Mechanical properties of pressure-frozen ice under triaxial compressive stress

Abstract

Artificial ground freezing is an effective method for underground constructions in deep alluvium. To study the compressive strength of frozen soil under high ground pressure and high hydraulic pressure, it is necessary to understand the mechanical behaviour of ice that is formed under triaxial compressive stress. A low-temperature triaxial test system was developed and used to study both formation and deformation of columnar ice under hydrostatic pressure. Cylindrical ice specimens 125 mm in height and 61.8 mm in diameter were prepared and tested under constant strain rates. At a strain rate of 5 × 10⁻⁵ s⁻¹, the peak axial stress showed a linear increase as the confining pressure increased from 2 to 30 MPa, while the peak deviatoric stress exhibited a slight decrease. At a confining pressure of 30 MPa, the peak deviatoric stress showed a logarithmic increase with the strain rate increasing from 5 × 10⁻⁶ to 5 × 10⁻⁴ s⁻¹, and the failure strain nearly doubled. A power law relationship between the time to failure and the strain rate was also observed. In this study, each test consistently demonstrated a ductile failure mode, with a noticeable reduction in cracking as the confining pressure increased. Due to the effect of the high confining pressure, crack propagation was suppressed, and an apparent recrystallization after peak stress was observed.

1. Introduction

1.1. Artificial ground freezing

Coal resources are abundant in China, with a total amount of 5.97 trillion tons. However, 53% of these resources are buried at depths exceeding 1000 m, and in the eastern and north-eastern regions, many valuable deposits are covered by deep alluvium (Cui, 1998). A vertical shaft, with a diameter of 5–15 m and a depth of several hundred metres, is typically constructed to serve as the primary passage for facilities and personnel entering and exiting the mine tunnel. Artificial ground freezing (AGF) is an effective method for constructing vertical shafts and other deep underground projects in water-bearing surface soil layers (Bai and others, 2020). The fundamental principle in ground freezing is to convert pore water into ice. Initially, a freezing station is constructed to continuously generate low-temperature brine. Subsequently, a series of holes is drilled radially around the proposed vertical shaft, penetrating through soil or rock layers. Refrigerant circulation pipes (referred to as ‘freezing pipes’) are then inserted into these holes, with each freezing pipe serving as the central point for freeze-up. As the freezing process advances, adjacent frozen soil (or rock) gradually intersects to form a closed circular structure, known as a ‘frozen wall’. This frozen wall not only withstands ground pressure but also prevents the inflow of groundwater, ensuring safe and efficient excavation. However, the mechanical properties of the ice-bearing soil/rock mass within the frozen wall are complex and not well understood, leading to inaccurate estimations of the design thickness of the frozen wall. On one hand, if the strength of the frozen soil body is underestimated, it will result in unnecessary construction costs. On the other hand, if the strength of the frozen soil body is overestimated, considering the creep behaviour of the frozen soil, it may undergo large deformations or even sudden instability and failure during its service period, leading to irreversible substantial losses.

Much research has been devoted to the mechanical properties of frozen soil and rock in the context of deep earth and cold region engineering construction (Lai and others, 2013; Shan and others, 2015; Gao and others, 2020; Kuang and others, 2022). As a binding agent, ice provides the cohesive force within frozen soil or rock. It is also critical to note the significant influence that ice lenses (or segregated ice) exert on the mechanical properties of permafrost (Shan and others, 2017; Wang and others, 2018). Figure 1 illustrates the application of AGF in shaft sinking and the formation of ice lenses in the freezing process.

Figure 1. Diagram of the artificial ground freezing (AGF) method for shaft sinking and the formation of ice lenses.

