Introduction

Principal stress rotation is a common phenomenon in soil structures, especially in foundations, embankments, and slope projects (Inamn et al. 2012; Shen et al. 2017). To elucidate the mechanical behavior of unfrozen soil under principal stress rotation, many experiments have been conducted applying hollow cylinder apparatuses (Shen et al. 1996; Gräbe 2002; Cai et al. 2015; Guo et al. 2013; Wang et al. 2017a; Wang et al. 2019a, b; Zhong et al. 2019; Chen et al. 2019a, b). For example, Shen et al. (1996) studied the dynamic characteristics of saturated sand under principal stress rotation by performing a series of cyclic rotational shear tests, which indicated that principal stress rotation noticeably lowered the dynamic strength of saturated sand. Wang et al. (2017a) designed experiments to investigate the dynamic characteristics of soft clay under multiaxial loading conditions including principal stress rotation, and concluded that the clay dynamic deformation significantly increased and its stiffness rapidly decreased under principal stress rotation. Additionally, similar experiments were performed by Wang et al. (2019a, b) and Zhong et al. (2019), who found that principal stress rotation promoted an increase in the residual strain and the pore water pressure of loess. Recently, Chen et al. (2019a, b), when discussing silt clay, suggested that liquefaction was uniquely related to the cyclic rotation of principal stress. All the aforementioned studies have confirmed that the effect of principal stress rotation on the strain, stiffness, pore water pressure, and strength of unfrozen soils cannot be ignored. Hence, soil improvement tests have been developed to enhance the structural stability of unfrozen soil in recent years (Firoozi et al. 2014, 2015, 2017a, b). However, few studies have focused on the dynamic characteristics of frozen soils under principal stress rotation.

Actually, frozen soils are special soils that are widely distributed in cold regions, which comprise approximately 24% of the land surface (Andersland and Ladanyi 1994; Esmaeili-Falak et al. 2017, 2019, 2020; Ren et al 2017; Tsytovich 1985; Liu et al. 2018). With economic development and the rapid expansion of living areas, many projects such as highways, railways, and dams have been built on frozen soil foundations; these projects are often disturbed by dynamic loading (Cheng 2003; Lai et al. 2000). Nevertheless, unlike studies on unfrozen soil, research on the dynamic characteristics of frozen soils is still in its infancy (Ma and Wang 2014; Simonsen et al. 2002). Some researchers have used triaxial apparatuses to investigate the mechanical deformation features of frozen soils under different conditions such as strain amplitude, confining pressure, frequency, temperature, and moisture level, under cyclic loading. Accordingly, some classic empirical formulas have been put forward to forecast variation of residual strain (Zhang et al. 2015, 2019; Wang et al. 2018; Li et al. 2019; Liu et al. 2019; Zhou et al. 2020). Other researchers have investigated variations in the stiffness of frozen soil using triaxial apparatuses and established corresponding empirical formulas to predict the variation of stiffness (Ling et al. 2013; Liu et al. 2016; Xu et al. 2020). Presently, the dynamic mechanical characteristics of frozen soils have been studied exhaustively using triaxial instruments, and many meaningful conclusions have been drawn. However, the triaxial apparatus could only realize axial stress and confining pressure loading; the direction of the major principal stress remained unchanged continuously in these experiments. However, in geotechnical projects, research into unfrozen soil and the rotation of major principal stress direction caused by dynamic loading has proved that principal stress rotation has a noticeable influence on the mechanical properties of soil. Therefore, understanding the dynamic characteristics of frozen soils by considering principal stress rotation is extremely important for the design, construction, and operation of projects in cold regions. Nevertheless, owing to the restrictions of the experimental instrument, few studies have paid attention to the dynamic mechanical characteristics of frozen soil under complex loading conditions, such as principal stress rotation; the mechanical deformation behavior of frozen soils in relation to principal stress rotation remains poorly understood.

