Study on the Pressure–Sinkage Process and Constitutive Model of Deep-Sea Sediment

Abstract

The driving performance of subsea mining vehicles is greatly affected by the pressure–sinkage characteristics of deep-sea sediment. Therefore, it is of great importance to analyze the microscopic properties of deep-sea sediment and establish the corresponding pressure–sinkage model for the safe operation of subsea mining vehicles. Hence, the present paper focuses on the physical properties of deep-sea sediment to provide a preliminary understanding of its pressure–sinkage process and evolution according to the solid–liquid two-phase flow characteristics and particle flow mechanism. In addition, the stress loading time and the rheological theory are applied in order to introduce a four-element model that describes the various pressure–sinkage stages that correspond to each stage of deep-sea sediment evolution. On this basis, the parameters of the pressure–sinkage constitutive model are determined by a specific calculation method. Moreover, a new pressure–sinkage constitutive model of deep-sea sediment that considers the time-variable mechanical properties is established in order to describe the full sinkage process. Finally, research results from the existing literature and experimental data are used to verify the rationality and correctness of the model. The results show that the proposed pressure–sinkage constitutive model is in good agreement with experimental data and is effective in describing the evolution of the mechanical properties and the trend in the sinkage rate of deep-sea sediment at various stages. A comparison with the Kelvin model indicates that the proposed pressure–sinkage constitutive model provides superior accuracy with the use of fewer parameters. Consequently, this study can provide a theoretical basis and technical support for the design of subsea mining vehicles.

1. Introduction

Deep-sea polymetallic nodules, commonly known as manganese nodules, are mainly distributed on the seabed at a depth of 4000–6500 m, and occur in surface sediments to a thickness of 0–25 cm. They are distributed in a planar manner, with an abundance of about 10–20 kg/m² [1]. Manganese nodules have various shapes, mainly including spherical, ellipsoidal, and conjoined nodules, along with small amounts of debris and plates, as shown in Figure 1. Deep-sea manganese nodule resources have become a strategic guarantee for sustainable economic development in the future [2,3,4], with the main manganese reserve areas being shown in Figure 2. The mining system has to withstand the extreme environment of the ultra-high deep-sea pressure of several kilometers, along with seawater corrosion and other problems, which makes mining technology extremely difficult and requires an extremely large mining system. At present, the most commercially promising deep-sea manganese nodule mining system is the hydraulic pipeline transportation system, which uses subsea mining vehicles to collect the manganese nodules, which are then pumped to a surface mining vessel through a 6000 m pipe [5,6,7,8]. Due to the particularity of the deep-sea sedimentary environment, deep-sea sediment has obvious stratification and vertical distribution characteristics in which the depth of the top sediments ranges from about 5 to 20 cm. At the same time, the top sediments have a higher water content, larger void ratio, greater consolidation coefficient, lower shear strength, lower penetration resistance, and more obvious rheological properties such as liquefaction flow, compared to those of ordinary land soil [9,10,11,12]. Hence, subsea mining vehicles can easily slip on, or even sink into, the deep-sea sediment, thereby affecting the stability of the vehicle and the mining efficiency. Therefore, studying the sinkage mechanism of deep-sea sediment, and establishing the corresponding pressure–sinkage constitutive model, are of great significance for the analysis of the shear strength and traction force of the subsea mining vehicles, the prediction of the driving resistance, and the establishment of design guidelines. However, due to the difficulty in obtaining the deep-sea sediment in situ owing to the extreme environment and high cost, a reasonable simulated soil is widely used as an experimental sediment for studying the mechanical properties of deep-sea sediment and subsea mining vehicles [13,14,15].

