Numerical Modeling of Thermal-Dependent Creep Behavior of Soft Clays under One-Dimensional Condition

Creep is a common phenomenon for soft clays. .e paper focuses on investigating the influence of temperature on the timedependent stress-strain evolution. For this purpose, the temperature-dependent creep behavior for the soft clay has been investigated based on experimental observations. A thermally related equation is proposed to bridge the thermal creep coefficient with temperature. By incorporating the equation to a selected one-dimensional (1D) elastic viscoplastic (EVP) model, a thermal creep-based EVPmodel was developed which takes into account the influence of temperature on creep. Simulations of oedometer tests on reconstituted clay are made through coupled consolidation analysis. .e bonding effect of the soil structure on compressive behavior for intact clay is studied. By incorporating the influence of the soil structure, the thermal creep EVPmodel is extended for intact clay. Experimental predictions for thermal creep oedometer tests are simulated at different temperatures and compared to that obtained from reconstituted clay. .e results show that the influence of temperature on the creep behavior for intact clay is significant, and the model, this paper proposed, can successfully reproduce the thermal creep behavior of the soft clay under the 1D loading condition.


Introduction
It is well known that soft clays exhibit time-dependent behavior due to their viscosity.e long-term settlement of these clays after the dissipation of excess pore water pressure, which is sometimes called creep deformation, has been an important issue in geotechnical engineering.e creep behavior of the soft clay has been investigated experimentally in [1][2][3][4][5].Based on that, some practical models have been developed [6][7][8][9].
Due to the deposition effect, interparticle bonds are usually formed in soft clays referred as the soil structure.When suffering loading for the soft clay, a significant progressive loss of bonds will happen.By comparing the compression curves in e − logσ '  v (void ratio versus vertical effective stress in the log scale) for intact and reconstituted samples, the large differences observed are induced by the bond elimination.In addition, the bond elimination effect on the creep behavior of soft clays has been studied in [10,11].
Soft clays are also subjected to the action of heat under many circumstances, for example, the nuclear waste isolation, heat energy storage, and geothermal development.Studies show that the creep behavior of soft clays is strongly related not only to the bonding structure but also to temperature.e creep behavior of these clays will be changed accordingly [12][13][14][15][16].However, the effect of temperature on the creep behavior was somehow showed, but not well documented.It will be nice if there exists a direct way for modeling the thermal creep behavior.
For this purpose, we focus on the 1D behavior which can bring fundamental features for more mechanical behavior.Firstly, the temperature dependency behavior of creep is studied based on the experimental observations.en, a thermally related EVP model for reconstituted clay is proposed by incorporating the effect of temperature.
Furthermore, the bonding elimination of the structure is incorporated into the thermal-based model.Finally, the prediction ability of the proposed model is shown by simulations at di erent conditions.e consolidation process of soil is usually divided into primary consolidation and secondary consolidation, and the boundary point is whether the excess pore pressure completely dissipates.Similarly, thermal secondary consolidation occurs for thermal consolidation after the dissipation of excess pore pressure [13].Figure 1 shows an idealized thermal consolidation test result.Actually, thermal secondary consolidation occurs throughout the thermal consolidation process, and the induced deformation only relates to time and temperature.

Temperature Dependency of Creep of Soil
us, thermal creep used in this paper is more suitable for the process.e slope of the linear portion of the thermal consolidation curve is the thermal creep coe cient ψ T , given in void ratio per log cycle.
A number of studies demonstrate that the temperature a ects signi cantly the thermal creep coe cient.For example, Figure 2 presents the evolution of the thermal creep coe cient with increasing temperature for intact Paci c illite conducted by Houston et al. [13].e values range from about 0.01 at 40 °C to about 0.06 at 200 °C.e increase in the rate of thermal creep deformation at the elevated temperatures was quite signi cant.Additionally, the creeptemperature tests on peats conducted by Fox and Edil [12] also show that creep dominates the consolidation process and temperature in uences the creep rate signicantly.
us, it can be concluded that it is necessary to account for the e ect of temperature on creep for soft intact clay.

