Numerical Analysis of Ice–Structure Impact: Validating Material Models and Yield Criteria for Prediction of Impact Pressure

Abstract

This study explores the application of numerical analysis and material models to predict ice impact loads on ships and offshore structures operating in polar regions. An explicit finite element analysis (FEA) approach was employed to simulate an ice and steel plate collision experiment conducted in a cold chamber. The pressure and strain history during the ice collision were calculated and compared with the experimental results. Various material model configurations were applied to the FEA to account for the versatile behavior of ice (whether ductile or brittle), its elastic-plastic yield criteria, and its dynamic strain rate dependency. In addition to the standard linear elastic-perfectly plastic and linear elastic-plastic relationships, this study incorporated the Crushable Foam and Drucker–Prager models, based on the specific ice yield criteria. Considering the ice’s strain rate dependency, collision simulations were conducted for each yield criteria model to compute the strain and reaction force of the plate specimens. By comparing the predicted pressures for each material model combination with the pressures from ice collision experiments, our study proposes material models that consider the yielding, damage, and behavioral characteristics of ice. Lastly, our study proposes a combination of ice material properties that can accurately predict collision force.

1. Introduction

Vessels and offshore structures operating in Arctic regions constantly face the risk of structural damage from collisions with ice. A notable incident took place in January 1994, when the tanker Overseas Ohio, weighing 10,000 tons and traveling at approximately 5 m/s, sustained bow damage due to an ice collision off the coast of Alaska, resulting in severe marine pollution [1]. In the design phase of Arctic ships and offshore structures, predicting ice collision force is crucial for identifying structures with impact-resistant stiffness to minimize the risk of hull damage. This study introduces a numerical procedure and methodology for estimating ice collision force and compares the results with experimental findings to assess the validity of the load prediction process. The determination of collision forces can be categorized into two main analytical frameworks: dynamic structural analysis and fluid-structure interaction analysis. Several studies [2,3,4] have employed dynamic structural analysis to calculate ice collision force. Additionally, methods utilizing fluid-structure interaction to simulate ice–structure collisions have been explored [5,6], along with dynamic structural analyses focusing on the damage to both ice and structures [7,8]. Various constitutive equations and damage models for ice have been applied to predict collision force [3,9,10]. Despite the numerous approaches to ice collision force analysis discussed in the literature, this topic remains a subject of ongoing investigation. This continued interest is due to the significant variability in material properties, including elastic and plastic deformation and ice fracture, as well as the irregular changes in contact area during collisions, which pose significant challenges to accurate simulation of ice damage and structural response. Different models have been applied to characterize the interactions between ice and other materials, including the Drucker–Prager model [5,11] and the Crushable Foam model [7,8]. However, variations in the ice-induced resistance faced by ships have been observed, highlighting the need to evaluate load predictions based on different yield criterion assumptions. Furthermore, studies using the discrete element method (DEM) have simultaneously considered the fragmentation of level ice during ship collisions, the formation and movement of ice blocks, and the interaction between ice blocks and ship structures [12,13,14]. Importantly, the DEM accounts for both ship and ice motion, including contact loading between individual ice blocks and the ship’s hull. Moreover, this approach represents the shape of ice as polygons or polyhedra and can simulate the mutual motion between ice and ship structure, as well as the ice fragmentation process. Models evaluating the impact of local ice collision force on ship structures, including various material properties and ice shapes, continue to attract considerable attention. Previous studies have conducted ice collision tests using various impact apparatuses. For example, Zhu et al. [7] and Cai et al. [8] conducted collision experiments with wedge-shaped ice specimens at speeds of 4 to 7 knots (approximately 2.06 to 3.61 m/s) to measure the loads applied to the impacted bodies. Additionally, Jang et al. [15] presented the results of collision experiments between hemispherical ice specimens and steel plates using a collision pendulum at controlled speeds (5 knots, approximately 2.57 m/s; 7 knots, approximately 3.61 m/s; and 10 knots, approximately 5.14 m/s) in a low-temperature chamber. These experiments accurately predict loads under experimental conditions, but they require scaling down the size of the ship and ice, and the experimental conditions are limited. In contrast, numerical simulation models do not account for size and shape limitations for ships and ice or collision speeds, but accuracy verification is essential. Therefore, numerical simulations must be performed to predict the ice collision force and to conduct analytical and experimental work for accuracy verification. The purpose of this study is to propose a procedure for numerically evaluating the local ice collision force and the response of steel plates by simulating simplified experiments. This procedure considers various factors, including yield criteria, material behavior, and strain rate effects on the material model of ice. To achieve this, several numerical simulations were conducted to reproduce the local collision force and structural deformation at the collision speeds realized in experiments. These simulations vary the material model to propose results and evaluate the impact of material properties on ice collision force predictions. This study also seeks to expand the range of environmental variables associated with ice collision, (e.g., collision speed) in existing empirical design formulas to demonstrate the applicability of numerical analysis for load prediction. By systematically examining the performance of proposed material models against experimentally observed responses, this research aims to develop the appropriate numerical model for ice collision force prediction.

2. Material Properties of Ice

The mechanical properties of ice, encompassing strength, and behavior, are profoundly influenced by several factors, including crystal structure, salinity, temperature, and strain rate [16,17]. These variables collectively contribute to a complex landscape for characterizing ice material properties and associated constants, making it an exceptionally challenging endeavor. Additionally, ice material properties, including elasticity, compressive and tensile strengths, and yield stresses, exhibit significant temperature dependence [18]. Notably, substantial variations in these properties occur within a temperature range of −10 °C to −20 °C. Modeling temperature-dependent ice behavior presents a substantial challenge, given the dynamic nature of yield surfaces as temperatures fluctuate [19]. Researchers have underscored the significance of temperature-dependent yield criteria, but achieving a consistent representation of temperature dependence across all material constants remains a formidable task.

Extensive studies [3,5,9,20,21,22,23] have explored the mechanical properties of ice, delving into topics such as constitutive relationships, fracture properties, and strain rate dependencies. These works underscore the need for models capable of capturing ice behavior under various conditions. Where high strain rates are prevalent, such as in structural-ice collisions, ice exhibits brittle behavior. Many studies have explored various plasticity models to find a suitable simulation model for specific ice–structure impact scenarios [21]. This literature provides a comprehensive account of the mechanical properties of ice, forming the foundational basis for deriving material properties and constitutive equations in our study.

It is crucial to acknowledge the challenges researchers face when defining distinct yield stress for ice, a material that significantly differs from metals in its response to stress and strain. The current literature presents a spectrum of yield or failure criteria, ranging from von Mises to Drucker–Prager, each with its unique advantages and limitations [24,25]. Among the phenomenological ice models, the Elastic-Plastic model, the Crushable Foam model, and the Ductile Damage model have been actively explored [3,22,23,26,27,28,29]. These models often accompany the adoption of Drucker–Prager [29,30] and von Mises [31] criteria to describe the ice failure process based on experimental data. Within the realm of ice behavior modeling, the Drucker–Prager and Crushable Foam models have emerged as prominent choices, valued for their effectiveness in simulating ice-crushing and deformation phenomena. The Drucker–Prager model effectively captures the failure envelope for materials such as ice, particularly under complex stress states characterized by pressure-sensitive yield. Conversely, the crushable foam model excels in simulating compressive and crushing behaviors of ice, especially at high strain rates. The Ductile Damage model, a recognized alternative, has also been considered in modeling high-velocity collisions, as demonstrated by Carney et al. [26] and Gagnon [22], who employed the Crushable Foam material model to simulate growler impacts on ships. These modeling approaches are integral components of commercial numerical codes such as ABAQUS and LS-Dyna [3,4,32], known for their computational efficiency.

