# Evaluating Ice Load during Submarine Surfacing and Ice Breaking

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## Abstract

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## 1. Introduction

## 2. Theoretical Calculation Model of Ice-Breaking Resistance

#### 2.1. Calculation Model of Ice-Breaking Resistance of Command Tower

#### 2.2. Calculation Model of Ice Resistance of Submarine Hull

## 3. Numerical Simulation

#### 3.1. Establishment of Numerical Model

#### 3.2. Numerical Simulation Results

#### 3.3. Comparison of Theoretical Model and Numerical Simulation Results

## 4. Results and Discussion

#### 4.1. The Influence of the Upper Area of the Control Tower

#### 4.2. The Influence of Initial Crack Length on Submarine Ice-Breaking Resistance

#### 4.3. The Influence of Ice Thickness on Submarine Ice-Breaking Resistance

#### 4.4. The Influence of Ice Bending Strength

#### 4.5. The Influence of Ice Elastic Modulus

#### 4.6. The Influence of Ice Friction Coefficient

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Zhang, J.; Wu, Q.; Zhao, X. Analysis of Polar Class Ship Development. Ship Eng.
**2016**, 11, 1–5. [Google Scholar] - Liu, R.W.; Xue, Y.Z.; Lu, X.K.; Cheng, W.X. Simulation of ship navigation in ice rubble based on peridynamics. Ocean. Eng.
**2018**, 148, 286–298. [Google Scholar] [CrossRef] - Suyuthi, A.; Leira, B.J.; Riska, K. A generalized probabilistic model of ice load peaks on ship hulls in broken-ice fields. Cold Reg. Sci. Technol.
**2014**, 97, 7–20. [Google Scholar] [CrossRef] - Lee, J.M.; Lee, C.J.; Kim, Y.S.; Choi, G.G.; Lew, J.M. Determination of global ice loads on the ship using the measured full-scale motion data. Int. J. Nav. Archit. Ocean.
**2016**, 8, 301–311. [Google Scholar] [CrossRef][Green Version] - Di, S.; Ji, S.; Xue, Y. Analysis of ship navigation in level ice-covered regions with discrete element method. Ocean. Eng.
**2017**, 35, 59–69. [Google Scholar] - Lubbad, R.; Loset, S. A numerical model for real-time simulation of ship–ice interaction. Cold Reg. Sci. Technol.
**2011**, 65, 111–127. [Google Scholar] [CrossRef][Green Version] - Huang, Y.; Guan, P.; Mu, Y.U. Study of the Sailing’s Moving Responses of An Icebreaker in Ice. Math. Pract. Theory
**2015**, 45, 149–160. [Google Scholar] - Liang, Y.; Ji, H.; Zhao, Q.; Wu, H.; Liu, Z. Technical progress of Russian submarine ice navigation tests. Mar. Equip./Mater. Mark.
**2021**, 29, 1–6. [Google Scholar] - Lewis, J.; Edwards, R. Methods for predicting icebreaking and ice resistance characteristics of icebreakers. SNAME Trans.
**1970**, 78, 213–249. [Google Scholar] - Kotras, T.; Baird, A.; Naegle, J. Predicting ship performance in level ice. Trans. Soc. Nav. Archit. Mar. Eng. SNAME
**1983**, 91, 329–349. [Google Scholar] - Lindqvist, G. A straightforward method for calculation of ice resistance of ships. In Proceedings of the International Conference on Port and Ocean Engineering under Arctic Conditions (POAC), Lulea, Sweden, 12–16 June 1989; pp. 722–735. [Google Scholar]
- Keinonen, A.; Robbins, I. Icebreaker Characteristics Synthesis; Report for the Transportation Development Centre; Report (TP12812E); Transport Canada: Ottawa, ON, Canada, 1996. [Google Scholar]
- Riska, K.; Wilhelmson, M.; Englund, K.; Leiviska, T. Performance of merchant vessels in the Baltic; Report 52; Helsinki University of Technology: Helsinki, Finland, 1997. [Google Scholar]
- Spencer, D.; Jones, S.J. Model-Scale/Full-Scale Correlation in Open Water and Ice for Canadian Coast Guard “R-Class” Icebreakers. J. Ship Res.
**2001**, 45, 249–261. [Google Scholar] [CrossRef] - Valanto, P. The resistance of ships in level ice. SNAME Trans.
**2001**, 109, 53–83. [Google Scholar] - Jeong, S.; Lee, C.; Cho, S. Ice Resistance Prediction for Standard Icebreaker Model Ship. In Proceedings of the Twentieth International Offshore and Polar Engineering Conference, Beijing, China, 20–25 June 2010; pp. 1300–1304. [Google Scholar]
- Li, L.; Gao, Q.; Bekker, A.; Dai, H. Formulation of Ice Resistance in Level Ice Using Double-Plates Superposition. J. Mar. Sci. Eng.
**2020**, 8, 870. [Google Scholar] [CrossRef] - Kozin, V.M.; Chizhumov, S.D.; Zemlyak, V.L. Influence of ice conditions on the effectiveness of the resonant method of breaking ice cover by submarines. J. Appl. Mech. Technol. Phys.
**2010**, 51, 398–404. [Google Scholar] [CrossRef] - Pogorelova, A.; Zemlyak, V.; Kozin, V. Moving of a submarine under an ice cover in fluid of finite depth. J. Hydrodyn.
**2019**, 31, 562–569. [Google Scholar] [CrossRef] - Sturova, I. The motion of a submerged sphere in a liquid under an ice sheet. J. Appl. Math. Mech.
**2012**, 76, 293–301. [Google Scholar] [CrossRef] - Zemlyak, V.; Pogorelova, A.; Kozin, V. Influence of peculiarities of the form of a submarine vessel on the efficiency of breaking ice cover. In Proceedings of the International Offshore and Polar Engineering Conference, Anchorage, AL, USA, 30 June–4 July 2013; pp. 1252–1258. [Google Scholar]
- Zemlyak, V.L.; Kozin, V.M.; Baurin, N.O.; Ipatov, K.I.; Kandelya, M.V. The study of the impact of ice conditions on the possibility of the submarine vessels surfacing in the ice cover. J. Phys. Conf. Ser.