Considering the lack of research in this area, we attempted to study the dynamic characteristics of frozen clay under principal stress rotation by applying frozen hollow cylinder apparatus (FHCA-300). We conducted experiments by controlling the axial stress and torsional stress in such a way that the major principal stress direction rotated while the deviatoric stress amplitude remained constant. The influence of CSRs and frequencies on dynamic characteristics of frozen clay was investigated under principal stress cyclic rotation. The variations in strains, hysteresis loops, resilient modulus, and damping ratios with respect to the loading cycle under different terms of CSR and frequency are discussed, respectively.

Experimental setup

A frozen hollow cylinder apparatus (FHCA-300) (Fig. 1) was applied to study the dynamic deformation features of frozen soil in this study. The FHCA-300 can individually exert different waveforms of axial force, torque, inner cell pressure, and outer cell pressure on specimens at negative temperatures. Therefore, multiaxial loading conditions, including principal stress rotation, can be achieved by controlling axial, radial, circumferential, and shear stresses. A detailed description of the instrument was provided by Guo et al. (2016a, b) and Lei et al. (2018). Additionally, based on the thin-wall assumption, Hight et al. (1983) established conversion formulas to calculate the stresses of the FHCA samples in each direction; they are listed as follows:

$${\sigma }_{z}=\frac{W-{p}_{0}s}{\pi \left({r}_{0}^{2}-{r}_{i}^{2}\right)}+\frac{{p}_{0}{r}_{0}^{2}-{p}_{i}{r}_{i}^{2}}{{r}_{0}^{2}-{r}_{i}^{2}},$$
(1)
$${\tau }_{z\theta }=\frac{{M}_{T}\left({r}_{0}+{r}_{i}\right)}{\pi \left({r}_{0}^{4}-{r}_{i}^{4}\right)},$$
(2)
$${\sigma }_{r}=\frac{{p}_{0}{r}_{0}+{p}_{i}{r}_{i}}{{r}_{0}+{r}_{i}},$$
(3)
$${\sigma }_{\theta }=\frac{{p}_{0}{r}_{0}+{p}_{i}{r}_{i}}{{r}_{0}+{r}_{i}},$$
(4)

where \({\sigma }_{z}\) represents the vertical stress, \({\sigma }_{\theta }\) represents the tangential stress, \({\sigma }_{r}\) represents the radial stress, \({\tau }_{z\theta }\) represents the shear stress, \(W\) represents the axial load, \({M}_{T}\) represents the torque, \({p}_{0}\) represents the outer confining pressure, \({r}_{0}\) represents the external radius, \({p}_{i}\) represents the inner confining pressure, \({r}_{i}\) represents the internal radius, and \(s\) represents the cross-sectional proportion of the axial stress rod.

Fig. 1
figure 1

Frozen hollow cylinder apparatus (FHCA-300)

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Material properties and specimen preparation

Clay collected from Beiluhe was used in this paper; this type of clay is widely distributed in the Qinghai–Tibet Plateau, China, which is an important foundation for numerous construction projects in cold regions. The particle distribution of the experimental soil is shown in Table 1. Table 2 shows the essential physical properties of the tested soil. According to the Unified Soil Classification System (ASTM D-2487 2017), the experimental soil can be categorized as lean clay (CL, i.e., clay of low plasticity).

Table 1 Particle size distribution of tested clay
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Table 2 Essential physical properties of tested clay
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Testing hollow cylinder samples, each 200, 50, and 60 mm in height, outer radius, and inner radius, respectively, were prepared using standard routine procedures. The clay was naturally open-air dried, pulverized, and sifted using a 2-mm screen. A certain amount of distilled water was poured into the weighed clay, after which the combination was mixed well by hand. Wet soil with a saturated water content of 19.8% was sealed for at least 24 h to ensure that the moisture was well distributed. Then, the required amount of wet soil was placed in a self-made hollow cylinder mold and compacted using the specimen-making machine to a height of 200 mm with a dry density of 1.77 g/cm3. The prepared room-temperature specimen was fitted into the pressure chamber of the FHCA-300 apparatus and quickly frozen to − 30 °C under a confining pressure of approximately 0.4 MPa to minimize frost heave deformation and water migration. Next, the temperature was adjusted to − 10 °C and maintained for approximately 12 h to ensure that it was well distributed throughout the specimen.

