A Fully Coupled Tribocorrosion Simulation Method for Anchor Chain Considering Mechano-Electrochemical Interaction

Abstract

This study aims at proposing a fully coupled numerical simulation method of tribocorrosion development on anchor chains during service life, where the mechano-electrochemical interaction is considered in a simplified way. The damage evolution can be realized by a user-defined UMESHMOTION FORTRAN subroutine, where both stress-accelerated corrosion and corrosion-accelerated wear can be considered. Based on this numerical method, the time-variant damage morphology of mooring chain can be obtained. Simulation results obtained by different damage evolution models are shown and compared, and some discussions on the simplified simulation method of reciprocating tribocorrosion are also presented. A systematic parametric study is carried out, and the key factors affecting the tribocorrosion of chain link are revealed. Finally, a modified design method is proposed, and it can be used for optimization of the design of marine anchor chains.

1. Introduction

The mooring cable is an important part of marine floating structures, which invariably consists of high-strength steel chains in the upper reaches and on the seafloor, as shown in Figure 1 [1,2,3]. The mooring chains exposed to a corrosive environment have periodic motion; therefore, reciprocating wear (in the contact region of inter-link as well as the touch-down point on the sea floor) and corrosion of the mooring chain occurs [4,5] and may cause premature failure of the mooring system. Due to the mechanical-chemical interaction, the tribocorrosion (refers to both wear and corrosion acting simultaneously) damage is much more severe than the simple addition of each factor acting separately. Typical damage morphology of mooring chain is shown in Figure 2.

Generally, the design of mooring chains includes a nominal allowance for corrosion loss, and the value is highly dependent on field experience [1]. However, the allowance for chain wear has not been considered explicitly. Hence, it is important to develop an approach so that the damage evolution can be accurately predicted. Even though some codes, such as API SPEC 2F [4] and DNVGL-OS-E302 [5], have addressed the design methods for mooring chains, an optimized design method to determine the allowance for tribocorrosion has not been proposed.

To obtain accurate tribocorrosion predictions, consideration should extend not only to corrosion and wear damage, but also the mechanical-chemical interaction. To date, many corrosion models have been proposed [6,7,8,9], however, most of them did not explicitly take the effect of applied stress into account. It has been proved by many studies that the existing stress promotes damage evolution and leads to early failure of engineering structures [10,11,12,13,14]. There are some works concerning mechano-electrochemical interaction on damage evolution [15,16,17,18,19,20,21]; in these works, the mechano-electrochemical model proposed by Gutman [15] was frequently used. It shows that the applied stress results in the equilibrium potential moving in the negative direction, leading to an increase in local current density [16,17,18,19,20], and the effect of plastic strain is more significant than elastic strain [16,21,22]. However, these methods required the calculation of electric field and stress distribution, leading to a high computational cost, and the time-dependent damage morphology cannot be obtained. More recently, a stress-corrosion model was proposed by Yang [23] based on the electrochemical test, where the stress acceleration effect was considered in an explicit way. Based on the study by Yang [23], Wang [24,25] proposed a simplified simulation method to predict the damage evolution of high-pressure pipe and H-piles.

For wear damage prediction, Archard’s [26] wear model and its modified version are the most widely used [27]. Based on the available research, Liu [28] evaluated the cumulative corrosion-wear damage of anchor chain by the finite element (FE) method. The wear-corrosion damage was calculated by the decoupling method, where a uniform corrosion rate was applied according to the available corrosion model, and the wear damage was calculated according to Archard’s wear model. A similar method was also adopted by Su [29]. However, the mechano-electrochemical interaction was not considered in these works.

Considering that mechanical-chemical interaction is important in tribocorrosion, whereas it has not been considered carefully, this work aims at proposing a simplified method for tribocorrosion simulation, where the mechanical-chemical interaction behavior can be explicitly considered, and providing some guides to optimize the design of anchor chain.

