One-Dimensional Strain Research of Coral Mud Based on a Modified Burgers Model Considering Stress History

Abstract

Coral mud is a special rock and sediment mass and is widely distributed in the South China Sea. Studying the deformation of coral mud is important for infrastructure development in the South China Sea. When choosing a model to describe the long-term deformation of coral mud, it is difficult for a simple nonlinear deformation model to accurately and universally describe the complex deformation processes of the sediment; a complex model is too time-consuming and difficult to apply to practical engineering. In this article, based on a classical element model, the Burgers model, certain elements are modified in combination with one-dimensional oedometer tests under normally consolidated situations. Then, combined with an unloading and reloading test, the modified Burgers model is further improved, and a modified Burgers model considering stress history is obtained. The modified Burgers model considering the stress history only has four parameters, all of which have practical physical significance, which makes the model easy to use. Different loading times and cyclic loading and unloading tests prove that the model has good stability and can, not only simulate the deformation characteristics of sediment, but can also provide good variation rules for the parameters.

1. Introduction

Corals grow widely in tropical oceans between the Tropic of Cancer and the Tropic of Capricorn. When coral groups die, their skeletons and shells accumulate together, forming inanimate marine rock and sediment over a long period of geological processes. Due to the different settling speeds of sediment particles with different sizes, various sediment layers are formed under the seabed. Such layers can be divided into a coral sand layer and a coral mud layer, according to the general particle size. There are several-meter-thick coral mud layers on the seabed of the South China Sea. It is of great significance to carry out one-dimensional loading, unloading, and reloading characteristic analyses of coral mud for reclamation projects and island and reef construction in the South China Sea.

The formation principle of coral mud and coral sand is the same, and both are light yellow, have extremely high calcium carbonate contents, and their particles have marked edges and corners. At present, there have been many studies on the static and dynamic characteristics of coral sand in the South China Sea [1,2]. However, few studies have been carried out on coral mud. In fact, although coral mud has the particle-size characteristics of clay, it has an extremely high permeability and many properties similar to those of coral sand [3]. Compared with coral sand, coral mud has cohesion and a high level of long-term deformation. Therefore, the study of coral mud is not only of practical engineering significance but also a link for a unified model of clay and sand.

The deformation of sediment under constant stress has an obvious nonlinear relationship with time, and many studies have been carried out on this nonlinear deformation. Terzaghi’s contribution to the principle of effective stress and consolidation theory [4,5] marked the beginning of modern soil mechanics. It is generally believed that the nonlinear deformation of a saturated sediment under the upper load consists of two parts; namely, a primary consolidation deformation due to the increase in effective stress and a secondary consolidation deformation due to the constant effective stress [6]. The increasing effective stress stage is usually called the primary consolidation stage, and the stage after this, where the effective stress is constant but the deformation continues to develop, is called the secondary consolidation stage. Through a large number of tests, scholars have found that for clay, the secondary consolidation deformation under the action of constant effective stress accounts for a high proportion of the total deformation, which cannot be ignored. Some tests have even proven that this part of deformation lasts for decades and continues to develop [7,8]. Ladd et al. [9] first asked whether there is secondary consolidation deformation in the primary consolidation stage when the excess pore-water pressure dissipates. There are two different assumptions about the secondary consolidation deformation of sediment in answer to this question; namely, hypothesis A and hypothesis B. Hypothesis A assumes that the secondary consolidation deformation only occurs after the primary consolidation stage, while hypothesis B considers that the secondary consolidation deformation occurs during the entire consolidation process. A large number of studies have proven that hypothesis B is more consistent with actual sediment deformation.