To-date, many constitutive models have been proposed and applied to study the pressure–sinkage behaviors of deep-sea sediment, which can be divided into several categories, including empirical models, component models, fractional-order derivative models, and so on [17,18,19]. The empirical model for describing the pressure–sinkage characteristics of deep-sea sediment is mainly based on the use of experimental data and Bekker and Reece’s theory of rolling resistance of free-rolling, towed, rigid wheels [20,21]. However, the empirical model has disadvantages in long-term prediction owing to the different timescales between the experimental study and the engineering project, with the experimental study usually having a much shorter duration. On the other hand, Bekker and Reece’s theory is based on terrestrial soils and cannot fully match the physical properties of deep-sea sediment. The component model is widely used to characterize the mechanical properties of deep-sea sediment by various combinations of standard components such as the Hooke spring, Newtonian sticky pots, and Saint-Venant slider [22]. The component model has the advantage of describing the quantitative relationship between the strain and stress on an object under the action of an external force. Such models include the Burgers model, Maxwell model, Kelvin model, Nishihara model, etc. [23,24,25]. The component model was originally used to analyze creep properties in geotechnical and hydraulic engineering, and was subsequently applied to analyze the pressure–sinkage properties of deep-sea sediment. Xu et al. used the generalized Kelvin–Hooke model and the generalized Burgers model to describe the combined compression–shear rheological properties of deep-sea sediment at various stress levels. The experimental results showed that both models successfully describe the variation in the rheological properties of deep-sea sediment with loading time, and the fitting results of the two models agreed well with the experimental data [26]. However, the component model has many parameters and requires complex mathematical calculations, thus making it difficult to determine the model parameters. Meanwhile, the fractional-order derivative model has been proposed to describe the mechanical properties of geotechnical materials between ideal solids and fluids, and uses the theory of fractional calculus to establish a new fractional component [23,27]. Fractional calculus has been considered a powerful tool for building time-dependent constitutive models due to the advantages of long-term history dependencies or memory effects. To analyze the traction creep properties of deep-sea sediment, Ma et al. proposed a new fractional-order viscoelastic model based on the classical Burgers model, in which the dashpot in the Burgers model was replaced by the Abel dashpot, and found that the Burgers model with fractional-order derivatives is superior to the unmodified Burgers model in reflecting the traction creep properties of deep-sea sediment [28]. Although the fractional-order derivative model has fewer parameters and higher accuracy, its fractional order is constant and, therefore, cannot determine the segmental characteristics of the pressure–sinkage process. The subsea mining vehicles rely on shearing sediment to provide traction during driving. At the same time, the mechanical properties of sediments are time-varying, i.e., there are nonlinear changes in the physical and mechanical parameters of the sediments under the combined action of ground pressure and time. Although the above studies have had some success in fitting experimental data, the pressure–sinkage process and the evolution of the mechanical properties of deep-sea sediment are still unclear. Moreover, there is no report describing other methods for determining the model parameters besides the least squares method.

The aim of the present study is to accurately describe the evolution of the mechanical properties of deep-sea sediment, and then propose a new pressure–sinkage constitutive model for deep-sea sediment on the basis of the time–stress superposition principle. First, the pressure–sinkage process and microscopic mechanism of deep-sea sediment are analyzed based on the physical properties of the deep-sea sediment and the particle flow mechanism, and a segmentation treatment is used to characterize the pressure–sinkage property of the deep-sea sediment. Then, the parameters of the pressure–sinkage constitutive model are determined via a specific calculation method that considers the influence of ground pressure and time on the pressure–sinkage process. In addition, a new pressure–sinkage constitutive model for deep-sea sediment that considers the time variability of the mechanical properties is established by using the time–stress superposition principle. Finally, the accuracy of the proposed model is verified by comparison with experimental data.

2. Analysis of the Deep-Sea Sediment Pressure–Sinkage Mechanism

2.1. Vertical Stratification Characteristics of the Deep-Sea Sediment

To describe the vertical stratification characteristics of the deep-sea sediment in polymetallic nodule exploration areas, in situ measurement data and a sampling experiment of deep-sea geotechnical mechanics are used herein to calculate the volume concentration at various buried depths according to Equation (1) [29]:

$$c_v=\frac{{\rho }_w}{(\eta {\rho }_s+{\rho }_w)}$$

(1)

where $c_v$ is the volume concentration of the deep-sea sediment, ${\rho }_w$ is the density of seawater (1.02372 g/cm³), $w$ is the water content, and ${\rho }_s$ is the dry density of the deep-sea sediment.

