## Abstract

Stress corrosion is a critical issue that leads to high costs in lost equipment and maintenance, affecting the operation and safety of aircraft platforms. Most aerospace structural components use the aluminum alloys 7xxx series, which contain Al, Cu, Zn, and Mg, due to the combined advantage of its high-strength and lightweight. However, such alloys, specifically AA7075-T4 and AA7075-T651, are susceptible to stress corrosion cracking when exposed to both mechanical stresses and corrosive environments. Stress corrosion cracking gives rise to a major technological challenge affecting aerospace systems as it leads to the degradation of mechanical properties. In addition, such corrosion presents an important yet complex modeling challenge due to the synergistic action of sustained tensile stresses and an aggressive environment. In light of this, we develop a finite element multiphysics model to investigate the interplay of mechanical loading and electrochemistry on the stress corrosion of aluminum alloys. The model includes a multiphysics coupling technique through which the kinetics of corrosion can be predicted in the presence of elastic and plastic deformation modes. The presented model provides useful information toward the kinetics of corrosion via tracking localized corrosion and stress distribution. Although the model is general, it has been made considering the characteristics of AA7xxx series, more specifically, taking AA7075.

## Introduction

Aluminum alloys are an essential part of aircraft structure restraining aircraft gas turbines. Specifically, they form nacelle, the housing that holds aircraft engines, as well as some parts of the fan section, including the fan blade and casing [1]. Aluminum alloys contribute to some extent toward safe turbine engine operation, as reinforcement rings for fan casings that prevent blade projection in the case of failure, as well as turbofans or turboshafts [24]. Aluminum alloys also represent relatively inexpensive and lightweight options with great machinability to fabricate complex aircraft components with high stiffness and fracture toughness. Aerospace aluminum grades generally draw in the AA2xxx and AA7xxx alloy series, extensively used in aircraft structures, fitting, gears, and shafts. Among these alloys, AA7075 shows high strength and therefore is one of the preferred options for vital aircraft parts [5,6].

7xxx series of aluminum alloys are well known to be highly susceptible to stress corrosion cracking (SCC) [710]. However, the initiation of SCC and the involved mechanisms are still poorly understood. Numerous studies over the last decades reveal that SCC of Al-Zn-Mg-Cu alloys is largely dependent on microstructure [11]. One mechanism for crack initiation is the transition from pit to crack that forms locally due to galvanic coupling between the intermetallic inclusions and aluminum matrix [11,12]. Another mechanism stems from the dissolution of precipitates on grain boundary followed by fast diffusion of produced hydrogen, which in turn leads to hydrogen embrittlement of the grain boundaries [11,13]. SCC results from a synergistic process between a chemical degradation of a material by localized corrosion and the combined effect of a mechanical stresses, residual or applied, leading to the initiation of one or more fragile cracks, and to their propagation within the ductile material, leading to possible failure. The integrity of the structure results from the accurate control of crack initiation, which is challenging to capture experimentally. The standard methods are primarily focused on the quantification of further crack propagation. However, the characterization of initiation stage events provides key factors that control subsequent propagation mechanics.

Computational modeling of microgalvanic corrosion of AA7075 has garnered a great attention recently. These models specifically focused on characterizing the galvanic corrosion between the microstructure heterogeneity and alloy matrix, utilizing several numerical methods, including finite element methods, boundary element methods, and peridynamics (which are detailed in Ref. [14] and the references therein). A numerical framework for modeling galvanic corrosion typically involves an electrochemical model applied over a meshed domain, an integration approach for the motion of the anodic front, and a remeshing scheme over the domain [15]. The main challenge in this kind of modeling arises from the difference in time scale between the different phenomena happening during the corrosion process such as ionic transport, motion of dissolving fronts, etc. In addition to developing a numerical framework, commercially available comsolmultiphysics allows for the modeling of galvanic corrosion [1618].

