# Microscale Modeling of Frozen Particle Fluid Systems with a Bonded-Particle Model Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- Flexibility in agglomerate generation, in which all particles and bonds can have their unique material or geometrical properties;
- Capability in mimicking the breakage behavior of agglomerate, such as the crack initiation, propagation, failure plane, etc.;
- Diversity in functional model usage, with numerous choices of rheological models in the particle-particle, particle-wall relationship, and solid bond models.

#### 1.1. Ice Rheology

^{−8}s

^{−1}to 10

^{−3}s

^{−1}, the compressive strength increases with an increase in strain rate, surpassing 10

^{−3}s

^{−1}compressive strength decreases with an increase in strain rate [29].

#### 1.2. Rheology of Frozen Soil

## 2. Materials and Methods

#### 2.1. Uniaxial Compression Test

#### 2.2. Specimen Preparation

#### 2.3. Investigated Parameter Space

^{−3}s

^{−1}was applied for a low strain rate, and 10

^{−2}s

^{−1}was used for a high strain rate. The strain rate is controlled by the compression speed, specified according to specimen height. The compression speed of 0.02 mm/s, which is two times larger than the minimal compression speed of the Texture Analyzer, was chosen as the low strain rate for all specimens. For high strain rate, compression speed is calculated by $specimenheight\times 0.01$. All experiments have been performed at a temperature of around −10 °C. The possibility of unfrozen water inside the agglomerate can be minimized with such a temperature.

#### 2.4. Ice Creep Behavior

#### 2.5. Fracture Patterns of Frozen PFS

^{−2}s

^{−1}). The fully saturated PFS samples can better transmit the pressure through the specimen. Thus, the cracks propagate from top to bottom, breaking the specimen into relatively large fragments containing numerous primary particles. In this case, the complete failure of the specimen occurred vigorously. In contrast, no large fragments were formed for PFS samples with 75% saturation. Only tiny pieces with several primary particles detached from the main structure during loading.

