Study on Influence of Confining Pressure on Strength Characteristics of Pressurised Frozen Sand

Abstract

In order to reveal the influence law of freezing pressure and confining pressure on the strength characteristics of frozen sand, with the self-developed high-pressure frozen soil triaxial instrument, the triaxial compression tests of frozen sand under different freezing pressures and confining pressures were carried out. The test results show that the freezing pressure will not change the stress–strain curve of the frozen sand. Similar to the confining pressure, the freezing pressure influences the strength of the frozen soil in two ways: strengthening and weakening. The threshold confining pressure resulting from the test was about 45 MPa. Through comparison of the initial elastic modulus with the secant elastic modulus at 0.5 times the strength, it is found that the initial elastic modulus is more appropriate to use in engineering calculations. The internal friction angle is greatly affected by the freezing pressure, and cohesion shows little change with the freezing pressure. Compared with the test results and other constitutive equations, it is found that the constitutive equation established in this paper considering the effect of freezing pressure can better describe the stress–strain relationship of the pressurised frozen sand.

1. Introduction

High-pressure crustal stress environment exists in cold regions and during an artificial freezing method. The construction design of deep underground engineering is especially very dependent on shear strength parameters of frozen soil [1,2]. Existing studies have shown that the shear strength of frozen soil exhibits a unique two-stage or three-stage development with an increase in stress, while the strength of conventional frozen soil tends to increase with an increase in pressure. Direct application of test parameters of the conventional frozen soil to deep frozen soil engineering will cause huge cost waste and engineering disasters [3,4]. Thus, it is very necessary to deeply study the mechanical properties of frozen sand under pressure.

Lai [5], Xu [6], Yang [7], Zhang [8], Yao [9], and other scholars also carried out non-pressure freezing of soil samples after K₀ consolidation, and then carried out triaxial shear tests for stress state recovery. Based on the computational analysis of strain energy theory, damage mechanics, and experimental constitutive theory, a huge number of strength fracture mechanisms of the frozen soil and the variation laws of mechanical properties of the frozen soil after consolidation were obtained. On the basis of analyzing variations of the shear strength of the frozen soil with confining pressure, Liang et al [10], provided an idea of dividing the shear strength into a basic shear strength and a special contribution shear strength, and proposed a new three-stage strength criterion, that is, the compressive strength of the frozen soil increases with the confining pressure in an early stage, and then weakens in a later stage.

It can be seen from the above research that the effect of high confining pressure on the mechanical properties of frozen sand is an issue worth considering. However, for most of the frozen sand tests at present, samples are prepared in molds, and batch freezing samples with uncontrollable confined pressures are used, and then transferred to a frozen soil triaxial instrument for destructive tests after a constant temperature [11,12,13]. This method of sample preparation is simple and suitable for a large number of tests, but it also has disadvantages that the freezing pressures and pore water pressures are difficult to control, and non-stress continuity tests have intensive interferences. Ma Wei, Wang Dayan et al. [14,15,16] studied the yield strength and deformation characteristics of frozen sand under low pressure (1–5 MPa) freezing through experiments, and confirmed the effect of consolidation stress (freezing pressure) on the mechanical properties of the frozen soil. The difficulty in the current test mainly lies in how to efficiently carry out a large number of pressure-freezing in situ destructive tests. Due to test requirements of constant temperatures of the samples, the triaxial test time of a single group of pressure-freezing sand oil with a conventional size (Φ39.1 mm, Φ61.8 mm) is as long as 2~3 d [17], the economic and time costs of conducting a large number of tests are relatively high, which is a main reason for a lack of research results on pressurized frozen soil nowadays. The single test time can be reduced by increasing the freezing efficiency and reducing the sample size. The method of increasing the cooling capacity and liquid nitrogen freezing to improve the freezing efficiency is limited by the upper limit of the circulating cooling capacity and the difficulty of precise temperature control. Therefore, it is more feasible to use small-sized samples to shorten the time of a single deep frozen soil mechanical test.

The main source of the size effect of the samples is an error of the sample process and the influence of particle gradation. Bragg [18], Mohammad [19], and Sitharam et al [20] conducted the uniaxial compression tests on frozen sand with a diameter of 28.7–61 mm at a temperature of −6 °C and found that for every 10 mm increase in the diameter of the sample, the compressive strength decreased by 0.24 MPa, and the failure strain increased by 0.014%, which was basically negligible. Wang [21] conducted a triaxial shear test for consolidation and drainage under different conditions, and explored the effects of grain size and confining pressure on the shear characteristics of coral sand in terms of critical confining pressure and shear strength, respectively, and found that the effect of grain size on strength mainly depends on the change of pore ratio. Skuodis [22] found that when the particle size of sand is small, its strength parameters are basically not affected by the sample size, and therefore it is feasible to use a small-sized frozen sand sample.

