Hypoplastic predictions of changes in undrained shear strength during episodic loading

Offshore foundations are subjected to a range of loads that vary depending on metocean and operational conditions. These loads affect the geotechnical properties and hence the foundation capacity over its lifetime, (Cocjin et al, 2014; Han et al., 2016; Zhou et al, 2019; O'Loughlin et al, 2020; Laham et al., 2021, 2023; Gourvenec, 2022; Kwa & White, 2023). Episodic loading involves loading that is interspersed with periods of consolidation. While loading under undrained soil conditions results in the development of excess pore-water pressures and reductions in soil strength, subsequent consolidation or dissipation of the generated excess pore-water pressures under sustained load can lead to soil densification and increases in shear strength (Schofield & Wroth, 1968; Andersen, 2015; Gourvenec, 2022). The effect of episodic loading patterns has been experimentally investigated in direct simple shear (DSS) tests on saturated, normally consolidated samples of kaolin clay (Laham et al., 2021, 2023). These DSS tests used a stacked ring configuration to impose a zero lateral strain boundary on samples. The episodic loading pattern applied consisted of stages of undrained shear to pre-failure stresses (i.e. less than the samples’ monotonic shear strength), interspersed with periods of consolidation as shown schematically in Fig. 1. Up to six episodes of shear consolidation were applied in the DSS tests performed in Laham et al. (2021, 2023). Numerical models including finite-element analyses can also be used in combination with laboratory tests to help predict soil–foundation response, once the constitutive model that is used in numerical models is calibrated and validated (Jostad et al., 2020; Ochmanski et al., 2021; Jerman et al., 2022). The increases in undrained strength (or decreases in void ratio) observed in episodic DSS tests (Laham et al., 2021) are not captured by the conventional modified Cam Clay model (MCCM) for episodes of undrained pre-failure shear-consolidation owing to the elastic assumptions within the MCCM yield surface (Gourvenec, 2022; Laham et al., 2023). Clay hypoplasticity can capture strengthening from consolidation through a decrease in void ratio, after pre-failure undrained shearing is applied. This paper explores the potential of clay hypoplasticity (Mašín, 2014) an alternative for numerically capturing the observed evolution of shear strength under episodic loading DSS conditions.

Hypoplasticity is a constitutive framework introduced by Kolymbas (1977). Clay hypoplasticity according to Mašín (2014) uses concepts from critical state soil mechanics as the critical stress-state locus that is defined according to Matsuoka & Nakai (1982). Similar to MCCM, clay hypoplasticity includes a critical state line (CSL) and normal compression line (NCL) in the mean effective stress, p′, against void ratio, e, plane. The NCL from Butterfield (1979) is included in the model, in order to define isotropic normally consolidated states, and follows a double-logarithmic relation:

The CSL is parallel to the NCL and is described by

Adopting the widely used intergranular strain concept extension (Niemunis & Herle, 1997; Mašín, 2014, 2019) for clay hypoplasticity improved model predictions for reductions in void ratio and increases in shear strength that were observed in the experiments. This is because the intergranular strain extension enables increased stiffness at small strain after loading reversals. It defines a specified small-strain range to improve the cyclic predictions of the model by preventing ratcheting, the over-accumulation of stresses and strains.

Table 1 summarises parameters of clay hypoplasticity (Mašín, 2014). The hypoplastic CSL and NCL parameters λ* and κ* were calibrated from an oedometric test on the same kaolin clay used in Laham et al. (2021) as shown in Figs 2(a) and 2(b). These parameters were fit to oedometric test data within double logarithmic ln σ_v′ against ln(1 + e) space, which is how the CSL and NCL are defined for hypoplasticity (Butterfield, 1979; Mašín, 2005) and so slightly differ from the CSL and NCL parameters in Laham et al. (2021) as they adopted the MCCM semi-logarithmic ln σ_v′ against 1 + e approach to define the CSL and NCL. The parameter N as fit to match the initial, normally consolidated states of the episodic tests in Laham et al. (2021). The critical friction angle φ_c was fit from the final stress ratios τ/σ_v′ measured in the DSS tests in Laham et al. (2021) as shown in Fig. 2(c) and described in Medicus et al. (2022). The parameter ν governs the shape of the undrained stress paths of hypoplasticity in Fig. 2(c) and has been adjusted to fit the experimental data of M62 and M90.

