A Pressure-Sinkage Model for Deep-Sea Sediments Based on Variable-Order Fractional Derivatives

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Introduction
Te deep sea contains rich mineral resources.Among them, polymetallic nodules have sizable reserves and are considered to be the most commercially promising resources [1][2][3].At present, the deep-sea mining system is mainly composed of four parts: the surface mining vessel, the pump pipe conveying system, the subsea mining vehicle, and the system collaborative control simulation platform, which cooperate to lift polymetallic nodules with a depth of 5,000-6,000 meters to the surface [4][5][6][7].Due to the complexity of the depositional environment, material source, and consolidation state, the mechanical properties of deepsea sediments are quite diferent from those of onshore soft clay, which are characterized by low strength, high compressibility, and low permeability [8][9][10].At the same time, subsea mining vehicles are the most critical equipment in pipeline hydraulic transportation systems.Subsea mining vehicles require adaptability and working reliability, and their stability directly determines the success or failure of mining.A schematic diagram of a subsea mining vehicle is shown in Figure 1.Subsea mining vehicles are in direct contact with the seafoor, relying on the shear resistance of the deep-sea sediments to pull the crawler.Terefore, it is important to analyze the mechanical properties of the deepsea sediments and their interaction mechanism with the subsea mining vehicle, establish the pressure-sinkage model on deep-sea sediments, and reveal the infuence of model parameters on the driving performance of the subsea mining vehicle.Tese factors provide a theoretical basis for the development and design of an undersea walking vehicle.However, due to the difculty of carrying out in-situ experiments at the current level of technology, researchers often use mixtures of bentonite and water instead of deepsea sediments to conduct experimental studies and obtain the mechanical properties of their interactions [11][12][13][14][15].
In recent years, domestic and foreign researchers have carried out a series of studies on the pressure-sinkage behaviours of deep-sea sediments [7,[16][17][18].Te Bekker bearing model has become the theoretical basis for analyzing the interaction characteristics between vehicles and the ground [19].Schulte et al. [11] proposed the empirical pressure-sinkage model according to the stress-strain characteristics of deep-sea sediments using the classical Bekker theory and also briefy introduced the calculation method of static load and the variation trend of pressure sinkage with time.Wang et al. [17] obtained the pressuresinkage curve of simulated sediments by carrying out pressure-sinkage tests and established a corresponding deep-sea sediment pressure-sinkage model based on the Bekker empirical model.According to the physicochemical properties of deep-sea sediments, Li [20] confgured simulated soils with similar physical properties and established the shear stress-displacement and pressure-sinkage relationship equations based on the Bekker theory and the Reece theory.Compared with the abovementioned static prediction methods, related scholars have also proposed component models that consider the rheological properties of deepsea sediments.Te component model mainly characterizes the viscoelastic-plastic properties of rock through diferent combinations of standard elements [21,22].Innovative to deep-sea mining research, Qi et al. [14,23] systematically studied the rheological properties of deep-sea sediments for the frst time and established the rheological constitutive equations of deep-sea sediment-simulated soils under various loads.At the same time, the traction-creep characteristics of deep-sea sediment simulated soil were analyzed using a fractional derivative and the Burgers creep model.Xu et al. [24] used the Kelvin-Hooke and Burgers rheological models to analyze the compressive creep behavior and compressive-shear creep behavior of the deep-sea sediment simulated soil, employing TableCurve-3D to determine various parameters.Te model-ftting results validate the experimental data, which can better refect its rheological properties.According to the elastic-plastic theoretical model, Wang et al. [25] proposed a deep-sea sediment shear stress-displacement relationship and verifed the accuracy of their model through the deep-sea sediment shear test.Te abovementioned constitutive models describe the mechanical properties of deep-sea sediments well, but due to diferences between deep-sea sediments and terrestrial soils, whether the abovementioned empirical models are applicable to deep-sea sediments remains to be verifed.At the same time, the deep-sea sediment pressure-sinkage process has obvious stage characteristics, and the fractional order model cannot describe the mechanical evolution of deep-sea sediments.
Historically, the variable-order fractional derivatives have attracted the attention of many researchers [26].Valério and Sá da Costa discussed the defnition of variable-order derivatives and their approximations under real complex numbers.Also, the variable-order approximations were obtained by using existing approximations for constant orders [27].Moghaddam and Sá da Costa developed fnite diference approach schemes for the variable-order and accuracy of the algorithm, which were verifed by numerical examples [28].Ortigueira et al.
modifed the previously proposed variable-order fractional derivatives and introduced a new approach based on the inverse Laplace transform [29].Almeida et al. systematically introduced the theory of fractional calculus and the calculus of variations, and they presented a new numerical tool for the solution of diferential equations involving Caputo derivatives of fractional variable order [30].Variable-order fractional derivatives have powerful properties for building nonlinear constitutive models and are thus widely used in various felds of natural science and engineering applications [31][32][33][34], such as biome analysis [35], mathematical modelling [36], infectious disease transmission [37], medical image denoising [38], and tumor growth analysis [39].Yet, no studies have established the pressure-sinkage model of deep-sea sediments based on the variable-order fractional diferential theory.
In order to analyze the mechanical properties of deep-sea sediments and their interaction mechanism with subsea mining vehicle tracks, this article establishes the pressuresinkage constitutive model for deep-sea sediments.Using the variable-order fractional order theory, the pressuresinkage constitutive model for deep-sea sediments was proposed according to the pressure-sinkage characteristic curve, and the mechanical properties of deep-sea sediments were revealed.Ten, by comparing with the experimental results from other literature, the new model is verifed to predict the pressure-sinkage process of deep-sea sediments under static and dynamic loads.Finally, a sensitivity study of model parameters is carried out using the control variable method, and the efects of grounding pressure and time on the law and mechanism of deep-sea sediment pressuresinkage are revealed.

