One-Dimensional Creep Consolidation Model for Peat Soil

Abstract

Peat soil exhibits significant creep deformation, and its consolidation law differs from that of soft soil. This study examines the strain characteristics of peat soils during three stages of consolidation using indoor one-dimensional creep consolidation tests. The results showed that the rebound deformation after the primary consolidation stage and the secondary consolidation stage is equivalent to the deformation seen during the primary consolidation stage, about 1.003 times. However, once the deformation stabilizes, the rebound deformation decreases to 0.32–0.85 times that of the deformation observed during the primary consolidation stage. The elastic and time-independent plastic strains of the peat soil showed two-stage linear changes with ${\ln \mathsf{\sigma }}_{\mathrm{z}}^{\prime }$. When the load was greater than the pre-consolidation pressure, the deformation modulus increases by approximately 2.10 and 1.56 times, respectively. On this basis, this study, for the first time, defines the creep rate according to the strain rate in the tertiary consolidation stage in the strain versus the time curve (${\epsilon }_z~t$). Based on the timeline, a one-dimensional creep consolidation model is established that can accurately predict the strain during the consolidation of the peat soil foundation. The results reveal distinct strain behaviors during each stage and improve the theoretical basis for the study of creep.

1. Introduction

Peat soil is a loose and water-rich aggregate of organic matter formed by the sedimentation of dead plants in swamps and lakes [1,2] and comprises about 5–8% of the total area of the Earth. Peat soil has a high water content [3,4], therefore, its compressibility is significantly higher than that of general soft soil [5]. The unique properties of peat soils present significant challenges in geotechnical engineering, particularly for infrastructure projects such as highways and railroads. In the process of building highways and railroads, according to the requirements of line smoothness, when the peat soil layer cannot be avoided, the shallow peat soil layer can be removed, but for the deep peat soil layer, replacement is difficult and costly. At this time, it is necessary to use stack preloading in order to reduce post-construction settlement. However, due to the unclear deformation properties and stress–strain relationships, the accuracy of settlement calculation for peat foundation is affected, which restricts engineering construction. Therefore, analyzing the settlement patterns of peat soil foundations contributes to the refinement of the soil mechanics theory, while also reducing construction costs and enhancing engineering quality.

The deformation of peat soil is significantly influenced by water and organic matter [6]. MacFarlane and Radforth [7] categorized peat soils as amorphous and fibrous peats and proposed that amorphous peat soil is primarily composed of colloidal particles, whereas fibrous peat soil primarily contains undecomposed plant fibers. According to this classification, Berry and Poskitt [8] proposed two constitutive models. Firstly, based on the rheological model proposed by Gibson and Lo [9], the nonlinear compression of the soil was simulated using a nonlinear spring, and the constitutive relationship of amorphous peat was established, as shown in Figure 1 and Equation (1). After the application of a load, spring P₁ experiences immediate compression, while, due to the existence of the Kelvin element, the compression of spring P₂ occurs at a more gradual pace, leading to deformation that will persist over a period of time, known as creep. Secondly, based on the pore characteristics of peat soil proposed by Adams [10], Berry and Poskitt [8] used the dual Terzaghi consolidation model to simulate the macropores and micropores (Figure 2) and established the constitutive relationship of fibrous peat.

$$\epsilon \left(t\right)=\Delta \sigma \left[a+b\left(1-e^{-\left(\lambda /b\right)t}\right)\right]$$

(1)

where $\Delta \sigma$ = increment in stress;

$t$ = time;

$a$ = consolidation coefficient;

$b$ = coefficient of secondary consolidation;

$\lambda /b$ = secondary consolidation rate factor.

Figure 1. Rheological model of secondary consolidation [9].

Figure 1. Rheological model of secondary consolidation [9].

Figure 2. Berry and Poskitt’s creep model of fibrous peat soil [8].

Figure 2. Berry and Poskitt’s creep model of fibrous peat soil [8].

The theoretical models can describe the deformation characteristics of peat soil in the primary and secondary consolidation stages; however, it cannot accurately describe the law of strain changes with time. Yin and Graham [11] proposed a timeline model for clay, in which the creep strain rate was constant in the normal consolidated state. The strain exhibited a linear relationship with the logarithm of the effective stress and creep strain rate. Den Haan [12] proposed the abc consolidation model of peat one-dimensional compression, assuming a constant relationship between the strain, the effective stress, and the creep strain rate. O’Loughlin [13] proposed that the abc model is not suitable for over-consolidated soils, because the equidistant lines of over-consolidated soils are neither parallel nor linear. Mesri and Ajlouni [3] and Mesri and Castro [14] proposed that peat soil conforms to the $C_{\alpha }/C_c$ law, that is, the ratio of the secondary consolidation coefficient to the compression coefficient is stable, and proposed that the range for peat soil is 0.05–0.07. Lv et al. [15] proposed a peat soil constitutive model that considered the influence of the degree of decomposition based on the triaxial test results and neural network theory. Madaschi and Gajo [16,17] conducted 1D consolidation tests on undisturbed and remolded samples with different organic matter contents and present a new rheological framework for describing the delayed behavior of geologic materials. Yamazoe et al. [18] applied incorporating a time-dependent model for peats to an evaluation of the residual settlement and validated the findings in laboratory tests. Wang et al. [19] proposed a hyperbolic model through data analysis, considering modulus reduction and damping versus shear strain relationships, to analyze the strain law of peat soil. Boumezerane [20] and Yang et al. [21] analyzed the constitutive relationship of peat soil based on the critical state theory. Long [1] proposed that, after calibration, SSC captured well the vertical settlement versus time behavior of the peat.

