International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Inter-relationship between physical-chemical processes and extreme value modelling Melchers, Robert E. Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1 Inter-Relationship Between Physical-Chemical Processes and Extreme Value Modelling  Robert E. Melchers  Professor of Civil Engineering, Centre for Infrastructure Performance and Reliability, The University of Newcastle, Australia.   ABSTRACT: Of considerable interest in various industries such as aerospace is the longer term safety of aluminium alloy structures and the effect of deterioration. Corrosion of aluminium alloys occurs mainly as pitting for which uncertainty and variability issues usually have led to maximum pit depth being considered a random variable that is also a function of time. An extreme value distribution is fitted to the statistical data obtained from multiple observations. Usually, the selection of the most appropriate model is based on the claim that one or other distribution is a ‘better fit’ to the data. This classical approach takes no account of prior knowledge of the underlying physicochemical process(es) that drive pitting behaviour. A more sophisticated approach uses such prior understanding. Recently it was shown that the linear model implied by the ‘pitting rate’ is too simplistic. Instead, a bi-modal model better represents both mass-loss as a function of exposure period and the evolution of maximum pit depth with time. This leads directly the possibility that one distribution may not be suitable for the whole range of pit depth data. These concepts are illustrated with examples.    1. INTRODUCTION  For applications such as pipeline walls, tank walls, ships plates and aircraft fuel tanks and exterior fuselage and the containment structures for nuclear waste storage long-term deterioration is of much interest. Achieving a high likelihood of reaching the expected service lives, ranging from 10 to 100,000 years, requires that likely future deterioration after a given period of exposure in a given environment can be predicted. Ideally prediction should be based on the best available information rather than be simply extrapolation from the past.  Herein pitting corrosion, mainly of aluminium alloys, is considered. Two issues are important: the development of the maximum depth of pitting as a function of time, and the estimation of the uncertainty in maximum pit depth at any point in time. Even for a given exposure environment and at a given point in time, the depths of pits can show considerable variability. This has long been recognized. In fact, the classical Extreme Value (EV) theory literature deals specifically with such problems. Further, pit depth uncertainty often is considered the archetypical extreme value problem (e.g. Galambos 1987).   Extreme value analysis conventionally relies heavily on data. For pitting, usually it is assumed (often unwittingly) that maximum pit depth (as measured on sub-areas of equal size) is an independent random variable (or for very large sample sizes can be considered so asymptotically) (Coles 2001). This permits the assumption that all deepest pit measurements are sampled from a homogenous population. Each pit depth is assumed equally likely to have been obtained during (random) sampling. The extreme value analysis proceeds by assigning each pit depth value a cumulative probability. A number of techniques are available for doing this (Galambos 1987, Coles 2001). Each pit depth value is then plotted on an extreme value plot against its assigned cumulative probability.    Largely as a result of the pioneering work of Aziz (1956), the Gumbel EV distribution (for the greatest value) has been accepted widely for representing the uncertainty in maximum pit 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2 depth. However, other distributions, such as the Weibull and the Generalized Extreme Value distribution, also have been used, always based on the claim that such a distribution provides a better ‘fit’ to the particular available data (Galambos 1987, Coles 2001). Other than considering whether a distribution for the maximum or for the minimum is more appropriate, consideration is seldom given to any underlying physico-chemical or other processes that might be involved.  In the light of modern thinking in probabilistic analysis, including the well-established (but in some frequentist circles still distasteful) concepts of Bayesian statistics and Bayesian up-dating (Lindley 1972), it is appropriate to consider how theoretical understanding of the processes underlying the pitting corrosion mechanisms can influence the choice of EV distribution and thus the probabilistic modelling of maximum pit depth. Conversely, it is of interest to see what can be deduced from an EV representation of pit depth data. Both of these matters are described herein with particular application to the pitting of aluminium. Some earlier discussion of this approach exists for mild steel (Melchers 2008).  