass of the alumina beads packed in the hourglass (6.72 g). 𝑡𝑡0 is the time when the hourglassis perpendicular to the direction of artificial gravity during its flip. 𝑡𝑡e is the time when the alumina beads are
depleted from the upper part of the hourglass or when they stop moving.
Figure 6 shows the mass flow rate results for different artificial gravities. The analysis revealed that themass flow rate decreased with the decreasing gravity. This trend was also observed in the simulation results.
However, a noticeable discrepancy was found between the experimental results and simulation findings inquantitative terms. This discrepancy suggests that the gravitational dependence of the adhesion force must be
considered to accurately represent the dynamic behavior of granular materials.
Back height

(a)

(b)

AoR

AoR

Front height

Depth: 15 mm
Front side

Back side

Figure 4. Angle of repose (AoR) measurement methods for (a) experiments and (b) simulations.

Figure 5. Results of the angle of repose.

Figure 6. Results of mass flow rate.

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4.

Adhesive force dependency on gravity

The previous section discussed the discrepancy between the experimental and simulation results for themass flow rate. Therefore, we aligned the two by adjusting the adhesion coefficient 𝑓𝑓adh . Figure 7 confirmsthat a reduction in the adhesive force was desirable as gravity decreased. Moreover, a greater adhesive force
must be provided when considering dynamic characteristics, such as the mass flow rate, compared with when
considering static characteristics, such as the AoR. From the above results, the adhesive force has conceivablegravity and velocity dependencies. Generally, from the law of conservation of mechanical energy, the velocityof an object falling freely at a certain distance is proportional to �𝑔𝑔. Therefore, the mass flow rate 𝑣𝑣m is also
considered proportional to �𝑔𝑔, as shown in Figure 6. However, individual differences in velocity are likely
to occur as the particle velocity increases. This leads to more opportunities for the particles to come into contact,
resulting in a relative increase in the contact area. As a result, van der Waals forces increase. The likelihood of
deformation during particle collisions owing to the faster velocity might also contribute to the increase in vander Waals forces. Furthermore, the increased probability of individual velocity differences may result in agreater electrostatic force. These considerations conclude that a higher particle velocity increases the adhesive
force. In contrast, in a state with static characteristics, such as the AoR, where the relative velocity of theparticles is zero, the adhesive force is neglected, and gravity dependency is barely observed (see Figure 7).
Therefore, we focus on the gravity and velocity dependencies of the adhesive force. According to Ozaki
et al.6), the relationship between the mass flow rate 𝑣𝑣m and gravity magnitude 𝐺𝐺 = 𝑔𝑔⁄𝑔𝑔E , where 𝑔𝑔E is thegravitational acceleration on Earth, can be expressed as
𝑣𝑣m = 𝛼𝛼𝐺𝐺𝛽𝛽+0.5 ,(3)𝐹𝐹 = 𝑎𝑎𝑎𝑎 + 𝑏𝑏,(4)

where 𝛼𝛼 and 𝛽𝛽 are coefficients that depend on granular material. These coefficients were obtained from theexperimental results. In the case of alumina beads, 𝛼𝛼 and 𝛽𝛽 were 15.41 g/s and 0.0836, respectively. The
dotted line in Figure 7 represents the linear fitting results for the adhesive force adjusted in the simulations.
The high value of the coefficient of determination R2 (0.980) suggests a relationship between the adhesion force
and the degree of gravity, as follows:

where 𝑎𝑎 and 𝑏𝑏 are the coefficients that depend on the granular material. 𝑎𝑎 and 𝑏𝑏 are 7.31 μN and 0.09 μN,
respectively. 𝑏𝑏 can be neglected as it is infinitesimally small. From Equation 3 and 4, the relationship among
the adhesive force 𝐹𝐹, mass flow rate 𝑣𝑣m , and gravity magnitude 𝐺𝐺 can be expressed as
𝐹𝐹 = 𝑎𝑎 �

𝑣𝑣m 2 −2𝛽𝛽
� 𝐺𝐺
.
𝛼𝛼

(5)

Thus, adhesive force should not be considered when dealing with static characteristics when no velocity ispresent, according to the above equation. However, when dealing with the dynamic characteristics, theinfluence of gravity must be considered. Furthermore, if 𝑣𝑣m is constant, a reduction in gravity increases theadhesive force, suggesting that the dominance of the adhesive force increases when gravity is low.

Figure 7. Gravitational dependency of adhesive force 𝐹𝐹. 𝐺𝐺 is the magnitude of gravity and R2 indicates thecoefficient of determination.