However, the Crushable Foam model of ABAQUS and LS-Dyna, as noted by Liu et al. [23] and Mokhtari et al. [32], exhibits significant limitations due to its inability to account for ice damage. To enhance the general Crushable Foam model, Liu et al. [23] added the Tsai–Wu yield surface and plastic flow theory to the Crushable Foam model and constitutive relationships. The failure criterion, which is based on effective plastic strain and hydrostatic pressure, was suggested to facilitate the simulation of damage evolution in the Crushable Foam model. The material model was simulated using LS-DYNA so that the contact pressures were investigated. Their results show that the proposed model (user-defined material subroutine VUMAT) produces reasonably good results. To address any shortcomings, several enhanced models have been proposed: (1) Damage models aim to accurately capture the initiation and progression of ice cracks and fractures. By utilizing continuum damage mechanics, they allow for a detailed simulation of damage initiation and growth, thus offering a more comprehensive understanding of ice behavior under stress [9,20,21]. (2) Elasto-Plastic or Viscoelastic-Plastic models, drawing on triaxial test data, present a more physically intuitive approach [4,5]. They excel in incorporating hardening and softening behaviors, essential for simulating the transition between ductile and brittle states in ice. Integrating these refined models significantly enhances our understanding and simulation of ice behavior, particularly concerning damage propagation.

Considering diverse material models of ice, our study aims to apply the wealth of data within an analytical framework that facilitates the comparison and analysis of different material property combinations, as detailed in

Table 1. Yield criteria and yield ratio of ice materials.

Material Behavior	Model	Reference
Yield criteria	Crushable Foam	Gagnon [3], Kim, et al. [4], Liu et al. [23]
	Drucker–Prager	Kajaste-Rudnitski & Kujala [5]
Damage model	Ductile Damage	Han, et al. [9], Kim, et al. [20], Jeon & Kim [21]
Strain rate effect		Tippmann [33]

. Despite the extensive body of literature addressing constitutive equations for ice and numerical procedures for ice impact scenarios, selecting appropriate material properties tailored to specific conditions remains an outstanding challenge. Moreover, determining material constants associated with chosen constitutive equations presents a complex endeavor. In this study, we place particular emphasis on selecting diverse material models to enhance our understanding and predictive accuracy of ice behavior under various impact scenarios. Our research seeks to incorporate the elastic-plastic deformation characteristics of ice, its fracture properties, and its strain rate sensitivity into numerical analyses, drawing inspiration from material models proposed in existing literature. Ultimately, our study endeavors to predict ice behavior in impact scenarios through the utilization of a composite plastic model, informed by insights and data derived from both experimental work and proposed viscoelastic deformation models in the literature.

2.1. Constitutive Equations for Ice Elasto-Plastic Behavior

In the initial phase of this study, we evaluated two distinct constitutive models to characterize the elastic-plastic behavior of ice. The first model, known as the Linear Elastic-Perfectly Plastic with Damage (LEPP) model, primarily emphasizes the elastic domain in ice behavior. The second model, the Linear Elastic-Plastic with Damage (LEP) model, assumes plastic deformation as the primary response. Load-displacement curves illustrating these models are depicted in Figure 1, showing a linear increase in behavior with applied load.

The LEPP model, which emphasizes the elastic domain, suggests that when the load is removed, the strain and stress are fully recoverable, indicating that no permanent deformation occurs until the yield strength (${\sigma }_y$) is reached. Since the plastic behavior is almost ignored, the material begins to damage as soon as the stress reaches the yield point (damage, D = 0). Therefore, we assume that the yield strength and ultimate strength (${\sigma }_u$) are similar (${\sigma }_y\approx {\sigma }_u)$. This constitutes a crucial aspect of the model, as it marks the onset of fracture or failure in the ice structure. Conversely, the LEP model assumes a lower yield stress threshold. Therefore, yielding occurs immediately upon loading, resulting in a negligible elastic phase. Consequently, after yielding, the ice does not return to its original state when the load is removed but continues to suffer damage until it reaches its ultimate strength, at which point damage propagation becomes more pronounced. The adequacy of these two material models is discussed by comparing their predicted behavior with experimental observations. Finally, this study aims to compare the predictive capabilities of these two material models with experimental results to determine the most appropriate description of ice behavior.

2.2. Yield Criteria Models of Ice

(1)

Crushable Foam yield model

The Crushable Foam (CF) model effectively represents the hardening characteristics associated with the volumetric strain of ice under compressive loading. The deformation of the model applying crushable foam is mostly non-recoverable when the load is removed, and this is why this material model is named Crushable Foam. This model has been successfully applied by researchers such as Gagnon and Derradji-Aouat [34], Kim et al. [4], Gagnon [3], and Gao et al. [35], and was also utilized by Obisesan and Sriramula [36] in their analyses. The Crushable Foam model mentioned in [3] is referred to as the isotropic Crushable Foam model related to MAT 63 (MAT_CRUSHABLE_FOAM) of LS-Dyna [37]. MAT 63 employs an elastic unloading mechanism up to a specified tension cutoff stress. Plasticity is incorporated through a return-mapping algorithm, which updates the stress while assuming elastic behavior until the yield stress is attained. On the one hand, the Crushable Foam model implemented in ABAQUS version 2016, applies plasticity theory to simulate the behavior of foam materials under load. The yield criterion is based on the Tsai–Wu surface, considering the von Mises stress and hydrostatic pressure. The plastic strain rate is calculated through a non-associated flow rule derived from a flow potential [32]. Meanwhile, the VUMAT subroutine developed by Liu et al. [23] based on the Tsai–Wu yield surface has been widely adopted for simulating the impact loads of glaciers and freshwater ice and is known to be superior to other plastic models. Mokhtari et al. [32] compared the efficiency and the pros and cons of the ABAQUS built-in CF model and the VUMAT model. Their results proved the efficiency of VUMAT as a material model for ice; however, this study selected only the ABAQUS built-in CF model for comparison purposes. This plasticity model is also based upon the Tsai–Wu yield surface, discussed in the previous section.

A key feature of the Crushable Foam model of ABAQUS built-in is the assumption that the material does not exhibit elastic recovery after yielding, resulting in permanent deformation. This model not only accounts for the failure stress associated with tension and compression but also considers the hardening characteristics. The yield surface ($F$) of the crushable foam with volumetric hardening model is defined by Equations (1)–(3) and is represented as an elliptical shape, as shown in Figure 2. Due to these characteristics, yielding occurs under the influence of hydrostatic compression pressures [38].

$$F\left(p,q\right)=\sqrt{q^2+{\alpha }^2{\left(p-p_0\right)}^2}-B=0$$

(1)

$$A=\frac{\left|p_c\right|+p_t}{2},B=q_{\max },p_0=\frac{p_c-p_t}{2},\alpha =\frac{B}{A}=\frac{3k}{\sqrt{\left(3k_t+k\right)\left(3-k\right)}}$$

(2)

$$k=\frac{{\sigma }_c^0}{p_c^0},k_t=\frac{p_c}{p_c^0}$$

(3)

