Chemomechanical finite element analysis for surface oxidation of Aluminum alloy

Abstract

Metals are prone to oxidation in high-temperature oxygen-containing environments, resulting in oxidative corrosion. This study proposes a fully coupled chemomechanical model and further develops a finite element method (FEM) to characterize the high-temperature oxidation process of metals. Then, we perform finite element analyses for surface oxidation of FeCrAlY alloy to verify the proposed chemomechanical model. Good agreements of FEM results with experimental observations suggest that the model can be used to predict the surface oxidation of metals. The numerical results also reveal the two-way coupling effects between chemical processes and mechanical stress. Furthermore, it is found that compressive stress could inhibit diffusion and chemical reaction in the oxide layer. In contrast, the stress distribution of the oxide layer can also be significantly influenced by compositional strain induced by the concentration change of diffusive species and growth strain induced by chemical reactions. In addition, we find that the barrier effect of alumina on diffusion can significantly slow down the growth of the oxide layer. This study provides an effective model for chemomechanical phenomena and may also shed light on the design of alloys in a request of resistance to oxidative corrosion.

Appendices

Appendix A: Concrete expressions of tangents

The partial derivatives of the residuals (50) are

$$\begin{aligned} \frac{{\partial {\mathbf{R}}_{{\mathbf{u}}} }}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }} & = - \int\limits_{V} {{\mathbf{B}}_{{\mathbf{u}}}^{{\mathbf{T}}} \frac{{\partial {\mathbf{\sigma }}}}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }}} dV, \hfill \\ \frac{{\partial {\mathbf{R}}_{{\mathbf{u}}} }}{{\partial {\boldsymbol{\upmu }}_{{\mathbf{{\rm M}}}}^{{\mathbf{e}}} }} & = - \int\limits_{V} {{\mathbf{B}}_{{\mathbf{u}}}^{{\mathbf{T}}} \frac{{\partial {\mathbf{\sigma }}}}{{\partial {\boldsymbol{\upmu }}_{{\mathbf{M}}}^{{\mathbf{e}}} }}dV} , \hfill \\ \frac{{\partial {\mathbf{R}}_{{\mathbf{u}}} }}{{\partial {\boldsymbol{\upmu }}_{{\mathbf{O}}}^{{\mathbf{e}}} }} & = - \int\limits_{V} {{\mathbf{B}}_{{\mathbf{u}}}^{{\mathbf{T}}} \frac{{\partial {\mathbf{\sigma }}}}{{\partial {\boldsymbol{\upmu }}_{{\mathbf{O}}}^{{\mathbf{e}}} }}dV} ,\; \hfill \\ \frac{{\partial {\mathbf{R}}_{{\mathbf{M}}} }}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }} & = \int\limits_{V} {aRT{\mathbf{N}}_{{\boldsymbol{\upmu }}}^{{\mathbf{T}}} \frac{{\partial \dot{\omega }}}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }}} dV,\;\;\; \hfill \\ \frac{{\partial {\mathbf{R}}_{{\mathbf{M}}} }}{{\partial {\boldsymbol{\upmu }}_{{\mathbf{M}}}^{{\mathbf{e}}} }} & = \int\limits_{V} {\left( {{\mathbf{N}}_{{\boldsymbol{\upmu }}}^{{\mathbf{T}}} \frac{{\mu _{M} - \mu _{M}^{t} }}{{\Delta t}}\frac{{\partial c_{M} }}{{\partial {\boldsymbol{\upmu }}_{{\mathbf{M}}}^{{\mathbf{e}}} }} + \frac{{c_{M} {\mathbf{N}}_{{\boldsymbol{\upmu}}}^{{\mathbf{T}}} {\mathbf{N}}_{{\boldsymbol{\upmu}}} }}{{\Delta t}} + aRT{\mathbf{N}}_{{\boldsymbol{\upmu}}}^{{\mathbf{T}}} \frac{{\partial \dot{\omega }}}{{\partial {\boldsymbol{\upmu}}_{{\mathbf{M}}}^{{\mathbf{e}}} }}} \right)dV,} \hfill \\ \frac{{\partial {\mathbf{R}}_{{\mathbf{M}}} }}{{\partial {\boldsymbol{\upmu}}_{{\mathbf{O}}}^{{\mathbf{e}}} }} & = \int\limits_{V} {aRT{\mathbf{N}}_{{\boldsymbol{\upmu}}}^{{\mathbf{T}}} \frac{{\partial \dot{\omega }}}{{\partial {\boldsymbol{\upmu}}_{{\mathbf{O}}}^{{\mathbf{e}}} }}dV,} \;\; \hfill \\ \frac{{\partial {\mathbf{R}}_{{\mathbf{O}}} }}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }} & = \int\limits_{V} {\left( {{\mathbf{N}}_{{\boldsymbol{\upmu}}}^{{\mathbf{T}}} \frac{{\mu _{O} - \mu _{O}^{t} }}{{\Delta t}}\frac{{\partial c_{O} }}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }} + \eta _{O} V_{m} {\mathbf{N}}_{{\boldsymbol{\upmu}}}^{{\mathbf{T}}} \frac{{\sigma _{{kk}} - \sigma _{{kk}}^{t} }}{{\Delta t}}\frac{{\partial c_{O} }}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }} + \frac{{c_{O} \eta _{O} V_{m} }}{{\Delta t}}{\mathbf{N}}_{{\boldsymbol{\upmu }}}^{{\mathbf{T}}} \frac{{\partial \sigma _{{kk}} }}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }}} \right.} \hfill \\ &\quad + \left. {bRT{\mathbf{N}}_{{\boldsymbol{\upmu }}}^{{\mathbf{T}}} \frac{{\partial \dot{\omega }}}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }} - RT{\mathbf{B}}_{{\boldsymbol{\upmu}}}^{{\mathbf{T}}} \frac{{\partial {\mathbf{j}}^{{\mathbf{O}}} }}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }}} \right)dV, \hfill \\ \frac{{\partial {\mathbf{R}}_{{\mathbf{O}}} }}{{\partial {\boldsymbol{\upmu}}_{{\mathbf{M}}}^{{\mathbf{e}}} }}& = \int\limits_{V} {\left( {{\mathbf{N}}_{{\boldsymbol{\upmu}}}^{{\mathbf{T}}} \frac{{\mu _{O} - \mu _{O}^{t} }}{{\Delta t}}\frac{{\partial c_{O} }}{{\partial {\boldsymbol{\upmu}}_{{\mathbf{M}}}^{{\mathbf{e}}} }} + \eta _{O} V_{m} {\mathbf{N}}_{{\boldsymbol{\upmu}}}^{{\mathbf{T}}} \frac{{\sigma _{{kk}} - \sigma _{{kk}}^{t} }}{{\Delta t}}\frac{{\partial c_{O} }}{{\partial {\boldsymbol{\upmu}}_{{\mathbf{M}}}^{{\mathbf{e}}} }} + \frac{{c_{O} \eta _{O} V_{m} }}{{\Delta t}}{\mathbf{N}}_{{\boldsymbol{\upmu}}}^{{\mathbf{T}}} \frac{{\partial \sigma _{{kk}} }}{{\partial {\boldsymbol{\upmu}}_{{\mathbf{M}}}^{{\mathbf{e}}} }}} \right.} \hfill \\ &\quad + \left. {bRT{\mathbf{N}}_{{\boldsymbol{\upmu }}}^{{\mathbf{T}}} \frac{{\partial \dot{\omega }}}{{\partial {\boldsymbol{\upmu}}_{{\mathbf{M}}}^{{\mathbf{e}}} }} - RT{\mathbf{B}}_{{\boldsymbol{\upmu}}}^{{\mathbf{T}}} \frac{{\partial {\mathbf{j}}^{{\mathbf{O}}} }}{{\partial {\boldsymbol{\upmu }}_{{\mathbf{M}}}^{{\mathbf{e}}} }}} \right)dV, \hfill \\ \frac{{\partial {\mathbf{R}}_{{\mathbf{O}}} }}{{\partial {\mathbf{\mu }}_{{\mathbf{O}}}^{{\mathbf{e}}} }} &= \int\limits_{V} {\left( {{\mathbf{N}}_{{\boldsymbol{\upmu}}}^{{\mathbf{T}}} \frac{{\mu _{O} - \mu _{O}^{t} }}{{\Delta t}}\frac{{\partial c_{O} }}{{\partial {\boldsymbol{\upmu }}_{{\mathbf{O}}}^{{\mathbf{e}}} }} + \frac{{c_{O} {\mathbf{N}}_{{\boldsymbol{\upmu }}}^{{\mathbf{T}}} {\mathbf{N}}_{{\boldsymbol{\upmu }}} }}{{\Delta t}} + \eta _{O} V_{m} {\mathbf{N}}_{{\boldsymbol{\upmu }}}^{{\mathbf{T}}} \frac{{\sigma _{{kk}} - \sigma _{{kk}}^{t} }}{{\Delta t}}\frac{{\partial c_{O} }}{{\partial {\boldsymbol{\upmu }}_{{\mathbf{O}}}^{{\mathbf{e}}} }}} \right.} \hfill \\ & \left. { + \frac{{c_{O} \eta _{O} V_{m} }}{{\Delta t}}{\mathbf{N}}_{{\boldsymbol{\upmu }}}^{{\mathbf{T}}} \frac{{\partial \sigma _{{kk}} }}{{\partial {\boldsymbol{\upmu }}_{{\mathbf{O}}}^{{\mathbf{e}}} }} + bRT{\mathbf{N}}_{{\boldsymbol{\upmu }}}^{{\mathbf{T}}} \frac{{\partial \dot{\omega }}}{{\partial {\boldsymbol{\upmu }}_{{\mathbf{O}}}^{{\mathbf{e}}} }} - RT{\mathbf{B}}_{{\boldsymbol{\upmu }}}^{{\mathbf{T}}} \frac{{\partial {\mathbf{j}}^{{\mathbf{O}}} }}{{\partial {\boldsymbol{\upmu }}_{{\mathbf{O}}}^{{\mathbf{e}}} }}} \right)dV, \hfill \\ \end{aligned}$$