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5.

Conclusion

Based on the Hourglass mission conducted on the ISS, we analyzed the magnitude of the adhesive force
acting between alumina beads using DEM. The results revealed that, when dealing with static characteristics,
the adhesive force is small and remains constant, regardless of gravity. In contrast, gravity and velocity
dependencies were observed in the dynamic characteristics of the particles. When gravity is low, the particle
velocity is low, and although the adhesive force is small, the dominance of gravity over the velocity becomessignificant. From these results, we conclude that a distinct difference in the adhesive force should be provided
whether dealing with the static or dynamic characteristics of granular materials. Specifically, in analysis ofstatic characteristics, such as in the selection of a landing site for spacecraft, there is no need to consider thegravity dependency of the adhesive force. Conversely, in the analysis of dynamic characteristics, such as in
the analysis of regolith dispersion during the takeoff and landing of spacecraft, the gravity dependency of theadhesive force must be considered, and the quantitative results obtained in this study can be applied. However,the boundary conditions of the particulate matter velocity that determine whether the adhesive force must be
considered remain unclear. In scenarios that require handling both the static and dynamic characteristics ofparticulate matter, such as the driving analysis of planetary exploration or human-pressurized rovers, carefulconsideration of the magnitude of the interparticle adhesive force is necessary. Moving forward, further
analyses will be conducted to determine the boundary velocity at which the magnitude of the adhesive force
should be considered. Additionally, we aim to verify the validity of a model that considers the gravity andvelocity dependency of the adhesive force, and to construct an adhesive force model that encompasses boththe static and dynamic characteristics of granular materials.
Acknowledgment
We express our special thanks to Prof. Ozaki of Yokohama National University and Prof. Kobayashi of
Ritsumeikan University for providing detailed soil test data. This study was supported by JSPS KAKENHI(grant number: JP23KJ1901).
References
1)
2)
3)

4)

5)
6)

7)
8)
9)

R.A. Kerr: Mars Rover Trapped in Sand, But What Can End a Mission? Science, 324 (2009) 998, DOI:
10.1126/science.324_998.
C. Immer, P. Metzger, P.E. Hintze, A. Nick and R. Horan: Apollo 12 Lunar Module exhaust plume impingement on
Lunar Surveyor III. Icarus, 211 (2011) 1089, DOI: 10.1016/j.icarus.2010.11.013.
D.E. Brownlee, W. Bucher and P. Hodge: Part A, Primary and Secondary Micrometeoroid Impact Rate on the Lunar
Surface: A Direct Measurement. In Analysis of Surveyor 3 material and photographs returned by Apollo 12, National
Aeronautics and Space Administration (1972).
H. Nakashima, Y. Shioji, T. Kobayashi, S. Aoki, H. Shimizu, J. Miyasaka and K. Ohdoi: Determining the angle ofrepose of sand under low-gravity conditions using discrete element method. J. Terramech, 48 (2011) 17, DOI:
10.1016/j.jterra.2010.09.002.
J.P. Marshall, R.C. Hurley, D. Arthur, I. Vlahinic, C. Senatore, K. Iagnemma, B. Trease and J.E. Andrade: Failures insand in reduced gravity environments. J. Mech. Phys. Solids, 113 (2018) 1, DOI: 10.1016/j.jmps.2018.01.005.
S. Ozaki, G. Ishigami, M. Otsuki, H. Miyamoto, K. Wada, Y. Watanabe, T. Nishino, H. Kojima, K. Soda, Y. Nakano,
M. Sutoh, T. Maeda and T. Kobayashi: Granular flow experiment using artificial gravity generator at International
Space Station. npj Microgravity, 9 (2023) 61, DOI: 10.1038/s41526-023-00308-w.
Ansys Inc.: DEM Technical Manual, Ansys Rocky 2023 R2 (2023).
J. Katagiri, T. Matsushima and Y. Yamada: Variations in shear behavior among specimens with different packing
patterns. Granular Matter, 16 (2014) 891, DOI: 10.1007/s10035-014-0530-4.
A. Afzali-Nejad, A. Lashkari and P.T. Shourijeh: Influence of particle shape on the shear strength and dilation ofsand-woven geotextile interfaces. Geotext. Geomembr., 45 (2017) 54, DOI: 10.1016/j.geotexmem.2016.07.005.
© 2023 by the authors. Submitted for possible open access publication under theterms and conditions of the Creative Commons Attribution (CC BY) license(http://creativecommons.org/li censes/by/4.0/).
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