1.2. Ice mechanics study

Following investigations in the 1950s and 1960s on ice deformation (Rigsby, 1951, 1957, 1960; Glen and Perutz, 1954; Glen, 1955; Langway, 1958; Glen and Jones, 1966), a multitude of studies on the triaxial mechanical properties of sea ice and fresh water ice have been conducted. Some employed constant strain rate tests, while others utilized constant stress tests (creep tests). Deformation of ice in a laboratory setting was best described by Glen's flow law, which takes strain rate, deviatoric stress, activation energy and temperature (Glen, 1955) into consideration. The impact of confining pressure and strain rate on strength was non-monotonic; furthermore, these two factors exhibited mutual influence (Goughnour and Andersland, 1968). At low temperatures and high strain rates, the fracture behaviour of ice showed a pressure dependence similar to that observed in brittle rocks, where strength increased with confining pressure (Nawwar and others, 1983; Cox and Richter-Menge, 1988; Richter-Menge, 1991; Gagnon and Gammon, 1995). However, at higher temperatures and pressures, strength was independent of pressures or decreased slightly with further increase in confinement (Jones, 1982; Durham and others, 1983; Kirby and others, 1985; Nadreau and Michel, 1986; Murrell and others, 1991; Rist and Murrell, 1994; Rist and others, 1994). Under a given confining pressure and temperature, a transition occurred from ductile to brittle behaviour upon raising strain rate (Schulson, 1990). Meanwhile, fine-grained ice was more prone to behaving in a ductile manner and had higher strength compared to columnar ice or coarse-grained ice (Cole, 1985; Mizuno, 1998). This could be due to its larger grain boundary area, resulting in a stronger capacity for energy absorption.

The brittle failure process has been well studied and can be described by the fault triggering mechanism incorporating both frictional sliding and splay cracking (Schulson and others, 1989, 1991; Gratz and Schulson, 1997; Schulson and Gratz, 1999; Schulson and Renshaw, 2022). The ductile failure process was mainly caused by micro-fracturing and recrystallization. Under high hydrostatic pressure combined with shear, a reduction of grain size was observed, indicative a deformation mechanism of very different nature than those operating at lower levels of pressure. (Meglis and others, 1999; Melanson and others, 1999; Barrette and Jordaan, 2001, 2003; Jordaan and others, 2005).

However, although there have been numerous studies on the mechanical response of ice under multiaxial compressive loading, there is still limited research available on ice formed in soils at high pressures (at depth), here referred to as pressure-frozen soil. First and foremost, it is necessary to develop a technique for freezing water specimens into ice under high confining pressure. Following this, triaxial tests of pressure-frozen ice should be conducted at different temperature, confining pressure, strain rate, etc. The ultimate goal of this research is to determine which findings from previous studies on sea ice or fresh water ice align with those on pressure-frozen ice, and which can be applied to deep-ground engineering and explore whether pressure-frozen ice possesses any unique properties.

In this paper, a novel method for preparing ice specimens under pressure of 30 MPa was firstly introduced. Then, constant strain rate (from 5 × 10⁻⁶ to 5 × 10⁻⁴ s⁻¹) triaxial tests were conducted at a temperature of −10℃, and the effect of confining pressure (from 2 to 30 MPa) and strain rate on mechanical properties of pressure-frozen ice were analysed. Finally, by combining observations of specimens after loading with thin sections taken at different loading stages, the failure mode of ice was initially discussed.

2. Apparatus and specimen preparation

2.1. Apparatus

A servo-controlled low-temperature triaxial testing system to grow and test ice under pressure was developed (Fig. 2). The system consisted of a triaxial cell inside which ice specimens are studied; a load frame housing the cell and providing axial load; oil, an oil pump, and a pressure volume controller to provide confining pressure; and a cooling system. The system is capable of imposing an axial load up to 400 kN and a confining pressure up to 64 MPa, with accuracies of 0.01 and 0.1%, respectively. The axial force was measured with a load cell. A 60 mm-thick insulation jacket was wrapped around the triaxial cell to prevent excessive inward heat flow from the surroundings. Two thermocouple rods were created to monitor the temperature of specimen and oil.

Figure 2. KTL-401 servo-controlled low-temperature triaxial testing system.

2.2. Fabrication of water specimen

The specimen space was constructed using two end caps and one heat-shrinkable tubing. One of the end caps was composed of inner and outer caps linked together through a threaded fastening method. The other end cap, located on the central axis, featured a water injection hole equipped with a closure bolt (Fig. 3a). An assembly cylinder was created, comprising an internal positioning column and four external split tubes. The column consisted of two sections: a long section matching the outer cap's inner diameter and a shorter section matching the water injection hole's inner diameter. Split tubes were partitioned equally along the circumference, each sharing the external size of the water specimen and the inner diameter of the outer cap (Fig. 3b).