**2017**, 919, 012004. [Google Scholar] [CrossRef][Green Version] - Ye, L.Y.; Wang, C.; Guo, C.Y. Peridynamic model for submarine surfacing through ice. Chin. J. Ship Res.
**2018**, 13, 51–59. [Google Scholar] - Wang, C.; Wang, J.; Wang, C.; Guo, C.; Zhu, G. Research on vertical movement of cylindrical structure out of water and breaking through ice layer based on S-ALE method. Chin. J. Theor. Appl. Mech.
**2021**, 53, 3110–3123. [Google Scholar] - Junzheng, Y.; Xianqian, W.; Chenguang, H. Multi-field coupling effect and similarity law of floating ice break by vehicle launched underwater. Chin. J. Theor. Appl. Mech.
**2021**, 53, 1930–1939. [Google Scholar] - Dempsey, J.P.; Defranco, S.J.; Adamson, R.M.; Mulmule, S.V. Scale effects on the in situ tensile strength and fracture of ice. Part I: Large grained freshwater ice at Spray Lakes Reservoir, Alberta. In Fracture Scaling; Springer: Dordrecht, The Netherlands, 1999; pp. 325–345. [Google Scholar] [CrossRef]
- Palmer, A.; Dempsey, J. Models of large-scale crushing and spalling related to high-pressure zones. In Proceedings of the Iahr International Symposium on Ice, Montreal, QC, Canada, 19–23 June 2002. [Google Scholar]
- Dempsey, J.P. Research trends in ice mechanics. Int. J. Solids Struct.
**2000**, 37, 131–153. [Google Scholar] [CrossRef] - Määttänen, M.; Hoikkanen, J. The effect of ice pile-up on the ice force of a conical structure. In Proceedings of the Iahr International Symposium on Ice, Espoo, Finland, 20–23 August 1990; pp. 131–153. [Google Scholar]
- von Bock und Polach, R.U.F.; Ettema, R.; Gralher, S.; Kellner, L.; Stender, M. The non-linear behavior of aqueous model ice in downward flexure. Cold Reg. Sci. Technol.
**2019**, 165, 102775. [Google Scholar] [CrossRef][Green Version] - Xu, B.; Guyenne, P. Nonlinear simulation of wave group attenuation due to scattering in broken floe fields. Ocean. Model.
**2023**, 181, 102139. [Google Scholar] [CrossRef] - Lu, W.; Lubbad, R.; Løset, S. In-plane fracture of an ice floe: A theoretical study on the splitting failure mode. Cold Reg. Sci. Technol.
**2015**, 110, 77–101. [Google Scholar] [CrossRef] - Jeong, S.Y.; Choi, K.; Kang, K.J.; Ha, J.S. Prediction of ship resistance in level ice based on empirical approach. Int. J. Nav. Archit. Ocean.
**2017**, 9, 613–623. [Google Scholar] [CrossRef] - Li, L.; Shkhinek, K. Dynamic interaction between ice and inclined structure. Mag. Civ. Eng.
**2014**, 45, 71–79. [Google Scholar] [CrossRef] - Xing, H.; Liu, Z.; Li, H. Calculation method of ice pressure of extruded ice plate in reservoirs based on the fracture mechanics. Adv. Sci. Technol. Water Resour.
**2013**, 33, 10–13. [Google Scholar] [CrossRef] - Tameroğlu, S. General Solution of the Biharmonic Equation and Generalized Levy’s Method for Plates. J. Struct. Mech.
**1986**, 14, 33–51. [Google Scholar] [CrossRef] - Dobrodeev, A. Ice resistance of ships in brash ice channel: Calculation method. Trans. Krylov State Res. Cent.
**2019**, 3, 11–21. [Google Scholar] [CrossRef] - Institute, C.A.R. Handbook of Stress Strength Factors; Science Publishers: New York, NY, USA, 1993. [Google Scholar]
- Ji, S.; Liu, H.; Xu, N.; Ma, H. Experiments on sea ice fracture toughness in the Bohai Sea. Adv. Water Sci.
**2013**, 024, 386–391. [Google Scholar] - Aksnes, V. A panel method for modelling level ice actions on moored ships. Part 1: Local ice force formulation. Cold Reg. Sci. Technol.
**2011**, 65, 128–136. [Google Scholar] [CrossRef] - Sawamura, J. 2D numerical modeling of icebreaker advancing in ice-covered water. Int. J. Nav. Archit. Ocean.
**2018**, 10, 385–392. [Google Scholar] [CrossRef] - Tan, X.; Su, B.; Riska, K.; Moan, T. A six-degrees-of-freedom numerical model for level ice-ship interaction. Cold Reg. Sci. Technol.
**2013**, 92, 1–16. [Google Scholar] [CrossRef] - Zhou, L.; Riska, K.; Ji, C. Simulating transverse icebreaking process considering both crushing and bending failures. Mar. Struct.
**2017**, 54, 167–187. [Google Scholar] [CrossRef] - Groves, N.C.; Huang, T.T.; Chang, M. Geometric Characteristics of DARPA SUBOFF Models; Report; David Taylor Research Center: Bremerton, WA, USA, 1989. [Google Scholar]
- Kjerstad, O.K.; Metrikin, I.; Løset, S.; Skjetne, R. Experimental and phenomenological investigation of dynamic positioning in managed ice. Cold Reg. Sci. Technol.
**2015**, 111, 67–79. [Google Scholar] [CrossRef] - Huang, L.; Tuhkuri, J.; Igrec, B.; Li, M.; Stagonas, D.; Toffoli, A.; Cardiff, P.; Thomas, G. Ship resistance when operating in floating ice floes: A combined CFD&DEM approach. Mar. Struct.
**2020**, 74, 102817. [Google Scholar] [CrossRef] - Han, D.; Paik, K.J.; Jeong, S.Y.; Choung, J. Prediction of the ice resistance of icebreakers using explicit finite element analyses with a real-time load control technique. Ocean. Eng.
**2021**, 240, 109825. [Google Scholar] [CrossRef] - Jeon, S.; Kim, Y. Numerical simulation of level ice-structure interaction using damage-based erosion model. Ocean. Eng.
**2020**, 220, 108485. [Google Scholar] [CrossRef] - Truong, D.D.; Jang, B.S. Estimation of ice loads on offshore structures using simulations of level ice-structure collisions with an influence coefficient method. Appl. Ocean. Res.
**2022**, 125, 103235. [Google Scholar] [CrossRef]