#### 2.6. Mechanical Behavior of Frozen PFS

#### 2.7. Bonded-Particle Model Approach

#### 2.8. Solid Bond Model Considering Creep Behavior

## 3. Result and Discussion

#### 3.1. Experimental Result

#### 3.2. Simulation Setup

#### 3.3. Comparison of Simulation and Experimental Results

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Arenson, L.U.; Springman, S.M.; Sego, D.C. The Rheology of Frozen Soils. Appl. Rheol.
**2007**, 17, 12147–1. [Google Scholar] [CrossRef] - Goodman, D.J.; Frost, H.J.; Ashby, M.F. The plasticity of polycrystalline ice. Philos. Mag. A
**1981**, 43, 665–695. [Google Scholar] [CrossRef] - Gold, L.W. The process of failure of columnar-grained ice. Philos. Mag.
**1972**, 26, 311–328. [Google Scholar] [CrossRef][Green Version] - Jellinek, H.H.G.; Brill, R. Viscoelastic Properties of Ice. J. Appl. Phys.
**1956**, 27, 1198–1209. [Google Scholar] [CrossRef] - Glen, J.W. The creep of polycrystalline ice. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1955**, 228, 519–538. [Google Scholar] [CrossRef] - Mellor, M.; Testa, R. Effect of Temperature on the Creep of Ice. J. Glaciol.
**1969**, 8, 131–145. [Google Scholar] [CrossRef][Green Version] - Hassan, M.F.; Lee, H.P.; Lim, S.P. The variation of ice adhesion strength with substrate surface roughness. Meas. Sci. Technol.
**2010**, 21, 075701. [Google Scholar] [CrossRef] - Nath, S.; Ahmadi, S.F.; Boreyko, J.B. How ice bridges the gap. Soft Matter
**2019**, 16, 1156–1161. [Google Scholar] [CrossRef] - Kellner, L.; Stender, M.; Polach, R.U.F.V.B.U.; Herrnring, H.; Ehlers, S.; Hoffmann, N.; Høyland, K.V. Establishing a common database of ice experiments and using machine learning to understand and predict ice behavior. Cold Reg. Sci. Technol.
**2019**, 162, 56–73. [Google Scholar] [CrossRef][Green Version] - Pernas-Sánchez, J.; Pedroche, D.; Varas, D.; López-Puente, J.; Zaera, R. Numerical modeling of ice behavior under high velocity impacts. Int. J. Solids Struct.
**2012**, 49, 1919–1927. [Google Scholar] [CrossRef] - Wang, C.; Hu, X.; Tian, T.; Guo, C.; Wang, C. Numerical simulation of ice loads on a ship in broken ice fields using an elastic ice model. Int. J. Nav. Arch. Ocean Eng.
**2020**, 12, 414–427. [Google Scholar] [CrossRef] - Long, X.; Liu, S.; Ji, S. Breaking characteristics of ice cover and dynamic ice load on upward–downward conical structure based on DEM simulations. Comput. Part. Mech.
**2020**, 8, 297–313. [Google Scholar] [CrossRef] - Yershov, E.D. General Geocryology; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar] [CrossRef]
- Harris, J.S. Ground Freezing in Practice; Thomas Telford Limited: London, UK, 1995. [Google Scholar]
- Wang, Y.; Chen, B.; Nie, C. Numerical Simulation of Nonlinear Fracture Failure Process of Frozen Soil. In Proceedings of the 2009 International Joint Conference on Computational Sciences and Optimization, Sanya, China, 24–26 April 2009; Volume 1, pp. 183–186. [Google Scholar] [CrossRef]
- Wang, Z.; Ma, L.; Wu, L.; Yu, H. Numerical simulation of crack growth in brittle matrix of particle reinforced composites using the xfem technique. Acta Mech. Solida Sin.
**2012**, 25, 9–21. [Google Scholar] [CrossRef] - Nishimura, S.; Gens, A.; Olivella, S.; Jardine, R.J. THM-coupled finite element analysis of frozen soil: Formulation and application. Géotechnique
**2009**, 59, 159–171. [Google Scholar] [CrossRef][Green Version] - Cuccurullo, A.; Gallipoli, D. DEM Simulation of Frozen Granular Soils with High Ice Content. National Conference of the Researchers of Geotechnical Engineering; Springer: Cham, Switzerland, 2020. [Google Scholar]
- An, L.; Ling, X.; Geng, Y.; Li, Q.; Zhang, F. DEM Investigation of Particle-Scale Mechanical Properties of Frozen Soil Based on the Nonlinear Microcontact Model Incorporating Rolling Resistance. Math. Probl. Eng.
