Experimental study on the influence of principal stress axes rotation on sand mechanical behavior and non-coaxiality effects

The soil around the foundations of offshore engineering experienced the principal stress axes rotation caused by the wave loadings. To investigate the influence of the principal stress axes rotation on the soil stress-strain relationship, a series of pure principal stress axes rotation tests under drained/undrained conditions were carried out with a hollow cylinder apparatus. The undrained test results show that the deviatoric stress, relative density, and starting point of rotation affect the mechanical properties of sand. Through the analysis of pore water pressure accumulation in the 1st cycle, it was observed that more pore pressure accumulates in Quadrants II and III, while relatively less accumulates in Quadrants I and IV. Under drained and undrained conditions, the non-coaxial behavior of the soil differs. The non-coaxiality can be significantly observed in the drained test, and the direction of strain increment tends to the stress increment direction when the cycle number increases.


Introduction
With the development of the world economy, the quantity and scale of ocean engineering projects, including marine oil and gas extraction and offshore wind power development, have been increasing year by year.The foundations of these ocean engineering structures are subjected to long-term wave loading.Within one cycle of wave loading, positive vertical pressure is exerted on the soil element when the wave crest acts above it, while negative vertical pressure is generated when the wave trough acts above it.In the intermediate region of wave action, the soil experiences horizontal shear stress.Consequently, as the wave loading propagates, the principal stress axes of the soil continuously rotate [1]   .
Currently, many researchers [2][3][4] have investigated the issue of principal stress axis rotation through laboratory experiments.Sivathayalan et al. [5] , employing hollow cylindrical torsional shear tests, examined the relationship between the undrained characteristics of sand under initial stress conditions and the direction of the principal stress axis.They observed that the rotation of the principal stress axis under constant shear stress leads to strain-softening responses.Cai et al. [6] conducted a series of hollow cylindrical torsional drained tests, and found that the deformation of sand accumulates with the rotation of the principal stress axis, demonstrating non-coaxiality.Liu et al. [7] investigated the influence of the middle principal stress coefficient on the strength and deformation of saturated soft clay under conditions of principal stress axis rotation.The experiments showed that with the increase of the middle principal stress coefficient, the critical stress ratio of the soil decreases.Qian et al. [8] conducted cyclic undrained tests on saturated soft clay with cyclic rotation of the principal stress axis, and analyzed the influence of the middle principal stress coefficient and generalized shear stress on deformation stiffness and non-coaxiality.Yang et al. [9] carried out undrained tests on saturated sandy soil with principal stress axis rotation.The results revealed that as the principal stress axis rotates, the strength of the sandy soil gradually weakens, and the degree of weakening is related to the compactness of the sand.The middle principal stress coefficient b has a significant impact on pore pressure response.
However, currently, these experiments do not take into account the influence of the rotation starting point in stress axis rotation tests, and there is a lack of comparison for non-coaxial characteristics under drained and undrained conditions.Therefore, we conducted a series of pure principal stress axis rotation tests using a hollow cylindrical apparatus under both drained and undrained conditions.The study aimed to investigate the effects of deviatoric stress, relative density, and rotation starting point on the mechanical properties of sandy soil and its non-coaxiality characteristics.

Hollow cylinder apparatus
As Fig. 1 shows, the Hollow cylinder apparatus from Shanghai Jiao Tong University (SJTU-HCA) was utilized [10] .The specimen is a hollow cylinder with an inner diameter of 60 mm, an outer diameter of 100 mm, and a height of 100 mm.The specimen is subjected to four loads, including axial force W, torque T, inner pressure Pi, and outer pressure Po, as illustrated in Fig. 2a.Fig. 2b depicts the stress state of the hollow cylindrical specimen.The α denotes the major principal stress angle, which is the angle between the first principal stress σ1 and the vertical direction.The variables σz, σθ, and σr represent the stresses in the axial, tangential, and radial directions, respectively, while σzθ represents the tangential stress induced by the torque T.