The isotach concept was first introduced by Sǔklje [10], which is used to describe the effective rate of the compressibility of clay. Based on the isotach concept, Crawford [11] carried out a long-term one-dimensional ${\mathit{K}}_0$ consolidation test for normally consolidated clay and determined that the logarithmic curves of the void ratio and vertical stress for different loading times are a group of approximately parallel curves. Bjerrum [12] proposed that the deformation process can be divided into instantaneous deformation and delayed deformation. Yin et al. [13] proposed the EVP model, by introducing the equivalent-time concept. Yao et al. [14] proposed the concept of an instant normal compression line on the basis of existing theories and experiments and introduced the conversion time into the yield equation of the unified hardening model, to establish the elasto–visco–plastic constitutive model under complex conditions. Feng et al. [15] produced the EVP model, considering unloading expansion in the long-term time-dependent behavior of Hong Kong marine deposits (HKMD), under the loading stage and unloading stage in a multistage loading oedometer test. In addition, with the development of computer technology, some scholars have also used the grey theory method to predict the long-term one-dimensional deformation of sediment, such as Weng et al. [16]. Due to the complex deformation characteristics of sediment, it is difficult to make an analogy with the deformation of other well-known materials. Some scholars have chosen to use basic element bodies in combination, to better simulate the deformation characteristics of sediment, such as in the Maxwell model [17] and Merchant model [18].

In engineering practice, it is found that some sediment layers have similar overconsolidation characteristics, which is called a quasi-overconsolidated sediment, although there is no unloading process in geological history. Bjerrum [19] stated that, due to the aging effect, quasi-overconsolidated sediment has certain overconsolidation characteristics, and its shear strength is also higher than that of normal consolidated sediments. Most current long-term nonlinear deformation models only consider the time–strain relationship under a certain constant stress. Some overconsolidation models only consider the historical maximum vertical stress and cannot comprehensively consider all stresses borne by the sediment in its history and the impact of the duration of each level of stress on the deformation. In this article, based on the deformation of coral mud in one-dimensional oedometer tests, some elements of the Burgers model are modified. Furthermore, based on the modified Burgers model, a modified Burgers model considering stress history is obtained by considering all stress histories and loading durations.

2. Material and Tests

2.1. Material

The coral mud used in this study was sampled from the seabed of the South China Sea and had a calcium carbonate content of more than 97%. The sediment properties are shown in

Table 1. Basic properties of the coral mud.

Specific Gravity	Liquid Limit	Plastic Limit	Clay Content	Silt Content
2.77	33.8%	23.0%	36.0%	64.0%

and Figure 1, which were measured using a hydrometer, liquid–plastic limit detector, and particle size distribution test. The color of the coral mud was white to light yellow; the particles had marked edges and corners; and the particle-size distribution was uneven. The scanning results of an SEM (electron microscope scanning) test of coral mud are shown in Figure 2, and the particle morphology was mainly striped and needle-shaped.

2.2. Tests Details

In this study, a multi-stage loading oedometer was employed to investigate the soil behavior in one-dimensional conditions. The sample was 20 mm high and 61.8 mm in diameter. Filter paper was attached to the top and bottom of the sample, and drainage stones were placed. During the experiments, the water level in the tank was kept higher than that in the sample, to ensure that the sample in the ring cutter was always saturated. The specimens were placed into the steel ring, and the internal surface of the ring had silicone grease applied to minimize the possible friction.

To study the influence of the sediment stress history and stress action time on deformation, three different specimens were prepared; namely, T1, T2, and T3. The water content, loading path, and duration of the three groups of tests are shown in

Table 2. Loading procedures of coral mud.

No.	Water Content	Duration	Loading Path
T1	38.84%	8 d	25 kPa–50 kPa–100 kPa–200 kPa–400 kPa–800 kPa–1600 kPa–3200 kPa
T2	41.17%	1 d	12.5 kPa–25 kPa–50 kPa–100 kPa–200 kPa–400 kPa–800 kPa–1600 kPa–3200 kPa(8d) –1600 kPa–800 kPa–400 kPa–200 kPa–100 kPa–200 kPa–400 kPa–800 kPa–1600 kPa–3200 kPa(8d) –1600 kPa–800 kPa–400 kPa–200 kPa–100 kPa–200 kPa–400 kPa–800 kPa–1600 kPa–3200 kPa(8d) –1600 kPa–800 kPa–400 kPa–200 kPa–100 kPa–300 kPa–400 kPa–800 kPa–1600 kPa–3200 kPa(8d)
T3	38.69%	10.5 min	12.5 kPa–25 kPa–50 kPa–100 kPa–200 kPa–400 kPa–800 kPa–1600 kPa–3200 kPa(8d)

. As the strain speed of the coral mud in a one-dimensional oedometer test generally decreased sharply 10 min after loading [16], the load duration of each load level for T1, T2, and T3 was 8 days, 1 day, and 10.5 min, respectively, to study the impact of different loading durations on the long-term nonlinear deformation of coral mud at 3200 kPa. In addition, T2 was also subjected to repeated unloading–reloading tests, to study the impact of stress history on the coral mud deformation.

