Influence of Principal Stress Orientation on Cyclic Degradation of Soft Clay Under Storm Wave Loading

Abstract

Coastal marine soft clays subjected to long-term storm wave loading often exhibit inclined initial principal stress orientation (α₀) and subsequent cyclic principal stress rotation (PSR). These stress states critically influence soil mechanical behavior and failure mechanisms, threatening offshore structural stability. This study employs hollow cylinder apparatus testing to investigate the undrained cyclic loading behavior of reconstituted soft clay under controlled α₀ and PSR conditions, simulating storm wave-induced stress paths. Results demonstrate that α₀ governs permanent pore pressure and vertical strain accumulation with distinct mechanisms, e.g., a tension-dominated response with gradual pore pressure rise at α₀ < 45° transitions to a compression-driven rapid strain accumulation at α₀ > 45°. Rotational loading with PSR significantly intensifies permanent strain accumulation and stiffness degradation rates, exacerbating soil’s anisotropic behavior. Furthermore, the stiffness degradation index tends to uniquely correlate with the permanent axial or shear strain, which can be quantified by an exponential relationship that is independent of α₀ and PSR, providing a unified framework for normalizing stiffness evolution across diverse loading paths. These findings advance the understanding of storm wave-induced degradation behavior of soft clay and establish predictive tools for optimizing marine foundation design under cyclic loading.

1. Introduction

Soft clays, widely distributed in coastal regions, have garnered significant attention in geotechnical engineering due to their prevalence in marine infrastructure projects. These soils are subjected to both static and cyclic loads (e.g., waves, wind, and earthquakes) during the service life of offshore structures. Under cyclic loading, soft clays exhibit progressive strength reduction, permanent strain accumulation, and stiffness degradation, ultimately compromising foundation stability [1,2,3,4]. The cyclic behavior of such soils is highly sensitive to stress loading parameters, including amplitude, path, and history. While cyclic direct simple shear (CDSS) tests have been extensively employed to investigate cyclic deformation under seismic loading [5,6,7], they inadequately replicate storm wave-induced loading conditions. Unlike transient seismic loads, storm waves impose long-term cyclic stresses characterized by continuous principal stress rotation (PSR) under near-constant deviatoric stress [8]. This discrepancy in loading paths limits the applicability of CDSS-derived results to wave-dominated marine environments. Furthermore, the inherent anisotropy of soft clays—arising from depositional processes—introduces directional dependence to their mechanical response under complex stress paths [9]. A systematic understanding of anisotropic soft clay behavior under wave-type cyclic loading with PSR is therefore critical for optimizing the safety and durability of marine foundations.

Conventional triaxial devices, while widely used to investigate soil behavior under cyclic loading, impose constraints by maintaining two principal stresses equal, thereby fixing the principal stress direction to vertical or horizontal orientations. This limitation restricts simulations to abrupt PSRs (e.g., 0° to 90° jumps) [10,11]. Although true triaxial apparatuses enable independent control of three principal stresses, their capability to replicate complex initial stress states remains limited [12,13]. To address these challenges, the hollow cylinder apparatus (HCA) has emerged as a critical tool, offering four degrees of freedom (axial force, torque, inner/outer pressures) to independently control stress parameters, including continuous PSR [14,15]. While HCA-based studies have advanced the understanding of shear strength and stiffness anisotropy under PSR [16,17,18,19,20,21], most investigations isolate either PSR effects or initial principal stress direction angles (α₀). Few studies integrate both factors, leaving rotational strain mechanisms under combined α₀ and PSR poorly resolved.