The next section describes the development of corrosion loss as a function of time and the development of models for maximum pit depth. Both functions are highly non-linear, reflecting the physicochemical principles underlying pitting corrosion. Attention is then turned to extreme value analysis of some of the few data sets available in the literature. These are considered in the light of the non-linear model for pitting corrosion to produce interpretations considerably different from the conventional notion that a Gumbel distribution best describes uncertainty in pit depth.  2. CORROSION OF ALUMINIUM  Aluminium and its alloys have excellent corrosion resistance. Even under longer-term exposures, they show little general and only moderate pitting corrosion, including in seawater and marine conditions (Vargel 2004). For this reason aluminium alloys are much used in the aerospace, defence and marine industries. As for other metals, various models for aluminium corrosion have been proposed. The simplest is the so-called ‘corrosion rate’, representing the average corrosion over a given period of time. In most practical applications, however, this is not representative of actual behaviour. The reason is that as the metal corrodes it develops protective oxide films that tend to reduce the instantaneous corrosion rate. As a result, particularly for corrosion in the atmosphere over many years (Ailor 1982, Vargel 2004), and also for pit depth development over much shorter time intervals, the so-called ‘power-law’ function commonly is adopted. It has the form    c(t) = AtB      (1)  where c is corrosion loss or pit depth, t is exposure time and A and B are ‘constants’. Normally these are obtained by fitting Eqn. (1) to data for a given site, and thus both A and B are site- and environment-specific and very sensitive to changes in data. Despite this, Eqn. (1) continues to be applied extensively (e.g. de la Fuenta et al. 2007, Sun et al. 2009). However, careful analysis of data has shown recently that Eqn. (1) usually is sub-optimal for describing the long-term corrosion of aluminium alloys (Melchers 2014).  Data for developing functions such as Eqn. (1) usually are obtained by exposing relatively small metal coupons, say 100 x 50 mm or larger, to the environment of interest and then determining the mass loss after a given period of exposure. Using many such coupons and assuming they are from the same batch of metal and exposed to essentially the same environment but for different lengths of time permits a plot of corrosion loss data against exposure period. Two such sets are shown in Fig. 1. These can be used to construct a function such as Eqn. (1) or a linear function (a ‘corrosion rate’) through the data. These are shown for each data set. It is clear that the actual trend in the data departs considerably from these two simple functions.  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3   Figure 1:Mass loss due to corrosion for pure aluminium with linear and power-law trend.    A better approach is to ask first what non-simple but smooth function would best fit through the data and then ask what might be the rationale for such a function. Fig. 2 shows the same data as in Fig. 1 but now with a ‘best fit’ smooth function (Stineman 1980) through the data sets. It is seen that both cases display a ‘bi-modal’ trend. Many similar examples exist, for a range of aluminium alloys and exposure conditions (Melchers 2014).         Figure 2: Same data as Fig. 1 but with best fit trend curves showing bi-modal trending.         On the basis of interpretation of data sets such as those in Fig. 2, it has been proposed that the longer term trend for corrosion of aluminium, in seawater, in fresh water, in marine and in other atmospheric corrosion exposures can be represented by a so-called ‘bi-modal’ function, shown in schematic form in Fig. 3. Such a function was first developed for the corrosion of steels in a variety of environments (Melchers 2003) and has been shown to be consistent with data for weathering steel, for low alloy steels, for high chromium steels and for cast irons. Fig. 3 also shows 5 phases, sequential in time t. Each phase corresponds to a different dominant corrosion mechanism as summarized in Fig. 3. Details are available (Melchers 2014).     Figure 3: Schematic bi-modal model for long-term corrosion loss and pit depth. In the idealized case the change from predominantly oxic to predominantly anoxic conditions occurs at ta.      