2.3. Tests Results

Figure 3a–c depicts the variations in vertical strain with time for the three sediment specimens (T1, T2, and T3), and Figure 3d depicts the unloading–reloading period of T2.

Figure 3a–c show that when the duration was different, the characteristics of the overall deformation curve was generally similar, but the time–strain relationship at 3200 kPa was different under the different durations. Figure 3d shows that during the unloading–reloading process, the deformation curves were generally parallel with the increase in unloading–reloading cycles. In other words, the long-term deformation of coral mud was affected by the loading path, duration, and unloading–reloading cycles.

3. Modified Burgers Model

3.1. Burgers Model

The strain of sediment after loading is generally divided into instantaneous strain and delayed strain. According to whether the strain can be recovered, the strain can be divided into elastic strain and plastic strain. Therefore, the total volumetric strain $\epsilon$ can be written as follows:

This is an example of Equation (1):

$$\epsilon ={\epsilon }_i^e+{\epsilon }_i^p+{\epsilon }_d^e+{\epsilon }_d^p,$$

(1)

where ${\epsilon }_i^e$ is the instantaneous elastic strain, ${\epsilon }_i^p$ is the instantaneous plastic strain, ${\epsilon }_d^e$ is the delayed elastic strain, and ${\epsilon }_d^p$ is the delayed plastic strain.

To accurately reflect the deformation characteristics of coral mud after loading, the Burgers model was selected on a theoretical basis in this study, and a sketch is shown in Figure 4. The Burgers model is composed of a Maxwell body and a Kelvin body in series. The model has four parameters, and each parameter has a clear physical meaning. The spring of the Maxwell body represents the instantaneous elastic strain after loading, while the other three elements jointly reflect the delayed strain.

The function of the Burgers model is

$$\epsilon =\frac{\sigma }{E_1}+\frac{\sigma }{{\eta }_1}t+\frac{\sigma }{E_2}\left(1-e^{-\frac{E_2}{{\eta }_2}}t\right),$$

(2)

where $\frac{\sigma }{E_1}$ is the instantaneous strain of coral mud after compression, ${\epsilon }_i^e$; $\frac{\sigma }{{\eta }_1}t$ is the nonrecoverable delayed plastic strain, ${\epsilon }_d^p$; and $\frac{\sigma }{E_2}\left(1-e^{-\frac{E_2}{{\eta }_2}}t\right)$ is the recoverable delayed elastic strain, ${\epsilon }_d^e$. It is obvious that the Burgers model ignores the instantaneous plastic strain ${\epsilon }_i^p$. That is, the three factors of the Burgers model represent the instantaneous elastic strain, delayed plastic strain, and delayed elastic strain. Therefore, the Burgers model can generally reflect the nonlinear deformation characteristics of sediment after loading.

3.2. Shortcomings of the Burgers Model

In combination with the deformation characteristics of coral mud under loading in Figure 3, it is not difficult to find problems with the Burgers model. At the moment of loading, the instantaneous strain of the sediment includes not only the instantaneous elastic deformation but also the instantaneous plastic strain. To more accurately describe the instantaneous strain of coral mud during unloading, it is necessary to modify the spring of the Maxwell body in the Burgers model. Taking the derivation of Equation (2) yields

$${\epsilon }^{\prime }=\frac{\sigma }{{\eta }_1}+\frac{\sigma }{{\eta }_2}{e}^{-\frac{E_2}{{\eta }_2}}t,$$

(3)

After loading, ${\epsilon }^{\prime }$ is always greater than 0; that is, the Burgers model maintains the feature of a monotonic increase, and the rate of strain increase finally tends to $\frac{\sigma }{{\eta }_1}$. This indicates that the Burgers model implies that the strain rate of sediment will tend to deform $\frac{\sigma }{{\eta }_1}$ endlessly and never stop, which is obviously inconsistent with the actual situation.