The symbols $p$ and $q$ denote the pressure stress $\left(-1/3trace\sigma \right)$ and the Mises stress respectively. $A$ is the size of the (horizontal) $p$-axis of the yield ellipse and $B$ is the size of the (vertical) $q$-axis of the yield ellipse with $B=\alpha A=\alpha (p_c^0+p_t)/2$. Hence, $A$ and $B$ represent the axes of the elliptical yield surface in the Crushable Foam model, respectively, with $p_0=(p_c^0-p_t)/2$ denoting the center of the yield ellipse. The yield strengths in hydrostatic compression and hydrostatic tension conditions are denoted by $p_c$ and $p_t$, respectively. The factor $k$ represents the ratio of yield in compression, whereas $k_t$ represents the ratio of yield under hydrostatic conditions. The initial yield stress in hydrostatic conditions is indicated by $p_c^0$, whereas ${\sigma }_c^0$ represents the uniaxial compressive yield stress, which is shown in Figure 2 as the point where the yield surface intersects the $q$-axis. As the deformation of the ice enters the plastic regime, the size and shape of the yield surface change; this change can be expressed by the relationship between the uniaxial compressive yield stress and the volumetric strain. These changes are described by a strain hardening model where the yield surface varies based on a line connecting $p_c$ and $p_t$ intersecting s with the $q$ axis. The evolution of the yield surface can be represented as a function of the volumetric plastic hardening, $p_c\left({\epsilon }_{vol}^{pl}\right)$, as expressed in Equation (4), which expresses the relationship between the volumetric plasticity and uniaxial stress. Volumetric hardening assumes that uniaxial plastic deformation (${\epsilon }_{axial}^{pl}$) is equivalent to volumetric deformation (${\epsilon }_{vol}^{pl}$). To implement the hardening of the Crushable Foam plastic model in ABAQUS, the change in the yield surface is reflected by the uniaxial compressive yield stress and plastic strain rate.

$$p_c\left({\epsilon }_{vol}^{pl}\right)=\frac{{\sigma }_c\left({\epsilon }_{axial}^{pl}\right)\left[{\sigma }_c\left({\epsilon }_{axial}^{pl}\right)\left(\frac{1}{{\alpha }^2}+\frac{1}{9}\right)+\frac{p_t}{3}\right]}{p_t+\frac{{\sigma }_c\left({\epsilon }_{axial}^{pl}\right)}{3}}$$

(4)

(2)

Drucker–Prager yield model

The Drucker–Prager model is known to be suitable for materials with internal friction, such as rock and concrete. This model effectively captures the strengthening of certain materials under compressive loads and is appropriate for describing the behavior of materials where the compressive strength is higher than the tensile strength. Noting the similarity between the behavior of ice and concrete, Kajaste-Rudnitski and Kujala [5], as well as Zhang et al. [11], have applied the Drucker–Prager approach to ice collision models. A key feature of the Drucker–Prager model is its use of a non-circular yield surface to adjust the yield stress values in both tensile and compressive scenarios. This model allows for the independent setting of the dilation and friction angles and, as shown in Figure 3, defines the yield surface through three invariants. The yield surface ($\psi$) of the Drucker–Prager model can be expressed as described in Equations (5)–(7).

$$F=t-p·tan\beta -d=0$$

(5)

$$d=\left(1-\frac{1}{3}tan\beta \right){\sigma }_c$$

(6)

$$t=\frac{1}{2}\left[1+\frac{1}{K}-\left(1-\frac{1}{K}\right){\left(\frac{r}{q}\right)}^3\right],$$

(7)

where $\beta$ represents the material’s angle of friction in the $p$–$t$ plane, $d$ denotes the material’s cohesion, $K$ is the ratio of the triaxial tensile yield stress to the triaxial compressive yield stress, $p$ is the equivalent pressure, and $q$ represents the von Mises stress. In other words, the yield condition of the Drucker–Prager model typically takes on a non-circular form, with the yield surface determined by the material’s angle of friction ($\beta$), cohesion ($d$), equivalent pressure ($p$), and the von Mises stress ($q$). The study by Kajaste-Rudnitski and Kujala [5] provides a detailed application of such a model.

(3)

Progressive damage model

The Elastic-Plastic model is characterized by a stress-strain relationship in which stress linearly increases with strain up to the yield point. Incorporating yield criteria such as the Crushable Foam and Drucker–Prager models into the elastic-plastic framework allows for a detailed simulation of ice behavior under various loading conditions. The Crushable Foam model effectively represents the crushing behavior of ice, while the Drucker–Prager criterion accounts for the pressure-dependent yielding typical of ice. Ductile damage, introduced post-yield, is responsible for the degradation in load-bearing capacity due to plastic deformation and damage accumulation. This model is crucial for predicting the entire failure process, capturing both the initial deformation and the subsequent failure. When ice impacts a structure, it initially resists according to the elastic-plastic behavior up to its yield limit. After yielding, the selected yield criteria dictate the continuation of plastic deformation. Concurrently, ductile damage tracking monitors the accumulation of damage, which intensifies with further plastic deformation.

The ultimate failure of materials in this model can be broken down into three phases. Initially, the material undergoes deformation following its elastic behavior. It then transitions into plastic deformation and ultimately into complete fracture. Figure 4 shows the uniaxial stress-strain curve assumed. Within this representation, segment $ab$ corresponds to linear elastic behavior, eventually reaching the yield strength denoted as ${\sigma }_Y$. The curve then progresses through $bc$, signifying stable plastic deformation, where $c$ denotes the onset point of damage initiation. It is at this point that stiffness degradation initiates. The scalar damage variable $D$ provides a continuum from the undamaged state ($D=0$) to complete rupture ($D=1$), offering a nuanced representation of the material’s loss of load-carrying capacity. The damage variable $D$ is defined in relation to the effective strain and its equivalent plastic strain (${\epsilon }^{pl}$), a characteristic length ($L)$, and the failure plastic strain ${\epsilon }_f^{pl}$ as described in Equation (8). $u^{pl}$ and $u_f^{pl}$ mean the equivalent plastic displacement and the equivalent plastic displacement at failure, respectively. Equation (9) describing the evolution of $D$ as an integral function of stress and plastic strain. ${\sigma }_c$ represents the critical compressive yield stress, while $G_f$ encapsulates the energy required for damage progression. In this expression, ${\sigma }_c$ denotes the uniaxial compressive yield stress, reflecting the material’s resistance to compressive forces. Selecting the values for $G_f$ is explained in the following sections.

$$D=\frac{L{\epsilon }^{pl}}{u_f^{Pl}}=\frac{u^{pl}}{u_f^{pl}}$$

(8)

$$D=1-exp\left(-{{\displaystyle \int }}_0^{u^{pl}}\frac{{\sigma }_c}{G_f}d{u}^{pl}\right)$$

(9)

It is assumed that fracture occurs when the plastic displacement ($u^{pl}$) reaches the plastic displacement at failure ($u_f^{pl}$). Alternatively, a fracture is assumed to occur when the energy per unit area ($G$) reaches the fracture energy per unit area or damage evolution energy ($G_f$). The relationship between $u_f^{pl}$ and $G_f$ is expressed as follows:

$$u_f^{Pl}=\frac{2G_f}{{\sigma }_Y}$$

(10)

where ${\sigma }_Y$ represents the initial yield stress in a uniaxial compression state. These enhancements provide an accurate and systematic description consistent with the theoretical approach. In this study, the damage model is characterized by deformation parameters, namely the fracture strain (${\epsilon }^{pl}$) and fracture energy ($G_f$), at the point of reaching ultimate strength under static conditions (strain rate = 0). The material parameter values for each damage model are elaborated in detail in Section 3.1 and Section 3.2.

2.3. Strain Rate Dependency of Ice: Yield Strength Ratio

This section examines the complex relationship between yield strength ratio and strain rate. Notably, higher strain rates typically lead to an increase in the yield strength of ice, as highlighted by Tippmann [33]. Moreover, Barette and Jordaan [16] reported that lower temperatures also contribute to an increase in the intrinsic strength of ice. It is also important to consider that both strain rate and yield stress tend to increase simultaneously as collision energy increases. Tippmann [33] explored these dynamics by investigating interactions between ice and structural elements, particularly in scenarios involving spherical ice specimens colliding with carbon fiber-reinforced plastic (CFRP) plates. Furthermore, in the context of high-velocity collisions, it is important to account for the strain rate dependence of the materials involved. This aspect has been addressed by adopting the model proposed by Tippmann [33]. Notably, the range of strain rates outlined by Tippmann [33] extends beyond the strain rate of this study, enabling the analysis of a broader spectrum of collision velocities. Based on observations from collision experiments with ice, it is evident that yield strength is dependent on strain rate. Tippmann [33] and Tippmann et al. [39] conducted studies focusing on this dependency, analyzing the relationship between strain rate and yield strength. Their findings demonstrated that as the strain rate increases, the compressive strength also shows an upward trend. During the plastic deformation of ice, yield occurs at a specific strain rate, after which the stress remains relatively constant. This phenomenon is particularly observed during high-speed ice collisions, making a perfect plasticity model appropriate, assuming a yield hardening coefficient of 0 [38,40]. Tippmann [33] and Tippmann et al. [39] compiled various research data and represented the relationship between strain rate and yield strength using a linear-logarithmic curve. In this study, this relationship, specifically the strain rate-yield ratio, was incorporated into the analysis. Figure 5 illustrates the relationship between compressive plastic strength and the yield strength ratio within a strain rate range of 0.1/s to 106/s, providing a visual representation of this curve and the relevant values. The yield strength ratio is normalized to the initial compressive plastic strength of 5.2 MPa.

