The tests employed the Hollow Cylinder Apparatus (HCA) manufactured by GDS Instruments Ltd., UK, to conduct drained coupled axial-torsional cyclic loading tests. The HCA facilitates coupled cyclic loading in both the axial and torsional directions. Axial and torsional loads are applied at the base of the specimen. Force transducers for measuring axial force and torque are located at the specimen top, while displacement transducers for measuring axial displacement and angular rotation are positioned at the specimen base. The maximum dynamic axial and torsional loads are 10 kN and 30 Nm, respectively, with a maximum loading frequency of 5 Hz for both modes. Confining pressures (outer cell pressure and inner cell pressure) and back pressure are applied and measured using standard pressure/volume controllers, with a maximum capacity of 2 MPa. Back pressure is applied at the specimen base, and pore pressure is measured at the specimen top. A detailed description of the HCA testing accuracy and the data processing procedures employed can be found in references (Chen et al., 2016; Zienkiewicz et al., 1980; Prasanna and Sivathayalan, 2021).

Figure 1a illustrates the stress state of a thin-walled specimen element in the HCA. The instrument’s control and loading parameters indicated in the figure are as follows: r _i, r _o = inner and outer radii of the specimen, respectively; u _i, u _o = inner radial displacement and outer radial displacement of the specimen during shearing, respectively; p _i = inner cell pressure, p _o = outer cell pressure; W = axial force; M _T = torque. The stress components acting on the thin-walled specimen element are depicted in Figures 1b,c: σ _z = axial stress; σ _r = radial stress; σ _θ = hoop (circumferential) stress; τ _zθ = shear stress in the plane perpendicular to the radial direction; σ ₁, σ ₂, σ ₃ = major, intermediate, and minor principal stresses of the element, respectively; α = orientation angle of the major principal stress; θ = torsional displacement (angle of twist) generated in the specimen during shearing. The relationships between these stress components are presented in Figure 1d.

All the marine sand was sourced from the site of the China Three Gorges New Energy Jiangsu Dafeng H8 - 2# 300 MW offshore wind farm project, which is situated in the southwestern part of the Yellow Sea, north of Maozhusha in Dafeng District, Yancheng City, Jiangsu Province. Specific details are presented in Figure 2. The offshore distance from the center of the site is 72 km, and the theoretical water depth ranges from 7.5 to 20.9 m. The sand predominantly consists of chalky sand with a shallow depth of 30 m. The sand exhibits a grey color, is saturated and loose, and contains quartz, feldspar, and a small quantity of clay in the surface layer. The primary minerals are quartz and feldspar, with occasional traces of shell debris and a small amount of clay particles. A significant amount of silt is mixed in the surface layer of some machine sites. The particles are angular, with a particle specific gravity (G _s) of 2.70, an average particle size of 0.15 mm, an inhomogeneity coefficient of 2.11, a coefficient of curvature of 0.97, a maximum porosity ratio (e _max) of 1.29, and a minimum porosity ratio (e _min) of 0.63. The gradation curve is depicted in Figure 3. The in - situ CPTu test revealed a cone tip resistance of 5.67 MPa, a sidewall friction resistance of 35.98 kPa, and a pore water pressure of 12.5 kPa. The in-situ shear wave velocity was approximately 150 m/s, and there seems to be an error in the repeated statement “the in-situ shear wave velocity was about 1.5 kPa” which is ignored as it likely contradicts the previous velocity value.

The specimen was a hollow cylinder with an outer diameter of 100 mm, inner diameter of 60 mm, and height of 200 mm. To ensure uniformity of specimen preparation, the layer-by-layer vibration compaction method was employed. The specimen was compacted in five layers. For each layer, the required mass of particles for each grain size fraction was calculated according to the gradation, weighed separately, uniformly mixed, and then slowly poured into the mold. Subsequently, each layer was compacted to the target layer height using a compaction hammer. The drop hammer had a mass of 1 kg and a free fall height of 15 cm. Saturation of the specimen was achieved through a three-step procedure: 1) CO₂ flushing: carbon dioxide (CO₂) was percolated through the specimen for 15 min to displace the air. 2) De-aired water percolation: de-aired water was then flushed from the bottom to the top of the specimen until no air bubbles were observed exiting the top. 3) Stepwise back-pressure saturation: back pressure was applied incrementally in stages. Following the stepwise back-pressure saturation, the pore pressure coefficient B was measured. A specimen was deemed saturated if the measured B-value exceeded 0.95. The saturated specimen was then subjected to isotropic consolidation under an initial effective confining pressure σ _3c' of 100 kPa.

