Spatially Structured Environmental Noise Controls Stability Thresholds in Fractional Corrosion Fields
Abstract
Since the perturbations of chloride concentration, wetting, dryness, oxygen supply, temperature, and contamination of the surface are unlikely to occur as uniform exposures in space, their organization through coating defects, pores, grains, inclusions, crevices, and exposure zones leads to the possibility that two environments of the same total variance may induce dissimilar corrosion impacts. The framework provides a fractional stochastic reaction-diffusion problem that translates the spatial structure of random exposure into the condition of the mean corrosion activity. In it, the material is modeled by a normalized interval with zero-flux boundary conditions, nonlocal transport is described by the spectral fractional Laplacian, while the environmental forcing acts on the Neumann stable modes. The separation of scales in the system leads to the reduced equation for corrosion activity in which the index of modal susceptibility $\Sr=\sum_{j\in\Jset}\alpha_j^2/(2D j^r)$ is the key to obtaining a deterministic term induced by the excitation of random stable modes. The set of numerical experiments includes 600 cases with $D\in\{0.5,1.0,1.5,2.0\}$, $r\in\{1.2,1.4,1.6,1.8,2.0\}$, $q\in\{1,2,3,4,5\}$, and $\alpha\in\{0.5,0.75,1.0,1.25,1.5,2.0\}$. Two classes of kinetics are considered. With cubic passivation--activation kinetics, the stability of passive state holds whenever $\Sr>1/3$, whereas in the case of logistic saturated damage, only those equilibria exist for which $\Sr\leq1/4$ in terms of the normalized carrying capacity. From the performed calculations, one finds that the contribution of low modes dominates threshold crossing: the first mode takes 75 out of 128 cases of cubic stabilization, while the fifth mode only contributes to two cases. The analysis confirms the importance of the spatial distribution of the environmental variability in the fractional corrosion problem.