**2018**, 2018, 2685709. [Google Scholar] [CrossRef][Green Version] - Cundall, P.A.; Strack, O.D.L. A discrete numerical model for granular assemblies. Géotechnique
**1979**, 29, 47–65. [Google Scholar] [CrossRef] - Dosta, M.; Dale, S.; Antonyuk, S.; Wassgren, C.; Heinrich, S.; Litster, J.D. Numerical and experimental analysis of influence of granule microstructure on its compression breakage. Powder Technol.
**2016**, 299, 87–97. [Google Scholar] [CrossRef] - Rybczyński, S.; Dosta, M.; Schaan, G.; Ritter, M.; Schmidt-Döhl, F. Numerical study on the mechanical behavior of ultrahigh performance concrete using a three-phase discrete element model. Struct. Concr.
**2020**, 23, 548–563. [Google Scholar] [CrossRef] - Beckmann, B.; Schicktanz, D.-I.K.; Reischl, D.-M.D.; Curbach, D.-I.E.M. DEM simulation of concrete fracture and crack evolution. Struct. Concr.
**2012**, 13, 213–220. [Google Scholar] [CrossRef] - Obermayr, M.; Dressler, K.; Vrettos, C.; Eberhard, P. A bonded-particle model for cemented sand. Comput. Geotech.
**2012**, 49, 299–313. [Google Scholar] [CrossRef] - Ouyang, Y.; Yang, Q.; Chen, X. Bonded-Particle Model with Nonlinear Elastic Tensile Stiffness for Rock-Like Materials. Appl. Sci.
**2017**, 7, 686. [Google Scholar] [CrossRef][Green Version] - Dosta, M.; Jarolin, K.; Gurikov, P. Modelling of Mechanical Behavior of Biopolymer Alginate Aerogels Using the Bonded-Particle Model. Molecules
**2019**, 24, 2543. [Google Scholar] [CrossRef][Green Version] - Dosta, M.; Skorych, V. MUSEN: An open-source framework for GPU-accelerated DEM simulations. SoftwareX
**2020**, 12, 100618. [Google Scholar] [CrossRef] - Gold, L.W. On the Elasticity of Ice Plates. Can. J. Civ. Eng.
**1988**, 15, 1080–1084. [Google Scholar] [CrossRef] - Petrovic, J.J. Review Mechanical properties of ice and snow. J. Mater. Sci.
**2003**, 38, 1–6. [Google Scholar] [CrossRef] - Haynes, F.D. Effect of Temperature on the Strength of Snow-Ice; U.S. Army Cold Regions Research and Engineering Laboratory: Hanover, NH, USA, 1978; Volume 78. [Google Scholar]
- Schulson, E.M. The structure and mechanical behavior of ice. JOM
**1999**, 51, 21–27. [Google Scholar] [CrossRef] - Weertman, J. CREEP DEFORMATION OF ICE. Annu. Rev. Earth Planet. Sci.
**1983**, 11, 215–240. [Google Scholar] [CrossRef][Green Version] - Naumenko, K.; Altenbach, H. Modeling of Creep for Structural Analysis; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2007. [Google Scholar] [CrossRef]
- Gold, L.W. Process of failure in ice. Can. Geotech. J.
**1970**, 7, 405–413. [Google Scholar] [CrossRef] - Arenson, L.U.; Johansen, M.M.; Springman, S.M. Effects of volumetric ice content and strain rate on shear strength under triaxial conditions for frozen soil samples. Permafr. Periglac. Process
**2004**, 15, 261–271. [Google Scholar] [CrossRef] - Arenson, L.U.; Almasi, N.; Springman, S.M. Shearing response of ice-rich rock glacier material. In Proceedings of the Eighth International Conference on Permafrost, Zurich, Switzerland, 21–25 July 2003; pp. 39–44. [Google Scholar]
- Taylor, D.W. Fundamentals of Soil Mechanics; LWW: Philadelphia, PA, USA, 1948. [Google Scholar]
- Arenson, L.U.; Springman, S.M. Triaxial constant stress and constant strain rate tests on ice-rich permafrost samples. Can. Geotech. J.
**2005**, 42, 412–430. [Google Scholar] [CrossRef] - Hooke, R.L.; Dahlin, B.B.; Kauper, M.T. Creep of Ice Containing Dispersed Fine Sand. J. Glaciol.
**1972**, 11, 327–336. [Google Scholar] [CrossRef][Green Version] - Ting, J.M.; Martin, R.T.; Ladd, C.C. Mechanisms of Strength for Frozen Sand. J. Geotech. Eng.
**1983**, 109, 1286–1302. [Google Scholar] [CrossRef] - Arenson, L.U.; Sego, D.C. The effect of salinity on the freezing of coarse-grained sands. Can. Geotech. J.
**2006**, 43, 325–337. [Google Scholar] [CrossRef] - Zhao, S.P.; Zhu, Y.L.; He, P. Recent progress in research on the dynamic response of frozen soil. In Proceedings of the Eighth International Conference on Permafrost, Zurich, Switzerland, 21–25 July 2003; pp. 1301–1306. [Google Scholar]
- Anderson, D.M.; Tice, A.R. Predicting Unfrozen Water Contents in Frozen Soils From Surface Area Measurements. Highw. Res. Rec.
