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

Authors: Qi-Yin Zhu; Ping Qi
Journal: Advances in Civil Engineering
 (2018)

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## Abstract

Creep is a common phenomenon for soft clays. The paper focuses on investigating the influence of temperature on the time-dependent 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 EVP model 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. The 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 EVP model 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. The 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.


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## Body

## 1. Introduction

It is well known that soft clays exhibit time-dependent behavior due to their viscosity. The 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. The creep behavior of the soft clay has been investigated experimentally in [1–5]. Based on that, some practical models have been developed [6–9].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 ine−logσv' (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. The creep behavior of these clays will be changed accordingly [12–16]. 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. Then, 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 different conditions.


## 2. Temperature Dependency of Creep of Soil

### 2.1. Experimental Observations

The 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. Thus, thermal creep used in this paper is more suitable for the process. The slope of the linear portion of the thermal consolidation curve is the thermal creep coefficient ψT, given in void ratio per log cycle.Figure 1
Definition of the thermal creep coefficient.A number of studies demonstrate that the temperature affects significantly the thermal creep coefficient. For example, Figure2 presents the evolution of the thermal creep coefficient with increasing temperature for intact Pacific illite conducted by Houston et al. [13]. The values range from about 0.01 at 40°C to about 0.06 at 200°C. The increase in the rate of thermal creep deformation at the elevated temperatures was quite significant. Additionally, the creep-temperature tests on peats conducted by Fox and Edil [12] also show that creep dominates the consolidation process and temperature influences the creep rate significantly. Thus, it can be concluded that it is necessary to account for the effect of temperature on creep for soft intact clay.Figure 2
Evolution of the thermal creep coefficient of Pacific illite [13].


### 2.2. Proposed Thermal Creep Equation

Based on the experiments, different expressions were given to describe the temperature-dependent behavior of the thermal creep coefficient. The relationship between thermal creep coefficient and temperature can be linearly functioned by Houston et al. [13]. That is,(1)ψT=A⋅T,where ψT is the thermal creep coefficient 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 ψT will be zero when the temperature decreases to zero and ψT 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 ψT, due to a change in soil temperature:(2)ψT=ψTrexpBT−Tr,where ψTr is the reference thermal creep coefficient under the reference temperature Tr, thermal relate parameter B is equal to 0.25 ± 0.02/°C for the peat in [12], and the parameter is independent of vertical effective 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 coefficient with temperature, which can be written as(3)ψT=ψTrTTrCT,where CT is the thermal relate parameter which can be correlated with the experiment results. This equation can overcome the deficiency exposed by Equations (1) and (2). Figure 3 plots the comparisons of the three equations describing the relationship between thermal creep coefficient and temperature. The point T = 20°C and ψ(20) = 0.006 is adopted as the reference. As indicated, three equations give different shape curves. Overall, the curve plotted from Equation (3) is more practical than the others.Figure 3
Comparisons of the equations describing the relationship between thermal creep coefficient and temperature.


## 2.1. Experimental Observations

The 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. Thus, thermal creep used in this paper is more suitable for the process. The slope of the linear portion of the thermal consolidation curve is the thermal creep coefficient ψT, given in void ratio per log cycle.Figure 1
Definition of the thermal creep coefficient.A number of studies demonstrate that the temperature affects significantly the thermal creep coefficient. For example, Figure2 presents the evolution of the thermal creep coefficient with increasing temperature for intact Pacific illite conducted by Houston et al. [13]. The values range from about 0.01 at 40°C to about 0.06 at 200°C. The increase in the rate of thermal creep deformation at the elevated temperatures was quite significant. Additionally, the creep-temperature tests on peats conducted by Fox and Edil [12] also show that creep dominates the consolidation process and temperature influences the creep rate significantly. Thus, it can be concluded that it is necessary to account for the effect of temperature on creep for soft intact clay.Figure 2
Evolution of the thermal creep coefficient of Pacific illite [13].


