Investigations on Flexural Strength of a Columnar Saline Model Ice under Circular Plate Central Loading

Abstract

The properties of ice strength have a significant impact on the design and safety of structures in ice-infested waters. To analyze the flexural strength of columnar saline model ice, we conducted circular plate center loading tests at the Small Ice Model Basin of the China Ship Scientific Research Center (CSSRC SIMB) in China. The tests involved varying the loading rate and ice temperature, and a numerical model was developed using FEM and LS-DYNA for validation and comparison. The results of the tests revealed the crack propagation process, stress distribution, load response, and failure mode of the model ice. The model ice displayed typical brittle failure, and the flexural strength was linearly related to ice temperature but not significantly correlated with loading rate. The porosity of the model ice affected the load response and time of failure but not the failure mode. The model ice with 7% porosity had a 7.8% reduction in load response compared to the nonporous model ice. This study provides a reliable method for measuring and analyzing the flexural strength of model ice. It also serves as a foundation for further research on the interaction between structures and ice sheets.

1. Introduction

In regions where ice is present, such as beneath a floating object, the ice sheet may fracture under vertical forces. This circumstance has gained significance in the fields of polar exploration and engineering [1]. The durability of the ice plays a crucial role in the design and assessment of structures regarding safety. The upward pressure exerted by the floating entity can trigger a range of failures, including shearing, bending, and flexural failure, which is the most frequently observed [2,3]. Conducting tests to determine flexural strength is essential for accurately predicting ice loads on structures and ensuring their safety. This information is critical in developing and assessing structures for floating objects under ice.

There exist three primary methods to assess ice strength: in situ cantilever tests, simply supported beam tests, and circular plate center loading tests [4,5,6]. An in situ cantilever test involves obtaining an ice beam directly from the ice layer and conducting the test while the ice remains in its natural environment [7]. Frederking and Timco executed in situ cantilever tests on model ice at the National Research Council Hydraulics Laboratory, discovering no significant relationship between loading rate and flexural strength [8]. Krupina also conducted in situ cantilever tests on sea ice in the Barents Sea, obtaining the distribution of sea ice flexural strength in the region [9]. Gang of CSSRC conducted an in situ cantilever beam test on model ice, revealing that the flexural strength of model ice remains constant as the loading rate changes but decreases with increased rewarming time [10]. The simply supported beam test can be split into three-point bending and four-point bending tests [11]. Ji et al. executed an indoor three-point bending test on the flexural strength of Bohai sea ice and found that the sea ice flexural strength had a linear relationship with the loading rate [4]. Gagnon conducted a four-point bending test on iceberg ice and found that the flexural strength of ice decreases with increased ice temperature [12]. Barrette et al. performed a four-point bending test on iceberg ice and found that the larger the average grain size inside the ice, the smaller its flexural strength [13]. Lastly, in circular plate center loading tests, icicles are cut from the ice layer and then shaped into disks according to predetermined thickness–diameter ratios. Krupina conducted an extensive series of circular plate center loading tests on sea ice situated in both southeastern and northeastern regions of the Barents Sea. The study aimed to investigate the effects of different ice properties, such as temperature, salinity, and brine volume, on flexural strength. Furthermore, the plate test outcomes were juxtaposed with those of the cantilever test, and a suitable correlation was established between the two [9]. Marchenko conducted circular plate center loading tests on sea ice in the northwest Barents Sea and found that flexural strength depends on ice temperature [6].

In conclusion, ice is a complex crystalline material, and its flexural strength is influenced by factors such as the structure, temperature, and external loading conditions of ice crystals [14].

