Experimental Study on the Characteristics of the Failure Strain Energy Density of Undisturbed Ice-Rich Frozen Clay

Abstract

Using the triaxial shear or compressive strength as a single index of the resistance of frozen soils to failure does not always meet frozen soil engineering requirements for the comprehensive evaluation of the resistance. In this study, triaxial compression experiments were carried out on undisturbed ice-rich frozen clay samples with various levels of water content under different confining pressures to study the characteristics of the failure strain energy density of the samples. The results indicate that as the confining pressure increased, the failure strain energy density first increased and then decreased. The failure strain energy density reached a maximum at a critical confining pressure of 2.00 MPa for 13.25–25.76% water content and 1.00 MPa for 26.02–45.82% water content. The failure strain energy density increased as the water content increased at low confining pressures (0.05–0.50 MPa) but then declined slightly at intermediate confining pressures (1.00–2.00 MPa). At a high confined pressure of 3.00 MPa, the failure strain energy density decreased overall as the water content increased. There were similarities and differences between the change characteristics of the compressive strength and the failure strain energy density. The failure strain energy density can be used as a supplementary reference index of the resistance of frozen soils to damage. The variation characteristics of the failure strain energy density of undisturbed frozen clay are essentially consistent with those of remolded frozen sandy soils. However, there are also clear differences between the characteristics of the failure strain energy density of these two types of frozen soil.

1. Introduction

Many engineered structures have been built on permafrost, such as some parts of the Qinghai–Tibet Railway and the China–Russia Crude Oil Pipeline [1,2]. Therefore, many studies have been carried out about the thermostability and mechanical characteristics of permafrost [1,3,4,5]. Permafrost regions often contain ice-rich frozen soils (a frozen soil with volumetric ice content above 20%) [1], the mechanical characteristics of which are very sensitive to changes in external conditions [6]. Therefore, it is important to research the mechanical characteristics of ice-rich frozen soils under different conditions for the maintenance of relevant engineered structures. The maximum deviatoric stress is normally taken as the shear or compressive strength to determine the mechanical failure characteristics of frozen soils. In triaxial compression tests, the deviatoric stress increases when the axial strain increases. The peak in the deviatoric stress is considered to be the shear or compressive strength. If a peak does not appear up to an axial strain of 20%, the deviatoric stress corresponding to an axial strain of 20% is taken as the shear or compressive strength [7], as shown in Figure 1.

As the triaxial shear or compressive strength is normally used as an index of the resistance of frozen soils to damage for frozen soil engineering, numerous experiments and theoretical studies have been carried out to determine the triaxial shear or compressive strength of frozen soils. Three stages have been identified where the confined pressure affects the strength of saturated frozen sand [8], and similar results have been obtained for saturated frozen sandy soil [9]. Some test results have demonstrated that the strength of frozen soil first rises and then declines when the confined pressure increases [10,11]. The effects of the confining pressure on the strength of an ice-rich rock glacier material were also studied [12]. Test results on frozen saturated sandy gravel indicated that as the confined pressure increased, the rate of increase in the strength gradually decreased [13]. However, some researchers found that changing the confining pressure had little or no effect on the strength of frozen soil [14,15]. It has been suggested that some of the differences in these test results can be attributed to the different ranges of confining pressure and water content that have been investigated [16,17]. The water content is an important factor affecting the strength properties and other parameters of frozen soil [18,19]. A general expression for the influence of the moisture content on the uniaxial compressive strength of frozen soil was presented in a previous study [18]. Some researchers have suggested that this general expression is considerably affected by the temperature and strain rate [20]. Experimental results for saturated frozen sand indicated that the uniaxial compressive strength decreased as the moisture content increased [21,22]. Similar results were found for frozen silty clay [23]. However, increasing the initial ice content from 30% to 304% caused the strength of frozen silty sand to increase nonlinearly and then gradually stabilize [24]. In some studies [25,26], different ranges of confining pressure and temperature were found to lead to different variation characteristics of the strength with the water content.