## 2.2. Proposed Thermal Creep Equation

Based on the experiments, different expressions were given to describe the temperature-dependent behavior of the thermal creep coefficient. The relationship between thermal creep coefficient and temperature can be linearly functioned by Houston et al. [13]. That is,(1)ψT=A⋅T,where ψT is the thermal creep coefficient 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 ψT will be zero when the temperature decreases to zero and ψT 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 ψT, due to a change in soil temperature:(2)ψT=ψTrexpBT−Tr,where ψTr is the reference thermal creep coefficient under the reference temperature Tr, thermal relate parameter B is equal to 0.25 ± 0.02/°C for the peat in [12], and the parameter is independent of vertical effective 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 coefficient with temperature, which can be written as(3)ψT=ψTrTTrCT,where CT is the thermal relate parameter which can be correlated with the experiment results. This equation can overcome the deficiency exposed by Equations (1) and (2). Figure 3 plots the comparisons of the three equations describing the relationship between thermal creep coefficient and temperature. The point T = 20°C and ψ(20) = 0.006 is adopted as the reference. As indicated, three equations give different shape curves. Overall, the curve plotted from Equation (3) is more practical than the others.Figure 3
Comparisons of the equations describing the relationship between thermal creep coefficient and temperature.


## 3. 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 effect needs to be selected as a base [6]. The adopted model is presented briefly in this part.Following the classic elastic viscoplastic approach, the total strain rate contains the elastic and viscoplastic strains rate, that is,(4)ε˙v=ε˙ve+ε˙vvp,where ε˙v represents the total strain rate and the superscripts “e” and “vp” denote the elastic and viscoplastic parts, respectively. The elastic strain rate is expressed as(5)ε˙ve=κ1+e0σ˙v′σv′,where e0 is the initial void ratio, σv′ is the effective vertical stress, and κ is the slope of recompression lines in e−lnσv′ space.For the viscoplastic strain rate, a one-dimensional formulation proposed by Kutter and Sathialingam [17] was adopted based on the creep coefficient ψ:(6)ε˙vvp=ψ1+e0τσv′σv′rλ−κ/ψ,where λ denotes the slope of the normal compression line in e−lnσv′ space; τ is the reference time, and it equals to the duration of each load increment in the oedometer test; and σv′r is the reference stress corresponding to the incremental time τ and increases with the development of the viscoplastic strain according to(7)σv′r=σp′rexp1+e0λ−κεvvp,where σp′r denotes the preconsolidation pressure.The above relationships have been suggested in [18–20] and validated in [6, 10].


### 3.2. Incorporation of the Thermal Effect

The effects of temperature on the stress-strain behavior of clay have been observed in the laboratory [12, 13], which are helpful to discuss the thermal effect 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. These 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 effective stress remains small [14, 23, 24] and the emphasis of this paper, the parameters λ, κ, and e0 will remain constant when clays suffering a change of temperature.The thermal creep coefficient can be incorporated into the basic EVP model directly. By substituting the parameterψ in Equation (6) by Equation (3), the 1D thermal-related viscoplastic strain rate changes to(8)ε˙vvp=ψTr1+e0τTTrCTσv′σv′rλ−κ/ψTrTr/TCT.Actually, the preconsolidation pressureσp′ also varies with temperature. The thermal-dependent behavior of σp′ has been studied from the oedometer tests or isotropic compression tests with variable temperatures. All of the results indicate that σp′ will decrease with an elevated temperature [25–29]. Based on the data collected from literature, Figure 4 plots the relationship between the normalized preconsolidation pressure and temperature. The regression analyses show that it is also reasonable to assume a linear relationship between logσp′/σp′r and logT/Tr, firstly proposed by [28]. Thus, the relationship between the preconsolidation pressure and temperature can be fitted by(9)σp′σp′r=TTrθ,where θ is a thermal parameter and σp′ and σp′r are the preconsolidation pressures at temperature T and the reference temperature Tr, respectively. The present model has no elastic limit, which is different from Perzyna’s overstress method [30].Figure 4
Temperature dependency of preconsolidation pressure.


### 3.3. Coupled Consolidation Analysis

In the following numerical simulations of the conventional oedometer tests, soil-water coupling analysis will be performed. Darcy’s law was adopted for the consolidation process:(10)−∂εv∂t=1+e0γw∂∂zk1+e∂u∂z,where z is the vertical depth, u is the excess pore pressure, and k is the hydraulic conductivity. Actually, k is also influenced 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. Thus, 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(11)k=k010e−e0/ck,where the initial hydraulic conductivity k0 is corresponding to e0 and the permeability coefficient ck can be easily measured from the oedometer test results by plotting e−logk.The above equations were implemented in finite element software PLAXIS Version 8, but in 1D, finite 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.