On this basis, the stress continuity triaxial compression test of the frozen sand under different freezing pressures and confining pressures was carried out by the self-developed high-pressure frozen soil triaxial instrument suitable for small-sized samples, and the variation laws of compressive strength, elastic modulus, cohesion, and internal friction angle with freezing pressure were obtained; on the basis of the test results, a hyperbolic constitutive model considering the freezing pressure effect was established, and it showed better accuracy when compared with calculation results of the tests and other models.

2. Experimental Section

2.1. Test Equipment

The size of the soil sample is Φ8 × 16 mm. The self-developed small-scale frozen sand triaxial test device is used; the pressure cylinder is made of 7075 aluminium alloy. The confining pressure and back pressure are controlled by a high-precision pressure volume controller, and the maximum pressure is 64 MPa, the pressure error is controlled within 0.01%, and the volume error is controlled within 0.01 mm³. The equipment is mainly composed of four parts: stepping motor, planetary reducer, load sensor, and pressure chamber. The maximum confining pressure is 70 MPa, the maximum axial force is 15 kN, and the loading rate is in the range of 10~7500 μm/min, as shown in Figure 1.

The sand sample is saturated in this paper. The density of dry sand shall be controlled during sample preparation [23], and the dry sand density is controlled during sample preparation. Due to limitations of the pressure chamber and the sample size, real-time temperature measurements of the samples cannot be performed. Therefore, the temperature calibration of the upper and lower ends of the samples under non-pressure conditions is carried out, as shown in Figure 2. It can be seen that the internal temperature uniformity of the sample is good during the freezing process, and it quickly drops from the normal temperature to −20 °C in the initial stage, and the stable time of the internal temperature of the sample is about 30 min. There exists a good linear relationship between the average temperature of the sample and the set temperature of the freezing box.

Referring to ATSM [24], it is necessary to correct the difference in deviatoric stress considering the thickness of the film, and the correction formula is as follows:

$$\Delta {\left({\sigma }_1-{\sigma }_3\right)}_m=\left(4E_m{t}_m{\epsilon }_1\right)/D_c$$

(1)

where $\Delta {\left({\sigma }_1-{\sigma }_3\right)}_m$ is the deviatoric stress caused by the film (MPa); $D_c$ and $h$ is the average diameter and height of the sample after consolidation (mm), $t_m$ is the thickness of the heat-shrinkable tube (mm); $E_m$ is the elastic modulus of the heat-shrinkable tube (MPa), ${\epsilon }_1$ is the axial strain of the heat shrinkable tube.

2.2. Test Preparation

In order to facilitate the comparison with other researchers, the Chinese ISO standard sand is used. The particle shape of the quartz sand is approximately spherical or ellipsoid. The basic physical parameters of the sample are shown in Figure 3 and

Table 1. Physical properties of the samples.

Parameter	GS	ω/%	Sr/%	e0
Value	2.71	24.5	100	0.687

.

At present, the average temperature of the freezing wall in China’s artificial freezing drilling projects has reached −25 °C. The use of a test temperature of −20 °C in the tests can better meet the actual engineering needs of freezing drilling. In this paper, the tests were carried out using equal pressure consolidation, and the specimens are shown in Figure 4. The test steps are as follows:

(1)

After the sample is saturated and sealed, the pore water pressure is maintained 10 kPa, the axial displacement is kept unchanged, and it is loaded to the test confining pressure at a rate of 0.05 MPa/s until the volume deformation rate is less than 0.5 mm³/h.

(2)

The cold bath is carried out with a constant temperature of 10 °C for 30 min to reach the same initial temperature, and the pore water pressure is set to the experimental value; while keeping the confining pressure and axial displacement unchanged, the temperature of cold bath circulation box is decreased to −20 °C, and the sample is frozen and kept at a constant temperature for about 4 h to complete the pressure-freezing of the sample.

(3)

The confining pressure is loaded to the test value at a rate of 0.05 MPa/s; then, while keeping the confining pressure unchanged, the axial loading rod is loaded at a rate of 0.5 mm/min until the axial displacement reaches 20%, during which the stress and strain are recorded.

3. Analysis of Test Results

3.1. Stress-Strain Curves

In order to study the effect of mechanical properties of compressive frozen sand with freezing pressure and confining pressure, several sets of compression tests were carried out with continuous freezing pressures of 0, 5, 10, 20, 30, and 40 MPa and confining pressures of 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, and 55 MPa and a loading rate of 0.5 mm/min. The stress–strain curves were plotted according to the freezing pressure classification in Figure 5.

It can be seen from the figure that the freezing pressure has no significant effect on the form of the stress–strain curve of the frozen sand, but the curve type varies greatly with the confining pressure. At 0 MPa, the stress–strain curves at different freezing pressures show an obvious strain-softening pattern, at 5 MPa and 10 MPa, the test curves approximate ideal elastoplasticity, and when the confining pressure is greater than 10 MPa, they all show a strain-hardening pattern.