Numerical frameworks addressing the influence of stress on galvanic corrosion kinetics have been presented for several materials/alloys, including steel [19,20] and magnesium alloys [1517]. However, the efforts are limited for stress corrosion cracking in aluminum alloys [21]. In light of this, a finite element (FE) multiphysics model was developed to investigate the mechano-electrochemical effect of the corrosion of aluminum alloys, AA7075. The elements of the multiphysics model were developed by combining electrokinetics, deformation, and material inelasticity. We begin by illustrating the effect of mechanical loading on corrosion parameters, e.g., anodic current density and potential, for an alloy with a preformed corrosion pit. We then investigate the growth of the pit under sustained tensile stress. The model will also be utilized to examine the common stress corrosion cracking issue that arises in the grain boundaries of AA7075-T651. The parametric input data for the model were taken from experimental investigations available in the literature. The presented model serves as a sufficiently reliable method for simulation and prediction of localized corrosion and SCC of aluminum alloys and can be implemented for other different types of materials.

## Materials

The chemical constituents of AA7075, given in Table 1, play a significant role in the high performance of the mechanical properties. However, these constituents make the alloy susceptible to SCC due to the formation of various compositions. These compositions differ based on the heat treatment performed on the alloy. For instance, the solution annealed AA7075-T4 possesses small precipitates within the grain identified as the s-phase, or $Al2CuMg$, and a very small size of metastable precipitates $η′,MgZn2$, see microstructure (Fig. 1(a)) [11]. The artificially aged alloy AA7075-T651, on the other hand, always possesses inclusions, dispersoids, and precipitates. The most common inclusion particles are the cathodic $Al7Cu2Fe$, and the anodic $Mg2Si$, and their size are in the order of 10 μm. The dispersoids are particles in the order of 0.2–0.5 μm and typically $Al12Mg2Cr$ or $Al7Cr$. The precipitates in AA7075-T651 alloys are typically a composition of $Mg(CuxZn(1−x))2$ (Fig. 1(b)) [22,23].

Fig. 1
Fig. 1
Close modal
Table 1

Chemical composition of the AA 7075 aluminum wires in wt %

AlSiFeCuMnMgCrZnZrTi
Bal0.0750.1691.230.0562.50.2035.720.0270.028
AlSiFeCuMnMgCrZnZrTi
Bal0.0750.1691.230.0562.50.2035.720.0270.028

The corrosion of AA7075 typically begins with the rupture of the oxide films, $Al2O3$, on the surface followed by the dissolution of metals underneath, either $Al+3$ or precipitates along the grain boundary $MgZn2$. The rupture of the film can occur due to either adsorption of halide ($Cl−$) on the surface [24], oxygen reduction reaction (ORR) of large nobler inclusions [22], or via applying stress. This will thereafter lead to the formation of a pit that progressively grows based on the passivization time of oxide film and conductivity of the corrosive environment.

The modeling of corrosion initiation is a significant task as the material may experience different chemical reactions at different locations on the surface. For example, ORR can take place around the nobler inclusion on the surface. The cathodic chemical reaction of ORR for AA 7075 is $O2+2H2O+4e−→4OH−$. To balance this cathodic reaction, fresh materials get corroded according to the anodic chemical reaction $Al→Al+3+3e−$. In addition, hydrogen emerges based on the chemical reaction $H2O+e−→H+OH−$. It is noteworthy that other common reactions may arise along the grain boundaries. These involve the anodic reaction of $MgZn2$ according to $Mg→Mg2++2e−$, which in return gets balanced by the cathodic hydrogen evolution according to the chemical reaction $2H++2e−→H2$. During the process of corrosion, homogeneous reactions also happen in the electrolyte. Some of these reactions have been reported in previous studies [18].