### 3.4. 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). The results are shown in Figure 5, and the parameters adopted for these simulations are listed in Figure 5(a). The 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.Figure 5
Simulations for 24 h oedometer tests: (a) compression curves at different temperatures; (b) compression behavior at each load increment for the test simulated atT = 80°C.
(a)
(b)Take the simulated test onT = 80°C; for example, the compression behavior for each load increment is presented in Figure 5(b). The simulated thermal creep coefficient ψ = 0.013 agrees well with that obtained from Equation (3) with Tr = 20°C and ψTr = 0.0065.


## 3.1. Adopted One-Dimensional Elastic Viscoplastic Model

First, a time-dependent stress-strain model without the thermal effect needs to be selected as a base [6]. The adopted model is presented briefly in this part.Following the classic elastic viscoplastic approach, the total strain rate contains the elastic and viscoplastic strains rate, that is,(4)ε˙v=ε˙ve+ε˙vvp,where ε˙v represents the total strain rate and the superscripts “e” and “vp” denote the elastic and viscoplastic parts, respectively. The elastic strain rate is expressed as(5)ε˙ve=κ1+e0σ˙v′σv′,where e0 is the initial void ratio, σv′ is the effective vertical stress, and κ is the slope of recompression lines in e−lnσv′ space.For the viscoplastic strain rate, a one-dimensional formulation proposed by Kutter and Sathialingam [17] was adopted based on the creep coefficient ψ:(6)ε˙vvp=ψ1+e0τσv′σv′rλ−κ/ψ,where λ denotes the slope of the normal compression line in e−lnσv′ space; τ is the reference time, and it equals to the duration of each load increment in the oedometer test; and σv′r is the reference stress corresponding to the incremental time τ and increases with the development of the viscoplastic strain according to(7)σv′r=σp′rexp1+e0λ−κεvvp,where σp′r denotes the preconsolidation pressure.The above relationships have been suggested in [18–20] and validated in [6, 10].


## 3.2. Incorporation of the Thermal Effect

The effects of temperature on the stress-strain behavior of clay have been observed in the laboratory [12, 13], which are helpful to discuss the thermal effect 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. These 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 effective stress remains small [14, 23, 24] and the emphasis of this paper, the parameters λ, κ, and e0 will remain constant when clays suffering a change of temperature.The thermal creep coefficient can be incorporated into the basic EVP model directly. By substituting the parameterψ in Equation (6) by Equation (3), the 1D thermal-related viscoplastic strain rate changes to(8)ε˙vvp=ψTr1+e0τTTrCTσv′σv′rλ−κ/ψTrTr/TCT.Actually, the preconsolidation pressureσp′ also varies with temperature. The thermal-dependent behavior of σp′ has been studied from the oedometer tests or isotropic compression tests with variable temperatures. All of the results indicate that σp′ will decrease with an elevated temperature [25–29]. Based on the data collected from literature, Figure 4 plots the relationship between the normalized preconsolidation pressure and temperature. The regression analyses show that it is also reasonable to assume a linear relationship between logσp′/σp′r and logT/Tr, firstly proposed by [28]. Thus, the relationship between the preconsolidation pressure and temperature can be fitted by(9)σp′σp′r=TTrθ,where θ is a thermal parameter and σp′ and σp′r are the preconsolidation pressures at temperature T and the reference temperature Tr, respectively. The present model has no elastic limit, which is different from Perzyna’s overstress method [30].Figure 4
Temperature dependency of preconsolidation pressure.


## 3.3. Coupled Consolidation Analysis

In the following numerical simulations of the conventional oedometer tests, soil-water coupling analysis will be performed. Darcy’s law was adopted for the consolidation process:(10)−∂εv∂t=1+e0γw∂∂zk1+e∂u∂z,where z is the vertical depth, u is the excess pore pressure, and k is the hydraulic conductivity. Actually, k is also influenced 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. Thus, 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(11)k=k010e−e0/ck,where the initial hydraulic conductivity k0 is corresponding to e0 and the permeability coefficient ck can be easily measured from the oedometer test results by plotting e−logk.The above equations were implemented in finite element software PLAXIS Version 8, but in 1D, finite 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.


