Unification of the deterministic and statistical approaches for predicting localized corrosion damage. I. Theoretical foundation

Abstract

In this paper, we provide an alternative, more general theoretical basis for damage function analysis (DFA), by drawing an analogy between the growth of a pit and the movement of a particle. In contrast to our previous formulation of DFA, which was developed specifically for enabling the damage function for localized corrosion to be calculated from the point defect model for passivity breakdown, the coupled environment pitting model for pit growth, and the theory of prompt and delayed repassivation, the new formulation readily incorporates any theories or models (deterministic or empirical) for these stages in the development of a pit. We show that the new formulation leads to the original expressions for the damage functions for active (living) and passivated (dead) pits, and hence for the differential and integral damage functions, as were obtained from the original theory. We also describe the unification of deterministic (damage function analysis, DFA) and empirical, statistical (extreme value statistics, EVS) methods for predicting the development of localized corrosion damage on metal surfaces. In particular, we have devised a means of estimating the central and scale parameters of EVS directly from DFA in a “first principles” manner, as well as from fitting the EVS distribution function to experimental data for short times, in order to predict the extreme value distributions at longer times. The techniques have been evaluated on EVS data for the pitting of manganese steel in CO₂-acidified seawater and for the pitting of aluminum in tap water. Finally, we outline the generalization of pit nucleation, as described by the point defect model, for external conditions that depend on time.

Introduction

The development of effective localized corrosion damage prediction technologies is essential for the successful avoidance of unscheduled downtime in industrial systems and for the successful implementation of life extension strategies. Currently, corrosion damage is extrapolated to future times by using various empirical models coupled with damage tolerance analysis (DTA). In this strategy, known damage is surveyed during each subsequent outage, and the damage is extrapolated to the next inspection period allowing for a suitable safety margin. It has been noted previously [1] that this strategy is inherently inaccurate and inefficient, and that in many instances it is too conservative. Instead, it has been suggested that damage function analysis (DFA) is a more effective method for predicting the progression of damage, particularly when combined with periodic inspection [1]. Although corrosion is generally complicated mechanistically, a high level of determinism has been achieved in various treatments of both general and localized corrosion, which can be used to predict accumulated damage in the absence of large calibrating databases [2].

It is important to note that the word “determinism” is used here to describe a model that contains a physically viable mechanism (i.e., one that accounts for the known properties of the system) and whose predictions along the system's evolutionary path are constrained by the natural laws. The great majority of models that are used in corrosion science and engineering do not satisfy the definition of determinism given above and indeed many are little more than sophisticated correlations. However, many statistical models, such as those contained within extreme value statistics (EVS), make no pretense of being mechanistically-based or of conforming to the constraints of determinism, yet they have proven to be of great utility in correlating experimental data in terms of relationships between the dependent and independent variables. Generally speaking, the traditional statistical approach that is used for extrapolating corrosion damage in time (and space) can be described as follows [3]. It is assumed that the distribution of the deepest pits on many specimens for a given observation time is described by one of the three classical extreme value distributions that are functions of two or three statistical parameters (central, u, scale, α, and a shape parameter, k). Empirical expressions are developed for parameters u, α, and k by assuming functional dependencies on time, t, which are then used to extrapolate the parameters to longer times (and often to larger area). Thus, in Ref. [3], it is assumed that u=u₀t^b and α=α₀t^b, with the same power, b, and that k=const, but there is no physical justification for the selection. In this approach, the fitting parameters u₀, α₀, b, and k are not related to the physical parameters of the corrosion system, such as pit propagation rate, repassivation constant, etc. Accordingly, in using such an approach, it is, in principle, impossible to predict corrosion damage for a system beyond the experimental field, particularly if the external conditions depend on time. Moreover, presentation of the central and scale parameters in the form of power functions renders impossible the prediction of damage when the depth of the deepest pit cannot increase beyond some critical value (e.g., due to repassivation of the pit) [4]. These models should not be termed “predictive”, since they all contain parameters (e.g., the central and scale parameters in the Gumbel Type I distribution function in EVS) for which there is no deterministic guidance as to their values and future time dependencies. This specific issue is addressed in the present paper.