Basic Theory of the Fractional Derivative
2.1.Fractional Order Calculus.Fractional order calculus is an extension of integer order calculus and represents an efective tool for studying derivatives of arbitrary order [40].Tere are several defnitions of fractional derivatives, such as Riemann-Liouville, Caputo, and Grunwald-Letnikov et al.Te Riemann-Liouville (RL) fractional derivative frst integral and then derivative operation sequence have certain advantages for model solving [41].Terefore, the Riemann-Liouville (R-L) fractional order calculus is chosen to defne and describe the pressure-sinkage properties of deepsea sediments in this article.Te Riemann-Liouville integral of β order is defned by where t is time, f(t) is the function in the integrable interval [0, t], Γ (β) is the gamma function, β is the fractional order greater than 0, and τ is the independent variable used for the Laplace transform.Correspondingly, the β order diferentiation of the function f(t) is defned as follows: 2 Mathematical Problems in Engineering where n � 1 if 0 < β < 1, and Γ (β) is the gamma function, which is defned as follows: , and the Laplace transform formula for the above fractional calculus is as follows: Te stress-strain relationship expressed in the fractional diferential form is as follows: where δ (t) is the stress, ε (t) is the strain, ξ is the coefcient of viscosity in the element (MPa•t β ), and t is time.When β � 0, let ξ � E. Equation ( 5) can be expressed as follows: where E is the modulus of elasticity of the spring element.In this case, the fractional order element represents the ideal elastic body (see Figure 2(a)).When β � 1, let ξ � α.Equation ( 5) can be expressed as follows: where α is the viscosity coefcient of the dashpot element and the fractional order element represents the ideal viscous body (see Figure 2(b)).When 0 < β < 1, the fractional order element is called the constant phase element, which can describe the nonlinear strain process of a viscoelastic body between ideal elastic and viscous bodies [23], as shown in Figure 2(c).In equation (1), the fractional order β (t) is a function of time, which implies a strong memory of not only the past history but also the fractional order.Equation ( 5) can also be written in integral form which is as follows: when δ (t) � δ 0 and the stress remains unchanged.Using the variable order operator in equation ( 1) and substituting the constant stress into equation ( 8), we can obtain the following equation: Using integration by parts on the right side, equation ( 9) can be expressed as follows: 2.2.Variable-Order Fractional Calculus.Variable-order fractional calculus evolves from fractional calculus, the diference being that for the former, the fractional order changes with the physical process and is a function of time (0 < β � μ (t) < 1).Te μ (t) order integral for the function f (t) is defned as [42] follows: where μ (t, τ) is the order of the fractional derivative, and its value varies with time t.In equation (11), the fractional order μ (t) is a function of time, which implies a strong memory of not only the past history but also the fractional order [42].Tus, we change the fractional order in equation ( 5) into a variable order function which is as follows: When the applied stress remains unchanged, by integrating both sides of equation ( 12) according to the variableorder fractional diferential operator, the constitutive equation of the variable-order fractional element can be obtained, namely, which is as follows: where μ k � μ (t) is the piecewise constant in the time domain, k � 1, 2, 3, . .., n, t k−1 ≤ t ≤ t k , and ξ k is the viscosity coefcient relative to μ k .Since the focus of this study is on the relationship between the ground pressure and pressure sinkage of deep-sea sediments, equation ( 13) is transformed into the following equation: where Z (t) is the amount of pressure sinkage over time and ζ k and λ k are the viscosity coefcients and segmentation constants of the components in the model, respectively; P 0 is the constant ground pressure.

