Analysis of the Effect of Applied Load on Crevice Corrosion Behavior

. A two-dimensional numerical model incorporating solid mechanics, electrochemistry, mass difusion, and ion migration processes is developed to investigate the load efect on the crevice corrosion. Te model is a transient model of crevice corrosion occurring in cracks of 304 stainless steel in a dilute NaCl solution, and the interaction between stress and electrochemical corrosion was considered. By solving the multiphysical coupling model in COMSOL, the efect of applied load on electrochemical corrosion in the crack tip region was calculated, and the local corrosion current density in the crack tip region with stress concentration within the crack was also calculated by using the Tafel relationship. Te distribution of Fe 2+ ion, Na + ion, CL − ion, and H and O 2 substance concentrations within the crack cavity is predicted by the equation analysis of substance transport. Te results show that metal oxidation is more clearly afected by plastic deformation, the rate of hydrogen evolution reaction increases with stress enhancement, and the oxygen absorption reaction is not afected by stress strain. Te distribution of iron ions, hydrogen, and oxygen within the crack is afected by the electrochemical reaction rate, and the distribution of iron ions, sodium ions, and chloride ions is afected by the electrolyte potential.


Introduction
Austenitic stainless steel 304 is widely used in the nuclear power industry for its excellent corrosion resistance and good machinability. It is mainly used to manufacture the key components in nuclear power plants such as the pressure vessel, core, in-vessel components, push rod, drive mechanism, circuit, piping and coolant pump, steam generator, heat exchanger, etc. [1] In the service process of these components, cracks and other defects could be inevitably induced by complex factors. When the crack length and width meet the conditions for the occurrence of crevice corrosion, corrosion media will enter into the crevice and result in crevice corrosion. Once crevice corrosion begins, the corrosion rate will rapidly increase and bring serious damage to the material and components [2].
It is reported that crevice corrosion could initiate due to the solution chemistry diference or electrode potential diference between the inside and outside of a crevice [3][4][5].
Kim et al. investigated the corrosion mass changes of canister candidate materials (Cu, Ni, Ti, and SS304) in an oxic groundwater solution. Te results show that the Ni and Ti electrodes did not demonstrate any signifcant changes in the presence of chloride ions, whereas the SS304 electrode was slightly increased compared to the absence of chloride ions [6]. By establishing a mechanical submodel, the reaction transport submodel, and a crack repair submodel, Meng et al. quantitively investigated the interconnected infuencing factors in practical engineering. Te calculation results show that compared with electrolyte concentration, the infuence of current density on crack closure rates are more obvious [7]. Meng et al. studied the localized corrosion of alloy 690 TT, and the results show that the scratch groove and the whole deformed region caused by scratching are considered as anode and correspond to the peak of tip current [8]. Unigovski et al. studied the corrosion creep of pure Mg and die-cast AZ91D alloy in borate and 3.5% NaCl solutions, and the results show that the efect of environment on the creep behavior of magnesium is mainly connected with plasticization of metal assisted by chemical reactions [9]. Te applied tensile stress, especially plastic stress, has a signifcant efect on the dissolution of steel [10][11][12][13][14][15].
In this paper, the efects of stress on crevice corrosion were studied by applying diferent displacement loads to the cracked stainless steel, and the local corrosion current density on the cracked surface was also calculated. By calculating the ion concentration distribution in the crack crevice, the infuence of ions concentration on the stress as well as the electrolyte potential was analyzed.

Crack Crevice Corrosion Process
2.1. Chemical Equilibrium Reaction. When corrosion occurs, the main consideration is the anodic and cathodic reactions in the crack tip area; the crack wall is considered as a passivated surface that does not react. Te main anodic reaction of the dissolution of the metal is described as (1), and the main cathodic reactions of the oxygen absorption reaction and the hydrogen evolution reaction are described as (2) and (3) separately [16]. Te hydrogen evolution reaction is described as the single electron production of hydrogen atoms rather than the double electron production of hydrogen molecules. Tis is mainly because hydrogen atoms are dissolved in solution, and their penetration can embrittle the steel and endanger its structural integrity.
Te ion difusion and electromigration occur during the crevice corrosion, so the solution in the crevice is considered static, and only Cl − , Na + , Fe 2+ , O 2 , and H will be considered in the mass transport process. As the length is much larger than the width of the crack crevice, the potential gradient and concentration gradient in the lateral direction can be considered zero, and only the electrochemical parameters in the direction of the length of the crevice and the change of ion concentration in the solution are considered. Moreover, the convection of ions is negligible due to the narrow crevice.

