Effect of principal stress direction on the instability of sand under the constant shear drained stress path

The grave risk posed by static liquefaction failure of slopes comprising loose, saturated fills is evidenced by over a century of documented case histories – for example, Calaveras water dam (Hazen, 1918; Jefferies & Been, 2015), Fort Peck water dam (Casagrande, 1965; Jefferies & Been, 2015), Aberfan colliery spoil tip (Bishop et al., 1966; Jefferies & Been, 2015), Stava tailings storage facility (Morgenstern, 2001), Nerlerk berm (Sladen et al., 1985; Jefferies & Been, 2015) and more recently the Fundão (Morgenstern et al., 2016), Cadia (Jefferies et al., 2019) and Feijão (Robertson et al., 2019; Arroyo & Gens, 2021) tailings storage facilities (TSFs). TSFs are earth structures that have often failed as a result of static liquefaction, owing to the loose saturated conditions of much of these hydraulically deposited fills. Water-retaining dams built from uncontrolled fills (e.g. hydraulically placed or uncompacted soil) have also undergone static liquefaction, such as Wachusett dam in 1907 (Olson et al., 2000), Calaveras dam in 1918 (Hazen, 1918), Sheffield dam in 1924 (Seed et al., 1969) and more recently the Edenville–Sanford dams in 2020 (France et al., 2022). Owing to the frequency of such events, several studies have been carried out to understand the mechanisms causing these failures and develop techniques to assess the susceptibility of materials to liquefaction.

Commonalities seen in earth structures that undergo static liquefaction failures are: (a) the presence of loose, saturated materials – that is, with a state parameter (Ψ) (Been & Jefferies, 1985) >−0·05 (Jefferies & Been, 2015), and thus having an instability stress ratio (η_IL) (Sladen et al., 1985; Lade, 1992) lower than the critical state friction ratio (M); (b) a stress path causing in situ stress conditions to reach η_IL, leading to triggering; and (c) a post-triggering brittle strength loss, such that the liquefied shear strength is lower than the static stresses, resulting in flow. In terms of stress paths causing elements of soil to reach the n_IL and thus trigger failure, while undrained loading is probably the most frequently conceptualised mechanism, it is common for field-scale failures to occur as a result of the stress conditions arising under drained stress paths. For example, drained conditions appear to have prevailed within the fills that were eventually brought to the point of triggering in all of the documented case histories attributed to ‘extrusion-type’ stress paths, such as Fort Peck dam (Casagrande, 1965; Jefferies & Been, 2015), Nerlerk berm (Sladen et al., 1985; Jefferies & Been, 2015), Fundão TSF (Morgenstern et al., 2016) and Cadia (Jefferies et al., 2019). Similarly, drained failure caused by increases in phreatic surface, referred to as constant shear drained (CSD) (Anderson & Sitar, 1995), led to static liquefaction triggering in the Aberfan tip (Bishop et al., 1966; Jefferies & Been, 2015), Wachusett dam (Olson et al., 2000), Stava TSF (Morgenstern, 2001) and Edenville–Sanford dam (France et al., 2022).

The concept of η_IL, or instability locus, adopted in much of the recent literature derives from an initial proposal by Lade (1992), where instability conditions are represented by a line passing from the origin of the effective stress space q–p′ with a slope equal to q/p′. The η_IL is located below M for loose contractive soils and above it for dense dilative soils (e.g. Chu et al., 2003; Junaideen et al., 2010), and it has been observed to depend on the state parameter at consolidation (Ψ₀) (e.g. Yang, 2002) or at peak deviatoric stress (Ψ_IL) (Imam et al., 2002; Chu et al., 2003). A relationship between Ψ and η_IL has been proposed by several researchers based on undrained triaxial testing (e.g. Imam et al., 2002; Yang, 2002; Chu et al., 2003), as schematically illustrated in Fig. 1.