Establishment of the Pressure-Sinkage Model
3.1.Deep-Sea Sediment Pressure-Sinkage Process.Due to the extreme high pressure and the particularity of the sedimentary environment, the microstructure of deep-sea sediments is dominated by the lamellar linking structure and the honeycomb focculation structure, which have superior water absorption.Terefore, its pressure-sinkage process has unique characteristics.Some researchers have investigated the deep-sea sediment pressure-sinkage process by simulating the crawler tooth plate of the subsea mining vehicle [23].Te deep-sea sediment pressure-sinkage process reveals obvious stages, as shown in Figure 3. Te result in Figure 3 illustrates that the sinkage depth of deep-sea sediments increases with time, with clear turning points.Details can be described as follows: (i) Stage I: Te sinkage depth occurs immediately when the load is applied to the deep-sea sediments, such that depth is directly related to the ground-specifc pressure.
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.

Construction of the Model.
Deep-sea sediment rheology is a complex, time-related process in which elastic, viscous, plastic, and other deformations coexist.Te pressuresinkage process explains the interaction mechanism between the subsea mining vehicle and deep-sea sediments.Exploring the pressure-sinkage characteristics of deep-sea sediments represents an important prerequisite for constructing the relationship between load stress and sinkage.Terefore, we defne the whole process of deep-sea sediment pressure sinkage and describe the changing characteristics of mechanical properties.According to the variable-order fractional derivative and the rheological model, a new fourelement pressure-sinkage constitutive model that describes the mechanical properties of deep-sea sediments is established, drawing on the modelling idea of classical element combination.Te pressure-sinkage model for deep-sea sediment is shown in Figure 4.
As shown in Figure 4, the pressure-sinkage model for deep-sea sediment is composed of basic mechanical elements in series.In the pressure-sinkage experiment, the deep-sea sediment will form an instantaneous deformation, independent of time, and the elastic element simulates elastic deformation in the rheological model, where E is the elasticity modulus of the deep-sea sediments.Te later stage is accompanied by viscous and plastic deformation and exhibits elastic recovery as well as some permanent deformation during the unloading process.Te values ζ 1 , ζ 2 , and ζ 3 are the viscosity coefcients at diferent stages, and λ 1 and λ 2 are the orders of fractional order diferentiation at different stages.According to the superposition principle, its constitutive model can be expressed as follows: where Z is the total pressure sinkage of the constitutive model for deep-sea sediments; t 1 , t 2 , and t 3 are the time of the diferent pressure-sinkage stages; P 1 , P 2 , P 3 , and P 4 are the ground pressures; and Z d , Z e , Z c , and Z s are the diferent pressure-sinkage periods.
One advantage of variable-order fractional calculus is that the diferent orders have a certain memory, because the variable order relates to time.We describe the pressuresinkage process of deep-sea sediments accordingly.When 0 t χ � f(t) exhibits the initial pressure sinkage.According to the structural features of the pressure-sinkage constitutive model for deep-sea sediments in Figure 4 and combined with equations 4 Mathematical Problems in Engineering ( 14)-( 16), the pressure-sinkage constitutive equation of this study evolves as follows:

Determination of Pressure-Sinkage Model Parameters.
From equation ( 17), the constitutive model of deep-sea sediments in the instantaneous pressure-sinkage and initial pressure-sinkage stages is described as follows: where E � p/Z(t) � p/Z(0), and equation ( 16) is written as a one-dimensional linear equation which is as follows: Using the infection point data of the deep-sea sediment pressure-sinkage experiment and solving equation (18), the coefcients of equation ( 19) (a 1 and b 1 ) can be calculated.Furthermore, E and ζ 1 are determined from a 1 and b 1 .Similarly, according to equation ( 17), the constitutive model of deep-sea sediments in the secondary pressure-sinkage stage can be expressed as follows: Taking the logarithm of both sides of equation (20), we obtain the following equation: where Z(t 1 ) � P/E + P/ζ 1 t 1 .
Te following relationships are assumed: Ten, equation ( 21) can be rewritten as a linear equation.
Based on the deep-sea sediment pressure-sinkage test data, equation ( 22) is solved to obtain a dataset of x and y.Fitting this dataset, the coefcient values of equation ( 23) (a 2 and b 2 ) can be calculated.Ten, λ 1 and ζ 2 are determined by a 2 and b 2 .
Similarly, the constitutive model of deep-sea sediments in the stable creep stage can be expressed as follows: Taking the logarithms of both sides of equation ( 25), we obtain the following equation: where Z(t 2 ) � P/E P E + P/ζ 1 t 1 + P/ζ 2 (t 2 − t 1 ) λ 1 /Γ(1 + λ 1 ).Te following relationships are assumed: Ten, equation ( 26) can be rewritten as another linear equation.
Based on the deep-sea sediment pressure-sinkage test data, equation ( 27) is solved to obtain a dataset of x and y.Fitting this dataset, the coefcient values of equation ( 28) (a 3 and b 3 ) can be calculated.Ten, λ 2 and ζ 3 are determined by a 3 and b 3 .