**1972**, 393, 12–18. [Google Scholar] - Istomin, V.; Chuvilin, E.; Bukhanov, B. Fast estimation of unfrozen water content in frozen soils. Earth’s Cryosphere
**2017**, 21, 116–120. [Google Scholar] [CrossRef] - Mellor, M.; Smith, J.S. Creep of Snow and Ice, Cold Regions Research and Engineering Laboratory, Vicksburg, U.S. Research Report. 1966. Available online: https://hdl.handle.net/11681/5879 (accessed on 20 November 2022).
- Lian, G.; Thornton, C.; Adams, M.J. A Theoretical Study of the Liquid Bridge Forces between Two Rigid Spherical Bodies. J. Colloid Interface Sci.
**1993**, 161, 138–147. [Google Scholar] [CrossRef] - Willett, C.D.; Johnson, S.A.; Adams, M.J.; Seville, J.P. Chapter 28 Pendular capillary bridges. Handb. Powder Technol.
**2007**, 11, 1317–1351. [Google Scholar] [CrossRef] - Nguyen, H.N.G.; Zhao, C.-F.; Millet, O.; Selvadurai, A. Effects of surface roughness on liquid bridge capillarity and droplet wetting. Powder Technol.
**2020**, 378, 487–496. [Google Scholar] [CrossRef] - Mindlin, R.D.; Deresiewicz, H. Elastic Spheres in Contact Under Varying Oblique Forces. J. Appl. Mech.
**1953**, 20, 327–344. [Google Scholar] [CrossRef] - Tsuji, Y.; Tanaka, T.; Ishida, T. Lagrangian numerical simulation of plug flow of cohesionless particles in a horizontal pipe. Powder Technol.
**1992**, 71, 239–250. [Google Scholar] [CrossRef] - Dosta, M.; Antonyuk, S.; Heinrich, S. Multiscale Simulation of Agglomerate Breakage in Fluidized Beds. Ind. Eng. Chem. Res.
**2013**, 52, 11275–11281. [Google Scholar] [CrossRef] - Iwamoto, T.; Murakami, E.; Sawa, T. A finite element simulation on creep behavior in welded joint of chrome-molybdenum steel including interaction between void evolution and dislocation dynamics. Technol. Mech. J. Eng. Mech.
**2010**, 30, 157–168. [Google Scholar] - Norton, F.H. The Creep of Steel at High Temperatures; McGraw-Hill B. Company, Incorporated: Columbus, OH, USA, 1929. [Google Scholar]
- Penny, R.K.; Marriott, D.L. Design for Creep; McGraw-Hill: New York, NY, USA; Columbus, OH, USA, 1971. [Google Scholar]
- Dosta, M.; Bistreck, K.; Skorych, V.; Schneider, G.A. Mesh-free micromechanical modeling of inverse opal structures. Int. J. Mech. Sci.
**2021**, 204, 106577. [Google Scholar] [CrossRef] - Descantes, Y.; Tricoire, F.; Richard, P. Classical contact detection algorithms for 3D DEM simulations: Drawbacks and solutions. Comput. Geotech.
**2019**, 114, 103134. [Google Scholar] [CrossRef][Green Version] - Leroy, B. Collision between two balls accompanied by deformation: A qualitative approach to Hertz’s theory. Am. J. Phys.
**1985**, 53, 346–349. [Google Scholar] [CrossRef] - El Shamy, U.; Zamani, N. Discrete element method simulations of the seismic response of shallow foundations including soil-foundation-structure interaction. Int. J. Numer. Anal. Methods Géoméch.
**2011**, 36, 1303–1329. [Google Scholar] [CrossRef] - Wang, X.; Yang, J.; Xiong, W.; Wang, T. Evaluation of DEM and FEM/DEM in Modeling the Fracture Process of Glass Under Hard-Body Impact. Int. Conf. Discret. Elem. Methods
**2017**, 188, 377–388. [Google Scholar] [CrossRef] - Nitta, K.-H.; Yamana, M. Poisson’s Ratio and Mechanical Nonlinearity Under Tensile Deformation in Crystalline Polymers; Intec: Rijeka, Croatia, 2012; pp. 113–132. [Google Scholar] [CrossRef][Green Version]
- Tan, Y.; Yang, D.; Sheng, Y. Study of polycrystalline Al2O3 machining cracks using discrete element method. Int. J. Mach. Tools Manuf.
**2008**, 48, 975–982. [Google Scholar] [CrossRef] - Lupo, M.; Sofia, D.; Barletta, D.; Poletto, M. Calibration of DEM simulation of cohesive particles. Chem. Eng. Trans.
**2019**, 74, 379–384. [Google Scholar] [CrossRef] - Cao, X.; Li, Z.; Li, H.; Wang, X.; Ma, X. Measurement and Calibration of the Parameters for Discrete Element Method Modeling of Rapeseed. Processes
**2021**, 9, 605. [Google Scholar] [CrossRef] - Daraio, D.; Villoria, J.; Ingram, A.; Alexiadis, A.; Stitt, E.H.; Munnoch, A.L.; Marigo, M. Using Discrete Element method (DEM) simulations to reveal the differences in the γ-Al2O3 to α-Al2O3 mechanically induced phase transformation between a planetary ball mill and an attritor mill. Miner. Eng.
**2020**, 155, 106374. [Google Scholar] [CrossRef] - Gu, X.; Zhang, J.; Huang, X. DEM analysis of monotonic and cyclic behaviors of sand based on critical state soil mechanics framework. Comput. Geotech.
**2020**, 128, 103787. [Google Scholar] [CrossRef]