Proposed ermal Creep Equation.
Based on the experiments, di erent expressions were given to describe the temperature-dependent behavior of the thermal creep coe cient.e relationship between thermal creep coe cient and temperature can be linearly functioned by Houston et al. [13].at is, where ψ T is the thermal creep coe cient under temperature T and A is the thermal relate parameter, which can be obtained by correlating the results as shown in Figure 2. Equation (1) describes ψ increasing linearly with temperature straightforwardly.However, we can observe that ψ T will be zero when the temperature decreases to zero and ψ T will be negative if the temperature still decreases and is unreasonable.
Based on the experimental results on peat, the following equation is used by Fox and Edil [12] to predict the value of ψ T , due to a change in soil temperature: where ψ T r is the reference thermal creep coe cient under the reference temperature T r , thermal relate parameter B is equal to 0.25 ± 0.02/ °C for the peat in [12], and the parameter is independent of vertical e ective stress and the magnitude of temperature change.Equation (2) describes that ψ increases nonlinearly with temperature.However, due to its mathematic structure, Equation (2) will deduce a very large number of ψ at a higher temperature, far beyond the normal range.
In this study, we propose a new equation to describe the nonlinear increase of the thermal creep coe cient with temperature, which can be written as where C T is the thermal relate parameter which can be correlated with the experiment results. is equation can overcome the de ciency exposed by Equations ( 1) and (2). Figure 3 plots the comparisons of the three equations 2 Advances in Civil Engineering describing the relationship between thermal creep coe cient and temperature.e point T 20 °C and ψ(20) 0.006 is adopted as the reference.As indicated, three equations give di erent shape curves.Overall, the curve plotted from Equation ( 3) is more practical than the others.

EVP Model considering the Temperature Effect on Creep
3.1.Adopted One-Dimensional Elastic Viscoplastic Model.First, a time-dependent stress-strain model without the thermal e ect needs to be selected as a base [6].e adopted model is presented brie y in this part.Following the classic elastic viscoplastic approach, the total strain rate contains the elastic and viscoplastic strains rate, that is, where _ ε v represents the total strain rate and the superscripts "e" and "vp" denote the elastic and viscoplastic parts, respectively.e elastic strain rate is expressed as where e 0 is the initial void ratio, σ v ′ is the e ective vertical stress, and κ is the slope of recompression lines in e − ln σ v ′ space.
For the viscoplastic strain rate, a one-dimensional formulation proposed by Kutter and Sathialingam [17] was adopted based on the creep coe cient ψ: where λ denotes the slope of the normal compression line in e − ln σ v ′ space; τ is the reference time, and it equals to the duration of each load increment in the oedometer test; and σ ′r v is the reference stress corresponding to the incremental time τ and increases with the development of the viscoplastic strain according to where σ ′r p denotes the preconsolidation pressure.e above relationships have been suggested in [18][19][20] and validated in [6,10].

Incorporation of the ermal E ect.
e e ects of temperature on the stress-strain behavior of clay have been observed in the laboratory [12,13], which are helpful to discuss the thermal e ect on the parameters in the above constitutive model.Increase and decrease in temperature may produce changes in the bonding of clay particles and the viscosity of absorbed water.
ese changes alternately produce more or fewer changes in compressibility.However, the experiments conducted in [15,21,22] show that the variations of λ and κ with temperature are negligible.Furthermore, a volume change due to elastic expansion of the clay particle will occur during the drainage of thermal consolidation.Considering that the strain under a constant e ective stress remains small [14,23,24] and the emphasis of this paper, the parameters λ, κ, and e 0 will remain constant when clays su ering a change of temperature.
e thermal creep coe cient can be incorporated into the basic EVP model directly.By substituting the parameter ψ in Equation ( 6) by Equation ( 3), the 1D thermal-related viscoplastic strain rate changes to Actually, the preconsolidation pressure σ p ′ also varies with temperature.e thermal-dependent behavior of σ p ′ has been studied from the oedometer tests or isotropic compression tests with variable temperatures.All of the results indicate that σ p ′ will decrease with an elevated temperature [25][26][27][28][29]. Based on the data collected from literature, Figure 4 plots the relationship between the normalized preconsolidation pressure and temperature.
e regression analyses show that it is also reasonable to assume a linear relationship between log(σ p ′ /σ ′r p ) and log(T/T r ), rstly proposed by [28].us, the relationship between the preconsolidation pressure and temperature can be tted by where θ is a thermal parameter and σ p ′ and σ ′r p are the preconsolidation pressures at temperature T and the reference temperature T r , respectively.e present model has no elastic limit, which is di erent from Perzyna's overstress method [30].soil-water coupling analysis will be performed.Darcy's law was adopted for the consolidation process:

Coupled Consolidation
where z is the vertical depth, u is the excess pore pressure, and k is the hydraulic conductivity.Actually, k is also in uenced by temperature.A decrease in pore water viscosity will happen with increasing temperature, which will result in an increase in the permeability of the soil.us, increasing temperature will speed up the consolidation process.Considering the emphasis of this paper, only the void ratio-dependent behavior of k is considered, and according to experimental results, the evolution of k can be expressed as k k 0 10 e−e 0 ( )/ck , (11) where the initial hydraulic conductivity k 0 is corresponding to e 0 and the permeability coe cient c k can be easily measured from the oedometer test results by plotting e − log k. e above equations were implemented in nite element software PLAXIS Version 8, but in 1D, nite element simulations can be established for modeling the primary consolidation and the creep process in an oedometer test.Details of coupled consolidation and creep analysis can be found in [19,31] and are not repeated here.

Simulated One-Dimensional Behavior.
In order to validate the proposed thermal-dependent EVP model, numerical simulations for the assumed oedometer creep tests were performed at three temperatures (T 20 °C, 50 °C, and 80 °C).e results are shown in Figure 5, and the parameters adopted for these simulations are listed in Figure 5(a).e simulated temperature behavior agrees with the common experimental phenomena on unstructured clay, as expected by the model's principle.For example, the simulated relationship between the preconsolidation pressure and temperature corresponds to the input value of θ.
Take the simulated test on T 80 °C; for example, the compression behavior for each load increment is presented in Figure 5(b).e simulated thermal creep coe cient ψ 0.013 agrees well with that obtained from Equation (3) with T r 20 °C and ψ T r 0.0065.

Thermal-Dependent EVP Model for
Intact Clay

Bonding E ect on Compression
Behavior.During the oedometer tests, the di erence of the compression curves obtained on intact and reconstituted clay is caused by bonding elimination as shown in Figure 6(a) for Wenzhou clay [32].e structures between soil particles for intact clay will be eliminated gradually during compressing.e tests conducted under variable temperatures show that the shape of the compression curves does not change with temperature, for instance, tests on intact Berthierville clay [25] and Linköping clay [28].us, we assume that the process of bonding elimination is thermal independent but only relates to the strain level.Figure 6(b) presents the schematic plot of the stress-strain curve at an arbitrary temperature T for soft intact clay.For a given viscoplastic strain level ε vp v , the bond elimination results in the current stress σ v ′ reaching point D for intact clay.Corresponding to the same viscoplastic strain, we de ne an intrinsic stress σ vi ′ on the reconstituted sample.We assume that the di erence between the current stress and intrinsic stress is due to the existing of the soil structure, based on which a bonding ratio χ σ v ′ /σ vi ′ − 1 can be proposed.us, the current stress σ v ′ during straining can be expressed as Initially, the bonding ratio χ χ 0 σ p ′ /σ pi ′ − 1.Following the increasing of strain, the bonds are broken gradually and χ decreases from its initial value χ 0 ultimately towards zero when the bonds are completely eliminated as plotting in Figure 6(a).According to the de nition, bonding ratio and the corresponding viscoplastic strain during compression is measured and plotted in Figure 6(c) for Wenzhou clay.Based on the results, we propose the following relationship to express the attenuation of bonding ratio: where the parameter ζ controls the rate of bonding elimination (ζ 8.0 for Wenzhou clay in Figure 6(c)).Actually, the intrinsic stress σ vi ′ in Equation ( 12) can be regarded as the reference stress as indicated in Equation ( 8), and the bonding ratio can be regarded as the scaling parameter.us, the present model is then composed of Equations ( 8), (9), and (12).Combining with the elastic strain rate in Equation ( 5), the stress-strain curve for a given temperature can be obtained.Reconstituted illite [26] Sulphide silty clay [27] Linköping clay [28] St-Roch-de-I'Achigan clay [29] Soft Bangkok clay [24] 1 θ e present model that combined creep and temperature involves a number of parameters which can be divided into four groups: (a) Parameters related to compressibility: initial void ratio (e 0 ), the intrinsic slope of the compression line (λ i ), and the slope of the recompression line (κ) .e values of λ i and κ can be measured from the oedometer tests on the reconstituted and intact samples, respectively.As the thermal expansion and contraction are neglected in this paper, e 0 can be measured initially at the reference temperature.(b) Parameter related to bonding elimination: the initial bonding ratio (χ 0 ) and the parameter ζ. e value of χ 0 can be measured from the oedometer tests on the intact and reconstituted samples conducted at the same temperature.It needs to point out that, for χ 0 , high-quality intact samples are needed.e parameter ζ representing the bonding elimination rate can be derived from Equations ( 12) and ( 13): where ε vp v is the volumetric viscoplastic strain corresponding to σ v ′ (Figure 6 Parameters related to preconsolidation pressure: the reference preconsolidation pressure (σ ′r p ) at the reference temperature T r and the thermal parameter (θ).σ ′r p can be obtained at the intersection of the compression curves for the reconstituted and intact samples, as shown in Figure 6(b).θ can be obtained directly from the oedometer tests on the reconstituted or intact samples at di erent temperatures.Wang et al. [33] investigated the value of θ for seven clays and summarized that θ varies from 0.125 to 0.194.Furthermore, θ can be obtained by the empirical correlation of liquid limit (w L ) expressed as us, θ can be obtained by correlating without carrying out the temperature-controlled tests.us, ten model parameters (e 0 , κ, λ, θ, T r , C T , ψ T r ,σ ′r p , χ 0 , and ζ) are required for the model, and all of the parameters can be determined straightforwardly from the temperaturecontrolled oedometer test.