Of the various drained triggers, the CSD stress path is a particularly important and dangerous mechanism for static liquefaction, as it is difficult to obviate the potential for a phreatic surface to increase at some point in the life of a slope comprising loose saturated soils, and pre-failure indicators (e.g. displacements) are likely to be minimal (e.g. Sasitharan et al., 1993; Reid et al., 2021a). CSD was first recognised as a potential trigger by Lindenberg & Koning (1981), later demonstrated in small-scale laboratory modelling of slopes (Eckersley, 1990) and element testing (Sasitharan et al., 1993), and subject to much subsequent study (e.g. Anderson & Riemer, 1995; Anderson & Sitar, 1995; Zhu & Anderson, 1998; Gajo et al., 2000; Chu et al., 2003; Daouadji et al., 2010; Junaideen et al., 2010). Chu et al. (2003) examined the instability locus from CSD tests, conducting tests on MT loose and WP dense Changi sand specimens proposing a relationship between η_IL and a modified state parameter, calculated from the void ratio at the instability state rather than the void ratio after consolidation prior to undrained shearing (i.e. Ψ₀). The concept of a modified state parameter proposed by Chu et al. (2003) for CSD stress-path tests is the same as that calculated at peak deviatoric stress in undrained tests, hence the symbol Ψ_IL is adopted in both stress paths. Fig. 2 provides an idealisation of a typical element test under the CSD stress path, including the idea of Ψ_IL proposed by Chu et al. (2003).

One area that has not received sufficient attention in the context of CSD triggering is the potential effects of cross-anisotropy on η_IL, as nearly all CSD elements tests carried out have been in either triaxial or plane strain devices under compression loading conditions (e.g. Chu & Wanatowski, 2008; Wanatowski et al., 2010), thus with direction of the major principal stress relative to the vertical axis (α) of 0°. Two exceptions are the triaxial extension CSD tests of Dong et al. (2016) and the direct simple shear (DSS) CSD tests of Reid & Fourie (2019) and Riveros & Sadrekarimi (2021). However, while such test formats allow some aspects of anisotropy to be investigated, they are not ideal, as triaxial extension tests do not produce realistic stress conditions for below-slope locations (areas most likely to trigger within a TSF) while the DSS does not provide full principal stress conditions for the element tested, thus limiting the insight gained.

Ladd (1991) has defined cross-anisotropy as being characterised by three components: inherent anisotropy, initial shear stress anisotropy and induced or evolving anisotropy. Inherent anisotropy arises from the soil structure at deposition characterised by microlevel (e.g. particle orientation and interparticle forces) and macrolevel (e.g. soils containing varve deposits, fissured clays and layering) fabrics. The initial shear stress anisotropy exists in materials with one-dimensional strain history, which is typical of soils initially consolidated at an at-rest coefficient (K₀) less than one and then subjected to shearing. Evolving anisotropy is induced by plastic strains due to shear and consolidation stresses developed below slope during staged construction (Ladd, 1991; Menkiti, 1995) and are common in structures like TSFs. Evolving anisotropy is thus dependent on rotation of principal stresses during consolidation.

The effect of cross-anisotropy on η_IL of reconstituted sands has been discussed by previous researchers on the basis of undrained torsional shear hollow cylinder (TSHC) tests sheared while maintaining a constant α (e.g. Shibuya, 1985; Uthayakumar, 1996; Yoshimine et al., 1998; Sivathayalan & Vaid, 2002; Reid et al., 2022a). The consensus of these studies is that η_IL decreases as α differs from zero, as illustrated in the example provided in Fig. 3 for materials exhibiting partial liquefaction or quasi-steady-state (QSS) behaviour. It is noted that partial liquefaction has been typically observed in TSHC tests carried out on reconstituted sands when prepared using the dry or wet pluviation techniques (e.g. Shibuya, 1985; Uthayakumar, 1996; Yoshimine et al., 1998; Sivathayalan & Vaid, 2002). However, as noted by Reid et al. (2022b), although there appears to be clear evidence of the effect of α on η_IL from undrained loading, there is a lack of data to provide reliable evidence as to the effect of α on CSD triggering. The purpose of the current work is to address this deficiency, by examining the effect of α on the η_IL under the CSD stress path. This was accomplished by means of a series of CSD tests using a TSHC device on specimens of sand prepared at various values of Ψ. A range of reconstitution methods (i.e. moist tamped (MT), dry pluviated (DP) and wet pluviated (WP)) was used to include examination of the effects of fabric on triggering. The critical state line (CSL) of the sand was also measured from triaxial compression testing to allow the results of this study to be interpreted in the context of Ψ, consistent with current tailings engineering practice (e.g. Morgenstern et al., 2016; Jefferies et al., 2019).