The variations in the physical properties of the deep-sea sediments with depth in the oceanic polymetallic nodule exploration area are plotted in Figure 3, where there is good correlation between each parameter and the burial depth. Specifically, the water content and void ratio decrease with increasing burial depth, while the wet density and volume concentration increase with increasing burial depth. In particular, the volume concentration exhibits a remarkable variation extent, along with an obvious stratification, such that the profile can be divided into the following three zones: (1) the suspension layer (0–3 cm), which is the supernatant water layer of deep-sea sediment, with a volume concentration of less than 9.5%, (2) the semi-liquid layer (3–20 cm), where the solid concentration is increasing, and the volume concentration is between 9.5% and 10.5%, and (3) the consolidated layer (20–35 cm), where the solid concentration remains stable at approximately 10.7%. The above analysis is consistent with the research conclusions of other scholars, thus indicating that the study of the vertical stratification characteristics of deep-sea surface sediments according to the variation in volume concentration has applicability [30,31].

2.2. Deep-Sea Sediment Pressure–Sinkage Process

The deep-sea sediment in the oceanic polymetallic nodule exploration area consists of clay soil with predominant silt, which has high sensitivity characteristics [32]. Therefore, the deep-sea sediment has a special pressure–sinkage mechanism and evolution law. In this paper, based on the change in the slope of the pressure–sinkage curve, the pressure–sinkage process of the deep-sea sediment is analyzed by using the available literature results and experimental data [33]. The analysis results are presented in Figure 4.

Here, the sinkage depth of the deep-sea sediment is seen to increase with time, and clear points of inflection can be identified. Before inflection point A, the sinkage depth increases sharply with time, showing a linear change, and sinkage appears in the instantaneous stage. Between inflection points A and B, the sinkage depth increases slowly with time, showing a nonlinear change. After inflection point B, the change in the amount of sinkage with depth is small. Therefore, the pressure–sinkage behavior of the deep-sea sediment can be divided into four components, as indicated in Equation (2):

$$Z=Z_d+Z_e+Z_c+Z_s$$

(2)

where $Z$ is the total pressure–sinkage, $Z_d$ is the instantaneous pressure–sinkage, $Z_e$ is the initial pressure–sinkage, $Z_c$ is the secondary pressure–sinkage, and $Z_s$ is the stable creep.

The variations in the depths of inflection points A and B with various ground pressure values are indicated in

Table 1. The depths of inflection points A and B under various ground pressures [33].

Inflection Point	P = 5 kPa	P = 10 kPa	P = 15 kPa	P = 20 kPa	P = 25 kPa
A (cm)	2.14691	3.85296	5.15954	6.13158	8.54775
B (cm)	2.94222	5.24755	6.51602	9.09158	11.15889

and Figure 5. Thus, the depths of both inflection points increase linearly with the increase in ground pressure.

An analysis of the plots in Figure 5 reveals that the relationship between the ground pressure and the depths of inflection points A and B can be expressed as Equation (3):

$$\{\begin{array}{l}{Z}_A=0.3016P+0.6437\\ Z_B=0.4055P+0.908\end{array}$$

(3)

where $Z_A$ and $Z_B$ are the depths of points A and B, respectively. Specifically, $Z_A$ is the dividing point between the initial and the secondary pressure–sinkage zones, while $Z_B$ is the dividing point between the secondary pressure–sinkage zone and the stable creep zone.

2.3. The Microscopic Mechanism of Deep-Sea Sediment Sinkage

The particle size analysis of the deep-sea sediment obtained from the west Pacific C-C district (152°30′–154°30′ E, 18°30′–20°0′ N) was performed using a micro-plus laser diffraction particle size analyzer (Malvern company, London, UK). The particle size range of the sample is 0.34–704 μm, the effective particle size (d₁₀) is 1.37–1.63 μm, and the average particle size (d₅₀) is 4.62–5.5 μm. The particle size distribution curves at various depths are presented in Figure 6.

The X-ray diffraction (XRD) analysis of the clay minerals in the surface sediments was performed on a fully-automatic (18 kW) rotating target X-ray diffractometer in the Marine Science Specialty Laboratory of the China University of Geosciences. The results show that the clay minerals in the sediment are dominated by illite (58.81–74.41%, with an average of 67.99%), followed by chlorite (15.24–29.71%, with an average of 18.76%). The content of kaolinite and montmorillonite is generally less than 10%, with the kaolinite content varying from 5.26% to 10.13% (average, 6.42%), and that of montmorillonite varying from 1.09% to 11.38% (average, 6.84%).

