Numerical Modeling of the Dynamic Response of an Elastoplastic Seabed Under Wave-Current Interactions

SHAN Zhigang, ZHU Zhipeng, WANG Dong, 3), *, and YE Guanlin

Numerical Modeling of the Dynamic Response of an Elastoplastic Seabed Under Wave-Current Interactions

SHAN Zhigang1), ZHU Zhipeng2), WANG Dong2), 3), *, and YE Guanlin4)

1) Zhejiang Huadong Construction Engineering Corporation Limited, Hangzhou 310014, China 2) MOE Key Laboratory of Marine Environment and Ecology, Ocean University of China, Qingdao 266100, China 3) Laboratory of Marine Geology, Qingdao National Laboratory for Marine Science and Technology, Qingdao266237, China 4) Department of Civil Engineering and State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

Wave-induced liquefaction of the seabed is a geohazard frequently encountered in shallow waters. Although widely dis- cussed, most studies paid attention to the seabed response under a single sequence of wave loading. However, the seabed suffers from repeated ‘wave loading–dissipation’ phases in a real ocean environment. In this study, a homogeneous sandy seabed model is estab- lished to investigate the mechanism of wave-induced liquefaction by considering the existence of currents. Finite element analyses are conducted by incorporating a kinematic hardening elastoplastic model into the commercial package Abaqus. The constitutive model is validated against centrifugal wave tests. Parametric studies are conducted to demonstrate the effects of relative densities, current, and wave-loading history on the seabed response. The predicted excess pore pressure, effective stress paths, and associated variation of relative density are discussed in detail. The results show that the densification of soils significantly enhances the resis- tance against liquefaction, which provides new insight into the mechanism of residual liquefaction during wave sequences.

seabed; liquefaction; wave; current; finite element methods; cyclic loading

1 Introduction

Wave-induced liquefaction in a sandy seabed is a threat to offshore developments. Interpretation of the potential mechanisms of wave-induced liquefaction is particularly important in the evaluation of the stability of offshore foundations. Two mechanisms were observed in field tests and laboratory tests, namely, momentary and residual liq- uefaction (Zen and Yamazaki, 1991; Sassa., 2006). The momentary liquefaction is induced when the oscilla- tory excess pore pressure exceeds the initial vertical effect- tive stress of the seabed (Zen and Yamazaki, 1990). Com- pared to the momentary liquefaction, the residual lique- faction is dominated by the plastic deformation of the soil skeleton and the accumulation of excess residual pore pressure. Similar to seismic scenarios, the residual liquefaction occurs as the residual excess pore pressure exceedsthe effective overburden pressure. Due to the accumulation of the residual excess pore pressure, residual liquefaction may be triggered even if the wave-induced momentary ex- cess pore pressure is much lower than the initial vertical effective stress. Therefore, the residual liquefaction is ex- plored here.

A number of physical model tests have been conducted in flumes or centrifuges to examine the response of the seabed subjected to waves. The pore pressures at different locations of the seabed were measured to indicate the oc- currence of liquefaction (Sumer., 2012), and high-speed cameras were used to quantify the fluctuation of the soil surface and the expansion of the liquefaction area (Sassa and Sekiguchi, 1999).

Except for experimental studies, theoretical and numeri- cal analyses based on Biot’s theory were conducted to re- produce the seabed response. The soil skeleton of the sea- bed was regarded as a linear elastic material in a large number of analyses by researchers such as Gao and Wu (2006), Ulker. (2009), and Sui. (2017), among others. However, the accumulation of pore pressures could not be reproduced. To overcome this disadvantage, the source term related to the accumulated pore pressure was incorporated into the analytical and numerical analyses for elastic soils. This method was originally presented by Seed and Rahman (1978), in which the oscillatory pore pressures and cyclic shear stresses were obtained from the poroelastic theory, and the residual pore pressures were calculated from an empirical equation depending on the number of wave cycles and the amplitude of the cyclic shear stress. The residual liquefaction of the seabed has been reproduced analytically (Jeng., 2007) and nu- merically (Sassa., 2001; Li., 2011; Sui., 2019) using similar methods. Although the accumulation of pore pressures was available, the real elastoplastic re- sponses were not captured owing to the assumption of elas- tic soils.