Although extensive studies have been carried out on the shear or compressive strength properties of frozen soil, the shear or compressive strength is only an index of the resistance of frozen soil to failure in terms of the stress magnitude. This approach does not always facilitate a comprehensive and detailed assessment of the resistance of frozen soil to failure for frozen soil practical engineering. The stress-strain curves of frozen soil may vary considerably while the shear or compressive strength remains unchanged. For example, the area of the graph enclosed by the stress-strain curve and the corresponding strain axis from 0 to the failure strain may change (the shaded region shown in Figure 1). The size of this area is directly related to the work exerted by the external forces from the initiation of the shear to the time at which the shear strength is achieved and corresponds to the strain energy density at this shear strength. The failure of frozen soil is also related to the energy input [27]. The energy input is another manifestation of the resistance of frozen soil to failure. The larger this area is, the more resistant the frozen soil is to damage. Studies on the damage characteristics of frozen soils have mostly been limited to remolded soil samples. However, some studies have shown both similarities and differences in the failure characteristics of remolded and undisturbed frozen soil samples [28,29]. Therefore, to address actual engineering needs, it is necessary to study the resistance of undisturbed frozen soil to failure considering the work exerted by the external forces. Considering the stress strength and failure strain energy density together provides a comprehensive and detailed assessment of the resistance of undisturbed frozen soil to failure, especially when the shear or compressive strength is fixed but the work the external forces exert changes. Du et al. determined the failure strain energy density characteristics of remolded frozen sandy soils [30,31]. Wang et al. studied the strain energy properties of frozen Genhe silty clay at the peak strength [27]. In this study, triaxial compression tests were performed on undisturbed ice-rich frozen clay samples with various levels of water content under different confining pressures to determine the variation characteristics of the failure strain energy density of undisturbed ice-rich frozen clay.

2. Experiment

2.1. Specimen Preparation

Undisturbed frozen clay samples were obtained from the permafrost region along the Qinghai–Tibet Railway at a sampling depth range of 0.0–5.0 m. Ten sampling points close to each other were chosen, and hollow cylindrical drill pipes were used to withdraw samples along a direction perpendicular to the ground level. To prevent disturbances to, and the melting of, the undisturbed soil samples, as well as the loss of water during transportation, appropriate measures for thermal insulation and protection of the samples were implemented. Figure 2 presents a representative grain size curve for the samples. The plastic limit is 18.80%, and the liquid limit is 36.50% for the samples. The cylindrical undisturbed soil samples were processed into standard samples (with a diameter of 61.8 mm and a height of 125.0 mm) along the sampling direction in a cryogenics laboratory. The end faces of the standard samples were flattened, and the sides of the standard samples were covered with rubber film. Two epoxy resin sample heads were fixed at both ends of each standard sample. Then, the standard samples were frozen for at least 48 h in a low-temperature thermostat at −30.0 °C. A total of 37 standard test samples were obtained. The sampling point numbers and depths of the standard test samples are presented in

Table 1. Sampling point numbers, sample depths, water contents and dry densities of the standard test samples.

Sampling Point Number	Sample Depth/m	Water Content/%	Dry Density/(g·cm−3)
10-1-2#	2.1	32.88	1.33
10-1-2#	2.3	16.84	1.84
10-1-2#	2.4	15.78	1.84
10-1-2#	2.5	18.12	1.74
10-1-2#	2.7	27.36	1.49
10-1-2#	2.9	19.81	1.70
10-1-3#	2.1	39.44	1.22
10-1-3#	2.3	22.34	1.62
10-1-3#	2.5	41.72	1.22
10-1-3#	3.0	23.23	1.59
10-1-3#N 3 m	2.0	35.50	1.28
10-1-3#N 3 m	2.3	31.66	1.41
10-1-3#N 3 m	2.5	25.76	1.50
10-1-3#N 3 m	3.6	40.92	1.24
10-1-3#N 3 m	3.8	37.50	1.26
10-1-3#N 4 m	2.6	33.37	1.35
10-1-3#N 4 m	2.9	32.89	1.36
10-1-3#N 4 m	3.2	27.82	1.49
10- 2-1#	2.1	22.56	1.61
10-2-1#	2.3	26.64	1.51
10-2-1#	3.6	45.82	1.07
10-2-1#	4.5	15.62	1.83
10-2-1#	4.7	17.37	1.80
10-2-2#	2.8	22.08	1.60
10-2-2#	3.2	24.72	1.58
10-2-3#	2.9	26.02	1.52
10-2-3#	4.3	13.29	1.94
10-2-3#	4.5	14.38	1.93
10-2-3#	4.8	13.25	1.95
10-3-1#	2.8	28.09	1.44
10-3-1#	3.0	26.33	1.46
10-3-1#	3.2	21.16	1.59
10-3-1#	3.8	19.03	1.76
10-3-1#	4.0	19.68	1.72
10-3-2#	3.2	24.02	1.53
10-3-2#	4.4	23.04	1.59
10-3-3#	3.4	30.29	1.43