Figure 3. The components of (a) end caps and (b) assembly cylinder.

During the fabrication process, the outer cap and end cap were placed separately at the ends of the positioning column. Split tubes were then assembled around the outside to form the cylinder assembly. A specific type of low-temperature-resistant adhesive with a slow curing speed was applied to the cap surfaces. Heat shrink tubing was applied over the entire structure, heated to tightly encase the caps and split tubes. Once the adhesive has fully cured, the positioning column and split tubes are removed through the outer cap. Finally, securing the inner cap to the outer cap completed the water specimen package. After filling the package with degassed distilled water through the water injection hole and tightening the closure bolt, the final cylindrical water specimen was obtained.

A mounting device (Fig. 4), comprising a fixed platform and a lifting platform, was designed to ensure the water specimen is not deformed due to the gravity exerted by the upper end cap and plunger during the insertion process. The lifting platform was equipped with a hanger and two hanger rods. The upper and lower end caps are securely attached to the plunger and baseplate, respectively. The plunger was threaded connected to hanger rods and suspended within the pressure vessel, with the baseplate resting on the lifting platform. Subsequently, the lifting platform elevates both the plunger and the baseplate in perfect synchrony, ensuring the insertion of the water specimen without any deformation. Following this step, the baseplate and pressure vessel were securely fastened together using eight equally spaced bolts to form the triaxial cell, which was then transported to the load frame. The final step involves connecting all the necessary conduits, completing the assembly of the test system.

Figure 4. Mounting device.

2.3. Preparation of pressure-frozen ice specimen

The oil was injected to the triaxial cell by the oil pump. After the injection process was completed, the water specimen was pressurized to 30 MPa at a speed of 0.2 MPa s⁻¹ by the pressure volume controller.

The cooling system consisted of circulation thermostats, and freezing channels located inside the upper, lateral and lower parts of the triaxial cell (Fig. 2). The freezing channels were connected to thermostats to circulate the ethyl alcohol at 2℃ (in the upper channel) and −30℃ (in the lower channel) in order to freeze the water specimen from bottom to top.

In the context of this study, both freezing and loading were conducted continuously under pressure. To ensure that the water specimen maintains a standard cylindrical shape after freezing, it is crucial to control the volume of oil during the freezing process. For example, if no measures are taken, the hydraulic oil will lose volume because of cold-induced contraction and the water specimen will expand along both axial and radial directions when freeze-up occurs, rather than forming a standard cylinder, and the specimen diameter will exceed that of the loading rod. Thus, in our work, oil was continuously replenished during the freezing process in order to compensate for the volume loss and force the specimen to expand only along the axial direction, by constraining the radial expansion. With the appropriate replenishment rate, the pressure was maintained at 30 MPa throughout the freezing process. After the freezing process was completed, the lateral channel was used to circulate the ethyl alcohol at −15℃ for 5 h, ensuring a uniform specimen temperature of −10℃.

Before commencing the formal experiments, several preliminary trials were conducted to investigate the temperature evolution during the freezing process and the uniformity of the specimen's temperature. Five thermocouples were evenly attached to a glass fibre rod and fixed along the central axis inside the specimen, while an additional four thermocouples were fixed outside the specimen (as shown in Fig. 5). They were used to measure the temperature of the specimen and hydraulic oil. Figure 6 displays the temporal variations of hydraulic oil temperature, specimen temperature and the volume of oil replenishment over time. Based on the temperature data of the oil, the replenishment amount at each moment can be calculated, approximated with a piecewise linear function for operational convenience (black dashed line in Fig. 6). Temperature data of the specimen indicated that, after 5 h of circulating the ethyl alcohol at −15℃ (in the lateral channel), the specimen's temperature stabilized at (−10 ± 0.3)℃. Among them, points A and E showed temperatures around −9.7 to −9.8℃, while points B, C and D showed slightly lower temperatures, approximately −10.3℃.