(57)

where the expressions of $\frac{{\partial {{\varvec{\upsigma}}}}}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }}$,$\frac{{\partial {{\varvec{\upsigma}}}}}{{\partial {{\varvec{\upmu}}}_{{\mathbf{M}}}^{{\mathbf{e}}} }}$,$\frac{{\partial {{\varvec{\upsigma}}}}}{{\partial {{\varvec{\upmu}}}_{{\mathbf{O}}}^{{\mathbf{e}}} }}$,$\frac{{\partial \dot{\omega }}}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }}$,$\frac{{\partial \dot{\omega }}}{{\partial {{\varvec{\upmu}}}_{{\mathbf{M}}}^{{\mathbf{e}}} }}$,$\frac{{\partial \dot{\omega }}}{{\partial {{\varvec{\upmu}}}_{{\mathbf{O}}}^{{\mathbf{e}}} }}$,$\frac{{\partial c_{M} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{M}}}^{{\mathbf{e}}} }}$,$\frac{{\partial c_{O} }}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }}$,$\frac{{\partial c_{O} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{M}}}^{{\mathbf{e}}} }}$,$\frac{{\partial c_{O} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{O}}}^{{\mathbf{e}}} }}$,$\frac{{\partial \sigma_{kk} }}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }}$,$\frac{{\partial \sigma_{kk} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{M}}}^{{\mathbf{e}}} }}$,$\frac{{\partial \sigma_{kk} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{O}}}^{{\mathbf{e}}} }}$,$\frac{{\partial {\mathbf{j}}^{{\mathbf{O}}} }}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }}$,$\frac{{\partial {\mathbf{j}}^{{\mathbf{O}}} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{M}}}^{{\mathbf{e}}} }}$, and $\frac{{\partial {\mathbf{j}}^{{\mathbf{O}}} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{O}}}^{{\mathbf{e}}} }}$ are as follows.

