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Article

Distribution Analysis of Local Ice Pressures in the Indentation Test at Various Velocities

1
Department of Mechanical Engineering, Dalian University of Technology, Dalian 116000, China
2
School of Ocean Science and Technology, Dalian University of Technology, Panjin 124000, China
*
Author to whom correspondence should be addressed.
J. Mar. Sci. Eng. 2022, 10(3), 433; https://doi.org/10.3390/jmse10030433
Submission received: 21 January 2022 / Revised: 9 March 2022 / Accepted: 15 March 2022 / Published: 16 March 2022
(This article belongs to the Section Ocean Engineering)

Abstract

:
When sea ice acts on vertical structures, there are much higher pressures in localized areas known as high-pressure zones (HPZs) than in other areas. The damage failure mode of sea ice varies with the sea ice velocity and affects the distribution of HPZs. In this study, an indentation test that drives ice sheet interaction with a vertical rigid plate (indentor) was designed, and a pressure sensor (consisting of 32 × 32 small pressure units of 100 mm2) was installed on the indentor face to measure the local ice pressure (LIP) at various velocities. The distribution of the LIPs along the ice thickness, the probability distribution of the LIPs and the distribution relationship of the LIPS in space and time were obtained from the measurement. The results show that the LIPs were mainly distributed in the middle of the sea ice, which is consistent with full-scale observations and previous research. Ductile failure of the sea ice results in a larger LIP distribution area than brittle failure at the same threshold kt (kt = σLcr). The probability distribution of the LIPs decreases exponentially with increasing pressure and follows a lognormal distribution. The maximum LIP appears at the peak moment of the global force when the sea ice failure mode is mainly ductile failure. However, the maximum LIP may not occur at the peak moment when the sea ice failure is mainly brittle failure and, instead can appear at any moment in the global force time history curve. The HPZ (which is larger than 7/8 times the maximum LIP) area is less than 2% of the nominal contact area at various velocities. The influence of the sea ice velocity on the spatial and temporal distribution of LIP is analyzed, and the results provide a reference for designing structures with local strength in ice regions.

1. Introduction

The sea ice failure mechanism is a complex process when it interacts with vertical structures. The failure mode is affected by the interaction velocity, sea ice properties and structural characteristics. Uniaxial compression tests on sea ice reveal brittle failure behavior at low strain rates (less than 10−4 s−1) and ductile failure behavior at high strain rates (greater than 10−2 s−1) [1,2]. Different failure modes directly affect the local ice pressures (LIPs). Observations show that the pressure is much higher in localized areas than elsewhere when sea ice acts on vertical structures [3,4,5,6], and it is necessary to clarify the local ice force when designing structures. Because effective models for the distribution characteristics of high-pressure zones (HPZs) (value, area size, number, location, etc.) are generally lacking, in structure design, the spatial average of the global force is typically used as the ice pressure that acts on the nominal contact surface [7,8].
The effect of HPZs on the structural local response is an important factor to consider. HPZs are highly random in both time and space and result in a significant maximum LIP [7]. The following factors may account for this situation: (1) the contact between the sea ice and the structure surface is non-simultaneous, and the sea ice failure is also non-simultaneous; and (2) the breaking strengths of the sea ice at local contact points are not necessarily the same. Based on experimental and field-measured ice force data, Sanderson [9] analyzed the trend of decreasing effective pressure with increasing contact area. Within addition to the data summarized by Sanderson, many scholars have studied the relationship between local ice pressure and local area and obtained their own results. Most of the results show that LIPs decrease with increasing contact area [10,11,12,13,14]. The relationship between LIP and contact area can be described by a power law. The maximum LIP occurs at a small point in the nominal contact area. When the width of the structure is much greater than the thickness of the sea ice, the failure of the sea ice in these two directions will differ due to different lateral ice restrictions. Under the action of the lateral limiting force, the extrusion and crushing characteristics are changed such that the HPZs are distributed in the center of the ice thickness, and the LIP near the free surface is lower. Some scholars have described this distribution of HPZs, which is mainly in the middle of the ice thickness, as line-like contact, and some methods to explain this characteristic have been proposed [3,5,15,16]. Jordaan [16] conducted a detailed analysis of the influence of the ice breaking process on LIPs and found that the ice has a triaxial stress state that varies from low confinement near the edges to high contact pressures near the center. Dempsey et al. found that the HPZs seem to be confined to the middle third of the ice thickness (ice thickness is less than 0.6 m) [3].
Considering that the failure mode of sea ice is necessary for studying the scale effect of sea ice, at different ice velocities, sea ice exhibits ductile failure or brittle failure. Sodhi proposed that the change in effective ice pressure is mainly affected by the failure mode of the sea ice; that is, whether the sea ice experiences ductile failure or brittle failure [5]. Full-scale tests and small-scale tests have shown that the failure mode of the sea ice, rather than the size of the contact surface, effectively reflects the interaction between the ice and the structure. At low loading speeds, the sea ice is mainly in ductile failure mode, with one or more cleavage cracks forming at different depths of the ice sheet, and the sea ice breaks into fragments and powder and is extruded from the upper and lower surfaces [17,18,19,20,21]. At high loading speeds, the sea ice is mainly in brittle failure mode, with the sea ice forming wedge-shaped crushing cracks. Then, the sea ice splits and spalls on the upper and lower surfaces due to the cracks [20,22,23]. This results in different pressure distributions. Taylor et al. proposed a probabilistic HPZ modes and a probabilistic fracture mechanics model for local pressures based on Japan Ocean Industries Association (JOIA) medium-scale test data [8,24]. The loading speed considered in their studies are 3 mm/s and 30 mm/s. More works about LIP spatial and time distributions relationship and area sizes at various velocities should be considered.
The above discussion indicated that the load speed is one of the main factors affecting the distribution of LIPs. The study discussed provides a reference for further research on HPZs. Based on the different failure modes of sea ice at various loading speeds, this paper studies the local ice pressure distribution of sea ice ductile failure at low ice velocities and brittle failure at high ice velocities. The effects of different failure modes on the distribution of local ice pressure on ice thickness and structure width are analyzed. Combined with the probability model, the distribution characteristics of the LIP are discussed. The spatial distribution characteristics of the HPZ are discussed at different ice failure models.