For waves applied in most previous studies, the tradi- tional linear or nonlinear wave theories were employed, but little attention was paid to the coexistence of waves and currents (Li., 2019). The pressures imposed on the seabed surface and the seabed response were changed by the currents (Hsu., 2009). Moreover, most existing studies were against a single sequence of wave loading, although the seabed is usually subjected to a series of wave sequences. The relative densities of sandy soils and their resistance to liquefaction have actually evolved progres- sively during repeated ‘wave loading–dissipation’ phases.

In this study, the seabed response under wave sequenceswith the presence of currents is investigated using the finite element approach. The CM model developed by Zhang. (2011) was adopted to mimic soil behaviors. The nu- merical results were thoroughly validated against existing centrifugal tests. The influences of relative densities, cur- rent, and wave-loading history on the seabed response are quantified, leading to the interpretation of the mechanism of residual liquefaction.

2 Methodology

The seabed subjected to the combined loading of waves and currents was simplified as a plane strain problem, as shown in Fig.1. The soil was regarded as an elastoplastic material described through the CM model proposed by Zhang. (2011). The commercial finite element pack- age Abaqus (Dassault Systèmes, 2014) was employed, and the CM model was incorporated into Abaqus through a user subroutine UMAT. Coupled effective stress–pore pressureanalyses for fully saturated porous medium were conducted, in which mixture theory was adopted to model the soil skeleton–pore fluid interaction behaviors of soils. The results proved that mixture theory is consistent with Biot’s theory in dealing with the mechanics of a deforma- ble porous medium (Coussy., 1998). The porous flow was governed by Darcy’s law. The soil was discretized through eight-node quadrilateral elements with reduced integration. The calculation was conducted through the im- plicit time integration scheme, and the time increment was set to be no larger than 1/40 of the wave period to retain computational accuracy.

Fig.1 Schematic of the numerical model.

2.1 Constitutive Model

The CM model proposed by Zhang. (2011) was based on the modified Cam-Clay model, incorporating a subloading surface (Hashiguchi and Ueno, 1977), super-loading surface (Asaoka, 1998), and rotating hard- ening law (Hashiguchi and Chen, 1998) to mimic the over-consolidation, soil structure, and stress-induced anisotro- py, respectively. A type of sand with different densities or structures can be described through a set of material prop- erties. A brief description of the CM model is presented here.

The current stress state is always located on the sub- loading surface, which can be expressed as

The variables involved in Eq. (1) are defined as

An associated flow rule is adopted

The consistency equation for the subloading yield sur- face can be obtained as follows:

The direction of plastic strain increment is

where Λ is the plastic multiplier, which can be determined as

where

where,, andbare three material properties that need to be calibrated through undrained and drained triaxial tests:controls the developing rate of the degree of over-con- solidationdue to the plastic strain,controls the devel- oping rate of structure*due to the plastic shear strain, andbcontrols the developing rate of stress-induced ani- sotropydue to the plastic shear strain.

The initial state variables are related to the initial void ratio0:

2.2 Loading and Boundary Conditions

Two wave theories were used in the subsequent simu- lations. The linear wave theory was used for the compar- ison with the centrifuge tests (Sassa and Sekiguchi, 1999), with the wave pressureand excess pore pressureon the seabed surface being calculated as

where0is the amplitude of the wave pressure;=2π/is a wave number, whereis the wavelength;=2π/is the circular frequency, whereis the wave period;is the horizontal ordinate, as shown in Fig.1; andis the time. The strong nonlinearity of waves in shallow water and the existence of currents need to be taken into account in prac- tical applications. Thus, an analytical solution of the third- -order approximation of the combing loading of waves and uniform currents proposed by Hsu. (2009) was used in the parametric study below. The accuracy of the ana- lytical solution was verified by Ye and Jeng (2012). The pressure fields and excess pore pressure boundary at the seabed surface can be calculated as

whereρis the density of seawater,is the gravitational acceleration,is the wave height,is the water depth, andis the current velocity, which is positive in the di- rection of the wave propagation. The dispersion relation of wave-current interactions is calculated by

When there is no current (,=0), the analytical so- lution is regressed to the approximation of the third-order Stokes wave.