Recalling Eqs. (33)–(35), we have

$$\begin{gathered} \frac{{\partial {{\varvec{\upsigma}}}}}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }} = {\mathbf{CB}}_{{\mathbf{u}}} - \eta_{O} V_{m} {\mathbf{C\delta }}\frac{{\partial c_{O} }}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }}, \hfill \\ \frac{{\partial {{\varvec{\upsigma}}}}}{{\partial {{\varvec{\upmu}}}_{{\mathbf{M}}}^{{\mathbf{e}}} }} = {\mathbf{C\delta }}\left(\frac{{L_{\beta } V_{m} }}{a}\frac{{\partial c_{M} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{M}}}^{{\mathbf{e}}} }} - \eta_{O} V_{m} \frac{{\partial c_{O} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{M}}}^{{\mathbf{e}}} }}\right), \hfill \\ \frac{{\partial {{\varvec{\upsigma}}}}}{{\partial {{\varvec{\upmu}}}_{{\mathbf{O}}}^{{\mathbf{e}}} }} = - \eta_{O} V_{m} {\mathbf{C\delta }}\frac{{\partial c_{O} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{O}}}^{{\mathbf{e}}} }}, \hfill \\ \frac{{\partial {\mathbf{j}}^{{\mathbf{O}}} }}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }} = - \frac{{D_{O} \nabla \mu_{O} }}{RT}\frac{{\partial c_{O} }}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }}, \hfill \\ \frac{{\partial {\mathbf{j}}^{{\mathbf{O}}} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{M}}}^{{\mathbf{e}}} }} = - \frac{{D_{O} \nabla \mu_{O} }}{RT}\frac{{\partial c_{O} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{M}}}^{{\mathbf{e}}} }}, \hfill \\ \frac{{\partial {\mathbf{j}}^{{\mathbf{O}}} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{O}}}^{{\mathbf{e}}} }} = - \frac{{D_{O} c_{O} }}{RT}{\mathbf{B}}_{{{\varvec{\upmu}}}} - \frac{{D_{O} \nabla \mu_{O} }}{RT}\frac{{\partial c_{O} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{O}}}^{{\mathbf{e}}} }}, \hfill \\ \frac{{\partial \dot{\omega }}}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }} = L^{A} L_{\beta } V_{m} \frac{{\partial \sigma_{kk} }}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }}, \hfill \\ \frac{{\partial \dot{\omega }}}{{\partial {{\varvec{\upmu}}}_{{\mathbf{M}}}^{{\mathbf{e}}} }} = L^{A} a{\mathbf{N}}_{{{\varvec{\upmu}}}} - L^{A} \frac{{\partial \mu_{{M_{a} O_{b} }} }}{{\partial {{\varvec{\upmu}}}_{{_{{\mathbf{M}}} }}^{{\mathbf{e}}} }} + L^{A} L_{\beta } V_{m} \frac{{\partial \sigma_{kk} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{M}}}^{{\mathbf{e}}} }}, \hfill \\ \frac{{\partial \dot{\omega }}}{{\partial {{\varvec{\upmu}}}_{{\mathbf{O}}}^{{\mathbf{e}}} }} = L^{A} b{\mathbf{N}}_{{{\varvec{\upmu}}}} + L^{A} L_{\beta } V_{m} \frac{{\partial \sigma_{kk} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{O}}}^{{\mathbf{e}}} }}. \hfill \\ \end{gathered}$$

(58)