Electrochemical Kinetic Equations.
During the occurrence of crevice corrosion, the potential of the electrolyte solution changes slowly and gradually reaches an equilibrium state; the ability to transport substances is related to the length of the crevice. According to Ohm's law and Laplace's equation, the potential in the crevice is controlled by where k is the conductivity of the electrolyte solution, µS·cm −1 ; w 2 is the width of the crevice, mm; and i is the current density in the crevice, A·m −2 .
For crevice corrosion, the current density is caused by the oxidation of metals, the reduction of oxygen, and the hydrogen evolution [17] and can be described by the Tafel expression according to equations (5)∼ (8).
where i 0 , i is the exchange current density of substance i, A·m −2 ; C O is oxygen concentration, mol·m −3 ; C O,ref is initial oxygen concentration, mol·m −3 ; ηi is activated on overpotential of i, V; A i is the Tafel slope for substance i, V; φ s is the external potential, V; φ l is the electrolyte potential, V; and E eq is the equilibrium potential, V.
Te additional chemical potential of atoms caused by dislocation in plastic metals will change the standard potential of metals [18] and could be expressed as equations (9)∼(11): where ΔE ae,eq and ΔE ap,eq are the shifts of the equilibrium potential of an anodic reaction under elastic and plastic deformations, respectively, V; E Fe,eq is the equilibrium potential of iron under the condition of no elastic-plastic deformation, V; ΔP is the external excess pressure experienced by the metal, and it will be denoted with the absolute value of the hydrostatic part of the stress tensor, and equals to hydrostatic pressure in Comsol calculations, Pa; V m is the mole volume of metal, m 3 /mol; F is the Faraday's number, c/mol; T is the absolute temperature, K; R is the gas constant; z is the ion valence; v a is an orientation-dependent factor; α is a coefcient; and N 0 is the initial density of dislocations prior to plastic deformation, m −2 . Te exchange current density in the cathodic reaction subject to elastoplasticity is calculated by the following equations [19]: where i 0 , i , ref is the exchange current density for substance i without external stress and strain, A·m −2 ; and σ Mises is von Mises stress.
2 Science and Technology of Nuclear Installations

Dilute Matter Transport Equation in Electrolyte Solution.
A mathematical model of the chemical and electrochemical environment in the crack crevice is established based on the mass transport equation of chemical substances in electrolyte solutions, which are considered diluted electrically neutral solutions. Te transport of substances in electrolyte solutions is controlled by three mechanisms: difusion, electromigration, and convection [20]. Considering the crack crevice is narrow, the convection in the crevice could be neglected, so the fux of the substance N i in the solution is given by where N i is fuxes of substances, mol·cm −2 ·s −1 ; u i is the mobility of substance i, C i is the concentration of substance i, mol·L −1 ; φ is the potential, V; and D i is the difusion coefcient of substance i, cm 2 ·s −1 . Te transport of substance i along the crack crevice can be given by the following equations [21]: where R i is the rate at which substance i is produced or consumed in a chemical reaction; i net is the current density of the reaction; Z i is the amount of charge transferred by substance i. Substituting (14) and (15) into (16), the specifc transport equation for substance i is obtained from following equation:

Material.
Te PLD-50 fatigue tensile tester was used to perform uniaxial tensile tests on plate tensile specimens of 304 austenitic stainless steel with a standard distance segment of 20 mm to obtain their uniaxial tensile mechanical properties. Te experiments were conducted at room temperature, and the specimens werestretched to fracture at a speed of 2 mm/min. Te obtained engineering stress-strain curve was converted using the formula to obtain the real stress-strain curve, as shown in Figure 1, and the obtained real stress-strain curve was simulated by adding the hardening function to the plastic deformation behavior in the model to ensure the accuracy of the simulation calculation. Electrochemical parameters are mainly from reference [19], and other parameters are from reference [22][23][24][25], Te main parameters to be used during the simulation are listed in Tables 1 and 2: 3.2. Geometric Models and Mesh. According to the geometric symmetry of the crack gap and the homogeneity of the solution, the shape of the specimen is simplifed as a rectangle with a length of 2W and a width of W, with a crack of depth L 1 and width L 2 on the left side of the specimen, and the crack tip is an arc with a radius equal to r. Te half-simplifed model of the specimen is shown in Figure 2(a), where region 1 is the crack crevice flled by NaCl solution, region 2 is the metal, and R 1 and R 2 are defned as analysis paths. Te overall free triangle mesh is used to divide the model, and the mesh is refned in the crack tip region to achieve high solution accuracy, as shown in Figures 2(b) and 2(c).