In addition to experimental studies, numerical simulations can also be utilized to investigate the flexural strength of ice. The discrete element method, the finite element method, and the peridynamics method are the most popular simulation methods used today. The discrete element method is suitable for the mechanics of discontinuous media [15], but its application for dynamic failure problems is limited due to result deviation in the continuum stage [16]. On the other hand, the finite element method has advantages in terms of fast calculation efficiency and high accuracy and is widely used to simulate ice–structure interaction [17]. However, the simulation accuracy of the finite element method depends on the material model chosen [18]. The peridynamics method analyzes the mechanics of a continuum with the help of the point method of matter and the idea of molecular dynamics. It constructs the object’s motion equation in the form of an integral [19]. However, compared with other numerical calculation methods, the peridynamics method also has the problem of low computational efficiency. When simulating the interaction between sea ice and structure, most scholars consider ice as a continuum, considering its elasticity and plasticity. For example, Li used an isotropic elastoplastic fracture model to simulate ice based on LS-DYNA [20]. Some scholars incorporated the effects of temperature and strain rate and constructed a numerical model of ice based on the multi-surface failure criteria of sea ice. Still, there were some differences between the test and simulation results [21]. Other scholars considered the existence of pores inside the ice, such as Von Bock und Polach et al., who applied LS−DYNA and selected the Lemaitre damage model to simulate ice and analyzed the dependence of the load response on the microstructure of ice [22]. To maintain the quality and energy of the model, some scholars used the node-splitting technique instead of the commonly used elemental erosion technique [23].

Despite the wealth of experimental and numerical data on in situ cantilever and simply supported beam tests, there has been limited research conducted on circular plate center loading tests. The International Towing Tank Conference’s mechanical properties test method for model ice also lacks relevant specifications for circular plate center loading tests, indicating the need for improvement in this area. As mentioned earlier, the flexural strength of ice is influenced by various factors, such as temperature and loading rate, and further experimental exploration is necessary. Additionally, there is a need for expanded research based on the material model, particularly regarding the influence of internal porosity on ice strength.

In this paper, the influence of loading rate and temperature on the flexural strength was explored by using a saline model ice under a circular plate center loading test, and the simulation of the porosity of ice was also realized to study the influence of the internal porosity of the model ice. The results of center loading on a circular plate of ice were discussed, and, in particular, the physical mechanisms of crack propagation were analyzed. This study provides a reference for the prediction of the ice’s mechanical characteristics.

2. Methods

Both laboratory experiments and numerical simulations were conducted in the present study, and the implementation of the two methods is briefly summarized below.

2.1. Laboratory Experiments

2.1.1. The Columnar Saline Model Ice

The SIMB of the CSSRC is an ice water tank with 8 m in length, 2 m in width, and 1 m in depth, as shown in Figure 1 [24]. The ice-making process used in the SIMB is similar to but partly improved from the method by the Evers [25,26] of the Hamburg Tank (HSVA) in Germany. Cooling fans are suspended from the roof, ceiling panels with tiny holes are installed underneath to exhaust cooling air, and circulating fans with guide plates are set on the side walls to suck the air used for cooling; thus, forced cooling air circulation is formed [27].

Model ice is made of sodium chloride solution, and the preparation process is divided into precooling, crystallization, and ice making. To control the density and to lower the strength, an underwater air bubbling system is used to produce tiny bubbles that are trapped in the model ice during the freezing process. Its mechanical properties can be adjusted by rewarming and underwater microbubble generation systems [10]. After rewarming, the temperature of model ice is usually around −0.8 °C.

Through a series of experimental measurements, the mechanical properties of model ice in the SIMB were statistically analyzed. The fresh model ice’s main characteristic parameters (−0.8 °C) in the SIMB are summarized in

Table 1. The main physical and mechanical performance parameters of model ice in SIMB [24].

Parameter	Value
Density/g·cm−3	0.89~0.91
Thickness/mm	10~100
Flexural strength/kPa	94.7
Compressive strength/kPa	80~150
Elastic modulus/MPa	250~450
Elastic modulus/flexural strength	800~1500

. With the continuous exploration of the experimenters, it was found that after adding bubbles to the saline model ice, the brittleness of the ice was greatly improved. The flexural fracture mode of the ice and the distribution ratio of the crushed ice flakes of various scales were consistent with natural sea ice, and the similarity could be maintained in terms of physical and mechanical properties and fracture behavior patterns [28].

2.1.2. Circular Plate Center Loading Tests

In a circular ice plate indentation test, vertical loads are exerted on the central area of the disk ice specimen under the circumferentially uniform support, as shown on the schematic diagram in Figure 2.