. Figure 3 shows two finished standard undisturbed samples. The ice in some of the original cracks can be seen at the sample surface.

2.2. Test Conditions

Prior to being tested, the standard test samples were removed from the low-temperature thermostat, placed in another thermostat and held at the test temperature for at least 48 h. Then, each sample was subjected to a triaxial compression test in a triaxial tester for frozen soil, where the axial loading direction was consistent with the sampling direction. The test was terminated when the axial strain reached 21%. The sample was removed from the tester, and the water content of the entire sample was measured (as the ice distribution was highly uneven in the undisturbed ice-rich sample, the measured water content (m (ice + unfrozen water) × 100%/m (dry soil)) corresponded to the mean water content). The dry density of a standard specimen before the test was calculated based on the measured water content. All obtained data are presented in Table 1. The water content of all the standard samples was in the range of 13.25–45.82%. The volumetric ice content of all the standard samples was higher than 20%, such that all the standard samples were considered to be ice-rich frozen soil. As the exterior surface of the standard sample was covered with rubber film and epoxy resin sample heads, the mean water content of the standard sample was assumed to be unchanged by the loading test. All the test data were automatically collected using a program at the State Key Laboratory of Frozen Soil Engineering in China. The loading test was performed under axial displacement control with a shear rate of 1.25 mm·min⁻¹, a temperature of −5.0 °C and confining pressures ranging from 0.05 MPa to 4.00 MPa. The test conditions were selected based on a previous study [31] for comparison purposes.

3. Results and Analyses

3.1. Characterization of Stress-Strain Behavior

The typical stress-strain curves for different levels of sample water content obtained under a confining pressure of 2.00 MPa are shown in Figure 4. The effects of the sampling points and sample depths on the test results were not considered. Some curves exhibit significant jumps at some points. This phenomenon may result from the inhomogeneity of the undisturbed samples and the presence of significant irregular cracks or defects in the samples.

Figure 4 shows that changes in the water content within the investigated range resulted in significant changes in the stress-strain curve, including most of the observed brittle failure (with a peak stress) and partly for the plastic failure (without a peak stress). The area of the graph enclosed by the curve and the corresponding abscissa axis from 0 to the failure strain was also considerably different for different levels of water content. Therefore, the water content is a very important factor affecting the failure characteristics of ice-rich frozen clay.

3.2. Failure Strain Energy Density Characteristics

Solids are deformed under the action of an exterior force, such that the action point of the force is displaced along the action orientation of the force, whereby the work is exerted by the force. During this process, the strain energy is stored by elastic deformation and dissipated by plastic deformation [32]. If the kinetic energy and other forms of energies can be neglected, the strain energy of a solid is numerically equal to the work performed by exterior forces according to the principle of work and energy [32]. The failure strain energy density corresponds to the strain energy density at the time of failure (considering both the elastic and plastic strain energy). The physical meaning of the failure strain energy density is the capability of a solid to absorb energy from the beginning of loading to failure [31]. Considering the mechanical characteristics of the cylindrical frozen soil samples determined during triaxial compression experiments, each sample was subjected to an axial deviatoric stress and a confined pressure during triaxial shear. The work exerted by these two forces on the sample during triaxial shear was calculated separately. The algebraic sum of the work exerted by the two forces is the total work exerted by the exterior forces. The total work divided by the volume of the sample is the strain energy density. The strain energy density at the shear strength corresponds to the failure strain energy density. The formula for the failure strain energy density is presented below.