Test cases
The test cases are categorized into three series, namely I, II, and III, as listed in Table 1.Series I and II involve undrained tests, examining the influences of deviatoric stress, relative density, and rotation starting point on soil mechanical properties.The rotation starting point can be considered as the result of consolidation history.Series III is a drained test, discussing the mechanical properties of the sand under drainage conditions.The undrained tests use standard Chinese ISO sand, while the drained tests employ marine sand from Pingtan, Fujian.
The stress path for the principal stress axes rotation tests is illustrated in Fig. 3.The specimen is initially saturated at point A, followed by incremental loading to point B with isotropic consolidation under inner and outer pressures both equal to 100 kPa.Subsequently, maintaining constant effective mean principal stress p', the deviatoric stress q is increased to the target values at Right point C or Left point D by anisotropic consolidation.After consolidation, the stress axis begins counterclockwise cyclic rotation with stress control, and the rotation period is 7200 s.Throughout the stress axis rotation process, the middle principal stress coefficient b, effective mean principal stress p', and deviatoric stress q are kept constant.The middle principal stress coefficient b = (σr-σθ)/(σz-σθ).The stress path in qp' space of test series I Fig. 5 illustrates the stress path in the (σz-σθ)/2p' -p' space, showing that as p' decreases, the (σzσθ)/2p' exhibits a decrease.In Case I-1, the (σz-σθ)/2p' change within the range of -0.5% to 0.5%, while in Case I-2, (σz-σθ)/2p' change within the range of -0.2% to 0.2%.The pore water pressure ratio (PWPR), defined as the pore pressure value divided by the initial effective mean principal stress, is used to measure the accumulation of pore pressure and concern the undrained strength of the soil.The pore water pressure ratio with the major principal stress angle α for test series I

Influence of relative density and consolidation history
Series II focused on the influence of the relative density and consolidation history.In Cases II-1 and II-2, the rotation starting point is the right end in (σz-σθ)/2p' -p' space, while in Cases II-3 and II-4, it is the left end.The stress path is shown in Fig. 7, and each case is approaching or has reached the CSL.
From the results of Cases II-1 and II-2, it can be observed that the high-density Case II-2 requires 10 cycles to reach the CSL, whereas the lower-density Case II-1 reaches the CSL in only 2.5 cycles.Similar findings are observed in Cases II-3 and II-4, where the higher-density Case II-4 requires 4 cycles to reach the CSL, while the lower-density Case II-3 is only 2 cycles.
Fig. 8 illustrates the stress paths in the (σz-σθ)/2p' -p' space for Series II.From Fig. 8a and 8b, it can be seen that when the rotation starting point is the right end, with the decrease in (σz-σθ)/2p' -p' exhibits a decrease followed by an increase trend.In the relatively lower-density II-1 test, after 2 cycles of rotation, (σz-σθ)/2p' -p' reaches 0.5%, while in the higher-density Case II-2, it takes 7 cycles to reach 0.5%.From Fig. 8c and 8d when the rotation starting point is the left end, with the decrease in (σz-σθ)/2p' -p' exhibits an increase followed by a decrease trend.In the relatively lower-density Case II-3, after 2 cycles of rotation, (σz-σθ)/2p' -p' stabilizes, while in the higher-density Case II-4, it takes 4 cycles to stabilize.9a, a cycle of counterclockwise stress axis rotation is divided into four quadrants: I, II, III, and IV.In quadrants I and II, the torque T is positive, while in quadrants III and IV, the torque T is negative.The axial force is positive in quadrants I and IV, while it is negative in quadrants II and III.The net increase in pore pressure Δu for a specific quadrant is defined as the pore pressure exiting the quadrant minus the pore pressure entering the quadrant.Figs.9b and 9c respectively depict the net increase in pore pressure per cycle for quadrants in Cases II-2 and II-4.
Comparing the pore pressure accumulation during the first cycle, all tests exhibit greater pore pressure accumulation in quadrants II and III, while the accumulation is relatively lower in quadrants I and IV.This indicates that the ability of the specimen to "resist" pore pressure accumulation is weaker when subjected to stress in quadrants II and III.This phenomenon is related to the inherent anisotropy of the specimen.The horizontal layering during the specimen preparation process results in an initial major principal stress angle of 0°, with the direction of the first principal stress in the vertical direction.During stress loading in quadrants II and III, the major principal stress angles are in the range of 45° to 90° and -90° to -45°, respectively.The direction of the first principal stress deviates toward the horizontal direction, making pore pressure accumulation more likely.
For Case II-2, which starts rotating from the right end, the maximum pore pressure accumulation occurs in quadrant II.For Case II-4, which starts rotating from the left end, the maximum pore pressure accumulation occurs in quadrant III.This demonstrates that the rotation starting point has an impact on pore pressure accumulation.
As explained in the previous Fig. 7, it is evident that Case II-2 requires 10 cycles to reach the CSL, while Case II-4 only needs 4 cycles.In Fig. 9, when N=8 for Case II-2 and N=2 for Case II-4, the net pore pressure increase is positive in quadrants II and IV, while it is negative in quadrants I and III.This phenomenon can be considered as an indication of specimen failure.