The material used in this study is a commercially available silica fine sand (SFS). A similar batch of this sand was used by Reid & Fourie (2019) to undertake testing in a modified DSS apparatus able to carry out CSD testing under dead-weight loading. SFS is a uniform fine sand containing predominantly quartz and traces of titanium and iron oxides. The sand has a specific gravity of 2·64, median particle size D₅₀ of 0·21 mm, a coefficient of uniformity C_u of 2·3 and rounded to subrounded particles. The particle size distribution (PSD) and typical particle morphology from scanning electron microscope (SEM) of the SFS are provided in Figs 4 and 5, respectively.

The CSL and dilatancy properties of the SFS were determined in a standard triaxial device manufactured by GDS Instruments Ltd (GDS) by means of isotropically consolidated drained (CID), undrained (CIU) and anisotropically consolidated undrained (CAU) triaxial compression tests.

The tests were carried out targeting a range of densities and adopting various preparation methods (i.e. MT, DP and WP). MT specimens were prepared in eight layers, applying an undercompaction ratio of 5% (Ladd, 1978), using SFS prepared at a gravimetric moisture content (GWC) of approximately 5%. DP and WP specimens were prepared following typical procedures such as those outlined by Wood et al. (2008). Testing was carried out using lubricated ends to promote sample uniformity, on specimens of 70 mm dia. and approximately 140 mm high. Void ratio was calculated adopting the end-of-test freezing method proposed by Sladen & Handford (1987). For all tests, saturation was achieved by flushing with deaired demineralised water and applying a sufficient back-pressure saturation to achieve a Skempton's coefficient B of 0·97 or higher.

Of the preparation methods adopted, only MT specimens could be prepared with Ψ₀ > 0, while DP and WP specimens were dense of the CSL. Loose specimens are preferred to enable the successful measurement of the CSL by promoting uniform sample deformation and lack of shear localisation at high strains, which is more likely to occur on samples dense of the CSL (Desrues et al., 1996; Reid & Fanni, 2022). For these reasons, MT specimens were used to infer the CSL, while DP and WP were used to determine stress- and state-dilatancy parameters.

Table 1 provides a summary of the states achieved after consolidation and end of test for all triaxial tests carried out, along with a commentary on the tests that were inferred to have reached critical state conditions. The stress paths of the loose and dense tests are shown as a state diagram in Fig. 6 along with the inferred CSL, indicating a quite incompressible CSL (i.e. λ_e = 0·008). The loose and dense tests are shown in terms of stress-dilatancy in Fig. 7, along with two dilatancy parameters determined from the tests – namely, the volumetric coupling parameter (N) (Jefferies & Been, 2015) and M_tc. These parameters were determined from both loose and dense tests using methods proposed by Shuttle & Jefferies (2016) and Bishop (1971), indicating N = 0·25 and M_tc = 1·21. The dense tests are also shown using the state-dilatancy framework proposed by Jefferies & Shuttle (2002) in Fig. 8, which provides a relationship between minimum dilatancy rate (D_min) and Ψ at D_min, indicating a state-dilatancy coefficient χ of 3·1. These values are consistent with those typically seen for sands (e.g. Jefferies, 1993; Jefferies & Shuttle, 2002; Jefferies & Been, 2015).

The TSHC tests were carried out in a dynamic electro-mechanical device manufactured by GDS, which allows testing of specimens with height of 200 mm, outer diameter of 100 mm and inner diameter of 60 mm. The cell, inner pressures and back-pressures are controlled by way of pressure pumps, while the torque and axial load are measured and controlled by a submersible multiaxial load cell. Shearing-induced excess pore pressure developed during undrained testing is measured by an external pore pressure transducer installed near the specimen. Axial (z) and torsional deformations (θ) are measured by an internal encoder, while the sample volume change (dV) and inner cavity volume change (dV_in) are tracked independently by the back-pressure and inner pressure pumps, respectively. Thus, the outer radius (u₀) and inner radius deformations (u_i) are calculated in this study from dV, dV_in and the height of the sample (H). At the top and bottom of the specimen an annular brass porous stone allows drainage. These porous stones include 1·5 mm high stainless steel ribs to prevent slippage between the porous stones and the specimen. These ribs are expected to cause negligible effects on the results of TSHC tests (Tatsuoka et al., 1986).

Inner and outer latex membranes approximately 0·3 mm thick were adopted in this study. Membrane stress corrections were applied on the basis of the methods outlined by Tatsuoka et al. (1986) but membrane penetration corrections were not applied, as they are considered negligible in materials with the SFS gradation (Frydman et al., 1973; Martin et al., 1978; Tokimatsu & Nakamura, 1986; Evans, 1992; Uthayakumar, 1996).