Under the condition of triaxial compression, the stress–strain curve of frozen sand shows elastic–plastic characteristics, and the stress at 15% strain can be regarded as the failure strength; in the uniaxial experiment, the failure strength is the peak stress on the stress–strain curve. The compressive strengths of frozen sand at different confining pressures are plotted according to the freezing pressure groupings as Figure 6.

It can be seen that the compressive strength of frozen sand generally increases with the increase of confining pressure. In this paper, the limit confining pressure is 45 MPa, and when the confining pressure is between 0 and 25 MPa, the slope of the strength envelope decreases continuously with the increase of the confining pressure, indicating that the impact of friction increases with the increase of the confining pressure. Comparing the change in strength at the same confining pressure shows that the freezing pressure affects the magnitude and rate of growth of the compressive strength of frozen sand.

3.2. Elastic Modulus

A great deal of research has been carried out by many scholars on how to calculate the elastic modulus of permafrost, and the following are the main ways of calculating the elastic modulus of permafrost at present:

Domestic and foreign scholars have carried out a lot of researches on calculation methods of the elastic modulus of frozen soil. Current calculation methods of the elastic modulus of frozen soil mainly include the following:

(1)

According to “Artificial frozen soil physics mechanics performance test” (MT/T 593.5-2011), in the triaxial shear test, the elastic modulus of the frozen soil is taken as the secant modulus corresponding to the axial strain at 0.5 times the frozen soil strength. Line modulus [25].

(2)

The secant stiffness in the initial stage of sample deformation is taken as the elastic modulus of frozen soil, and the strain less than 0.5%~1% is generally regarded as the elastic stage [26].

(3)

Referring to the calculation methods of the elastic modulus of non-frozen soil, the sample is subjected to a four-stage cyclic loading and unloading test until the sample is destroyed, and the straight slope at the endpoint of the hysteretic loop is taken as the elastic modulus of the frozen soil.

According to the stress–strain curve in Figure 5, the initial elastic modulus (strain 1%) and the secant elastic modulus at 0.5 times the strength (0.5) of the frozen soil under different freezing pressures and confining pressures are calculated as

Table 2. Calculated values of elastic modulus.

Confining Pressure/MPa	Frozen Pressure/MPa
	0	5	10	20	30	40	0	5	10	20	30	40
	$0.5{\mathit{\sigma }}_{\mathit{c}}/\mathbf{MPa}$						Initial Elastic Modulus/MPa
0	651	625	602	533	566	1013	769	909	909	613	556	1754
5	572	1187	1344	1029	884	1233	1515	1176	1316	1333	980	1316
10	1397	1253	1407	1533	689	2126	1471	1340	1852	1786	1142	1695
15	1375	1135	1797	1417	1058	1124	1111	1449	1887	1389	1266	1163
20	741	1921	1821	1367	1304	1050	1042	1603	1754	2119	1613	1631
25	907	764	2432	1313	1259	1196	1832	1786	1852	1923	1250	1282
30	934	990	970	1366	1033	933	1408	1351	1429	1389	1724	1527
35	1165	1044	1137	1467	1498	1187	1042	1698	1587	1818	1852	2128
40	1646	1537	955	1664	1269	1377	1215	1250	1124	2041	1639	2041
45	1065	939	1022	1471	1105	1300	1678	1862	1923	1887	1389	2128
50	1141	1089	993	1362	1415	1441	1756	1887	1709	1961	1887	1667
55	857	1567	1897	2044	1202	1373	1852	1961	1961	2174	1961	1786

.

Distributions of the elastic modulus of the drawn frozen sand are shown in Figure 7 and Figure 8.

It can be seen from the figures that the elastic moduli obtained through the two calculation methods have a certain discreteness. When the confining pressure increases from 0 MPa to 55 MPa, the elastic modulus variation range calculated by the 0.5${\sigma }_{\mathrm{c}}$ method is about 1000–1600 MPa, and the elastic modulus variation rate per unit of confining pressure is between 18.2–29.0 MPa. When the initial elastic modulus is employed for calculation, the elastic modulus variation range is about 800–1200 MPa, and the elastic modulus variation per unit of confining pressure is about 14.5–21.8 MPa. Therefore, during each of the confining pressure stages, the initial elastic modulus is more stable than the secant modulus, and a large instability of the secant modulus indicates that the frozen sand under pressure is more sensitive to the change of confining pressure during a large deformation period. Therefore, when the freezing pressure is not considered, it is more reliable to use the initial elastic modulus of frozen sand in engineering calculations.

According to the calculation results, the elastic modulus of the frozen sand under the confining pressure of 0–55 MPa can be divided into three stages according to the freezing pressure:

(1)

0 ≦ ${\sigma }_f$ ≦ 20 MPa. At this stage, due to consolidation and compaction, the initial elastic modulus of the frozen sand increases with the increase of the freezing pressure; when the freezing pressure is equal to 20 MPa, discreteness of the secant modulus increases significantly, reason for this may be that, the soil skeleton is damaged due to a freezing pressure of 20 MPa, and during the large deformation stage, development of the damage leads to an increase of the discreteness of the sample, resulting in a large difference in the elastic modulus.