Seminal work by Ishihara and Towhata [22] demonstrated that PSR drives significant plastic strain accumulation, with the strain increment direction deviating from stress orientation. Subsequent studies [23,24] confirmed that strain evolution, stiffness degradation, and pore pressure generation are intrinsically linked to principal stress direction. Stiffness degradation, a critical factor in ground movement prediction, exhibits strong strain dependency; small-strain stiffness (critical for geotechnical design) rapidly diminishes at medium to large strains, revealing nonlinear stress–strain behavior [25]. Empirical models often describe this degradation using hyperbolic functions normalized by elastic modulus [26,27]. Idriss et al. [28] found that the plasticity index dominates degradation trends under both monotonic and cyclic loading, while cyclic degradation is quantified via degradation indices with logarithmic expressions [2]. Notably, while energy-based approaches demonstrate potential in unifying pore pressure and stiffness degradation predictions for high-OCR clays [29], their applicability to soft clays under principal stress rotation (PSR) remains underdeveloped. Complementing this gap, recent HCA investigations on natural marine clays subjected to continuous PSR [15] have refined axial/torsional degradation indices, revealing consistent anisotropic degradation patterns wherein torsional degradation exceeds axial degradation. This evidence establishes the necessity to resolve multidirectional degradation mechanisms under complex loading paths. Gasparre et al. [30] utilized HCA to demonstrate stiffness uniqueness across loading paths, yet most studies focus on simplified conditions, neglecting PSR-coupled effects. However, existing PSR studies predominantly address level ground scenarios with zero initial shear stress, overlooking conditions prevalent in sloping terrains or beneath gravity structures—where non-zero initial shear stresses and principal stress axis inclinations are inherent. This gap undermines predictions for marine environments, where storm wave loading imposes cyclic PSR on seabed soils subjected to different α₀. Although Gasparre et al. [30] and Yang et al. [31] observed accelerated stiffness degradation under PSR compared with non-rotational loading path, current constitutive models and simulations remain anchored to simplified stress paths. Systematic experiments integrating PSR and α₀ effects are thus essential to unravel stiffness degradation mechanisms under storm wave loading and furnish robust parameters for numerical models, ultimately enhancing the safety of marine foundations.

This study conducted a series of undrained cyclic HCA tests on saturated soft clay from Hangzhou, China, under controlled principal stress orientations and prescribed stress paths, with particular emphasis on the effects of α₀ and PSR on strain development and stiffness behavior. Both the non-rotational and rotational loading test results revealed a robust correlation between the stiffness degradation index and permanent strains, which remained invariant to loading path and principal stress orientation. This finding holds significant implications for accurately evaluating the cyclic degradation of soft clays under complex cyclic loading conditions.

2. Sample Preparation and Test Procedures

The natural soft clay investigated in this study was obtained from Hangzhou, located in southeastern China. The particle size distribution curve and index properties of the clay are provided in Figure 1 and

Table 1. Index properties of the tested soft clay.

Index Property	Value
Specific gravity, Gs	2.68
Initial density, ρ0 (g/cm3)	1.78
Water content, w (%)	28.9
Plastic limit, wP (%)	23.7
Liquid limit, wL (%)	41.5
Plastic index, IP	17.8
Unit weight of soil, γ (kN/m3)	18.6
Initial void ratio, e0	1.131–1.149

, respectively. According to the ASTM D2487-17 [32], the soft clay was defined as lean clay. To address inherent heterogeneity in natural clay and ensure sample reproducibility, reconstituted specimens were prepared via one-dimensional consolidation of clay slurry. The slurry was initially mixed with distilled water to achieve a water content of 80% (approximately twice the liquid limit [33]) and consolidated in a large cylindrical oedometer under incremental vertical stress up to 50 kPa until the settlement rate fell below 0.1 mm/day. A custom trimming device was used to fabricate hollow cylindrical samples (height = 200 mm, outer diameter = 100 mm, inner diameter = 60 mm), minimizing stress non-uniformity inherent to hollow cylinder tests [15]. Prior to testing, samples were mounted on the HCA base pedestal with internal/external latex membranes and radial filter paper drainage. Full saturation was ensured by applying a 350 kPa backpressure for 24 h, achieving B-values ≥ 0.96.

Cyclic tests were conducted using a dynamic HCA developed by GDS Instruments (Global Digital Systems, Hampshire, UK). The system independently controls axial force (W), torque (M_T), inner cell pressure (p_i), and outer cell pressure (p₀), enabling precise application of vertical stress (σ_z), shear stress (τ_zθ), circumferential stress (σ_θ), and radial stress (σ_r). This capability permits direct manipulation of the three principal stresses and the orientation angle of the major principal stress axis. A detailed description of the apparatus and stress–strain calculation methodology is provided by Pan et al. [21].