Predictions on Wenzhou Clay.
e thermal creep coe cient ψ is also in uenced by the bonding elimination process during straining.In order to evaluate the model ability to reproduce the behavior of soft intact clay, simulations were performed for the thermal oedometer tests with di erent temperatures.e input ordinary parameters are the same as those given in Table 1.For intact clay, two Advances in Civil Engineering parameters χ 0 3.5 and ζ 8 were used.ree temperatures with T 20 °C, 50 °C, and 80 °C are adopted in the simulations.Figure 7 shows typical results of the thermal oedometer tests in natural intact clay simulated by the new model: (1) In Figure 7(a), the compression curves in di erent temperatures are similar to the oedometer test on intact Wenzhou clay (Figure 6(a)).e preconsolidation pressure is in uenced signi cantly by   7(b), the typical curves of strain versus time were reproduced for natural intact clay, where the thermal creep coe cient ψ can be obtained at the end of each load; here, only the simulation results for T 80 °C are presented.(3) In Figure 7(c), the measured ψ at three temperatures were plotted versus the normalized applied stress by the preconsolidation pressure.Here, the preconsolidation pressure is di erent for the three tests.e value ψ for intact clay increases rapidly with the applied stress, and when the stress reaches the preconsolidation pressure, ψ reaches a peak value and then decreases.e di erence between these curves is due to the combined in uence of temperature and bonding elimination.(4) In Figure 7(d), the measured ψ for intact and reconstituted clay are compared for the test at T 80 °C.e di erence between the two curves is due to the existence of bonding and its elimination.Advances in Civil Engineering di erent temperatures for both intact and reconstituted samples.e clay properties are as follows: liquid limit w 55%, e 0 1.977, and σ p ′ 55 kPa for the intact sample at the reference room temperature T 20 °. e other parameters for the model were correlated with thermal oedometer tests results.Figure 8(a) shows the compression tests on intact and reconstituted samples at T r 20 °. e parameter λ i 0.18 is determined based on the results of the reconstituted sample.

Predictions on Utby
e parameter κ 0.003 is correlated from the recompression curve.Bonding ratio χ 0 54 is obtained based on the method described above.e value of ζ 15 was determined by selected a stress-strain point in Figure 8(a) and using Equation (14).Figure 8(b) presents the variation of the thermal creep coe cient with temperature.Adopting ψ T at T 20 °as a reference, the predicted values of ψ T for T 5 °agree well with the experimental results with C T 0.31.Furthermore, an average value of ψ T r 0.02 is used based on the experimental results at T 20 °.With Equation ( 15), θ 0.15 is adopted in the simulation.To simulate the long-term creep test, the permeability of soil was taken as 4 × 10 −6 m/h estimated from the consolidation curves of vertical strains versus time.e value of c k was equal to e 0 /2, as suggested by Tavenas et al. [35] based on observations on soft marine clays.All of the parameters are summarized in Table 1.
Figure 9 shows the comparison between experimental and simulation results.e predicted compression behavior at di erent temperatures shows good agreement with the experimental results for the values of the preconsolidation pressures and for the shape of the compression curves (Figure 9(a)).Furthermore, the predicted thermal creep coe cient also agrees well with the experimental results.For the values of vertical stress equal to the preconsolidation pressure, the thermal creep coe cient reaches to the maximum value (Figure 9(b)).It can be concluded that it is necessary to account for the coupling e ect of temperature and destructuration for accurate predictions of the thermal compression behavior of the soft clay.