3. Ice Material Calibration

In the previous section, various material properties of ice were described, focusing on plastic deformation relationships, yield conditions, and the strain rate dependency of yield strength. As reported in previous studies, different models are employed for different material properties, reflecting the complex and uncertain nature of the deformation properties of ice. In this section, considering the diversity of material models mentioned in Section 2, we present the selection process for material models and material constants that can account for the dynamic compressive deformation characteristics of ice.

• Plastic deformation relationships: Models such as the Linear Elastic-Perfectly Plastic (LEPP) and Linear Elastic-Plastic (LEP) models define how materials behave when they exceed the elastic limit. While the LEPP model assumes that no deformation occurs beyond the yield point, the LEP model describes post-yield hardening or softening. These material plastic properties are applied to the analyses to verify their results.
• Yield criteria: The Crushable Foam (CF) and Drucker–Prager (DP) models define the conditions under which materials reach yield. The CF model simulates the collapse of cellular materials under compression, whereas the DP model models the yield of pressure-dependent materials. Given the sensitivity of ice to pressure and density, the appropriateness of these conditions was investigated.
• Strain rate dependence: In dynamic events such as collisions, strain rate affects the yield stress and failure modes of the material. Results are analyzed based on whether or not strain rate dependence is considered.

In this study, the combination of these three models was applied to material modeling efforts and compared with experimental data to select the most appropriate model. This approach helps to reduce the complexity and uncertainty associated with material modeling and allows for more accurate predictions of ice behavior in real-world scenarios. Four models were adopted by combining (1) two plastic constitutive equations and (2) two yield conditions. Additionally, (3) the effects of strain rate are included in all eight material model combinations, as summarized in

Table 2. Material property combination for numerical simulation.

Models	Constitutive Equation	Yield Criteria
LEPP-CF	Linear elastic-perfectly plastic	Crushable Foam
LEPP-DP	Linear elastic-perfectly plastic	Drucker–Prager
LEP-CF	Linear elastic-plastic	Crushable Foam
LEP-DP	Linear elastic-plastic	Drucker–Prager

. Determining material constants for the ICE material models, including elasticity coefficients, yield stresses, hardening coefficients, and yield strength ratios, typically requires performing tensile tests, bending tests, or triaxial compression tests. In some cases, these constants can be indirectly estimated through theoretical analysis. In this research, however, these values were derived by performing quasi-static compression experiments on ice and simulating these experiments using finite element analysis. A detailed description of this process is provided in Section 3.1 and Section 3.2. Additionally, the compression strength ratio as a function of strain rate, as measured by Tippmann [33] in high-speed collisions with CFRP, was used to assess the dynamic deformation properties of ice with strain rate dependence. The values required to define the strain rate dependence are provided in Figure 5 of the previous section, with the lower limit of the compression strength ratio curve being applied.

3.1. Determination of Material Model Constants

This section discusses the process of determining the material constants required for the elastic constitutive equations (i.e., linear elastic deformation relations) for ice, including the LEPP and LEP models, as well as the yield conditions required for the CF and DP models. The constants that must be defined for each material model are summarized below:

• The LEPP model: This model requires the determination of two fundamental material constants, Young’s modulus ($E$) and the yield stress (${\sigma }_Y$). Young’s modulus is conventionally obtained from tensile tests, whereas the yield stress is the stress point at which the material begins to undergo plastic deformation.
• The LEP model: Similar to LEPP, the LEP model requires the determination of Young’s modulus ($E$) and yield stress (${\sigma }_Y$). In addition, it requires the consideration of additional constants associated with the hardening model and the hardening coefficient or plastic flow rule.
• The CF model: This model requires the definition of material constants that govern the compressive behavior of foams. Key constants include density, initial strength (${\sigma }_i$), and hardening coefficient. The determination of these constants is facilitated by the analysis of stress-strain curves derived from compression tests on specimens.
• The DP model: The DP model requires a set of constants related to the angle of internal friction (β), material cohesion (d), drag parameters such as the dilatancy angle (ψ), and constants related to the hardening rule. Typically, these constants are obtained from triaxial compression tests and should be considered in conjunction with stress paths.

For material constants that are difficult to measure or calculate directly, assumptions were made based on existing literature. Specifically, values from Tippmann [33], Kajaste-Rudnitski and Kujala [5], Zhang et al. [11], Kim et al. [4], and Han et al. [9] were used to set the density and Poisson’s ratio of ice to 900 kg/m³ and 0.3, respectively. Undetermined material constants for the CF and DP models were hypothetically varied and then incorporated into finite element analyses, ultimately yielding material constants consistent with experimental results. It should be noted, however, that the material models applied to quasi-static compression simulations, which primarily represent low rates of deformation, are tailored to exhibit characteristics specific to quasi-static compression. The constants that define the yield criteria, including parameters associated with damage initiation and stiffness reduction after initiation, are taken from the works of Obisesan and Sriramula [36] and Jeon and Kim [21]. The determined parameters for the yielding models are summarized in

Table 3. Parameters of yield surface (LEPP model).

LEPP	Crushable Foam (CF) (Obisesan and Sriramula [36])		Drucker–Prager (DP) (Jeon and Kim [21])
Parameter	Compression Yield Stress Ratio	Hydrostatic Yield Stress Ratio	Friction Angle	Flow Stress Ratio	Dilation Angle
Value	1.49	1.79	36 deg	1	12 deg

. Additionally, the undetermined material constants specific to each material model were refined to best match the experimental results through simulation analyses simulating ice compression tests, as described in Section 3.2. The material constants defined in the ice compression simulations are as follows:

• The LEPP model: The apparent elastic modulus was assumed to be 0.14 GPa. The yield stresses for the CF and DP models were set at 4.8 MPa and 3 MPa respectively. The material damage model parameters included a failure strain of 1.0 × 10⁻⁴ and an energy release rate of 15.0 J.
• The LEP model: The elastic modulus, influenced by stress and strain rate, was set to 9 GPa. Hardening constants for this model are detailed in

  Table 4. Hardening coefficients derived from stress-strain (LEP model).

  LEP	Crushable Foam (CF)		Drucker–Prager (DP)
  Hardening Value	Stress	Strain	Stress	Strain
  	0.5 MPa	0	0.5 MPa	0
  	3.3 MPa	0.019	3 MPa	0.025
  	3.3 MPa	0.5	3 MPa	0.5

  for both the CF and DP models, and failure strains were designated as 0.019 and 0.025, respectively, with a damage evolution energy of 15.0 J.

These material constants were carefully tuned to accurately simulate the ice compression behavior under the chosen material models.