Recently, experimental investigations of SCC [11] on the alloys AA7075-T4 and T651 have shown that AA7075-T4 is more susceptible to SCC than T651, and stress corrosion failure of T4 was due to the growth of a preformed pit on the surface, which lead to microcracks. The pit originated due to the galvanic coupling between the s-phase inclusions and the adjacent aluminum matrix. In addition, stress corrosion failure of T651 was observed to be originated from the dissolution of the grain boundary, which contained a continuous seam of stable $MgZn2$. In view of this, we provide a numerical framework that aid in characterizing some of these experimental observations. Note that numerical modeling for such alloys was developed previously addressing just the galvanic coupling between the aluminum matrix and the cathodic nobler inclusions, for example, see [18,25]. Hence, the aim of the presented model is to aid in characterizing the corrosion kinetics of materials (or alloys) stemming from the synergistic action due to sustained tensile stresses and aggressive environment. The details of the model are presented in the next section.

## Model Development

For AA7075-T4, pit growth is well regarded as the initial step that can set in motion a destructive corrosion process. This is because the stresses at the bottom of the pit can lead to a runaway damage process through an interplay of electrochemical processes and inelasticity. However, not all pits may ultimately lead to destructive failure or appreciable loss of strength. We took a canonical elliptical pit on a material immersed under a corrosive liquid (Fig. 2(a)). This is a general enough configuration to give us the necessary verification with other studies. For AA7075-T651, the dissolution of grain boundary in the presence of a corrosive environment can be characterized based on the galvanic coupling with the cathodic reaction of adjacent aluminum grains. Here, we assume a thin rectangle of $MgZn2$ coherently enclosed by two longitudinal grains, a typical grain microstructure of AA7075-T651 [23,26]. Note that we also assume that the orientation of grain boundary can be varied to address its influence on dissolution rate under mechanical loading, see Fig. 2(b).

Fig. 2
Fig. 2
Close modal
The governing equations for mechanics are the balance laws of force equilibrium $∇·τ=0$ ($τ$ being stress) and charge equilibrium $∑ziCi=0$ (zi is charge number and Ci the concentration of the ith species). The electrochemical transport equations for a given species are given by the Nernst–Planck equation
$∂Ci∂t=−Di∇2Ci−ziFui∇·(ci∇ϕ)+∇·(ciV)$
(1)

where Ci is the concentration of species i, and similarly D refers to the diffusion, F represents Faraday's constant, u is the mobility, $ϕ$ is the electric potential measured with respect to the standard calomel electrode, and V refers to the electrolyte velocity. If one assumes that the electrolyte is mixed well (i.e., no gradient in the concentration of ionic species) and incompressible, the steady-state of Nernst–Planck equation (i.e., $∂Ci∂t=0$) can be simplified to the straightforward Laplacian equation $∇2ϕ=0$.

The boundary conditions for the mechanical problem consists of remote traction (load) in the horizontal direction and no other external stress, see Fig. 2. This condition is typical of plane stress problems occurring widely in the aerospace and pipeline industry that allows for a good benchmark for initial verification. The boundary condition on the electrolyte is of great interest. It is typically reasonable to assume that far-field electric fields are negligible, leading to the so-called isolated boundary condition or $∂ϕ∂n=0$. The boundary condition on the electrolyte/metal interface is governed by the empirical Tafel equation where the current density at the anodic and cathodic sites are respectively given by
$ja=j0aeηaβa and jc=j0ceηcβc$
(2)
Here, ja is the charge-transfer current density for anode (analogous explanations for cathode), $j0a$ refers to the anode exchange current density, βa is the Tafel slope for the anode, and $η=ϕ−ϕeq$ is the activation potential with $ϕeq$ being the standard equilibrium potential of the electrode for the given concentrations and stress. The Tafel parameters are typically obtained experimentally via polarization curves. These current densities are used to relate to the electrolyte potential at a site via $j=−σc∂ϕ∂n$, with σc being the conductivity of the electrolyte. The corrosion rate of the anode is related to the evolution of current density according to Faraday's law given by
$CR=MzFρja$
(3)

where M is the molar mass of the corroding metal and ρ is its density, z is the number of electrons lost by the corroding surface.