## 3.4. 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). The results are shown in Figure 5, and the parameters adopted for these simulations are listed in Figure 5(a). The 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.Figure 5
Simulations for 24 h oedometer tests: (a) compression curves at different temperatures; (b) compression behavior at each load increment for the test simulated atT = 80°C.
(a)
(b)Take the simulated test onT = 80°C; for example, the compression behavior for each load increment is presented in Figure 5(b). The simulated thermal creep coefficient ψ = 0.013 agrees well with that obtained from Equation (3) with Tr = 20°C and ψTr = 0.0065.


## 4. Thermal-Dependent EVP Model for Intact Clay

### 4.1. Bonding Effect on Compression Behavior

During the oedometer tests, the difference of the compression curves obtained on intact and reconstituted clay is caused by bonding elimination as shown in Figure6(a) for Wenzhou clay [32]. The structures between soil particles for intact clay will be eliminated gradually during compressing. The 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]. Thus, 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 εvvp, the bond elimination results in the current stress σv′ reaching point D for intact clay. Corresponding to the same viscoplastic strain, we define an intrinsic stress σvi′ on the reconstituted sample. We assume that the difference between the current stress and intrinsic stress is due to the existing of the soil structure, based on which a bonding ratio χ=σv′/σvi′−1 can be proposed. Thus, the current stress σv′ during straining can be expressed as(12)σv′=1+χσvi′.Figure 6
Bonding effect on the compressibility and the evolution of bonding ratio: (a) compression curves for intact and reconstituted Wenzhou clay atT =20°C; (b) illustration of the bonding elimination with viscoplastic strain; (c) evolution of bonding ratio with viscoplastic strain.
(a)
(b)
(c)Initially, the bonding ratioχ=χ0=σp′/σpi′−1. Following the increasing of strain, the bonds are broken gradually and χ decreases from its initial value χ0 ultimately towards zero when the bonds are completely eliminated as plotting in Figure 6(a). According to the definition, 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:(13)χ=χ0e−ζεvvp,where the parameter ζ controls the rate of bonding elimination (ζ = 8.0 for Wenzhou clay in Figure 6(c)). Actually, the intrinsic stress σvi′ 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. Thus, 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.


### 4.2. Model Parameters

The 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 (e0), the intrinsic slope of the compression line (λi), and the slope of the recompression line (κ). The values of λi 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, e0 can be measured initially at the reference temperature.(b)
Parameter related to bonding elimination: the initial bonding ratio (χ0) and the parameter ζ. The value of χ0 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 χ0, high-quality intact samples are needed. The parameter ζ representing the bonding elimination rate can be derived from Equations (12) and (13):(14)ζ=−ln1χ0σv′σvi′−11εvvp,where εvvp is the volumetric viscoplastic strain corresponding to σv′ (Figure 6(b)) at an arbitrary temperature. Thus, ζ can be calculated by selecting a point on the postyield curve.(c)
Parameters of creep: reference thermal creep coefficient (ψTr) and thermal parameter (CT). ψTr can be measured directly from the oedometer tests on reconstituted at a reference temperature (Tr). CT can be obtained by correlating ψTr with temperature.Parameters related to preconsolidation pressure: the reference preconsolidation pressure (σp′r) at the reference temperature Tr and the thermal parameter (θ). σp′r 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 different 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 (wL) expressed as(15)θ=0.1072+0.0008wL.Thus,θ can be obtained by correlating without carrying out the temperature-controlled tests.Thus, ten model parameters (e0, κ, λ, θ, Tr, CT, ψTr,σp′r, χ0, and ζ) are required for the model, and all of the parameters can be determined straightforwardly from the temperature-controlled oedometer test.