When the Burgers model is under incomplete unloading conditions, it is assumed that the vertical stress at time ${\mathit{t}}_0$ is $\sigma$ unloaded to ${\sigma }^{\prime }$. The unloading equation is obtained as follows:

$$\begin{array}{c}\hfill \epsilon =\frac{\sigma }{E_1}+\frac{\sigma }{{\eta }_1}t+\frac{\sigma }{E_2}\left(1-e^{-\frac{E_2}{\eta 2}t}\right)-\frac{\sigma -{\sigma }^{\prime }}{E_1}-\frac{\sigma -{\sigma }^{\prime }}{{\eta }_1}\left(t-t_0\right)-\frac{\sigma -{\sigma }^{\prime }}{E_2}\left(1-e^{-\frac{E_2}{\eta 2}\left(t-t_0\right)}\right)\\ \hfill =\frac{{\sigma }^{\prime }}{E_1}+\left(\frac{{\sigma }^{\prime }}{{\eta }_1}t+\frac{\sigma -{\sigma }^{\prime }}{{\eta }_1}{t}_0\right)+\left(\frac{{\sigma }^{\prime }}{E_2}+\frac{\sigma }{E_2}\left(e^{-\frac{E_2}{{\eta }_2}\left(t-t_0\right)}-e^{-\frac{E_2}{{\eta }_2}t}\right)-\frac{{\sigma }^{\prime }}{E_2}{e}^{-\frac{E_2}{{\eta }_2}\left(t-t_0\right)}\right),\end{array}$$

(4)

At the moment of incomplete unloading, the instantaneous elastic strain partially recovers; the rate of delayed plastic strain decreases, but still develops; and the delayed elastic strain partially recovers after unloading. Superimposing the strain changes of these three parts, the incomplete unloading curve is as shown in Figure 5. The strain first decreases and finally tends to increase slowly and never stops, which is also inconsistent with the actual situation.

At the moment of incomplete unloading, i.e., ${\sigma }^{\prime }=0$ in Equation (4), the unloading equation is as follows:

$$\epsilon =\frac{\sigma }{{\eta }_1}{t}_0+\frac{\sigma }{E_2}\left(e^{-\frac{E_2}{{\eta }_2}\left(t-t_0\right)}-e^{-\frac{E_2}{{\eta }_2}t}\right),$$

(5)

Taking the derivation of Equation (5) yields:

$${\epsilon }^{\prime }=\frac{\sigma }{{\eta }_2}\left(e^{-\frac{E_2}{{\eta }_2}t}-e^{-\frac{E_2}{{\eta }_2}\left(t-t_0\right)}\right)<0,$$

(6)

When unloading completely, the instantaneous elastic strain can be completely recovered in an instant; the delayed plastic strain is maintained at $\frac{\sigma }{{\eta }_1}{t}_0$ and does not decrease with unloading; and the delayed elastic strain will recover slowly. Therefore, when t tends to infinity, the final strain also tends to $\frac{\sigma }{{\eta }_1}{t}_0$.

3.3. Modified Method

The instantaneous strain related to $E_1$ includes the instantaneous elastic strain and instantaneous plastic strain, i.e., $E_1=E_1^e+E_1^p$. It is difficult to distinguish the two parts of the strain only from the loading test; therefore, the unloading test is needed for analysis. As the Burgers model only considers the instantaneous elastic strain, it is necessary to modify the spring of the Maxwell body; during the loading process, the modified Burgers model still retains the form of a spring; during unloading, only the instantaneous elastic strain can be recovered; thus, $E_1=E_1^e$. Therefore, $E_1$ is no longer equal before and after continuous unloading and reloading.

Long-term indoor tests [8] and practical engineering have proven that the deformation of sediment may “never stop”, but it can be clearly observed that this speed will tend to zero indefinitely. Therefore, it is unreasonable that the delayed plastic strain rate in the Burgers model tends to a fixed value; that is, the dashpot element of the Maxwell body is unreasonable.