Research has shown that the deformation of peat soil lasts long and is characterized by viscoelasticity and plasticity. Based on this, some constitutive equations for peat soil have been proposed. Firstly, the constitutive relationship established through Hooke springs, Kelvin elements, etc., can accurately calculate the final strain of peat soil; however, due to the single parameters, its deformation process cannot be accurately described. Secondly, models based on mechanical properties have been the main research direction in recent years. However, due to the lack of consensus on the division of the consolidation process of peat soil, and the fact that most calculation parameters are consistent with soft soil, such models cannot accurately describe the deformation principle of peat soil, and the adaptability of these calculation methods is poor.

In this research, one-dimensional consolidation tests were carried out on four varieties of peat soils in Dali, Yunnan. Utilizing the three-stage consolidation method, the mechanical properties of deformation at varying stages were suggested. Based on this, the elastic line and reference timeline in the timeline model were redefined, and a creep model for peat soil was established. The boundary conditions for the consolidation of peat soil were examined, and a semi-analytical solution for the model was proposed. The computed results were in agreement with the measured data. This model elucidates the deformation characteristics of peat soil, unveils its foundation settlement law, and offers a theoretical foundation for creep research.

2. Materials and Methods

Peat soil samples were collected from three sites in Dali City, Yunnan Province, China. Sampling site ① is located in Xihu Village, Eryuan County, Dali City, at 26°0′55″ north latitude, 100°3′12″ east longitude, and an altitude of approximately 1970 m. After manual excavation of the peat soil layer, the sampling platform was trimmed, and a thin-walled metal sampling bucket (diameter 10 cm, height 20 cm) was used for sampling, as shown in Figure 3.

Sampling site ② is located in Dajiankou Village, Jianchuan County, Yunnan Province, at 99°34′30″ east longitude and 26°30′ north latitude, as shown in Figure 4a. There was a natural stream flowing on the west side of the sampling location, and there was a black peat soil layer approximately 0.5 m below the surface. The sampling method used was the same as that used for Sampling site ①.

Sampling site ③ is located in Dajiankou Village, Jianchuan County, Yunnan Province, at 99°34′12″ east longitude and 26°30′ north latitude. Figure 4b shows the sampling points, which were on a plain covered with vegetation. There was a black peat soil layer approximately 1 m below the surface, which was rich in plant roots. The sampling method used was the same as that used for Sampling site ①.

Because the sampling position is lower than the horizontal plane, the peat soil was almost saturated. After the original samples were removed, they were wrapped in plastic to ensure air isolation and to prevent water loss. The main soil parameters were measured according to CN-JTG 3430-2020 [22] and are listed in

Table 1. Soil parameters and loading sequence.

Specimen	Loss on Ignition (%)	Water Content (%)	Wet Density (g/cm3)	Specific Gravity	Load Sequence (kPa)
Peat	70.5–90.2	502.0–717.3	0.88–1.09	1.71–1.72	Each type of specimen is loaded with the following loads: 12.5–25–50–100 a 12.5–25–50–100–200 a 12.5–25–50–100–200–400 a 12.5–25–50–100–200–400–800 a
Highly organic peaty soil	40.6–59.4	261.9–611.9	0.91–1.11	1.81–1.84
Medium organic peaty soil	25.1–39.7	203.6–1100.0	0.97–1.18	1.69–2.01
Low organic peaty soil	12.08–23.73	54.5–222.7	0.83–1.43	2–2.08

^a Last level of loading sequence is the target load; and the loading times of parallel pattern are listed in Table 2.

.

The consolidation test assessed the compressive behavior of the soil samples under various loads and lateral constraints using lever loading. After unpacking the soil taken to the laboratory, we used a ring knife (with Vaseline coating on the inner wall) to cut the undisturbed sample in a direction perpendicular to the formation. The WG type single lever consolidation instrument was used for step-by-step loading, and the load sequence is shown in Table 1. The choice of load sequence primarily aims to maintain a loading ratio of 1, and also to encompass the foundation loads. This study was conducted according to the peat soil one-way consolidation test method proposed by Peng et al. [23]. The sample height was 30 mm, the ambient temperature was controlled to be similar to the formation temperature, and, simultaneously, the sample was soaked in 1% thymol solution.