## 4.1. Bonding Effect on Compression Behavior

During the oedometer tests, the difference of the compression curves obtained on intact and reconstituted clay is caused by bonding elimination as shown in Figure6(a) for Wenzhou clay [32]. The structures between soil particles for intact clay will be eliminated gradually during compressing. The 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]. Thus, 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 εvvp, the bond elimination results in the current stress σv′ reaching point D for intact clay. Corresponding to the same viscoplastic strain, we define an intrinsic stress σvi′ on the reconstituted sample. We assume that the difference between the current stress and intrinsic stress is due to the existing of the soil structure, based on which a bonding ratio χ=σv′/σvi′−1 can be proposed. Thus, the current stress σv′ during straining can be expressed as(12)σv′=1+χσvi′.Figure 6
Bonding effect on the compressibility and the evolution of bonding ratio: (a) compression curves for intact and reconstituted Wenzhou clay atT =20°C; (b) illustration of the bonding elimination with viscoplastic strain; (c) evolution of bonding ratio with viscoplastic strain.
(a)
(b)
(c)Initially, the bonding ratioχ=χ0=σp′/σpi′−1. Following the increasing of strain, the bonds are broken gradually and χ decreases from its initial value χ0 ultimately towards zero when the bonds are completely eliminated as plotting in Figure 6(a). According to the definition, 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:(13)χ=χ0e−ζεvvp,where the parameter ζ controls the rate of bonding elimination (ζ = 8.0 for Wenzhou clay in Figure 6(c)). Actually, the intrinsic stress σvi′ 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. Thus, 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.


## 4.2. Model Parameters

The 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 (e0), the intrinsic slope of the compression line (λi), and the slope of the recompression line (κ). The values of λi 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, e0 can be measured initially at the reference temperature.(b)
Parameter related to bonding elimination: the initial bonding ratio (χ0) and the parameter ζ. The value of χ0 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 χ0, high-quality intact samples are needed. The parameter ζ representing the bonding elimination rate can be derived from Equations (12) and (13):(14)ζ=−ln1χ0σv′σvi′−11εvvp,where εvvp is the volumetric viscoplastic strain corresponding to σv′ (Figure 6(b)) at an arbitrary temperature. Thus, ζ can be calculated by selecting a point on the postyield curve.(c)
Parameters of creep: reference thermal creep coefficient (ψTr) and thermal parameter (CT). ψTr can be measured directly from the oedometer tests on reconstituted at a reference temperature (Tr). CT can be obtained by correlating ψTr with temperature.Parameters related to preconsolidation pressure: the reference preconsolidation pressure (σp′r) at the reference temperature Tr and the thermal parameter (θ). σp′r 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 different 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 (wL) expressed as(15)θ=0.1072+0.0008wL.Thus,θ can be obtained by correlating without carrying out the temperature-controlled tests.Thus, ten model parameters (e0, κ, λ, θ, Tr, CT, ψTr,σp′r, χ0, and ζ) are required for the model, and all of the parameters can be determined straightforwardly from the temperature-controlled oedometer test.


## 5. Calibration and Validation

### 5.1. Predictions on Wenzhou Clay

The thermal creep coefficientψ is also influenced 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 different temperatures. The input ordinary parameters are the same as those given in Table 1. For intact clay, two parameters χ0 = 3.5 and ζ = 8 were used. Three temperatures with T = 20°C, 50°C, and 80°C are adopted in the simulations.Table 1
Values of model parameters and state variables for selected clays.
clay
λ
Λ
κ
e
0
σ
p
′
r kPa
χ
0
ζ
θ
T
r
ψ
T
r
C
T
k (m/h)
c
k
Wenzhou
0.39
0.037
1.92
30
3.5
8
0.2
70
0.006
0.5
4 × 10−6
0.6
Utby
0.18
0.03
1.98
55
54
15
0.15
20
0.02
0.31
2 × 10−6
0.99
Tokyo bay
0.36
0.05
2.4
79
0.4
7
0.17
20
0.031
—
1 × 10−7
1.2Figure7 shows typical results of the thermal oedometer tests in natural intact clay simulated by the new model:(1)
In Figure7(a), the compression curves in different temperatures are similar to the oedometer test on intact Wenzhou clay (Figure 6(a)). The preconsolidation pressure is influenced significantly by the temperature as presented in Figure 5 on reconstituted clay.(2)
In Figure7(b), the typical curves of strain versus time were reproduced for natural intact clay, where the thermal creep coefficient ψ can be obtained at the end of each load; here, only the simulation results for T = 80°C are presented.(3)
In Figure7(c), the measured ψ at three temperatures were plotted versus the normalized applied stress by the preconsolidation pressure. Here, the preconsolidation pressure is different for the three tests. The 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. The difference between these curves is due to the combined influence of temperature and bonding elimination.(4)
In Figure7(d), the measured ψ for intact and reconstituted clay are compared for the test at T =80°C. The difference between the two curves is due to the existence of bonding and its elimination.Figure 7
Predictions of the thermal oedometer test on Wenzhou clay with different temperatures: (a) compression curves at different temperatures; (b) strain-time curves for the case withT = 80°C; (c) evolution of the thermal creep coefficient with normalized vertical stress; (d) comparisons of the thermal creep coefficient between intact and reconstituted clay.
(a)
(b)
(c)
(d)