2. Numerical Model and Simulation Scheme

In this section, the numerical model and simulation scheme are presented. The simulation was conducted using the general-purpose FEM software ABAQUS [30], where the mechano-electrochemical interaction on the damage evolution was realized by a user-defined FORTRAN subroutine, UMESHMOTION. A schematic plot of the numerical model is shown in Figure 3a. Due to the symmetry of geometry and load, a half model was established, where the end of the upper chain was fixed and the lower chain was constrained as a rigid body to improve the computational efficiency. The boundary conditions of the lower chain were applied on a reference point (RP); all degrees of freedom of this point are constrained except for the translational degree of freedom along the Y-axis and the rotational degree of freedom about the X-axis. It should be noted that the boundary conditions used in this work are indeed stricter than those in practical applications. A more consistent numerical model of the chain link can be obtained by making the horizontal displacement free on just one side of the model. The simplified boundary conditions were used in this work for the following reasons: (1) only axial load was applied in the numerical model, so there is no lateral displacement since no lateral load was applied; (2) the convergence of numerical simulation can be improved when the horizontal displacement is constrained; otherwise, the lateral stiffness can only be provided by contact, which may bring a convergence problem. The contact between two chain links was simulated by the surface-to-surface contact algorithm, and the friction coefficient was set as 0.3. As tribocorrosion is a surface damage evolution process, the mesh size in the contact region should be fine enough to accurately predict the surface stress distribution. Mesh independence was determined before further analysis, and 20 elements were used in the circumferential direction of the chain link every 90 degrees in the contact area. The bias ratio of mesh size was 5 in this study, a finer mesh was used in the contact region, and a coarser mesh was used in the other domain; the total element number of each part was 11,500. An eight-node linear brick element (C3D8) was adopted to discretize the geometric model (Figure 3b).

2.1. Material Parameters

The material of Q235 steel with a bi-linear elastoplastic constitution model was adopted in this study, and all of the values of material parameters used can be seen in

Table 1. Material parameters of the numerical model obtained from Yang [23] and Yu [31].

Material	E (GPa)	σy (MPa)	σ0.2 (MPa)	σu (MPa)	Elongation
Q235 [23,31]	206.1	235	269	476	26 ± 1%

. The typical diameter dimension of mooring chain used in buoys is 14 mm, and therefore, that value was used in the numerical analysis.

2.2. Stress-Corrosion Evolution Model

It is well-known that the presence of stress enhances the corrosion because of the effect of mechanical strain on the electrochemical thermodynamic activity of the steel and the corrosion scale. On one hand, a mechanical deformation can lead to a redistribution of electrochemical heterogeneities and increase the area for cathodic reaction [15]. In addition, an increase in slip steps, micro-cracks and surface defects generated during plastic deformation reduces the activation energy of hydrogen evolution [16]. On the other hand, the effect of applied strain on the corrosion scale has been illustrated by Xu [17], who indicated that the tensile strain could expand the porous corrosion scale and enhance the entry of corrosive species.

To date, there are some works focusing on the damage evolution of corroded structures subject to tensile stress, such as the works conducted by Xu [16], Sun and Cheng [18,19,20]. However, the methods used were complicated. Both the stress distribution and electric field should be calculated to obtain the local corrosion current density, and then the corrosion rate determining the local damage evolution can be calculated based on Faraday’s law. In this work, a more simplified method is proposed based on the test data of Yang [23], and the stress-corrosion (SC) model can be written as [24]:

$$\begin{array}{l}\mathrm{mean}\left(\lg v_{cr}\right)=a\exp \left(b\frac{\sigma }{{\sigma }_y}\right)\\ \mathrm{std}\left(\lg v_{cr}\right)=c{\left(\frac{\sigma }{{\sigma }_y}\right)}^2+d\end{array}$$

(1)

where v_cr is the corrosion rate (mm/year), a, b, c and d are empirical parameters determined by electrochemical experiment, σ and σ_y are applied stress and the yield stress of materials, respectively, and Std is the standard deviation. The values of each parameter are shown in

Table 2. Model parameters [23,24].

Model Parameters	a	b	c	d
Value	−0.69455	−0.27111	−0.0207	0.01993

. For numerical analysis, the stress σ was replaced by Von Mises (σ_Mises). As the stress corrosion test was conducted in NaCl solution with a mass fraction of 3.5%, the simulation results represent the wear-corrosion behavior of chain link in sea water.