3.2. Finite Element Analysis Simulating Ice Compression Experiments

This study successfully conducted quasi-static compression tests on ice specimens using compression testing equipment, which enabled the validation of specific material constants. Building on these experiments, finite element analysis (FEA) was then performed. Throughout this process, multiple FEA iterations were undertaken to determine previously undefined material constants. According to the quasi-static compression test results reported by Jang et al. [15], it was observed that ice specimens, configured as square columns with cross-sectional dimensions of 100 mm × 100 mm and a height of 250 mm, failed under loads ranging from approximately 25 to 32 kN when compressed by 5 mm (Figure 6). The quasi-static compressive force was set to move at a speed of 1.0 mm/s during the experiments. The experiments were conducted using a low-temperature chamber at $-20℃$. The temperature of the chamber was controlled using liquid nitrogen. During the compression tests, a 15 mm thick sponge was affixed between the ice specimen and the compression jig. The purpose of this sponge was to ensure the uniform transmission of the compression load to the ice surface. Therefore, it can be considered that the compression load is effectively transferred to the ice only after the deformation of the sponge reaches approximately 15 mm. In other words, the deformation of the sponge is estimated to be 15 mm, and it is observed that the ice starts to compress after the deformation of the sponge is completed. When the compression deformation reaches approximately 5 mm, fracture and damage to the ice specimen are observed as shown in Figure 6.

For the finite element modeling, these specimens were represented using 39,375 solid elements (C3D8R), with each element having a grid size of 4 mm. The compression plate in the model was assumed to be a rigid body (Figure 7). The setup for the analysis model was as follows:

Contact conditions: The lower surface of the ice specimen was fixed in place. On the upper surface, which came into contact with the rigid compression plate, a hard node-to-surface frictionless contact condition was applied. This setup ensured the prevention of any penetration between the specimen and the plate.

Compression conditions: The experimental conditions were mirrored in the finite element analysis. Considering the quasi-static compressive load controlled by the forced displacement to the specimen, the test conditions were applied to the numerical model (refer to Figure 8). We excluded the compression of the sponge and focused solely on the compression zone of the ice. We observed that the ice specimen experienced failure when the imposed displacement reached approximately 5 mm in the numerical model as was observed in the experiments.

The quasi-static compression experiments on ice were reproduced using finite element analysis models to calculate the load and displacement values leading to failure. Verification was achieved through a comparison with experimental results. One of the primary objectives of this study was to optimize the material constants for each finite element model to accurately reproduce the average failure load of 28 kN, as observed in the experiments. This involved finely tuning various material constants, including yield stress, hardening coefficient, and parameters of the damage model. The aim was to ensure that the predicted load-displacement relationship derived from the FEA closely matched the experimental data, maintaining an error margin of less than 1.5%.

In the experimental setup, the reaction force corresponding to the displacement was measured. In contrast, the total compression load acting on the bottom surface of the ice was calculated via FEA. Through iterative processes, the material constants were precisely adjusted to align with experimental observations. The refined load-displacement relationships for each material model are depicted in Figure 8.

Table 5. Predicted force and displacement for material model combinations through simulation.

FEA Result	LEPP-CF	LEPP-DP	LEP-CF	LEPP-DP
Peak force [kN]	28.2	27.9	27.8	28.2
Displacement [mm]	5.02	5.1	4.98	4.92

summarizes the compression loads at specific points, confirming that the material constants deduced in this study effectively replicate the ice compression behavior observed in the experiments.

4. Simulation Model for the Ice–Structure Collision Experiment

In this study, the analysis of ice–structure collisions is divided into two aspects: ice damage and the dynamic response of the structure. Explicit finite element analysis was conducted to investigate these aspects. The commercial software ABAQUS/Explicit version 2016 was used to simulate the nonlinear dynamic response during the collision process between the two objects. Solid elements (C3D8R) were chosen to represent the yielding and failure behavior of the ice, whereas the structure was represented by shell elements (S4R). The main features of this finite element analysis model are explained below:

• Ice yield criteria: The CF and DP models were used to model the yielding and failure behavior of ice, as described in Section 2.1.
• Contact conditions: Contact conditions were established to account for the interaction between the two objects and to calculate the load transferred from the ice to the structure. For the contact between ice elements and plates, hard contact and tie conditions are applied. The tie conditions were applied to ice and holder, and steel plate and jig, respectively.
• Element with reduced integration: Reduced integration was used to establish the stiffness matrix to enhance the computational efficiency, along with the selection of the hourglass control.
• Time increment: In accordance with the Courant–Friedrichs–Lewy (CFL) condition, the time increment ($\mathsf{\Delta }t$) for the explicit finite element analysis was determined. The time increment was calculated using the formula: $\mathsf{\Delta }t\le L_{min}/c_p$, where $L_{min}$ is the size of the smallest element within the mesh. The term $c_p$ represents the material’s dilatational wave speed (P-wave speed), which is the function of the material’s mechanical properties. For isotropic material, $c_p$ was calculated using the formula $c_p=\sqrt{E\left(1-\nu \right)/\rho \left(1+\nu \right)\left(1-2\nu \right)}$, where $E$ is the elastic modulus, $\nu$ is the Poisson’s ratio, and $\rho$ is the density of the material.

Considering that the yield strength of the steel plate also increases at high strain rates, the strain-rate dependency of steel is applied to the material properties of the steel plate. This study utilizes the results from strain-rate dependency experiments conducted by Choung (2020) at −20 °C, with strain rates ranging from 0 to 200/s. While both the Johnson–Cook and Cowper–Symonds models account for the strain rate dependency of steel, the Cowper–Symonds model was chosen for its specific focus on strain rate effects, without considering temperature-related softening. This model is particularly suited for capturing the increased resistance of steel to deformation under dynamic conditions. The dynamic flow stress (${\sigma }_d$), representing the stress required to continue deforming the steel at a specific strain rate under dynamic conditions, is calculated as a function of the strain rate. This relationship is given by Equation (11):

$${\sigma }_d=\left(1+{\left({\dot{\epsilon }}_p/D_d\right)}^{-q_d}\right)\times {\sigma }_s$$

(11)

where ${\sigma }_s$ is the static flow stress or the yield stress of the steel under quasi-static or very low strain rate conditions, ${\dot{\epsilon }}_p$ is the plastic strain rate, and $D_d$ and $q_d$ are the constants determined through the high-speed tensile tests. Using results from the quasi-static and high-speed tensile tests at −20 °C, a dynamic hardening curve reflecting the strain rate effect was incorporated, as shown in Figure 9. The parameters of the Cowper–Symonds model ($D_d$ and $q_d$) were estimated from the curves of the dynamic tensile test. The density, elastic modulus, and Poisson’s ratio of the steel plate were assumed to be 7800 $\mathrm{kg}/{\mathrm{m}}^3$, 200 $\mathrm{GPa}$, and 0.3, respectively.

Both quasi-static material models and dynamic material models of ice, which account for strain rate effects discussed in the previous section, were also applied in the analysis. Analyses were conducted for each yield model considered, and the obtained strain and ice load on the plate were compared with the experimental values. This study encompasses four distinct material models: the LEPP, LEP, CF, and DP models. These models are integrated into the FEA to assess and compare the fidelity of ice collision simulations across each material model. This comparative analysis aims to identify the most accurate model for simulating the behavior of ice–structure impact.

4.1. Overview of the Ice–Structure Collision Simulation Model

In this study, the collision pendulum experiment conducted by Jang et al. [15] was chosen for numerical simulation. Figure 9 (left) shows the ice specimen, the impactor, and the specimen holder used in the experiment. During the ice–structure impact experiment, biaxial rosette strain gauges were affixed at the center of the specimen’s backside to accurately measure horizontal (${\epsilon }_x$) and vertical (${\epsilon }_y$) strains on the steel plate. Concurrently, high-pressure pre-scale films (HS grade, Fujifilm) were applied to the contact surface to detect and quantify the reaction force ($R$) during collisions. These films, responsive to pressures between 50 and 150 MPa, exhibit a color change proportional to the pressure magnitude, allowing for precise determination of pressure distribution, contact area, and collision force. The changes in color intensity were meticulously quantified using an Epson Perfection V370 scanner (Epson America Inc., Long Beach, CA, USA) coupled with FPD-8010E software ver. 2.7.0.1 (FujiFilm Co. Ltd., Valhalla, NY, USA). In parallel, FEA involved calculating the principal strains on the specimen’s backside to validate and correlate with the experimentally measured strains. The reaction force ($R$) is calculated by summing the contact loads at all nodal points on the specimen’s front side, ensuring the prediction of reaction force ($R$).