### 5.2. Predictions on Utby Clay

Li et al. [34] presented a set of the long-term oedometer tests on Utby clay (6 m depth) with different temperatures for both intact and reconstituted samples. The clay properties are as follows: liquid limit w = 55%, e0 = 1.977, and σp′ = 55 kPa for the intact sample at the reference room temperature T = 20°. The 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 Tr = 20°. The parameter λi = 0.18 is determined based on the results of the reconstituted sample. The parameter κ =0.003 is correlated from the recompression curve. Bonding ratio χ0 = 54 is obtained based on the method described above. The 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 coefficient with temperature. Adopting ψT at T = 20° as a reference, the predicted values of ψT for T=5° agree well with the experimental results with CT = 0.31. Furthermore, an average value of ψTr = 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. The value of ck was equal to e0/2, as suggested by Tavenas et al. [35] based on observations on soft marine clays. All of the parameters are summarized in Table 1.Figure 8
Determination of parameters from the thermal oedometer test: (a) compression parameters atT = 20°; (b) the thermal creep coefficient based on the experiment on the reconstituted sample.
(a)
(b)Figure9 shows the comparison between experimental and simulation results. The predicted compression behavior at different 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 coefficient also agrees well with the experimental results. For the values of vertical stress equal to the preconsolidation pressure, the thermal creep coefficient reaches to the maximum value (Figure 9(b)). It can be concluded that it is necessary to account for the coupling effect of temperature and destructuration for accurate predictions of the thermal compression behavior of the soft clay.Figure 9
Predictions of the thermal oedometer test on intact Utby clay (depth = 6 m) with different temperatures: (a) compression curves; (b) creep coefficient.
(a)
(b)


### 5.3. 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. The sample which is suffered 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 e0 = 2.4. From the experimental results on the sample consolidated at room temperature (Figure 10(a)), the parameter λi = 0.36, κ =0.05, and σp′ = 79 kPa can be obtained. In addition, the reference ψTr = 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). χ0 = 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 correlated based the compression curve, and the consolidation-related parameters k = 1 × 10−7 m/h, ck = 1.2. All of the parameters are collected in Table 1.Figure 10
Determination of parameters from the oedometer tests for the sample consolidated at room temperature: (a) compression curves; (b) creep coefficient versus vertical stress.
(a)
(b)Figure11(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. The 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.Figure 11
Predictions of the oedometer test on Tokyo bay clay consolidated at high temperatures: (a) compression curves; (b) creep coefficient versus normalized vertical stress.
(a)
(b)