The controlled loading path traces an ellipse in the [τ _zθ, (σ _z-σ _θ)/2] stress space. This path achieves a stress variation pattern characterized by continuous rotation of the principal stress axes, where the normal deviatoric stress, shear stress, and resultant generalized deviatoric stress (composed of both components) all undergo cyclic variations. As illustrated in Figure 4, the shape of this elliptical stress path is primarily defined by three key parameters: the major axis (b), the ratio of minor axis to major axis (a/b), and the inclination angle (β). Once these three characteristic parameters are specified, the stress path is uniquely defined. When a = 0, the ellipse degenerates into an inclined straight line. When a = b, the ellipse evolves into a circle.

To systematically investigate the influence of density state and stress path on the dilative behavior of saturated Nanjing fine sand, the relative density (D _r) of the sand was categorized into three levels: 35%, 50%, and 70%, corresponding to loose, medium-dense, and dense states, respectively. For specimens sharing the same D _r value, cyclic elliptical stress paths with different cyclic stress ratio (CSR = q _cyc/σ _3c'), a/b, β were applied. The loading frequency was maintained at 0.1 Hz. The loading frequency was maintained at 0.1 Hz. This frequency was selected for three primary reasons to ensure both representativeness and experimental validity: 1) It falls within the central range of typical wave loading frequencies (0.05–0.2 Hz) observed in offshore environments, thereby representing common storm-wave conditions; 2) It ensures fully drained conditions throughout the cyclic tests, allowing pore water to freely flow without generating excess pore pressure, which is crucial for accurately isolating the mechanical response of the soil skeleton; and 3) It is compatible with the dynamic performance characteristics of the HCA system and aligns with the study’s focus on long-term cumulative deformation rather than transient dynamic response. Where q cyc = τ 2 + σ v − σ θ 2 / 4 | max is the cyclic deviatoric stress amplitude. In coupled axial-torsional cyclic loading tests, q_cyc is conventionally employed to represent the deviatoric stress amplitude, describing the variation in the magnitude of the deviatoric stress. Its physical meaning corresponds to the major axis length of the elliptical stress path. The specific experimental program is summarized in Table 1. Specimen ID were designated according to the following naming convention: E-D _r-a/b-β-CSR.

To elucidate the mechanism of volumetric strain accumulation under complex cyclic loading, a detailed analysis of the stress-strain response is first presented. The following typical results not only demonstrate the apparatus’s capability to replicate prescribed stress paths but also provide the foundational data for understanding the subsequent decomposition of volumetric strain into reversible and irreversible components, as well as the influence of stress path.

Figure 5 presents typical test results of marine sand under cyclic elliptical stress paths, including: comparison of measured versus theoretical stress paths in the [τ _zθ, (σ _z-σ _θ)/2] stress space; time histories of axial strain (ε _z), shear strain (γ _zθ), and excess pore water pressure (u _e); hysteretic loops of σ _z versus ε _z; hysteretic loops of τ _zθ versus γ _zθ; time history of volumetric strain (ε _vd). The following observations can be drawn from Figure 5: (a) The HCA instrument demonstrated its capability to accurately simulate the prescribed coupled cyclic stress paths; (b) The σ _z-ε _z relationship exhibited distinct hysteretic loops, these loops were initially unclosed, indicating significant plastic deformation occurred during the early loading cycle; (c) as the number of cycles (N) increased, the hysteretic loops transitioned from unclosed to closed, and their enclosed area progressively decreased. This evolution signifies the soil approaching a critical state; (d) throughout the test, the ε _z development with increasing N comprised two distinct phases: a phase of rapid growth followed by a phase of stable accumulation. This behavior is primarily attributed to changes in the irreversible dilative component during the initial N; (e) during the coupled axial-torsional cyclic loading, the u _e value of the specimen consistently remained within the range of 0 ∼ 1 kPa. No significant accumulation of u _e was observed, indicating that pore water could freely flow into or drain out of the specimen.