According to the method proposed by Casagrande, the pre-consolidation pressures of the four peaty soils were 94.3, 53.9, 60.2, and 148.4 kPa. The loads in this study were 100, 200, 400, and 800 kPa. According to the three-stage division method of peat soil consolidation proposed by Feng et al. [24] (Figure 5), the consolidation process of peat soil can be divided into the primary consolidation stage with fast linear deformation, the secondary consolidation stage with variable speed deformation, and the tertiary consolidation stage with slow linear deformation growth. Multiple parallel samples were loaded simultaneously, and, after each stage, one sample was removed and placed in a sealed bag, and its rebound deformation was measured after 7 d. In accordance with Feng et al. [24], the completion times of the primary and secondary consolidation stages of the peat soil used in this study are listed in

Table 2. Completion time of consolidation stages.

Load Value (kPa)	Consolidation Stages	Completion Time of Different Stages (min)
		Peat	Highly Organic Peaty Soil	Medium Organic Peaty Soil	Low Organic Peaty Soil
100	primary consolidation stage	94.6	97.8	46.8	95
	secondary consolidation stage	3901.1	3018.5	2600.8	1800
200	primary consolidation stage	85.4	92.3	42.8	70
	secondary consolidation stage	3174.0	3572.4	1652.4	1350
400	primary consolidation stage	93.1	88.8	47.1	90
	secondary consolidation stage	2955.0	3534.3	1659.9	1325
800	primary consolidation stage	86.0	87.2	51.8	127
	secondary consolidation stage	3560.5	3430	1474.5	1215

, and the deformation process ends when the deformation is less than 0.01 mm/d. According to Table 2, the primary consolidation stage lasts between 42.8 min and 94.6 min, while the secondary consolidation stage lasts much longer than the primary stage. The duration of the secondary consolidation stage decreases as the load increases and the organic matter content decreases, with a minimum of 1215 min and a maximum of 3901 min.

3. Results and Discussion

3.1. Deformation Properties of Peat Soil at Different Stages

The 7 d rebound deformation of the peat soil after loading to the end of the three consolidation stages was measured ($T_P:$ Unloading after the primary consolidation stage; $T_s:$ Unloading after secondary consolidation stage; $T_t:$ Unloading after the third consolidation stage), and the deformation properties of each stage of peat soil consolidation were analyzed, considering the deformation curve at 100 kPa load as an example, as shown in Figure 6.

Figure 6 shows that the 7 d average rebound rates of the four peat soils in the primary and secondary consolidation stages were similar and were greater than the rebound rates in the third consolidation stage. The same rate indicates the same rebound deformation. The ratio of the rebound deformation of the four peat soils to the compression in the primary consolidation stage is expressed as Equation (2). The calculation results are presented in Figure 7.

$$R_{b,i}={\displaystyle \frac{d_{re,i}}{d_{p,i}-d_{0,i}}}$$

(2)

where $R_{b,i}$ is the ratio of the rebound of the i-th sample 7 d after unloading to the compression of the last level of loading in the primary consolidation stage;

$d_{re,i}$ = rebound deformation of the i-th sample after unloading for 7 d (mm);

$d_{0,i}$ = deformation of the i-th sample before the last load level was applied (mm);

$d_{p,i}$ = deformation of i-th sample at end of the primary consolidation stage under the last load (mm).

Figure 7 shows that the $R_b$ at the end of the primary and secondary consolidation stages was close to 1, and the average value of the four peat soils was 1.003, indicating that the rebound deformation after the primary and secondary consolidation stages was almost equal to that of the primary consolidation stage deformation. The range of $R_b$ after the third consolidation stage was 0.32–0.85, indicating that the rebound deformation of the sample in the tertiary consolidation stage was smaller than that of the compression deformation in the primary consolidation stage.

Therefore, the primary consolidation stage of the peat soil produces elastic deformation that can be recovered, whereas plastic deformation occurs during the secondary consolidation stage. In the third consolidation stage, the deformation lasts longer and is primarily plastic. Owing to the decomposition of the organic matter and the discharge of bound water [4,25], the peat soil skeleton is destroyed, and partially recoverable elastic deformation is transformed into non-recoverable plastic deformation.

3.2. Viscoelasticplastic Constitutive Relation of Peat Soil

Graham and Yin [26] and Yin and Graham [11] proposed the theory of using equivalent time to calculate viscous strain. However, due to the particularity of the deformation process of peat soil, the deformation parameters of peat soil are not applicable to the timeline theory. Therefore, the stress and strain of the peat soil were analyzed here.

The consolidation process of peat soil can be divided into the primary consolidation stage of elastic deformation, the secondary consolidation stage of plastic deformation (can be divided into time-independent and time-dependent parts), and the tertiary consolidation stage of viscous deformation. The strain can be expressed by Equations (3)–(5).

$${\epsilon }_z={\epsilon }_z^e+{\epsilon }_z^{sp}+{\epsilon }_z^{tp}$$

(3)

$${\epsilon }_z^{vp}={\epsilon }_z^{sp}+{\epsilon }_z^{tp}$$

(4)

$${\epsilon }_z^{ep}={\epsilon }_z^e+{\epsilon }_z^{sp}$$

(5)

where ${\epsilon }_z^e$ = elastic strain (recoverable strain after unloading);

${\epsilon }_z^{sp}$ = time-independent plastic strain (unrecoverable strain unaffected by loading time);

${\epsilon }_z^{tp}$ = viscous strain (unrecoverable strain affected by loading time);

${\epsilon }_z^{vp}$ = viscoplastic strain (unrecoverable strain after unloading);

${\epsilon }_z^{ep}$ = time-independent elastic–plastic strain.