2.3. Wear and Corrosion Accelerated Wear Model

The volume loss (V, mm³) of the material under a normal force (F, N) over a sliding distance (s, mm) can be described by Archard’s wear model [26]:

$$V=\frac{K}{H}Fs$$

(2)

where K is the dimensionless wear coefficient, and H is the hardness of the material (N/mm²). Dividing both sides of Equation (2) by the contact area (A, mm²) and test time (t) yields the following equation:

$$v_w=\frac{h_w}{t}=\frac{V}{At}=\frac{K}{H}\frac{F}{A}\frac{s}{t}=\frac{K}{H}P{V}_{slip}$$

(3)

where v_w is the wear rate (mm/s), h_w is the wear depth (mm), P is the contact pressure (N/mm²), and V_slip is the slip rate (mm/s).

It has been pointed out that the corrosion-enhanced erosion is related to the degradation of surface hardness of target materials, and the surface hardness reduces with increasing anodic current density [32,33]. The relative hardness degradation (△Hv/Hv₀) is just a linear function of the logarithm of the anodic current density, i_A, as shown in Figure 4. As the erosion resistance of the target material is obviously reduced with the decrease in hardness [32,33,34], the effect of hardness degradation should be considered. An approximation of the dependency of hardness degradation on the anode current density can be written as follows:

$$\frac{\Delta H\mathrm{v}}{H\mathrm{v}}=m\ln \left(i_A\right)+n$$

(4)

where m = −0.011, m = −0.0824 are constants; both values were obtained based on the test data of Guo [32] and Lu [33]. Based on the coupling model shown in Equation (4), the effect of corrosion on erosion can be taken into account.

Note that the anodic current density, i_A (mA/cm²), is currently unknown; the value can be calculated based on Faraday’s law and the corrosion rate (v_cr) obtained in Section 2.2:

$$i_A=\frac{nF\rho }{M}{v}_{cr}$$

(5)

where M is mole mass of steel (56 g/mol), n is charge number (n = 2 for steel), F is Faraday’s constant (96,485 C/mol), and ρ is the density of steel (7850 kg/m³). In this way, the wear-corrosion interaction can be considered, and the hardness of the surface is:

$$H=H_0+\Delta H\mathrm{v}=H_0+H_0\left[m\ln \left(i_A\right)+n\right]$$

(6)

where H₀ is the original hardness of the steel.

To improve the computation efficiency, the wear results are scaled. Similar acceleration factors were employed in earlier studies [35,36], as the wear coefficient is nearly a constant. Expressing Equation (3) in an infinitesimal form with respect to the sliding distance at a node ‘i’ yields

$$v_{w,i}={\displaystyle \sum _{j=1}^n{S}_f\frac{K}{H}{P}_j}{V}_{slip,j}$$

(7)

where S_f is the scaling factor, H is the surface hardness based on Equations (1), (4) and (6), P_j and V_slip,j are the contact stress and the slip rate at the j-th increment step, respectively, and n is the total increment number. The coefficients of the wear model are shown in

Table 3. Model parameters [28].

Material	K (Dimensionless)	H0 (GPa)	Cycle of Motion, T (s)	Angle of Reciprocating Motion, θ (°)	Ring Diameter, d (mm)	Friction Coefficient
Value	1 × 10−6	2.5	12	1	14	0.3

.

2.4. Linear Damage Accumulation Rules Considering Mechano-Electrochemical Interaction

The total tribocorrosion damage is treated in a linear damage accumulation way, which means the total damage rate v_t can be obtained as follows:

$$\begin{array}{l}{v}_t=v_{cr}+v_w\\ \{\begin{array}{l}{v}_{cr,i}={10}^{a\exp \left(b\frac{{\sigma }_{Mises,i,j}}{{\sigma }_y}\right)}\\ v_{w,i}=S_f\frac{K}{H\left(i_A\right)}{P}_j{V}_{slip,j}\end{array}\end{array}$$

(8)

where v_i is the corrosion or wear rate at node i, and the subscript j is the j-th incremental step. It is worth noting that the unit of time should be kept consistent when such a method is applied.