Photographic images of the analyzed sediments are presented in Figure 7. The deep-sea sediment particles are relatively loosely arranged, and the pores are essentially filled with seawater. Thus, the sediment is classified as saturated soil and has the characteristics of solid–liquid two-phase flow. Due to the poor permeability of this soil, the pore water pressure therein will rise sharply when driven over by a subsea mining vehicle. The pore water in the soil passes through the following four stages: (1) fully drained, (2) partially drained, (3) completely undrained, and (4) viscous. With the increase in time, the rheological properties of the deep-sea sediments no longer satisfy the properties of a Newtonian fluid, and the relationship between the amount of sinkage and specific ground pressure during pressure–sinkage must be characterized using non-Newtonian fluid dynamics. In particular, the yield stress can be used to represent the viscosity of the fluid, which is an important parameter for describing the characteristics of solid–liquid two-phase flow [34,35]. The yield stress is related to factors such as volume concentration, particle size, and clay properties. Among these, the volume concentration has the greatest influence, exhibiting an exponential relationship with the yield stress [36]. Therefore, the yield stress (τ) of deep-sea sediment can be expressed as Equation (4):

$$\tau =k{e}^{rC}$$

(4)

where $c_v$ is the volume concentration of the deep-sea sediment, and k and r are parameters related to the particle size and clay properties thereof.

According to the above analysis, the particle size distribution of the deep-sea sediment is similar to that of laterite nickel ore with the same rheological properties [37]. Therefore, without considering the mineral composition and microstructure, the rheological test results of laterite nickel ore are used herein to determine the values of the parameters $k$ and $r$ [38]. Using the relationship between the yield stress and volume concentration of laterite nickel ore, $k$ and $r$ were obtained as 0.0245 and 0.2966, respectively. Substituting these values into Equation (4) gives the expression for the yield stress (τ) in Equation (5):

$$\tau =0.0245e^{{}_{0.2966C_i}}$$

(5)

The volume concentration and density of the deep-sea sediments at various depths are related by Equation (6):

$$C_i=\frac{{\rho }_i-{\rho }_w}{{\rho }_s-{\rho }_w}$$

(6)

where $C_i$ and ρ_i are the volume concentration and density at sinkage depth i, ρ_w is the density of seawater and ρ_s is the dry density of the deep-sea sediment.

Based on the above formulas and calculation results, the relationship between the yield stress and volume concentration of the deep-sea sediment is shown in Figure 8. The deep-sea sediment is predominately zeolite clay and pelagic clay, and exhibits various rheological properties during sinkage, transitioning from a Newtonian fluid to a non-Newtonian fluid to the limiting concentration [39]. When the volume concentration of the deep-sea sediment is lower than 10%, the yield stress is almost zero; when the concentration reaches more than 20%, the yield stress increases more quickly, thereby indicating that the resistance increases with the volume concentration. Based on the above formulas and calculation results, the relationship between the yield stress and volume concentration of the deep-sea sediment is shown in Figure 8.