Testing procedures

This experiment mainly focused on investigating the mechanical deformation characteristics of frozen clay with different amplitudes and frequencies under complex dynamic loading conditions. The cyclic stress ratio (CSR) reflects the influence of amplitude on the mechanical deformation characteristics of frozen clay, which can be expressed as follows (Wang et al. 2020a):

$$CSR=\frac{{q}_{cyc}}{{2p}_{0}}$$
(5)

where \(CSR\) is the cyclic shear stress ratio, \({q}_{cyc}\) is the cyclic shear stress amplitude, and \({p}_{0}\) is the confining pressure.

With reference to the static test study of Chen et al. (2019a, b), the specimen with CSRs of 0.225, 0.30, 0.375, 0.45, and 0.525 were applied, and failure was determined by an axial strain of 20%. The loading frequencies of the FHCA tests were selected as 0.1, 0.5, 1, and 2 Hz. The ground temperature of the Qinghai–Tibet Plateau in the cold season ranges from − 15 to − 1 °C (Yi et al. 2017); because this study aimed to study the dynamic characteristics of cryogenic frozen soil, a lower temperature of − 10 °C was selected as the test temperature. The detailed experimental conditions are listed in Table 3.

Table 3 Contents of test
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For the combined axial–torsional tests, initial static deviating stress was applied to the specimens for the purpose of simulating static loading on the foundation, after which axial dynamic coupling and torque loading were performed on the samples. All tests were suspended for 1200 s or till the axial strain reached 20%. Figure 2a shows a comparison of the measured and calculated axial loads, and torque waveforms; Fig. 2b shows the tested and calculated stress paths.

Fig. 2
figure 2

The comparison of measured and calculated for the frozen hollow cylinder apparatus experiments: a axial stress and torque loading waveform; b stress path

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Test results and discussion

Strain development

In cyclic triaxial loading tests, the axial strain is always used to depict the deformation features of frozen samples. However, for combined axial–torque cyclic loading, the shear strain obviously increased because the torsional shear stress was considered under principal stress rotation. Hence, the impact of torsional shear strain on the deformation features of frozen specimens could not be neglected. According, the generalized shear strain was introduced in place of the axial strain to depict the deformation features of the frozen sample; calculation of this strain is listed below (Shen et al., 1996):

$${\gamma }_{g}=\frac{\sqrt{2}}{3}{\left(\frac{9}{2}{\varepsilon }_{z}^{2}+\frac{3}{2}{\gamma }_{r}^{2}\right)}^{{~}^{1}\!\left/ \!\!{~}_{2}\right.}$$
(6)

where \({\gamma }_{r}\) represents the torsional shear strain and \({\varepsilon }_{z}\) represents the axial strain.

Figure 3a presents an enlarged picture of the generalized shear strain relative to the loading cycle for CSR = 0.225 and f = 2 Hz. Figure 3a shows that the total generalized shear strain consisted of irrecoverable plastic and recoverable elastic strain, where the irrecoverable plastic strain is the discrepancy between the total strain and the recoverable elastic strain. The cumulative plastic strain gradually increased with the increasing loading cycle, whereas the elastic strain slightly increased with increasing loading cycles at the beginning of the loading phase and later stabilized. The growth rate of the cumulative plastic strain initially decreased and subsequently tended to become stable.

Fig. 3
figure 3

Strain versus loading cycle: a enlarged picture of the generalized shear strain versus loading cycles for CSR = 0.225 and f = 2 Hz; b variation of the cumulative plastic strain versus loading cycle under different CSRs; c variation of the cumulative plastic strain versus loading cycle under different frequencies