2. Crush Failure Characteristics of Sea Ice

There are bubbles, brine and impurities in sea ice, which cause defects in the sea ice microstructure. In the uniaxial compression test, the sea ice shows ductile failure characteristics when the strain rate is less than 10−4 s−1. The sea ice shows brittle failure characteristics when the strain rate is greater than 10−2 s−1. The sea ice also shows characteristics of the ductile-brittle transition at the transition rate. The compressive strength of the sea ice generally reaches its maximum value in this range [1,2,25]. At a low loading rate, the sea ice releases strain energy through creep expansion deformation. With increasing loading rate, the internal microcracks expand rapidly and fracture, showing a transition from ductile failure to brittle failure.

2.1. Ductile Failure

At a low strain rate (less than 10−4 s−1), the initial cracks are generated in the sea ice (as shown in Figure 1a,b). With continuous loading, the sea ice deforms, and the microcracks develop into wing cracks that are not always parallel to the loading direction. The horizontal tensile stress promotes the vertical growth of cracks. Due to the slow increase in stress, the stress concentration at the crack tip can be released by creep relaxation, preventing it from growing, and thus brittle failure does not occur (as shown in Figure 1a,b). The experimental results are similar to those of Chen et al. [26].
Figure 1c,d show that multiple wing cracks form in the sea ice during the later stages of loading. Creep behavior dominates, exhibiting ductile failure characteristics. The ice specimen expands and deforms macroscopically. Based on the ductile failure characteristics of the sea ice, when the interaction velocity between sea ice and the vertical structure is low, many microcracks are produced in the near-field ice in front of the structure. The internal cracks are evenly distributed, and the sea ice breaks into small fragments or powder. In the direction of the ice thickness, the broken ice may extrude from the upper and lower free surfaces in fluid form. Under the influence of the lateral limiting force, the HPZs are distributed close to the center of the ice thickness (Figure 1c,d).

2.2. Brittle Failure

At a high strain rate (greater than 10−4 s−1), the initial microcrack development is similar to that at a low strain rate. Microcracks appear during the initial stage and develop into wing cracks. Due to the high strain rate, the stress concentration at the tip of the wing cracks cannot be completely released by creep. Due to stress concentration, the stress path of the specimen develops mainly around the crack, forming a yield surface (as shown in Figure 2a,b). Wing cracks will develop into primary cracks macroscopically along the crack to the free surface, splitting the ice (Figure 2b). The experimental results are similar to those of Chen et al. [26].
Figure 2c,d shows that as the interaction velocity between the sea ice and the structure increases, the near-field ice in front of the structure mainly experiences brittle failure, with some sea ice breaking into blocks (near the free surface), and some breaking into fragments or powder (near the center). The broken ice at the center extrudes from the upper and lower free surfaces. The HPZs are mainly distributed in the central area, with the probability of HPZs decreasing farther from the center. The effective contact area in brittle failure is smaller than that in ductile failure.
The analysis revealed that along the direction of the sea ice thickness, the ice pressure is concentrated in the center of the ice thickness. The above discussion indicated that the ice velocity affects the sea ice failure mode during the sea ice/structure interaction, and the distribution of LIPs in the direction of the ice thickness differs when the sea ice failure mode changes. To further clarify the distribution of LIPs when the sea ice acts on structures at various velocities, an indentation test that drives ice sheet interactions with a vertical rigid plate (indentor) was designed. This test can be used to measure the LIP values and distributions.