A detailed discussion of the material models associated with this simulation was discussed in Section 2, Section 3.1 and Section 3.2. Correspondingly, the finite element model (FEM) mirrors the geometry of the ice and steel plate specimens, as well as the specimen holder. Figure 10 (right) illustrates the apparatus of the pendulum, which incorporates the impactors. The dynamic behavior of collision and the steel plate’s response are simulated using an explicit scheme of FEM. The Finite element mesh representing the collision behavior and conditions are shown in Figure 10. The collision velocities tested in the experiment—specifically 2.5 knots (approximately 1.29 m/s), 3.5 knots (approximately 1.81 m/s), and 5 knots (approximately 2.57 m/s)—were applied in the FEA, spanning a duration of 0.05 s.

Figure 11 and Figure 12 illustrate the configurations of the ice specimen and holder, as well as the steel plate and its corresponding jig, respectively. The ice specimen is modeled as a hemisphere with a 250 mm height and a 300 mm diameter, represented by solid elements in the simulation. The total weight of the ice holder, including the pendulum mass, was defined as 150 kg. The steel plate, with dimensions of 1000 × 500 mm and a thickness of 20 mm, was modeled using shell elements. Additionally, the jig for supporting the steel plate was assumed as a rigid body, measuring 200 × 500 × 50 mm. At the interfaces where the ice specimen contacts the holder and the steel plate contacts the jig, node-to-surface contact with hard contact and frictionless conditions were implemented. In these contact conditions, the steel plate specimen, which possesses higher rigidity, was designated as the master surface. Additionally, tie contact conditions were established between the ice specimen and its holder, and between the steel plate specimen and its jig.

Table 6. Summary of geometry and finite element model.

	Ice Specimen	Ice Holder	Steel Plate	Steel Jig
Geometry	$\mathrm{Dia}.$= 300 mm (Figure 10)	$\mathrm{Dia}.$= 300 mm (Figure 10)	1000 × 500 × 20 mm (Figure 11)	200 × 50 × 500 mm (Figure 11)
Finite element	Shell Element (S4R)	Rigid Element (R3D4)	Solid Element (C3D8R)	Rigid Element (R3D4)
Contact	Hard contact with steel plate (penalty contact, frictionless)	Tie condition with ice	Hard contact with ice (penalty contact, frictionless)	Tie condition with steel plate

summarizes the shapes and conditions of each component employed in the analysis. A critical aspect of our analysis involved assessing mesh convergence. During the analysis process, mesh convergence was examined, and the effect of mesh size on the collision force was investigated by varying the ice grid size from 2 to 7 mm. The results indicated that a biased mesh size of 2–5 mm was the most appropriate.

4.2. Mesh Convergence Test

The FEA of ice impact was followed by a convergence test of reaction force with respect to mesh size. The convergence assessment, aligned with Tippmann’s [33] methodology, focused on the maximum impact force exerted by the ice on a flat plate. The mesh size for the ice specimen comprised uniform grids of 7, 5, and 3 mm, and a biased mesh that was refined near the collision surface. This biased mesh ranged from 2 mm at the impact interface up to 5 mm in the radial direction of the sphere. Figure 13 illustrates the mesh configurations and vertex counts for the ice specimens at different mesh sizes. Figure 14 presents the load history for each mesh size, revealing variations in the peak force tail across different meshes. This trend, similar to Tippmann’s findings, led us to select mesh sizes based on the convergence of peak values despite post-peak discrepancies. Figure 14 displays the relationship between mesh size and the maximum load, with the peak values and computational times detailed in

Table 7. Mesh sensitivity study of ice collision simulation.

Mesh Size	Max. Force [kN]	Normalized Force	CPU Time
7 mm	23.69	0.948	0 H 25 min
5 mm	24.41	0.976	1 H 15 min
3 mm	24.96	0.999	3 H 10 min
2 to 5 mm	24.99	1.000	2 H 05 min

. The analysis found that smaller mesh sizes tend to converge toward 25 $\mathrm{kN}$. A mesh of 5 mm showed a peak collision load deviation within 5%, and the biased mesh structure (2–5 mm) enhanced time efficiency by over 30% compared to a uniform 3 mm mesh. Consequently, considering time efficiency, a biased mesh (2–5 mm) was adopted.

4.3. Collision Analysis Reflecting Quasi-Static Compressed Material Model

The four material models introduced in Section 3.1 were applied to the analyses for specific collision velocities. However, this section did not include the dynamic strain rate effect of ice. Four separate analyses corresponding to a collision velocity of 10 knots (approximately 5.14 m/s) were conducted, with the predicted shapes and stresses of the specimen deformation presented in Figure 15, Figure 16 and Figure 17. The top portions of each figure display the results using the LEPP model coupled with the CF and DP yield models. The bottom portions of each figure represent the results calculated with the LEP model of the ice material coupled with the CF and DP yield models, respectively. Interesting results were obtained when the CF yield model was applied to the LEPP model. In this case, the analysis results reveal the generation of large fragments due to spalling following the collision. On the other hand, the remaining three analysis models produced smaller fragments, which were more consistent with crushing phenomena. Significant differences in the observed loads during collision were noted depending on the characteristics of the analyzed fragments. Particularly, the model where large fragments were generated due to spalling exhibited significant deviations from the actual experimental results. These results underscore the importance of material model selection and its influence on collision analysis.

• Predicted strains on steel plate

Figure 18 and Figure 19 depict the strain history in both the horizontal (${\epsilon }_x$) and vertical (${\epsilon }_y$) directions of the steel plate specimen across the collision velocities. The maximum strains calculated for each material model at distinct collision velocities are summarized in

Table 8. Comparison of maximum strain (${\epsilon }_x$) between experiment and FEA.

Collision Velocity (Kts)	Experiment (Avg.)	LEPP-CF	LEPP-DP	LEP-CF	LEP-DP
5 (2.57 m/s)	6.44 $\times {10}^{-4}$	3.77 $\times {10}^{-4}$(41.4%)	3.59 $\times {10}^{-4}$(44.3%)	3.44 $\times {10}^{-4}$ (46.7%)	5.59 $\times {10}^{-4}$(13.2%)
7 (3.61 m/s)	8.85 $\times {10}^{-4}$	4.72 $\times {10}^{-4}$(46.6%)	6.19 $\times {10}^{-4}$(30.1%)	4.65 $\times {10}^{-4}$ (47.4%)	7.22 $\times {10}^{-4}$(18.4%)
10 (5.13 m/s)	1.08 $\times {10}^{-3}$	5.26 $\times {10}^{-4}$(51.3%)	5.87 $\times {10}^{-4}$(45.6%)	5.37 $\times {10}^{-4}$ (50.3%)	9.35 $\times {10}^{-3}$(13.4%)
Difference	-	46.5%	40.0%	48.1%	15.0%

and

Table 9. Comparison of maximum strain (${\epsilon }_y$) between experiment and FEA.