When the plate is thin enough, it belongs to a typical plate–shell structure according to the theory of shell mechanics [29]. The maximum bending moment of the circular plate is at the center of the circular plate with radius R when the load q is uniformly distributed in a circle of radius r. By integrating the deflection caused by the load on the torus plane with radius b and width db (Figure 2b), the deflection of the center of the plate can be obtained, and then the bending moment M_max at the center of the plate can be determined [28]:

$$\begin{gathered} M_{\max }=q{\displaystyle {\int }_0^r\left(\frac{1-v}{4}\frac{R^2-b^2}{R^2}-\frac{1+v}{2}\ln \frac{b}{R}\right)}bdb \\ =\frac{F}{4\pi }\left[(1+v)\ln \frac{R}{r}+1-\frac{(1-v)r^2}{4R^2}\right] \end{gathered}$$

(1)

where F represents the total load πr²q. ν = 0.33, is the Poisson’s ratio [30]. The relationship between the bending moment and stress is:

$$\sigma =\frac{6M}{h^2}$$

(2)

In the above equation, h is the thickness of the plate, so the maximum tensile stress in the plate, that is, the flexural strength σ_f, is:

$${\sigma }_f=\frac{3F}{8\pi h^2}\left[4-(1-v){\left(\frac{r}{R}\right)}^2-4(1+v)\ln \left(\frac{r}{R}\right)\right]$$

(3)

The relationship between the central bending moment and curvature obtained from the pure bending hypothesis is the basis of the above derivation, and the effect of shear forces on the flexural strength in a plane parallel to the plate surface is not considered. The value of σ_f so far is only an approximate solution, and its accuracy depends on the ratio of the thickness of the disk to the outer diameter [31]. When the thickness–diameter ratio is less than 0.15, the effect of shear force on the bending strength is negligible [6].

2.1.3. Test Conditions

The test was carried out in a small cold laboratory adjacent to the ice water basin. The flexural strength of ice is affected by a variety of factors [1], and the loading rate and ice temperature were selected as the control factors of the present test. Different test conditions are listed in

Table 2. Summary of plate specimen size and testing conditions in laboratory experiments.

Diameter/Thickness (mm/mm)	Specimen Number	Loading Rate (mm/min)	Ice Average Temperature (°C)	Laboratory Temperature (°C)	Measurements
140/20	20	100, 150, 200, 250, 300	−0.8	−0.8	Peak force Failure time
140/20	15		−2.0	−2.0
140/20	15		−4.0	−4.0
140/20	15		−6.0	−6.0
140/20	15		−8.0	−8.0

. The peak force and failure time of the circular plate indentation test are two important measurements. The average temperature of the newly made model ice was about −0.8 °C, the density was 0.91 g·cm⁻³, and the salinity was 2.75 ppt.

The test process was divided into three phases: the test preparation stage, the plate specimen sampling and storage freezing stage, and the measurement and data recording stage. In the test preparation stage, the small cold laboratory was first cooled to reach the target air temperature, and the high-speed camera, electronic universal testing machine, thermometer, and other related instruments were precooled. In the plate sampling and storage freezing stage, a total of 80 plate specimens were drilled from the flat model ice sheet. The diameter of the ice plate specimen was 140 mm, and the thickness was 20 mm (Figure 3). It produced a thickness–diameter ratio of less than 0.15, agreeing with the calculation requirements. After sampling, 20 newly made model ice specimens were placed in a small cold laboratory for test measurement, and the rest were stored in a refrigerator for subsequent tests. In the last test and measurement stage, the loading rate of the electronic testing machine was first adjusted according to the requirements of the test conditions. Then the force curve of the ice specimen from the initial bearing to the flexural failure was recorded. The high-speed camera was also used to capture the failure details of the ice specimen. The thickness and temperature of model ice were quickly measured and recorded after failure.

In the data recording stage, the force curve of the ice specimen from the initial loading to the flexural failure was recorded with a force measurement system, and the flexural strength was obtained according to the peak force. At the same time, the failure details of the ice specimen were documented with a high-speed video camera. The overall layout scheme of the instruments is shown in Figure 4, and the parameters of the test instrument are shown in

Table 3. Test instrument parameter table.

Instrument	Accuracy
Thermometer	0.01 °C
Electronic testing machine	1 mm/min
High-speed cameras	Frame rate: 2800
Sensor	0.01 N
Vernier calipers	0.01 mm
Ice density measurement instruments Salinity meter	0.01 g/cm3 ±3% (FS)

.