Most component models are modified by adding more components. However, every additional element means that at least one parameter is added to the model equation. For a four-element model such as the Burgers model, the equation of the model is sufficiently complex. Adding elements does not improve the problem of infinite development and also increases the complexity and difficulty of the solution. Therefore, only the element properties are modified. In combination with the strain characteristics shown in Figure 3, this paper considers modifying the dashpot into a nonlinear dashpot; that is, the viscosity coefficient of the dashpot is no longer constant but increases with time and tends to infinity, which is also consistent with the real strain of the sediment under loading. The modified Burgers model expression is as follows:

$$\epsilon =\frac{\sigma }{E_1}+\frac{\sigma }{{\eta }_1}\left(1-e^{-t}\right)+\frac{\sigma }{E_2}\left(1-e^{-\frac{E_2}{{\eta }_2}t}\right),$$

(7)

For the first level analysis of T1, as shown in Figure 6, the predicted value of the modified Burgers model tends to be a straight line, which is more reasonable than the original Burgers model, which develops rapidly and constantly destroying.

3.4. Model Stability Verification

The fitting parameters may vary with the increase or decrease in the data points involved in the model calculation. To verify the stability of the modified Burgers model, the data of the first stage of T1 with different time lengths were substituted into the modified Burgers model for fitting, and the changes in the model parameters obtained are shown in Figure 7.

With the increase in time involved in the model calculation, $E_1$ is relatively stable, ${\eta }_1$ and $E_2$ decrease slightly, and ${\eta }_2$ greatly increases. Namely, the instantaneous strain and delayed plastic strain are relatively stable. Although $E_2$ and ${\eta }_2$ jointly reflect the delayed elastic strain, this part of the strain finally approaches $\frac{\sigma }{E_2}$; that is, it is only related to $E_2$, while ${\eta }_2$ determines the speed tending to the maximum strain.

Although the time of substitution into the model is different, only the parameter ${\eta }_2$ in the modified Burgers model is significantly affected; that is, it only affects the speed at which the sediment tends to the maximum strain. The model can still accurately grasp the total strain of the sediment. The modified Burgers model has good stability.

4. Modified Burgers Model Considering the Stress History

At present, when analyzing a compression test of sediment under different loads, most methods analyze the time–strain relationship under each level of load. These analysis methods are fragmented and cannot take into account the influence of the historical loading, unloading, and reloading processes of the sediment [20]. Figure 3 shows that the time–strain relationship under each load level is related to all previous load levels; therefore, this study extends the modified Burgers model.

As shown in Figure 8, when the sediment is subjected to the first level load, ${\sigma }_1$, the time–strain relationship is:

$$\epsilon ={\epsilon }_1=\frac{{\sigma }_1}{E_{1-1}}+\frac{{\sigma }_1}{{\eta }_{1-1}}\left(1-e^{-t}\right)+\frac{{\sigma }_1}{E_{2-1}}\left(1-e^{-\frac{E_{2-1}}{{\eta }_{2-1}}t}\right),$$

(8)

where $E_{1-1}$, ${\eta }_{1-1}$, $E_{2-1}$, and ${\eta }_{2-1}$ are the first level model parameters, which are obtained by fitting the time–strain relationship under the first level load.

At time $t_1$, the load increases from the first level load ${\sigma }_1$ to the second level load ${\sigma }_2$. At this time, the second level load ${\sigma }_2$ is divided into ${\sigma }_1$ and ${\sigma }_2-{\sigma }_1$. That is, the total strain is the sum of the strain produced by ${\sigma }_1$ in the whole continuous process and the strain produced by ${\sigma }_2-{\sigma }_1$ from $t_1$. Then, the time–strain relationship is

$$\begin{array}{c}\hfill \epsilon ={\epsilon }_2=\frac{{\sigma }_1}{E_{1-1}}+\frac{{\sigma }_1}{{\eta }_{1-1}}\left(1-e^{-\left(t+t_1\right)}\right)+\frac{{\sigma }_1}{E_{2-1}}\left(1-e^{-\frac{E_{2-1}}{{\eta }_{2-1}}\left(t+t_1\right)}\right)\\ \hfill +\frac{{\sigma }_2-{\sigma }_1}{E_{1-2}}+\frac{{\sigma }_2-{\sigma }_1}{{\eta }_{1-2}}\left(1-e^{-t}\right)+\frac{{\sigma }_2-{\sigma }_1}{E_{2-2}}\left(1-e^{-\frac{E_{2-2}}{{\eta }_{2-2}}t}\right),\end{array}$$