Non-coaxial characteristics
The soil has different non-coaxial characteristics between drained and undrained tests.For the undrained test, during the 1st cycle of stress axis rotation in Fig. 10a, the strain increments are relatively small, and the anisotropy at different major principal stress angles is evident.When the major principal stress angle is in the range of 45° 90° (quadrants II), the strain increment is larger.However, according to the 12th cycle of stress axis rotation in Fig. 10b, the anisotropy at different major principal stress angles diminishes, and the strain increments are generally larger, with noncoaxial angles averaging around 30°.
For the drained test, during the 1st cycle of stress axis rotation in Fig. 10c, the strain increments are relatively large, with non-coaxial angles averaging around 30°.However, by the 13th cycle of stress axis rotation in Fig. 10d, the strain increments decrease compared to the 1st cycle, and the direction of strain increments tends to align with the direction of stress increments.
Overall, in undrained conditions, with an increase in the number of cycles N, there is not a significant change in the angle between strain increments and stress increments.In drained conditions, as the N increases, the direction of strain increments tends to align with the direction of stress increments.

Conclusion
This study conducted principal stress axes rotation test with undrained and drained conditions on saturated sandy soil and has drawn the following conclusions.1) Deviatoric stress, relative density, and rotation starting point influence the mechanical characteristics of the sandy soil.In pure principal stress axis rotation tests, higher deviatoric stress and lower relative density make the soil more inclined to failure.The rotation starting point has an impact on the evolution of pore pressure accumulation.
2) Analyzing the pore pressure accumulation in different quadrants during the undrained test, in the 1st cycle of stress axis rotation, it was observed that more pore pressure accumulates in Quadrants II and III, while relatively less accumulates in Quadrants I and IV.
3) Under drained and undrained conditions, the non-coaxial behavior of the soil differs.In undrained conditions, with an increase in the number of cycles N, there is not a significant change in the angle between strain increments and stress increments.In drained conditions, as the N increases, the direction of strain increments tends to align with the direction of stress increments.
The experiments in this study used ISO standard sand with significant shear dilation characteristics.However, this is insufficient for studying the anisotropy of soil under the rotation of the stress principal axis.In the future, it is necessary to supplement the experiments with more types of sand and clay.

Fig 3 .
Fig 3.The stress path of pure principal stress axes rotation tests

Fig 4 .
Fig 4.The stress path in qp' space of test series I Fig.5illustrates the stress path in the (σz-σθ)/2p' -p' space, showing that as p' decreases, the (σzσθ)/2p' exhibits a decrease.In Case I-1, the (σz-σθ)/2p' change within the range of -0.5% to 0.5%, while in Case I-2, (σz-σθ)/2p' change within the range of -0.2% to 0.2%.The pore water pressure ratio (PWPR), defined as the pore pressure value divided by the initial effective mean principal stress, is used to measure the accumulation of pore pressure and concern the undrained strength of the soil.Fig.6depicts the relationship between PWPR and the major principal stress angle in Series I tests.In the later stages of Case I-1, the PWPR value quickly reaches 0.6, indicating that the specimen is approaching failure.In contrast, the PWPR value in Case I-2 increases mainly in the first cycle, reaching approximately 0.1, and then stabilizes in the second cycle.

2 Fig 5 . 2 Fig 6 .
Fig 4.The stress path in qp' space of test series I Fig.5illustrates the stress path in the (σz-σθ)/2p' -p' space, showing that as p' decreases, the (σzσθ)/2p' exhibits a decrease.In Case I-1, the (σz-σθ)/2p' change within the range of -0.5% to 0.5%, while in Case I-2, (σz-σθ)/2p' change within the range of -0.2% to 0.2%.The pore water pressure ratio (PWPR), defined as the pore pressure value divided by the initial effective mean principal stress, is used to measure the accumulation of pore pressure and concern the undrained strength of the soil.Fig.6depicts the relationship between PWPR and the major principal stress angle in Series I tests.In the later stages of Case I-1, the PWPR value quickly reaches 0.6, indicating that the specimen is approaching failure.In contrast, the PWPR value in Case I-2 increases mainly in the first cycle, reaching approximately 0.1, and then stabilizes in the second cycle.

Fig 7 . 4 Fig 8 .
Fig 7. The stress path in qp' space of test series Ⅱ

4 Fig. 9
Fig. 9 Pore water pressure accumulation in each quadrant

Fig. 10
Fig. 10 Stress path and strain increment undrained and drain tests

Table 1 .
Cases of pure principal stress axes rotation tests