Recognition that the overall trend for aluminium corrosion mass loss has a bi-modal functional relationship immediately provides a new framework for interpreting other data sets. For example, Sun et al. (2009) reported mass losses for the thin (80 micron thick) layers of aluminium cladding used for protection of steel sheeting. These were exposed to 3 different atmospheres in China for up to 20 years. Figs. 4 and 5 show the mass losses reported for aluminium alloys 2024 and 7075 at 1, 3, 6, 10 and 20 years. Consistent with conventional wisdom, Sun et al. fitted linear and power-law 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4 functions through each data set. However, it is clear that the data exhibit considerable deviation from these simple functions. In contrast, the data sets appear to trend consistently with the bi-modal function. This is shown by ‘best fit’ trend constructed through each data set (Figs. 4 and 5).    Figure 4: Best fit and interpreted trends  for 2024 aluminium.     Figure 5: Best fit and interpreted trends for 7075 aluminium.      In Figs. 4 and 5 the best fit trend has been modified to give a so-called subjective, or ‘interpreted’ trend. This shows that with only small departures from the best fit trend a bi-modal function is equally possible, with the trend still passing through all observed data - the only interpretation is in the trend between the data points.  The practical importance of the form of the longer-term trend is illustrated in Fig. 6. It shows two different interpretations of data. One is a linear trend through the data without regard to underlying corrosion principles and extrapolated to predict likely future corrosion loss. The other, fitted to the same data, is based on a direct extension of the bi-model model. There is considerable difference in the projected rate of future corrosion between these two approaches.    Figure 6: Extrapolation from different interpretations of the same data.    For statistical analysis, there are important implications arising from the progressive changes in corrosion processes in the bi-modal model (Fig. 3). Only data from any one corrosion phase can be considered as drawn from a homogenous population. Conversely, data sampled at different points in time over an experiment extending over a long period of time, most likely are from the different corrosion phases in the bi-modal model. It is unlikely that these data represent a homogeneous set. They therefore should not be treated as such in an EV analysis.    12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5 3. PIT DEPTH TRENDS  As noted earlier, for maximum pit depth two matters are of interest – the development of the maximum depth of pitting with time and the uncertainty in pit depth at any point in time. For most metals an intimate relationship exists between corrosion loss as measured by mass loss (Figs. 1-2, 4-6) and the extent and depth of localized or pitting corrosion. This is the case also for aluminium (Vargel 2004). It is therefore not unexpected that the few data sets that exist for pitting corrosion of aluminium, and that are consistent and coherent, support the notion that maximum depth of pitting follows a bi-modal trend. One example is shown in Fig. 7. Other, similar, cases exist (Melchers 2014).        Figure 7: Trend curves through maximum depth pit data.       The trend curves shown through the data points were obtained by the ‘best fit’ routine (Stineman 1980). No interpretation was required. Both trends show the distinctive, high upswing in corrosion rate after about 8 years exposure, corresponding to the beginning of the second mode of the bi-modal model. An immediate consequence is that data collected for estimating maximum pit depth also is subject to the constraint on homogeneity. This has been noted also for pitting in steels (Melchers 2008).   4. PIT DEPTH UNCERTAINTY Very few data sets in the open literature have sufficient data points for extreme value analysis of pitting. This includes aluminium. Aziz (1956) plotted maximum depth pit data for 2S-O aluminium pipe exposed to Kingston tap water for up to one year on a Gumbel plot (Fig. 8a). As is well-known, a Gumbel plot consists of one axis showing the reduced variate w that represents the cumulative probability also given by the function φ(y1). It defines the probability that the pit depth y < y1. Best-fit lines were constructed through each complete data set (Fig. 8a). Each line represents the Gumbel EV distribution that fits the corresponding data set.    