Collision Velocity (Kts)	Experiment (Avg.)	LEPP-CF	LEPP-DP	LEP-CF	LEP-DP
5 (2.57 m/s)	4.10 $\times {10}^{-4}$	1.76 $\times {10}^{-4}$(57.1%)	2.01 $\times {10}^{-4}$(50.9%)	1.78 $\times {10}^{-4}$(56.5%)	2.96 $\times {10}^{-4}$(27.9%)
7 (3.61 m/s)	3.96 $\times {10}^{-4}$	2.58 $\times {10}^{-4}$(34.9%)	2.51 $\times {10}^{-4}$(36.6%)	1.81 $\times {10}^{-4}$(23.1%)	2.90 $\times {10}^{-4}$(26.8%)
10 (5.13 m/s)	3.74 $\times {10}^{-4}$	3.09 $\times {10}^{-4}$(17.3%)	2.34 $\times {10}^{-4}$(37.4%)	3.87 $\times {10}^{-4}$(3.4%)	3.80 $\times {10}^{-4}$(1.5%)
Difference	-	36.4%	41.7%	38.1%	18.7%

, as well as depicted in Figure 20. It was observed that the Linear Elastic-Perfectly Plastic (LEPP) and Linear Elastic-Plastic (LEP) models, when integrated with the Drucker–Prager (DP) yield criterion, outperformed their counterparts utilizing the Crushable Foam (CF) model. Notably, the LEP model demonstrated the most accurate performance, aligning closely with the experimental data and maintaining an error rate within 15%.

• Ice collision force prediction results

Figure 21 presents the collision force history applied to the plate specimen for various collision velocities. Figure 22 and

Table 10. Comparison of maximum force between experiment and FEA.

Collision Velocity (Kts)	Experiment (Avg.)	LEPP-CF	LEPP-DP	LEP-CF	LEP-DP
5 (2.57 m/s)	32.4	16.7 (48.5%)	20.1 (37.9%)	19.3 (40.3%)	27.5 (15.1%)
7 (3.61 m/s)	45.3	31.8 (29.8%)	25.6 (43.5%)	21.4 (52.8%)	36.9 (18.5%)
10 (5.13 m/s)	70.4	24.99 (64.5%)	36.6 (48.0%)	23.9 (66.1%)	52.8 (25.0%)
Difference	-	47.6%	43.1%	53.1%	19.6%

summarize the maximum collision force in conjunction with the strain rate for different material model combinations. Similar to the strain rate prediction results, the DP yield model outperformed the CF model in terms of accuracy. Particularly, the combination involving the LEP model exhibited a deviation of 22.1% from the experimental results. Among the material properties obtained from quasi-static compression experiments (W/O strain rate effect), this combination proved to be the closest to the experimental results. These results emphasize the significance of material model selection in collision-related analyses and highlight the importance of choosing a model that accurately predicts responses under various conditions.

4.4. Collision Analysis Reflecting Yield Ratio according to Strain Rate

This section presents the results of collision analysis, including the dynamic material properties of ice as a function of strain rate, as well as the models introduced in Section 4.1. Initially, the four material models introduced earlier were applied to calculate collision force and strain rates. The analysis was conducted for a collision velocity of 10 knots (approximately 5.14 m/s), and the calculated deformation shapes of the specimens are presented in Figure 23, Figure 24 and Figure 25. Figure 23 illustrates the failure pattern of ice predicted by combining the LEPP ice material model with the CF and DP yield models. Figure 24 illustrates the plastic strain distribution on the ice surface after the collision, whereas Figure 25 displays the Mises equivalent stress in the plate specimen. The analysis that combines LEP with the CF yield model does not consider the yield ratio in relation to strain rate. However, the analysis in Section 4.1 confirmed that the strain rate on the collision surface falls within the 20–30 mm/mm range. Based on the relationship between strain rate and yield strength ratio presented in Figure 5, the yield stress ratio could be estimated as 2.5. Additionally, the analysis that combined the LEPP model with the DP yield model illustrated the formation of spalling phenomena and the generation of large, fragmented particles.

• Predicted strains on steel plate

The strain rate histories in the width $\left({\epsilon }_x\right)$ and height $\left({\epsilon }_y\right)$ directions of the plate specimen for various collision velocities are presented in Figure 26 and Figure 27, respectively. Figure 28 illustrates the maximum strain rates as a function of collision velocity for each material model. Detailed strain rate values are summarized in

Table 11. Comparison of max. strain (${\epsilon }_x$) between the experiment and FEA (with strain rate effect).

Collision Velocity (Knots)	Experiment (Avg.)	LEPP-CF	LEPP-DP	LEP-CF	LEP-DP
5 (2.57 m/s)	6.44 $\times {10}^{-4}$	3.30 $\times {10}^{-4}$(48.8%)	4.82 $\times {10}^{-4}$(25.2%)	7.98 $\times {10}^{-4}$(23.9%)	7.55 $\times {10}^{-4}$(17.3%)
7 (3.61 m/s)	8.85 $\times {10}^{-4}$	4.72 $\times {10}^{-4}$(46.6%)	1.07 $\times {10}^{-3}$(21.3%)	1.08 $\times {10}^{-3}$(21.7%)	8.26 $\times {10}^{-4}$(6.7%)
10 (5.13 m/s)	1.08 $\times {10}^{-3}$	5.26 $\times {10}^{-4}$(51.3%)	1.73 $\times {10}^{-3}$(29.6%)	1.24 $\times {10}^{-3}$(14.5%)	1.06 $\times {10}^{-3}$(2.0%)
Difference	-	48.9%	30.2%	20.0%	8.7%

and

Table 12. Comparison of max. strain (${\epsilon }_y$) between the experiment and FEA (with strain rate effect).

Collision Velocity (Knots)	Experiment (Avg.)	LEPP-CF	LEPP-DP	LEP-CF	LEP-DP
5 (2.57 m/s)	4.10 × 10−4	1.83 $\times {10}^{-4}$(55.4%)	1.95 $\times {10}^{-4}$(52.4%)	4.31 $\times {10}^{-4}$(5.03%)	3.32 $\times {10}^{-4}$(19.0%)
7 (3.61 m/s)	3.96 × 10−4	2.59 $\times {10}^{-4}$(34.6%)	3.78 $\times {10}^{-4}$(4.6%)	5.30 $\times {10}^{-4}$(33.9%)	3.92 $\times {10}^{-4}$(0.9%)
10 (5.13 m/s)	3.74 × 10−4	3.09 $\times {10}^{-4}$(17.4%)	4.67 $\times {10}^{-4}$(24.8%)	3.36 $\times {10}^{-4}$(10.0%)	4.39 $\times {10}^{-4}$(17.4%)
Difference	-	35.8%	27.3%	16.3%	12.4%

. Consistent with the results of the previous section, the DP yield model exhibited higher accuracy than the CF model. Among these models, the material model associated with the LPE model, which emphasizes plastic behavior, exhibited a lower deviation from the experimental results, with differences of 8.7% and 12.4%.

• Ice collision force prediction results

Figure 29 illustrates the history of collision force applied to the plate specimen as a function of collision velocity. The maximum strain rates for each material model are summarized in Figure 30 and

Table 13. Comparison of max. forces of the experiments and FEA (with strain rate effect).

Collision Velocity (Knots)	Experiment (Avg.)	LEPP-CF	LEPP-DP	LEP-CF	LEP-DP
5 (2.57 m/s)	32.4	17.8 (45.1%)	23.2 (28.4%)	47.6 (46.9%)	36.5 (12.7%)
7 (3.61 m/s)	45.3	28.7 (36.6%)	56.1 (23.8%)	49.9 (10.2%)	39.9 (11.9%)
10 (5.13 m/s)	70.4	31.8 (54.8%)	78.2 (11.1%)	61.5 (12.6%)	61.8 (12.2%)
Difference	-	45.5%	21.1%	23.2%	12.3%

. The DP yield model accurately predicted the ice loads applied to the plate specimen, outperforming the CF model. Furthermore, the material model associated with the LEP model, which accentuates plastic behavior, showed a deviation of 12.2% from the experimental results. This indicates a better fit compared to models that do not account for strain rate effects.