The observed stress-strain responses, particularly the evolution of hysteretic loops and the development of axial strain, are direct manifestations of the underlying “intrinsically consistent physical mechanism” mentioned in the introduction, which couples shear deformation with volumetric change. The following decomposition of total volumetric strain into reversible and irreversible components allows for a more direct examination of this coupling effect under complex stress paths. The cyclic volumetric strain is defined as half the difference between the maximum and minimum volumetric strain (ε _vd) values within a single cycle (Xu et al., 2025), and the accumulated volumetric strain is defined as the average of the maximum and minimum ε _vd values within a cycle. As evident in Figure 5, ε _vd comprises a reversible cyclic strain component (ε _vd,re) and an irreversible accumulated strain component (ε _vd,ir). This decomposition aligns with the experimental findings of Zhang, (2000), Duku et al. (2008), and Wichtmann et al. (2005). The reversible component (ε _vd,re) is elastic in nature and does not induce changes in the effective stress state of the soil. For soil dynamics characteristics, the focus is often on permanent deformations and accumulated pore water pressure, which are intrinsically linked to the irreversible accumulated strain component (ε _vd,ir). Therefore, the primary investigation centers on the influence of density state and stress path on ε _vd,ir for the saturated sand. Key experimental findings as following: 1) ε _vd,ir is irrecoverable, representing plastic deformation. ε _vd,ir consistently manifested as monotonic volume compression; 2) By convention, ε _vd,ir is defined as negative during dilation and positive during contraction; 3) The increasing rate of ε _vd,ir was highest during the initial stages of cyclic loading. Furthermore, the maximum rate of change within each half-cycle consistently occurred at the point of shear stress reversal; 4) The increasing rate of ε _vd,ir progressively diminished with increasing N, asymptotically approaching zero as the soil reached the critical state.

To further investigate the differences in ε _vd increments of saturated marine fine sand under various cyclic stress paths induced by wave loading, supplementary tests were conducted on saturated marine fine sand with D _r = 50%. These included pure cyclic compression-tension loading test and pure cyclic torsional loading test, both controlled at a CSR of 0.20. Figure 6 presents the relationships between ε _vd and N for the specimens under pure compression-tension loading, pure torsional loading, and coupled axial-torsional loading. The following key observations are drawn from the figure: consistent with the behavior observed under coupled axial-torsional loading, ε _vd,ir under both pure compression-tension loading and pure torsional loading increased with N. Furthermore, the increasing rate of ε _vd,ir first decrease rapidly as N increases, and then gradually approach 0. At an identical CSR, the development of ε _vd,ir with increasing N was essentially consistent between the two simple stress paths (pure compression - tension and pure torsion). The magnitude of ε _vd,ir developed under the complex stress path (coupled axial-torsional loading) was significantly greater than that developed under either simple stress path. This finding indicates that previous experimental studies based solely on simple stress paths (either pure compression-tension or pure torsion) underestimated the development of ε _vd,ir in saturated marine sand. Additionally, ε _vd,re under pure compression-tension loading was slightly greater than that observed under coupled axial-torsional loading. The value of ε _vd,re under pure torsional loading remained essentially zero. This behavior arises because ε _vd,re is induced by the periodically varying mean effective stress, which is present in pure compression-tension loading but absent in pure torsion (where mean effective stress remains constant).

Conventionally, in the estimation of site settlement induced by earthquake ground motions, the accumulated volumetric strain at N = 15, denoted as (ε _vd,ir)₁₅, is used to approximate the settlement for an earthquake magnitude of M = 7.5 (Duku et al., 2008). This convention provides a well-established and widely understood reference point for characterizing strain accumulation under cyclic loading. Beyond its seismic origin, the parameter (ε _vd,ir)₁₅ has proven to be an effective and stable normalization benchmark in various cyclic loading studies, as it captures a representative stage of strain development while minimizing the influence of early-cycle variability. Furthermore, in view of the effectiveness and applicability of (ε _vd,ir)₁₅ in the normalizing the volumetric strain accumulation characteristics of soil subjected to cyclic loading (Duku et al., 2008; Yee et al., 2014; Tokimatsu and Seed, 1987), the accumulated volumetric strain at the 15th cycle (ε _vd,ir)₁₅ is analyzed herein to investigate the ε _vd,ir development in marine sand subjected to wave-induced complex cyclic loading. Figure 7 presents the relationship curves between the (ε _vd,ir) _N /(ε _vd,ir)₁₅ and N for various test conditions. The results demonstrate that: for specimens with identical D_r, the (ε _vd,ir) _N /(ε _vd,ir)₁₅ values under different complex stress paths are distributed within a relatively narrow band as N increases. This observation indicates that the influence of stress path on (ε _vd,ir) _N /(ε _vd,ir)₁₅ is relatively minor. Consequently, the analysis can focus primarily on the relationship between (ε _vd,ir)_N/(ε _vd,ir)₁₅ and N. The (ε _vd,ir) _N /(ε _vd,ir)₁₅ be reasonably and simply expressed as a logarithmic function of N: f N = ε vd , ir N / ε vd , ir 15 = C 2 ⁡ ln C 1 N + 1 (1) where C ₁ and C ₂ are fitting parameters.