The elastic strain of the peat soil primarily occurred in the primary consolidation stage because the skeleton of the soil was destroyed in the tertiary consolidation stage, and the elastic strain was not equal to the rebound. Therefore, the elastic strain was calculated by accumulating the strain in the primary consolidation stage, as shown in Equation (6) and Figure 8.

$${\epsilon }_z^e={\displaystyle \sum }_{i=1}^n{\epsilon }_{zi}^e$$

(6)

where ${\epsilon }_z^e$ = elastic strain;

${\epsilon }_z^e$ = elastic strain of the i-th load.

Figure 8. Schematic of strain accumulation curve.

Figure 8. Schematic of strain accumulation curve.

The secondary consolidation stage of the peat soil produces plastic strain, including time-independent plastic strain and time-dependent plastic strain (viscous strain). The time-independent plastic strain was calculated using the strain accumulation method. The rate of viscous strain ($R_{ter}$) is the slope of the stress–strain curve during the third consolidation stage. According to Figure 5, $R_{ter}$ changed insignificantly with time during the loading process. Therefore, the time-independent plastic strain can be calculated by subtracting the viscous strain from the strain in the secondary consolidation stage, as expressed in Equation (7).

$${\epsilon }_{zi}^{sp}={\epsilon }_{szi}-R_{ter,i}\left(T_{sec,i}-T_{pri,i}\right)$$

(7)

where ${\epsilon }_{zi}^{sp}$ = time-independent plastic strain of the i-th load;

${\epsilon }_{szi}$ = strain in secondary consolidation stage of the i-th load;

$R_{ter,i}$ = viscous strain rate of the i-th load (/min);

$T_{pri,i}$ = completion time of the primary consolidation stage for the i-th load (min);

$T_{sec,i}$ = completion time of the secondary consolidation stage for i-th load (min).

Figure 9 shows the strain curves of the peat, highly organic peaty soil, and medium organic peaty soil. The consolidation process of low organic peaty soil is divided into two stages when the load is less than the pre-consolidation pressure. At this time, the strain is divided into the strains in the primary and secondary consolidation stages. When the load is greater than the pre-consolidation pressure, it is divided into three stages [27]. Therefore, the strains in low organic peaty soils must be discussed separately.

Figure 9 shows that both the elastic and time-independent plastic strains of the peat soil exhibit a two-stage linear relationship with $\ln {\sigma }_z^{\prime }$. When the load is less than the pre-consolidation pressure, the elastic and time-independent plastic strains increase with the increase in $\ln {\sigma }_z^{\prime }$ at a relatively low rate $\left(\kappa , \zeta \right)$. The strain in the primary and secondary consolidation stages of the low organic peat soil also have a linear relationship with $\ln {\sigma }_z^{\prime }$. When the load is greater than the pre-consolidation pressure, the strain change law of the low organic peat soil is consistent with that of the other peat soils. At this time, the strain modulus of the peat soil increases significantly, and its slope is represented by ${\kappa }^{\prime }, \zeta ’$. Therefore, the elastic strain can be expressed as follows:

$${\epsilon }_z^e={\epsilon }_{z0}^e+\kappa \ln \left({\displaystyle \frac{{\sigma }_z^{\prime }}{{\sigma }_{zref}^{\prime }}}\right)$$

(8)

where ${\sigma }_{zrefi}^{\prime }$ is the reference effective stress (kPa). When the load is less than the pre-consolidation pressure, it is considered as the initial effective stress ${\sigma }_{z0}^{\prime }$; and when the load is greater than the pre-consolidation pressure, it is considered as the pre-consolidation stress ${\sigma }_{z{P}_c}^{\prime }$;

${\epsilon }_{z0}^e$ = strain value when ${\sigma }_z^{\prime }={\sigma }_{zref}^{\prime }$;

$\kappa$ = slope of ${\epsilon }_z^e~\ln {\sigma }_z^{\prime }$. When the load is greater than the pre-consolidation pressure, $\kappa$ is considered as $\kappa$’.

The time-independent plastic strain can be expressed as follows:

$${\epsilon }_z^{sp}={\epsilon }_{z0}^{sp}+\zeta \ln \left({\displaystyle \frac{{\sigma }_z^{\prime }}{{\sigma }_{zref}^{\prime }}}\right)$$

(9)

where ${\epsilon }_{z0}^{sp}$ = strain value at ${\sigma }_z^{\prime }={\sigma }_{zref}^{\prime }$;

$\zeta$ = slope of ${\epsilon }_z^{sp}~\ln {\sigma }_z^{\prime }$. When the load is greater than the pre-consolidation pressure, $\zeta$ is considered as $\zeta$’.

