Spectral Noise Filtering and Mean-Field Thresholds in Fractional Corrosion Reaction–Diffusion Dynamics
Volume 1 (2022) Issue 2,
DOI:
https://doi.org/ 10.71448/jcm2022v1i24
Submission Received: 15 Aug 2022
Revised: 02 Nov 2022
Accepted: 28 Nov 2022
Published: 30 Dec 2022
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Abstract
Corrosion that varies from place to place is frequently initiated by perturbations which are not acting directly on the spatial average of the surface, coating, pore structure, or reinforcing interface but which are able to shift the spatial average of a field through nonlinear response. Here, we develop a closed fractional stochastic reaction-diffusion framework for that process. We define a normalized corrosion-activity field on $[0,\pi]$, with Neumann boundary conditions, where fractional transport is represented by the operator $(-\Delta)^{r/2}$, for $1<r\leq 2$, and environmental variability is accounted for by finite-dimensional Wiener forcing on the stable modes of cosine functions. The critical quantity is the mean, whereas the forced high-frequency modes quickly relax due to strong diffusion. The effect of averaging the stationary fluctuations of the stable modes is the spectral weighting of their variance
$\mathcal{S}_{r}=\sum_{j\in\mathcal{J}}\alpha_j^2/(2Dj^r)$, which plays the role of deterministic modification of the reaction law. With the case of cubic passivation and activation kinetics, the passive state is stable if $\mathcal{S}_{r}>1/3$. In logistic damage growth kinetics with carrying capacity $K$, nonzero sustained steady-states appear exclusively if $\mathcal{S}_{r}\leq K^2/4$. The analysis relies on $r\in\{1.2,1.6,2.0\}$, one forced mode $q\in\{1,2,3,4,5\}$, $D\in\{0.5,1.0,1.5,2.0\}$, and $K=1$, except if indicated. Our findings suggest that modal positioning is as important as the amplitude, since lower frequency modes have larger leverage on the mean-field due to weak fractional diffusion damping relative to higher frequency modes. This model therefore provides a clear cut-off value for the interpretation of corrosion pattern resulting from variations in chlorides, defects in coatings, porosity, and wet-dry cycling.
Keywords
corrosion activity,stochastic reaction--diffusion,fractional Laplacian,chloride heterogeneity,spectral averaging,passivation threshold