Regardless of the wave theory employed, the lateral boundaries of the seabed are constrained in the horizontal direction and free in the vertical direction, whereas the bottom boundary is fixed. Except for the seabed surface, all the boundaries are impermeable. The central zone of soil is referred to as the real response, and the lateral boundaries have to be sufficiently far from the central. In the trial calculations, the boundary effect can be ignored when the seabed length is larger than 2.

3 Validation

Sassa and Sekiguchi (1999) performed a series of cen- trifugal wave tank tests to investigate the wave-induced pore pressures in fine Leighton Buzzard sand. The sand bed in the centrifuge was 200mm wide and 100mm deep and saturated by silicone oil of 50cSt. The tests were un- der an acceleration of 50g. The specific gravitysof the Leighton Buzzard sand was 2.65. The mean grain size was 0.15mm. The maximum void ratiomaxand mini- mum void ratiominwere equal to 1.07 and 0.64, respec- tively. The average relative densityDof the seabed was 40% corresponding to an average initial void ratio0of 0.898. The unit weight of the silicone oil was 470kNm−3, and the submerged unit weight of soilwas equal to 425kNm−3. The earth pressure coefficient at rest0was ta- ken as 0.52. The permeability of soilkwas 1.5×10−4ms−1(Sassa and Sekiguchi, 2001). Linear waves were generat- ed in the tests, with=510mm and=90.9ms. Wave pressures on the seabed surface were calculatedEq. (12). The excess pore pressure and its residual component were referred as Δand Δu, respectively.

Three groups of tests were used to validate the reliabil- ity of the CM model: I) Tests P5-1 and P6a-1 were with a single-wave sequence, with0=5 and 6kPa, respectively. II) Test P6b-1 was composed of a wave sequence (0=6kPa) and subsequent pore pressure dissipation without anywaves. III) Tests P7-1, P7-2, P7-3, and P7-4 were featured with four repeated ‘wave loading–dissipation’ phases. Group III was designed to observe the re-liquefaction of soils, with0=7kPa applied in the loading phases, and excess pore pressures allowed to fully dissipate in each dissipation phase. All three groups were simulated, al- though the laboratory testing data of the Leighton Buzzard sand were not provided by Sassa and Sekiguchi (1999).

Table 1 Soil material properties and initial state variables for validation

For group I, the excess pore pressures predicted (solid line) and measured (residual component only; dotted line) are compared in Fig.2. The predicted and measured curves indicate the onset of liquefaction. However, the measured excess pore pressure experienced a slow growth period at the early stage, whereas the predicted excess pore pres- sure had no slow growth period and rapidly increased. A similar discrepancy was also observed between the pre- dicted and measured results of flume tests reported by Sumer. (2012). The physics behind this slow growth of pore pressure is complicated. A possible reason is that, in a narrow range of wave height around the critical wave height, an amount of wave cycle (too small to induce the rapid accumulation of pore pressure at initial) is needed to start the rearrangement of soil grains (Sumer., 2012). For this case, the critical wave pressure amplitude was set to 4.7kPa, as suggested by Sassa and Sekiguchi (1999), which is close to0in group I. Apparently, the CM model cannot describe these complicated mechanics. However, this discrepancy only exists for cases with a narrow range of wave height around the critical wave height. For cases with large wave heights, the applicability of the CM model will be proven in subsequent experiments.

Fig.2 Comparison between the predicted excess pore pressure and measured residual pore pressure for cases with a sin- gle-wave sequence (Sassa and Sekiguchi, 1999).

For group II, the accumulations and dissipations of ex- cess pore pressures at three depths with effective vertical stress of 10.6, 29.7, and 39.4kPa are demonstrated in Fig.3. The numerical results are in good agreement with the test data during the wave-loading phase, whereas the pre- dicted dissipation of excess pore pressure is moderately quicker than the measured.