According to Equations (8) and (9), the time-independent elastic–plastic strain can be expressed as Equation (10). The strain diagram of the peat soil is shown in Figure 10.

$${\epsilon }_z^{ep}={\epsilon }_{z0}^{ep}+\left(\kappa +\zeta \right)\ln \left({\displaystyle \frac{{\sigma }_z^{\prime }}{{\sigma }_{zref}^{\prime }}}\right)={\epsilon }_z^{ep}+\lambda \ln \left({\displaystyle \frac{{\sigma }_z^{\prime }}{{\sigma }_{zref}^{\prime }}}\right)$$

(10)

$$\lambda =\kappa +\zeta$$

(11)

Generally, the viscous strain of soft soil is calculated according to the secondary consolidation coefficient in the $d-\log t$ curve. For peat soil, the viscous strain rate is the strain rate of the tertiary consolidation stage and can be expressed as Equation (12).

$${\epsilon }_z^{tp}=\eta \left(t-t_{ref}\right)=\eta t_e$$

(12)

$$t=t_{ref}+t_e$$

(13)

where ${\epsilon }_z^{tp}$ = viscous strain;

$\eta$ = slope of the strain curve ${\epsilon }_z~t$ in the tertiary consolidation stage (viscous strain stage), as shown in Figure 11;

$t_{ref}$ = reference time for the onset of viscous strain (min);

$t_e$ = creep duration, that is, the equivalent time (min).

Therefore, the modified nonlinear rheological model of the peat soil can be expressed as shown in Figure 12.

Figure 11 shows that the viscous strain of the peat soil grows at a stable low rate for a long time, and its creep rate hardly changes. Therefore, the difference between the nonlinear rheological model of the peat and the soft soils is that its viscous strain rate (${\dot{\epsilon }}_{tz}^{tp}$) may be equal to the viscous strain rate (${\dot{\epsilon }}_{tzref}^{tp}$) of the normal consolidation line, that is, ${\dot{\epsilon }}^{tp}\ge {\dot{\epsilon }}_{ref}^{tp}$ on the upper side of the normal consolidation line, with ${\dot{\epsilon }}^{tp}\le {\dot{\epsilon }}_{ref}^{tp}$ on the lower side, as shown in Figure 12.

According to Equations (3)–(13), the viscoelastic–plastic constitutive equation for the peat soil is as follows:

$${\epsilon }_z={\epsilon }_z^e+{\epsilon }_z^{sp}+{\epsilon }_z^{tp}={\epsilon }_z^{ep}+{\epsilon }_z^{tp}={\epsilon }_{z0}^{ep}+\lambda \ln \left({\displaystyle \frac{{\sigma }_z^{\prime }}{{\sigma }_{zref}^{\prime }}}\right)+\eta t_e$$

(14)

Equation (14) is the deformation calculation formula for peat soil, which was obtained based on the timeline model proposed by Yin and Graham. Parameters $t_e$, ${\epsilon }_{z0}^{ep}$, and ${\sigma }_{zref}^{\prime }$ are discussed in Section 3.2.1 and Section 3.2.2, whereas $\kappa$, $\lambda$, and $\eta$ are determined from Figure 9.

3.2.1. Calculation Method for ${\mathrm{t}}_{\mathrm{e}}$

The reference time $t_e$ was calculated according to the method proposed by Yin and Graham [11] by substituting the parameters of the peat soil. The loading path is instantaneously loaded from points $n$ to ${\left(n+1\right)}^{\prime }$, and it then creeps to $\left(n+1\right)$, as shown in Equation (15).

$$t_{{e,\left(n+1\right)}^{\prime }}={\displaystyle \frac{1}{\eta }}\left({\epsilon }_{z,n}-{\epsilon }_{z0}^{ep}\right)+{\displaystyle \frac{\kappa }{\eta }}\ln \left({\displaystyle \frac{{\sigma }_{z,\left(n+1\right)}^{\prime }}{{\sigma }_{z,n}^{\prime }}}\right)-{\displaystyle \frac{\lambda }{\eta }}\ln \left({\displaystyle \frac{{\sigma }_{z,\left(n+1\right)}^{\prime }}{{\sigma }_{zref}^{\prime }}}\right)$$

(15)

Because point $n$ is instantaneously loaded to ${\left(n+1\right)}^{\prime }$, $t_{{e,\left(n+1\right)}^{\prime }}=t_{e,n}$, and the equivalent time $t_{e,\left(n+1\right)}$ at point $\left(n+1\right)$ can be obtained by adding the equivalent time $t_{e,\left(n+1\right)}$ of point $n$ to $\Delta t$, where $\Delta t$ is the duration of viscoplastic deformation, as shown the Equation (16).

$$t_{e,\left(n+1\right)}=t_{e,n}+\Delta t$$

(16)

Therefore,

$$t_{e,\left(n+1\right)}=\Delta t+{\displaystyle \frac{1}{\eta }}\left({\epsilon }_{z,n}-{\epsilon }_{z0}^{ep}\right)+{\displaystyle \frac{\kappa }{\eta }}\ln \left({\displaystyle \frac{{\sigma }_{z,\left(n+1\right)}^{\prime }}{{\sigma }_{z,n}^{\prime }}}\right)-{\displaystyle \frac{\lambda }{\eta }}\ln \left({\displaystyle \frac{{\sigma }_{z,\left(n+1\right)}^{\prime }}{{\sigma }_{zref}^{\prime }}}\right)$$

(17)

According to a previous analysis, $\Delta t$ is the sum of the secondary and tertiary consolidation stages.