Closer inspection of the data sets for one month and one year exposures shows that each could be considered to be composed of two sub-sets, for shallower and deeper pits respectively. These are marked with curved lines in Fig. 8(a). Insight about the shape of these curved lines can be obtained by changing the pit depth (horizontal) axis to a lognormal scale and thereby changing the Gumbel plot to a Frechet plot (Fig. 8b), following, empirically, the earlier proposal for pitting in steels (Melchers 2008). Straight trend lines have been added. These provide a reasonable fit to the respective sub-sets of data. Translating these straight lines back to the Gumbel EV plot Fig. 8(a) gives the curves shown. The net result is that two extreme value distributions, one for shallower pits and one for the deeper pits, provide a better description of the statistics of maximum pitting. Importantly, for extrapolation not the whole data set but only the upper data set, for the deeper pits, is relevant.   Aluminium grade 2S-O is a soft aluminium alloy that was used extensively for pipe-work in aircraft. It has good resistance to pitting corrosion in chloride environments but is less resistant to fresh water. This can be seen in Fig. 8(a) that shows considerable pit depths (300-400 µm) already after one-week exposure. Data for grade 65ST aluminium alloy shows similar overall characteristics as in Fig. 8(a) but the maximum pit depths are about 25% lower.  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  6  (a)    (b)  Figure 8: Gumbel and Frechet EV distribution representation of maximum depth pit data.    Another example is shown in Fig. 9(a) for pitting inside pipes conveying fresh water at unspecified velocities v and 2v for pipelines of 75 mm and 50 mm nominal diameter respectively. The straight lines represent Gumbel EV distributions fitted to the whole data set in each case. However, in each case the data set clearly is not linear and shows two very distinct subsets. These are marked with the two curves. Some insight about the nature of these two curves can be obtained by plotting all the data on a Frechet plot (Fig. 9b). On this plot there are two distinct linear trends for each data set. This indicates that one Frechet EV distribution is appropriate for smaller maximum depth pits and a different one for the deeper pits. Translating these straight lines back to Fig. 9(a) produces the curves shown. Despite some scatter remaining about the straight lines shown in Fig. 9(b) for the deeper pits, extrapolation from this subset would be much more appropriate than extrapolation from the one trend fitted to all the data.    (a)    (b)  Figure 9: (a) Gumbel and (b) Frechet plots for maximum pit depths in aluminium pipes. Data from Aziz (1956). Note 1 mil = 0.0254 mm = 25.4 microns.    Finally, to show that the above developments are not restricted to the pitting of aluminium or to low alloy steels (Melchers 2008), consider the maximum pit depth data reported by Provan and Rodriquez (1989) for stainless steel samples immersion exposed to the very aggressive water known in the paper-making industry as ‘white 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  7 water’. It contains thiosulfates, sulfates and chlorides at solution pH = 4.5. Multiple measurements of pit depth were made at each exposure period. Only data for 16 and 64 day exposures are considered here. Gumbel lines are drawn through each data set as shown on Fig. 10. However, in view of the above trends, a much better interpretation considers the data in each set as composed of shallower and deeper pits, with the deeper sub-set best represented by a curvilinear trend rather than a straight line. Because the original data were not directly reported it is not possible to obtain a Frechet plot but the sub-set of data for the deeper pits appears to be consistent with the Frechet EV distribution shown in the examples above.    Figure 10. Gumbel plot for maximum pi depths for of stainless steel exposed to ‘white water’. The straight light black lines are the conventional Gumbel lines.     5. DISCUSSION   The examples given above demonstrate that a single EV distribution may not be appropriate to represent the complete range of values obtained from observations of extreme events. The examples given here, for pitting of aluminium and of stainless steel extend earlier results for pitting of mild steel in seawater (Melchers 2008). That earlier work showed that the change in trend seen also in the above examples appeared, for mild steel, to be associated with the change from the first to the second mode of the bi-modal model (Fig. 3). This cannot be ascertained for the data shown in Figs. 8-10 as the pitting corrosion trends as a function of exposure period was not reported for these cases. Irrespective, it is clear from statistical considerations alone that in each case the change in the trend for maximum pit depth on the Gumbel or on the Frechet plots represent a change in the applicable distribution function. It also follows that this is the result of different statistical populations being applicable.  