# Study on the Constitutive Equation and Mechanical Properties of Natural Snow under Step Loading

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Sample Preparation and Experimental Procedure

#### 2.1. Sample Preparation

^{3}, 400 kg/m

^{3}, 450 kg/m

^{3}, 500 kg/m

^{3}, and 550 kg/m

^{3}were created using different masses of snow. To prevent the snow samples from sublimating during storage and to maintain integrity, they were wrapped tightly in cling film and stored in a refrigerator at a constant temperature of −25 °C. As the tests were carried out at different temperatures, the snow samples were stored in a refrigerator at the required temperature for 24 h prior to each test to ensure uniform temperature throughout the entire sample. Before the test began, the mass and height of the snow samples were measured using an electronic scale and tape measure to calculate the actual density, ensuring that the error between it and the experimental density was less than ±20 kg/m

^{3}. Through measuring the mass and volume of snow, the natural density of fresh snow was derived to be 128 kg/m

^{3}. The above operations can reduce the error to the tolerable range.

#### 2.2. Test Procedure

## 3. Constitutive Model

#### 3.1. Maxwell Model

#### 3.2. Parameter Calculation

^{−1}); t is the time (s); E is the elastic modulus of snow (MPa); and $\eta $ is the viscosity coefficient of snow (MPa·s).

## 4. Results

#### 4.1. Effect of Loading Rate on Elastic Modulus and Viscosity Coefficient of Snow

^{3}and 500 kg/m

^{3}snow samples at a constant temperature of −15 °C with step loading rates ranging from 5 to 50 N/s to observe the effect of loading rate on the elastic modulus and viscosity coefficient. As shown in Figure 5a, the elastic modulus is more scattered for the 500 kg/m

^{3}snow than for the 400 kg/m

^{3}snow. With increasing loading rate, the value in snow elastic modulus is significantly greater at high densities than at low densities, which indicates that the elastic modulus gradually increases with increasing density. At the same time, it is found that density is a factor that affects the degree of change in elastic modulus, with a greater impact on the elastic modulus with loading rate at higher densities. For example, the elastic modulus of the 400 kg/m

^{3}snow with a range of 5 to 50 N/s is 3.39 to 9.33 MPa, and the difference is 5.94 MPa. The elastic modulus of the 500 kg/m

^{3}snow with a range of 5 to 50 N/s is 7.18 MPa to 26.98 MPa, and the difference is 19.80 MPa. However, the results for the elastic modulus of snow show no clear tendency for dependence on the loading rate.

^{3}snow sample and from 2.57 to 7.27 GPa·s for the 400 kg/m

^{3}snow sample, showing that the results are more scattered for the high-density snow. The average value of the viscosity coefficient for 500 kg/m

^{3}snow is larger than for 400 kg/m

^{3}snow, which suggests that the viscosity coefficient tends to increase gradually with an increase in density. Density also influences the change degree of the viscosity coefficient from Figure 5b. The difference in viscosity coefficients is 4.70 GPa·s for snow samples at 400 kg/m

^{3}, and 29.72 GPa·s for snow samples at 500 kg/m

^{3}. It is shown that the higher the density, the more intense the change in viscosity coefficient. When the loading rate is less than 10 mm/min, the mean values of the viscosity coefficients at both densities tend to decrease with an increase in loading rate. However, the mean values of viscosity coefficients at the two densities show different trends with increasing loading rate. The mean value of snow viscosity coefficient tends to be a constant at 400 kg/m

^{3}, while the mean value of viscosity coefficient tends to increase, then decrease, and then increase again at 500 kg/m

^{3}.

#### 4.2. Effect of Temperature on Elastic Modulus and Viscosity Coefficient of Snow

^{3}snow samples is lower than that of the 500 kg/m

^{3}snow samples, which reverses the previous results. It is found that the average value of the elastic modulus first decreases and then increases with decreasing temperature above 500 kg/m

^{3}and fluctuates in a very narrow band with a density at 400 kg/m

^{3}. This concludes that the temperature has an impact on the change of the elastic modulus in the density range above 500 kg/m

^{3}. However, more experiments need to be conducted at densities below 500 kg/m

^{3}to uncover the exact density range within which temperature has an impact.