In the absence of stress, the overpotential is given by $η=ϕ−ϕeq$. However, as stresses are applied, they begin to influence the electrochemical variables. The primary mechanism for this is believed to be the presence of mechanical deformation, which engenders a shift in the electrode standard equilibrium potential of the anodic surface, which in turn accelerates the corrosion rate of the material. This is reflected in the expression for overpotential at the anodic surface, according to Gutman's theory [27], that is now $η=ϕ−ϕeq0−Δϕeqe−Δϕeqp$. The term $Δϕeqe=ΔPVmzF$ refers to the shift in the electrode potential of the anodic surface due to the elastic deformation, $ΔP$ is the hydrostatic pressure, and Vm is the molar volume of the anodic electrode. The shift in the potential of the anodic electrode due to plastic deformation $Δϕeqp$ is governed by the plastic strain and its expression is written as $Δϕeqp=RTzFln(α¯νN0εp+1)$, where ν is an orientation-dependent factor [27], $α¯$ is an aluminum material constant that relates the dislocation to the evolving plastic strain, and N0 is the initial dislocation density, which represents the effect of material processing. More interestingly, the cathodic current density has been previously shown to be altered during the corrosion process due to the application of stress according to the semi-empirical formula $jc=j0ce−τvmVm6Fβc,$ where τvm is von Mises stresses of the electrode [19]. These empirical equations have been used in the context of steel corrosion; however, we hypothesize a more general validity at this time due to thermodynamics arguments in their derivation.

Note that solid mechanics simulations, implying an isotropic elastic-plastic material model and small plastic strains, are conducted to compute the evolution of hydrostatic stress and plastic strain fields. The yield criterion of von Mises is considered with a kinematic hardening model. The expression for the isotropic hardening model is $σys=σyo+σhεpe$, where σys represents the yield stress function of the alloy, $σy0$ is the initial yield stress, and $σhϵpe$ refers to the hardening function with ϵpe being the effective plastic strain [28]. Note that the hardening function is obtained from stress–strain data of aluminum alloys from the literature. Hence, in the presence of stress, the overall electrode potential is expressed as [27]
$ϕeq=ϕeq0−ΔPVmzF−RTzFln(α¯νN0εp+1)$
(4)

and used within the Laplacian equation illustrated earlier to couple the mechanical deformation with the electrochemical process. The numerical implementation of the coupling is performed utilizing comsol software [29]. Note that comsol software does not have a direct coupling method for solving multiphysics coupling problems presented here. However, we considerably modified the code to achieve this, and then we were able to perform the coupling following several steps, explained clearly below.

In this work, two test cases were simulated. The first case addresses the evolution of a corrosion pit located on the surface of AA7075-T4. The chemical reaction considered for this case is the anodic reaction of corrosion for aluminum $Al→Al+3+3e−$, while the cathodic counter-reaction stems from the evolution of hydrogen inside the pit due to the reduction of water according to $H2O+e−→H+OH−$ [11,30]. The dimensions utilized for the geometry are given in Fig. 2(a) and are based on experimental observations from the literature [18,31]. For this case, the initial simulation conducted for coupling the mechanics and electrokinetics of the corrosion pit was performed through sequential coupling of mechanics and current distribution. We performed a mechanical study under different loading conditions and evaluated the stresses, after which the current distribution was calculated. The second simulation was a time-dependent one illustrating the growth of the preformed corrosion pit. For such a complicated time-dependent multiphysics coupling problem, we introduced the concept of a deformed geometry interface in the modeling to account for the changing shape of an object (the corrosion pit). At each time-step, the interaction loop generated for solving the problem was performed via tracking the deformed geometry, and then mechanical stresses field evaluations were calculated through the current distribution along the corrosion pit. The input parameters used for the simulations were taken from experimental observation. We used available data for electrolyte of 3.5% NaCl, with $σc=5 S/m$ [32]. The electrochemical parameters used for evaluating the Tafel equations are illustrated in Table 2 [30]. In addition, the elastic mechanical properties of AA7075-T4 utilized for the simulation are the modulus of elasticity E =67.8 GPa, Poisson's ratio $vR=0.33$ and the initial yield strength $σy0=337$ MPa while the plastic properties were incorporated into the simulation via using data from stress–strain curve adapted from [28,33]. The other input parameters were chosen as $ν=0.45$ [27], $α¯=1.67×1011cm−2$ [27], $N0=1×109cm−2$ [34], $Vm=10 cm3/mol,M=26.3982 g/mol$, and $ρ=2.7 g/cm3$ and z =3 for aluminum.