The interaction mechanism between subsea mining vehicles and deep-sea sediment is shown in Figure 9, and the changes involved in the micromechanism of deep-sea sediment compaction are shown in Figure 10. From the analysis of the deep-sea sediment sinkage mechanism and mechanical evolution law, the pressure–sinkage process of the deep-sea sediment can be divided into the instantaneous pressure–sinkage stage (Figure 10a), the initial pressure–sinkage stage (Figure 10b), the secondary pressure–sinkage stage (Figure 10c), and the stable creep stage (Figure 10d). The instantaneous pressure–sinkage stage involves the shear deformation of the deep-sea sediment under the action of subsea mining vehicles, which is mainly due to the fact that the crawler tracks dislodge the surface sediments, and the sediments under the crawler do not experience direct sinkage. With the increase in the action time, the initial pressure–sinkage and secondary pressure–sinkage stages mainly occur in the semi-liquid layer of the deep-sea sediment. There are numerous biological components (such as diatoms, radiolarians, and other siliceous bioclastic organic matter, etc.) in the sediments of the semi-liquid layer, thus resulting in the unique physicochemical properties, with a loose and porous structure and strong water absorption capacity [40,41,42]. Among the various stages, the initial pressure–sinkage stage involves elastic change, wherein the shear strength at each point is proportional to the sinkage rate. At this time, the fluid properties of the sediments are similar to those of clear water, the yield stress is approximately zero, and the particles exhibit the characteristics of Newtonian bodies. The main reason for this is that the amount of free water in the sediment is very large, and the distance between particles is great, thereby resulting in a very small force between the solid phases. By contrast, the secondary pressure–sinkage stage presents a nonlinear change trend. Parts of the sediments in the semi-liquefied layer undergo shear damage, and the deep-sea sediment begins to show the characteristics of a non-Newtonian fluid due to increased volumetric concentrations of deep-sea sediment. The main reason for this is that the excess pore water pressure in the sediments continues to dissipate under the action of the subsea mining vehicle, and the particles are connected to each other to form flocs under the influence of flocculation; moreover, the particles are subjected to collisional friction between layers under the lubrication of free water, which increases the yield stress. The stable creep stage is related to the translational movement of deep-sea sediment particles due to the dense contact friction between particles, such that the flocculant structure of the sediments is reconstructed. This eventually leads to the formation of a dense network space structure, which has strong stability and yield stress.

3. Establishing the Deep-Sea Sediment Pressure–Sinkage Constitutive Model

3.1. Construction of the Model

Deep-sea sediment pressure–sinkage behavior has the characteristics of elasticity, viscosity, and plasticity, and is related to the ground pressure and the amount of time. The entire process of pressure–sinkage can be sequentially understood as the instantaneous elastic, viscoelastic, and viscoplastic effect. According to the stress loading time and the rheological model, the Burgers model can be expressed as a one-dimensional state [43]. Therefore, the correspondence between the combined components of the Burgers model and each pressure–sinkage stage is analyzed in order to examine the change characteristics of the entire pressure–sinkage process. The pressure–sinkage constitutive model of deep-sea sediment is shown in Figure 11.

Assuming that a constant ground pressure is applied to the model at time t = 0, the relational expressions of the four stages of the deep-sea sediment pressure–sinkage constitutive model are as follows:

• The instantaneous pressure–sinkage stage. At the moment of loading, the deep-sea sediment produces an instantaneous elastic response (Z_d), which can be described by the spring element in the pressure–sinkage constitutive model, given by Equation (7):

$$Z_d=\frac{P}{E_1}$$

(7)

where P is the constant ground pressure, and E₁ is the elastic modulus in this stage.

2.

The initial and secondary pressure–sinkage stages. Due to the low elastic modulus in these two stages, viscous fall will occur, thereby increasing the volume concentration. In addition, the deep-sea sediment will be consolidated and hardened, so its sinkage rate will gradually decrease. This stage can be modeled by the spring element and the Newton dashpot element according to Equation (8):

$$Z_e{}_c=\frac{P}{E_2}\left[(1-\exp \left(\frac{-E_2}{{\mathsf{\eta }}_1}t\right)\right]$$

(8)

where E₂ is the elastic modulus, and η₁ is the viscosity coefficient in this stage.

3.

The stable creep stage. With the passage of time, the volume concentration of the deep-sea sediment becomes higher, and the sinkage rate gradually becomes stable. This behavior can be simulated by the Newton dashpot element, as given by Equation (9):

$$Z_{\mathrm{s}}=\frac{P}{{\mathsf{\eta }}_2}t$$

(9)

where ${\mathsf{\eta }}_1$ is the viscosity coefficient at this stage. Combining Equations (7)–(9), gives the pressure–sinkage constitutive model used in this study as Equation (10):

$$Z=\frac{P}{E_1}+\frac{P}{E_2}\left[1-\exp \left(\frac{-E_2}{{\mathsf{\eta }}_1}t\right)\right]+\frac{P}{{\mathsf{\eta }}_2}t$$

(10)

3.2. Determination of the Pressure-Sinkage Constitutive Model Parameters

Examination of Equation (10) shows that the instantaneous elastic modulus $E_1$ is only affected by the ground pressure. The instantaneous sinkage depth can be obtained from the pressure–sinkage experimental data, and the instantaneous elastic modulus $E_1$ is given by Equation (11):

$$E_1=\frac{P}{Z_{e,t\to 0^{+}}}$$

(11)

where $Z_{e,t\to 0^{+}}$ is the instantaneous pressure–sinkage depth at time $t\to 0^{+}$.