2.2. Numerical Modeling

2.2.1. Numerical Model of Ice Material

A big advantage of the finite element method is that many contact algorithms allow the coupling of ice and structural models. Therefore, in this paper, the finite element solver LS-DYNA was used to establish a model ice failure model. The circular plate center loading tests were used to vertically load the central area of the disk ice specimen under the circumferentially uniform support. The contact between the indenter and the ice disk specimen occurred during loading, and the explicit nonlinear finite element method can be used to solve such contact problems. The calculation method used by the nonlinear finite element program is the explicit integration method [32].

When using the finite element method to simulate the model ice mechanical test, it is necessary to determine the material model parameters that match the macroscopic characteristics of the material. Standard model ice constitutive models include the elastoplastic model, elastic brittleness model, etc. According to Karr and Choi [33], model ice materials are considered isotropic in their undeformed state. This assumption was adopted in this paper, so the isotropic elastoplastic fracture model (*MAT_ISOTROPIC_ELASTIC_FAILURE) in LS-DYNA was selected to simulate the model ice. The failure criterion for the material is the Von Mises yield criterion [34].

The material parameters of the model ice are shown in

Table 4. Model ice material parameters.

Material Properties	Value
Density/g·cm−3	0.92
Shear modulus /MPa	76.9
Plastic hardening modulus /MPa	94.7
Yield stress/kPa	83
Bulk modulus/MPa	75
Plastic failure strain Failure pressure/kPa	0.05 −110

. The specific parameters of density, plastic hardening modulus, and plastic failure strain were obtained by the model ice mechanical test [1]. According to the relevant ice mechanics numerical simulation, the ranges of other material parameters were obtained, and finally, the specific parameters were given by trial and error.

According to von Bock und Polach [22], the air pores inside the model ice can be simulated by deleting elements by using a random algorithm, and the number of deleted elements depends on the porosity. This is another advantage of numerical simulation as compared with the experiments in Section 2.1, because ice porosity is difficult to control according to predetermined values, although an underwater air bubbling system has been placed already. The so-called K file of the model ice numerical model was employed and set the random number function, which was used to delete a part of the model ice elements according to the input value of porosity. The flow chart is shown in Figure 5. There are several key points in this program. First, read the K file, defining each line of information in it as a string. Determine whether each line string represents the coordinate information of the cell; if so, proceed to the next step, and vice versa, and output to a K file. Secondly, enter the sampling rate, combine the random function to obtain a set of unit numbers that need to be deleted, correspond them one-to-one with the read unit string, replace the unit numbers that need to be deleted with spaces, and output them to a new K file.

The corrected ice model with a different porosity is shown in Figure 6. Since the elements used in this model were hexahedral, the resulting pore shape was also hexahedral. In fact, model ice with different porosities differs in the number of pores, but the size of the individual pores does not change, which is the same as Von Bock und Polach [22]. The intervals between pores are random in Figure 6 and may be different from the actual internal pore distribution of ice because the brine channels of actual sea ice tend to be vertical and continuous, but the pores in numerical models cannot guarantee continuity. In addition, the distribution of pores is different from the actual sea ice, the pores of the actual sea ice tend to exist only in the interior, and the surface is continuous. Still, the pores in this paper also exist on the surface. The resulting difference in loading force is hard to evaluate at present, but it is supposed to be ignorable because the given porosity is small (≤7%).

2.2.2. Numerical Simulation of Circular Plate Center Loading Tests

In the numerical calculation, the ice specimen was loaded similarly to that shown in Figure 2. During modeling, the annular support below the ice specimen was set as a fixed boundary, the cone indenter had only the degrees of freedom in the Z direction, and the load was applied downward on the upper surface of the ice specimen at a constant loading rate. The main parameters of the numerical model are shown in

Table 5. Main parameters of the numerical model of the circular plate center loading test.

Ice Specimen Radius (mm)	Ice Specimen Thickness (mm)	Ring Support Outer Diameter (mm)	Ring Support Inner Diameter (mm)	The Radius of the Lower Surface of the Tapered Indenter (mm)
70	20	70	60	5

. To simulate the fragmentation phenomenon of the model ice, a fine mesh was used in the central area of the model ice, and a large gradient mesh was used in the outer part, as shown in Figure 7a. Through the circular plate center loading tests, the numerical calculation condition was determined. Combining this with the pore simulation principle of Section 2.2.1, a model ice with a porosity of 3% can also be obtained, as shown in Figure 7b.