Initial Value and Boundary Conditions.
To achieve the assumption of fast reaction kinetics, an external potential φ s on the anode surface at the crack tip of the model was set to a constant potential; the equilibrium potential was set to a constant value; and the main condition of the electrode reaction was set to thermodynamic equilibrium. Te slit wall

Parameters
Values Science and Technology of Nuclear Installations 3 and the crack tip were considered a passivated surface and an electrode surface with activity separately. To exclude other factors from infuencing the crack tip and slit wall, it is assumed that the boundary fux is zero and no hydrogen fows in and out of the model boundary location. Te concentration of the electrolyte solution at the initial moment in the crack crevice is 0.01 mol/L sodium chloride solution; the initial pH of the electrolyte solution is 6.8; according to the Faraday's law, the oxygen reduction reaction will produce oxygen fux at the crack crevice opening and to ensure the authenticity of the simulation results, the oxygen concentration content in the atmospheric concentration is used as the initial value of the oxygen concentration variable. Diferent displacement loads are applied to the upper boundary of the model.

Corrosion of Cracked Surfaces.
For a clearer and more precise analysis of electrochemistry afected by stress-strain, the case without applied load was added to the calculation, when the metal does not produce any elastic-plastic change and the corrosion inside the crack is not afected by the load, and the results of this calculation were set as a control group. Figure 3(a) shows the Mises stresses generated on the crack profle with diferent applied displacement loads. Te stress has maximum value at a crack tip (θ � 0°) and decreases gradually along the crack contour on both sides. When the applied displacement load d � 0.05 mm, there is no stress change in the crack profle from 0°to 59°, and the stress is 355.3 MPa. When d � 0.08 mm, the stress change from 22°to 66°is smaller and the same at 355.3 MPa. Te stresses in all other cases of displacement load are gradually increasing. Figure 3(b) shows the equivalent plastic strain at the crack contour. It can be seen that, at d � 0.02 mm, the stress is small compared to the yield stress, and no signifcant plastic strain occurs, while the other three loads produce large plastic. Figure 4(a) shows the local corrosion current density distribution of the hydrogen evolution reaction on the surface of the crack contour, which refects the corrosion velocity. When the applied load is 0, the local current corrosion keeps consistent around the crack tip contour; when the specifed load displacement d increases to 0.02 mm, even though the crack tip has only elastic deformation and no plastic deformation, according to Figure 3, the corrosion current of hydrogen evolution reaction still has a large change compared to the initial condition of no applied load, and its growth trend is similar to the distribution of von Mises stress, which indicates that the increase in corrosion current density is mainly afected by stress. When the applied load d increased to 0.11 mm, the current density is signifcantly enhanced by the stress and has a peak of −0.1 A/m 2 at the crack tip. Te results show that the corrosion strength increases with increasing stress.     Science and Technology of Nuclear Installations Figure 4(b) shows the distribution of the local corrosion current density over the crack contour for the iron oxidation reaction. When the boundary designation displacement is 0.02 mm, the increase of corrosion current density is not obvious compared to initial condition, but the corrosion current density increased dramatically when the load increased to 0.05 mm, 0.08 mm, and 0.11 mm. By comparing Figure 3, it can be seen that when the load displacement is 0.02 mm, the equivalent plastic deformation produced is small, even though there is also an obvious stress concentration on the crack tip. In the case of the other three applied displacement loads, the plastic deformation is large, so it is indicated that anodic galvanic corrosion is mainly infuenced by plastic deformation. Te larger the boundary displacement load, the stronger the corrosion current on the surface of the crack. As the crack tip has the maximum plastic deformation, the corrosion current density is much enhanced here and decreased with increased angle along the crack contour. When d � 0.11 mm, a larger equivalent plastic strain is generated, i Fe is signifcantly enhanced by 137.49% around 0°-73°than the other regions. Figure 4(c) shows the distribution of the local corrosion current density of oxygen absorption reaction on the crack contour. Te calculation results show that the local current density of oxygen absorption corrosion seems unchanged under diferent loads. Even though the increased load displacement increased the oxygen absorption and corrosion velocity, the oxygen consumption increased correspondingly. Due to the low level of dissolved oxygen inside the crack, the consumed oxygen at the crack tip could only be supplied by the difusion of