Since the changes in the elastic modulus $E_2$ and the viscosity coefficient ${\mathsf{\eta }}_1$ are affected by the ground pressure and time, the sinkage rate (${Z^{\prime }}_{deci}$) at any time in the initial and secondary pressure–sinkage stages can be obtained by taking an arbitrary time point $t_i$, as in Equation (12):

$$Z^{\prime }{}_{deci}=\frac{P}{{\mathsf{\eta }}_{1i}}\exp (-\frac{E_{2i}}{{\mathsf{\eta }}_1{}_i}{t}_i)$$

(12)

Combining Equations (7) and (8) gives the constitutive model for the first three pressure–sinkage stages at any time $t_i$ as Equation (13):

$$Z_{deci}=\frac{P}{E_1}+\frac{P}{E_{2i}}\left[(1-\exp (-\frac{E_{2i}}{{\mathsf{\eta }}_{1i}}{t}_i)\right]$$

(13)

Further, Equations (12) and (13) can be combined to give the elastic modulus $E_{2i}$ and the viscosity coefficient ${\mathsf{\eta }}_{1i}$ at an arbitrary time $t_i$ in the first three pressure–sinkage stages as Equation (14):

$$Z_{deci}-\frac{P}{E_1}=\frac{P{t}_i}{-\ln \left(\frac{Z^{\prime }{}_{deci}}{P}{\mathsf{\eta }}_{1i}\right){\mathsf{\eta }}_{1i}}\left[1-\frac{Z^{\prime }{}_{deci}}{P}{\mathsf{\eta }}_{1i}\right]$$

(14)

Taking the logarithm of both sides of Equation (14), and using the Taylor series expansion, gives Equation (15):

$$\begin{array}{ll}\ln \left[\frac{Z^{\prime }{}_{deci}}{P}{\mathsf{\eta }}_{1i}\right]& =\ln \left(\frac{Z^{\prime }{}_{deci}}{P}\right)+\ln {\mathsf{\eta }}_{1i}\\ & =\ln \left(\frac{Z^{\prime }{}_{deci}}{P}\right)+\left\{\left({\mathsf{\eta }}_{1i}-1\right)+0\left[{\left({\mathsf{\eta }}_{1i}-1\right)}^2\right]\right\}\end{array}$$

(15)

Substituting Equation (15) into Equation (14) then gives Equation (16):

$$Z_{deci}-\frac{P}{E_1}=\frac{P{t}_{\mathrm{i}}}{-\left[\ln \left(\frac{Z^{\prime }{}_{deci}}{P}\right)+{\mathsf{\eta }}_1{}_i-1\right]{\mathsf{\eta }}_{1i}}\left[1-\left(\frac{Z^{\prime }{}_{deci}}{P}\right){\mathsf{\eta }}_{1i}\right]$$

(16)

The viscosity coefficient ${\mathsf{\eta }}_{1i}$ can then be obtained by inserting the experimental data into Equation (16), after which the elastic modulus $E_{2i}$ can be determined by inserting the viscosity coefficient ${\mathsf{\eta }}_{1i}$ into Equation (12):

$$E_{2i}=-\frac{{\mathsf{\eta }}_{1i}}{t_i}\ln \left(\frac{Z^{\prime }{}_{deci}{\mathsf{\eta }}_{1i}}{P}\right)$$

(17)

During the stable creep phase of the deep-sea sediment at a certain randomly selected time, the relationship in Equation (18) exists:

$$Z_{decsi}=\frac{P}{E_1}+\frac{P}{E_{2\infty }}+\frac{P}{{\mathsf{\eta }}_{2i}}{t}_i$$

(18)

where $Z_{decsi}$ is the amount of sinkage at time $t_i$.

