A probabilistic computational framework for the prediction of corrosion-induced cracking in large structures

Giuseppe Meazza

Stress corrosion and SCC mechanism

The stress corrosion process, as shown in Fig. 2, starts from the local breakage of the passive film. Under a corrosive environment, the metal is corroded and produce cations (\({M}^{+}\)) into the electrolyte as well as the electrons in the electrode. During service life, complex mechanical loading conditions typically occur in different locations. However, mechanical loading changes the chemical potential of the electrode (metal), affecting the corrosion process. Stress concentration at the tip of the pitting corrosion amplifies the influence from the mechanical stress. In order to properly quantify the contribution from mechanical load, a generalized potential is introduced.

Figure 2
figure 2

Schematics of stress corrosion.

Generalized chemical potential and kinetics in corrosion system with mechanical stress

The generalized chemical potential in the electrochemical system is expressed as

$$\mu = {\mu }_{t}+{\mu }_{el}+{\mu }_{gr}+{\mu }_{me},$$


where \(\mu\) is the well-defined chemical potential, \(\mu\) usually consists of thermal potential \({\mu }_{t}\), electrical potential \({\mu }_{el}\), gradient energy density or gradient potential \({\mu }_{gr}\), and mechanical energy density or mechanical potential \({\mu }_{me}\).

The influence of mechanical deformation on corrosion process originates from the potential change due to mechanical deformation on the surface of the solid28,29. According to Gibbs–Duhem equation30, the differential of chemical potential dependence is described as

$$\sum_{i}{N}_{i}d{\mu }_{i}=-SdT+VdP$$


where \(N\) is molarity, \(V\) is the volume, \(S\) is the entropy, \(T\) is the temperature and \(P\) is the pressure, calculated with hydrostatic stress by \(P=-\frac{1}{3}\sum_{k=1}^{3}{\sigma }_{kk}\).

Assuming the metal response is linearly elastic31, the linear expression for \({\mu }_{me}\) by integrating Eq. (2) follows expression in mechanochemistry area by Gutman28,29

$$\mu _{{me}} = \int\limits_{{P_{1} }}^{{P_{2} }} {V(P)} dP \approx \Delta PV_{m} ,$$


where \({P}_{1},{P}_{2}\) are the initial and end pressure and \({V}_{m}\) is the molar volume.

The thermal potential is expressed as

$${\mu }_{t}=g\left(\overline{c }\right)+{c}_{0}RT\left({\overline{c} }_{+}\mathrm{ln}{\overline{c} }_{+}+{\overline{c} }_{-}\mathrm{ln}{\overline{c} }_{-}\right)+\sum_{i}{c}_{i}{\mu }_{i}^{\Theta },$$


where \({\varvec{c}}=\left\{c, {c}_{+},{c}_{-}\right\}\) is a set of concentrations for the metal atom, metal cations, and electron, respectively. Further, \(\overline{{\varvec{c}} }\) is defined as the set of dimensionless concentrations as \(\left\{\overline{c }=\frac{c}{{c}_{s}},{\overline{c} }_{+}=\frac{{c}_{+}}{{c}_{0}},{\overline{c} }_{-}=\frac{{c}_{-}}{{c}_{0}}\right\}\), where \({c}_{s}\) is the site density of the metal iron and \({c}_{0}\) is the bulk concentration of electrolyte solution, \(R\) is molar gas constant, and \(T\) is the temperature. The double-well function \(g\left(\overline{c }\right)=W{\overline{c} }^{2}{\left(1-\overline{c }\right)}^{2}\) is used to describe the transition between electrode \(\left(\overline{c }=1\right)\) and electrolyte \(\left(\overline{c }=0\right)\). \(W\) represents barrier height of corrosion. \({\mu }_{i}^{\Theta }\) is the reference chemical potential of spices \(i\).

The electric potential \({\mu }_{el}\) is expressed as

$${\mu }_{el}={\rho }_{e}\phi ,$$


where \(\phi\) is the electrostatic potential, \({\rho }_{e}\) is the charge density which can be expressed as \({\rho }_{e}=F{z}_{i}{c}_{i}\) where \(F\) is Faraday’s constant, \({z}_{i}\) is the valence and \({c}_{i}\) is the concentration of species \(i\).

The interfacial potential \({\mu }_{gr}\) is given by taking partial derivative of the interface energy as6

$${\mu }_{gr}={g}^{^{\prime}}\left(\overline{c }\right)-\kappa {\nabla }^{2}\overline{c },$$


where \(\kappa\) is interface coefficient.