## 5.1. Predictions on Wenzhou Clay

The thermal creep coefficientψ is also influenced 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 different temperatures. The input ordinary parameters are the same as those given in Table 1. For intact clay, two parameters χ0 = 3.5 and ζ = 8 were used. Three temperatures with T = 20°C, 50°C, and 80°C are adopted in the simulations.Table 1
Values of model parameters and state variables for selected clays.
clay
λ
Λ
κ
e
0
σ
p
′
r kPa
χ
0
ζ
θ
T
r
ψ
T
r
C
T
k (m/h)
c
k
Wenzhou
0.39
0.037
1.92
30
3.5
8
0.2
70
0.006
0.5
4 × 10−6
0.6
Utby
0.18
0.03
1.98
55
54
15
0.15
20
0.02
0.31
2 × 10−6
0.99
Tokyo bay
0.36
0.05
2.4
79
0.4
7
0.17
20
0.031
—
1 × 10−7
1.2Figure7 shows typical results of the thermal oedometer tests in natural intact clay simulated by the new model:(1)
In Figure7(a), the compression curves in different temperatures are similar to the oedometer test on intact Wenzhou clay (Figure 6(a)). The preconsolidation pressure is influenced significantly by the temperature as presented in Figure 5 on reconstituted clay.(2)
In Figure7(b), the typical curves of strain versus time were reproduced for natural intact clay, where the thermal creep coefficient ψ can be obtained at the end of each load; here, only the simulation results for T = 80°C are presented.(3)
In Figure7(c), the measured ψ at three temperatures were plotted versus the normalized applied stress by the preconsolidation pressure. Here, the preconsolidation pressure is different for the three tests. The 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. The difference between these curves is due to the combined influence of temperature and bonding elimination.(4)
In Figure7(d), the measured ψ for intact and reconstituted clay are compared for the test at T =80°C. The difference between the two curves is due to the existence of bonding and its elimination.Figure 7
Predictions of the thermal oedometer test on Wenzhou clay with different temperatures: (a) compression curves at different temperatures; (b) strain-time curves for the case withT = 80°C; (c) evolution of the thermal creep coefficient with normalized vertical stress; (d) comparisons of the thermal creep coefficient between intact and reconstituted clay.
(a)
(b)
(c)
(d)


## 5.2. Predictions on Utby Clay

Li et al. [34] presented a set of the long-term oedometer tests on Utby clay (6 m depth) with different temperatures for both intact and reconstituted samples. The clay properties are as follows: liquid limit w = 55%, e0 = 1.977, and σp′ = 55 kPa for the intact sample at the reference room temperature T = 20°. The 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 Tr = 20°. The parameter λi = 0.18 is determined based on the results of the reconstituted sample. The parameter κ =0.003 is correlated from the recompression curve. Bonding ratio χ0 = 54 is obtained based on the method described above. The 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 coefficient with temperature. Adopting ψT at T = 20° as a reference, the predicted values of ψT for T=5° agree well with the experimental results with CT = 0.31. Furthermore, an average value of ψTr = 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. The value of ck was equal to e0/2, as suggested by Tavenas et al. [35] based on observations on soft marine clays. All of the parameters are summarized in Table 1.Figure 8
Determination of parameters from the thermal oedometer test: (a) compression parameters atT = 20°; (b) the thermal creep coefficient based on the experiment on the reconstituted sample.
(a)
(b)Figure9 shows the comparison between experimental and simulation results. The predicted compression behavior at different 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 coefficient also agrees well with the experimental results. For the values of vertical stress equal to the preconsolidation pressure, the thermal creep coefficient reaches to the maximum value (Figure 9(b)). It can be concluded that it is necessary to account for the coupling effect of temperature and destructuration for accurate predictions of the thermal compression behavior of the soft clay.Figure 9
Predictions of the thermal oedometer test on intact Utby clay (depth = 6 m) with different temperatures: (a) compression curves; (b) creep coefficient.
(a)
(b)


## 5.3. 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. The sample which is suffered 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 e0 = 2.4. From the experimental results on the sample consolidated at room temperature (Figure 10(a)), the parameter λi = 0.36, κ =0.05, and σp′ = 79 kPa can be obtained. In addition, the reference ψTr = 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). χ0 = 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 correlated based the compression curve, and the consolidation-related parameters k = 1 × 10−7 m/h, ck = 1.2. All of the parameters are collected in Table 1.Figure 10
Determination of parameters from the oedometer tests for the sample consolidated at room temperature: (a) compression curves; (b) creep coefficient versus vertical stress.
(a)
(b)Figure11(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. The 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.Figure 11
Predictions of the oedometer test on Tokyo bay clay consolidated at high temperatures: (a) compression curves; (b) creep coefficient versus normalized vertical stress.
(a)
(b)


## 6. Conclusions

The 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. The determination of the model parameters is straightforward. Numerical simulations have been conducted to examine the predictive ability of the model for the soft clay.Experimental predictions have carried out the thermal oedometer tests at different temperatures. The bonding elimination effect on the evolution of the thermal creep coefficient has been highlighted by comparing predictions with and without considering bonding elimination.The 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.



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*Source: 9827673-2018-10-17.xml*