Assuming that the pressure–sinkage parameters at any time $t_i$ remain unchanged, the critical time $t_j$ is selected as time $t_i$. The value of the viscosity coefficient ${\mathsf{\eta }}_2$ can then be determined from Equation (19):

$${\mathsf{\eta }}_{2i}=\frac{P}{Z_{decsi}-Z_{decsj}}(t_i-t_{\mathrm{j}})$$

(19)

where $Z_{decsj}$ is the amount of sinkage at time $t_j$ during the stable creep phase of the deep-sea sediment.

4. Validation of the Deep-Sea Sediment Pressure–Sinkage Constitutive Model

4.1. Experimental Research on Deep-Sea Sediment

Pressure–sinkage tests were performed by Xu Feng et al. [26] using model sediments based on the mechanical characteristics and mineral components of deep-sea sediments in a C-C poly-metallic nodule exploration area in the Pacific Ocean. Using montmorillonite and diatomite as the main raw materials, sediments with various moisture contents were prepared. The wet density of the sediment was 1.315 g/cm³, the water content was 165.6%, the cohesion was 6.2 kPa, and the friction angle was 1.72°. The measuring plate was a standard cuboid with a length of 450 mm, a width of 450 mm, and a height of 500 mm. Pressure–sinkage experiments were performed under various ground pressures (5–25 kPa) by applying a specific stress to the prepared soils to obtain the sinkage vs. loading time curves shown in Figure 12. Here, the sinkage depth of the model deep-sea sediment is seen to increase with increasing ground pressure. In addition, the loading time has a large influence on the sinkage depth, especially during the initial and secondary pressure–sinkage stages, such that the sinkage depth increases with increased loading time under a given ground pressure.

With further processing, the curves of sinkage rate against time at ground pressures of 5 kPa and 20 kPa were obtained (Figure 13). In each case, the sinkage rate is seen to decrease sharply and then more gradually with time, eventually stabilizing at close to zero. This indicates a transition from decaying sinkage to stable creep, such that each curve can be divided into the following three regions: (1) the initial pressure–sinkage region, in which the sinkage rate decreases rapidly with time, (2) the secondary pressure–sinkage region, in which the sinkage rate decreases more gradually with time, and (3) the stable creep region, in which the sinkage rate is close to zero.

4.2. Determination of the Pressure–Sinkage Parameters Based on the Experimental Data

Based on the above experimental results, the instantaneous elastic modulus $\left(E_1\right)$ is obtained under various ground pressures using Equation (11), and the results are plotted in Figure 14. Here, $E_1$ is seen to vary in direct proportion to $P$, giving the relationship expressed by Equation (20):

$$E_1=0.00527P+3.546$$

(20)

Under the action of pressure, hydrodynamic force, and disturbance by subsea mining vehicles, the pressure–sinkage parameters of the deep-sea sediment gradually change with time and with specific ground pressure. Therefore, the principle of time–stress superposition can be used to analyze the evolution of the mechanical properties of the deep-sea sediment. The elastic modulus ($E_2)$ and the viscosity coefficients $\left({\mathsf{\eta }}_1 \mathrm{and} {\mathsf{\eta }}_2\right)$ have a functional relationship with ground pressure and time, as summarized by Equation (21):

$$\left(E_2,{\mathsf{\eta }}_1,{\mathsf{\eta }}_2\right)=f\left(P,t\right)$$

(21)

Further, it is assumed that $E_2$, ${\mathsf{\eta }}_1$ $\mathrm{and}$ ${\mathsf{\eta }}_2$ can be expressed as a function of the product of ground pressure and time during pressure–sinkage, as per Equation (22):

$$\left(E_2,{\mathsf{\eta }}_1,{\mathsf{\eta }}_2\right)=f\left(Pt\right)$$

(22)

To illustrate the relationship in Equation (22), and to fully characterize the deep-sea sediment sinkage curves, the stable creep stage duration under various ground pressure conditions is selected, and the two-dimensional correlation diagrams shown in Figure 15 are obtained. Here, it can be seen that while the specific trends for E₁, η₁, and η₂ are distinct, each pressure–sinkage parameter exhibits a consistent trend under various ground pressure conditions. Thus, η₁ and η₂ increase gradually with increasing Pt, while E₂ gradually decreases with the increase in Pt. These trends reveal that the elastic coefficient gradually decreases, the viscosity coefficient increases, and the deep-sea sediments exhibit complex rheological properties, with near-linear to nonlinear characteristics, during the pressure–sinkage process.