A rigid body model was selected to simulate the annular support and cone indenter, and the material parameters were defined using the keyword *MAT_RIGID, as shown in

Table 6. Ring support and indenter material parameters.

Material Properties	Value
Density/g·cm−3	7.83
Elastic modulus/GPa	207
Poisson’s ratio	0.33

.

LS-DYNA provides a variety of contact algorithms for explicit analysis, divided into single-sided contact, point-to-face contact, and face-to-face contact [35]. The contact between the subglacial surface of the model and the upper surface of the annular support had a large contact area and symmetrical shape, so the surface contact was selected. Since the ring support was modeled using shell elements, the Automatic Contact Algorithm in surface contact was chosen. It can consider the influence of element thickness, allowing contact to appear on both sides of the shell element, making it more accurate when calculating contact forces. The contact between the upper surface of the model ice and the lower surface of the cone table indenter also adopted surface contact, and the erosion contact algorithm in surface contact was selected because the ice breakage effect of the erosion contact simulation model is good. Erosion contact was employed to control time steps. It automatically invoked negative volume failure criteria for all solid elements in the model, which circumvents procedural errors due to negative volumes by removing solid elements that produce negative volumes.

Different test conditions in numerical simulations are listed in

Table 7. Summary of plate specimen size and testing conditions in numerical simulations.

Diameter/Thickness (mm)	Specimen Number	Loading Rate (mm/min)	Ice Porosity (%)	Measurements
140/20	4	150	0	Peak force Failure time
140/20	4		3
140/20	4		5
140/20	4		7

. Compared with the test conditions in laboratory experiments (Table 2), the impact of ice temperature was ignored. In fact, the influence of temperature on the properties of ice materials is difficult to achieve by numerical simulation. Therefore, to reduce the influence of temperature on the results, the input parameters of the ice material in the numerical model were the average mechanical parameters of the model ice at a specific temperature (−0.8 °C). Similarly, in numerical simulation, the flexural strength as an input quantity, and the influence of external parameters’ (porosity) changes on the flexural strength cannot be reflected, and the influence of external parameters on bending performance can only be analyzed through the change in load response.

3. Results and Analysis

The test results and the numerical simulation results were compared from the two aspects of the time history curve and damage phenomenon.

3.1. Time History Curve and Failure Mode of Model Ice

The numerically calculated time history curve was compared with the model ice mechanics test time history curve in Figure 8. During the bending process of the ice specimen, both time history curves showed linear changes without an obvious yield stage and increased sharply from 0, reaching the maximum value Pmax within only 0.8 s, and it can be seen that the failure mode of the ice specimen was an obvious brittle failure. The flexural strength σ_f of the model ice can be calculated from Pmax according to Equation (3). The peak force of the two was similar, the loading time difference was 0.01 s, and the absolute error did not exceed 5 N, so the rationality of the calculation model can be verified from the changing trend of the time history curve. However, due to differences in material properties and test conditions, the downward trend of the test curve and the numerical curve was not the same. The downward trend of the test curve was faster because in the model test, when the sensor read the maximum force peak, the indenter stopped pressing down and upward, and the numerical calculation did not simulate this behavior, but this did not affect the numerical model’s simulation of the mechanical properties of the model ice.

A high-speed camera was used to observe the state of the model ice at the four moments A, B, C, and D in Figure 8, and the flexural failure process of the model ice was captured, as shown in Figure 9. It is clear that the model ice broke along the diameter, and the crack propagation was rapid, with an expansion time of less than 0.06 s. The model ice had no prominent yield stage, and there was no obvious plastic deformation at the failure location. It can be deduced that the failure of the model ice belongs to a brittle failure. Compared with the experimental phenomenon in the breakthrough loads of floating ice sheets carried out by Sodhi [36], the model ice disk did not have holes in the middle area but just failed cracks along the radial direction. This may be because of the different constraints.

The experiment could not observe the failure pattern of the bottom during the model ice fracture failure, but numerical simulations supplemented this part of the study. By observing the bottom of the model ice at the three moments E, F, and G in Figure 8, it was found that the model ice started to crack from the center of the bottom (Figure 10). The crack extended firstly along the radius direction to the bottom boundary, and then extended along the thickness direction to the top surface until the model completely failed, as shown in Figure 10. The flexural failure of the model ice was essentially the tensile failure of the bottom of the model ice, consistent with the flexural failure of the sea ice proposed by Lainey [37]. It indicates that the numerical model of the model ice can better reflect the crack propagation during model ice breakage.