The TSHC device used in this study can undertake dynamic testing at a maximum frequency of 5 Hz with axial load and torque applied by an electro-mechanical system. This device can control axial load and torque at an accelerated rate more responsively than a conventional static electro-mechanical load frame when set to maintain a constant load or stress. Most static load frames have an absolute maximum rate of loading of less than 2 mm/s with limited acceleration profiles, while a dynamic electro-mechanical system is capable of higher speeds and, most importantly, higher acceleration rates. Hence, this device is well suited to investigate the instability of geomaterials under stress-controlled conditions where rapid rates of displacement occur at the initiation of instability.

In this study, instability is identified when a sudden decrease in deviatoric stress is accompanied by a rapid increase in octahedral shear strain (γ_oct). This approach is similar to that adopted by Sasitharan et al. (1993) in their seminal work on the instability of sand under the CSD stress path performed using a dead-weight system, and more recently by Morgenstern et al. (2016) and Reid & Fourie (2019).

The general stress conditions within the TSHC are presented in Fig. 9, while the equations adopted to define the stresses and strains in the TSHC are provided in Table 2.

It is noted that, as indicated in Table 2, the deviatoric stress is defined in this study using two formulations – that is the deviatoric stress formulation typically used in triaxial testing (⁠$q13$⁠) and the octahedral formulation (⁠$q$⁠) that takes into consideration the effect of the intermediate principal stress (⁠$σ2$⁠). In axisymmetric conditions $q13=q$ as $σ2$ = $σ3$ (i.e. intermediate principal stress ratio b = 0). The distinction above is required as the code implemented in the software to target a specific stress-path controls $q13$ and not $q$ as discussed subsequently. Nevertheless, the η_IL is calculated in this study from the octahedral deviator stress, consistently with current practice involving non-axisymmetric conditions (Chu & Wanatowski, 2008; Wanatowski et al., 2010; Jefferies et al., 2019; Reid et al., 2022b; Shuttle et al., 2022).

Ten undrained strain-controlled and 25 stress-controlled CSD TSHC tests were carried out under a range of α and b values held constant during shearing. Five of the undrained TSHC tests were on very loose MT specimens, while the remaining five undrained TSHC tests were DP specimens prepared to the loosest possible condition using this method. The undrained strain-controlled tests were carried out as they allow a clearer assessment of the instability behaviour of materials than stress-controlled tests, therefore providing further insight on the mechanism of instability of the SFS from CSD tests. Each series of undrained tests (MT or DP) was prepared targeting similar densities, although a range of Ψ₀ of up to 0·03 was seen in the tests within a given set.

The TSHC CSD specimens were prepared at a range of densities to enable investigating the effect of Ψ on η_IL of the SFS, on MT, DP and WP specimens. Only one TSHC CSD test was carried out on WP at α = 45°, b = 0·2 as, similarly to the triaxial tests, WP specimens could not be prepared in a sufficiently loose state to exhibit contractive behaviour.

As the scope of this study was to compare the instability of the sand from triaxial compression conditions to stresses typically developing below a slope, undrained and CSD TSHC tests were carried out at α = 0° and b = 0 (i.e. triaxial compression conditions) and α = 22·5° and 45° with b = 0·2 (a reasonable estimate of below-slope conditions – e.g. Wanatowski (2005)). Undrained TSHC tests were carried out also at α = 30° and b = 0·2 on both MT and DP specimens because initial testing on the DP specimen at α = 22·5° indicated dilative behaviour. The additional tests were therefore added to assess the approximate α value at which the DP specimens transitioned from strain hardening to strain softening.

Undrained TSHC tests were also conducted at α = 45° and b = 0·5 in order (a) to assess stress–strain non-uniformities across the wall of the hollow cylinder following criteria typically adopted in TSHC testing (i.e. criterion 1 based on Nakata et al. (1998), and criterion 2 based on Yoshimine et al. (1998); see Table 3 for relevant formulas) and (b) to provide a preliminary indication of the effect of b on η_IL of the SFS. It is noted that non-uniformities in the TSHC are negligible at α = 45° and b = 0·5. Therefore, the results of this test represent a reference condition that may provide some insights on the validity of this study (e.g. Hight et al., 1983; Vaid et al., 1990).

The undrained tests were carried out at a constant horizontal shear strain rate $γ˙zθ$ of 2% per hour, while CSD tests were undertaken by decreasing the cell and inner pressures at a rate of 5 kPa p′ per minute – a rate used in another study involving CSD stress-path tests on sands with similar drainage path lengths (Chu et al., 2012).