According to previous formulation of electrochemical reaction kinetics, the reaction rate, \({R}_{e}\) of corrosion is expressed as the difference between forward \(\left({S}_{1}\to {S}_{2}\right)\) and backward \(\left({S}_{2}\to {S}_{1}\right)\) reactions in a form of Butler–Volmer equation as

$${R}_{e}={k}_{0}\left(\mathrm{exp}\left[\frac{-\left({\mu }_{t}^{ex}-{\mu }_{1}\right)}{RT}\right]-\mathrm{exp}\left[\frac{-\left({\mu }_{t}^{ex}-{\mu }_{2}\right)}{RT}\right]\right),$$


where \({\mu }_{1}\) and \({\mu }_{2}\) refers to the total chemical potential at state 1 and state 2 respectively, \({\mu }_{t}^{ex}\) is the activation barrier, \({k}_{0}\) is the reaction constant.

Based on the potentials Eqs. (1–6) we defined previously, we can get the potential expressions of initial state 1 (electrode) and later state 2 (electrolyte) by different components in the corrosion reaction, \(M\to {M}^{n+}+n{e}^{-}\)

$${{\mu }_{1}=\mu }_{M}={\mu }_{M}^{t}+{\mu }_{M}^{me}+{\mu }_{M}^{gr}={\mu }_{gr}+{\mu }_{M}^{\Theta }+{\mu }_{me},$$


$${\mu }_{2}={\mu }_{{M}^{n+}}^{t}+{n\mu }_{el}=RT\mathrm{ln}{a}_{{M}^{n+}}+{\mu }_{{M}^{n+}}^{\Theta }+nF{\phi }_{s}+nRT\mathrm{ln}{a}_{e}+{n\mu }_{e}^{\Theta }-nF{\phi }_{e},$$


where \({\phi }_{s}\) and \({\phi }_{e}\) are the electrostatic potential in the solution and the electrode respectively, \({a}_{M}\), \({a}_{{M}^{n+}}\) and \({a}_{e}\) are the activities of the components. The activity for electrons is unity assuming that the electrolyte solution is dilute. The interfacial potential difference is \(\Delta \phi ={\phi }_{e}-{\phi }_{s}\). At the equilibrium, the potential difference \(\Delta \mu ={\mu }_{2}-{\mu }_{1}=0\) according to the Nernst equation

$$\Delta {\phi }^{eq}=\frac{{\mu }_{{M}^{n+}}^{\Theta }+{n\mu }_{e}^{\Theta }-{\mu }_{M}^{\Theta }+RT\mathrm{ln}{\overline{c} }_{+}+{\mu }_{me}-{\mu }_{gr}}{nF}.$$


Outside equilibrium, the reaction is driven by the overpotential, \(\eta\), which is defined as

$$\eta =\Delta \phi -\Delta {\phi }^{eq}.$$


Substituting Eq. (10) with Eq. (9), \(\eta\) is expressed as

$$\eta =\Delta \phi -\frac{{\mu }_{{M}^{n+}}^{\Theta }+{n\mu }_{e}^{\Theta }-{\mu }_{M}^{\Theta }}{nF}-\frac{RT\left(\mathrm{ln}{\overline{c} }_{+}\right){-\mu }_{gr}}{nF}+\frac{{\mu }_{me}}{nF},$$


where the second term represents the standard potential difference between reactants and products, the third term expresses the concentration overpotential, and the third term is the influence of mechanical elastic energy density. The total overpotential can be separated into activation overpotential \({\eta }_{a}=\Delta \phi -\frac{{\mu }_{{M}^{n+}}^{\Theta }+{n\mu }_{e}^{\Theta }-{\mu }_{M}^{\Theta }+{\mu }_{me}}{nF}\), concentration overpotential \({\eta }_{c}=-\frac{RT\left(\mathrm{ln}{\overline{c} }_{+}\right){-\mu }_{gr}}{nF}\).

The excess electrochemical potential in the transition state is defined as32

$${\mu }_{t}^{ex}=RT\mathrm{ln}{\gamma }_{t}+{\mu }_{me}^{ex}+\left(1-\alpha \right){\mu }_{M}^{\Theta }+\alpha \left({\mu }_{{M}^{n+}}^{\Theta }+{n\mu }_{e}^{\Theta }\right),$$


where \({\gamma }_{t}\) is the activity coefficient at the transition state, \({\mu }_{me}^{ex}\) represents the mechanical potential at the transition state, and \(\alpha\) is an approximate constant ranging from zero to one called symmetry factor.