Saturated samples were first isotropically consolidated to an initial mean effective stress p₀′ = 100 kPa, followed by anisotropic consolidation to an initial static deviatoric stress q₀ = 20 kPa with fixed principal stress orientation angles (α₀ = 0°, 22.5°, 45°, 67.5°, and 90°). Post-consolidation, the cyclic deviatoric stress of q_cyc = 20 kPa was applied under two distinct stress paths: (1) non-rotational (linear) loading: fixed α₀ with deviatoric stress linearly increased to 40 kPa and then reduced to 20 kPa (Figure 2a); (2) rotational loading: cyclic PSR with Δα varying between ±10° based on different α₀ through a combination of vertical and torsional cyclic loading (Figure 2b). All specimens were subjected to 3000 loading cycles (N = 3000) at a frequency of 0.1 Hz under undrained conditions unless failure occurred earlier.

3. Test Results and Discussion

3.1. Pore Pressure Generation

Figure 3 depicts the typical effective stress paths of samples under non-rotational and rotational cyclic loading conditions with principal stress orientations of α₀ = 0° and 22.5°. The stress paths exhibit a consistent developmental trend, with minor variations attributed to differing α₀ values. Specifically, a progressive leftward shift in the stress path occurs due to gradual excess pore water pressure (PWP) accumulation during undrained cyclic loading, ultimately stabilizing the stress path at a position significantly distant from the zero effective stress state. Comparing Figure 3a,c demonstrates that rotational loading accelerates pore pressure development more rapidly than non-rotational conditions. For the same loading path, the cumulative rate of effective stresses increases significantly as α₀ increases from 0° to 22.5°, irrespective of rotational loading application.

Figure 4 provides a more comprehensive view of the evolution trend of the permanent pore pressure ratio ($r_u^p$) applying at the end of each stress cycle as loading proceeds for different α₀. Unlike transient pore pressure fluctuations, $r_u^p$ captures the minimum pore pressure value per cycle, reflecting cumulative material changes during cyclic loading [29]. This metric provides a robust representation of long-term sample behavior by filtering transient effects, making it particularly suited for predicting cyclic degradation. As illustrated in Figure 4a, for the specified conditions of α₀ = 0° and 90°, the effective stress exhibits minimal fluctuations within a confined range. This is attributable to the predominantly unaltered nature of τ_zθ, which consequently results in a negligible increase in the pore pressure ratio, thereby maintaining its overall stability. For other conditions, initially, the pore pressure gradually increases but then accelerates as the effective stress becomes progressively diminished. A stress state with PSR is distinguished from a triaxial pure compression or tension test stress state in that the combined action of vertical stress and shear stress results in the pore–water pressure continuously changing with increasing α₀ [34]. Specifically, as α₀ increases from 0° to 45°, the PWP buildup accelerates progressively. However, beyond 45°, further increases in α₀ lead to a decline in PWP accumulation, resulting in enhanced resistance to cyclic failure.

3.2. Cyclic and Permanent Strain Development

Figure 5 presents the vertical stress–strain relationships of samples under a non-rotational loading path. All samples exhibit strain softening behavior, characterized by hysteresis loops progressively shifting toward increasing vertical strain (rightward drift) with cyclic progression. As depicted in Figure 5a–d, increasing α₀ induces a transition in vertical strain (ε_z) from compression-dominant to extension-dominant regimes. The critical angle α₀ = 45° correlates with the observed inflection in $r_u^p$ (Figure 4), governed by loading angle dependency. Representative hysteresis loops at specific cycle numbers (N = 1, 10, 100, and 1000) further elucidate cyclic degradation patterns. At low N (N < 10), tightly spaced loops reflect limited strain accumulation. As N increases, the vertical strain evolves gradually yet persistently, accompanied by hysteresis loop widening and incremental strain magnitude escalation, signaling accelerated strain development rates.