MT specimens were prepared in ten layers of equal volume, applying an undercompaction ratio of 5% using SFS prepared at a gravimetric moisture content (GWC) of approximately 5%. DP and WP specimens were prepared using similar procedures as those used to prepare the triaxial specimens and following general steps as reported by Vaid et al. (1990). After preparation, an isotropic stress of 50 kPa was applied and saturation carried out using a procedure that depended on the initial condition of the specimens. MT and DP specimens were saturated by flushing with deionised water, followed by raising of the back-pressure over a period of 1 h, while the WP specimen was directly back-pressure saturated following application of a suction of 50 kPa (as this sample was already in a near-saturated condition owing to its method of preparation). A Skempton's coefficient B of 0·97 or higher was achieved in all tests carried out.

Two different stress paths were adopted to investigate the effect of the consolidation path on the instability of the SFS: stress path A and stress path B. A description of these stress paths is provided in Table 4 and illustrated schematically in Figs 10 and 11 in p′–q–α space. Stress path B better simulates the likely development of in situ stresses under the staged construction of embankments and TSFs, as it involves a progressive rotation of the principal stress under drained loading conditions (Jardine & Smith, 1991). Consolidation paths similar to stress path A and stress path B have been used in other TSHC programmes (e.g. Vaid et al., 1990; Wijewickreme & Vaid, 1993; Uthayakumar, 1996; Yoshimine et al., 1998; Zdravković & Jardine, 2001), and therefore despite stress path B being more realistic it was considered useful to include stress path A for comparison purposes.

The η_IL inferred from the TSHC undrained tests carried out on MT and DP specimens is illustrated in e–p′, q–γ_oct (stress–strain) and q–p′ plots in Figs 12, 13 and 14. It is noted that in this study instability in undrained tests is considered only for tests showing strain-softening behaviour, including those tests on DP specimens exhibiting QSS behaviour, while the instability of DP specimens displaying strain-hardening behaviour is taken at the point of maximum mobilised η.

The states at instability are also summarised in Table 5, along with the non-uniformity at η_IL calculated following the criteria proposed by Nakata et al. (1998) and Yoshimine et al. (1998). Based on these criteria, stress and strain non-uniformities within the wall of the hollow cylinder specimens at strains relevant to the mobilisation of η_IL are expected to be small on all tests undertaken, thus indicating that the results of the tests are reliable.

The results of the tests on MT specimens indicate a progressive decrease of η_IL with increasing α, while DP specimens transition from strain-hardening to strain-softening behaviour from α = 30° or greater, as tests carried out at lower α indicate strain-hardening behaviour. Comparing the degree of anisotropy of MT and DP specimens by using as reference stress condition the results of the tests carried out at α = 0° and b = 0, DP presents a greater degree of anisotropy than MT, as at α = 45° the η_IL of the SFS decreases by more than 30% for MT but more than 50% for DP. The higher anisotropy observed in DP specimens is consistent with the microscope images of Yang et al. (2008), where it was seen that DP specimens presented a stronger anisotropic microstructure than specimens prepared using the MT method. Further observations on the anisotropy of MT and DP specimens are made subsequently in this paper based on the results of CSD TSHC tests carried out at various Ψ.

The effect of b on the strain softening of both MT and DP specimens can be appreciated by comparing the tests carried out at α = 45° and b = 0·2 and b = 0·5, respectively. These tests clearly show a lower η_IL with an increasing b for both MT and DP specimens, as well as higher contractiveness and lower liquefied strengths in DP specimens.

The difference between the Ψ₀ and Ψ_IL is generally small (i.e. ranging between +0·002 and −0·004). This small difference is clearly an effect of the flat CSL of the SFS tested in this study and consistent with data reported in the literature (e.g. Chu et al., 2003, 2012).

The results of the TSHC CSD tests carried out on MT specimens are illustrated in Fig. 15 as e–p′ plots (top row), q–p′ plots (middle row) and q–γ_oct plots (bottom row), while the testing on DP and WP specimens is provided in Fig. 16, as e–p′ plots, and in Fig. 17, as q–p′ and q–γ_oct plots. The states at instability are summarised in Table 6 along with the non-uniformity at onset of instability calculated following Nakata et al. (1998) and Yoshimine et al. (1998) criteria. Consistently with the undrained TSHC tests, stress and strain non-uniformities in the TSHC CSD tests are small, as both criteria are generally met. Only two tests recorded a marginally higher non-uniformity coefficient based on criteria 1 and 2 (i.e. TSHC8 on MT at α = 45° and TSHC1 on WP at α = 45°, which indicated dilative behaviour).

