# Experimental Investigation of Uniaxial Compressive Strength of Distilled Water Ice at Different Growth Temperatures

^{*}

## Abstract

**:**

^{−6}s

^{−1}to 10

^{−2}s

^{−1}. It is found that the uniaxial compressive strength of ice is a power function of strain rate and a linear relationship with the −1/2 power of grain size. Combined with the relationship between strength and grain size and the relationship between grain size and temperature, it is deduced that the peak compressive strength has a logarithmic relationship with the growth temperature. In addition, it shows that the growth temperature affects the strength of ice by controlling the grain size.

## 1. Introduction

^{−1}. Kim indicated that the uniaxial compressive strength of ice is almost constant when the strain rate is high enough [16]. Timco and Frederking [17] studied the mechanical properties of freshwater ice and summarized the quantitative expressions of compressive strength and strain rate of ice in a ductile regime. Bonath, Sinha and Chen also discussed the relationship between strain rate and stress for different types of ice [18,19,20]. A series of studies have shown that the maximum compressive strength of ice appears in a special strain rate range, under which ice performs a transition state from ductility to brittleness, which is called the ductile-to-brittle transition regime. Qi’s results show that the ductile-to-brittle transition regime appears in the strain rate range of 10

^{−4}s

^{−1}to 10

^{−3}s

^{−1}[21]. However, some studies found that the compressive strength reaches a maximum at a larger strain rate. Deng [22] found that the ductile-to-brittle transition regime of ice at the test temperature of −18 °C was in the range of 10

^{−4}s

^{−1}to 10

^{−2}s

^{−1}through a uniaxial compression test. Schulson [23] obtained the peak compressive strength at the strain rate of 10

^{−2}s

^{−1}.

## 2. Methods

#### 2.1. Preparation of Distilled Water Ice and Test Ice Samples

#### 2.2. Physical Properties Measurement

#### 2.2.1. Ice Density Measurement and Porosity Calculation

^{3}); $S$ is salinity (‰); $T$ is ice temperature (°C); ${\rho}_{i}$ is the density of pure ice (g/cm

^{3}); ${F}_{1}\left(T\right)$ and ${F}_{2}\left(T\right)$ are cubic polynomials about temperature. The research object of this study is distilled water ice with a salinity of 0, thus the porosity is the air volume fraction. Equation (1) can be simplified as:

#### 2.2.2. Ice Crystal Structure Measurement

^{2}); $n$ is the number of grains in the section.

#### 2.3. Uniaxial Compression Test

#### 2.3.1. Test Devices

#### 2.3.2. Test Principle and Procedure

^{−1}) is controlled by adjusting the displacement rate $\dot{X}$ (mm/s) of the indenter. It can be calculated by combining the original length ${L}_{0}$ (mm) of the sample as follows:

## 3. Results

^{−6}s

^{−1}to 10

^{−2}s

^{−1}calculated by Equation (7).

#### 3.1. Results of Uniaxial Compressive Strength Test

#### 3.1.1. Deformation Types

#### 3.1.2. Stress-Strain Curves Correction

^{−5}s

^{−1}as an example, Figure 7 shows the correction method of the stress-strain curve when ice shows ductile behavior at a low strain rate.

#### 3.2. Physical Properties and Crystal Structure of Distill Water Ice

^{3}increasing with the decrease in temperature. The ice density measured in this study is relatively large because ice is frozen from distilled water without salt and other impurities. Because there is almost no disturbance of air and water flow, which makes the ice crystal growth more uniform and stable, the icing process is static. In this way, the bubble content is low. The average porosity of distilled water ice grown at different temperatures is calculated by Equations (3) and (4) combined with the average density of ice samples. The results are shown in Table 2.

## 4. Discussion

#### 4.1. Stress vs. Strain Rate

^{−6}s

^{−1}to 10

^{−2}s

^{−1}, the uniaxial compressive strength first increases and then decreases. The strain rate range can be divided into ductile and brittle regimes according to the mechanical behavior of ice. The uniaxial compressive strength of ice reaches a maximum at the ductile-to-brittle transition point. The uniaxial compressive strength of ice at a strain rate higher than 10

^{−2}s

^{−1}was not obtained due to the limitation of test conditions. Some researchers have studied the uniaxial compressive strength of ice under a high strain rate. Test results from Jones [42] show that the uniaxial compressive strength of freshwater ice and low salinity sea ice does not decrease in the strain rate range of 10

^{−1}s

^{−1}to 101 s

^{−1}but continues to increase. Wu et al. [17] conducted uniaxial compression tests on distilled water ice and lake ice in the range of strain rate 80 s

^{−1}to 600 s

^{−1}and the uniaxial compressive strength changed little.

^{−1}. The fitting curves of the relationship between uniaxial compressive strength and strain rate in the ductile regime are shown in the dashed lines in Figure 11 and the empirical coefficient and judgment coefficient are given in Table 3.