Verification of the Deep-Sea Sediment
Pressure-Sinkage Model  Mathematical Problems in Engineering deep-sea sediments [24].A series of pressure-sinkage experiments with diferent ground pressures were carried out, and the pressure-sinkage curve of simulative soil for deepsea sediments in the range of ground pressure P � 5 ∼ 25 kPa was obtained.Because the design value of the ground pressure of the subsea mining vehicle is 5 kPa, the experimental axial pressure range should exceed the design value.As shown in Figure 5, the deep-sea sediments experience instantaneous strain at the moment of loading, and its magnitude increases directly with ground pressure.After the instantaneous elasticity, the test curve is attenuated and stable, and the sinkage depth gradually steadies to a constant value with the increase in action time.When the ground pressure continues to increase, the action time to enter the stable creep stage increases.Under the same ground pressure, the sinkage depth of the deep-sea sediment simulant gradually increased with increasing action time.
In this study, P � 5 kPa and P � 20 kPa.Pressuresinkage test data were selected; the parameter of the pressure-sinkage constitutive model was ftted and analysed, and the pressure-sinkage model for deep-sea sediments under the static load experiment was established.Te sinkage rate curve of simulated soil for deep-sea sediments was acquired, and the sinkage rate appears in Figure 6.
Figure 6 demonstrates that when the ground pressure is between 5 kPa and 20 kPa, the sinkage rate of the deep-sea sediments gradually decreases with time, and the sinkage rate gradually stabilizes and eventually reaches zero.Te pressure-sinkage curve for deep-sea sediments is nonlinear and exhibits obvious stage characteristics.Te greater the ground pressure, the greater the variation range of the pressure-sinkage curve, indicating that the higher the ground pressure, the greater the deep-sea sediment sinkage depth.Tis occurs because the contact area of soil particles becomes larger, the pores of the soil expand, and fuidity increases under the action of the ground pressure.According to the trend of the pressure-sinkage and sinkage rate curves, the values of t 1 and t 2 at P � 5 kPa and P � 20 kPa can be determined, respectively.When P � 5 kPa, t 1 � 3.2273 and t 2 � 32.8951; when P � 20 kPa, t 1 � 8.9441 and t 2 � 67.076.With the increase in ground pressure, the values of t 1 and t 2 will gradually increase.Based on the pressure-sinkage experiment, the deep-sea sediments pressure-sinkage curves evolved from the proposed model, then the model parameters were ftted and analysed using equation (17).Te ftting results of the pressure-sinkage experiment appear in Table 1.Table 1 exhibits that, during the sinkage of deep-sea sediments, the fractional order decreases gradually with time.Te change of pores under external force represents an important manifestation of the pressure-sinkage of deep-sea sediments.Under external force, the pore water of deep-sea sediments discharges continuously, the soil viscosity increases, and the evolution of mechanical properties is roughly the same as the movement law of pore water.
As shown in Figure 7, the pressure-sinkage curves calculated by the pressure-sinkage constitutive model, presented in the article at diferent ground pressures, are consistent with the experimental data of deep-sea sediments.Te proposed pressure-sinkage model based on the variable-order fractional derivative can accurately describe the sinkage process of deepsea sediments under static loads and can fully refect the evolution law and mechanism of deep-sea sediments under the diferent pressure-sinkage stages.In order to further verify the accuracy of the proposed model, we used R 2 , MAPE, and MFE to evaluate the model ftting curve and the ftting accuracy.Te ftting correlation coefcient R 2 is greater than 0.98.From the calculated MFE values in Table 1, when P � 5 kPa, the predicted value of the model is generally lower than the experimental data (negative deviation).When P � 20 kPa, the predicted value of the model is generally higher than the experimental data (positive deviation).Te MAPE values of the pressure-sinkage prediction results at P � 5 kPa and P � 20 kPa are 1.51% and 2.47%, respectively.In addition, the comparisons between the variable order model and the constant fractional order model indicate that the pressuresinkage constitutive model presented in the article is reliable.
For the subsea mining vehicle in China, the length, width, and height are 9.2 m, 5.2 m, and 3 m, respectively, and the driving speed is 1 m/s.Terefore, it is necessary to focus on the pressure-sinkage process and change the law of the subsea mining vehicle in the frst ten seconds.Combining equation ( 17) and the measured values in the above article, the sinkage depth of simulated soil for deep-sea sediments in the frst ten seconds is calculated (presented in Figure 8 with measured values).Te error between the calculated and measured values is small, and the results illustrate that the proposed model achieves accurate predictions for the frst ten seconds of subsea mining vehicle operation.

