Encyclopedia of Ocean Engineering

Living Edition
| Editors: Weicheng Cui, Shixiao Fu, Zhiqiang Hu

Flexural Strength of Sea Ice

  • Anliang WangEmail author
  • Shunying Ji
Living reference work entry
DOI: https://doi.org/10.1007/978-981-10-6963-5_301-1



Flexural strength of sea ice is the maximum bending stress that the sea-ice material can take. This definition is drawn from the traditional material mechanics. Once the sea-ice strength is surpassed, the failure and fracture of sea-ice plate are followed. The flexural strength is closely connected with many natural phenomenon and engineering properties of sea ice. For example, the breakup, rafting, and ridge-building are often observed in polar area and seasonal ice-covered region, when the ice covers are under the actions of wave, current, and wind. In the cold region, the jacket platform with ice-breaking cones, artificial island, slope protection, and ice breaker suffer from the collision of the ice cover. These situations generally cause the bending failure mode of sea ice, and the ice loads of the structures are subsequently determined by the flexural strength.

In essence, the flexural strength of sea ice is induced by the momentum parallel to the ice cover. The momentum generally causes the longitudinal stresses, compressive, and tensile, which distribute linearly with perpendicular distance from the neural axis in a pure and homogeneous elastic material. Note that the fissure appears at the surface with the zone of critical stress to be very thin, and the fissure propagates in a stress gradient. As we know that sea ice is a complex material with multiphase, anisotropy and not pure elasticity, the effective moduli for tension and compression are not equal, when sea ice takes the bending momentum. Therefore, the sea-ice mechanical properties evidently perform inhomogeneous. Unlikely to the classic material mechanics, the flexural strength of sea ice does not equal to the uniaxial tensile strength.

Scientific Fundamentals

Test Types

In general, the flexural strength of sea ice is acquired through the tests in laboratory or in situ. According to the loading types, there exist three test patterns: three-point beam test, four-point beam test, and cantilever beam test, as shown in Fig. 1. Cantilever beam test can only be carried out in situ, and the other two are applicable both in situ and in laboratory.
Fig. 1

Three types of flexural strength tests for sea ice: (a) cantilever beam, (b) three-point beam, and (c) four-point beam

Cantilever beam test is a full-scale measurement of the strength and the load is applied at the free end of sea-ice sample until the sample breaks at the base of the beam. Three-point beam test and four-point beam test are performed based on the cubical samples cut off from the ice cover, and the loading points are on the center point and middle two of the quantile points, respectively. Cantilever beam test is much better than the other two on measuring the real strength condition of sea ice, because the sea-ice state is maximally preserved in situ without significant movement (Timco and Weeks 2010). Specifically, cantilever beam test facilitates a relatively easy way to perform on a large beam and can keep the natural temperature gradient in the ice sheet. However, cantilever beam test is highly consuming on many aspects such as time and labor, so it is hard to guarantee enough number of the test for the mechanical property studies of sea ice. Besides, the result of the test is influenced by the variation of physical properties with ice depth, which is brought about by the ice growth history. Stress concentration at the corners of the beam root is another concern for the test. The root of the cantilever may be not rigid enough, especially when the beam is much wider with reference to its depth. The stress concentration sometimes takes place at the corners of the beam root, which could introduce extra error for the measurement of the flexural strength and even fail the measurement.

Otherwise, the three- and four-point beam test is most widely used because of its controllability and feasibility, and its failure position is anticipated much easier than the four-point beam test. More often, the three-point beam test is preferred to be performed in a laboratory on smaller samples of ice cut from the ice cover. Therefore, most of the studies on the relationship between the sea-ice flexural strength and its influencing factors are based on the three-point beam test. However, the broken location of the three-point beam test is often at the midpoint of the beam, where takes the maximum moment. As a result, the failing location may not be the weakest of the sea-ice sample.

Whereas, four-point bending tests preserve a section of the beam, where the moment is constant and the shear stress is zero. This distribution is able to guarantee the failure happens at the weakest point of the sea-ice sample. Therefore, the four-point test is highly recommended for sea ice, although it is a little more complex than the three-point test.

Stress concentration is another concern to carry out three- and four-point beam test. During the test, the local indentation effects sometimes happen at the loading and supporting points. This has a negative effect on the accuracy of the flexural strength. To avoid stress concentration or local indentations at these points, the test apparatus is suggested to have cylinder supports, which should have enough stiffness comparing with sea-ice sample (ITTC 2014).

