An elastic–plastic ice material model for ship-iceberg collision simulations
Introduction
More voyages across the Arctic are likely to be possible in the future due to global warming, however, ships are vulnerable to collisions with icebergs in this region. As a result, ship-iceberg collisions are currently a hot topic of research. Accurate collision scenario predictions are necessary for designing ship structures to ensure that ships maintain a sufficient safety level. It is crucial to establish iceberg material models that can be used for realistic representations of iceberg impact loads in ship-iceberg collisions and structural response predictions.
One approach to predicting ship-iceberg impact loads during the ship structure design stage is to use ice class rules, such as the ‘Finnish-Swedish Ice Class Rules’ (FSICR, 2008) and International Association of Classification Societies (IACS) (IACS, 2011). These rules can be used to calculate the pressure within a ship-iceberg contact area based on parameters that are associated with the ice classification and considered ship structure. Although these references provide a convenient method for predicting collision loads, the current class rules cannot be used to predict ice loads for all collision situations. Considering such limitations, many researchers have adopted numerical simulations to examine ship-iceberg collisions. Ralston (1977) used plasticity theory to describe the ice crushing failure mode, providing a new approach to study the mechanical behavior of ice. Jebaraj et al. (1992) used the finite element method to simulate ship-ice interactions. The ice was considered to be elastic using the ‘Tsai-Wu’ failure criterion. A failure reference number was adopted to initialize the element failure. In their work, the relationship between the impact velocity and ice failure mode was discussed. They reported that the ice would crush rather than bend under high impact velocity conditions. Jordaan (2001) evaluated the physical properties involved in the interaction of an iceberg with an offshore structure; ranked data from ship rams were used to predict the ice load. von Bock und Polach and Ehlers (2013) proposed a Lemaitre damage model to simulate model-scale ice. The parameters of the material model were based on experimental data. Considering the difference between actual and model-scale ice, it is unclear whether this model is suitable for ship-ice simulations. Using the ‘Tsai-Wu’ yield function and an empirical failure criterion, Liu et al. (2011b) presented a plastic material model to simulate the ice behavior in ship-iceberg collisions. In addition, the plastic material model was successfully applied to the integrated analyses of ship side and bow collisions (Liu et al., 2011a). The failure criterion developed in this paper is based on the one proposed by Liu. Lubbad and Løset (2011) proposed a real time simulation program to simulate ship navigation in a broken ice field. The ship-ice contact area was calculated using a discrete method, and the nominal contact area was used to replace the actual contact area. Jia et al. (2009) used an isotropic elastic–plastic constitutive material model (including material hardening) to represent the ice material during ship-ice interactions. The material data were derived from the results of experiments. Gagnon (2011) proposed a crushable foam material model to simulate ice with a melted layer. The temperature in the contact surface that was measured in the experiment decreased with time, which is primarily because the melted ice ‘absorbs’ the heat created by the high-pressure conditions. In Gagnon׳s model, Poisson׳s ratio was set to zero to simulate viscous fluid flow. Although there has been substantial work on ice material modeling, no previous material models have perfectly represented all ice characteristics.
A clear understanding of iceberg mechanics in ship-iceberg collisions forms the basis for representing ice by a constitutive material model in ship-iceberg collision simulations using the finite element (FE) method. Iceberg mechanical properties strongly depend on the surrounding environmental conditions, which contribute to the complexity of an iceberg׳s material characteristics. For example, the ice in the Baltic Sea and around Russia can be very different with each other due to the high levels of fresh water pouring into the marginal seas (Timco and Weeks, 2010). Therefore, icebergs in different regions should be treated separately. The icebergs considered in this paper are located in the Arctic. Apart from environmental conditions, ice has complex components because of its formation and growth processes. Sea ice generally consists of solid ice, brine sells, gas and pores. As ice grows, the percentage and arrangements of these components change substantially, leading to different ice properties (Cole, 2001). Therefore, ice in different stages or ages should be studied separately. According to the World Meteorological Organization (WMO, 1970), the development of ice cover can be briefly separated into six main stages, i.e., new ice, nilas, pancake ice, young ice, first-year ice and old ice. Fig. 1 shows these ice types and the different stages in their development. Crystal arrangement determines whether ice properties are orthotropic or isotropic. Regarding first-year ice, crystals grow faster in the vertical direction than in the transverse direction because of the vertical heat flow. Therefore, first-year ice possesses a typical orthotropic property. Unlike first-year ice, old ice is conventionally treated as an isotropic material (Sanderson, 1988). The icebergs in the Arctic belong to the old ice category; thus, the material model should be isotropic. Other factors, such as ice thickness, also influence ice mechanics. For thin ice, bending and cracking are the dominating failure modes, crushing is the major failure mode for thick ice. Moreover, because of the high homologous temperature of ice, it is a strain-rate-dependent material. If the strain rate is low (mm/s level), creep and micro-cracking dominate the ice behavior; this ice can be treated as a viscous elastic material (Jordaan, 2001). At high strain rates, ice has a typical brittle failure mode. In reality, bergy bits collide with ships at relatively high speeds and correspondingly high strain rates (>10−3 /s), therefore, ice can be represented by a linear-elastic constitutive material model (Schulson, 2001). Considering the aforementioned points, an isotropic elastic–plastic material model is proposed to simulate the icebergs in the Arctic. In addition to the above discussion, many other aspects contribute to the mechanical properties of ice. However, proposing a general material that can capture the mechanics of ice in all aspects is a challenging task. Therefore, the main efforts in this study are to simulate the characteristics of ice loads during ship-iceberg interactions.