Figure 5. Layout of temperature measurement points.

Figure 6. Temperature of oil and specimen and the volume of oil replenishment over time.

Two preliminary experiments were conducted to determine if the specimen dimensions meet the requirements for triaxial testing. The average diameters of upper, medium and lower parts of specimens were 61.71, 61.73 and 61.92 mm, respectively, while the average height was 125.32 mm, meeting the standard size of 61.8 mm in diameter and 125 mm in height (Fig. 7a). Thin sections showed that the specimen was bubble-free, and the columnar ice had a grain size of 5.1 ± 0.6 mm (Figs 7b, c). The density remained stable, ranging from 0.915 to 0.916 g cm⁻³.

Figure 7. Pictures of (a) ice specimen, (b) thin section under natural light and (c) thin sections viewed through crossed polaroids.

3. Tests and results

3.1. Stress and strain measures

Axial stress σ ₁ is defined by

(1)$$\sigma _1 = \displaystyle{F \over A}, \;$$

where F is the axial force, acting between the loading rod and the specimen, directly measured by a load cell. A is the initial circular cross-sectional area, which is equal to 2998.1 mm².

Radial stress σ ₃ is equal to the confining pressure. Deviatoric stress is defined as the difference between the axial stress and the radial stress, which is represented by σ ₁–σ ₃.

Axial true strain ε ₁ is defined by

(2)$$\varepsilon _1 = \int {\displaystyle{{{\rm d}\delta _{\rm s}} \over h}} , \;$$

where δ _s is the deformation of the specimen and h is the instantaneous length of the specimen.

The deformation of a test specimen is not directly measured but is calculated by

(3)$$\delta _{\rm s} = \delta -\delta _{\rm f}, \;\;\;\delta _{\rm f} = ( {4.64 \times {10}^{{-}2}} ) + ( {7.2 \times {10}^{{-}3}} ) F-( {1 \times {10}^{{-}5}} ) F^2 + ( {1 \times {10}^{{-}8}} ) F^3, \;$$

where δ is the displacement of the loading device and is measured during a test. δ _f is the deformation of the load frame, and was obtained through a calibration using a 304 stainless steel specimen with the same dimensions as the ice specimen and a compression stiffness of 194 GPa. Based on this, a formula for calculating the deformation of load frame δ _f from the axial force F is obtained through polynomial fitting, where δ, δ _s and δ _f are measured in millimetres (mm), F is measured in kilonewtons (kN), and the units of the coefficients are determined by the power of F.

3.2. Experimental scheme

Figure 8 shows the schematic diagram of loading and unloading. Once the freezing was completed, the confinement was immediately adjusted from the freezing pressure to the confining (testing) pressure. To prevent damage, a maximum rate of 0.5 MPa min⁻¹ was used. Here, the freezing pressure σ _f is 30 MPa, representing the pressure applied during the freezing process. σ _c means the confining pressure during triaxial tests, ranging from 2 to 30 MPa. Axial force was applied in a constant strain rate, ranging from 5 × 10⁻⁶ to 5 × 10⁻⁴ s⁻¹, and the loading was stopped when the axial strain reached 5%. After tests, the deviatoric stress was firstly unloaded at a rate of 0.05 mm min⁻¹, and then, the axial stress and radial stress were synchronously unloaded to zero at a rate of 0.33 MPa min⁻¹. Finally, the oil was drained, and the specimen was removed from the cell.

Figure 8. Schematic diagram of loading and unloading.

3.3. Test results

Test results are summarized in Table 1, and axial stress–axial strain curves are shown in Figure 9. The peak point on the stress–strain curve corresponds to the peak axial stress σ _1p, where the difference of peak axial stress and confining pressure is the peak strength (or peak deviatoric stress). The axial strain at the highest point is regarded as the failure strain ε _1p. The deviatoric stress at 5% axial strain corresponds to the residual strength. The stress–strain curves consistently demonstrated a pattern of strain-softening. In the initial stage, stress and strain exhibited a linear relationship. After reaching the peak, the ice entered the softening stage where the stress decreased, exhibiting a noticeable post-peak inflection point. Ultimately, the specimen reached the plastic flow stage, where the stress remained nearly constant.