3. Indentation Test of Sea Ice Acting on a Structure

The indentation test is a small-scale experiment. The test system includes a pressure sensor (model number: MF-3232, produced by Suzhou LEANSTAR company, Suzhou, China, consisting of 32 × 32 small pressure units of 1 cm2) on the face of the indentor, and the local ice pressure (LIP) was measured at various velocities. The distribution of the LIPs along the ice thickness, the probability distribution of the LIPs and the distribution of the LIPs in space and time were obtained from these measurements.

3.1. Indentation Test Device

Figure 3a shows a schematic diagram of the indentation test device, and consist of the followings: 1. rigid plate (indentor), which is fixed to the ground and has a contact width of 500 mm with the ice sheet; 2. thin-film pressure transducer, which is attached to the indentor surface (the acquisition area is 320 mm × 320 mm (the nominal contact width (NCW)), the acquisition frequency is 25 Hz, the acquisition accuracy is 0.001 MPa) (Figure 3c); 3. ice sheet, with a size of 1750 mm × 1000 mm, freezes the ice in the ice box; 4. support platform, which holds the ice box; 5. ice box, which is used to freeze the ice and has a size of 1750 mm × 1000 mm × 200 mm, with a notch (520 mm × 550 mm) in the middle of the ice box so that the ice can act on the indentor and a slide rail installed at the bottom and installed on the support platform; 6. thermocouple sensor, which is used to measure the temperature of the ice sheet during ice freezing; and 7. hydraulic actuator, which has a maximum thrust of 50 kN and drives the ice sheet interaction with the indentor. Sea salt was used to prepare high-salinity seawater in the ice box to reduce the ice strength. The brine water was frozen by rapidly cooling to −20 °C, and then rewarmed to bring the internal temperature of the ice sheet to −7 °C or −8 °C (Figure 3b,d).
To simulate the different failure characteristics of the sea ice, the actuator was set to five different loading velocities (Vload: 1 mm/s, 4 mm/s, 7 mm/s, 15 mm/s, and 30 mm/s), causing the ice sheet to act on the indentor at a corresponding velocity. The thin-film pressure transducer was attached to the indentor surface so that it was in direct contact with the ice sheet. The ice pressure values at different positions of the indentor were obtained by direct measurements. After the loading test, the specimens for the uniaxial compression test were cut from the ice sheet to determine the compression strength. The main test parameters are listed in Table 1.

3.2. Local Ice Pressures and Global Ice Force

Because the ice sheet is frozen in the ice box, the front of the ice sheet is smooth and flat. Simultaneous crushing occurs when the ice plate interacts with the structure during the initial stage. The global ice force data shows that the first peak force is much larger than the subsequent peak (as shown in Figure 4). As loading progresses, the subsequent crushing section is no longer flat, the crushing after the first peak exhibits non-simultaneous crushing characteristics, and the value of the peak force decreases. Therefore, the data on the ice force before the first peak crushing stage were removed in the following analysis. Due to the different ice velocities, the interaction times differ. The same sea ice breaking length after the first peak at different velocities was selected as the interaction time period for the data analysis.
By comparing the total ice sheet loading length at different velocities, the interaction times when the ice sheet breaking length is approximately 210 mm after the first peak were selected. The loading length is less than 210 mm when Vload is 1 mm/s, so the measured data at 60–180 s were extracted. For the other loading velocities, the loading time period with a loading length of approximately 210 mm was selected (as shown in Figure 5). The start time in the selected period was recorded as Tstr, and the end time was recorded as Tend (as shown in Table 2). Some measured data on the ice force and pressures were obtained in the selected period (including the peak value of the global ice force Fmax, the maximum LIP pLmax, the average LIP pLave, and the standard deviation of LIP pLstd, as shown in Table 2).
The global ice force curve in Figure 5 exhibits a typical periodic change in the form of triangular waves when the loading velocity is less than 7 mm/s. In the loading stage of the ice force, the loading curve shows the characteristics of a linear increase and unloading to the minimum value immediately after reaching the peak value (as shown in Figure 5a,b). At loading velocities of 15 mm/s and 30 mm/s, there are several small loading-to-unloading processes during a complete loading-to-unloading cycle on the global ice force curve. The global ice force curves are mostly in the form of wavy lines during a complete loading-to-unloading period (as shown in Figure 5c,d). The differences in the loading curves at different velocities are mainly due to the different failure modes of the ice sheet. Ductile failure at low velocities results in more contact between the ice and the structure, and brittle failure at high velocities results in more frequent non-simultaneous failure of the local area and local unloading throughout the loading process.