As can be seen from the above, for the given D _r, (ε _vd,ir)₁₅ is the key parameter characterizing the volumetric strain accumulation characteristics of saturated marine sands. Figure 8 presents the relationship between (ε _vd,ir)₁₅ and CSR for saturated marine sands. The figure reveals that CSR significantly influences the development of (ε _vd,ir)₁₅. For specimens with given a/b, β, and D _r, (ε _vd,ir)₁₅ increases rapidly with increasing CSR. However, for specimens sharing the same D_r, identical CSR, and identical β, the magnitude of (ε _vd,ir)₁₅ increases with increasing a/b. Furthermore, the influence of a/b cannot be normalized into a unified pattern across the CSR vs. (ε _vd,ir)₁₅ relationship curves. This behavior stems from the fact that CSR solely characterizes the variation in deviatoric stress amplitude (represented by the semi-major axis length of the ellipse) but fails to adequately capture changes in the shape of the stress path, such as the minor-to-major axis ratio (a/b). The deformation of the specimen is essentially induced by the coupled action of the axial stress amplitude and the shear stress amplitude. For coupled axial-torsional cyclic loading tests sharing the same b value but differing in a/b, the applied stress paths differ, consequently leading to distinct shearing effects (e.g., differences in non-coaxiality).

Figure 9 illustrates the influence of β on (ε _vd,ir)₁₅ at a constant CSR. As shown, β has no significant effect on (ε _vd,ir)₁₅; that is, once CSR falls below a specific threshold value, (ε _vd,ir)₁₅ becomes virtually independent of β. For marine sand across different D _r values, this threshold CSR value is approximately 0.20. This finding aligns with Xu et al. (2014), who likewise observed no significant influence of the elliptical inclination angle β on the cyclic resistance.

The preceding experimental results demonstrate the limitations of CSR when used to analyze coupled cyclic loading tests, specifically its inability to adequately represent the effective stress level experienced by soil elements undergoing continuous principal stress rotation. To address this, (Huang et al., 2015). [29] proposed the term Equivalent cyclic stress ratio (ESR) to characterize the magnitude of cyclic stress under complex stress paths, ESR serves as a more comprehensive index than CSR, as it integrates both the axial and shear stress amplitudes over the entire elliptical path, thereby better representing the effective cyclic stress level under conditions of principal stress rotation. ESR is defined as Equation 2: ESR = q equ / σ 3 c ′ q equ = 1 T ∫ 0 T q t dt = 1 T ∫ 0 T τ 2 + σ v − σ θ 2 2 dt (2)

Where, T represents the cyclic loading period, |q(t)| denotes the distance from any point on the stress path to the origin. Thus, q _equ physically represents the average intensity of cyclic loading, defined as the mean value of the maximum cyclic shear stress experienced by the soil element. For elliptical stress paths, the equivalent cyclic stress q(t) and q _equ can be explicitly expressed as Equation 3: q t = τ d · ⁡ sin 2 π T t 2 + σ d 2 · ⁡ cos 2 π T t + φ 2 q equ = 1 T ∫ 0 T q t d t = 0.64 ∼ 1 q cyc (3)

Where, τ _d is the cyclic shear stress amplitude, σ _d is the cyclic axial stress amplitude. For elliptical stress paths where q _equ cannot be obtained through direct integration, numerical integration methods may be employed for its evaluation.

Figure 10 illustrates the relationship between (ε _vd,ir)₁₅ and ESR for specimens with various D _r. For given D _r, regardless of whether the stress paths are identical, the (ε _vd,ir)₁₅ values of saturated sand with different ESR values are distributed within a narrow band. Moreover, (ε _vd,ir)₁₅ increases linearly with rising ESR, demonstrating a strong correlation. This indicates that ESR can serve as a characteristic parameter to effectively describe complex stress paths involving principal stress axis rotation. At the same D _r, ESR can ideally characterize the (ε _vd,ir)₁₅ under different stress paths: ε vd , ir 15 = C 3 ESR − ESR t (4) where C ₃ are fitting parameter. In addition, as shown in Figure 10, saturated marine sand exhibits a distinct volumetric threshold cyclic stress ratio ESR_t, When ESR is below ESR_t, no ε _vd,ir develops; once ESR exceeds this threshold, the ε _vd,ir increases rapidly with further increases in ESR. The pore pressure ratio under undrained conditions shares the same physical basis as the volumetric strain under drained conditions. Therefore, ESR_t is of significant importance for understanding and addressing geodynamic problems induced by cyclic loading, such as seismic excitation, ocean wave action, and pile-driving vibrations. Ivsic (2006) and Park et al. (2015) suggested that the threshold cyclic stress ratio can be determined based on the threshold shear strain (γ _tv). For saturated sand, the γ _tv ranges from 0.014% to 0.023%, corresponding to ESR_t = 0.05 to 0.06.