From Eqs. (26) and (23), we have

$$\begin{gathered} c_{M} = c_{M\max } \exp \left(\frac{{\mu_{M} - \mu_{M}^{0} }}{RT}\right), \hfill \\ c_{O} = c_{O\max } \exp \left(\frac{{\mu_{O} - \mu_{O}^{0} + \eta_{O} V_{m} \sigma_{kk} }}{RT}\right). \hfill \\ \end{gathered}$$

(59)

Then, we have

$$\begin{gathered} \frac{{\partial c_{M} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{M}}}^{{\mathbf{e}}} }} = \frac{{c_{M} {\mathbf{N}}_{{{\varvec{\upmu}}}} }}{RT}, \hfill \\ \frac{{\partial c_{O} }}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }} = \frac{{\eta_{O} V_{m} c_{O} }}{RT}\frac{{\partial \sigma_{kk} }}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }}, \hfill \\ \frac{{\partial c_{O} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{M}}}^{{\mathbf{e}}} }} = \frac{{\eta_{O} V_{m} c_{O} }}{RT}\frac{{\partial \sigma_{kk} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{M}}}^{{\mathbf{e}}} }}, \hfill \\ \frac{{\partial c_{O} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{O}}}^{{\mathbf{e}}} }} = \frac{{c_{O} {\mathbf{N}}_{{{\varvec{\upmu}}}} }}{RT} + \frac{{\eta_{O} V_{m} c_{O} }}{RT}\frac{{\partial \sigma_{kk} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{O}}}^{{\mathbf{e}}} }}. \hfill \\ \end{gathered}$$

(60)

Owing to

$$\sigma_{kk} = \frac{E}{1 - 2\upsilon }\left\{ {\varepsilon_{kk} - 3\left[ {\eta_{O} V_{m} (c_{O} - c_{O0} ) + \frac{{L_{\beta } }}{a}(c_{M0} - c_{M} )V_{m} } \right]} \right\},$$

(61)

we have

$$\begin{gathered} \frac{{\partial \sigma_{kk} }}{{\partial {\mathbf{u}}^{{\mathbf{e}}} }} = \frac{{RTE{\mathbf{M}}_{{\mathbf{u}}} }}{{3E\eta_{O}^{2} V_{m}^{2} c_{O} + RT(1 - 2\upsilon )}}, \hfill \\ \frac{{\partial \sigma_{kk} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{M}}}^{{\mathbf{e}}} }} = \frac{{3EL_{\beta } V_{m} c_{M} {\mathbf{N}}_{{{\varvec{\upmu}}}} }}{{3aE\eta_{O}^{2} V_{m}^{2} c_{O} + aRT(1 - 2\upsilon )}}, \hfill \\ \frac{{\partial \sigma_{kk} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{O}}}^{{\mathbf{e}}} }} = - \frac{{3\eta_{O} V_{m} c_{O} E{\mathbf{N}}_{{{\varvec{\upmu}}}} }}{{3\eta_{O}^{2} V_{m}^{2} Ec_{O} + RT(1 - 2\upsilon )}}, \hfill \\ \end{gathered}$$

(62)

where

$${\mathbf{M}}_{{\mathbf{u}}} = \left[ {\begin{array}{*{20}c} {\frac{{\partial N_{1} }}{\partial x}} & {\frac{{\partial N_{1} }}{\partial y}} & \cdots & \cdots & {\frac{{\partial N_{m} }}{\partial x}} & {\frac{{\partial N_{m} }}{\partial y}} \\ \end{array} } \right].$$

(63)

Finally, from Equation (32),

$$\frac{{\partial \mu_{{M_{a} O_{b} }} }}{{\partial {{\varvec{\upmu}}}_{{\mathbf{M}}}^{{\mathbf{e}}} }} = - \frac{{\tilde{c}_{M} }}{{1 - \tilde{c}_{M} }}{\mathbf{N}}_{{{\varvec{\upmu}}}} .$$

(64)

Appendix B: Material parameter estimations

2.1 B.1: Chemical potential boundary conditions of oxygen ions [44]

Since the experiments of Tolpygo et al.[3, 4] were perform in air at atmospheric pressure, we take $p_{{O_{2} }} = 0.21bar = 0.021MPa$.