(2)

20 < ${\sigma }_f$ ≦ 30 MPa. The elastic modulus obtained by the two calculation methods both decreased to varying degrees, and the initial elastic modulus showed greater discreteness than the secant modulus. That above illustrated that the damage of the soil skeleton under a freezing pressure of 30 MPa is more serious than that under the freezing pressure of 20 MPa, so that the strengthening effect of the confining pressure on deformation capacity is greatly weakened, thus showing a greater discreteness.

(3)

30 < ${\sigma }_f$ ≦ 40 MPa. At this stage, with the increase of freezing pressure, sand fracturing and fragmentation increase, and the rearrangement of the structure intensifies, showing better structural properties, and therefore the elastic modulus continues to increase.

Since the discreteness of the secant modulus is significantly smaller than the initial elastic modulus under high freezing pressure, when the freezing pressure needs to be considered in the engineering calculation, it is more reliable to select the initial elastic modulus when the freezing pressure is less than 20 MPa, while the secant modulus obtained through the 0.5 ${\sigma }_{\mathrm{c}}$ calculation will be chosen under a higher freezing pressure.

3.3. Internal Friction Angle and Cohesion

The basic theory of soil mechanics shows that the triaxial compressive strength of frozen soil consists of the cohesive force c and the frictional force $\tau$. After the stress transformation, it is obtained that:

$${\tau }_f=\frac{1}{2}({\sigma }_1-{\sigma }_3)=\frac{1}{2}({\sigma }_1+{\sigma }_3)\sin \phi +c\cos \phi$$

(2)

where ${\sigma }_1$ and ${\sigma }_3$ are the principal stresses; ${\tau }_f$ is the shear strength of the specimen; $\tau$ is the frictional force due to the normal stress in the shear plane.

The Mohr–coulomb shear surface and the strength wrap are plotted as shown in Figure 9.

Where line ${\tau }_f$ is the test stress circle tangent wrap and $k_f$ is the Mohr damage wrap (maximum shear stress linkage), the average positive and partial stresses are expressed as:

$$\overline{p}=\frac{1}{2}({\sigma }_1+{\sigma }_3);\overline{q}=\frac{1}{2}({\sigma }_1-{\sigma }_3)$$

(3)

In this way, the stress state of the frozen sand is well integrated with the stress Mohr circle in the $\overline{p}$-$\overline{q}$ plane. $\overline{p}$ is the horizontal coordinate of the Mohr circle, $\overline{q}$ is the radius of the Mohr circle, and the point on the $\overline{p}$-$\overline{q}$ plane determines the size and position of the Mohr circle, thus Equation (3) can be transformed into:

$$\overline{q}=\overline{p}\sin \phi +c\cos \phi$$

(4)

The $\overline{p}$-$\overline{q}$ scatter plots (critical state/maximum shear stress state) for triaxial compression tests of frozen sand at different freezing stresses are plotted as Figure 10.

As can be seen from the graph, there is a roughly linear trend at each point as the circumferential pressure increases. A linear fit was made to each point at different freezing pressures, and the variation of cohesion and internal friction angle with freezing pressure was calculated for the 0–55 MPa and 0–30 MPa envelope pressure ranges, respectively, as shown in

Table 3. Fitted strength parameters (0–55 MPa).

Frozen Pressure/MPa	Confining Pressure/MPa
	0~55 MPa			0~30 MPa
	$\mathbf{tan}\mathit{a}$	$\mathit{\phi }$	$\mathit{c}$	$\mathbf{tan}\mathit{a}$	$\mathit{\phi }$	$\mathit{c}$
0	0.262	15.17	7.65	0.35	20.24	5.19
5	0.29	16.62	7.40	0.37	21.96	5.02
10	0.30	17.88	7.13	0.39	23.52	4.81
20	0.33	19.39	6.98	0.43	25.59	4.75
30	0.31	17.88	7.54	0.43	25.21	4.89
40	0.35	20.37	7.41	0.43	25.34	5.05

.

As can be seen in Table 3, both internal friction angle and cohesion show the same trend in the freezing pressure range of 0–20 MPa, but there are some differences in the values of both. The reason for this difference is due to the variability of frozen soil under high confining pressure, the source of which may be the fragmentation and reorganisation of particles, the melting of ice under force and friction, shear shrinkage, etc. When the fitting was carried out in the range of 0~55 MPa surrounding pressure, the correlation coefficient of multiple groups was less than 0.9, so it did not have a good linear relationship; the value of R² of each group in the range of 0–30 MPa was greater than 0.98, and the linear correlation was good. The variation curves of the two parameters are plotted according to Table 3 as shown in Figure 11.