Table 1. Summary of test results

Figure 9. Axial stress–axial strain curves at (a) different confining pressure and at (b) different strain rates.

4. Discussion

4.1. Effect of confining pressure

At a strain rate of 5 × 10⁻⁵ s⁻¹ (Fig. 10), the peak axial stress showed a linear increase with confining pressure. At a confining pressure of 2 MPa, the strength measured 15.2 MPa, and it increased to 41.7 MPa when subjected to a confining pressure of 30 MPa. The axial stress and confining pressure exhibited a linear relationship with a slope of 0.90, less than unity, which means the peak deviatoric stress slightly decreases as the confining pressure increased from 2 to 30 MPa. Previous experiments on polycrystalline ice have showed that at a temperature of (−11.8 ± 0.9)°C and a strain rate of 5.5 × 10⁻⁵ s⁻¹, the strength of polycrystalline ice (fine-grained ice) under confining pressures of 10, 20 and 30 MPa is reported to be 6–7 MPa. However, the strength of columnar-grained ice in this study is around 10 MPa (Jones, 1982). This could be attributed to a higher proportion of grain boundary area in polycrystalline ice, making it more prone to softening. Besides, the failure strain in all tests was <1%, with the range of 0.64–0.90%, and the average of 0.76%. The average residual strength was 4.11 MPa, with a minimum of 3.08 MPa occurring at a confining pressure of 2 MPa, and a maximum of 4.68 MPa occurring at a confining pressure of 10 MPa.

Figure 10. Dependence of peak axial stress and peak deviatoric stress on confining pressure at a strain rate of 5 × 10⁻⁵ s⁻¹.

Currently, with reference to the construction of shafts using artificial freezing as an example, the depth of soil and water-rich rock penetration has surpassed 750 and 950 m, respectively (Dai and Wang, 2019). Theoretical estimates suggest that the water pressure within the soil layer nears 8–10 MPa, alongside an approximate vertical ground stress of 20 MPa. Considering the localized pressure increase resulting from freezing expansion during freezing operations, the underground water pressure involved will exceed 10 MPa. In the rock layers, water pressure spans from 15 to 25 MPa, while vertical stress may exceed 65 MPa. The findings of this study reveal that the peak axial stress in ice under a confining pressure of 30 MPa is 2.7 times higher than that under a confining pressure of 2 MPa. This implies that, for deep underground construction projects involving high-pressure frozen soil, rock or ice excavation, the utilization of more effective drilling and blasting methods or more robust equipment is warranted to overcome the increasing strength associated with depth or overburden pressure. In addition, the confinement during ice growth, the freezing pressure, could have affected the ice strength. The freezing pressure may decrease the porosity and air volume of the ice, thus increasing the strength, but it is also possible that any air in the pores is under pressure causing local tensile stresses within the ice. Further, as the freezing pressure was not removed before the strength tests, any stresses formed in the ice during the freezing process were not able to undergo any relaxation before measuring the strength.

4.2. Effect of strain rate

At a confining pressure of 30 MPa, the peak deviatoric stress showed a linear increase with the logarithm of strain rate. When the strain rate increased from 5 × 10⁻⁶ to 5 × 10⁻⁴ s⁻¹, the strength of ice increased from 6.15 to 23.87 MPa, and the failure strain nearly doubled from 0.64 to 1.38% (Fig. 11). This serves as a reminder that when using machinery such as the tunnel boring machine for excavating through ice or frozen soil layers, it is crucial to carefully control the excavation speed, as excessively high rates can lead to accelerated wear and tear on components.

Figure 11. Dependence of peak deviatoric stress and failure strain on strain rate at a confining pressure of 30 MPa.

In addition, the time from the start of loading to reaching peak strength is defined as the time to failure t _f. It was found that the failure time and the strain rate have a power function relationship (Fig. 12), that fits the following equation:

(4)$$t_{\rm f} = 3.48\dot{\varepsilon }^{{-}0.85}.$$

Figure 12. Dependence of the time to failure on strain rate.