Instability boundaries in e–p′ space for each set of tests carried out at a specific $α$ are presented in Figs 15 and 16 along with the stress path of each test during the CSD stage. The instability boundaries represent the locus of points at onset of instability for tests carried out at different consolidated void ratios and it is relevant to the locus in e–p′ space at a specific constant deviatoric stress – that is, $q$ ≈ 64 kPa at $b$ = 0 and $q$ ≈ 58 kPa at b = 0·2. The tests indicate that these boundaries are moving towards the right of the p′ axis as α increases, thus suggesting that $ηIL$ decreases with increasing $α$⁠. The instability boundary of DP at α = 45° shifts further towards the right of the p′ axis compared to the tests on MT at α = 45°, with p′ at instability ranging between 70 kPa and 80 kPa for Ψ_IL of approximately −0·02 and −0·04, respectively, indicating that fabric anisotropy plays a significant role in the instability of the SFS. On the contrary, within this range of Ψ_IL, MT specimens indicate dilative behaviour independently of α, as also evident from examination of the $ηIL$ reported in Table 6 and following discussions.

The critical friction ratio from triaxial compression conditions (i.e. M_tc) is illustrated in the q–p′ plots as a reference to assist in investigating the mechanism of collapse from the various tests. However, it is acknowledged that, as reported in other studies (e.g. Matsuoka & Nakai, 1974; Van Eekelen, 1980; Wanatowski, 2005; Jefferies & Shuttle, 2011), the critical friction ratio (commonly defined as M for general stress conditions) decreases with increasing b, hence its value when b = 0·2 may be lower than the value reported in this study relevant to axisymmetric triaxial compression conditions (i.e. b = 0).

All tests with $ηIL<Mtc$ exhibited liquefaction behaviour, indicated by the near-instant drop in deviatoric stress and sudden development of excess pore pressure and γ_oct. In these tests collapse occurred at γ_oct of generally less than 1%. The tests that failed at $ηIL>Mtc$ exhibited a slower failure (although with progressive octahedral shear strain rate, $γ˙oct$⁠, increase) and negative volumetric strains typical of drained tests on dilative soils. In these tests instability was considered to have been reached when the $γ˙oct$ increased and the deviatoric stress dropped from its target constant value, an approach consistent with that used in other studies – for example, Chu et al. (2003, 2012) and Rabbi et al. (2019). Typical $γ˙oct$ patterns during the CSD stage are provided in Fig. 18 for loose and dense MT specimens at α = 0°.

Consistent with its dense state of Ψ₀ = −0·08, the test carried out on the WP specimen at α = 45° and b = 0·2 indicates dilative behaviour.

Figure 19 provides a typical overview of the axial strain (⁠$εz$⁠), horizontal shear strain (⁠$γzθ$⁠) and volumetric strain (⁠$εv$⁠) developing during the CSD stage on MT and DP specimens prepared in a loose state for α = 0°, 22·5° and 45° conditions. The strains shown in these figures suggest that at α = 0° the $εz$ is predominant (as expected from previous triaxial compression CSD tests). At α = 22·5° both $εz$ and $γzθ$ are concomitantly increasing until sudden collapse, while at α = 45° the $γzθ$ is the prevailing strain. This pattern of strains may provide insight regarding the displacements likely to be developing in a TSF below slope when subjected to the CSD stress path and may be critical for successful implementation of displacement monitoring technology. However, it is important to recognise that even with the use of current monitoring technology (e.g. remote sensing), there may not be actionable warning, as at the onset of collapse the strains resulting from the CSD trigger mechanism are generally very low (i.e. in the order of 1% or less) and could be difficult to detect accurately or discern from other strains occurring naturally in a TSF (e.g. due to consolidation). Of note also is that even the sample slightly dense of critical (i.e. DP) showed very small vertical strains prior to initiation of failure.