Based on these results, a regression analysis is performed herein using the logistic algorithm in Matlab, with Pt as the independent variable, and the parameters $E_2$, ${\mathsf{\eta }}_1,{\mathrm{and} \mathsf{\eta }}_2$ as dependent variables. Thus, after introducing the estimated equation for each pressure–sinkage parameter into Equation (11), the pressure–sinkage constitutive model for deep-sea sediment considering the time variability of the mechanical properties is established as Equation (23):

$$Z=\frac{P}{E_1\left(P\right)}+\frac{P}{E_2\left(Pt\right)}\left[1-\exp \left(-\frac{E_2\left(Pt\right)}{{\mathsf{\eta }}_1\left(Pt\right)}t\right)\right]+\frac{P}{{\mathsf{\eta }}_2\left(Pt\right)}t$$

(23)

4.3. Credibility Analysis of the Pressure–Sinkage Constitutive Model

In this section, a fitting analysis of the deep-sea sediment experimental data under various ground pressures is performed in order to verify the rationality of the pressure–sinkage constitutive model. The prediction results of the as-proposed model are compared with the experimental results, and with those of the Kelvin model, in Figure 16. Here, the as-proposed pressure–sinkage constitutive model fully describes the evolution law of the deep-sea sediment in each sinkage stage. Although some of the model data have certain deviations from the experimental data, the model has good consistency with the experimental data, and the fitting correlation coefficient (R²) is greater than 0.95. Furthermore, when P = 5 kPa, the predicted value is generally higher than the experimental value, thereby showing a positive deviation. When P = 25 kPa, however, the predicted value is generally lower than the experimental value, showing a negative deviation. The mean square error (MSE) values of the model prediction results are 0.014 and 0.048 under P = 5 kPa and P = 25 kPa, respectively, which further indicates that the proposed model can fit the experimental data well. Furthermore, the as-proposed pressure–sinkage constitutive model exactly captures the amount of sinkage of the deep-sea sediment with fewer parameters than the Kelvin model. In summary, the proposed model can better predict the pressure–sinkage of deep-sea sediment, and thus provides a reference for the design of subsea mining vehicles.

5. Conclusions

Herein, the mechanisms of pressure–sinkage and microscopic evolution of deep-sea sediment are investigated in order to construct a relevant constitutive model according to the physical properties of the deep-sea sediment. Thus, drawing on the stress loading time and the rheological model theory, a four-element model was introduced that can describe the rheological characteristics of the entire process of deep-sea sediment evolution under external load in terms of the distinct pressure–sinkage stages. The parameters of the model were then determined via a novel calculation method, and an improved pressure–sinkage constitutive model for deep-sea sediment that considers the time variability of the mechanical properties was established by using the time–stress superposition principle. Finally, the accuracy of this model was evaluated using experimental data. The following conclusions were drawn:

• The deep-sea surface sediments have obvious vertical stratification characteristics, which can be separated into the following three sections: (i) the suspension layer, (ii) the semi-liquid layer, and (iii) the consolidated layer. According to the relationship between the sinkage depth and time, the pressure–sinkage process can be divided into four stages, thereby revealing the microscopic mechanism and rheological characteristics of the deep-sea sediments during pressure–sinkage.
• The pressure–sinkage is greatly affected by the ground pressure and time, according to the model parameters that were determined via the new calculation method. Based on the physical mechanism and time–stress superposition principle, a new pressure–sinkage constitutive model for deep-sea sediment considering time-variable mechanical properties was established.
• The pressure–sinkage constitutive model agrees well with the results of the experimental tests, which verifies that the model can effectively describe the four stages of the pressure–sinkage process of the deep-sea sediment. The accuracy of the model reaches 95%, which can provide an important theoretical basis and technical support for the optimal design of deep-sea mining systems. In addition, the pressure–sinkage constitutive model of deep-sea sediment not only has important academic value for enriching and developing soil mechanics, but also has great engineering significance and provides a design basis for project fields such as subsea cable trenching, marine pile foundation stability, deep-sea space station, etc.