#### 4.2. Stress vs. Grain Size

#### 4.3. Stress vs. Growth Temperature

## 5. Conclusions

- The growth temperatures were set at −5 °C, −10 °C, −15 °C, −20 °C, −25 °C, −30 °C and −35 °C. The ice crystal structure in this study is columnar. Ice density ranges from 900 kg/m
^{3}to 920 kg/m^{3}and increases with decreasing temperature. Grain size increases with the increases in ice depth, and the average grain size range is 2–7 mm, which roughly decreases with the decrease in temperature. - The uniaxial compressive strength of ice at different strain rates is obtained, which first increases with the increase in strain rate, and then decreases with the increase in strain rate after reaching the peak value. The results show that ice is ductile at a low strain rate and brittle at a high strain rate. The relationship between uniaxial compressive strength and strain rate is a power function.
- Compared with previous studies, it is found that the peak compressive strength of ice gradually increases with the decrease in grain size. The results show that the relationship between the peak compressive strength and the—1/2 power of grain size is a linear function.
- Referring to the previous research, three functional forms of linear, logarithmic and polynomial have been proposed. The experimental results of this study are fitted in three functions to reveal the mathematical relationship. In addition, by summarizing the relationship between growth temperature and grain size, combined with mathematical requirements and physical facts, it is finally determined that the relationship between ice peak compressive strength and growth temperature conforms to the logarithmic function. This result explains that the growth temperature affects the compressive strength of ice by controlling the grain size.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Ice preparation and cryogenic laboratory temperature control panel. (

**a**) Plastic foam condensation tanks filled with distilled water in a cryogenic laboratory. (

**b**)Temperature control panel.

**Figure 6.**Stress-strain curves and deformation types of uniaxial compression on distilled water ice grown at different temperatures. (

**a**) The strain rate is 10

^{−5}s

^{−1}, at which ice samples perform ductile deformation. (

**b**) The strain rate is 10

^{−3}s

^{−1}, at which ice samples perform brittle deformation. (

**c**,

**d**) are ductile and brittle deformation respectively.

**Figure 9.**Thin sections of distilled water ice grown at −15 °C photographed in polarized light. (

**a**) is the horizontal section at the depth of 0 cm (top) and (

**b**) is the vertical section of 8–16 cm.

**Figure 11.**Test data of uniaxial compressive strength versus strain rate and fitted curves for distilled water ice samples grown at different temperatures.

**Figure 13.**Peak stress versus growth temperature and curves fitted with different function types. Solid squares are test data. Dashed lines are fitted curves. (

**a**–

**c**) are linear, logarithmic and polynomial function respectively.

**Figure 14.**Grain size versus growth temperature and curves fitted with different function types. Solid squares are test data. Dashed lines are fitted curves. (

**a**) is a logarithmic function and (

**b**) is a linear function. Fitting equations and correlation coefficients are given.

T (°C) | $\dot{\mathit{\epsilon}}$ (s^{−1}) | ${\mathit{\sigma}}_{\mathit{m}\mathit{a}\mathit{x}}\left(\mathbf{MPa}\right)$ | ${\mathit{\epsilon}}_{\mathit{m}\mathit{a}\mathit{x}}(\%)$ |
---|---|---|---|