Table 2

Electrochemical Tafel parameters utilized for pit growth in AA7075-T4 [30]

Cathodic hydrogen evolutionAnodic oxidation of Al
$ϕeq0$−0.65 V$ϕeq0$−0.9 V
j0$1.19×10−4 A/cm2$j0$9.2×10−5A/cm2$
Cathodic hydrogen evolutionAnodic oxidation of Al
$ϕeq0$−0.65 V$ϕeq0$−0.9 V
j0$1.19×10−4 A/cm2$j0$9.2×10−5A/cm2$

The second test case addresses the dissolution of the grain boundary in AA7075-T651. For such an alloy, it is widely accepted that magnesium gets corroded along the grain boundary due to the cathodic reaction of neighboring aluminum grains [15,3537]. Accordingly, the chemical reaction considered is the anodic reaction of magnesium $Mg→Mg+2+2e−$ while the cathodic counter-reaction stems from the production of $OH−$ on the surface $O2+2H2O+4e−→4OH−$ [11,30]. The dimensions utilized for the simulations are given in Fig. 2(b) [23]. The simulation steps for this case are similar to the previous case. However, the input parameters used for the simulations were taken from different experimental observations [10]. The electrochemical data were obtained from polarization curves presented in experimental investigations on the common intermetallic phases in aluminum alloys, Table 3 [38]. The experiments were performed in a corrosive medium of 0.1 M NaCl (i.e., $σc=1.26 S/m$) [18,39] and a near-neutral PH ($≈6$). In addition, for the mechanical properties of the grains, we assume low-angle grain boundaries [40]. Such an assumption allows us to write the elastic constants for the two grains as $C11=107.3 GPa, C12=60.9 GPa$, and $C44=28.3 GPa$ [41]. The plastic properties for the Al grains are not needed as the grain boundary will yield faster than the grains. For the grain boundary, we utilized the mechanical properties for magnesium alloy, AE44. These properties are the modulus elasticity E =45 GPa, Poisson's ratio $vR=0.33$ and the initial yield strength $σy0=142$ MPa, and we assumed a linear isotropic hardening with isotropic tangent modulus of 3.5 GPa [17]. The rest of input parameters were chosen as $ν=0.45$ [27], $α¯=1.67×1011cm−2$ [27], $N0=4×109 cm−2$ [34], $Vm=7.13 cm3/mol, M=24.305 g/mol$, and $ρ=1.82 g/cm3$ and z =2 for magnesium [17].

Table 3

Electrochemical Tafel parameters for dissolution of $MgZn2$ and cathodic ORR of aluminum in AA7075-T651 [38]

Cathodic oxygen reduction reactionAnodic oxidation of Mg
$ϕeq0$−0.7 V$ϕeq0$−1.25 V
j0$4×10−7A/cm2$j0$1×10−3A/cm2$
Cathodic oxygen reduction reactionAnodic oxidation of Mg
$ϕeq0$−0.7 V$ϕeq0$−1.25 V
j0$4×10−7A/cm2$j0$1×10−3A/cm2$