The flexural strength of sea ice is not considered as a basic material property. The distribution of the stress field is not uniform inside the sample for the test of flexural strength. The flexural strength of sea ice is more complex than the classic mechanics, and there needs to be extra assumptions to explain the test results. However, because it is a necessary parameter for numerical simulation and ice engineering projects, the flexural strength of sea ice is widely measured.

Influencing Factors of Flexural Strength

Many physical properties of sea ice influence the flexural strength such as salinity, porosity, and temperature. As a deduced parameter from salinity and temperature of sea ice, the brine volume is widely adopted to reflect the physical properties of sea ice, when the relationship between flexural strength and physical properties of sea ice is investigated. Sea ice is also a rate-dependent material, thus many researchers arguably identify stress rate as another parameter to influence the flexural strength of sea ice. In addition, the scale effect of the flexural strength is observable, so the strength deriving from the small scale test cannot be directly applied to the sea-ice process of the large scale.

Sea-ice temperature is a key element to determine the flexural strength. On microscale, the lower sea-ice temperature is, the denser ice crystal conglomerates. As is embedded on the macroscale, the flexural strength becomes stronger. Many studies suggest that sea-ice flexural strength is approximately a linear function of its temperature. The flexural strength increases with the decrease of the sea-ice temperature. Contrarily, sea-ice salinity sustains a positive relationship with the flexural strength. The higher salinity of sea ice prone to make more porosity inside of the sea ice, the distribution of these defaults inside the sea ice lower the flexural strength. However, the significance of the linear relationship between the flexural strength and the sea-ice salinity is low, so the influence of the salinity on the flexural strength is more complex. Alternatively, the sea-ice brine volume is computed from the sea-ice temperature and salinity, and it thus can combine these two influencing factor into one variable by (Frankenstein and Garner 1967):
$$ {v}_{\mathrm{b}}={S}_{\mathrm{i}}\left(\frac{49.185}{\left|{T}_{\mathrm{i}}\right|}+0.532\right) $$
where vb, Si, and Ti denotes brine volume, salinity, and temperature of sea ice. This equation is validated only when the ice temperature is −22.9 ℃ ≤ Ti ≤ − 0.5℃. Therefore, the brine volume tend to be selected to probe into the influence of sea-ice physical properties on the flexural strength.
Many flexural strength tests have been carried out to investigate the relationship between the brine volume and the flexural strength. Most of the researchers reported that the flexural strength of sea ice has a negative exponential relationship to square root of brine volume. The others concluded that the flexural strength of sea ice is a linear function of square root of brine volume. Although the form of the equation is given, the fitting parameters are a little different from different researchers and different experimental sites. Timco and O’Brien (1994) combined 2500 reported measurements of the flexural strength of freshwater ice (1556 beams) and first-year sea ice (939 beams), as shown Fig. 2. They gave the exponential correlation between the flexural strength and the square root of brine volume as:
$$ {\sigma}_{\mathrm{f}}=a{e}^{-c\sqrt{v_{\mathrm{b}}}} $$
where σf and vb denotes flexural strength and brine volume of sea ice, respectively. a and c are the parameters through the fitting line.
Fig. 2

Flexural strength versus the square root of the brine volume for first-year sea ice (Timco and O’Brien 1994)

This equation is most representative to calculate sea-ice flexural strength just based on the temperature and salinity of sea ice. Basically, its data comes from a large number of researchers and thus significantly compromises the fitting parameters of a variety of geographic locations, in both polar and seasonal ice-covered region. It is worthwhile to point out that there may be remarkable difference for some certain regions from the fitting parameters (as shown by Fig. 3), but the exponential correlation has been confirmed to be effective to fit the flexural strength and brine volume of sea ice (Ji et al. 2011).
Fig. 3

Influence of brine volume on the flexural strength (Ji et al. 2011)