**Figure 6.**Distribution Map of von Mises equivalent stress in the ice sheet at different moments when the submarine is out of the water and crossing the ice.

Parameters (Unit) | Symbol | Value |
---|---|---|

Density [kg/m${}^{3}$] | $\rho $ | 900 |

Modulus of elasticity [GPa] | E | 1.8 |

Shear modulus [GPa] | G | 0.72 |

Poisson’s ratio | $\nu $ | 0.25 |

Bending strength [MPa] | ${\sigma}_{b}$ | 0.36 |

Compression strength [MPa] | ${\sigma}_{b}$ | 1.08 |

Shear strength [MPa] | $\tau $ | 0.54 |

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**MDPI and ACS Style**

Li, L.; Meng, X.; Bekker, A.; Makarov, O.; Wang, W.; Zhang, T. Evaluating Ice Load during Submarine Surfacing and Ice Breaking. *J. Mar. Sci. Eng.* **2023**, *11*, 736.
https://doi.org/10.3390/jmse11040736

**AMA Style**

Li L, Meng X, Bekker A, Makarov O, Wang W, Zhang T. Evaluating Ice Load during Submarine Surfacing and Ice Breaking. *Journal of Marine Science and Engineering*. 2023; 11(4):736.
https://doi.org/10.3390/jmse11040736

**Chicago/Turabian Style**

Li, Liang, Xiangbin Meng, Alexander Bekker, Oleg Makarov, Wei Wang, and Tao Zhang. 2023. "Evaluating Ice Load during Submarine Surfacing and Ice Breaking" *Journal of Marine Science and Engineering* 11, no. 4: 736.
https://doi.org/10.3390/jmse11040736