Meanwhile, the damage phenomenon of the model ice was compared between numerical simulations with the picture captured by the high-speed camera in Figure 11. When the ice specimen failed, under the action of the cone indenter, there was a significant deflection change at the center position of the model ice, and the crack propagation was more consistent. It can be considered that the numerical simulation results and the experimental results have good similarity, and the conclusions obtained can confirm each other’s results. The impact factors on model ice flexural strength are discussed in two different ways, and these conclusions are considered to apply to both methods.

The numerical simulation also studied the stress distribution of the model ice at the time of cracking, and the stress distribution is shown in Figure 12. The stress on the top was 249.6 kPa, and the stress on the bottom was 71.3 kPa. The stress on the top was about 3.5 times the stress on the bottom. In fact, under the vertical action of the indenter, the stress on the top was caused by extrusion, and the stress on the bottom was caused by tension. Figure 10 shows that the destruction of the model ice occurred firstly on the bottom surface because the model ice is a brittle material: its tensile strength is much lower than its compressive strength [38]. Thus, the failure usually occurs at the stretch of the bottom surface.

3.2. Impact Factors on Flexural Strength

3.2.1. The Effect of Loading Rate

Discussions in this section are based on model test results because the behavior of the indenter was not fully represented in numerical simulations, considering that this may interfere with the conclusions drawn for the variable loading rate. Figure 13 shows a typical time history curve for model ice at different loading rates (−0.8 °C). Under different loading rates, the loading force of the model ice failure did not change much. When the loading force increased to the peak point, the time history curve decreased rapidly, and the downward trend was basically the same, which further indicates that the failure of the model ice was a brittle failure. It can be seen from the figure that the time difference between the peaks of the time history curves at different loading rates was obvious, followed by the insignificant change in the peak of the time history curve.

Based on experimental data, the correlation between loading rate and flexural strength was analyzed using the Pearson correlation coefficient and the Spearman correlation coefficient. Among them, the Pearson correlation coefficient can reflect the degree of linear correlation between two random variables; the Spearman correlation coefficient measures the strength of monotonicity between variables. Through calculation, it was found that the Pearson correlation coefficient between the flexural strength and loading rate of the model ice was 0.33, and the p-value was greater than 0.05, indicating that there was no significant linear correlation between the loading rate and the flexural strength of the model ice. The Spearman correlation coefficient between the flexural strength and loading rate was 0.37, and the p-value was greater than 0.05, indicating that there was no significant monotonic correlation between the loading rate and the flexural strength of the model ice, as shown in Figure 14. Therefore, it can be judged that there was no obvious correlation between the ice flexural strength and the loading rate of the model. In fact, in the other four test groups at temperatures of −2 °C, −4 °C, −6 °C, and −8 °C, the relationship between loading rate and flexural strength also showed no significant correlation, and thus was not shown here. This agrees with the conclusion of Frederking et al. in an ice basin [8]. Similarly, in polar sea ice observations, it has been found that the loading rate has less influence on the flexural strength of the ice [39].

3.2.2. The Effect of Ice Temperature

As mentioned above, since it is difficult to show the influence of temperature on the mechanical properties of ice materials in numerical simulations, this section is also based on the experiments. This section quantifies the relationship between model ice flexural strength and model ice temperature. The temperature range of the model ice was −0.8 °C to −9 °C, as can be seen from Section 3.2.1.

According to the statistical test data, the Pearson correlation coefficient between ice temperature and flexural strength was −0.89, and the p-value was 3.15 × 10⁻²⁴. The Spearman correlation coefficient was −0.78, and the p-value was 7.1 × 10⁻¹³. Curve fitting to the experimental data was conducted, with a 95% confidence band, as shown in Figure 15, and the regression equation is as follows:

$${\sigma }_f=192.24-204.36·T$$

(4)