**Figure 1.**Texture Analyzer system equipped with a climate chamber: (

**Left**) Climate chamber coupled with Texture Analyzer; (

**Right**) CAD design.

**Figure 2.**Two exemplary samples of frozen PFS: (

**Left**) aggregate with alpha-alumina primary particles at 75% saturation level; (

**Right**): glass bead PFS with 100% saturation.

**Figure 3.**Polycrystalline ice specimen subjected to constant force load for 240 s under three different forces.

**Figure 4.**Representative stress-strain curve of polycrystalline ice during creep experiment (primary loading phase).

**Figure 5.**The fracture pattern for agglomerates at the strain rate is 10

^{−2}s

^{−1}: (

**a**) 100% saturation, alpha-alumina PFS (Supplementary material: Video S1); (

**b**) 75% saturation, glass PFS (Supplementary material: Video S2).

**Figure 6.**Representative stress-strain characteristics for different frozen PFS under a high strain (HS) rate of 10

^{−2}s

^{−1}and low strain (LS) rate of 10

^{−3}s

^{−1}.

**Figure 8.**Average Young’s modulus and breakage stress of different PFS: (

**a**) Young’s modulus; (

**b**) Breakage stress.

**Figure 9.**Representative agglomerate with diameter 10 mm, height 16 mm (

**upper**part: agglomerates in complete form;

**lower**part: agglomerates’ internal structure).

**Figure 10.**Experimental and simulation results for 100% saturation level for high strain (HS) rates (10

^{−2}s

^{−1}) and low strain (LS) rates (10

^{−3}s

^{−1}).

**Figure 11.**Comparison of experimental and simulation results for 100% saturation at −10 °C: (

**a**) Young’s modulus; (

**b**) Breakage stress.

**Figure 12.**Comparison of experimental and simulation results for 75% saturation at −10 °C: (

**a**) Young’s modulus; (

**b**) Breakage stress.

Stiffness | Shape | Surface Roughness | Particle Size (mm) | ||||
---|---|---|---|---|---|---|---|

Soft | Hard | Spherical | Non-Spherical | Ra | Rz | ||

Polyethene | X | X | 12.808 | 50.723 | 1.8 | ||

Glass bead | X | X | 1.767 | 11.462 | 1.65 | ||

Alpha-alumina | X | X | 49.262 | 187.453 | 1.72 | ||

Quartz sand | X | X | 13.416 | 49.623 | 0.5 |

**Table 2.**Mechanical behavior overview concerning saturation levels, temperatures, strain rates, and particle properties.