The corresponding chemical potential of oxygen molecules in an ideal gas is

$$\mu_{{O_{2} }} = \mu_{{O_{2} }}^{0} + RT\ln \left( {\frac{{p_{{O_{2} }} }}{{p^{0} }}} \right),$$

(65)

where p⁰ is the reference pressure, taken as 1 bar.

We assume that equilibrium conditions prevail between the diatomic oxygen molecule O₂ in the gas and the oxygen ion O in the material:

$$\frac{1}{2}O_{2} \leftrightarrow O.$$

(66)

At equilibrium, the chemical potential of the gas must equal the chemical potential of the oxygen ions in the solid:

$$\mu_{O} = \frac{1}{2}\mu_{{O_{2} }} = \frac{1}{2}\left[ {\mu_{{O_{2} }}^{0} + RT\ln \left( {\frac{{p_{{O_{2} }} }}{{p^{0} }}} \right)} \right] = \mu_{O}^{0} - 0.78RT.$$

(67)

2.2 B.2: Reference chemical potentials and chemical reaction proportionality coefficient

As discussed in Sect. 4.1, the weighted sum of the reference chemical potential and the chemical reaction proportionality coefficient will affect the degree of reaction at equilibrium and the stress in the oxide layer. The degree of reaction at equilibrium increases with the decrease in the weighted sum of the reference chemical potential. Since the oxidation reaction of metals is usually irreversible, on the premise of ensuring the robustness of the program, the weighted sum of the reference chemical potential should be as small as possible to make the reaction degree as high as possible in the steady state. On the other hand, since plasticity and creep are not considered in our model, it is impossible to accurately describe the stress evolution in the high-temperature oxidation process of metal. However, by changing the parameters, we try to make the calculated average stress and the experimentally measured residual stress in the oxide layer at the same level. From Fig. 4b, the values of two chemical reaction proportionality coefficients L_β correspond to the same stress level, but combined with Fig. 4a, the increase of L_β will significantly reduce the reaction degree at equilibrium. Although the chemical reaction degree may be increased by reducing the weighted sum of the reference chemical potential, a too small weighted sum of the reference chemical potential will destroy the robustness of the program. Based on the above considerations, we take the smaller L_β corresponding to the given stress.

By trial and error, we take $\mu_{Al}^{0} /(RT) = \mu_{O}^{0} /(RT) = - \mu_{{Al_{2} O_{3} }}^{0} /(RT) = 2$, and L_β = 0.005. Under such parameters, the conversion of Al can reach about 99.1% (there will be slight changes at different temperatures) in the steady state, and the biaxial compressive stress in the oxide layer is about 4.5GPa at 1473 K and 1573 K, which is in approximate agreement with the experimental data.

B.3 Diffusion coefficient reference value, reaction rate coefficient reference value, diffusion activation energy, reaction activation energy.

We assume that the produced aluminum oxide is dense, so D_O0,2 = 0, and the diffusion activation energy in the oxide is not required because the diffusion coefficient in the oxide is zero no matter how large the diffusion activation energy in the oxide is taken.

The reference value of the diffusion coefficient in the matrix, diffusion activation energy in the matrix, the reference value of reaction rate coefficient, and reaction activation energy are obtained by comparing with the experimental curve of oxide layer thickness growth. The specific values are given in Table 1.