3.2.2. Calculation Method for ${\mathsf{\epsilon }}_{\mathrm{z}0}^{\mathrm{ep}},{\mathsf{\sigma }}_{\mathrm{zref}}^{\prime }$

If the initial elastic–plastic strain ${\epsilon }_{z0}^{ep}$ is considered zero, the initial reference stress ${\sigma }_{zref}^{\prime }$ is a point on the normal consolidation line, assuming that point $m$ is on the normal consolidation line, according to Equation (9). Equation (18) can then be obtained as follows:

$${\epsilon }_{mz}^{ep}=\lambda \ln \left({\displaystyle \frac{{\sigma }_m^{\prime }}{{\sigma }_{zref}^{\prime }}}\right)$$

(18)

The elastic–plastic deformation under any load can be calculated using Equation (9) and substituting it into Equation (17) to solve the initial reference stress.

3.3. One-Dimensional Nonlinear Rheological Consolidation Model

Yu [28] proposed the use of the initial effective stress to represent the stress state of a foundation at different depths. Based on this, a one-dimensional nonlinear rheological consolidation equation for peat soil was derived, assuming that the foundation is as shown in Figure 13.

The strain in the peat soil can be calculated using Equation (14), and ${\sigma }_{zref}^{\prime }$ can be considered the initial effective stress, that is, the self-weight stress, which can be calculated as follows:

$${\sigma }_{z0i}^{\prime }={\displaystyle \frac{h_i}{2}}\left(\gamma -{\gamma }_w\right)+{\displaystyle \sum }_1^{i-1}{h}_i\left(\gamma -{\gamma }_w\right)$$

(19)

where $h_i$ = thickness of the soil layer, (m);

${\sigma }_{z0i}^{\prime }$ = initial effective stress of the i-th layer (kPa);

$\gamma$ = gravity of the soil;

${\gamma }_w$ = gravity of the water.

Assuming that the initial elastic–plastic strain is zero, Equation (14) can be simplified as follows:

$${\epsilon }_z=\lambda \ln \left({\displaystyle \frac{{\sigma }_z^{\prime }}{{\sigma }_{zref}^{\prime }}}\right)+\eta t_e$$

(20)

Mesri and Ajlouni [3] proposed that the void ratio and permeability coefficient of peat soil satisfy the $e-\log k_v$ relationship, as follows:

$$e-e_0=C_k\log {\displaystyle \frac{k_v}{k_{v0}}}$$

(21)

where $e$ = void ratio;

$e_0$ = initial void ratio;

$k_v$ = vertical permeability coefficient (cm/s);

$k_{v0}$ = vertical permeability coefficient when the void ratio is $e_0$, (cm/s);

$C_k$ = permeability coefficient variation index, that is, the slope of the $e-\log k_v$.

The consolidation equation was established according to Darcy’s law, as follows:

$${\displaystyle \frac{1}{{\gamma }_w}}{\displaystyle \frac{\partial }{\partial z}}\left(k{\displaystyle \frac{\partial u}{\partial z}}\right)=-{\displaystyle \frac{\partial {\epsilon }_z}{\partial t}}$$

(22)

where $u$ = excess pore water pressure, (kPa);

Therefore, Equation (20) is modified as follows:

$${\epsilon }_z=\lambda \ln \left({\displaystyle \frac{{\sigma }_z^{\prime }}{{\sigma }_{z0i}^{\prime }}}\right)+\eta t_e$$

(23)

According to Equation (21), the relationship between the strain and permeability coefficient is as follows:

$${\epsilon }_z=-{\displaystyle \frac{C_k}{\left(1+e_0\right)\ln 10}}\ln {\displaystyle \frac{k_v}{k_{v0}}}$$

(24)

According to Equations (23) and (24), the permeability coefficient can be calculated as follows:

$$k_v=k_{v0}{\left({\displaystyle \frac{{\sigma }_z^{\prime }}{{\sigma }_{z0i}^{\prime }}}\right)}^{{\displaystyle \frac{\lambda }{A}}}·e^{{\displaystyle \frac{\eta t_e}{A}}}$$

(25)

where $A=-C_k/\left[\left(1+e_0\right)\ln 10\right]$.

The $k_{v0}$ in the formula can be calculated simultaneously with the $e-\log \sigma$’ relationship proposed by Mesri and Rokhsar [29] or calculated by substituting $t_e=0$ in Equation (24).