Predictions on Tokyo Bay
Clay.Tsuchida et al. [36] conducted the oedometer tests on Tokyo bay clays which are consolidated at room temperature (25 °) and at high temperature (75 °), respectively.e sample which is su ered consolidating at a high temperature and cooled after the completion consolidation will behave like the lightly aged clay.Adopting the sample consolidated at room temperature as a reference, the properties are as follows: liquid limit w 78% and e 0 2.4.From the experimental results on the sample consolidated at room temperature (Figure 10(a)), the parameter λ i 0.36, κ 0.05, and σ p ′ 79 kPa can be obtained.In addition, the reference ψ T r 0031 is averaged for the vertical stress larger than preconsolidation pressure (Figure 10(b)).Also, the parameter θ 0.17 can be calculated by Equation (15).χ 0 0.2 is obtained by the increase of preconsolidation pressure due to the cooling of the sample after high temperature consolidation.Similarly, ζ 7 is  All of the parameters are collected in Table 1.
Figure 11(a) shows that the predicted compressibility of Tokyo bay clay consolidated at a high temperature agrees well with the experiment.Furthermore, the predicted thermal creep coefficient has the same shape with experiment.
e predicted values are a little smaller than the experimental results (Figure 11(b)).Overall speaking, the model can reproduce well the thermal creep behavior for the soft clay.Experimental predictions have carried out the thermal oedometer tests at different temperatures.

Conclusions
e bonding elimination effect on the evolution of the thermal creep coefficient has been highlighted by comparing predictions with and without considering bonding elimination.
e results demonstrate that the proposed model can well reproduce temperature-dependent creep behavior of soft intact clay under the one-dimensional loading condition.Future work will be done to extend the proposed model to three-dimensional general stress space.

Figure 3 :
Figure 3: Comparisons of the equations describing the relationship between thermal creep coe cient and temperature.
(b)) at an arbitrary temperature.us, ζ can be calculated by selecting a point on the postyield curve.(c) Parameters of creep: reference thermal creep coe cient (ψ T r ) and thermal parameter (C T ).ψ T r can be measured directly from the oedometer tests on reconstituted at a reference temperature (T r ).C T can be obtained by correlating ψ T r with temperature.

Figure 5 :
Figure 5: Simulations for 24 h oedometer tests: (a) compression curves at di erent temperatures; (b) compression behavior at each load increment for the test simulated at T 80 °C.

Figure 6 :
Figure 6: Bonding e ect on the compressibility and the evolution of bonding ratio: (a) compression curves for intact and reconstituted Wenzhou clay at T 20 °C; (b) illustration of the bonding elimination with viscoplastic strain; (c) evolution of bonding ratio with viscoplastic strain.
Clay.Li et al.[34] presented a set of the long-term oedometer tests on Utby clay (6 m depth) with

Figure 7 :
Figure 7: Predictions of the thermal oedometer test on Wenzhou clay with di erent temperatures: (a) compression curves at di erent temperatures; (b) strain-time curves for the case with T 80 °C; (c) evolution of the thermal creep coe cient with normalized vertical stress; (d) comparisons of the thermal creep coe cient between intact and reconstituted clay.

Figure 8 :eFigure 9 :
Figure 8: Determination of parameters from the thermal oedometer test: (a) compression parameters at T 20 °; (b) the thermal creep coe cient based on the experiment on the reconstituted sample.

Figure 10 :Figure 11 :
Figure 10: Determination of parameters from the oedometer tests for the sample consolidated at room temperature: (a) compression curves; (b) creep coe cient versus vertical stress.
e temperature-dependent behavior of creep for soft intact clay has been investigated based on experimental observations from experimental results.A thermally related equation is proposed to bridge the thermal creep coefficient with temperature.By incorporating the equation to a selected one-dimensional EVP model, a thermal creep-based EVP model is developed taking into account the temperature dependency of creep.e determination of the model parameters is straightforward.Numerical simulations have been conducted to examine the predictive ability of the model for the soft clay.

Table 1 :
Values of model parameters and state variables for selected clays.