For group III, as depicted in Fig.4, the numerical re- sults are in fair agreement with the experimental data in all sequences. The calculated and measured excess pore pressures accumulated slower with the wave-loading his- tory. By undergoing liquefaction three times, soil with depth=−70mm was no longer liquefied in the subse- quent wave loading. The comparison shows that the in-fluence of wave load history on the seabed response can be captured reasonably.

4 Results and Discussion

4.1 Progressive Liquefaction of the Seabed

Fig.3 Comparison between the predicted excess pore pressure and measured residual pore pressure for test P6b-1 with the dissipation phase (Sassa and Sekiguchi, 1999).

Fig.4 Comparison between the predicted excess pore pressure and measured residual pore pressure for test P7 under four wave sequences (Sassa and Sekiguchi, 1999).

The distribution of excess pore pressures along the sea- bed depthat different wave cycles/is plotted in Fig.5. Periodic shear stresses were generated on the seabed sur- face due to wave propagation, resulting in the accumula- tion and re-distribution of pore pressures within the sea- bed. With the application of the waves, liquefaction takes place quickly under the seabed surface, followed by down-ward expansion. The liquefaction depth was propagated to −1, −2, and −4.5m after 12, 15, and 20 cycles, respectively. The process of liquefaction is clearly shown through the effective stress path in Fig.6. The CSL and initial0line of soils in the-space are also plotted in Fig.6. The CSL (=), as defined in the modified Cam-Clay model, refers to the failure state of soils.Initial0line (=3(1–0)/(1+20)) refers to the initial0consolidation state of the soil before any wave loading. The deviator stressand mean effective stressat=1.2m are decreased progres- sively, leading to a stress state from the0consolidation line to the CSL. In the first eight cycles, the values ofandreduce markedly, indicating the rapid accumulation of residual excess pore pressure, which is shown at=1.2m in Fig.5 as well. Then the reduction ofandslows down and the stress path eventually cycles around the CSL. This kind of stress path loop, called ‘cyclic mobility’, is a typical characteristic of liquefied sand reported widely. The CM model is able to describe the cyclic mobility and the post-liquefaction behaviors of sand; therefore, the ex- pansion of the liquefaction zone is captured reasonably well.

Fig.5 Predicted residual excess pore pressure profile with depth z.

Fig.6 Prediction for the effective stress paths of soil at z=1.2m.

4.2 Influence of Relative Densities

The seabed with various relative densities may have dif- ferent dynamic responses when subjected to wave loading. Parametric studies were conducted to investigate the ef- fects of relative densities on the residual response. The parameters used are listed in Table. 2. The residual excess pore pressure profile with various relative densities at/=15 are plotted in Fig.7. For the case withD=40%, the residual excess pore pressure was quickly accumulated. The liquefaction zone was expanded to −3.3m at/=15. For the dense sand (D=60%), the development rate of the residual excess pore pressure reduced remarkably, and liquefaction did not occur at any depths of the seabed. For very dense sand (D=80%), the seabed response is nearly pure elastic, with residual excess pore pressure close to zero. The sandy seabed with a low relative density is prone to liquefaction, andthe liquefaction zone can be extended to a deep area. The sandy seabed with a high relative den- sity is difficult to liquefy.

Table 2 Soil properties used for relative densities

4.3 Influence of Current

Usually, waves and currents simultaneously appear, and wave-induced pressure may be changed significantly by the existence of currents. In this study, the current veloci- ties ranged between −2ms−1and 2ms−1.

The residual excess pore pressures at=1.2m against various current velocities are demonstrated in Fig.8. For the cases with=1 and 2ms−1, liquefaction occurred after 14.4 and 13.3 wave cycles, respectively. For the case without current, the liquefaction zone was expanded to a depth of=1.2m after 16.4 wave cycles. However, for the two cases with opposing currents, liquefaction was not triggered within 20 wave cycles, which suggests that the opposing current may significantly weaken the lique- faction potential.

Fig.7 Predicted residual excess pore pressure profile with various relative densities.

Fig.8 Predicted liquefaction potential with various current conditions at z=1.2m.