Data in the range of perimeter pressure of interest in engineering calculations (0–30 MPa, corresponding to a depth of burial of around 0–1500 m) were selected for analysis. In this range, the cohesive force has a smaller rate of change with freezing pressure than the angle of internal friction, with a maximum variation of 8%, while the maximum change in angle of internal friction is 26.4% from Figure 11a. The angle of internal friction increases from 20.24° to 25.59° at freezing pressures less than 20 MPa, with a rate of increase of 0.27. Whereas the cohesive force decreases from 5.19 MPa to 4.75 MPa, at a rate of 0.02. It can be assumed that the freezing pressure mainly affects the internal friction angle of the frozen sand and has little effect on cohesion which can be seen in Figure 11b.

4. Improved Hyperbolic Principal Structure Model

The principal equation of soil is a mathematical expression that reflects the mechanical properties of the soil and is a mathematical tool that describes the quantitative relationship between soil deformation processes and boundary conditions. For strain softening of soil, Lai [27] proposed an improved Duncan–Zhang model to describe the strain-softening and strain-hardening deformation of frozen sand by fitting triaxial test data to unpressurised frozen sand:

$${\sigma }_1-{\sigma }_3=\frac{{\epsilon }_1}{m+n{\epsilon }_1+l{\epsilon }_1{}^2}$$

(5)

where m, n, l are the fitting parameters associated with the convergence values of the peak partial stress ${\left({\sigma }_1-{\sigma }_3\right)}_{\mathrm{p}}$ and strain-hardened partial stress limits ${\sigma }_{\mathrm{p}}$.

When the freezing pressure is greater than 5 MPa, the stress–strain curve during uniaxial compression decreases faster in the initial phase after the peak due to the compacting effect of the confining pressure on the sand before freezing. It then remains largely constant, reaching residual strength earlier, at which point the Duncan–Zhang model is no longer applicable to the post-peak deformation phase. A new improved Duncan–Zhang model based on the consideration of the residual strength of ice after strain softening is proposed by Shan [28]:

$${\sigma }_1-{\sigma }_3=\frac{a{\epsilon }_1{}^2+{\epsilon }_1}{b+c{\epsilon }_1+d{\epsilon }_1{}^2}$$

(6)

where a, b, c, and d are the fitting parameters, degenerating to a modified Duncan–Zhang model when a = 1 and to a generalised hyperbolic model when a = 1 and d = 0.25c².

Equation (6) tends to fall more slowly after the material reaches its peak stress. Since Equation (6) assumes that the residual strength of ice tends to be constant as strain tends to infinity, unlike the uniaxial experiments in this paper where the partial stress reaches the critical state point more quickly. Equation (6) is only applicable to describe the stress–strain relationship before the critical state point, as shown in Figure 12.

The stress–strain curve under triaxial stress can be fitted well using the above model. For uniaxial compression tests, the stress–strain curve can be fitted by dividing it into two hyperbolic sections.

When ${\epsilon }_1\le {\epsilon }_{\mathrm{b}}$, the first stage of the specimen includes an elastic section and a yield section, and Equation (6) is used to describe the stress–strain curve of the first stage:

$${\sigma }_1-{\sigma }_3=\frac{d_1{\epsilon }_1{}^2+{\epsilon }_1}{a_1+b_1{\epsilon }_1+c_1{\epsilon }_1{}^2}$$

(7)

The equations for each model parameter solution can be obtained:

$$\{\begin{array}{l}{a}_1=\frac{1}{E_0}\\ \frac{d_1}{c_1}={({\sigma }_1-{\sigma }_3)}_r\\ a_1+2a_1{d}_1{\epsilon }_{\mathrm{p}}+b_1{d}_1{\epsilon }_{\mathrm{p}}^2-c_1{\epsilon }_{\mathrm{p}}^2=0\\ ({\sigma }_1-{\sigma }_3{)}_{\mathrm{p}}=\frac{d_1{\epsilon }_{\mathrm{p}}{}^2+{\epsilon }_{\mathrm{p}}}{a_1+b_1{\epsilon }_{\mathrm{p}}+c_1{\epsilon }_{\mathrm{p}}{}^2}\end{array}$$

(8)

When ${\epsilon }_1>{\epsilon }_{\mathrm{b}}$, the structure of the frozen sand is basically lost, and the sand is broken under the action of local concentrated stress; at this stage, the sand is mainly broken and rearranged. With the increase of strain, the deviatoric stress of the sample remains basically unchanged after the rapid decrease, and the fitting deviation is larger by using Equation (7), so another related equation is introduced to describe:

$${\sigma }_1-{\sigma }_3={\left(\frac{d_2{\epsilon }_1{}^2+{\epsilon }_1}{a_2+b_2{\epsilon }_1+c_2{\epsilon }_1{}^2}\right)}^{-1}$$

(9)