^{3}, the viscosity coefficient of snow varies from 3.36 to 7.77 GPa·s, and the difference is 4.41 GPa·s. At the density of 500 kg/m

^{3}, the viscosity coefficient of snow varies from 11.04 to 21.78 GPa·s, and the difference is 10.74 GPa·s. The mean value of the viscosity coefficient generally decreases and then increases with decreasing temperature. Furthermore, it is noted that the density of the snow samples affects the degree of variation in the mean viscosity coefficient with temperature. We also analyze the changes in the average value of viscosity coefficient under three densities. It is found that the minimum viscosity coefficient of snow at the density of 400 kg/m

^{3}is 3.98 GPa·s at −15 °C, and the maximum value is 6.64 GPa·s at −20 °C, with a difference of 2.66 GPa·s. The minimum viscosity coefficient of snow at the density of 500 kg/m

^{3}is 12.71 GPa·s at −15 °C, the maximum value is 20.13 GPa·s at −20 °C, and the difference is 7.42 GPa·s. Similarly, the minimum viscosity coefficient of snow at a density of 550 kg/m

^{3}is 22.24 GPa·s at −10 °C, and the maximum value is 33.68 GPa·s at −20 °C, with a difference of 11.44 GPa·s. It can be observed that from the above results, the viscosity coefficient of high-density snow samples fluctuates significantly more than that of low-density snow samples as the temperature decreases. Previous research results have shown that the viscosity coefficient can vary with temperature by four to five orders of magnitude, which was concluded via uniaxial compression tests [31].

#### 4.3. Effect of Density on Elastic Modulus and Viscosity Coefficient of Snow

^{2}) is 0.76, indicating a good correlation between both.

^{3}).

^{3}, the viscosity coefficient of snow varies from 1.38 to 31.22 GPa·s, which has a relatively poor repeatability compared with the elastic modulus. Furthermore, as the snow density gradually increases, the dispersion of the viscosity coefficient also increases. Taking the results for 350 kg/m

^{3}and 550 kg/m

^{3}as examples, the viscosity coefficient of the 350 kg/m

^{3}snow is from 1.38 to 3.75 GPa·s, and the difference is 2.37 GPa·s. The viscosity coefficient of the 550 kg/m

^{3}snow is from 13.68 to 31.22 GPa·s, and the difference is 17.54 GPa·s. This indicates that the difference in the viscosity coefficient of snow becomes larger with increasing density, emphasizing the need to consider multiple measurements and calculate the average value for accurate results in practical applications. The functional relationship between the viscosity coefficient and density is obtained by the least square method, according to Equation (10). The relationship between the viscosity coefficient and the density of snow is an exponential function, and the coefficient of determination (R

^{2}) is 0.78, indicating a good correlation between both.

^{3}).

^{2}values of both fitted equations are above 0.70, indicating that the fitted curves are closer to the actual data and the derived constitutive equations are more accurate.

^{3}are taken in the article; σ is the compressive strength of snow (MPa); and $\dot{\epsilon}$ is the strain rate (s

^{−1}).

## 5. Discussion

#### 5.1. Elastic Modulus

^{3}; 1 × 10

^{−4}~2 × 10

^{−3}s

^{−1}). A weak correlation between normalized Young’s modulus and strain rate was found, indicating that the elastic modulus of snow increases with strain rate. This result was attributed to the relationship between the elastic modulus of ice and the strain rate. In conclusion, this discrepancy may be attributed to differences in the microstructure of the snow samples, as well as variations in test conditions and methods. To compare and verify these results, standardized test methods for evaluating the viscoelastic properties of snow are required.

^{−3}~2 × 10

^{−2}s

^{−1}), which is much higher than the elastic modulus value of other data. Köchle and Schneebeli [40] utilize X-ray microcomputer tomography to construct the three-dimensional microstructure of snow and calculate the elastic modulus D using finite element simulation, ranging from 0.3 to 1000 MPa. This value is also higher than the elastic modulus obtained using quasi-static measurements at the same density. This is because the quasi-static measurement of the elastic modulus inevitably takes the viscous strain into account, which leads to the measured elastic modulus not representing the true value.