2.5. UMESHMOTION Subroutine Realization

Based on the material parameters and damage evolution model mentioned above, a user defined FORTRAN subroutine was developed to assess the time-variant damage morphology of chain link. For this analysis, two simulation steps are required, which are shown in Figure 5, and the flowchart is presented in Figure 6.

Firstly, a user-defined Python program is developed to establish the chain link model; the material parameters, loading conditions and the boundary conditions are also defined, and then the initial stress distribution can be obtained.

Secondly, the tribocorrosion damage is obtained. In this step, an Arbitrary Lagrangian-Eulerian (ALE) meshing domain should be defined, which corresponds to the domain suffering tribocorrosion damage. The motion of element node in the ALE domain is controlled by the user-defined UMESHMOTION FORTRAN subroutine. The corrosion rate at each node can be obtained based on the SC model defined by Equation (1); so, as the surface stress distribution, slip rate are known, then the damage evolution is simulated by adjusting the node coordinates based on the calculated local damage. Subsequently, the new corrosion morphology can be obtained, and such new damaged morphology is used as the geometry model for the calculation of stress distribution at the next increment. It should be noted that the tribocorrosion only occurs in the contact region. The advantage of this method is that the geometry of interest can be modified continuously by ALE meshing, and the damage evolution can be visualized continuously.

Because both SC model and wear coefficient are derived from elastic test conditions, the stress is limited to elastic range, which means a cutoff value of stress (σ_y), including von Mises stress and contact stress, is used in the FORTRAN subroutine to avoid unrealistic stress acceleration effect.

3. Model Validation

Before conducting further parametric studies, a basic model validation was carried out to confirm the reliability of the numerical simulation, where a uniform corrosion rate (0.2 mm/year) was applied, and the simulation results were compared with theoretical values (see Figure 7), indicating that the simulation results match well with the theoretical values. A small deviation between the numerical and the theoretical value was mainly caused by the axial force-induced deformation.

4. Results and Discussion

The time-variant damage morphology of chain link was calculated by the simulation method presented in Section 2, and the results are shown herein. The evolution of the geometry profile under different damage evolution models was investigated to reveal the effect of mechanical-chemical interaction.

4.1. Comparison of Different Damage Evolution Models

Even though most of the engineering structures are in an elastic deformation stage during operation, local plastic flow may still occur in the chain contact region [2]. Preliminary simulation results indicated that the predicted contact stress given by elastic analysis was too high, which was far beyond the yield stress of the material; therefore, elastoplastic analysis was carried out in this study.

Figure 8 and Figure 9 show the damage evolution of chain link by using different damage evolution models; it is clear that the damage evolution will be underestimated if the mechanochemical interaction is not considered, even though the exposure time is short. The volume loss caused by wear is small, as it is limited to the contact region; however, the local thickness loss at the contact center is much higher if stress-accelerated corrosion and wear are considered. As shown in Figure 8b, the actual damage is underestimated for the uniform corrosion damage evolution model, which is commonly used in practical engineering, and it should be responsible for the premature failure of the mooring system. The fast increase in the displacement U₂ (along Y axis) at the beginning of the tribocorrosion is due to the small contact area. As the tribocorrosion proceeds, the surface of the chain link at the contact center is gradually worn down (Figure 9d); the increasing contact area results in the decrease in contact stress, hence the damage rate decreases. The displacement U₂ shown in Figure 8b also includes the force-induced deformation, which becomes severe as the damage evolves under a constant operational load. Damage morphology shown in Figure 9 also indicates that the damage in the shoulder region and the contact region is more severe. Detail displacement shown in Figure 8b indicates that the corrosion-enhanced wear is marginal under low operational load (F/F_test = 0.1); in this case, only a small discrepancy can be observed when reversing the motion occurs (see dotted circle in Figure 8c). The underlying reason is that the corrosion-induced hardness degradation is small, for a 0.2 mm/year corrosion rate, the anodic current density, i_A, is about 0.017 mA/cm², and the hardness degradation is about 157.4 MPa, which has little impact on the wear rate. The results shown in Figure 8 also indicate that the stress-enhanced corrosion is the most significant. One should notice that such conclusion is valid for the model parameter combination shown in Table 1, Table 2 and Table 3; different phenomena may be observed when a decrease in corrosion rate and increase in wear rate occur.