Adopting the instability concept of Lade (1992) and the modified state parameter idea proposed by Chu et al. (2003) for CSD tests, trends of η_IL against Ψ specific to the SFS can be determined and the effect of cross-anisotropy from the CSD stress path investigated. Figs 20 and 21 present the results of the tests carried out at $α=0deg$ and $b=0$ (axisymmetric triaxial compression conditions) as η_IL plotted against Ψ and the normalised η_IL/M_tc plotted against Ψ, respectively. Included in these figures and summarised in Table 7 are the η_IL and η_IL/M_tc relationships from previous studies carried out in standard triaxial compression apparatus (Imam et al., 2002; Yang, 2002; Chu et al., 2003; Jefferies et al., 2019; Reid et al., 2021b, 2022c). It is noted that the data from Imam et al. (2002) are from testing carried out on various sands (i.e. Ottawa sand, Toyoura sand, Syncrude sand, Fraser River sand and Erksak sand), hence the normalisation of η_IL was calculated assuming an average M_tc (equal to 1·25) between those reported by Imam et al. (2002) for these sands. The state parameter relevant to this study reported in Figs 20 and 21 is the Ψ_IL, while both Ψ₀ and Ψ_IL are used for the data reported from the literature. Comparing the η_IL of the SFS with relationships reported in these studies, it appears that the η_IL of the SFS lies near typical literature data, and closer to the data from Chu et al. (2003) for tests carried out on Changi sand. In addition, the stress path during consolidation (i.e. stress path A and stress path B) did not appreciably influence the observed trend between η_IL and Ψ. The TSHC undrained tests lie reasonably close to the TSHC CSD tests, indicating that the η_IL from undrained and CSD tests may be unique, similar to observations presented by Chu et al. (2003) for triaxial compression tests.

Figure 22 summarises the results carried out at various $α$ on MT and DP specimens plotting η_IL against Ψ. This figure indicates a clear effect of cross-anisotropy on the CSD stress path of the SFS, where the η_IL reduces not only with increasing Ψ but also with increasing α. No appreciable influence on the instability of the SFS at α > 0° can be observed comparing the results of the tests that were consolidated following stress path A against those that followed stress path B, as also noted on tests carried out at α = 0°. Consistent with the undrained TSHC testing, contractive behaviour was observed in DP when α = 45° and dilation at α = 0° and 22·5°. The testing also indicates that fabric anisotropy has a remarkable effect on the η_IL of SFS, as within the range of Ψ_IL achieved on DP (i.e. −0·04 and −0·02) MT shows dilative behaviour at $α$ = 45°, indicating a less anisotropic fabric of MT than DP at these states. The undrained TSHC tests on MT at α = 22·5° lie near the locus inferred for TSHC CSD tests, while the η_IL from the undrained TSHC on loose MT and DP specimens sheared at α = 45° lies slightly below the locus inferred from the CSD tests.

Consistent with the results of the undrained tests, the CSD test series indicates that the difference between the state parameter after consolidation and that at instability is generally small (i.e. ranging generally between +0·004 and +0·008).

The results of the tests provided in Fig. 22 are synthesised in Fig. 24 as normalised reduction in $ηIL/M$ with Ψ_IL. Fig. 24 was produced normalising $ηIL$ by a critical friction ratio (M) that takes cognisance of the effect of b based on analytical equations available in the literature (Matsuoka & Nakai, 1974; Van Eekelen, 1980; Jefferies & Shuttle, 2011). These relationships are illustrated in Fig. 23. Along with these relationships is the Mohr–Coulomb failure criterion calculated for a constant critical friction angle (ϕ_cr) of 30° (i.e. M_tc = 1·21). Finally, Fig. 23 includes the data reported in Jefferies & Shuttle (2011) based on plane-strain triaxial testing carried out by Cornforth (1961) on Brasted sand and the M reduction coefficient used in this study to correct the M_tc. An M_tc reduction of 10% at b = 0·2 is adopted in this study, which is assumed based on the analytical relationships and the data from Cornforth (1961). Adopting this normalisation, Fig. 24 clearly shows that $α$ has an appreciable effect on the MT specimens prepared at various initial void ratios, with the instability of the SFS decreasing with increasing $α$⁠. Perhaps more interesting is the clear effect that fabric- or inherent-anisotropy (i.e. comparing the MT vs DP test series) has on the instability of SFS when the direction of $α$ is 45°, which shows a significant decrease in $ηIL/M$ even when the sand was prepared at negative Ψ₀.