Experimental Research on Deep-Sea Sediment Pressure-Sinkage under Dynamic Load.
A pressure-sinkage experiment of simulated soil for deep-sea sediments under different dynamic load conditions was carried out by Ma et al. [23].Te dynamic load experiment of simulated soil for deep-sea sediments includes weight drops on the measuring plate from a certain height.Ten, according to the law of conservation of mass, the relationship between the impact height of the object and its kinetic energy is established.Te ground pressure is controlled at 5 kPa, 10 kPa, 15 kPa, 20 kPa, and 25 kPa, and the design velocity varies between 0.84 m/s, 1.19 m/s, 1.46 m/s, 1.68 m/s, and 1.88 m/s.When the ground pressure P � 5 kPa, the dynamic load pressuresinkage curve of the simulated soil for deep-sea sediments at diferent walking velocities is shown in Figure 9.
In this study, the experimental data of v � 0.84 m/s and v � 1.68 m/s were selected, the parameters of the pressuresinkage prediction model were ftted and analysed, and the deep-sea sediment pressure-sinkage model under the dynamic load experiment was established.Te sinkage rate curve of simulated soil for deep-sea sediments was acquired, and the sinkage rate is plotted in Figure 10.
As shown in Figures 9 and 10, when v = 0.84 m/s and v = 1.68 m/s, the dynamic load pressure-sinkage curve of the 6 Mathematical Problems in Engineering simulated soil for deep-sea sediments has the same trend as the static load pressure-sinkage curve.Four stages occur in total: (1) Te instantaneous pressure-sinkage stage has nothing to do with time, and deep-sea sediments will form an instantaneous sinkage at the moment of loading; (2) Te initial pressure-sinkage stage, in which the sinkage rate exhibits a decaying characteristic and the rate changes greatly; (3) Te secondary pressure-sinkage stage, where the strain rate exhibits a decaying and stable characteristic; (4) Te stable creep stage, in which the sinkage rate is smaller and remains constant.
According to the trend of the pressure sinkage and the sinkage rate curves, the t 1 and t 2 values when v � 0.84 m/s and v � 1.68 m/s can be determined.When v � 0.84 m/s, t 1 � 39.81182 and t 2 � 55.1132; when v � 1.68 m/s, t 1 � 13.5833 and t 2 � 45.2296.As the walking velocity increases, the values of t 1 and t 2 gradually decrease, and the deep-sea sediment pressure-sinkage process enters the stable creep faster with time.After ftting and analyzing the model parameters under dynamic loading with the above formula, the calculated model parameters and prediction efect evaluation results are shown in Table 2. Deep-sea sediments belong to saturated soils, with the increase in sinkage time, pore water gradual discharge, and viscous response.Te fractional order is consistent with the change in viscosity and represents the evolution of mechanical properties.Te results of this study are the same as those of other researchers [43].
Figure 11 exemplifes that the instantaneous pressuresinkage stage after loading is described correctly by the proposed constitutive model.Meanwhile, the secondary pressure-sinkage stage and the stable creep stage are also expressed correctly.Te correlation coefcients (R 2 ) of the ftted curves are all above 0.97.In addition, the proposed constitutive model has fewer parameters, and the physical meaning of the model parameters is clear, which refects the  According to the piecewise function in equation ( 15), the pressure-sinkage process of the subsea mining vehicle in the frst ten seconds under dynamic load conditions is obtained, as captured in Figure 12.Te calculated value of the proposed constitutive model possesses consistency with the measured value and accurately predicts the pressure-sinkage changes of the subsea mining vehicle in the frst ten seconds under dynamic load conditions.(15) to quantitatively analyse the infuence of the ground pressure P on the proposed constitutive model.Te ground pressure level varies between 2 kPa, 5 kPa, 8 kPa, and 11 kPa.Te results of the pressure-sinkage tests at diferent ground pressure levels are shown in Figure 13.

Parameter Sensitivity Analysis of the
Te sinkage depth of deep-sea sediments directly increases with ground pressure levels.However, ground pressure has no signifcant efect on the sinkage rate.Higher ground pressure levels result in greater sinkage depth in the instantaneous, secondary, and initial pressure-sinkage stages, increasing the delay in entering the stable creep stage.In addition, with the increase in the ground pressure level, the form of the change curve of the secondary pressuresinkage stage of the deep-sea sediments gradually changes to the exponential type.

Efect of the Secondary Pressure-Sinkage Time t 2 .
To analyse the infuence of secondary pressure-sinkage time t 2 on the mechanical properties of deep-sea sediments, the control variable method helps obtain the pressure-sinkage curves under diferent secondary pressure-sinkage times t 2 , as shown in Figure 14.Mathematical Problems in Engineering As shown in Figure 14, the secondary pressure-sinkage time t 2 gradually increases from 27.5 to 42.5.Te frst two stages are unrelated to secondary pressure-sinkage time t 2 ; they do not change with time, but the stable creep stage will gradually increase in duration with increasing secondary pressure-sinkage time t 2 .In addition, the sinkage depth under diferent secondary pressure-sinkage times t 2 will change.As the secondary pressure-sinkage time t 2 increases, the sinkage depth of the secondary pressure-sinkage stage continues to deepen, but the sinkage rate gradually decreases.