The following solution to the parameters in Equation (9) can be seen according to Equation (12). The type of relational curve in Equation (9), where the critical state point $({\epsilon }_r,{({\sigma }_1-{\sigma }_3)}_r)$ is introduced, when ${\epsilon }_1={\epsilon }_r$, there is

$$({\sigma }_1-{\sigma }_3{)}_r={\left(\frac{d_2{\epsilon }_r{}^2+{\epsilon }_r}{a_2+b_2{\epsilon }_r+c_2{\epsilon }_r{}^2}\right)}^{-1}$$

(10)

Taking the derivative of Equation (7) with respect to ${\epsilon }_1$ gives the expression for the tangential modulus of elasticity as:

$$E_t=\frac{d_2({\sigma }_1-{\sigma }_3)}{d{\epsilon }_1}=\frac{c_2{\epsilon }_1{}^2-b_2{d}_2{\epsilon }_1{}^2-2b_2{d}_2{\epsilon }_1+a_2}{{(d_2{\epsilon }_1{}^2+{\epsilon }_1)}^2}$$

(11)

When ${\epsilon }_1={\epsilon }_r$, there is

$$E_r=\frac{c_2{\epsilon }_r{}^2-b_2{d}_2{\epsilon }_r{}^2-2b_2{d}_2{\epsilon }_r+a_2}{{(d_2{\epsilon }_r{}^2+{\epsilon }_r)}^2}=0$$

(12)

That is

$$c_2{\epsilon }_r{}^2-b_2{d}_2{\epsilon }_r{}^2-2b_2{d}_2{\epsilon }_r+a_2=0$$

(13)

According to the continuity of the stress–strain curve, at the point of interruption ${\epsilon }_{\mathrm{b}}$, there is:

$$\{\begin{array}{l}{\frac{c_2{\epsilon }_1{}^2-b_2{d}_2{\epsilon }_1{}^2-2b_2{d}_2{\epsilon }_1+a_2}{{(d_2{\epsilon }_1{}^2+{\epsilon }_1)}^2}|}_{{\epsilon }_1\to {\epsilon }_{\mathrm{b}}}={\frac{b_1{d}_1{\epsilon }_1{}^2+2a_1{d}_1{\epsilon }_1+a_1-c_1{\epsilon }_1{}^2}{{(a_1+b_1{\epsilon }_1+c_1{\epsilon }_1{}^2)}^2}|}_{{\epsilon }_1={\epsilon }_{\mathrm{b}}}\\ {{\left(\frac{d_2{\epsilon }_1{}^2+{\epsilon }_1}{a_2+b_2{\epsilon }_1+c_2{\epsilon }_1{}^2}\right)}^{-1}|}_{{\epsilon }_1\to {\epsilon }_{\mathrm{b}}}={\frac{d_1{\epsilon }_1{}^2+{\epsilon }_1}{a_1+b_1{\epsilon }_1+c_1{\epsilon }_1{}^2}|}_{{\epsilon }_1={\epsilon }_{\mathrm{b}}}\end{array}$$

(14)

To simplify Equation (14), assume that

$$\{\begin{array}{l}A=\frac{b_1{d}_1{\epsilon }_{\mathrm{b}}{}^2+2a_1{d}_1{\epsilon }_{\mathrm{b}}+a_1-c_1{\epsilon }_{\mathrm{b}}{}^2}{{(a_1+b_1{\epsilon }_{\mathrm{b}}+c_1{\epsilon }_{\mathrm{b}}{}^2)}^2}\\ B=\frac{d_1{\epsilon }_{\mathrm{b}}{}^2+{\epsilon }_{\mathrm{b}}}{a_1+b_1{\epsilon }_{\mathrm{b}}+c_1{\epsilon }_{\mathrm{b}}{}^2}\end{array}$$

(15)

That is

$$\{\begin{array}{l}\frac{B^2{c}_2{\epsilon }_{\mathrm{b}}{}^2-B{b}_2\left({\epsilon }_{\mathrm{b}}+2\right)\frac{a_2+b_2{\epsilon }_{\mathrm{b}}+c_2{\epsilon }_{\mathrm{b}}{}^2-B{\epsilon }_{\mathrm{b}}}{{\epsilon }_{\mathrm{b}}}+B^2{a}_2}{{(a_2+b_2{\epsilon }_{\mathrm{b}}+c_2{\epsilon }_{\mathrm{b}}{}^2)}^2}=A\\ d_2=\frac{a_2+b_2{\epsilon }_{\mathrm{b}}+c_2{\epsilon }_{\mathrm{b}}{}^2-\mathrm{B}{\epsilon }_{\mathrm{b}}}{B{\epsilon }_{\mathrm{b}}{}^2}\end{array}$$

(16)