The pressure-area relationship is generally used to illustrate ice mechanics in ship-ice interactions. Sanderson (1988) presented a plot of the data from literature and field measurements to represent the pressure-area relationship for ship-ice interactions. Masterson et al. (2007) summarized a series of test data and proposed a new pressure-area curve for local pressure. This pressure–area relationship is recommended according to the International Organization for Standardization (ISO) standards (ISO/CD, (2010)). Other scholars, e.g., Chai and Lawn (2007), have made substantial progress in specifying the governing discipline behind Masterson׳s pressure–area relationship. Palmer et al. (2009) discussed the pressure–area curve using a fracture mecanics approach. According to his explanation, if ice is idealized as an elastic-brittle material, in which its strength can be defined by linear elastic fracture mechanics (LEFM), the area effect can be determined. Currently, the pressure–area relationship is a cornerstone in ice mechanics and is widely used to represent collision loads.
In conclusion, an isotropic elastic–plastic material model is proposed in this paper to simulate iceberg impact loads during ship-iceberg collisions in the Arctic, especially for the Abnormal Limit State (ALS) conditions (ISO/CD, (2010)). Calibration of the proposed ice material model was performed by comparing the pressure–area relationship with that recommended by ISO. Afterwards, a sensitivity analysis of the material parameters was performed to estimate the effects of their properties. Integrated numerical simulations of ship side collisions and ship bow collisions were also conducted using the proposed material model. This model was incorporated into the LS-DYNA finite element code (Hallquist, 2007) using a user-defined subroutine. Therefore, ship-iceberg collisions under ALS conditions can be predicted solely via numerical simulations using the proposed material model. The results can provide technical support for ship structure design, especially in the case of shipping in ice-covered regions.
Section snippets
Isotropic elastic-perfectly plastic ice material model
Due to the physical complexity of ice, it is challenging to create a material model that completely captures the behavior of icebergs during collisions. As a result, the primary objective of ice modeling is to simulate the characteristics of iceberg impact loads, contact areas and failure criteria. An isotropic elastic–perfectly plastic material model was developed in this study to simulate ice impact loads under ALS conditions. The remainder of this section addresses the development of this
Calibration of the isotropic elastic–perfectly plastic material model
The elastic–perfectly plastic material model was developed for the ALS format. Therefore, the material model should be calibrated against the ALIE curve. For this purpose, a numerical simulation of the rigid plate-iceberg collision was performed; the results of this simulation are discussed in this section. The calculated pressure–area curve was compared with the ISO recommendation for calibration. McKenna (2005) assumed that the mean iceberg model shape could be represented by a sphere. As a
Numerical simulations
According to the damage survey conducted by the Finnish and Swedish maritime administrations (Hänninen, 2005), most of the damage caused by floes collisions in the Baltic Sea is small indents on plates in the bow and midship hull area. Therefore, comprehensive tanker side-iceberg and ship bow collision analyses were performed, using the isotropic elastic–perfectly plastic ice model proposed in this paper. First, the collision scenario parameters are presented. Then, the simulation results are
Conclusions
This paper proposes a specially developed isotropic elastic–perfectly plastic material model for icebergs in the Arctic. Based on the criterion proposed by Liu et al. (2011b), a new failure criterion is presented in the material model to simulate ice failure behavior. This failure criterion depends on the effective plastic strain and hydrostatic pressure. The material model was used for the Abnormal Limit State conditions; therefore, it was calibrated using the design pressure–area curve
Acknowledgments
The work contained in this paper was part of a joint-research project between State Key Laboratory of Ocean Engineering in Shanghai Jiao Tong University and Department of Shipping and Marine Technology in Chalmers University of Technology. Moreover, the Natural Science Fund of China (Grant no. 51239007) supported this collaborative work. The authors greatly appreciate both of these sources of support.
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Postal address: State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, No.800 Dongchuan Road, Shanghai, China.