For samples subjected to non-rotational loading paths with principal stress orientations of α₀ = 0° (fully vertical) and α₀ = 90° (fully circumferential), the shear stress–strain responses are negligible. Consequently, Figure 6 focuses on samples with α₀ = 22.5°, 45°, and 67.5°, which exhibit pronounced shear behavior. As shown in Figure 6a,c, the shear strain remains below 4% even after 1000 loading cycles. In contrast, the α₀ = 45° case (Figure 6b) displays markedly wider hysteresis loop intervals at N = 10 and N = 100, with shear strain exceeding 9% within the first 100 cycles. This rapid strain accumulation triggers cyclic shear failure prior to N = 1000, underscoring the critical role of principal stress orientation in shear-induced instability.

The evolution of vertical and shear strains under the rotational path with different α₀ are shown in Figure 7 and Figure 8, respectively. The observed responses are consistent with those shown in Figure 5 and Figure 6, exhibiting strain-softening behavior under cyclic loading. Notably, the rate of strain accumulation is most pronounced at α₀ = 45°, a finding that aligns well with previous results from undrained rotational cyclic tests on sand and clay [35,36]. However, a key distinction is evident under rotational loading conditions, which induce significantly higher cumulative vertical and shear strains compared with non-rotational loading. For example, Figure 8c shows that under rotational loading at α₀ = 45°, the shear strain exceeds 11% by N = 100—substantially higher than the 9% observed under the non-rotational loading path (Figure 6b). This result underscores the importance of investigating the impact of rotational loading, such as that induced by storm wave action, on soil behavior. Moreover, a direct comparison between Figure 7 and Figure 8 reveals that the shear strain evolves more rapidly than vertical strain under rotational loading. This suggests that the mechanical response of the samples is governed by shear-dominated failure mechanisms.

Figure 9 and Figure 10 summarize the evolution of strain components—vertical strain (ε_z), circumferential strain (ε_θ), and shear strain (γ_zθ)—under non-rotational and rotational loading paths, respectively, including permanent strains (${\epsilon }_z^p$, ${\epsilon }_{\theta }^p$, ${\gamma }_{z\theta }^p$). As depicted in Figure 9a,e, ε_z remains negligible even at high cycle numbers, indicating high failure resistance for α₀ = 0° and 90° cases. In contrast, tests with α₀ = 22.5°to 67.5° exhibit progressive vertical strain accumulation that stabilizes with increasing N. The circumferential strain shows minimal variation, though its accumulation direction depends on α₀: tensile-dominated for α₀ = 0° and 22.5° (Figure 9a,b) versus compressive-dominated for α₀ = 67.5° and 90° (Figure 9d,e). Notably, pure cyclic shear at α₀ = 45° (Figure 9c) triggers rapid ${\gamma }_{z\theta }^p$ growth to 14%, culminating in failure at N = 180, highlighting heightened shear susceptibility.

Rotational loading (Figure 10) replicates these trends but amplifies terminal ${\epsilon }_z^p$ and ${\gamma }_{z\theta }^p$, reflecting greater plastic strain accumulation compared with non-rotational paths. Shear strain peaks at α₀ = 45°, decreasing beyond this threshold—a pattern aligning with permanent pore pressure ratio evolution (Figure 4). These findings underscore the critical role of principal stress rotation in accelerating strain localization and failure. The comparison between Figure 9 and Figure 10 signifies that the development of permanent strain is influenced by both α₀ and the continuous PSR.

In order to provide further elucidation on the strain evolution during cycling, the permanent strain envelopes obtained at N = 100 under non-rotational and rotational loading paths are plotted in Figure 11. As shown, the asymmetrical envelope reveals that the induced strain initially increases and then decreases with increasing α₀, reaching its maximum at α₀ = 45°. The variations reflect significant anisotropy induced by initial principal stress orientation. In addition, the persistent PSR serves to aggravate this anisotropic behavior, as evidenced by increasingly irregular and asymmetric strain envelopes. From a microstructural perspective, larger PSR angles amplify soil anisotropy and vertical cumulative plastic strain by altering particle-scale contact mechanics [37].