Saturation Level | Strain Rate | Smooth Particles (Polymer and Glass) | Rough Particles (Sand and Alpha-Alumina) |
---|---|---|---|

100% | Low | Mostly brittle with failure just after the yield point | Dilatant with slight strain softening or hardening |

High | Brittle failure | Brittle behavior with failure just after yield or brittle failure | |

75% | Low | Brittle failure | Dilatant with vast strain softening |

High | Brittle failure | Brittle failure |

**Table 3.**Main agglomerates properties used for simulation of different types of PFS with 100% saturation level.

Parameter | Polyethene/Glass/ Alpha-Alumina (Spherical) | Sand (Non-Spherical) |
---|---|---|

Particle diameter (mm) | 1.8/1.7/1.65 | 0.5 |

Bond diameter (mm) | 1.0 | 0.3 |

Particle density (kg/m^{3}) | 960/2500/3960 | 2640 |

Particle Young’s modulus (GPa) | 0.8/72.3/150 | 72 |

Particle Poisson’s ratio (-) | 0.36/0.22/0.22 | 0.2 |

$\mathrm{Maximal}\text{}\mathrm{bond}\text{}\mathrm{generation}\text{}\mathrm{distance}\text{}{L}_{gen}^{max}$ (mm) | 0.7 | 0.2 |

Numbers of particles (-) | ≈230 | ≈11,200 |

No. of bonds (-) | ≈1100 | ≈66,000 |

Porosity (-) | 0.44 | 0.42 |

Particle-wall sliding friction (-) | 0.45/0.45/0.45 | 0.45 |

Particle-wall rolling friction (-) | 0.05/0.05/0.05 | 0.5 |

Particle-particle sliding friction (-) | 0.45/0.4/0.45 | 0.45 |

Particle-particle rolling friction (-) | 0.05/0.05/0.05 | 0.5 |

Restitution coefficient (-) | 0.1 | 0.1 |

**Table 4.**Material properties for ice bonds used for modeling different agglomerates at different loading rates.

Primary Particles | ||||
---|---|---|---|---|

Polyethene | Glass | Alpha-Alumina | Natural Sand | |

Normal and shear strengths | ||||

- -
- High strain rate (MPa)
| 3.5 | 4.2 | 20 | 20 |

- -
- Low strain rate (MPa)
| 6 | 2.7 | 20 | 20 |

Creep parameter A (-) | 0.1 | 0.1 | 0.3 | 0.1 |

Creep factor m (-) | 0.1 | 0.1 | 0.16 | 0.1 |

Parameter | Polyethylene/Glass/Alpha-Alumina (Spherical) | Sand (Non-Spherical) |
---|---|---|

Bond diameter (mm) | 0.75 | 0.22 |

$\mathrm{Maximal}\text{}\mathrm{bond}\text{}\mathrm{generation}\text{}\mathrm{distance}\text{}{L}_{gen}^{max}$ (mm) | 0.01 | 0.01 |

Number of particles | ≈230 | ≈11,200 |

Number of bonds | ≈550 | ≈34,000 |

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**MDPI and ACS Style**

Chan, T.T.; Heinrich, S.; Grabe, J.; Dosta, M. Microscale Modeling of Frozen Particle Fluid Systems with a Bonded-Particle Model Method. *Materials* **2022**, *15*, 8505.
https://doi.org/10.3390/ma15238505

**AMA Style**

Chan TT, Heinrich S, Grabe J, Dosta M. Microscale Modeling of Frozen Particle Fluid Systems with a Bonded-Particle Model Method. *Materials*. 2022; 15(23):8505.
https://doi.org/10.3390/ma15238505

**Chicago/Turabian Style**

Chan, Tsz Tung, Stefan Heinrich, Jürgen Grabe, and Maksym Dosta. 2022. "Microscale Modeling of Frozen Particle Fluid Systems with a Bonded-Particle Model Method" *Materials* 15, no. 23: 8505.
https://doi.org/10.3390/ma15238505