According to the one-way consolidation test, the volume compressibility $m_v$ can be calculated as follows:

$$m_v={\displaystyle \frac{\partial {\epsilon }_z}{\partial {\sigma }_z^{\prime }}}$$

(26)

Considering Equations (23) and (26), $m_v$ can be calculated as follows:

$$m_v={\displaystyle \frac{\partial {\epsilon }_z}{\partial {\sigma }_z^{\prime }}}={\displaystyle \frac{\lambda }{{\sigma }_z^{\prime }}}+\eta {\displaystyle \frac{1}{d{\sigma }_z^{\prime }}}={\displaystyle \frac{\lambda }{{\sigma }_z^{\prime }}}+\eta {\displaystyle \frac{1}{{\dot{\sigma }}_z^{\prime }}}$$

(27)

where ${\dot{\sigma }}_z^{\prime }$ = rate of stress change.

Based on the assumption of complete saturation and incompressibility of water, ${\dot{\sigma }}_z^{\prime }$ is calculated as follows:

$${\sigma }_z^{\prime }\left(t\right)=\sigma \left(t\right)-u\left(t\right)=\left(q\left(t\right)+{\sigma }_{z0i}^{\prime }\right)-u\left(t\right)$$

(28)

where $\sigma \left(t\right)$ = stress over time, (kPa);

$u\left(t\right)$ = pore water pressure over time (kPa);

$q\left(t\right)$ = uniformly distributed force over time (kPa).

Substituting Equations (25), (27), and (28) into Equation (22), the one-dimensional nonlinear creep consolidation equation for the peat soil is calculated as follows:

$${\displaystyle \frac{k_v}{{\gamma }_w}}·{\left({\displaystyle \frac{e^{\eta t_e}}{\sigma {\prime }_{z0i}^{\lambda }}}\right)}^{{\displaystyle \frac{1}{A}}}·{\displaystyle \frac{\partial }{\partial z}}\left[{\left(q\left(t\right)+{\sigma }_{z0i}^{\prime }-u\left(t\right)\right)}^{{\displaystyle \frac{\lambda }{A}}}·{\displaystyle \frac{\partial u}{\partial z}}\right]={\displaystyle \frac{\lambda }{q\left(t\right)+{\sigma }_{z0i}^{\prime }-u\left(t\right)}}·\left({\displaystyle \frac{\partial u}{\partial t}}-{\displaystyle \frac{\partial q}{\partial t}}\right)-\eta$$

(29)

According to Figure 13, the boundary conditions are as follows:

$$\mathrm{when} z=0, u=0;$$

(30)

$$\begin{gathered} \mathrm{when} z=\mathrm{H}, u=0 \left(top surface is permeable\right) \\ \mathrm{or} {\displaystyle \frac{\partial u}{\partial t}}=0 \left(\mathrm{top} \mathrm{surface} \mathrm{is} \mathrm{impermeable}\right) \end{gathered}$$

(31)

$$\mathrm{when} t=0,u=q\left(0\right)$$

(32)

3.4. Semi-Analytical Solution

For discretization of formation thickness and time, the following conditions are assumed:

(1) The stratum with thickness $H$, as shown in Figure 13, is divided into $n$ layers on average, assuming that the internal properties of each layer are uniform, and denotes each soil layer with subscript i.

(2) The average soil layer is assumed to satisfy Terzaghi’s one-dimensional consolidation theory (Equation (33)) per unit of time.

$$C_{vi}{\displaystyle \frac{{\partial }^2{u}_i}{\partial z^2}}={\displaystyle \frac{\partial u_i}{\partial t}}$$

(33)

where $C_{vi}$= consolidation coefficient of the i-th soil layer at a certain moment, $C_{vi}=k_{vi}/\left({\gamma }_w{m}_{vi}\right)$;

$u_i$ = pore water pressure of the i-th soil layer at a certain moment.

In the time period from $t_{n-1}$ to time $t_n$, the effective stress is approximately constant, and ${\sigma }_{zi}^{\prime }\left(t_{n-1}\right)$ can represent the stress in the specific time period. The effective stress in that time period can be calculated as follows:

$${\sigma }_{zi}^{\prime }\left(t_{n-1}\right)=q\left(t_{n-1}\right)+{\sigma }_{z0i}^{\prime }-u_i\left(t_{n-1}\right)$$

(34)

The equivalent time at time $t_{n-1}$ can be calculated as follows:

$$t_{ei}\left(t_{n-1}\right)=\Delta t_{\left(n-1\right)}+{\displaystyle \frac{1}{\eta }}{\epsilon }_{z,n}+{\displaystyle \frac{\kappa }{\eta }}\ln \left({\displaystyle \frac{{\sigma }_z^{\prime }\left(t_{n-1}\right)}{{\sigma }_z^{\prime }\left(t_{n-2}\right)}}\right)-{\displaystyle \frac{\lambda }{\eta }}\ln \left({\displaystyle \frac{{\sigma }_z^{\prime }\left(t_{n-1}\right)}{{\sigma }_{zi}^{\prime }}}\right)$$

(35)

where $\Delta t_{\left(n-1\right)}$ = time interval from $t_{n-1}$ to $t_n$.