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Figure 3b and c show the variation in the cumulative plastic strain versus the loading cycle under different values of CSRs and frequencies. The result observed in Fig. 3b clearly shows that the cumulative plastic strain remained stable with an increase in the loading cycle at CSR = 0.225. At a large CSR, the cumulative plastic strain significantly increased as the loading cycle increased (Fig. 3b). Additionally, the cumulative plastic strain increased with an increase in CSR at a certain loading cycle. When the CSR increased from 0.225 to 0.300, 0.300 to 0.375, 0.375 to 0.450, and 0.450 to 0.525, the final cumulative plastic strain increased by 2.42%, 2.31%, 4.05%, and 1.25%, respectively (Fig. 3b). The increase in the cumulative plastic strain associated with the CSR can be attributed to a decrease in the strength of the sample. A higher CSR resulted in a lower sample strength, and a larger cumulative plastic strain was generated. Similar results were also obtained under cyclic triaxial loading (Wang et al. 2018; Li et al. 2019; Zhang et al. 2015; Zhou et al. 2020).

Figure 3c depicts the variation features of the cumulative plastic strain versus the loading cycle for different loading frequencies. Figure 3c shows that when the frequency remains constant, an increase in the loading cycle leads to an increase in the cumulative plastic strain. For a given loading cycle, a smaller loading frequency corresponded to a larger cumulative plastic strain. This indicates that the cumulative plastic strain rate decreased as the frequency increased. Previous experimental studies have also obtained similar results (Zhang et al. 2015; Guo et al. 2016a, b; Wang et al. 2019a, b). A possible reason for this result is that for a given cycle number, the cyclic loading lasted longer at a lower frequency, and the densification phenomenon was more pronounced. Hence, the frozen samples had greater cumulative strains (Wang et al. 2019a, b). Additionally, if the duration of loading time was the same, a higher frequency would have resulted in a larger final cumulative plastic strain (Fig. 3). When the frequency increased from 0.1 to 0.5 Hz, 0.5 to 1 Hz, and 1 to 2 Hz, the final cumulative plastic strain increased by 0.90%, 0.37%, and 1.49%, respectively. This suggested that the variation in loading cycles had a pronounced impact on the development of cumulative plastic strain during the same period; that is to say, the loading rate had a notable influence on the cumulative characteristics of the final strain. The failure characteristics of the frozen clay samples at different frequencies are shown in Fig. 4. As shown in Fig. 4, during the hollow cylinder test, the frozen clay samples not only experienced axial compression but also generated shear cracks in horizontal and vertical directions. This result indicated that the shear stress had a significant effect on the strength of the specimen when the principal stress rotation was considered.

Fig. 4
figure 4

The failure characteristics of frozen clay samples at different frequencies

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Evolution of the hysteresis loop

A complete hysteresis loop contains a large amount of information, and the degrees of non-closure, inclinations, and areas of the hysteresis loops represent plastic strain, stiffness, and dissipated energy, respectively.

Effects of CSR on the hysteresis loop

Figure 5 presents the typical axial stress–strain hysteresis loops of frozen soils under different conditions for CSR = 0.525 and N = 100. Figure 5a shows that the areas and inclination degrees of the hysteresis loops significantly increased with the increase in loading cycles under CSR = 0.525, which indicates that the stiffness and dissipated energy increased as the loading cycle increased. However, previous studies demonstrated that the stiffness of frozen soil increased and the dissipation energy decreased as the loading cycle increased owing to cyclic densification (Wang et al. 2018; Zhang et al. 2019). Evidently, the results obtained in this study are inconsistent with the previous study, which can be attributed to the effect of torsional stress; the damage it caused increased under principal stress rotation. Additionally, it differed from unfrozen soil samples in that the stiffness of frozen soil samples increased under cyclic rotational shear tests (Wang et al. 2017b; Zhou and Gong 2001). It was assumed that the cementing force of the porous ice improved the stiffness of the frozen specimen, whereas the accumulated pore water pressure resulted in the stiffness of the unfrozen specimen. Furthermore, it can be observed from Fig. 5b that as the CSR increased, the plastic strain of each cycle and the area of the hysteresis loop gradually increased, but the slope of the hysteretic loop decreased. This is because frozen samples exhibit a higher rigidity against deformation at low stress amplitudes.