The following calibration procedure was applied to determine the intergranular strain parameters. The values for R, χ and m_rat were set to the recommended values for soft clays as outlined in Mašín (2019). The parameters A_g and n_g quantify the dependency of small-strain shear modulus on mean stress with G₀ = σ*A_g(p′/σ*) (Wroth & Houlsby, 1985; Mašín, 2014). In this study A_g = 4500 and n_g = 0·5 were selected according to the relationship in Hardin & Black (1968, 1969), Benz (2007) of

where p_r′ = 100 kPa is a reference pressure equal to atmospheric pressure. The parameter β_r was selected such that the predicted final strengths matched the final strengths measured in one-way cyclic episodic centrifuge testing on offshore foundations, which were less than 2–3 times the initial monotonic undrained strength (Cocjin et al., 2014, 2017; Boukpeti & White, 2017; Zhou et al., 2019; O'Loughlin et al., 2020). Further details on the calibration procedure can be found in Mašín (2014, 2019); Kadlĭček et al. (2022) and all parameters selected in this study are within the typical range of values that have been adopted for soft clays in Mašín (2014, 2019) and Kadlĭček et al. (2022).

This section presents results from element test simulations where episodic undrained shearing followed by consolidation is applied with clay hypoplasticity. The hypoplastic simulations have been carried out with the Incremental Driver by Niemunis (2017) using the User MATerial subroutine (umat) of clay hypoplasticity (Mašín, 2014). The results are compared with experimental episodic DSS tests from Laham et al. (2021) where episodes of undrained pre-failure shear followed by consolidation were also applied, see schematic illustration in Fig. 1. The intergranular strain extension (Mašín, 2014) has been included in all the simulations.

Figures 3 and 4 compare the experimental results from the episodic DSS tests (Laham et al., 2021) with the hypoplastic model. The simulated stress paths were matched with the episodic DSS experiments. Samples were first subjected to oedometric normally consolidated conditions and are close to oedometric normal compression line, which is parallel to the NCL, c.f. Mašín (2019); Kadlĭček et al. (2022). The sample was then sheared undrained to a stress ratio τ/σ′_v0 and then consolidated up to while σ′_v = σ′_v0 while τ is kept constant. The undrained shear-consolidation half-cycles are repeated n = 6 times, followed by a final monotonic undrained shear stage to the critical state (n = 7).

In this simulation, an initial consolidation pressure of σ′_v0 = 63 kPa and stress ratio τ/σ′_v0 = 0·18 were applied for comparison with experiment E63 067, as shown in Fig. 3. Figures 3(a) and 3(b) show that hypoplasticity well predicted the final undrained shear strength, which is similar to the measured shear strength and higher than the simulated monotonic undrained strength of a comparable sample at σ′_v0 = 63 kPa (shown by the dashed orange line). The τ/σ′_v stress loops in Fig. 3(b) are also well predicted, overlapping with the experimental data (shown with grey lines). The shapes of the experimental and simulated final monotonic undrained shear paths to failure differ and this is because the simulations over-predicted the reduction in void ratio to a more over-consolidated state, particularly after the first half-cycle Figs 3(c) and 3(d). The over-predicted reduction of void ratio is a result of the large strain (or stress) amplitudes applied during shearing, which exceeded the small-strain range governed by the parameter R. However, this over-prediction in the reduction of void ratio did not result in an over-prediction of shear strength in Figs 3(a) and 3(b).

An initial consolidation pressure of σ′_v0 = 72 kPa and stress ratio τ/σ′_v0 = 0·18 was applied in this simulation for comparison with the episodic DSS experiment E72 070 shown in Fig. 4. The trends in the simulated response relative to the experimental results are very similar to observations made for E63 067 and the final predicted strength was similar to the measured strength.

Within the critical state soil mechanics framework, undrained shear strength increases, s_u,i+1/s_u,i, from consolidation per half-cycle and can be calculated by comparing the change in void ratio before and after each consolidation stage (Schofield & Wroth, 1968):

Equation (3) is based on the semi-logarithmic formulations for the gradient of NCL and CSL, λ, used in conventional MCCM and resulted in a good fit for final measured strength in the episodic DSS tests in Laham et al. (2021). Equation (3) can be adjusted for hypoplasticity to allow for the different gradients of the NCL and CSL in double logarithmic lnp′ or lnσ_v′ against ln(1 + e) space.

Equations (3) and (4) are not mathematically equivalent because the parameters λ and λ* cannot be directly converted between one another but they can be fitted for a selected stress range (void ratio range, respectively) as discussed for example in Medicus (2020). A relationship λ = λ*(1 + e) proposed in Vermeer & Neher (2019), can also be used to approximate λ* for a given void ratio. For the range of void ratios measured in the DSS episodic tests of e = 1·1–1·3 and given a λ = 0·22 this resulted in a small variation in λ* = 0·1 of λ* = 0·104–0·096.

To find these increases in strength per half-cycle, a reference initial undrained strength s_u0 achieved from a monotonic undrained shear test is required, conducted by Laham et al. (2021) or by way of clay hypoplastic simulations.