oxygen via the small crack opening crevice. However, the supply of oxygen could not meet the increased oxygen absorption velocity, so the oxygen absorption reaction seems to be the same with diferent load conditions, and the reaction is controlled by the difusion of oxygen difusion in the crevice. Figure 5 shows the concentration distribution of Fe 2+ in the crevice under diferent load condition. According to (17), the distribution of Fe 2+ is afected by the dissolved metal, its difusion under a concentration gradient, and the electrolyte potential. As metal Fe dissolve as Fe 2+ at the crack tip, the concentration of Fe 2+ is the highest at the crack tip, which leads to the concentration gradient and drives the Fe 2+ difusion along the crevice to the crack mouth, thus a gradually decreased concentration towards the crack mouth. However, the concentration diferences around the crack contour are very small due to the tiny crack tip. With the increase of the load displacement, the concentration of Fe 2+ increases along the crevice, and that is because the dissolve velocity of metal increases with the load displacement. Figure 6 shows the change of electrolyte potential under diferent load conditions. Te larger the external load, the lower the electrolyte potential at the crack tip, which blocks the Fe 2+ difusion towards the crack mouth; however, the efects of electrolyte potential between both ends to block the difusion are smaller than the driving force caused by the concentration gradient. Te combined efects of potential and concentration gradient lead to the Fe 2+ distribution as shown in Figure 6(a). Figure 7 shows the distribution of hydrogen concentration in the crevice under diferent loads. As the local corrosion current density increases with the increase in load, the hydrogen generation velocity at the crack tip also increases with the increased load, and the hydrogen concentration decreases towards the crack mouth, as shown in Figure 7. Since the hydrogen is electrically neutral, its distribution within the crevice is mainly infuenced by electrochemical corrosion as well as the difusion of the substance. Compared with Fe 2+ , hydrogen has a higher difusion coefcient and is not afected by the electrolyte potential, and its difusion to the crevice mouth is faster. Figure 8 shows the oxygen concentration distribution in the crevice under diferent loads, and the distribution is only infuenced by the concentration gradient and electrochemical corrosion rate. Te oxygen on the crack surface is consumed rapidly, and the oxygen on the crack mouth difuses freely towards the crack tip, which leads to a gradual decrease of oxygen concentration towards the crack tip, as shown in Figures 8(a) and 8(b), the oxygen concentration along the crack contour has the minimum value at the crack tip and increases towards the crack wall, which indicates a faster oxygen consumption at the crack tip than on both sides. However, the oxygen concentration near the crack tip is so small that the corrosion current densities have little diference with diferent applied loads, so there is no signifcant diference in the oxygen distribution in the crevice. Figure 9 shows the concentration distribution of Na + and Cl − within the crevice, which is only afected by the concentration gradient and electrolyte potential. Since the electrolyte potential decreases from the crack mouth to the    Science and Technology of Nuclear Installations crack tip and their diferences with the increase in applied load, Na + with a positive charge will migrate towards the inside of the crevice, and Cl − with a negative charge will migrate towards the crack mouth. Tus, a gradual increase of Na + and a gradual decrease of Cl − along the crevice towards the crack tip could be found. And with increased load displacement, the concentration diferences at both ends increase.

Conclusions
(1) When the electrochemical reaction occurs within the crack, the iron oxidation reaction in the anode is mainly afected by plastic deformation, and the corrosion rate accelerates with the increase of the equivalent plastic strain. Within the electrolyte, Fe dissolves into Fe 2+ at the crack tip and then difuses to the crack mouth. Te difusion process of Fe 2+ is resisted by electrolyte potential. Te dissolution velocity of Fe increases with increased load. (2) Te hydrogen evolution reaction velocity increases with the increase in applied load and leads to a gradual decrease in concentration trend towards crack mouth. (3) Due to the low dissolved oxygen, the oxygen absorption reaction at the crack tip is not afected by the load condition and it is controlled by the oxygen difusion in the crevice. Te oxygen concentration decreases towards the crack tip. (4) Infuenced by the electrolyte potential, Cl − migrates towards the crack mouth and Na + migrates towards the crack tip.

Data Availability
Te data used to support the fndings of this study were calculated according to the fnite element method, and they are included in the article. Te parameters used in the calculation model were cited from the references listed. Science and Technology of Nuclear Installations 9