For both test cases simulated in this work, we utilized a linear triangular mesh type. In addition, mesh sensitivity analysis was performed and a maximum element size of 0.2 $μ$m was sufficient to achieve convergence in quantifying the current density and stresses along the corrosion pit. Similarly, a mesh resolution at the grain boundary ranging from 0.1 nm to 2 nm provided a convergence in the corrosion depth along with the elastic mismatch due to the bond with grains. The solver selected for the solution was a direct “multifrontal massively parallel sparse” [29]. Note that as the deformed geometry highly depends on the mesh quality, the mesh resolution mentioned above was also sufficient to obtain FE convergence in the growth of corrosion pit and dissolution of the grain boundary. The simulations for the two test cases presented in this work were performed on Intel(R) Xeon(R) W-2145 CPU @ 3.70 GHz with 8 cores and installed RAM 64 GB. For case 1, the computation time for the growth of the corrosion pit of AA7075-T4 for a period of 10 h took 4 min when there was no mechanical loading. However, once mechanical loading was present, the computation time was 12 h due to the difficulty of convergence. For the case of the dissolution of grain boundary of AA7075-T651, the computation time was approximately 2 and 13 min for the case of no stress and presence of stress, respectively.

The verification of the coupled model was performed by comparing it with existing computational studies of pit-to-crack transition in literature for steel [20]. The numerical framework considered for the steel pipeline was built upon experimental observations. The stability of our code was ensured, and the verification of results is discussed in Appendix  A. It is also noteworthy that as the passive film may play a vital role in resisting corrosion of aluminum alloys, the polarization curves obtained from the literature for the two cases illustrated in this work did not show any passivization [30,38]. Hence, passivization role was neglected in the numerical framework presented here.

## Results and Discussions

### Simulation on Pit Growth of AA 7075-T4.

To illustrate the effect of elastic and plastic deformation on the anodic current density and corrosion potential of the preformed pit, we applied displacement boundary conditions to the right edge of the alloy. These boundary conditions are expressed as longitudinal engineering strain equal to 0%, 0.2%, 0.4%, and 0.6%. The von Mises stresses along the corrosion pit length are shown in Fig. 3(a). Here, the length of the pit was normalized by its width. The dashed line in the figure illustrates the instant of yielding of the alloy. Clearly, a longitudinal strain higher than 0.2% was needed to observe inelasticity along the corrosion pit. This can also be seen in Fig. 3(b), where plastic strain is only observed for the cases of 0.4% and 0.6% longitudinal strains. In addition, both figures show that the highest stress is located at the center of the pit. This implies that the center of the pit would experience more localized corrosion. This is obviously clear in the corrosion potential along the pit, illustrated in Fig. 3(c) for the four cases of deformation selected for this simulation. Corrosion potential increases with the increasing stress, yet more importantly, it is found to be significantly affected by plastic strain. We do not observe a noticeable change in the potential for the case of 0.2% in which the alloy does not experience plastic deformation. A similar trend is also depicted when looking at the anodic current density, which controls the corrosion rate as presented earlier by Faraday's law, see Fig. 3(d). We again notice that the anodic current density is highly affected by the plastic strain. Locations of higher plastic strain will be mostly affected by localized corrosion in the alloy (here, the center of the pit represents the location of the highest stress). Elastic deformation has a slight effect on the corrosion rate of AA7075 and hence may be neglected, indicating that stress corrosion cracking initiation requires yielding of the material. In addition, areas of high stresses give rise to higher localized corrosion rates that may lead to crack initiation. Such a conclusion was also reported in the literature [11,19,42].