$1\times {10}^{-6}$ | 1.95 | 0.51 | |

$1\times {10}^{-5}$ | 3.52 | 0.64 | |

−5 | $1\times {10}^{-4}$ | 5.88 | 0.71 |

$1\times {10}^{-3}$ | 1.54 | 0.18 | |

$1\times {10}^{-2}$ | 1.30 | 0.26 | |

$1\times {10}^{-6}$ | 2.08 | 0.49 | |

$1\times {10}^{-5}$ | 3.98 | 0.72 | |

−10 | $1\times {10}^{-4}$ | 6.39 | 0.71 |

$1\times {10}^{-3}$ | 2.11 | 0.22 | |

$1\times {10}^{-2}$ | 1.75 | 0.27 | |

$1\times {10}^{-6}$ | 1.82 | 0.40 | |

$1\times {10}^{-5}$ | 4.51 | 0.67 | |

−15 | $1\times {10}^{-4}$ | 6.08 | 0.54 |

$1\times {10}^{-3}$ | 2.91 | 0.29 | |

$1\times {10}^{-2}$ | 2.29 | 0.27 | |

$1\times {10}^{-6}$ | 4.61 | 0.63 | |

$1\times {10}^{-5}$ | 4.67 | 0.79 | |

−20 | $1\times {10}^{-4}$ | 5.49 | 0.77 |

$1\times {10}^{-3}$ | 3.46 | 0.36 | |

$1\times {10}^{-2}$ | 1.75 | 0.48 | |

$1\times {10}^{-6}$ | 4.34 | 0.73 | |

$1\times {10}^{-5}$ | 5.81 | 0.87 | |

−25 | $1\times {10}^{-4}$ | 5.13 | 0.52 |

$1\times {10}^{-3}$ | 3.52 | 0.27 | |

$1\times {10}^{-2}$ | 1.22 | 0.34 | |

$1\times {10}^{-6}$ | 3.07 | 0.83 | |

$1\times {10}^{-5}$ | 7.17 | 0.94 | |

−30 | $1\times {10}^{-4}$ | 7.45 | 0.87 |

$1\times {10}^{-3}$ | 3.64 | 0.38 | |

$1\times {10}^{-2}$ | 1.23 | 0.31 | |

$1\times {10}^{-6}$ | 4.70 | 1.09 | |

$1\times {10}^{-5}$ | 8.65 | 0.99 | |

−35 | $1\times {10}^{-4}$ | 7.02 | 0.74 |

$1\times {10}^{-3}$ | 3.63 | 0.48 | |

$1\times {10}^{-2}$ | 2.18 | 0.22 |

Growth Temperature (°C) | Average Density (kg/m^{3}) | Average Porosity (‰) |
---|---|---|

−5 | 906 | 12.75 |

−10 | 907 | 12.42 |

−15 | 915 | 4.47 |

−20 | 916 | 4.14 |

−25 | 918 | 2.72 |

−30 | 919 | 2.40 |

−35 | 919 | 3.16 |

**Table 3.**The fitting coefficients and coefficients of determination ${R}^{2}$ of uniaxial compressive strength tests on distilled water ice.

${\mathit{T}}_{\mathit{g}}$ (°C) | Ductile Regime | Brittle Regime | ${\mathit{\sigma}}_{\mathit{c}\mathit{p}}$ (MPa) | ||||
---|---|---|---|---|---|---|---|

B | n | R^{2} | B | n | R^{2} | ||

−5 | 38.798 | 0.214 | 0.58 | 0.225 | −0.35 | 0.78 | 5.09 |

−10 | 60.236 | 0.243 | 0.70 | 0.460 | −0.246 | 0.76 | 5.34 |

−15 | 71.611 | 0.264 | 0.77 | 0.991 | −0.184 | 0.77 | 5.75 |

−20 | 16.837 | 0.095 | 0.73 | 0.248 | −0.34 | 0.88 | 6.70 |

−25 | 12.001 | 0.071 | 0.75 | 0.332 | −0.293 | 0.85 | 5.96 |

−30 | 38.262 | 0.172 | 0.71 | 0.553 | −0.277 | 0.63 | 7.55 |

−35 | 24.741 | 0.109 | 0.60 | 0.370 | −0.32 | 0.89 | 8.50 |

${\mathit{T}}_{\mathit{g}}$ (°C) | Average Grain Size (mm) | ${\mathit{\sigma}}_{\mathit{c}\mathit{p}}$ (MPa) |
---|---|---|

−5 | 5.86 | 5.09 |

−10 | 6.05 | 5.34 |

−15 | 4.85 | 5.75 |

−20 | 5.20 | 6.70 |

−25 | 4.02 | 5.96 |

−30 | 4.93 | 7.55 |

−35 | 2.37 | 8.50 |

Items | Schulson, 1990 | Nixon, 1996 | Cole, 1987 | Present Study |
---|---|---|---|---|

${\sigma}_{0}$ | 3.67 | 3.52 | −0.89 | −1.81 |

${k}_{c}$ | 8.27 | 26.4 | 12.25 | 17.91 |

R^{2} | 0.44 | 0.56 | 0.45 | 0.54 |

**Table 6.**Regression and correlation coefficients for peak stress vs. growth temperature fitting functions.

Function Type | Fit Coefficients | R^{2} | |||
---|---|---|---|---|---|

A | B | C | D | ||

Linear | 0.106 | 4.290 | / | / | 0.85 |

Logarithmic | 1.527 | 2.096 | / | / | 0.71 |

Polynomial | −0.0002 | −0.0103 | −0.217 | 4.16 | 0.91 |

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**MDPI and ACS Style**

Zhang, Y.; Qian, Z.; Lv, S.; Huang, W.; Ren, J.; Fang, Z.; Chen, X.
Experimental Investigation of Uniaxial Compressive Strength of Distilled Water Ice at Different Growth Temperatures. *Water* **2022**, *14*, 4079.
https://doi.org/10.3390/w14244079

**AMA Style**

Zhang Y, Qian Z, Lv S, Huang W, Ren J, Fang Z, Chen X.
Experimental Investigation of Uniaxial Compressive Strength of Distilled Water Ice at Different Growth Temperatures. *Water*. 2022; 14(24):4079.
https://doi.org/10.3390/w14244079

**Chicago/Turabian Style**

Zhang, Yujia, Zuoqin Qian, Song Lv, Weilong Huang, Jie Ren, Ziwei Fang, and Xiaodong Chen.
2022. "Experimental Investigation of Uniaxial Compressive Strength of Distilled Water Ice at Different Growth Temperatures" *Water* 14, no. 24: 4079.
https://doi.org/10.3390/w14244079