4. Results and Discussion

4.1. Distribution Area of LIPs

To study the distribution law of HPZs along the ice thickness, a statistical analysis of LIPs in the time period discussed above (from Tstr to Tend) is performed in this section. σL is a threshold value, it is used to describe the area of LIP greater than this value. The value of σL was determined as a multiple the average uniaxial compression strength (σcr). This multiple is represented by the threshold coefficient kt (kt = σLcr, taken as 0.5, 0.7, 1.0, 1.5, 2.0, 2.5, 3.0). Taking the threshold σL as the limiting condition, the LIPs at each time and each location measured by the sensor were counted. When the LIP value is greater than or equal to the threshold limit σL, the count is increased by one; otherwise, the count remains constant and is expressed by the function N(kt).
N j ( k t ) = N u m ( P j L σ L )     ( j = 1 , 2 , 3 , R )
Z j ( k t ) = N j ( k t ) M s     ( j = 1 , 2 , 3 , R )
where PjL is the LIP measured by the jth local pressure sensor; Nj is the number of PjL greater than or equal to σL; R is the total number of local pressure sensors in the nominal contact area (NCA); Zj is the probability of occurrence at different locations of the contact surface when the LIP is greater than or equal to the threshold coefficient kt; and Ms is the total number of LIPs at each contact location measured by the sensor from Tstr to Tend. The probability contour maps of the pressure distribution area in the NCA were determined for various velocities. The color bar indicates the value of Zj (as shown in Figure 6).
According to the probability contour maps of the LIP on the contact surface, the HPZs at various loading velocities and threshold coefficients kt are distributed near the center of the ice thickness. The blue to red areas in the figure indicates the probability of the local pressure exceeding σL, ranging from low to high probabilities. The contour map shows that closer to the center, the probability distribution of the LIPs is higher (as shown in Figure 6. The area of all the colored areas in Figure 6 is defined as AL, and the whole NCA is defined as ANCA. When the same threshold coefficient kt is used, the local pressure distribution area accounts for a large portion of the NCA at lower loading velocities (1 mm/s, 4 mm/s and 7 mm/s), while the local pressure distribution area AL accounts for a small portion of the NCA at higher loading velocities (15 mm/s and 30 mm/s). According to the analysis in Section 2, the sea ice mainly exhibits ductile failure at low loading velocities. The strain during failure is large, which increases the contact between the ice sheet and the indentor and causes the failure to be more simultaneous. At a high loading velocity, the main failure mode of the ice sheet is brittle failure, which results in a more irregular contact surface and makes the failure more non-simultaneous. The ratio of the LIP area to the NCA under different thresholds can be calculated (Equation (3)):
q ( k t ) = A L ( k t ) A NCA
where q is the area ratio coefficient, and kt is the threshold coefficient.
The analysis of the variation law of the area ratio coefficient q versus the threshold coefficient kt is shown in Figure 7, and it can be observed that the area ratio coefficient q changes linearly with the threshold coefficient kt at various velocities. When the coefficient kt = 1.0, the coefficient q is approximately 0.5 at a low loading velocity and approximately 0.25 at a high loading velocity. This result indicated that HPZs subjected to pressures greater than σcr represent 1/2 of the NCA at low velocities (Figure 7a) and approximately 1/4 of the NCA at high velocities (Figure 7b). The HPZs are generally located near the centerline of the ice. Furthermore, HPZs subjected to pressures greater than 3 times σcr represent less than 2% of the NCA (Figure 8a) at low velocities, and HPZs subjected to pressures greater than 2.5 times σcr represent less than 1% of the NCA (Figure 8b) at high velocities. Dempsey et al. found that the HPZs appear to be confined to the middle third of the ice thickness [3]. The red dashed box in Figure 7a,b indicates that HPZs represent approximately 1/3 of the NCA. The corresponding threshold coefficient kt is 1.5 at low loading velocities and 0.7 at high loading velocities. That is, an HPZ pressure greater than 1.5 times the average uniaxial compressive strength σcr is located in the middle third of the ice thickness at a low loading velocity, and an HPZ pressure greater than 0.7 times the average uniaxial compressive strength σcr is located in the middle third of the ice thickness at a high loading velocity. The distribution area of HPZs is determined by the threshold values and failure models.
Because the area ratio coefficient q versus threshold coefficient kt varies linearly, a linear fitting formula (Equation (4)) was used to fit and analyze the data in Figure 7. The fitting parameters a and b are shown in Table 3. According to the above discussion, the HPZs are generally distributed in the centerline of the ice thickness. The distance of the HPZ edge from the ice thickness centerline can be calculated by Equation (5):
q ( k t ) = b k t + a   ( k t > 0 , q 0 )
W d = ± h i c e 2 q ( k t ) = ± h i c e 2 ( b k t + a )
where Wd is the distance of the HPZ edge from the ice thickness centerline when the threshold coefficient is kt and hice is the ice thickness.
According to R2 value (as shown in Table 3), it can be seen that the sea ice has a good fit when ductile failure occurs, and the R2 is relatively small when brittle failure occurs, which indicates that the distribution of high-pressure area is more concentrated when ductile failure occurs.
The above discussion shows that the LIP distribution varies with the velocity, and the sea ice failure mode leads to the difference in the LIP distribution. The distribution area of the HPZs during brittle failure is smaller than that during ductile failure.