Coupling Equation (10) and Equation (13), it is easy to obtain:

$$\{\begin{array}{l}({\sigma }_1-{\sigma }_3{)}_r=\frac{a_2+b_2{\epsilon }_r+c_2{\epsilon }_r{}^2}{\frac{c_2{\epsilon }_r{}^2+a_2}{b_2{\epsilon }_r+2b_2}{\epsilon }_r+{\epsilon }_r}\\ d_2=\frac{c_2{\epsilon }_r{}^2+a_2}{b_2{\epsilon }_r{}^2+2b_2{\epsilon }_r}\end{array}$$

(17)

This gives the system of equations for the parameters $a_2$, $b_2$ and $c_2$ as:

$$\{\begin{array}{l}\frac{B^2{c}_2{\epsilon }_{\mathrm{b}}{}^2-B{b}_2\left({\epsilon }_{\mathrm{b}}+2\right)\frac{a_2+b_2{\epsilon }_{\mathrm{b}}+c_2{\epsilon }_{\mathrm{b}}{}^2-B{\epsilon }_{\mathrm{b}}}{{\epsilon }_{\mathrm{b}}}+B^2{a}_2}{{(a_2+b_2{\epsilon }_{\mathrm{b}}+c_2{\epsilon }_{\mathrm{b}}{}^2)}^2}=A\\ ({\sigma }_1-{\sigma }_3{)}_r=\frac{a_2+b_2{\epsilon }_r+c_2{\epsilon }_r{}^2}{\frac{c_2{\epsilon }_r{}^2+a_2}{b_2{\epsilon }_r+2b_2}{\epsilon }_r+{\epsilon }_r}\\ \frac{c_2{\epsilon }_r{}^2+a_2}{b_2{\epsilon }_r{}^2+2b_2{\epsilon }_r}=\frac{a_2+b_2{\epsilon }_{\mathrm{b}}+c_2{\epsilon }_{\mathrm{b}}{}^2-\mathrm{B}{\epsilon }_{\mathrm{b}}}{B{\epsilon }_{\mathrm{b}}{}^2}\end{array}$$

(18)

The values of the parameters $a_2$, $b_2$, and $c_2$ can be found by substituting the relevant values from the experimental results and then substituting them into Equation (17) to obtain $d_2$. For segment fitting, the main considerations are the location of the breakpoint and the continuity of the curve at the breakpoint. In the segmented fit of this paper, the continuity of the curve function at the segmented points is considered by Equation (14), and the parameters $a_2$, $b_2$, and $c_2$ can be found from the experimental data by substituting A, B, ${\epsilon }_r$, $({\sigma }_1-{\sigma }_3{)}_r$, ${\epsilon }_{\mathrm{b}}$, etc., then substituting into Equation (17) to obtain the value of $d_2$. The peak and critical points are selected separately below as intermittent points to compare the continuity of the curves.

When the peak point is used as the boundary, the stress–strain curve obtained from the test is not strictly speaking an ideal hyperbola in the $({\sigma }_1-{\sigma }_3)\sim {\epsilon }_1$ coordinate after the peak point, and there is a section of the curve after the peak stress point where the initial elastic mode is small and cannot be fitted well. Therefore, the point between the peak and critical state points was chosen as the intermittent point. It was found that the strain range corresponding to the peak stress–critical point in the uniaxial compression test was basically in the range of 2.5–5%. Taking the uniaxial experiment with 10 MPa freezing pressure as an example, the peak strain (2.5%) and 3% strain were selected as the intermittent point for fitting, and the effect is shown in Figure 13.

As can be seen from Figure 13, the fit is better in the first stage when the peak point is used as the intermittent point, the fitted curve in the second stage differs significantly from the test curve after the peak strain, and the two curves are discontinuous at the peak point when equal weighting is used for the fit. The main reason for this is that the strain- softening curve for uniaxial compression tests on frozen sand is not an ideal hyperbolic curve after peak strain. Defining the tangential modulus on the curve in terms of Et, it can be seen that the tangential modulus of the test after peak strain increases gradually from 0 and then decreases. For the strain-softening curves in Figure 5, the tangential modulus values are all in the late stages of increase when the strain is 3%, and the remaining segments of the curve can be fitted well with a hyperbola. Therefore, for Equation (18), an interruption point of ${\epsilon }_{\mathrm{b}}$ = 3% is appropriate.