According to the permeability continuity condition of the boundary of the multilayer foundation,

$$u_i=u_{i+1},k_{vi}{\displaystyle \frac{\partial u_i}{\partial z}}=k_{v\left(i+1\right)}{\displaystyle \frac{\partial u_{\left(i+1\right)}}{\partial z}},\left(i=1,2,3,4\dots \dots \right)$$

(36)

Xie and Pan [30] and Yu [28] proposed a solution method for a multilayer linear elastic foundation, according to which the solution to Equation (33) can be calculated as follows:

$$u_i={\displaystyle \sum }_{m=1}^{\infty }{C}_m{g}_{mi}\left(z\right)e^{-{\beta }_{m}t},\left(i=1,2,3,4\dots \dots \right)$$

(37)

$${\beta }_m={\lambda }_m^2{C}_{v1}/H^2$$

(38)

where $C_m$, $g_{mi}\left(z\right)$, ${\beta }_m$, and ${\lambda }_m$ = undetermined coefficients.

The calculation parameters are listed in

Table 3. Calculation parameters of semi-analytical solution.

	$\mathit{\lambda }$ a	$\mathit{\lambda }\prime$ b	$\mathit{\Psi }\prime$	${\mathit{k}}_{\mathit{v}} \left(\mathbf{c}\mathbf{m}/\mathbf{s}\right)$
Peat	0.051	0.116	0.116	9.42 × 10−5
Highly organic peaty soil	0.090	0.134	0.134	3.32 × 10−5
Medium organic peaty soil	0.111	0.153	0.153	8.45 × 10−5
Low organic peaty soil	0.020	0.100	0.100 c	8.43 × 10−5

^{a, b} When the load is less than the pre-consolidation pressure, we use $\lambda$, and when the load is greater than the pre-consolidation pressure, we use ${\lambda }^{\prime }$; ^c creep rate ${\Psi }^{\prime }$ of low organic peaty soil is 0 when the load is less than the pre-consolidation pressure and takes the value shown in Table 3 when the load is greater than the pre-consolidation pressure.

. The equivalent time $t_e\left(t_n\right)$ is calculated using Equation (35), and the effective stress is taken as the sum of the load and self-weight stress, and is calculated as follows:

$${\sigma }_{zi}^{\prime }=P+{\sigma }_{z0i}^{\prime }$$

(39)

where $P$ = Load (kPa).

Figure 14 shows the strains of the four typical peat soils calculated using the semi-analytical method. As shown in the figure, the nonlinear rheological consolidation model proposed in this study accurately predicted the strain curve of the peat soil. A small amount of error mainly occurs in the secondary consolidation stage, indicating that the calculated strain curve is not sufficiently smooth, and the error increases with the decrease in the organic matter content. The main reason for this error is that the calculation of the time-independent elastic–plastic and viscous strains is controlled by different moduli. In the calculation, the two moduli changed significantly during the secondary consolidation stage; therefore, errors occurred. Because this error only had a minor impact on the analysis, it was not investigated.

4. Conclusions

In this study, the characteristics of strain at different stages of peat soil consolidation were analyzed, the relationship between the strain and stress was explored, and a one-dimensional creep consolidation model of peat soil was established. The main conclusions are as follows.

(1) During the consolidation of the peat soil, the primary consolidation stage mainly produces elastic deformation, whereas the secondary consolidation stage mainly produces plastic deformation. After unloading, following the initial two stages, the rebound deformation is 1.003 times that of the deformation observed during the primary consolidation stage. In the tertiary consolidation stage, which lasts longer and has a slower deformation rate, viscous deformation mainly occurs. The rebound deformation decreases to 0.32 to 0.85 times that of the primary consolidation stage.

(2) Both the elastic and the time-independent plastic strains of the peat soil showed two-stage linear changes with $\ln {\sigma }_z^{\prime }$. When the load is greater than the pre-consolidation pressure, the deformation modulus increases by approximately 2.10 and 1.56 times, respectively.

(3) A time-dependent viscoelastoplastic constitutive model of peat soil is proposed. The elastic line and the reference time line in the rheological model of peat soil are defined by the staged strain accumulation curve, and the viscous strain rate of peat soil is defined by the slope of the ${\epsilon }_z~t$ curve.

(4) Based on the stress–strain relationship, a one-dimensional creep consolidation model of peat soil was established and verified using a semi-analytical solution. The model can accurately predict consolidation deformation and provide a theoretical basis for modeling peat soil foundation.

The one-dimensional creep consolidation model proposed in this paper clarifies the deformation characteristics of peat soil and uncovers the settlement patterns of the foundation. This model is not only directly applicable to engineering practices, but also offers theoretical approaches for software simulation and fresh insights for the investigation of high-creep soft soil. However, as the tests in this study only focus on four typical peat soils in the Yunnan region, further research is needed to explore the applicability to peat soils with different regions and compositions. Meanwhile, the anisotropy of peat soil can be further studied.