(9)

where $E_{1-1}$, ${\eta }_{1-1}$,$E_{2-1}$, and ${\eta }_{2-1}$ are obtained by fitting Equation (8), and $E_{1-2}$, ${\eta }_{1-2}$, $E_{2-2}$, and ${\eta }_{2-2}$ are the second-level model parameters, which are obtained by fitting the time–strain relationship under the second-level load.

Then, the time–strain relationship for the i-th level load is

$$\begin{array}{c}\hfill \epsilon =\sum _1^{n=i}{\epsilon }_i=\sum _1^{n=i}\frac{{\sigma }_n-{\sigma }_{n-1}}{E_{1-n}}+\sum _1^{n=i}\frac{{\sigma }_n-{\sigma }_{n-1}}{{\eta }_{1-n}}\left(1-e^{-\left(t-{\sum }_1^{k=n-1}{t}_k\right)}\right)\\ \hfill +\sum _1^{n=i}\frac{{\sigma }_n-{\sigma }_{n-1}}{E_{2-n}}\left(1-e^{-\frac{E_{2-i}}{{\eta }_{2-i}}\left(t-{\sum }_1^{k=n-1}{t}_k\right)}\right),\end{array}$$

(10)

The time–strain relationship thus obtained can reflect the entire stress history that the sediment has been subjected to and the duration of each level of loading. In the fitting process of each level, only four parameters need to be solved, which ensures the model is not too complex.

The experimental data of T1 were substituted into the calculation, as shown in Figure 9. The modified Burgers model considering the stress history can well represent the time–strain relationship of sediment under different loads.

5. Discussion

The test data of T1, T2, and T3 were substituted into the model calculation.

5.1. Loading Period

For the T1, T2 (only the first 9 levels of loading), and T3 tests, the time–strain relationship was substituted into the modified Burgers model considering the stress history, and the model parameters with different durations of each level of load were obtained, as shown in Figure 10.

5.1.1. Parameters

In the loading period, the four parameters showed an overall upwards trend, and they generally showed similar laws. At the beginning, these four parameters had little difference, but the difference gradually became obvious when increasing the loading, indicating that the different durations of previous loadings cumulatively affected the time–strain relationship of subsequent loading.

The longer the duration of each load level, the greater the $E_1$, and ${\eta }_1$, $E_2$ and ${\eta }_2$ are smaller. With an increase in the duration of the previous loading, the deformation of the sediment is greater, and the compressibility of the sediment decreases during the subsequent reloading; therefore, $E_1$ increases. Figure 10b shows that the greater the duration of the previous loading, the greater ${\eta }_1$; that is, the greater the ultimate strain rate of the sediment. As the duration of sediment increases, the delayed strain of the sediment inevitably increases; therefore, $E_2$ is smaller. The decrease in ${\eta }_2$ indicates that the maximum delayed strain of the sediment becomes lower.

5.1.2. Proportion of Strain

The strain of a sediment under loading can be divided into instantaneous strain, delayed plastic strain, and delayed elastic strain. For T1, T2, and T3, the proportion of each level of strain in the loading period of these three parts is shown in Figure 11.

By comparing the proportions of the three parts, it can be seen that the instantaneous strain was smaller and close together in the loading process, except that the first level was larger. By comparing T1, T2, and T3, it can be seen that with the duration at each level, the proportion of instantaneous strain related to $E_1$ in the total strain of each level increased. This is because, with the decrease in the duration, the duration of delayed strain was shorter, and the amount of delayed strain was also reduced, so that the proportion of instantaneous strain was correspondingly higher.