Fig.9 illustrates the pore pressure profiles with various current velocities after 15 wave cycles. For the cases with current velocities of 0, 1, and 2ms−1, the liquefaction zone was expanded to −1, −2.5, and −4m, respectively, while the opposing currents effectively prevented the liquefac- tion.

4.4 Influence of the Wave-Loading History

Large-amplitude waves result in the accumulation of ex- cess pore pressures in the sandy seabed, but soil becomes re-consolidated during the subsequent dissipation of pore pressures. Soil may be re-liquefied or remain stable during the next sequence of waves, and the extension of the liq- uefaction zone depends on the wave-loading history. In this study, four sequences of wave loadings were applied. A sufficient long interval was set between two wave se- quences to allow the excess pore pressure to dissipate com- pletely.

Fig.9 Predicted residual excess pore pressure profile with various current conditions.

To further investigate how the wave-loading history in- fluences the soil response, the stress path at=1.2 and 4.2m during the second wave sequence is plotted in Fig.11. Different from the first sequence, the stress paths at the commencement of the second sequence are both above the0line. This is because the stress-induced anisotropy becomes larger when soil suffers from liquefaction and consolidation. The reductions ofandin each wave cycle of the second sequence are much smaller, and the cyclic amplitude ofremains almost constant.at=1.2m decreased from 6kPa to 2.8kPa, whereasat=4.2m decreased from 19.3kPa to 9.1kPa. At the end of the se- cond wave sequence, the stress paths at both depths did not reach the CSL, and the tendency was also observed in the third and fourth wave sequences.

Fig.12 shows the relative density of soil,D, at=1.2 and 4.2m under four wave sequences. At=1.2m, the origi- nal value ofDis 41.3%, and it was increased to 48.7% after the first consolidation phase. However, the variation ofDin the next three ‘wave loading– dissipation’ phases was not remarkable, whereDwas increased to 51.1%,52.6%, and 53.8% at the end of each phase. A similar tendency was observed at=4.2m: the original value ofDwas 41.8%, and its values after the four consolidation phases were 52.9%, 55.0%, 56.3%, and 57.4%. Clearly, the densification of sands was attributed to the dissipation of excess pore pressure. The soil strength and resistance against liquefaction were enhanced by the densification, which is consistent with the ‘crust’ layer frequently ob- served on the sandy seabed (Tian., 2019).

Fig.10 Predicted excess pore pressure in terms of the wave-loading history.

Fig.11 Predicted stress path under the second wave-loading stage.

Fig.12 Predicted relative density under four wave sequences.

5 Conclusions

In this study, finite element analyses incorporating the CM model were conducted to investigate the dynamic re- sponse of a sandy seabed under combined wave and cur- rent loadings. The CM model reasonably reproduced the post-liquefaction behaviors of sand, and the reliability of the numerical analyses was verified through a comparison with centrifuge tests. The influences of relative densities, currents, and wave-loading history were analyzed in de- tail. The main conclusions are given below:

1) A loose seabed is prone to liquefaction, whereas a dense seabed may not liquefy. The increase in relative den- sities may remarkably limit the expansion of liquefaction zones.

2) The following current co-existing with the waves tended to accelerate the process of liquefaction and en- large the liquefaction depth. By contrast, the opposing cur- rent may effectively reduce the liquefaction potential of the seabed.

3) The seabed response evolved progressively during repeated ‘wave loading– dissipation’ phases. The liquefied soil became densified with the dissipation of excess pore pressures. The peak value of the residual pore pressure ac- cumulated under the subsequent wave loading significantly decreased. The results suggest that the long interval be- tween two wave sequences is helpful in resisting lique- faction.

Acknowledgements

This paper was supported by the National Natural Science Foundation of China (Nos. U1806230 and 42025702), and the Key Science andTechnology Plan of PowerChina Huadong Engineering Corporation (No. KY2018-ZD-01).

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(May 18, 2021; revised August 23, 2021; accepted November 9, 2021)

© Ocean University of China, Science Press and Springer-Verlag GmbH Germany 2023

Corresponding author. E-mail: dongwang@ouc.edu.cn

(Edited by Xie Jun)