where σ_f is the model ice flexural strength in kPa, and T is the model ice temperature. It is clear from Figure 15 that in the temperature range from −0.8 °C to −9 °C, the model ice flexural strength decreased linearly with ice temperature, consistent with previous results of sea ice. Ding et al. measured the flexural strength of Bohai sea ice, and the experimental data showed that the flexural strength increased with the decrease in ice temperature, and the relationship between the two was linear [40]. It is worth noting that in the −2 °C to −6 °C range, there are some data points that deviate from the 95% confidence band, possibly due to errors in the measurement of ice temperature. Marchenko [6] tested the flexural strength of sea ice in the northwestern part of the Barents Sea and obtained the relationship between flexural strength in MPa and temperature and salinity, as follows:

$${\sigma }_f=0.236-0.095·S-0.0134·T$$

(5)

where S is the salinity of sea ice. The empirical formulas for the flexural strength of Barents Sea ice also show ice temperature has an approximate linear relationship with the flexural strength if a constant or average salinity is given. However, it should be especially noted that the model results and the results of the flexural strength measurement in marine conditions in absolute values cannot coincide, as well as other strength characteristics.

3.2.3. The Effect of Ice Porosity

Since ice crystal observation experiments are labor-intensive and time-consuming, and the expected porosity is difficult to control artificially, this section is based on numerical simulation results. The relationship between porosity, the model ice load response, and the model ice failure mode were studied through four sets of numerical simulations with different porosities, as listed in Table 7. As mentioned earlier, the flexural strength was used as a parameter input to the numerical model, so the influence of porosity on the bending properties of the model ice cannot be discussed in this section through the flexural strength but only by the load response. It was found that the load response of the model ice flexural failure was in the range of 24.44 N~26.5 N, as shown in Figure 16. The greater the porosity, the smaller the load response of the model ice, and the earlier the time for failure to occur. Statistics show that the load response of model ice with 7% porosity was 0.04 s earlier than model ice without porosity. In addition, the peak load decreases gradually with increasing porosity, and this conclusion is similar to Wang’s conclusions about sea ice at Prydz Bay, East Antarctic [41].

The final failures of model ice at different porosities are also shown in Figure 17. A crack extending along the diameter appeared on the surface of the model ice under different porosities. There was a significant deflection change at the center of the model ice, and there was a gap in the thickness direction. This indicates that within the range of porosity < 7%, porosity had little effect on the failure mode of the model ice. Compared with nonporous model ice, model ice with pores was more likely to develop microcracks around the pores or further expand the pores. There was a certain crush failure near the indenter, but the failure mode of the model ice was still dominated by flexural failure.

4. Conclusions

In this paper, both laboratory experiments and numerical modeling were carried out to investigate the flexural strength of a columnar saline model ice under circular plate central loading. The influence of different factors on the flexural strength of model ice was analyzed and the crack propagation law of the model ice flexural fracture failure was found. The main conclusions are as follows:

• The experimental and numerical results were compared from two aspects including the time history curve and damage phenomenon, and their results agree well; these could reflect the flexural strength characteristics of the model ice and confirm each other’s results.
• According to the time history curve of the ice specimen from the initial bearing to the flexural failure, it was found that the ice specimen had no obvious yield stage. The high-speed camera observed no obvious plastic deformation at the failure location. The model ice began to crack from the center of the bottom surface, and the crack extended along the radius direction to the lower surface boundary, then extended along the thickness direction to the top surface until complete failure. The failure process of the model ice was judged to be a typical brittle failure.
• There was no significant correlation between the loading rate and the flexural strength. A significant linear correlation between the model ice temperature and the flexural strength was explored, and the flexural strength of the model ice increases continuously with the continuous decrease in the model ice temperature in the range from −0.8 °C to −9 °C. The larger the porosity, the smaller the load response of the model ice, and the earlier the time of failure. Compared with the nonporous model ice, the load response of model ice with 7% porosity was reduced by 7.8%, and the failure time was 0.04 s earlier. Within the range of 7%, ice porosity had little effect on the failure mode of the model ice.

This paper provided a feasible means to measure and predict the mechanical properties of model ice and realized the simulation of the internal pores of the model ice, forming a preliminary foundation for the research of the mechanism of ice loading under the vertical interaction between the structure and the ice cover. In the future, the results obtained by different methods such as circular plate center loading tests, in situ cantilever tests, and simply supported beam tests will be enriched and compared with each other, to establish a systematic ice flexural strength measurement and analysis technology. In addition, the numerical model will continue to be optimized and improved to reflect the influence of temperature and brine, especially the internal structure and anisotropic characteristics of the model ice.