Note that the instantaneous strain includes the instantaneous elastic strain and the instantaneous plastic strain; that is, $E_1$ is not the instantaneous elastic modulus. It is difficult to distinguish the instantaneous elastic strain and the instantaneous plastic strain only from the loading test; therefore, this needs to be analyzed using the unloading–reloading test.

5.2. Unloading and Reloading Period

The time–strain relationship of the whole loading–unloading–reloading process of T2 was brought into the modified Burgers model considering the stress history, and the model parameters were obtained as shown in Figure 12.

5.2.1. $E_1$

One of the most remarkable characteristics of sediment deformation is nonlinearity. In the commonly used nonlinear elastic model of sediment, it is generally assumed that the elastic parameters change with the stress state. The relationship between the vertical strain and $E_1$ is shown in Figure 13. There is no mapping relationship between the height and elastic modulus. During the loading period, $E_1$ gradually and slowly increases; indicating that, with increasing vertical stress, the sediment is continuously compressed, and the compressibility gradually decreases. In the reloading period, the relationship of the vertical strain and $E_1$ is closer to a parallel line.

5.2.2. ${\eta }_1$

The variation dispersion of ${\eta }_1$ is relatively large, and there are two very small negative values; namely, the first and second unloadings from 3200 kPa to 1600 kPa. In this paper, although the strain rate of the nonlinear dashpot under constant vertical stress decreased slowly, it was always greater than 0, and its viscosity coefficient could not be negative. In combination with Equation (9), when the vertical stress was unloaded from 3200 kPa to 1600 kPa, the resilience of the elastic element was smaller than the continuous compression of the plastic element, which led to the overall strain of the sediment showing the characteristic of a continuous increase in a short time. At this time, the model cannot accurately identify these mutually canceling strains, which leads to temporary distortion of the parameters.

5.2.3. $E_2$ & ${\eta }_2$

$E_2$ reflects the total amount of delayed elastic strain. The larger the value of $E_2$, the smaller the total amount of delayed elastic strain. Figure 12 shows that $E_2$ increases gradually in the process of loading and reloading and decreases gradually in the process of unloading. During each (re)loading–unloading period, the peak value of $E_2$ appeared when unloading from 3200 kPa to 1600 kPa, and with the increase in loading–unloading times, the $E_2$ under the corresponding load increases.

${\eta }_2$ reflects the speed of the delayed elastic strain. The larger the value of 2, the slower the sediment will reach the maximum delayed plastic strain. Section 3.4 shows that ${\eta }_2$ has a certain instability; therefore, it can generally only reflect the actual situation. Similarly to $E_2$, the peak value of ${\eta }_2$ in each (re)loading–unloading period occurred when unloading from 3200 kPa to 1600 kPa, and with the increase in (re)loading–unloading times, ${\eta }_2$ increases. Figure 12 shows that, compared with the loading period, the value of ${\eta }_2$ corresponding to the unloading–reloading period was larger; that is, it takes longer to reach the maximum delayed plastic strain.

5.2.4. Proportion of Strain

Figure 14 shows that in the period of the four (re)loading and three unloading cycles, the strain during the first loading cycle is much larger than that during the subsequent reloading and unloading period, and the strain during the subsequent reloading and unloading cycles is generally the same. During each cycle of unloading–reloading, when increasing the loading pressure, the total strain will be concave.

6. Conclusions

• Based on one-dimensional oedometer tests of coral mud, this study modified some elements of the Burgers model and obtained a modified Burgers model that can better reflect the time–strain relationship under constant vertical stress;
• Based on a one-dimensional (re)loading–unloading test of coral mud, a modified Burgers model considering the stress history was proposed in this study. The model can describe the time–strain relationship under different action durations and different loading and unloading stress histories. The number of parameters at each load level of the model is four, which is convenient for calculation and retains the physical meaning;
• The parameters of the modified Burgers model considering stress history were analyzed, and the strain characteristics of sediment under different durations and repeated (re)loading and unloading conditions were obtained. It was found that the model can simulate the strain characteristics of coral mud, and the parameters have a good variation law.