The following segmental principal structure equations were obtained to describe the mechanical properties of uniaxial and triaxial compression tests on compressed frozen sand.

$$\{\begin{array}{ll}{\sigma }_1-{\sigma }_3=\frac{d_1{\epsilon }_1{}^2+{\epsilon }_1}{a_1+b_1{\epsilon }_1+c_1{\epsilon }_1{}^2}& 0<{\epsilon }_1<3\%\\ {\sigma }_1-{\sigma }_3={\left(\frac{d_2{\epsilon }_1{}^2+{\epsilon }_1}{a_2+b_2{\epsilon }_1+c_2{\epsilon }_1{}^2}\right)}^A& 3\%<{\epsilon }_1\end{array}$$

(19)

In the triaxial test, $a_1=a_2$, $b_1=b_2$, $c_1=c_2$, $d_1=d_2$. Four typical sets of test results are selected to validate the model. First get ${\epsilon }_r$, $({\sigma }_1-{\sigma }_3{)}_r$, $E_0$, ${\epsilon }_{\mathrm{p}}$, $({\sigma }_1-{\sigma }_3{)}_{\mathrm{p}}$ from the curve, determine $a_1$, $b_1$, $c_1$, $d_1$ from Equation (8), and substitute into Equation (7) to obtain the prediction model for the triaxial compression condition. The four parameters and ${\epsilon }_{\mathrm{b}}$ are then substituted into Equation (15) to obtain the values of A and B, substituting ${\epsilon }_r$, $({\sigma }_1-{\sigma }_3{)}_r$ to obtain $a_2$, $b_2$, $c_2$, $d_2$. The resulting stress–strain predictions for uniaxial compression conditions were obtained and the fitted curves are plotted in Figure 14.

For a visual comparison of the three models, a comparison of the Lai’ model, the Shan model, and the experimental data is also given in Figure 14. From the fitted curves, it can be seen that the new model can accurately describe the partial stress–strain curves of frozen sand materials under different freezing pressures and different surrounding pressures, and the fitted correlation coefficients reach above 0.99, and the predicted results are in good agreement with the experimental results. For the strain-softening model, the segmented model in this paper can better fit the rapid decline and slowdown phase of the partial stress after the peak strain. For the strain hardening curve, the accuracy of the three instants is basically the same when described by the model with A = 1. It can be seen from the

Table 4. Model parameters under different conditions.

	Lai’ Model			Shan’ Model				Model Calculation
	m	n	l	a	b	c	d	a1/a2	b1/b2	c1/c2	d1/d2
I	0.00048	0.088	0.166	621.5	−3.31	0.044	5181	9.663/221.9	−0.102/0.08	0.002/−0.01	43.7/1929
II	0.00069	0.013	0.008	0.951	0.043	0.0009	59.18	−2.03/6.36	0.043/21.64	0.0003/961	32.08/4029
III	0.00061	0.069	0.180	18.62	−0.082	0.002	174.3	5.13/578.5	−0.30/1.33	0.0015/−0.11	33.96/1540
IV	0.00072	0.009	0.0248	0.41	0.012	0.001	23.8	0.41/131.6	0.004/0.91	0.0006/0.109	37.4/8090

, the physical meaning of the parameters is clearer when the Lai’ model is used to fit the small strain phase of frozen sand, and more accurate when the segmented model is used to describe the decay of partial stresses occurring in the small strain phase.

5. Conclusions

In this paper, a large number of pressure–freezing–saturated sand triaxial compression tests were carried out under the stress continuity test methods of “consolidation–freezing–loading”, and the variation laws of strength parameters of the saturated frozen sand under different freezing pressures (0–40 MPa) and different confining pressures (0–55 MPa) were obtained. The constitutive equation considering the effect of freezing pressure was established based on the test results. The main conclusions of the test are as follows:

(1)

The freezing pressure will not change the stress–strain curve of frozen sand. With the increase of the freezing pressure, the yield strain of the sample increases gradually. The effect of freezing pressure on the strength of frozen soil has two aspects of strengthening and weakening. The compaction of particles caused by low freezing pressure increases the strength of the frozen soil; after freezing pressure exceeds 20 MPa, initial damage of sand particles will occur, and the damage will be exacerbated by a higher confining pressure, showing a decrease in strength. In a freezing pressure range of 40 MPa, the maximum increase in strength can reach 43%.

(2)

The strength o frozen sand increases roughly parabolically with the confining pressure, and the growth rate is affected by the freezing pressure. The limit confining pressure value in this paper was about 45 MPa; a phenomenon of strength weakening of frozen sand under different freezing pressures was observed under a high confining pressure, but the strength of the frozen sand under pressure did not decrease.

(3)

Both the initial elastic modulus and the secant elastic modulus at 0.5 times the strength of frozen sand under pressure show a fluctuating growth trend with the increase of the freezing pressure; when the freezing pressure is greater than 20 MPa, the secant elastic modulus is more stable than the initial elastic modulus. It is more appropriate to select the initial elastic mould within the depth range of 1500 m commonly used in engineering calculations.

(4)

The internal friction angle of the frozen sand is greatly affected by the freezing pressure, and the cohesion changes little with the freezing pressure; when the freezing pressure increases from 0 MPa to 20 MPa, the internal friction angle increases by 26.4%; when the freezing pressure is greater than 20 MPa, the internal friction angle remains basically unchanged at 25°.

(5)

The established constitutive equation considering the effect of freezing pressure can better describe the stress–strain relationship of frozen sand under pressure when the confining pressure increases from 0 MPa to 55 MPa.