$${\upsilon }_{{\epsilon }_{\mathrm{f}}}={\displaystyle {\int }_{ 0}^{ {\epsilon }_{\mathrm{f}}}\frac{{\sigma }_1-{\sigma }_3}{1-\epsilon }}\mathrm{d}\epsilon$$

(1)

A detailed derivation of this formula has been presented by Du et al. [31]. In Equation (1), ${\sigma }_1$ is the axial stress; ${\sigma }_3$ is the confined pressure; $\epsilon$ is the axial strain; ${\epsilon }_{\mathrm{f}}$ is the axial failure strain (the axial strain corresponding to the strength); and ${\upsilon }_{{\epsilon }_{\mathrm{f}}}$ is the failure strain energy density. To facilitate the calculation of the integral term in Equation (1), the ordinate (the deviatoric stress: ${\sigma }_1-{\sigma }_3$) of the stress‒strain curve is converted to ($\frac{{\sigma }_1-{\sigma }_3}{1-\epsilon }$) while maintaining the abscissa as the axial strain. Therefore, the integral of the converted curve from 0 to the failure strain is the failure strain energy density. As a result, the convexity of the converted curve (the abscissa is from 0 to the failure strain) has a considerable impact on the determined failure strain energy density. Figure 5 is a comparison of the curves before and after the conversion is performed. The position of the stress peak is changed by the conversion. Therefore, when using the converted curve to calculate the failure strain energy density, integration is performed from 0 to the failure strain value. The strain at the stress peak of the converted curve should not be taken as the upper integration limit.

The failure strain energy density was calculated under different conditions based on a geometric interpretation of the integration of the converted stress‒strain curve. The failure strain energy density properties of undisturbed ice-rich frozen clay are analyzed below.

3.2.1. Influence of the Confined Pressure on the Failure Strain Energy Density

The measured water content of all the standard samples presented in Table 1 was used to classify the standard samples into four water content ranges, i.e., 13.25–19.81%, 21.16–25.76%, 26.02–28.09% and 30.29–45.82%. The smallest possible ranges that contained a sufficient number of data points were chosen to enable the change trend to be analyzed. The results of a previous study [23] on the influence of the moisture content on the strength were also considered in determining these ranges. Figure 6 shows how the failure strain energy density changed with the confined pressure. The influence of the water content was not considered within each water content range when studying the influence of the confined pressure on the failure strain energy density.

Figure 6 shows that the failure strain energy density increased to a maximum and then decreased when the confined pressure increased. The maximum failure strain energy density occurred at a critical confining pressure. The critical confining pressure was 2.00 MPa for 13.25–25.76% water content. The critical confining pressure was 1.00 MPa for 26.02–45.82% water content. The critical confining pressure decreased when the water content increased. The amplitude of the change in the failure strain energy density with the change in the confined pressure was smaller for 30.29–45.82% water content than for 13.25–28.09% water content. These variation characteristics are essentially in agreement with the results of Du et al. [31] for remolded frozen sandy soils. However, remolded frozen sandy soils were found to have a larger critical confining pressure than undisturbed frozen clay.

3.2.2. Influence of the Water Content on the Failure Strain Energy Density

The variation in the failure strain energy density with the water content is shown in Figure 7. Due to the very limited number of data points for each confining pressure condition (0.05 MPa, 0.20 MPa, and 0.50 MPa), a change trend for the individual confining pressures could not be identified. Moreover, Figure 6 indicates that the confined pressure had a small effect on the failure strain energy density for this range of confined pressure. Hence, all the data points within the confined pressure range of 0.05–0.50 MPa are shown in the same diagram (Figure 7a). In Figure 7a, the effect of the confining pressure was not considered.

Figure 7a shows that at low confining pressures (0.05–0.50 MPa), the failure strain energy density increased when the water content increased. However, it then declined slightly at intermediate confining pressures (1.00–2.00 MPa), as shown in Figure 7b,c. Figure 7d shows that at a high confining pressure of 3.00 MPa, the failure strain energy density decreased overall when the water content increased. A comparison of the changes in the failure strain energy density with the water content under various confined pressures shows that the raise in confined pressure tended to reduce the water content at the maximum failure strain energy density. These results are essentially consistent with those of Du et al. [31] for remolded frozen sandy soils. However, the water content at the maximum failure strain energy density determined in this study was different from that reported by Du et al. [31]. In addition, remolded frozen sandy soils were found to have a considerably higher failure strain energy density than undisturbed frozen clay for water contents lower than 30%.