4.5. Evaluation of Suitability for Each Combination of Ice Material Models

In this section, we compare the analysis results from Section 4.3 and Section 4.4 with experimental data, summarizing the suitability of the analysis for each combination of material models. Below, the accuracies of the strain rate and collision force predictions for each ice material model are summarized. The strain rates for each material model were compared to identify models that closely match the experimental results. Figure 31 presents a comparison of the accuracy of the maximum strain rates in the width and height directions of the plate specimen as a function of collision velocity.

Table 14. Comparison between the experimental and FEA-derived maximum strain x (${\epsilon }_x$).

Collision Velocity (Knots)	LEP-CF	LEP-DP	LEP-CF + Strain Rate Effect	LEP-DP + Strain Rate Effect
5 (2.57 m/s)	3.44 $\times {10}^{-4}$(46.7%)	5.59 $\times {10}^{-4}$(13.2%)	7.98 $\times {10}^{-4}$(23.9%)	7.55 $\times {10}^{-4}$(17.3%)
7 (3.61 m/s)	4.65 $\times {10}^{-4}$(47.4%)	7.22 $\times {10}^{-4}$(18.4%)	1.08 $\times {10}^{-3}$(21.7%)	8.26 $\times {10}^{-4}$(6.7%)
10 (5.13 m/s)	5.37 $\times {10}^{-4}$(50.3%)	9.35 $\times {10}^{-3}$(13.4%)	1.24 $\times {10}^{-3}$(14.5%)	1.06 $\times {10}^{-3}$(2.0%)
Avg. Difference	48.1%	15.0%	20.0%	8.7%

and

Table 15. Comparison of maximum strain y (${\epsilon }_y$) from experiment and FEA.

Collision Velocity (Knots)	LEP-CF	LEP-DP	LEP-CF + Strain Rate Effect	LEP-DP + Strain Rate Effect
5 (2.57 m/s)	1.78 $\times {10}^{-4}$(56.5%)	2.96 $\times {10}^{-4}$(27.9%)	4.31 $\times {10}^{-4}$(5.03%)	3.32 $\times {10}^{-4}$(19.0%)
7 (3.61 m/s)	1.81 $\times {10}^{-4}$(23.1%)	2.90 $\times {10}^{-4}$(26.8%)	5.30 $\times {10}^{-4}$(33.9%)	3.92 $\times {10}^{-4}$(0.9%)
10 (5.13 m/s)	3.87 $\times {10}^{-4}$(3.4%)	3.80 $\times {10}^{-4}$(1.5%)	3.36 $\times {10}^{-4}$(10.0%)	4.39 $\times {10}^{-4}$(17.4%)
Avg. Difference	38.1%	18.7%	16.3%	12.4%

provide a comparison of the calculated and experimental values for the horizontal strain $\left({\epsilon }_x\right)$ and vertical strain $\left({\epsilon }_y\right)$ for each analysis combination. Here, we observed that dynamic material properties combined with the DP model (LEP-DP) accurately simulated ${\epsilon }_x$, with an 8.7% deviation, and ${\epsilon }_y$, with a 12.4% deviation. In summary, our findings demonstrated that considering dynamic effects yields better strain rate predictions, and when considering both width and height directions, the combination of the dynamic material properties of ice with the DP model is deemed the most appropriate model.

Subsequently, we turned our attention to the maximum collision force exerted by the ice on the plate specimen. The maximum collision force applied by the ice to the plate specimen was compared for different ice material property models. Figure 32 compares the predicted collision force for each material model combination with the experimental data.

Table 16. Comparison between the experimental and FEA-derived maximum force.

Collision Velocity (Knots)	LEP-CF	LEP-DP	LEP-CF + Strain Rate Effect	LEP-DP + Strain Rate Effect
5 (2.57 m/s)	19.3 (40.3%)	27.5 (15.1%)	47.6 (−46.9%)	36.5 (−12.7%)
7 (3.61 m/s)	21.4 (52.8%)	36.9 (18.5%)	49.9 (−10.2%)	39.9 (11.9%)
10 (5.13 m/s)	23.9 (66.1%)	52.8 (25.0%)	61.5 (12.6%)	61.8 (12.2%)
Avg. Difference	53.1%	19.6%	23.2%	12.3%

provides a comprehensive summary of the differences between experimental and analytical collision force for each material model combination. The inclusion of dynamic strain rate effects to the LEP material model with the DP yield condition (LEP-DP with strain rate effect) accurately predicts collision force with a 12.2% deviation, demonstrating high accuracy in both strain rates and collision force.

Overall, upon comparing the different numerical analysis configurations analyzed in this study, our findings demonstrated that the LEP material model, combining the DP yield condition and strain rate effects, best matches the experimental results.

Table 17. Comparative summary of the experimental and FEA-derived results.

FEA Result	Material Model	5 kts (2.57 m/s)	7 kts (3.61 m/s)	10 kts (5.13 m/s)	Mean Diff.
Max. strain x (${\epsilon }_x$) (Difference with experiment)	LEP-DP	5.59 × 10−4 (13.2%)	7.22 × 10−4 (18.4%)	8.26 × 10−4 (13.4%)	15.0%
	LEP-DP + Strain rate effect	7.55 × 10−4 (17.3%)	8.26 × 10−4 (6.7%)	1.06 × 10−3 (2.0%)	8.7%
Max. strain y (${\epsilon }_y$) (Difference with experiment)	LEP-DP	2.96 × 10−4 (27.9%)	2.90 × 10−4 (26.8%)	3.80 × 10−4 (1.5%)	17.7%
	LEP-DP + Strain rate effect	3.32 × 10−4 (19.0%)	3.92 × 10−4 (0.9%)	4.39 × 10−4 (17.4%)	12.4%
Max. force (Difference with experiment)	LEP-DP	27.5 (15.1%)	36.9 (18.5%)	52.8 (25.0%)	19.6%
	LEP-DP + Strain rate effect	36.5 (12.7%)	39.9 (11.9%)	61.8 (12.2%)	12.3%

summarizes the comparison between both strain rates and collision force values obtained through experiments and numerical modeling. Notably, the strain rate effects in the analysis results were very similar to the experimental results, with deviations of 8.7% for ${\epsilon }_x$, 12.4% for ${\epsilon }_y$, and 12.2% for collision force. Therefore, our findings confirmed that the ice material properties formulated by combining the DP yield condition and strain rate-yield strength ratio within the LEP material model best approximate the results of the ice collision experiments.

5. Conclusions

This study reviews various material models and numerical simulation methodologies relevant to the impact of ice on structures, with a focus on developing a sophisticated numerical simulation model capable of accurately capturing the pressure characteristics induced by ice impacts. Using dynamic explicit finite element analysis, this study incorporated the elastic-plastic material properties of ice specimens, the effect of strain rate, yield condition, and the fracture behavior of materials. Initial analyses were conducted to validate necessary material constants for each ice material model, serving as a foundation for subsequent finite element simulations of ice impact scenarios. These simulations were iteratively refined to align closely with empirical impact load. The analysis process entailed exploring combinations of various material models. These included the Linear Elastic-Perfectly Plastic relationship, the Linear Elastic-Plastic relationship, the Crushable Foam model, and the Drucker–Prager model, along with considering the strain rate dependency of ice. Collision simulations were instrumental in computing the strain and reaction forces of plate specimens, which were then compared with values measured in collision experiments. Our findings suggest that a material model which incorporates the Drucker–Prager yield criterion and a dynamic yield ratio within a Linear Elastic-Plastic framework most accurately reflects the experimental results. In addition, this study emphasizes the need for future experiments across a wider range of velocities and with thinner plate materials to advance our understanding of ice impacts on structures and the potential for plastic deformation. This study sought to validate the feasibility of simulating experiments with different material models of ice, which are essential for the analysis of ice–structure interactions. The validation of this numerical model enhances the predictive capabilities for offshore structural loads and allows for the derivation of more realistic ice material models through empirical comparisons.