4.2. Probability Distribution of LIPs

The probability distribution of the LIPs in the time period discussed in Section 3 (from Tstr to Tend) is determined in this section. The LIP value Pl measured by each local pressure sensor and the total number Ms of NCAs in the whole time period were obtained. The occurrence probability of Pl within each pressure value range was calculated at intervals of 0.01 MPa. Pi is the calculated value in the i-th step. P1 is 0, and Pi+1 = Pi + 0.01. The number of Pi < PlPi+1 measured by the local pressure sensor is Ni, and the probability of Pri can be calculated by Equation (6):
Pr i ( P i < P l P i + 1 ) = N i M s ( i = 1 , 2 , 3 )
The probability distribution histograms of the pressure value were determined for various loading velocities. The probability of LIPs decreases exponentially with increasing LIP. The probability of LIPs at various loading velocities follows a lognormal distribution (Figure 9). The lognormal function in Equation (7) was used to calculate the corresponding parameters of μs and σs at various loading velocities. The red lines in Figure 9 are the lognormal fitting lines based on Equation (7).
p ( x ) = A s p s ( x ) = A s x σ s 2 π e ( ln x μ s ) 2 2 σ s 2
p(x) is fitting equation. ps(x) is probability density of LIP, As is the cumulative coefficient, x is LIP value. Because the selected interval is 0.01 MPa, p(x) is the area that reflects the composition of the y-axis and x-axis values, Since the area covered by the equation and x-axis is 0.01, in order to make the cumulative probability equal to 1, the coefficient As is 0.01. The parameters of the function at various velocities are shown in Table 4.
Table 4 shows that the μs ranges by approximately 0.04 MPa and σs ranges by approximately 0.3 MPa at various velocities. This indicates that the probability of LIP on the interaction surface between the ice sheet and indentor is similar across the whole action time at various velocities. When Vload is 1 mm/s, σs is greater than that under the other three loading velocities, which indicating that the local pressure fluctuates greatly. When Vload is 1 mm/s, μs is greater than that under the other three loading velocities. At this time, the non-simultaneous failure of ice sheet is obvious, the overall mean pressure is large and R2 value is smaller than others. When Vload are 1, 7, 15 mm/s, the R2 value is high, and the curve has a good fitting, which indicates that its distribution characteristics accord with lognormal distribution.
There is little difference in the local fatigue damage caused by LIPs at various velocities. The number of different pressures acting on the local area decreases exponentially with increasing pressure. The fatigue damage caused by the maximum LIP accounts for a small proportion of the total damage.