#### 5.2. Viscosity Coefficient

^{3}at −7 °C to −10 °C and found a positive correlation between the viscosity coefficient and density. It was concluded that the viscosity coefficient gradually increased with increasing density through triaxial compression tests and uniaxial compression creep tests from [15,35].

^{3}. Additionally, the tendency of the elastic modulus to change with density is only indirectly affected at higher temperatures (−5 °C and −10 °C). The effect of temperature on the elastic modulus of snow is relatively complex [46]. Furthermore, the test also shows that the viscosity coefficient of snow increases exponentially with increasing density, first decreasing and then increasing with temperature. Limited by the test conditions, the variable range set in this test was relatively small. For example, the temperature range was only −5~−20 °C, the density range was only 350 kg/m

^{3}~550 kg/m

^{3}, and the loading rate range was only 5~50 N/s. Although the results showed that the elastic modulus and viscosity coefficient varied with temperature and density. A reasonable physical model could not be built to describe the response of elastic modulus and viscosity coefficient to different influencing factors. Therefore, the subsequent temperature and density range should be expanded in an effort to explore its physical phenomenon, which is beneficial for the application of snow in different aspects, for example, in the construction of snow roads in cold regions, runways, and polar infrastructure. The constitutive equations are used to describe the deformation behavior of snow under different loading conditions, which can provide important mechanical parameters and technical support for the design, construction, and maintenance of these snow-related projects.

## 6. Conclusions

- The elastic modulus of natural snow increases exponentially with increasing density. Temperature has a certain influence on the elastic modulus of snow. The elastic modulus first decreases and then increases with decreasing temperature, and this relationship is more obvious at high densities. There is no correlation between the elastic modulus and the loading rate.
- Density is an important factor in the change in the viscosity coefficient of snow. The viscosity coefficient increases exponentially as density increases. The snow viscosity coefficient is affected by temperature, that is, the viscosity coefficient first decreases and then increases as the temperature decreases. The loading rate is weakly correlated with the viscosity coefficient of snow.
- A new constitutive equation considering snow density is derived by introducing the functional relation between elastic modulus, viscosity coefficient, and the density of snow based on the Maxwell model.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Compacting tools and snow sample: (

**a**) Compaction equipment, from left to right, the protective cylinder, the compactor hammer, and the compaction cylinder are followed; (

**b**) sample.

**Figure 2.**WDW–100 universal testing machine: 1. compression device; 2. cryogenic test chamber; 3. control system.

**Figure 5.**The relationship between the viscoelastic properties and loading rate: (

**a**) the elastic modulus; (

**b**) the viscosity coefficient.

**Figure 6.**The relationship between the viscoelastic properties and temperature: (

**a**) the elastic modulus; (

**b**) the viscosity coefficient.

**Figure 7.**The relationship between the viscoelastic properties and density: (

**a**) the elastic modulus; (

**b**) the viscosity coefficient.

**Figure 8.**The relationship between the elastic modulus and density from different studies: (A) Uniaxial compression, −25 °C [37]; (B) triaxial compression, −20 °C < T < −2 °C [38]; (C) triaxial compression, −12 °C [41], published in [39]; (D) finite element simulation, the data fit curve [40]; (E) uniaxial compression and tension, −25 °C < T < −12 °C [31]; (F) uniaxial compression under step loading, −5 °C (in this study).

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**MDPI and ACS Style**

Han, H.; Yang, M.; Liu, X.; Li, Y.; Gao, G.; Wang, E.
Study on the Constitutive Equation and Mechanical Properties of Natural Snow under Step Loading. *Water* **2023**, *15*, 3271.
https://doi.org/10.3390/w15183271

**AMA Style**

Han H, Yang M, Liu X, Li Y, Gao G, Wang E.
Study on the Constitutive Equation and Mechanical Properties of Natural Snow under Step Loading. *Water*. 2023; 15(18):3271.
https://doi.org/10.3390/w15183271

**Chicago/Turabian Style**

Han, Hongwei, Meiying Yang, Xingchao Liu, Yu Li, Gongwen Gao, and Enliang Wang.
2023. "Study on the Constitutive Equation and Mechanical Properties of Natural Snow under Step Loading" *Water* 15, no. 18: 3271.
https://doi.org/10.3390/w15183271