Fig. 5
figure 5

Axial stress–strain hysteresis loops of frozen soils under different conditions: a CSR = 0.525; b N = 100

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Figure 6 shows the torsional stress–strain hysteresis loops under different conditions for CSR = 0.525 and N = 100. As shown in Fig. 6a, it can be found that the hysteresis loop is almost closed, implying that the specimens nearly exhibited viscoelastic behavior under the action of the torsional component. The areas of the hysteresis loops decreased as the loading cycle increased, whereas the slope of the hysteresis loops increased as the loading cycle increased. This reveals that the torsional stiffness increased and the torsional energy dissipation decreased with an increase in the cyclic loading cycle. This is because the vertical stress was greater than the torsional stress, which resulted in a larger strength of frozen samples along the torsional direction under the cyclic rotational shear test. Therefore, the torsional stiffness increased and the torsional energy dissipation decreased with an increase in the cyclic loading cycle. Figure 6b presents the typical torsional stress–strain hysteresis loops under different conditions of CSR for N = 100. Figure 6b depicts that as the CSR increased, the area of the torsional stress–strain hysteresis loops gradually increased, and the slope of hysteretic loops slightly decreased.

Fig. 6
figure 6

Torsional stress–strain hysteresis loops of frozen soils under different conditions: a CSR = 0.525; b N = 100

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Effects of frequency on hysteresis loop

Figure 7 shows the axial stress–strain hysteresis loops for the frequency f = 0.1 Hz, and loading cycle N = 100. Figure 7a shows that the unloading and loading curves are dissimilar owing to nonlinearity and hysteresis, which resulted in a large plastic strain. For a given frequency, the plastic strain initially decreased and subsequently stabilized as the loading cycle increased, whereas the area and inclination of the hysteresis loops slowly increased with increasing loading cycles. Figure 7b shows the axial stress–strain hysteresis loops under different frequencies for N = 100. As shown in Fig. 7b, the lower frequency corresponded to a greater degree of non-closure of the hysteresis loop. When the frequency increased and reached a certain value, the hysteresis loop almost closed. Additionally, as the frequency increased, the areas of the hysteresis loops decreased, and the slopes of the hysteresis loops increased. This indicates that the degradation of stiffness increased with decreasing loading frequency, resulting in the dissipative and plastic strains increasing with a decrease in the loading frequency (Wang et al. 2019a, b).

Fig. 7
figure 7

Axial stress–strain hysteresis loops of frozen soils under different conditions: a f = 0.1 Hz; b N = 100

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Figure 8 shows the torsional stress–strain hysteresis loops under different conditions for the frequency f = 0.1 Hz, and loading cycle N = 100. Figure 8a shows that the variation in the torsional hysteresis loops significantly differed from that in the axial hysteresis loops at the same frequency. As the loading cycle increased, the areas of the torsional hysteresis loops initially increased and then slowly decreased, whereas the inclination of the torsional hysteresis loops initially decreased and subsequently increased. These results indicated that the torsional strain increased in the initial stage, resulting in a decrease in the stiffness of the specimen and an increase in the dissipative energy. In the subsequent loading stages, the hardening of frozen samples also resulted in an increase in stiffness. This can be interpreted as follows: In the initial loading stage, the torsional stiffness and deformation were dependent on the deformation of the ice in the horizontal direction. The torsional stress acting on the solid particles led to a rupture between the particles and the cemented ice; the failure gradually expanded along the shear stress direction. As the dynamic loading continued, the frictional slide process stabilized, and the frozen samples harden under cyclic densification (Zhou et al. 2020; Lai et al. 2014). In addition, Fig. 8b shows the torsional stress–strain hysteresis loop under different frequencies for N = 100. As shown in Fig. 8b, at a certain loading cycle, the areas of the hysteresis loops decreased as the frequency increased, whereas the slopes of the hysteresis loops slightly increased as the frequency increased. This means that the degradation of the stiffness increased as the loading frequency decreased.