4.3. Spatial and Time Distribution Relationship

To study the spatial distribution of the LIP, LIP values were obtained at different moments during the loading and unloading processes of the global force. Figure 10a,b shows the selection method at low loading velocities (1 mm/s and 7 mm/s). Five time points of a complete loading and unloading cycle were selected: a. the initial loading moment; b. a moment during loading; c. the peak force moment; d. a moment during unloading; and e. the complete unloading moment. Figure 10c,d shows the selection method at high loading velocities (15 mm/s and 30 mm/s). The five time points during a complete loading and unloading cycle were: a. the initial loading moment; b. a small peak moment during loading; c. the peak force moment; d. a valley moment during unloading; and e. the complete unloading moment.
Figure 10a,b shows that the fluctuation of the global ice force curve in a single cycle is very small and tends to be loaded in a straight line. The global ice force curve presents a regular sawtooth shape, and the sea ice is periodically damaged at low velocities. At a low loading rate, the ice sheet shows ductile characteristics and a large strain, allowing sufficient contact on the width of the structure. The local failure of the ice sheet tends to be more simultaneous.
Figure 10c,d shows that the global ice force curve shows randomness and has no obvious periodicity. At a high loading rate, the ice sheet shows brittle failure. In the loading stage, there is a short time when the ice force declines in the curve, and failure occurs in some local areas. The failure section of the ice sheet is more irregular, and the failure in the contact width tends to be non-simultaneous. The sheet is continuously destroyed during the whole cycle on the contact width. The HPZs are concentrated in smaller local areas.
The LIP data at five moments (a, b, c, d, e) were extracted at various velocities. Table 5 shows the maximum LIP at five corresponding times (a, b, c, d, e) at each loading velocity. pLmax is the maximum LIP from Tstr to Tend and pTmax is the maximum value at the five moments (a, b, c, d, e).
At low loading velocities (1 mm/s and 7 mm/s), Table 5 shows that pTmax appears at the peak moment of the global ice force in the loading cycle (moment c). The maximum pressure at the initial moment (moment a) and the complete unloading moment (moment e) are less than half of the value at the peak force moment (moment c). At high loading velocities (15 mm/s and 30 mm/s), Table 5 shows that pTmax appears at a small peak moment during loading (moment b). pTmax does not appear at the peak force moment time in the loading cycle (moment c). The maximum pressure at the initial moment (moment a) and the complete unloading moment (moment e) are less than half of the value at the peak force moment (moment c). The fluctuations of the local maximum ice pressure at the four moments in the whole loading cycle are small, and the range of the maximum ice pressure is within 20% (beginning at moment b). The occurrence of pTmax is affected by the failure mode of the ice sheet. When the failure mode of the ice sheet is mainly ductile failure, pTmax occurs at the peak force. pTmax can occur at any moment of the global ice force when the ice is mainly brittle failure.
From the heat figures of the LIP at five moments during the loading cycle (Figure 11), the spatial distribution characteristics of the LIP in the loading cycle can be analyzed.
Figure 11a,b show that the spatial distributions of LIPs at low velocities (1 mm/s and 7 mm/s) are similar. At the moments of loading (moment a) and unloading (moment e), the pressure distribution on the NCA is mainly in the blue pressure range (less than 1/8 times pTmax). At the center of the ice thickness, there is a partial LIP range distributed at 2/8 to 3/8 times pTmax. The pTmax value for the whole cycle occurs at moment c. The LIP distribution color in the figure is mainly green near the centerline, indicating that the LIP value is approximately 1/2 times the pTmax; at the same time, there is a small amount of red in the figure of moment c. This result indicates that the area of the LIP value greater than 7/8 times the pTmax is less than 2% of the NCA. At a given moment on the contact surface between the ice and the structure, the number of HPZs is greater than the number of other local zones, which generally occurs in 1–3 pressure acquisition units. The above variation characteristics are shown in the width direction of the NCW, which indicates that the ice sheet fails at most positions along the NCW at the peak time (moment c).
At high velocities (15 mm/s and 30 mm/s), a higher pressure was also distributed near the center of the ice thickness. In contrast to low loading velocities, the LIP distribution color is mainly green in the NCA from moment a to moment e at high loading velocities. A small number of red areas are distributed in the NCA from moment b to moment e. This indicates that there are HPZ values greater than 1/2 times pTmax distributed throughout the cycle period and a few HPZ values greater than 7/8 times pTmax (1–3 pressure acquisition units, less than 2% of the NCA) (as shown in Figure 11c,d)). There are some areas with alternating green and blue colors throughout the cycle period. For example, at a loading velocity of 30 mm/s, multiple blue areas at moment a become green areas at moment b and then become blue areas again at moment c. The above phenomena show that the sea ice in some areas fails at moments a and c, and fails in some areas at moment b along the width direction of the indentor at high velocities indicating non-simultaneous failure mode of the ice sheet.

5. Conclusions

The failure mode of the ice sheet is affected by the ice velocity. The sea ice experiences mainly ductile failure at low ice velocities and brittle failure at high ice velocities. Theoretical analysis of the interaction between the ice sheet and the vertical structure shows that the distribution of local ice pressures in the direction of the ice thickness differs under different failure modes. The indentation test verified the influence of the failure mode on the local ice pressure distribution at various velocities.
At low loading velocities, the global ice force curve presents a regular sawtooth shape, and the ice sheet has the same periodic loading-to-unloading characteristics at low velocities. In addition, the global ice force curve increases linearly during the loading stage. At high loading velocities, there are many small loading and unloading processes during a complete loading-to-unloading cycle, with most of the loading curves taking the form of random wavy lines.
The threshold coefficient kt is taken as a multiple of the average uniaxial compressive strength σcr. When the coefficient kt = 1.0, the coefficient q is approximately 0.5 at low loading velocities and approximately 0.25 at high loading velocities. The area ratio coefficient q changes linearly with the threshold coefficient kt at various velocities. HPZs subjected to pressures greater than three times σcr represent less than 2% of the NCA at low velocities, and HPZs subjected to pressures greater than 2.5 times σcr represent less than 1% of the NCA at high velocities. The HPZs are distributed in a smaller area during brittle failure than during ductile failure at the same kt. The probability distribution of the LIPs in local areas follows a lognormal distribution, and the ice velocity has little effect on the distribution probability.
At low loading velocities, as loading progresses during a loading cycle, the local extreme ice pressure increases. The local extreme ice pressure reaches its maximum at the moment that the global ice force reaches its peak. At high loading velocities, the pTmax in a loading cycle may appear at any moment of the global ice force. The maximum local ice pressure fluctuation at different moments is small throughout the loading cycle. The distribution of HPZs (greater than 7/8 times the pTmax) accounts for less than 2% of the NCA. Only a few local areas are affected by highest LIP on the NCA.