Specifically, increased rotation angles promote horizontal particle displacement in the sample’s central zone, enhancing horizontal contact interactions while reducing and unevenly distributing vertical contacts. This imbalance inhibits the horizontal separation of vertical normal contact forces, accelerating anisotropic evolution. Consequently, redistributed contact forces directly contribute to increased vertical cumulative plastic strain.

3.3. Stiffness Degradation Behavior

The highly nonlinear stress–strain behavior, indicating the rapid degradation of plastic stiffness, is a prominent characteristic of soils. Under undrained cyclic loading, the stiffness quantified using the equivalent Young’s (E_eq) and shear (G_eq) modulus is examined by calculating the slope of the backbone curve from the hysteretic vertical and shear stress–strain loop [38,39,40]. Figure 12a illustrates the degradation of E_eq with cycle number in non-rotational loading paths, excluding the case of α₀ = 45°. The α₀ = 0° test exhibits the maximum initial E_eq (~82 MPa) and modest degradation, while α₀ = 90° shows a parallel curve with slightly lower values. Progressively lower initial E_eq values and accelerated degradation rates occur at α₀ = 22.5–67.5°, highlighting the sensitivity of stiffness degradation to principal stress orientation. Figure 12b reveals parallel degradation trends for G_eq at α₀ = 22.5° and 45°, positioned above and below the α₀ = 67.5° curve, respectively, indicating near-constant directional degradation rates. Notably, degraded stiffness decreases and then recovers with increasing α₀, demonstrating non-monotonic dependence of E_eq and G_eq on stress orientation. Accelerated degradation at α₀ = 45° likely results from alignment of elevated shear stresses with weak bedding planes.

Under rotational loading (Figure 12c,d), PSR universally amplifies stiffness degradation, with E_eq and G_eq declining more rapidly than in non-rotational cases. This accelerated degradation is further illustrated in Figure 13, which emphasizes the evolution of the equivalent modulus for specified numbers of cycles. The width of the band formed by equivalent modulus values at different cycle counts quantifies degradation severity: wider bands correspond to more pronounced stiffness reduction. Band width variation with orientation angle and loading path aligns with prior conclusions, demonstrating maximum anisotropy at α₀ = 45°. Experimental studies [22,41,42] indicated that the principal directions of the strain increment and the stress do not coincide under PSR. This non-coincidence is called non-coaxiality, and the non-coaxiality has negligible variation with the change of σ_z in contrast with τ_zθ from the viewpoint of micro-mechanics. Combined with Figure 5, Figure 6, Figure 7 and Figure 8, when α₀ rotates in the range of 45° to 90°, the torsional shear stress component τ_zθ decreases and the non-coaxiality becomes stronger. In contrast, when α₀ is in the range of 0° to 45°, where τ_zθ increases, the non-coaxiality becomes weaker. The degree of non-coaxiality becomes larger at a higher stiffness degradation level. It may be due to the fact that the strain increment becomes less coaxial as the equivalent modulus decreases [43,44].

The degradation index δ, first proposed by Idriss et al. [28], quantifies the stiffness behavior of soft clay as the ratio of Young’s or shear moduli at cycle N to their initial values at N = 1. This study adopts δ_E and δ_G for equivalent Young’s and shear moduli, respectively. Figure 14 compares δ_E–N and δ_G–N correlations across α₀ values, revealing generally linear relationships that can be matched by the following semi-logarithmic expression:

$$\delta ={\delta }_1-alnN$$

(1)

where a (α₀-dependent) governs degradation rate, and δ₁ represents the degradation index at N = 1 with a default value of 1.0. Non-rotational loading at α₀ = 0° and 90° exhibits slower degradation due to alignment with inherent vertical stiffness anisotropy. Conversely, accelerated degradation occurs at α₀ = 22.5°, 45°, and 67.5° due to the disruption of the anisotropic fabric caused by non-coaxial shear deformation [15]. The parameter a derived from the experimental data increases at low α₀, peaks at α₀ = 45° for δ_G, then declines sharply toward α₀ = 90°. A comparison of Figure 14a,c reveals that PSR universally intensifies stiffness degradation, and the parameter a in the semi-log model effectively captures the interplay between loading path and α₀, highlighting its utility in modeling anisotropic soil behavior under multidirectional cyclic loading.