3.2.3. Comparison of the Characteristics of the Triaxial Compressive Strength and the Failure Strain Energy Density

The triaxial compressive strength and failure strain energy density are indexes of the resistance of frozen soils to failure based on the stress magnitude and work exerted by exterior forces, respectively. The failure strain energy density corresponds to the strain energy density at the compressive strength. Therefore, there are similarities and differences between the compressive strength and the failure strain energy density. To develop a comprehensive and detailed method for assessing and distinguishing the resistance of frozen soils to failure, it is necessary to compare the change characteristics of the compressive strength and failure strain energy density. Figure 8 is a comparison of the variation characteristics of the compressive strength and failure strain energy density under the same conditions. This comparison is performed for changes in the confining pressure in Figure 8a,b, and for changes in the water content in Figure 8c,d.

Figure 8a shows that the triaxial compressive strength hardly changed with increased confining pressure. As the confined pressure increased, the failure strain energy density first increased to a maximum and then declined, as shown in Figure 8b. In such a case, considering the compressive strength and failure strain energy density together enables a detailed comparison and analysis of the resistance of frozen soil to failure because a large failure density should produce a great resistance to failure. Figure 8c,d indicate that increasing the water content caused the strength and the failure strain energy density to first increase and then decrease. These findings suggest similarities and differences in the variation characteristics of the strength and failure strain energy density. Du et al. [31] also found such similarities and differences for remolded frozen sandy soils.

4. Discussion

The formula for the strain energy density provided in Material Mechanics I [33] is only applicable to the linear elastic stage of a material. The failure strain energy density corresponds to the strain energy density (considering both the elastic and plastic strain energy) at the time of failure. However, the stress-strain curve of frozen soil cannot be characterized simply by a linear elastic stage from initial loading to failure. Therefore, in order to calculate the failure strain energy density, a calculation formula for the strain energy density must be re-derived based on the principle of work and energy. At the linear elastic stage of the stress-strain curve, the strain energy density equals the product of the stress and the corresponding strain multiplied by a factor of one half [33]. Although the stress-strain curve of a frozen soil sample from initial loading to failure is not completely linear and the failure strain energy density is affected by the convexity of the stress-strain curve, the failure strain energy density can be reasonably approximated by the product of the stress strength and the failure strain multiplied by a factor of one half. Therefore, a high stress strength is often accompanied by a high failure strain energy density. Some variation characteristics of the failure strain energy density are in agreement with those of the stress strength. Some mechanisms that explain these variation characteristics can be found in the research results on the stress strength characteristics of frozen soil. Some stress-strain curves exhibit a large stress strength but a low failure strain, leading to a low failure strain energy density. Therefore, there are also some differences in the variation characteristics of the failure strain energy density and stress strength.

5. Conclusions

The results of triaxial compression experiments on undisturbed ice-rich frozen clay samples with various levels of water content under different confining pressures were used to determine the variation characteristics of the failure strain energy density, and the following conclusions were reached.

(1)

Increasing the confined pressure caused the failure strain energy density to first increase and then decrease. The failure strain energy density reached a maximum at a critical confining pressure of 2.00 MPa for 13.25–25.76% water content and of 1.00 MPa for 26.02–45.82% water content. Increasing the water content caused the critical confined pressure to decline.

(2)

At low confined pressures (0.05–0.50 MPa), the failure strain energy density increased when the water content increased, but then declined slightly at intermediate confined pressures (1.00–2.00 MPa). At a high confined pressure of 3.00 MPa, the failure strain energy density declined overall with increasing the water content. With increasing confined pressure, the water content at the maximum failure strain energy density decreased.

(3)

Similarities and differences were observed in the change characteristics of the triaxial compressive strength and failure strain energy density. A detailed assessment of the resistance of frozen soil to failure can be performed using the failure strain energy density as a supplementary reference metric.

(4)

The variation properties of the failure strain energy density of undisturbed frozen clay and remolded frozen sandy soils were found to be essentially the same. However, some clear differences in the failure strain energy density of these two soil types were also found.