Author Contributions

Conceptualization, G.W., Q.Y. and D.Z.; Data curation, G.W., Y.F. and X.P.; Formal analysis, D.Z., Y.F., X.P. and R.D.; Funding acquisition, D.Z.; Methodology, G.W. and D.Z.; Writing—original draft, G.W.; Writing—review & editing, G.W. and D.Z. All authors have read and agreed to the published version of the manuscript.

Funding

National Natural Science Foundation of China (grant number 52071055); The Innovation Team of Colleges and Universities 353 in Liaoning Province (grant number LT2019004).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (grant number 52071055) and the Innovation Team of Colleges and Universities 353 in Liaoning Province (grant number LT2019004).

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Crack growth at a low loading speed and the theoretical crushing process in front of the vertical structure. (a) Schematic of internal crack propagation, (b) Uniaxial compressive test at a strain rate of 10−4 s−1, (c) Initial crack formation in the sea ice in front of the structure, (d) Local ice pressure and ice fragmentation extrusion.
Figure 1. Crack growth at a low loading speed and the theoretical crushing process in front of the vertical structure. (a) Schematic of internal crack propagation, (b) Uniaxial compressive test at a strain rate of 10−4 s−1, (c) Initial crack formation in the sea ice in front of the structure, (d) Local ice pressure and ice fragmentation extrusion.
Jmse 10 00433 g001
Figure 2. Crack growth at low loading speeds and the theoretical crushing process in front of the vertical structure. (a) Schematic of internal crack propagation, (b) Uniaxial compressive test at a strain rate of 10−4 s−1, (c) Initial crack formation in the sea ice in front of the structure, (d) Local ice pressure and ice fragmentation extrusion.
Figure 2. Crack growth at low loading speeds and the theoretical crushing process in front of the vertical structure. (a) Schematic of internal crack propagation, (b) Uniaxial compressive test at a strain rate of 10−4 s−1, (c) Initial crack formation in the sea ice in front of the structure, (d) Local ice pressure and ice fragmentation extrusion.
Jmse 10 00433 g002
Figure 3. Indentation test device and schematic. (a) device schematic; (b) device and frozen ice sheet; (c) thin-film pressure transducer; (d) test load.
Figure 3. Indentation test device and schematic. (a) device schematic; (b) device and frozen ice sheet; (c) thin-film pressure transducer; (d) test load.
Jmse 10 00433 g003
Figure 4. Total forces when Vload equals 4 mm/s and 7 mm/s.
Figure 4. Total forces when Vload equals 4 mm/s and 7 mm/s.
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Figure 5. Selected global ice force curves. (a) Vload is 1 mm/s, (b) Vload is 4 mm/s and 7 mm/s, (c) Vload is 15 mm/s, (d) Vload is 30 mm/s.
Figure 5. Selected global ice force curves. (a) Vload is 1 mm/s, (b) Vload is 4 mm/s and 7 mm/s, (c) Vload is 15 mm/s, (d) Vload is 30 mm/s.
Jmse 10 00433 g005aJmse 10 00433 g005b
Figure 6. Probability contour maps of the LIP distribution area on the NCA at various velocities. (a) Vload is 1 mm/s and kt is 1.0; (b) Vload is 1 mm/s and kt is 1.5; (c) Vload is 7 mm/s and kt is 1.0; (d) Vload is 7 mm/s and kt is 1.5; (e) Vload is 15 mm/s and kt is 1.0; (f) Vload is 15 mm/s and kt is 0.7; (g) Vload is 30 mm/s and kt is 1.0; (h) Vload is 30 mm/s and kt is 0.7.
Figure 6. Probability contour maps of the LIP distribution area on the NCA at various velocities. (a) Vload is 1 mm/s and kt is 1.0; (b) Vload is 1 mm/s and kt is 1.5; (c) Vload is 7 mm/s and kt is 1.0; (d) Vload is 7 mm/s and kt is 1.5; (e) Vload is 15 mm/s and kt is 1.0; (f) Vload is 15 mm/s and kt is 0.7; (g) Vload is 30 mm/s and kt is 1.0; (h) Vload is 30 mm/s and kt is 0.7.
Jmse 10 00433 g006aJmse 10 00433 g006b
Figure 7. Area ratio coefficient q versus threshold coefficient kt. (a) Vload of 1 mm/s, 4 mm/s and 7 mm/s; (b) Vload of 15 mm/s and 30 mm/s.
Figure 7. Area ratio coefficient q versus threshold coefficient kt. (a) Vload of 1 mm/s, 4 mm/s and 7 mm/s; (b) Vload of 15 mm/s and 30 mm/s.