Fig. 8
figure 8

Torsional stress–strain hysteresis loops of frozen soils under different conditions: a f = 0.1 Hz; b N = 100

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Evolution of resilient modulus

The dynamic resilient modulus and damping ratio of the frozen soil play a decisive role in engineering design in cold regions, where they are calculated based on the features of the hysteresis loops. According to the calculation method of resilience modulus and damping ratio of the frozen soil, the resilient modulus and damping ratio of frozen clay under principal stress rotation were calculated (Ling et al. 2013; Zhang et al. 2019). The results are discussed below.

Effect of CSR on resilient modulus

Figure 9 displays the variations in the axial and torsional resilient moduli versus the loading cycle at different CSRs. Figure 9a shows that the variations in the axial resilient modulus can be divided into three phases for a given CSR, including a rapid decrease phase, a slow increasing phase, and a gradual stabilization phase. Under the same loading cycle, a smaller CSR corresponded to a larger axial resilient modulus. Figure 9b displays that the variation in the torsional resilient modulus is consistent with that of the axial resilient modulus. It was verified that a higher stress amplitude (larger CSR) led to specimen softening for a given loading cycle (Wang et al. 2020b). Additionally, it differed from other conclusions; the resilient modulus of frozen soil in this study decreased in the initial loading stage under principal stress rotation (Li et al. 2019; Zhang et al. 2018; Zhou et al. 2020; Ling et al. 2012). This may be explained by the weakening of the cementation of ice in the horizontal bedding plane, which resulted in poor interlocking, easy sliding with torsional stress, and consequent stiffness attenuation in the initial loading stage.

Fig. 9
figure 9

Variation in the axial and torsional resilient moduli versus loading cycle under different CSRs: a axial resilient modulus; b torsional resilient modulus

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Effect of frequency on resilient modulus

Figure 10 displays the variation in the axial and torsional resilient moduli versus the loading cycle under different frequencies. Figure 10a shows that the axial resilient modulus increased as the loading cycle increased for a given frequency, but its growth rate decreased with increasing frequency. Under the same loading cycle, the resilient modulus increases with the rising of frequency. Figure 10b shows that with an increase in the loading cycle, the torsional modulus quickly decreased during the first few loading cycles, and subsequently increased. Under the same loading cycle, the torsional resilient modulus increased with increasing frequency. As mentioned previously, the lower frequency vibration produced lower stiffness for the frozen specimen because it was conducive to the propagation of micro-cracks or macro-shear zones at low frequencies, which led to a decrease in resilient modulus. Therefore, it can be concluded that the frequency had a notable effect on the resilient modulus under multiaxial loading conditions, including principal stress rotation. These results agreed with the previous conclusions obtained by Wang et al. (2019a, b) and Guo et al. (2016a, b), but differed from those of Lekarp et al. (2000).

Fig. 10
figure 10

Variation in the axial and torsional resilient moduli versus loading cycle under different frequencies: a axial resilient modulus; b torsional resilient modulus

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Evolution of damping ratio

Effect of CSR on damping ratio

Figure 11 shows the variation in axial and torsional damping ratios versus the loading cycle at different CSRs. The results show that the axial damping ratio was relatively stable at the low CSR, whereas the axial damping ratio weakly increased at the high CSR. At the same loading cycle, the axial damping ratio increased with increasing CSRs, as the growth rate gradually decreased. It is noteworthy that the same previous studies have confirmed degradation of the axial damping ratio versus the loading cycle under cyclic triaxial test (Ling et al. 2013; Zhang et al. 2019). This result indicates that the principal stress rotation accelerated the weakening of the specimen structure, where the stiffness of samples decreased and the ability to absorb dissipation energy increased. Different from the variation in the axial damping ratio, the torsional damping ratio gradually decreased with increasing loading cycle. The greater the CSR, the more obvious attenuation of the torsional damping ratio versus the loading cycle. It can be explained that principal stress rotation induced anisotropy in the specimen, the stiffness of the specimen increased along the shear direction, and the ability to absorb dissipation energy decreased.