Discussion
Te fractional order theory has always been a research hotspot.Te variable-order fractional derivative is an extension of the fractional derivative, and the fractional order is a function of time.It can be closer to the mechanical properties of viscoelastic materials and can characterize their time-dependent mechanical properties.
In this study, a new four-element pressure-sinkage model was established by introducing variable-order fractional derivatives into the modelling idea of classic element combination to describe the full pressure-sinkage regions of deep-sea sediments.Trough ftting a curve to experimental results, the proposed model's accuracy reached 97%.We found that the fractional order gradually decreased with time during the sinkage of deep-sea sediments, implying that the mechanical properties of deep-sea sediments have changed.Te fractional order is consistent with the process of pore water and deep-sea sediment skeleton, which preliminarily reveals the gradual transformation process of sediments from an elastic state to a viscous state during sinkage.

Conclusion
Te mechanical properties of deep-sea sediments and their interaction mechanism with subsea mining vehicle tracks are analyzed better, and the pressure-sinkage constitutive model is established for deep-sea sediments.In this study, a four-element pressure-sinkage model is established by introducing variable-order fractional order into the modelling idea of classic element combination, and the constitutive equations of diferent pressure-sinkage stages are constructed.Ten, by comparing with experimental results from other literature, the proposed model is verifed to predict the pressure-sinkage process of deep-sea sediments under static and dynamic loads.Finally, through sensitivity analysis of the model parameters, the infuence of the ground pressure and time on the pressure-sinkage law and mechanism for deep-sea sediments is revealed.Te following conclusions may be drawn: (1) According to the curve of deep-sea sediment sinkage rate, the pressure-sinkage process is divided into four stages, and the time-dependent mechanical properties of deep-sea sediments during sinkage emerge.

Figure 1 :
Figure 1: Te schematic diagram of a subsea mining vehicle.

Figure 6 :Figure 7 :
Figure 6: Te sinkage rate of the deep-sea sediments under static load for (a) P � 5 kPa and (b) P � 20 kPa.

Figure 8 :
Figure 8: Comparison of calculated and measured values under static load.

Table 1 :
Parameters and errors of the pressure-sinkage model for deep-sea sediments under static load.

Deep- Sea Sediment Pressure-Sinkage Model
Consequently, the ground pressure and action time are the main factors of deep-sea sediment pressure sinkage and exhibit crucial infuence on the construction of the pressuresinkage constitutive model for deep-sea sediments.Terefore, this article will use the control variable method to conduct sensitivity analysis on the above model parameters, study the infuence of ground pressure P and secondary pressure-sinkage time t 2 on deep-sea sediments, and analyse the evolution of model parameters on the dynamic characteristics of pressure sinkage.Te model parameters are selected from the experimental data under static load, and the specifc parameters are as follows:P � 5 kPa, E 0 � 3.572 MPa, ζ 1 � 30.826MPa•S, λ 1 � 0.677, ζ 2 � 50.912MPa•S λ1 , λ 2 � 0.437, and ζ 3 � 189.12 MPa•S λ2 .5.1.Efect of the Ground Pressure P. Based on the experimental data, the above ftted parameters E 0 � 3.572 MPa, ζ 1 � 30.826MPa•S, λ 1 � 0.677, ζ 2 � 50.912MPa•S λ1 , λ 2 � 0.437, and ζ 3 � 189.12 MPa•S λ2 are input to equation

Table 2 :
Parameters and errors of the pressure-sinkage model for deep-sea sediments under dynamic load.
Te Riemann-Liouville theory, variable-order fractional derivatives, and rheological mechanics theory support the establishment of a new pressure-sinkage constitutive model for deep-sea sediments.(2) Te pressure-sinkage constitutive model in this article is validated under static and dynamic loads.Trough experimental data verifcation, the results demonstrate that the proposed model is optimal for simulating the pressure-sinkage process of deep-sea sediments.Te model's accuracy reached 95%, which is satisfactory for engineering applications.(3) Te sensitivity analysis of the main infuencing parameters in the proposed model, including ground pressure P and the secondary pressuresinkage time t 2 , illustrates that ground pressure and time are directly related to the sinkage depth and rate.