Appendix C: Analytical solution of the isotropic plane stress-free equilibrium swelling with chemical reaction

Making use of Eq. (33) under the plane strain conditions, we have

$$\begin{gathered} \sigma_{1} = \sigma_{2} = \frac{E}{(1 + \upsilon )(1 - 2\upsilon )}\left\{ {\varepsilon_{0} - (1 + \upsilon )\left[ {\eta_{O} V_{m} (c_{O} - c_{O0} ) + \frac{{L_{\beta } }}{a}(c_{M0} - c_{M} )V_{m} } \right]} \right\}, \hfill \\ \sigma_{3} = \frac{E}{(1 + \upsilon )(1 - 2\upsilon )}\left\{ {2\upsilon \varepsilon_{0} - (1 + \upsilon )\left[ {\eta_{O} V_{m} (c_{O} - c_{O0} ) + \frac{{L_{\beta } }}{a}(c_{M0} - c_{M} )V_{m} } \right]} \right\}. \hfill \\ \end{gathered}$$

(68)

Then, application of the stress-free condition in the x₁-x₂ plane gives

$$\varepsilon_{1} = \varepsilon_{2} = \varepsilon_{0} = (1 + \upsilon )\left[ {\eta_{O} V_{m} (c_{O} - c_{O0} ) + \frac{{L_{\beta } }}{a}(c_{M0} - c_{M} )V_{m} } \right],$$

(69)

and

$$\sigma_{kk} = \sigma_{3} = - E\left[ {\eta_{O} V_{m} (c_{O} - c_{O0} ) + \frac{{L_{\beta } }}{a}(c_{M0} - c_{M} )V_{m} } \right].$$

(70)

At equilibrium state, the diffusion of species O reaches steady state, so the diffusion Eq. (34) yields

$${\mathbf{j}}^{{\mathbf{O}}} = - \frac{{D_{O} c_{O} }}{RT}\nabla \mu_{O} = {\text{constant}}{.}$$

(71)

Since the prescribed boundary condition $\mu_{O}^{*}$ is homogeneous around the block, one can conclude that μ_O is homogeneous throughout the block,

$$\mu_{O} = \mu_{O}^{*} .$$

(72)

Therefore, c_M is also a constant at equilibrium. Here, it can be seen that only two undetermined variables exist, that is, c_M and c_B.

The steady-state chemical reaction rate is zero. Thus, application of the expression of the chemical reaction rate in Eq. (35) gives

$$\dot{\omega } = L^{A} \left( {a\mu_{M} + b\mu_{O} - \mu_{{M_{a} O_{b} }} + L_{\beta } V_{m} \sigma_{kk} } \right) = 0.$$

(73)

Substituting Eq. (70) into Eqs. (72) and (73), and combining with Eq. (53), we have

$$\begin{aligned} & {}[\mu_{M}^{0} + RT\ln (\tilde{c}_{M} ) + b\mu_{O}^{*} - \mu_{{M_{a} O_{b} }}^{0} - RT\ln (1 - \tilde{c}_{M} )] \hfill \\ & \quad - L_{\beta } \left[ {\tilde{c}_{M} V_{m1} + (1 - \tilde{c}_{M} )V_{m2} } \right]^{2} \left[ {\tilde{c}_{M} E_{1} + (1 - \tilde{c}_{M} )E_{2} } \right]\left[ {\eta_{O} (c_{O} - c_{O0} ) + \frac{{L_{\beta } }}{a}(c_{M0} - c_{M} )} \right] = 0, \hfill \\ \end{aligned}$$

(74)

$$\begin{gathered} \mu_{O}^{0} + RT\ln (\tilde{c}_{O} ) \hfill \\ + \eta_{O} \left[ {\tilde{c}_{M} V_{m1} + (1 - \tilde{c}_{M} )V_{m2} } \right]^{2} \left[ {\tilde{c}_{M} E_{1} + (1 - \tilde{c}_{M} )E_{2} } \right]\left[ {\eta_{O} (c_{O} - c_{O0} ) + \frac{{L_{\beta } }}{a}(c_{M0} - c_{M} )} \right] - \mu_{O}^{*} = 0. \hfill \\ \end{gathered}$$

(75)

Combining Eqs. (74) and (75) provides the implicit solution to the concentration of M and O with proper parameters and boundary conditions of μ_O, which can be solved by the multivariate Newton method. Then, all the other variables can be derived, such as ε₀ and σ₃.