4.2. Effects of Applied Load and Moving Angle

Figure 10 shows the effect of operational load on damage evolution; a fast increase in volume loss is observed as the applied load increases from 0.1F_test to 0.3F_test (F_test = 0.014D² (44 − 0.08D), unit: kN [4]. D is the diameter of the chain link), while the increase in volume loss rate slows down as the load further increases. The reason is that a “cutoff value” of the von Mises stress (equal to the yield stress) was used in the FORTRAN subroutine to avoid an unrealistic stress acceleration effect. In addition, the increase in applied load has a great impact on the displacement U₂ at the contact center due to significant plastic deformation and a higher damage rate (see Figure 11c). The simulation results shown in Figure 11 indicate that the corrosion-enhanced wear is marginal; for the wear material parameters shown in Table 3, such an effect can be ignored.

Considering that the corrosion-enhanced wear is not significant, only stress-corrosion and wear were simulated in the following part. The volume loss and displacement U₂ at the contact center under different moving angles are shown in Figure 12. Again, due to the local damage characteristic of wear, a change of parameters has little impact on the volume loss. However, the damage depth at the contact center varies considerably under different loading conditions. The simulation results also indicate that the wear damage can be well-controlled by applying either wear-resistant material or motion inhibition measures.

4.3. Effect of Wear Coefficient

Case studies with different wear coefficients were carried to investigate the effect of the wear coefficient on damage evolution. The results are shown in Figure 13. The corrosion-enhanced wear is not considered, so the hardness is not affected. Generally, the wear coefficient is closely related to the hardness; however, such dependence is not discussed in this study. The results shown in Figure 13 indicate that there is no significant additional volume loss due to the increase in wear coefficient, since wear only occurs in local contact areas. However, a significant local material loss is observed when materials with low wear resistance are used, and a significant wear scar (see Figure 1 and Figure 2) appears after long-term service.

One should notice that the tribocorrosion simulation framework proposed in this work still has some limitations. The prediction accuracy is guaranteed when the model coefficients are validated by test data. Nonetheless, the simulation method proposed in this work is universal. Moreover, the model is only validated in the elastic range. However, local plastic flow may still occur in the chain contact region [2], and the corresponding wear damage and corrosion damage may be underestimated in this case [15,16,17,18,19,20,21]. To predict the damage under plastic flow conditions, the model needs to be further improved.

4.4. A Modified Chain Link Design Method

A modified design method for chain link is also proposed in this study based on the method mentioned in Section 2. By applying this method, an accurate corrosion allowance can be determined. First, a basic dimension is determined (see Figure 14a), which must satisfy the minimum strength law (F_test). Then, the material deposition is simulated, and the required allowance dimension can be obtained by considering design load, environment conditions (wave period and moving angle) and service life.

The representative configuration and the volume change are shown in Figure 14 and Figure 15, respectively. The unsymmetric stress distribution is caused by the motion of the ring. The simulation results show that the deposition rate is approximately linear under the loading condition used in this study. However, it is noteworthy that the exposure surface is becoming larger over time, and the approximately linear increase in material deposition volume indicates a decrease in the deposition rate.

5. Conclusions

In this work, a numerical simulation method of tribocorrosion was proposed, where the mechanical-chemical interaction was considered in a simplified way. A user-defined Python program and FORTRAN subroutine were developed to establish the novel numerical model. Based on the present study, the following conclusions can be drawn:

• The mechano-electrochemical interaction of tribocorrosion can be considered by the proposed simplified method, and the time-variant damage morphology can be obtained.
• Ignoring mechanochemical interaction leads to significant underestimation of damage, especially for stress-accelerated corrosion, and also leads to unsafe design. Additionally, as the hardness reduction caused by corrosion is very limited, the corrosion-accelerated wear is marginal, which can be ignored.
• Reducing service load is beneficial to reduce tribocorrosion, and the recommended operational load is 0.3 times the test load.
• The proposed modified design method can be used to obtain an optimized configuration and allowance dimension for chain link.