Fig. 11
figure 11

Variation in the axial and torsional damping ratios versus loading cycle under different CSRs: a axial resilient modulus; b torsional resilient modulus

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Effect of frequency on damping ratio

Before analyzing the damping ratio under different frequencies, it should be explained that when the frequency was 2 Hz, the variation range of the damping ratio was large and the law was poor. Here, the variation of damping ratio was analyzed when the frequency was 0.1, 0.2, and 1 Hz. Figure 12 represents the variation in axial and torsional damping ratios versus the loading cycle at different frequencies. It is clear that at a certain frequency, the axial damping ratio decreased first and then increased with an increase in the loading cycles. At a certain loading cycle, the axial damping ratio decreased with an increase in frequency. Moreover, the torsional damping ratio increased first and then fluctuated in a certain range with an increase in loading cycles. This result confirms the stiffness of the specimen increased with an increase in the frequency and the ability to absorb dissipation energy simultaneously decreased. In addition, it can be found that at the initial stage of the cyclic loading, the frozen sample was dense along the axial direction, the stiffness increased, and the ability to absorb dissipation energy decreased. However, owing to the shortening and particle rearrangement along the shear direction, the stiffness decreased and the ability to absorb dissipation energy increased. With the continuous loading, structural damage would occur, where the ability to absorb dissipation energy increased.

Fig. 12
figure 12

Variation in the axial and torsional damping ratios versus loading cycle under different frequencies: a axial resilient modulus; b torsional resilient modulus

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Conclusions

This study presented the results of a cyclic rotational shear experiment, considering principal stress rotation by using an FHCA-300. It focused on investigating the variation rules of accumulative plastic strains, hysteresis loops, and dynamic resilient moduli at different values of CSRs and frequencies. According to the results obtained from the experiment, the main findings of this study are summarized as follows:

  1. 1.

    The frozen hollow cylinder apparatus (FHCA-300) accurately achieved the designed circular stress path, considering principal stress rotation under multiaxial loading conditions. The accumulative plastic strain increased with the increasing loading cycle, and a larger CSR and a higher frequency resulted in a larger final accumulative plastic strain.

  2. 2.

    As the loading cycle increased, the areas and inclination of the axial hysteresis loops slightly increased. This suggested that the stiffness and energy dissipation of the specimen simultaneously increased with principal stress cyclic rotation under dynamic loading. As the loading cycle increased, the area of the torsional hysteresis loop may have increased in the initial stage and then decreased, while the inclination degree of the torsional hysteresis loop may have decreased in the initial stage and then increased. For a given loading cycle, as the CSR increased, the areas of the axial and torsional hysteresis loops significantly increased, and their inclination degrees decreased. Furthermore, a lower frequency led to a larger area and a smaller inclination degree for the hysteresis loop. This indicates that stiffness increased with a decrease in the CSRs and frequencies.

  3. 3.

    Considering the principal stress rotation, the structure of the specimen experienced a transition from initial local compaction to uniform cumulative deformation; the resilient modulus decreased. Afterward, the specimen experienced cyclic densification, the stiffness of the specimen increased, and the resilient modulus gradually increased, finally stabilizing.

  4. 4.

    The damping ratio increased with an increase in CSR, while the damping ratio decreased with an increase in frequency. When the rotation of principal stress was considered, the damping ratio of frozen soil would also increase owing to the shear failure and structural damage under cyclic dynamic loading.

  5. 5.

    The frozen specimen underwent axial compression and shear failure under principal stress rotation. Hence, there were two distinct variation trends of the damping ratio: decreasing and increasing. This indicated that the strength of the foundation soil could have been overestimated in traditional dynamic triaxial analysis; this is not conducive to engineering design safety principles such as seismic safety.

Notably, the hollow torsional shear test under the rotation of the principal stress was designed to simulate dynamic loading on the foundation. However, there are still some practical situations that cannot be considered. For example, the present experimental work is a specific case from one site, and the effects of principal stress rotation on dynamic strains, dynamic moduli, and damping ratios were not quantified by comparisons with dynamic triaxial tests. Moreover, an experimental study on pure principal stress rotation under the three-dimensional dynamic coupling loading of axial stress, torque, and confining pressure has not been conducted.