The reaction rate can be expressed by substituting Eqs. (8–12) into Eq. (6)

$$r=\frac{{k}_{0}}{{\gamma }_{t}}\mathrm{ exp}\left(-\frac{{\mu }_{me}^{ex}}{RT}\right)\times \left\{\mathrm{exp}\left(\frac{{\mu }_{gr}+\left(1-\alpha \right){\eta }_{a}}{RT}\right)-{\overline{c} }_{+}\mathrm{exp}\left(\frac{-\alpha nF{\eta }_{a}}{RT}\right)\right\}.$$


The influence of the interfacial potential concentration gradient at the interface on the corrosion process is usually small comparing to other components in the total chemical potential33. A nonlinear relationship for phase transforming is proposed by Liang et al.6,34 as

$$r={-L}_{\sigma }\left({g}^{^{\prime}}\left(\overline{c }\right)-\kappa {\nabla }^{2}\overline{c } \right)-{L}_{\eta }\left(\mathrm{exp}\left(\frac{\left(1-\alpha \right){\eta }_{a}}{RT}\right)-{\overline{c} }_{+}\mathrm{exp}\left(\frac{-\alpha nF{\eta }_{a}}{RT}\right)\right),$$


where \({L}_{\sigma }=\frac{{k}_{0}}{RT{{\gamma }_{t}c}_{s}}\mathrm{exp}\left(-\frac{{\mu }_{\mathit{me}}^{\mathit{ex}}}{\mathit{RT}}\right)\mathrm{exp}\left(\frac{\left(1-\alpha \right){\eta }_{a}}{RT}\right)\) represents interfacial mobility and \({L}_{\eta }=\frac{{k}_{0}}{{\gamma }_{t}}\mathrm{exp}\left(-\frac{{\mu }_{{\mu }_{me}}^{ex}}{RT}\right)\) represents a reaction coefficient. In the work, we assume the reaction coefficient is a constant value which will be calibrated in Section “Calibration of the phase-field model”.

Governing equations for corrosion with stress

A continuous order parameter \(\xi\) is introduced to describe the diffuse interface in the proposed phase field model. The order parameter physically corresponds to the dimensionless concentration of the metal, as \(\xi =\overline{c }\). The dimensionless concentration \(\overline{c }=1\) in the metal and \(\overline{c }=0\) in the electrolyte solution.

In this model, we consider the order parameter’s evolution is driven by electrochemical reaction rate \(r\). Thus, the driving force can be clearly divided into two parts: the interface energy and the electrode reaction. To describe the electrochemical reaction kinetics at the diffuse interface, an interpolating function \({h}^{\mathrm{^{\prime}}}\left(\xi \right)=30{\xi }^{2}{\left(1-\xi \right)}^{2}\) is introduced. Notice that the order parameter changes from one to zero for phase evolution in corrosion process. The phase evolution has a negative relationship with reaction rate \(r\). Therefore, the governing equation for the phase evolution is

$$\frac{\partial \xi }{\partial t}={-L}_{\sigma }\left({g}^{^{\prime}}\left(\overline{c }\right)-\kappa {\nabla }^{2}\overline{c } \right){-L}_{\eta }{h}^{^{\prime}}\left(\xi \right)\left(\mathrm{exp}\left(\frac{\left(1-\alpha \right){\eta }_{a}}{RT}\right)-{\overline{c} }_{+}\mathrm{exp}\left(\frac{-\alpha nF{\eta }_{a}}{RT}\right)\right).$$


Equation (14) indicates that the mechanical deformation changes the total chemical potential with the mechanical potential \({\mu }_{me}\) (in term \({\eta }_{a}\)). If \({\mu }_{me}>0\), the mechanical contribution has a positive influence on the corrosion process; if \({\mu }_{me}<0\), the mechanical contribution has a negative influence on the corrosion process.

The metal atom is considered as fixed except during diffusion process. The electrochemical reaction provides the source term which depends on the corrosion (metal consumption) process. The diffusion can be described with the Nernst-Plank equation as

$$\frac{\partial {\overline{c} }_{+}}{\partial t}=\nabla \cdot \left({D}^{eff}\nabla {\overline{c} }_{+}+\frac{{D}^{eff}{\overline{c} }_{+}}{RT}nF\nabla \phi \right)-\frac{{c}_{s}}{{c}_{0}}\frac{\partial \xi }{\partial t},$$


where \({D}^{eff}\) represents the effective diffusion coefficient as \({D}^{eff}={D}^{e}h\left(\xi \right)+{D}^{s}(1-h(\xi ))\), where \({D}^{e}\) and \({D}^{s}\) are the diffusion coefficients for metal cation in the electrode and electrolyte, respectively.