Figure 5 shows the normalised shear strength τ/σ′_v0 achieved at the end of the monotonic tests M62 and M90 and the episodic tests E63 067 and E72 070. The τ/σ′_v0 achieved from the monotonic simulations (as indicated by an M in Fig. 5 and by orange dashed lines) were similar to the monotonic experiments (shown in grey in Figs 5(a) and 5(c)). The final simulated strength ratios s_u/σ′_v0 for the episodic tests were also similar to the experiments E63 067 and E72 070. In Fig. 6, Equations (3) and (4) are compared to the measured experimental final undrained strengths in the episodic tests, which are normalised by the measured undrained strengths from M62 and M90 in Laham et al. (2021) presented in Figs 5(a) and 5(c), and with crosses in Fig. 6. The dots in Fig. 6 represent the calculated increases in strength per half-cycle from using Equation (3) that adopts the semi-logarithmic MCCM formulations and the open circles are for the proposed alternative double-logarithmic approach for hypoplasticity (Equation (4)).

The calculated changes in strength from Equations (3) and (4) (i.e. comparing dots against open circles in Fig. 6 are very similar, which suggests that Equation (4) is just as suitable for calculating changes in strength as Equation (3)). The grey-coloured set of dots and open circles correspond to the calculated gain in undrained strength from experimentally measured changes in void ratio, while the black-coloured set of dots and open circles represent the calculated gain in undrained strength from simulated void ratios using hypoplasticity. The black-coloured data set lies above the grey data set owing to the over-prediction in the reduction in void ratio per half-cycle in the simulations. This translates to a higher calculated gain in strength and final strength that is closer to the experimental measured strength when normalised by s_u,M62 as this is less than s_u,M90 which are marked with crosses in Fig. 6. The variation between the simulated and experimentally derived changes in undrained strength in Fig. 6 are reasonable when compared to the variation in the experimentally measured final monotonic strength of the samples M62 and M90 shown in Figs 5 and 6.

Clay hypoplasticity with the intergranular strain extension can be used to capture general trends in the evolution of void ratio and undrained strength that are observed in episodic DSS experiments on normally consolidated kaolin clay. While clay hypoplasticity tends to over-predict densification, or decreases in void ratio, the increase in shear strength can be well predicted with the model. The differences between the predicted and experimental undrained strength changes per episode are comparable to variations in the measured experimental undrained strength. Therefore, clay hypoplasticity has potential to be more widely applied in finite-element applications to explore soil-strengthening responses from whole-life episodic undrained shear and consolidation loading conditions relevant to offshore foundations.

This research was funded in part by the Austrian Science Fund (FWF) 10.55776/V918 and 10.55776/P28934. G.M. is funded by the FWF. For open access purposes, the author has applied a CC BY public copyright license to any author accepted manuscript version arising from this submission. This research is also supported by the Royal Academy of Engineering (RAEng) under the Research Fellowship Programme. K.K. is supported by the RAEng Research Fellowship scheme and also by the BritInn- Academic Network Britain-Innsbruck. Some data used are available from the corresponding author by request. The authors thank the developers of the Incremental Driver (Niemunis, 2017) and of the umat of clay hypoplasticity (Mašín, 2014), that they kindly provide their resources on the platform SoilModels.com (Gudehus et al., 2008).

D

stretching tensor, D is the symmetric part of the velocity gradient, which is approximately equivalent to the strain rate $ε˙$⁠. For rectilinear extensions $D=ε˙$

e

void ratio

e₀

initial void ratio

e_c

stress-dependent critical void ratio

M

critical stress ratio q/p′under axisymmetric triaxial compression, M = 6 sinφ_c/(3 − sinφ_c)

N

value of ln(1 + e) at p′ = 1 kPa on the normal compression line (NCL) defined in ln p′ against ln(1 + e) plane; parameter of hypoplasticity

p′

mean effective stress

q

deviatoric stress

R, β_R, χ, A_g, n, m_rat

parameters of hypoplasticity that govern the small-strain (the so-called intergranular strain) extension

s_u

undrained shear strength

s_u/σ′_v0

strength ratio

s_u0

initial undrained shear strength measured from monotonic direct simple shear (DSS) test

κ*

gradient of the swelling line in ln p′ against ln(1 + e) plane, parameter of hypoplasticity

λ

gradient of the NCL, critical state line (CSL) in ln p′ against 1 + e plane

λ*

gradient of the NCL, CSL in ln p′ versus ln (1 + e) plane, parameter of hypoplasticity

ν

parameter of hypoplasticity that controls the ratio of bulk-to-shear modulus

σ*

reference stress σ* = 1 kPa

σ′_v

vertical effective stress

σ′_v0

vertical effective consolidation stress

τ

shear stress

τ/σ′_v0

shear stress ratio

φ_c

critical friction angle; parameter of hypoplasticity