As revealed by Jardine et al. [25] and Clayton [39], the stiffness mobilized within the strain range 0.001–0.1% is relevant to a wide range of soil–structure interaction problems. The Mechanistic-Empirical Pavement Design Guide (AASHTO [45]) clearly states that permanent strain decreases with increasing equivalent modulus. The conclusion of this guide can be corroborated by combining the strain development (Figure 9 and Figure 10) and the stiffness degradation behavior (Figure 12) in this study. Despite its significance, few studies have rigorously quantified this linkage so far [46,47,48]. Figure 15 therefore establishes the relationship curve between ${\epsilon }_z^p$ (${\gamma }_{z\theta }^p$) and δ_E (δ_G), revealing an inverse correlation where permanent strain decreases with increasing equivalent modulus. The semi-log plot exhibits linear trends described by:

$${\epsilon }_z^p=e^{b-c{\delta }_E}\mathrm{o}\mathrm{r}{\gamma }_{z\theta }^p=e^{b-c{\delta }_G}$$

(2)

where b and c = intercept and slope of the line in the semi-log plot, respectively. Specifically, parameter b exhibits a substantial discrepancy, signifying that ${\gamma }_{z\theta }^p$ is marginally less sensitive to the degradation index than ${\epsilon }_z^p$. While δ_E and δ_G quantify the cumulative damage under cyclic loading through modulus degradation, the exponential model directly relates this damage to permanent strain, providing a theoretical framework for long-term settlement prediction, consistent with findings by Cai et al. [2].

To verify the existence of this relationship, the data of cyclic loading tests on soft clay samples under undrained conditions from Wang et al. [15] and Zhu et al. [4] is used. Figure 16 shows the relationship between ${\epsilon }_z^p$ and δ_E in their studies. It can be seen that similar exponential correlations exist and Equation (2) is also applicable to both results (the correlation coefficients are 0.95 and 0.99, respectively).

4. Conclusions

A series of undrained cyclic tests were performed on reconstituted soft clay under controlled principal stress orientations and prescribed stress paths, with particular emphasis on the effects of initial principal stress orientation (α₀) and principal stress rotation (PSR) on pore pressure development, permanent strain accumulation, and stiffness degradation behavior. The major conclusions are summarized as follows:

• Samples at α₀ = 0° and 90° maintain stable pore pressure ratios with negligible permanent vertical, circumferential, and shear strain accumulation even at high cycle numbers. Conversely, orientations between 22.5 and 67.5° exhibit progressive degradation, with α₀ = 45° triggering rapid shear strain development and earliest failure.
• The α₀-dependent strain development demonstrates the significant anisotropy of the tested soft clay. Furthermore, compared with non-rotational cases, samples subjected to rotational loading exhibit higher generated pore pressure and more pronounced accumulation of permanent strain, underscoring that PSR significantly amplifies the anisotropic response of soft clay.
• Consistent with observations in permanent strains, PSR accelerates stiffness degradation, evidenced by more rapid declines in equivalent Young’s and shear moduli compared with non-rotational loading. Moreover, minimum stiffness values occur at α₀ = 45°, where the degradation rate peaks due to non-coaxial shear disruption of anisotropic fabric. This peak response is systematically exacerbated by PSR across all orientations.
• The stiffness degradation index tends to uniquely correlate with the permanent axial or shear strain, which can be quantified by an exponential relationship that is independent of loading path. This correlation characterizes anisotropic effects from varying principal stress orientations by linking the cumulative damage on cyclic Young’s and shear moduli to permanent strain, providing a unified framework for long-term settlement prediction.

Finally, the HCA tests outlined above provide a basis for evaluating cyclic degradation of soft clays under complex cyclic loading conditions. They help address critical gaps in understanding the mechanical response of coastal marine soft clays under long-term storm wave loading. However, further investigation by conducting in situ tests or laboratory tests on undisturbed soil samples should be undertaken to further predict the stiffness softening and verify the proposed model. Additional laboratory testing on a wider range of geomaterials is also required to test the general applicability of the model proposed in this study.