The mechanical equilibrium equation is expressed with stress tensor \(\sigma ={A}^{e}{\varepsilon }^{eq}\) as

$$\mathrm{div}\left(\sigma \right)=0,$$


where the body force is neglected.

It is obvious that fully corroded metal cannot support stress or strain. The equivalent elastic strain tensor considering solid–liquid interface is modified as follows:

$${\varepsilon }^{eq}=p\left(\overline{c }\right)\left\{{\varepsilon }_{ij}^{e}\right\}=p\left(\overline{c }\right)\left\{\frac{1}{2}\left(\frac{\partial {u}_{i}}{\partial {x}_{j}}+\frac{\partial {u}_{j}}{\partial {x}_{i}}\right)\right\}\left(i,j=\mathrm{1,2},3\right),$$


where \({u}_{i}\) and \({u}_{j}\) are displacement components, \(p\left(\overline{c }\right)\) is an interpolation function to smooth the discontinuity in the interface and it also satisfies \(p\left(0\right)=0\) and \(p\left(1\right)=1\). Combining Eqs. (17) and (18), we can get the governing equation for mechanical equilibrium as

$$\mathrm{div}\left({A}^{e}p\left(\xi \right)\left\{\frac{1}{2}\left(\frac{\partial {u}_{i}}{\partial {x}_{j}}+\frac{\partial {u}_{j}}{\partial {x}_{i}}\right)\right\}\right)=0.$$


Pit-to-crack transition

Two different SCC initiation criteria are implemented based on the corrosion morphology and the applied mechanical load.

One criterion is based on the Von Mises yield criterion or Tresca yield criterion (which give the same results under a plane strain assumption). The crack is assumed to initiate when the stress reaches the Von Mises yield criterion. Note that this is an approximate conservative criterion for crack initiation

$${\sigma }_{1}-{\sigma }_{2}=2\sqrt{{\left(\frac{{\sigma }_{x}-{\sigma }_{y}}{2}\right)}^{2}+{{\tau }_{xy}}^{2}}<2Y/\sqrt{3},$$


where \(Y\) is uniaxial yield stress, \({\sigma }_{y}\) is zero under a uniaxial stress state, \({\sigma }_{x}\) is the normal stress at the tip, and \({\tau }_{xy}\) is the shear stress at the tip. Both \({\sigma }_{x}\) and \({\tau }_{xy}\) at the tip position can be approximated with the stress concentration equation35

$${\sigma }_{x}={\sigma }_{norm}\left(1+2\sqrt{\frac{a}{\rho }}\right), {\tau }_{xy}={\tau }_{norm}\left(1+2\sqrt{\frac{a}{\rho }}\right) ,$$


where \(a\) represents corrosion depth and \(\rho\) represents the curvature at the bottom of corrosion pits or potential crack tip position. Both the curvature radius \(\rho\) and corrosion depth \(a\) are calculated from the corrosion morphology. From the Eqs. (20), (21), we can notice that the larger corrosion depth and sharper interfaces are more likely to initiate cracking.

The other SCC initiation criterion is the Tsujikawa–Kondo criterion27, which compares the corrosion growth velocity with the crack propagation velocity. When the crack propagation speed surpasses the corrosion growth speed, the crack is assumed to initiate. One implementation is based on that the driving force from mechanical stress are separated from that from electrode dissolution18. The corrosion speed at pit tip and mouth area of the model are calculated as \({v}_{tip}\) and \({v}_{mouth}\) based on the displacement of the interface at pit tip location and the mouth top location, respectively. A parameter \({K}_{v}=\left({v}_{tip}-{v}_{mouth}\right)/{v}_{mouth}\) is used to evaluate the portion of driving force from mechanical stress. \({v}_{tip}\) represents the crack propagation speed and \({v}_{mouth}\) represents the corrosion speed in the crack initiation stage. When the \({v}_{tip}\) is twice as large as the \({v}_{mouth}\), implying the mechanical driving force is much larger than the corrosion without stress, the crack is assumed to initiate. Both criteria are implemented and compared as given in Fig. 3. Both the approximated Von Mises stress at the tip and parameter \({K}_{v}\) are plotted over time. The SCC initiation time is identified when the criterion is met, i.e., the Von Mises stress reaches the yield stress, or \({K}_{v}\) reaches two. The proposed SCC initiation criterion is an indicator about the time and location of a possible SCC initiation. The overall trend for these two parameters is similar while the criterion based on the approximated stress is more conservative and stable than that based on corrosion growth speed.

Figure 3
figure 3

Comparation between two crack initiation criteria.

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