Fig. 3
Fig. 3
Close modal

We now investigate the growth of the pit for the deformation case of 0.6% strain (for which the plastic strain at the center of the pit $≈$ 2%). Based on the initial current distribution results presented earlier, the anodic current density at the center of the pit is approximately $2.28A/m2$. This implies that the uniform corrosion rate in the absence of stress can be estimated as 0.27 $μm/h$. However, stresses will make the corrosion rate vary at each time-step. To illustrate the growth of the pit with time, we present the growth of the pit at different times, 0, 5, and 10 h, see Fig 4(a). We notice that the pit clearly widens with time, but no crack was noticeable for this simulation. This could be due to the isolation of the grain boundary role. For AA7075-T4, the embrittlement of grain boundary was suggested [11] as the authors observed intergranular microcracks outgoing from pit, although no dissolution was observed along the grain boundary. Hence, we hypothesize that a model incorporating grain boundary integrity evolution would be critical. In addition, during the growth of the pit, stresses will increase, leading to a varying corrosion rate as opposed to the case of no stress, where corrosion will happen at a constant rate. This is here ensured by the increase of the von Mises stress at the center of the pit when plotted against the time intervals of simulation, see Fig. 4(b). Interestingly, we notice a deviation in the stress at the period of approximately 6 h. This sudden increase in stress is attributed to the change of the shape of the pit as it starts to change from an elliptical shape to almost a U-shape due to the assumption that only the pit experiences corrosion growth and not the upper surface of the alloy. The stress afterwards starts to increase in a similar fashion to the first 5 h. Note that an increase in stresses will also be accompanied by an increase in the anodic current density, as presented earlier in our model [27]. The temporal variation of anodic current density is depicted in Fig. 4(c). We notice that the trend of both stresses and the anodic current density is linear, where for other materials (e.g., steel), the trend was exponential [19], implying that the corrosion rate will increase in an exponential fashion. These results suggest that the model presented is reliable and can be utilized to predict the growth of corrosion pits, as the results coincide with observations in the literature [11].

Fig. 4
Fig. 4
Close modal

### Simulation on Grain Boundary Dissolution of AA 7075-T651.

We begin with an illustration of the anodic dissolution of the grain boundary in both cases of galvanic corrosion with no stresses and the presence of constant stresses for the transient corrosion (i.e., growth of corrosion of grain boundary with time). For the case of no stress, a contour plot showing the potential of the electrolyte along with the corrosion depth is presented in Fig. 5(a) at time 5 s. Note that the time scale is very small because the length scale considered for the sample is small, nm. We notice that the entire width of the grain boundary corrodes, and the final shape is flat due to the uniform corrosion. We then applied a 0.18% longitudinal elastic strain on the sample that resulted in a plastic strain of 3.6% at the junction (location of elastic mismatch between grains and grain boundary) and 0.8% at the center of the grain boundary. This amount of plastic strain was sufficient to engender variation in electrochemical variables [17]. The contour plots of stress distribution and electrolyte potential are also given in Fig. 5(a). The figure highlights the increase in the potential distribution as compared to the case of no stress. The corrosion depth also increases at the galvanic junction, implying high localized corrosion due to the increase in hydrostatic stress and plastic strain. The fast dissolution of the grain boundary, specifically when stresses are present, will lead to evolution of hydrogen that may accelerate grain boundary dissolution [15,3537] or fast diffusion of hydrogen, which in turn causes failure due to hydrogen embrittlement [11,13]. To thoroughly investigate the influence of stresses on grain boundary dissolution, we plot the variation of the anodic current density along the normalized width of the grain boundary and compare it with the case of no stress, see Fig. 5(b). The figure shows no change in the anodic current for the case of no stress, while a v-shape of distribution of the current due to again the high localization of plastic strains near the junctions. A similar trend is also observed for the corrosion depth of grain boundary for both cases of stress and no stress, see Fig. 5(c).

Fig. 5
Fig. 5
Close modal

Moving to the effect of grain boundary orientation on corrosion behavior of AA7075, we look at how stresses alter the galvanic corrosion between the Al grains and the dissolution of Mg on grain boundary. We specifically investigated the case of applying a 0.18% longitudinal elastic strain on the sample, as illustrated in the initial results mentioned above. In reality, not all neighboring grains are bonded in a vertical fashion [23]. Typically, the neighboring grains are oriented differently, which puts the grain boundary at an angle with respect to a vertical axis, see Fig. 2(b). Here, we vary the orientation angle from 0 deg to 50 deg and plot the corrosion depth of the grain boundary (calculated by averaging the corrosion depth along the width of the grain boundary). In the absence of stress and at a period of 5 s, we notice that grain boundary orientation does not significantly alter the corrosion depth brought about the galvanic corrosion. We roughly see the same values of corrosion depth at different orientation angles Fig. 6. When constant stress is present, we do not see any effect on corrosion depth at low angles <40 deg. We, however, notice a sharp increase in the corrosion depth at a 50 deg angle, which is attributed to the increase in stresses around the galvanic junction. A similar experimental observation was also concluded for AA7075-T651 [11], where the authors showed a crack initiated due to the dissolution of the grain boundary, which was oriented at an angle of 45 deg. This again implies that areas of high localized stresses serve as an indication of localized corrosion that may eventually lead to brittle failure due to intergranular cracking. In addition, the results presented in this work highlight the effect of stress-induced localized corrosion at different time intervals. Hence, the model developed here can serve as a foundation for corrosion behavior of stressed alloys and can help in characterizing stress corrosion cracking initiation.