The results of the tests provided in Fig. 22 are further synthesised in Fig. 25 as normalised reduction in $ηIL/M$ with $α$⁠, using as a reference condition for the normalisation the testing carried under triaxial compression conditions. Included in Fig. 25 are the data presented by Reid (2020) and of an available TSHC programme on reconstituted sands for which contractive behaviour was observed (Uthayakumar & Vaid, 1998; Sivathayalan & Vaid, 2002; Shibuya et al., 2003), as well as those recently reported by Reid et al. (2022a) in a study carried out on in situ block samples of tailings retrieved from three TSFs (i.e. gold, copper and platinum tailings). As the literature data are reported from testing carried out at constant b to enable a direct examination of only the effect of α on the instability of the various materials, the data shown in Fig. 25 relevant to this study have been normalised assuming the M for b = 0·2 based on the M_tc reduction coefficient shown in Fig. 23 (as per data synthesis used to develop Fig. 24). The effect of anisotropy of the SFS is reported for two Ψ_IL conditions, namely, +0·04 and −0·01, for both MT and DP fabrics and is calculated from the η_IL locus reported in Fig. 22. The results of this study are consistent with the previously cited studies included in Fig. 25, clearly showing that the magnitude of the effect of cross-anisotropy on η_IL of the SFS is within the range seen for other reconstituted sand specimens in undrained shearing. Further, the effect of fabric anisotropy on the instability of the SFS is evident comparing the normalised η_IL at α = 45° of DP against MT, where the reduction of normalised η_IL calculated for the MT fabric is approximately 15%, while for DP it is approximately 40%, a result consistent with the undrained TSHC tests carried out in this study and data in the literature.

This paper outlines results of CSD tests carried out in a TSHC device at various α values. The effect of cross-anisotropy was clearly evident as the η_IL reduces with increasing α. In addition, fabric anisotropy plays a crucial role in the instability of the SFS, consistent with data reported in other studies (e.g. Yang et al., 2008). Indeed, the tests carried on reconstituted SFS prepared using the DP method indicate a greater reduction in η_IL than MT specimens when α ≥ 30°, thus clearly indicating the importance of fabric anisotropy on the instability of the SFS. The pattern of strains observed under CSD loading was seen to be a function of α, with $εz$ dominant at α = 0°, while at α = 45° failure occurs with a progressive and sudden increase of $γzθ$⁠. These strains may provide insight regarding the displacements likely to be developing on a TSF below slope when subjected to the CSD stress path and may be critical for useful implementation of a displacement monitoring technology.

The results of this study indicate that cross-anisotropy plays a crucial role in the instability of sands under the CSD stress path, and discarding its effect on this trigger mechanism may lead to unconservative slope stability assessments. In this context, the TSHC apparatus provides insights on some crucial aspects relevant to the design of safer TSFs, for example: (a) the stress conditions that might cause a sand that is presumed to be dilative based on triaxial compression tests to behave contractively; (b) the reduction of η_IL with α; and (c) insight on the effect and magnitude of fabric anisotropy on η_IL.

Data generated or analysed during this study are available at https://doi.org/10.26182/826h-yg87.

Material used for testing in this study is available from the corresponding author upon reasonable request.

B

Skempton's pore pressure coefficient

b

intermediate principal stress ratio

C_u

coefficient of uniformity

D

dilation rate

D_min

minimum dilatancy rate

D₅₀

median particle size

dV

sample volume change

dV_in

inner cavity volume change

e

void ratio

e_c

critical state void ratio

H

initial sample height

K_c

effective principal stress ratio = σ′₃/σ′₁

M

critical friction ratio general stress conditions

M_T

torque

M_tc

critical friction ratio triaxial compression

N

volumetric coupling parameter

N₁, N₂

non-uniformity torsional shear hollow cylinder coefficients criteria 1 and 2

P_i

inner pressure

P_o

outer pressure

p′

mean effective stress

q, q₁₃

deviator stress: octahedral and axisymmetric

r

radial deformation

r_o, r_i

outer and inner radii

u_o, u_i

outer and inner radial displacements

W

axial load

z

axial deformation

α

direction of major principal stress relative to the vertical axis

Γ, λ_e

semi-log critical state line parameters: intercept at 1 kPa and slope

γ′_oct

octahedral shear strain

$γ˙zθ$

horizontal shear strain rate

η

stress ratio, q/p′

η_IL

stress ratio at instability

Θ

torsional shear hollow cylinder angle of rotation

θ

torsional deformation

$σ2$

intermediate principal stress

χ

state-dilatancy parameter

Ψ

state parameter

Ψ₀

state parameter at consolidation

Ψ_IL

state parameter at instability