The dependence of the flexural strength on strain (stress) rate is seldom studied and remained as a debatable question (Timco and Weeks 2010; Timco and O’Brien 1994). The dependence on stress rate is confused, and it is not easy to calculate how much degree the observed trends are dominated by the sea-ice physical changes or by the imperfections of test technique. Until now, the agreement with this question has been hardly reached between different researchers. Some researchers insisted that the stress rate has very little effect on flexural strength (Frederking and Timco 1983; Timco and Weeks 2010). The others argued that the flexural strength of sea ice slightly increases with increasing of the stress rate (Ji et al. 2011). One of the main reasons for this inconsistency is inaccuracy of the stress rate. The samples completely crack very soon from the beginning to the end of the loading, approximately within one second (Timco and Weeks 2010). If the sampling frequency is not high enough, the average stress rate is computed less accuracy than that in compression test, where the rate-dependence of compression is clearly observed. However, the flexural strength shows a decreasing trend with the stress rate a little more clearly, when the salinity and temperature of sea ice maximally keep uniform.

This trend enlightens the combination of the brine volume and stress rate to study their coupling effect on the flexural strength of sea ice. Many studies have been focused on the influence of a single factor on the flexural strength of sea ice. In fact, these influential factors are coupled together. Furthermore, the specified factor such as salinity and temperature is hardly controlled to be uniform during the test of sea ice. This causes the inaccuracy of the dependence of the flexural strength on a single factor with the other factors to be constant. Thus, the flexural strength as a multivariate function of temperature, salinity, and stress rate has been proposed. The flexural strength of sea ice can be simulated as a two-parameter exponential equation with the consideration of both brine volume and stress rate (Ji et al. 2011) in Fig. 4. This scheme overcomes the disadvantages of the single factor, but there still need to be more experimental data to determine the fitting equation and its parameters.
$$ {\sigma}_{\mathrm{f}}=a{\mathrm{e}}^{\left(c+d\dot{\sigma}\right)\sqrt{v_{\mathrm{b}}}} $$
with a = 2.61, c = −5.58, and d = 2.09.
Fig. 4

Curve-fitted flexural strength surface (a), flexural strength contour (b), and stress rate contour (c)

The influence of the scale on the flexural strength is seldom studied, but it is verified by many phenomenon from the large-scale monitor of sea ice such as satellite images (Marsan et al. 2004). For laboratory scale, the sea-ice samples are small volumes, which make the test results hard to be applied into the large-scale condition. Generally, the flexural strength decreases as the scale become large (Parsonsa et al. 1992; Weiss and Dansereau 2017). For the flexural strength of laboratory, the refined samples could be too small to enclose large quantity of ice crystals. This shortage makes the data not competent to account for grain size effects. Moreover, small samples are likely to reduce the randomness of the distribution of sea-ice flaws. To some degree, the cantilever beam test overcomes this shortage, because it is carried out in situ and maximally preserves the natural flaws of sea ice (Parsonsa et al. 1992). Therefore, we could observe that the flexural strength from the cantilever tests tend to be lower than other measurement methods. However, the conversion equation between different scales has not been given, so much more test data especial from full-scale tests are eagerly to be incorporated to study scale effects.

Loading direction should be another factor influencing the flexural strength. In practice, the direction of the moment is perpendicular to the ice cover, when the interactions between sea-ice covers and engineering structures take place under the current and wave. Thus, the influence of the loading direction on the flexural strength has been paid less attention.

Shortage of Strain Measurement of Sea-Ice Test

As we mentioned, the accurate measurement of strain rate is a key problem to study the relationship between flexural strength and strain rate. It is not so easy to get the real strain rate, because the length of the time is very short from the beginning of the loading to the break of sea-ice sample. In traditional ice mechanics, displacement actuators, strain gauges, and extensometers are used to measure the deformation of specimen (Timco and Weeks 2010). Those measurements need some operation on the surface of sea-ice sample, and it is likely to bring the local damage and cause the stress concentration. Alternatively, the equivalent displacement is widely adopted to study the relationship based on the displacement from an indenter (Wang et al. 2019). The loading surface of the specimen and indenter on it move together and thus have the same displacement all the time. Therefore, the strain of the specimens can be deduced from the displacement of indenter without considering the deformation of the rig itself. This method provides only one value to represent the overall deformation of the specimen, even under the assumption that the loading system has enough stiffness compared with that of sea-ice specimen. For sea-ice material, the local variation in deformation cannot be ignored because of the existence of brine pockets and air bubbles inside of sea ice.