Fig. 6
Fig. 6
Close modal

## Conclusion

In this work, a FE multiphysics model was established, combining electrokinetics and material inelasticity to simulate and predict the stress corrosion behavior of aluminum alloys, AA7075. The equations from electrokinetics, deformation, and inelasticity were coupled using comsol Multiphysics software. The code was used to simulate the growth of corrosion pit on AA 7075-T4 sample and dissolution of grain boundaries on AA 7075-T651. Here, using available material and electrochemical properties, a pit to crack for AA7075-T4 was not observed, clearly underscoring the missing physics. We hypothesize that a model incorporating grain boundary integrity evolution would be critical in properly modeling this transition. In addition, the presence of stresses resulted in an acceleration of the grain boundary's dissolution. More importantly, the model predicts localized dissolution around areas of material mismatch that serve as locations of crack initiation.

The model presented here is fundamental and addresses the anodic dissolution of either corrosion pit or grain boundaries due to the galvanic coupling. However, in cases where the degradation of the grain boundary influences the growth of the pit, advanced models are needed that can be built upon our model. In addition, our model does not address the effect of hydrogen embrittlement that arises due to the corrosion products. In such case, the presented model can be used addressing the degradation rate of material due to the anodic dissolution and hydrogen diffusion may be incorporated using mass-balance law.

Overall, this work highlights the complex interaction that arises due to the synergistic action of sustained mechanical deformation and electrochemical process. The numerical framework presented may be useful for characterizing the corrosion of structural alloys in the presence of mechanical deformation. In particular, the model provides fundamental key factors to the initiation of stress corrosion cracking.

## Acknowledgment

This work was supported by the Defense Advanced Research Projects Agency (DARPA) under agreement number HR00112020043.

### Appendix A: Computational Verification—Steel Pipeline Stress Corrosion

Here, we verify the numerical framework presented for the synergistic action due to sustained tensile stresses and aggressive environment. The stability of the code was ensured, and the results verified for an archetypical pit-to-crack transition from an existing computational model on a steel pipeline [20] in a solution with a PH$≈6.8$. The geometry was a planar beam of length 2 m and thickness of 19.1 mm, and the pit geometry was 8 mm width and 2 mm depth with the load being imposed on the right edge. The parametric input conditions for the model were different compared to the presented case of aluminum alloy. Due to the versatility of the model presented here, we used it for the corrosion of steel. Specifically, the chemical reactions taken place along the corrosion defect (pit) were the dissolution of steel according to the anodic reaction $Fe→Fe2++2e$ and the hydrogen evolution cathodic reaction $2H2O+2e→H2+2OH$. The results of our code, specifically the contour plots of the von Mises stress distribution around the defect geometry for the period of 0, 10, and 20 years are illustrated in Fig. 7(a). We notice a similar distribution of stresses when comparing the results of our simulation with the results from the literature, see Fig. 7(b). Both simulations observed a crack-like flaw at the period of 20 years around the center of the defect geometry. In addition, we provide a temporal variation of von Mises stress at the center of defect geometry resulted from the simulation procedure considered in this work and literature, see Fig. 7(c). We again notice a good agreement in the results with the example used for verification. As such, the model presented here is reliable for testing the growth of corrosion of the aluminum alloy.

Fig. 7
Fig. 7
Close modal

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