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Figure 8. Less than 2% of HPZs located in NCA. (a) kt is 3 and Vload is 1 mm/s; (b) kt is 2 and Vload is 15 mm/s.
Figure 8. Less than 2% of HPZs located in NCA. (a) kt is 3 and Vload is 1 mm/s; (b) kt is 2 and Vload is 15 mm/s.
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Figure 9. Local pressure versus probability. (a) Vload is 1 mm/s; (b) Vload is 7 mm/s; (c) Vload is 15 mm/s; (d) Vload is 30 mm/s.
Figure 9. Local pressure versus probability. (a) Vload is 1 mm/s; (b) Vload is 7 mm/s; (c) Vload is 15 mm/s; (d) Vload is 30 mm/s.
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Figure 10. (a) Selected period from the global force when Vload is 1 mm/s; (b) five selected moments when Vload is 1 mm/s (a, b, c, d, e); (c) selected period from the global force when Vload is 15 mm/s; (d) five selected moments when Vload is 15 mm/s (a, b, c, d, e).
Figure 10. (a) Selected period from the global force when Vload is 1 mm/s; (b) five selected moments when Vload is 1 mm/s (a, b, c, d, e); (c) selected period from the global force when Vload is 15 mm/s; (d) five selected moments when Vload is 15 mm/s (a, b, c, d, e).
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Figure 11. Distribution of local pressures in a period. (a) Vload is 1 mm/s; (b) Vload is 7 mm/s; (c) Vload is 15 mm/s; (d) Vload is 30 mm/s.
Figure 11. Distribution of local pressures in a period. (a) Vload is 1 mm/s; (b) Vload is 7 mm/s; (c) Vload is 15 mm/s; (d) Vload is 30 mm/s.
Jmse 10 00433 g011
Table 1. Parameters of the indentation test.
Table 1. Parameters of the indentation test.
SpecimenS
(‰)
hice
(mm)
Tice
(°C)
Vload
(mm/s)
σcr
(MPa)
Failure
0516-v12655−9.310.553ductile
0531-v13142−8.810.351ductile
0602-v72860−870.287mixed
0606-v43065−9.340.362ductile
0606-v153365−7.2150.221brittle
0609-v153363−8.2150.336brittle
0611-v302965−8300.358brittle
0724-v302344−7.0300.232brittle
S—Salinity of ice; hice—Average thickness of the ice sheet; Tice—Average ice temperature; Vload-Loading velocity; σcr—Average uniaxial compression strength of the ice sheet.
Table 2. Measured global ice force and local ice pressure data.
Table 2. Measured global ice force and local ice pressure data.
SpecimenTstr
(s)
Tend
(s)
Fmax
(kN))
pLmax
(MPa)
pLave
(Mpa)
pLstd
(Mpa)
0516-v1601806.331.660.2040.255
0531-v1601802.471.230.0960.112
0602-v722525.461.230.1230.121
0606-v422755.381.260.1210.127
0606-v158222.390.390.0420.037
0609-v158221.460.670.0780.071
0611-v3012202.920.750.1580.151
0724-v3012201.890.560.0790.099
Table 3. The fitting parameters.
Table 3. The fitting parameters.
FailureabR2
Ductile0.741−0.2550.950
Brittle0.44−0.1740.828
Table 4. Parameters of the lognormal distribution at various velocities.
Table 4. Parameters of the lognormal distribution at various velocities.
Vload
(mm/s)
μs
(MPa)
σs
(MPa)
R2
10.0851.570.961
70.0851.280.962
150.0721.320.963
300.111.370.836
Table 5. The maximum LIP at five moments.
Table 5. The maximum LIP at five moments.
Vload
(mm/s)
Maximum Pressure (MPa)
abcdepTmaxpLmax
10.4641.341.6351.3450.8041.6351.66
70.440.7540.9150.7740.3060.9151.23
150.270.3690.3360.2940.3170.3690.67
300.3690.5540.5340.520.530.5540.75
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Wang, G.; Yue, Q.; Zhang, D.; Fu, Y.; Peng, X.; Dong, R. Distribution Analysis of Local Ice Pressures in the Indentation Test at Various Velocities. J. Mar. Sci. Eng. 2022, 10, 433. https://doi.org/10.3390/jmse10030433

AMA Style

Wang G, Yue Q, Zhang D, Fu Y, Peng X, Dong R. Distribution Analysis of Local Ice Pressures in the Indentation Test at Various Velocities. Journal of Marine Science and Engineering. 2022; 10(3):433. https://doi.org/10.3390/jmse10030433

Chicago/Turabian Style

Wang, Guojun, Qianjin Yue, Dayong Zhang, Yanghua Fu, Xin Peng, and Rui Dong. 2022. "Distribution Analysis of Local Ice Pressures in the Indentation Test at Various Velocities" Journal of Marine Science and Engineering 10, no. 3: 433. https://doi.org/10.3390/jmse10030433

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