Digital image correlation (DIC) method provides a new concept to obtain strain (rate) from the sequential images. This method has been applied to the deformation measurements of other materials. Based on the DIC, full-field displacement and strain can be accurately obtained by comparing the digital images of specimen surfaces for the initial and deformed states in Fig. 5. Wang et al. (2019) used the DIC to investigate the uniaxial compressive strength and fracture mode for sea ice and lake ice. Moreover, based on the DIC technique, the full-field deformation is accurately deduced, and the damage process of the specimen is clearly obtained at high spatiotemporal resolution.
Fig. 5

Digital image correlation applied into the three-point beam test to get the deformation field

With the assistance of high-speed camera, the strain can be computed on enough frequency even within a very short time from the loading to the break of sea-ice sample. The DIC technique will facilitate the measurement of sea-ice strain (rate). Then the study of relationship between flexural strength and strain rate will become more accurate. Meanwhile, digital images can be used to capture the local conditions of sea-ice specimens and depict the failure characteristics during the loading process (Ji et al. 2020). This will extend the ability to further explore the complex mechanical behaviors of sea ice.

Discrete Element Method for Simulation of Flexural Strength

Sea ice is composed of solid ice, brine, and gas. This multiphase makes its mechanical properties behave much heterogeneously. Therefore, sea ice is isotropic with granular structures as well as is anisotropic with columnar structures. In turn, these differences are reflected to the mechanical properties of the ice (Timco and Weeks 2010). The flexural strength of sea ice is such a mechanical property and depends on many influencing factors such as temperature, salinity, porosity, and grain structure. Consequently, the mechanical properties of sea ice are complex. Experimental investigations in situ or in laboratory do not satisfy the requirement of relationship between flexural strength and its many influencing factors.

Numerical method leverage us another tool to investigate the mechanical properties of sea ice. Considering the multiphase and anisotropic mechanical properties of sea ice, discrete element method (DEM) is being developed to accomplish the goal and has got a better performance. DEM is based on the discontinuous mechanics model, which approaches the real physical composition of sea ice more closely than the traditional numerical methods such as finite element method and finite different method. Several works have been done to look into the failure process of the three-point beam test and to simulate the mechanical behavior of sea ice in response to ocean waves (Xu et al. 2012).

The principal of the DEM was first proposed in the late of 1970s. Since then, it has been developed to simulate the failure process of brittle material such as sea ice (Long et al. 2018; Bateman et al. 2019). For the DEM, sea ice is approximated as an assemblage of the bonded particles, of which each is dominated by Newton’s second law to simulate the mechanical behavior. On microscale, the main parameters of the particle model include normal and shear stiffness, normal and shear bonding strength, particle diameter, sample size, and interparticle friction coefficient, as shown in Fig. 6. When the interparticle stress surpasses a critical threshold, the bonds between particles break. The flexural strength of sea ice can be obtained depending on these micro-parameters in the DEM.
Fig. 6

Parallel bond model between two spherical particles

During the simulation using DEM, the particle size, bonding strength, and friction coefficient are the major parameters to influence the flexural strength of sea ice. In order to simulate complex mechanical behavior during the experiment of flexural strength, it is essential to develop an effective method to bridge the gap between the micro-parameters and the macro-strength in the DEM. In general, these parameters are calibrated and validated according to that of sea-ice strength, as measured in a series of flexural and compressive tests. The assemblage of particles is subjected to the load and then the adjustment of bending stress accomplishes the block fractures in Fig. 7. The interparticle bond strength limits are adjusted until the critical failure stress determined for the numerical sample falls within the ranges confirming to the data from the test of flexural strength.
Fig. 7

Failure processes of three-point bending test (Long et al. 2018)

After filling the gap, the relationship between the flexural strength and physical properties (salinity and temperature) can be checked out. With the development of parallel computation of GPU, the number of the particles can easily reach the million level. As a result, the influencing of strain rate on the strength is able to be figured thoroughly, which is not so easy to be obtained for tests in situ or in laboratory. However, there need to be more developments to improve the theoretical basis of DEM numerical approach. Furthermore, the mechanism of more micro-parameter effects on macro-behaviors of mechanics should be analyzed in detail in future works.



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© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Marine Disaster Forecasting and Warning DivisionNational Marine Environmental Forecasting CenterBeijingChina
  2. 2.State Key Laboratory of Structural Analysis for Industrial EquipmentDalian University of TechnologyDalianChina

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  • Shun-Ying Ji

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