As previously noted [1], [5], the evolution of localized corrosion damage is a birth (nucleation)–growth (propagation)–death (repassivation) process, in which these stages (birth, growth, and death) occur sequentially for a single event, but tend to occur in parallel for a large ensemble of events. These basic features classify the system as being “progressive”, because new, damaging events nucleate as existing events live and die. Accordingly, any models that are incorporated into DFA must recognize the progressive nature of the evolution of damage. The progressive nature of the process is captured in the “damage function” (DF), which is the histogram of event frequency versus increment in depth (Fig. 1).

The predicted DFs shown in Fig. 1 for aluminum in chloride solutions [6] represent the distribution in depth of the damaging process (e.g., pitting and/or stress corrosion cracking) for different observation times. The principal value of the DF is that it provides ready definitions of “failure” and the “time to failure”, with the latter being the observation time at which the upper extreme exceeds the critical dimension, L_cr. Thus, with regard to Fig. 1, failure would not have occurred after one and two years of exposure, but it would have occurred after three. Clearly, prediction of the evolution of the DF with time is the necessary requirement for calculating the time at which a system will fail due to localized corrosion and hence for calculating the failure time. However, the DF also allows one to predict the extent of localized corrosion damage as being the integrated area under the distribution that defines the DF. For many applications, the extent of damage is an important quantity, depending upon the definition of “failure”. For example, corrosion-induced roughening of the surface of a ball bearing will often cause the bearing to fail in a time that is related to the extent of roughening rather than to the maximum depth.

The authors have established the theoretical basis of DFA over the past decade [1], [2], [5], [6], and its utility in predicting localized corrosion damage in a number of systems has been well demonstrated [1], [5], [6]. In the present work, we develop a new method of calculating the DF, by drawing an analogy between the growth of pits and the transport of particles. This alternative treatment, which is outlined below, leads to the same result as our previous analysis in the simplest case, but it also provides a foundation for quantitatively describing arbitrarily complicated systems, particularly for incorporating multiple modes of damage (pitting, stress corrosion cracking, and corrosion fatigue) into DFA, or for investigating cases when the external conditions (temperature, corrosion potential, electrolyte composition, etc.) depend on time.

While considerable advances have been made in the deterministic prediction of corrosion damage over the past two decades, a strictly deterministic description of the propagation of corrosion damage in a majority of real systems continues to be a major theoretical challenge that is only slowly yielding to scientific inquiry. The difficulties inherent in this task not only reflect the lack of information for various kinetic parameters for the nucleation, repassivation, and propagation of pits, but they also include the frequently poorly defined statistical nature of some corrosion processes, even though the underlying causes are deterministic in nature. For example, we do not adequately understand the statistical nature of the corrosion cavity growth rate in terms of the physico-electrochemical origin of the distribution or its mathematical form, although if this information were available then, at least in principle, the inclusion of the mechanism in a deterministic model is feasible. There exist many possible reasons for a distribution in cavity growth rate, including, example, the spatial distribution in anodic and cathodic sites on the corroding surface or because of microstructural and microchemical non-uniformity of the metal [7], [8]. In this paper, we report the first attempt to demonstrate the application of DFA to the case where a distribution exists in the corrosion cavity propagation rate.

For example, it will be shown from DFA that, with certain simplifying assumptions, the distribution of the deepest pit among “identical” specimens must be described by extreme value statistics (EVS). A combination of DFA and traditional statistical analysis (SA) offers significant advantages over purely statistical/empirical approaches that are not based upon deterministic principles. Thus, DFA allows us to express the fitting parameters for SA in terms of values for the physical parameters of the component deterministic model(s) (e.g. pit nucleation and propagation rates, repassivation constants, etc.) that can be validated by independent experiment. Accordingly, it becomes possible to predict the statistical fitting parameters as the external conditions change with time.