The power exponent is close to that of previous uniaxial tests which were carried out for columnar ice with grain size of 4–5 mm at −10℃, where it was found to be −0.82 (Sinha, 1982). Within the ductile range, a lower strain rate provides more time for the evolution of internal damage of the ice specimen, ultimately may leading to a lower strength.

4.3. Failure mode

The stress–strain curves of all tests in this paper exhibited ductile behaviour, with stress showing a gradual softening after reaching the peak point, rather than a sudden decrease. However, there were still some differences in cracking. At a strain rate of 5 × 10⁻⁵ s⁻¹, with confining pressures of 2 MPa and 10 MPa, numerous fishbone-like and wing-shaped cracks were observed, and a higher confining pressure resulted in reduced cracking. When the confining pressure was 30 MPa, even if the strain was loaded to 5%, the ice was still transparent and clear, without visible cracks. With the increase of strain rate, although a small number of fine microcracks appeared, it was still almost clear (Fig. 13). This suggests that confining pressure plays a role in limiting crack nucleation and growth, which is consistent with previous observations of polycrystalline ice (Rist and Murrell, 1994). In the earlier creep tests of multiyear sea ice at confining pressure of 30 MPa, strain rate of 10⁻² s⁻¹, and temperature of −10℃, it was also observed that cracking was completely inhibited (Sammonds and others, 1998).

Figure 13. Crack distribution when loaded to 5% strain.

Thin sections of specimens of axial strain ε ₁ = 0, 0.5, 1.5% at a confining pressure of 30 MPa and a strain rate of 5 × 10⁻⁵ s⁻¹ were made (Fig. 14). It was found that the columnar grains essentially maintained their initial appearance both before loading and when loaded to a strain of 0.5%, and the recrystallization was observed when the strain reached 1.5%. This reflects that under high confining pressure, the strain-softening behaviour of ice may be primarily driven by dynamic recrystallization (DRX). In the future, more experiments should be taken to study if there are clear relationships between DRX and confining pressure or strain rate.

Figure 14. Thin sections of different loading stage at a confining pressure of 30 MPa and a strain rate of 5 × 10⁻⁵ s⁻¹.

5. Conclusions

Using a self-developed low-temperature triaxial test system, cylindric columnar ice specimens were made for constant strain rate triaxial tests. The main conclusions can be summarized as follows:

1. (1) A method for preparing pressure-frozen ice was described. By freezing the water specimen from bottom to top and replenishing the oil during freezing process, cylindric columnar ice specimens can be obtained, with the size of 125 mm in height and 61.8 mm in diameter, and its density is close to pure fresh water ice. Both freezing and testing were conducted continuously under pressure.

2. (2) Constant strain rate triaxial tests of 30 MPa pressure-frozen ice at −10℃ in the range of confining pressure of 2–30 MPa and strain rate of 5 × 10⁻⁶ to 5 × 10⁻⁴ s⁻¹ were conducted. At a constant strain rate of 5 × 10⁻⁵ s⁻¹, the peak axial stress showed a linear increase with confining pressure, while the peak deviatoric stress exhibited a slight decrease.

3. (3) At a confining pressure of 30 MPa, the peak deviatoric stress showed a linear increase with the logarithm of strain rate, and the failure strain has nearly doubled, when the strain rate increased from 5 × 10⁻⁶ to 5 × 10⁻⁴ s⁻¹. It was also found that the time to failure and the strain rate have a power law relationship.

4. (4) Each set of tests exhibited a ductile failure mode, although with some variations. The failure mode of ice includes macro-scale crack propagation and micro-scale recrystallization. Cracking was suppressed by high confining pressure. Recrystallization occurred after the specimen reached peak strength, and its correlation with confining pressure and loading rate warrants further investigation.

5. (5) The focal point of deep-earth research lies in recreating (or simulating) high-pressure environments in the laboratory. The freezing and testing method proposed in this paper, along with its characteristic of maintaining controlled stress states throughout the entire process, facilitates the conduction of triaxial tests under different stress paths. For example, confining pressure reduction tests can reflect lateral unloading conditions in wells (or boreholes), while axial pressure reduction tests can reflect bottom unloading conditions.