Figs. 8-10 show that for any given exposure period and exposure condition, the deepest pits follow a different statistical distribution to the less deepest pit as measured on individual samples or sample areas. This indicates that at any time there are at least two populations of deepest pits – the population of very deep ones and another of the less deep ones. Since the probability distributions associated with these are different, this indicates immediately that there are differences in the pitting mechanics involved. To understand why this might be so, it is necessary to go back to observations in corrosion science. Usually it is assumed that pits, once initiated, grow in depth in a continuous manner and as a smooth (if non-linear) function of time. That would make all pits part of a homogeneous population and thus all sample measurements of maximum pit depth, samples from such a homogeneous population. However, pitting in practice is seldom of a continuous, smooth nature. Apart from the changes in the development of pit depth corresponding to the bi-modal corrosion model (Fig. 3), pitting for steel has been shown to be a discontinuous process, with early pits reaching a limited depth owing to a limit on the electrochemical potential available to drive the pitting process (e.g. Turnbull 1993). Lateral growth of pits is well-known and coalescence of pits has been observed, at least for steel (Melchers 2008). Coalescence of pits leaves plateaus of corroded metal and these have been observed to permit renewed pitting. There are similar observations for the longer term pitting of aluminium alloys 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  8 (Sun et al. 2009). No attention was drawn to this fact by these authors. The result is that at any given time, there are at least two, if not more, stages in the pitting process on the surface of a metal, each of which constitutes a different statistical population. Further, the change from phase 2 to phase 3 of the bi-modal model represents a substantial change in behaviour and this, too, can be expected to have an influence on the observed maximum depths of pits.     The above shows that there is a strong link between the evolution and the changing nature of the corrosion (and pitting) process and the representation of uncertainty. In the past invariably it was assumed that pitting corrosion followed one continuous evolutionary process, such as represented by Eqn. (1). One consequence is that all pit depth data were assumed generated from a homogenous population. Deviations from the ideal straight line on an extreme value plot (e.g. Gumbel plot) invariably were dismissed as the result of data or experimental errors or a natural consequence of variability. However, such an explanation is disingenuous if the same type of ‘deviation’ from a straight line on the Gumbel plot is seen repeatedly, in different data sets, for steel, for stainless steel and for different grades of aluminium. A proposal for the appearance of the Frechet EV distribution has been given (Melchers 2008) but this remains to be verified for aluminium.  6. CONCLUSION   The modelling of uncertainty the deterioration of structural materials can be improved if use is made of basic principles underlying physicochemical processes rather than using data alone. As for steel, the corrosion of aluminium follows a sequence of different corrosion phases in a bi-modal overall form as a function of exposure time. Each phase represents a different process and thus a different population for statistical purposes. This should have a direct influence on the interpretation of extreme value data. In particular, it is shown that at any point in time, the depth of deeper pits is more consistent with the Frechet than with the classical Gumbel distribution. In turn this can have a major effect on estimation of exceedence probabilities for maximum pit depth and for extrapolation of probability estimates. The underlying physico-chemical reason(s) for the appearance of the Frechet EV distribution remains to be explored.     ACKNOWLEDGEMENTS  The support of the Australian Research Council in providing funding for a DORA Fellowship (2014-2016) is much appreciated.   7. REFERENCES  Ailor W.H. (Ed.) (1982) Atmospheric Corrosion, Wiley, New York.  Aziz P.M. (1956). Application of the statistical theory of extreme values to the analysis of maximum pit depth data for aluminum, Corrosion, 12(10) 495t-506t.      Coles, S. (2001). An introduction to the modelling of extreme values, Springer.  de la Fuente D., Otero-Huerta E. and M. Morcillo M. (2007) Studies of long-term weathering of aluminium in the atmosphere, Corros. Sci. 49: 3134-3148. Galambos, J. (1987). The asymptotic theory of extreme order statistics, 2nd Ed, Krieger, Malabar, FL. Lindley, D.V. (1972) Bayesian Statistics, A Review, Society of Industrial and Applied Mathematics. Melchers R.E. (2003). Modeling of marine immersion corrosion for mild and low alloy steels, Corrosion 59(4) 319-334.     Melchers, R.E. (2008). Extreme value statistics and long-term marine pitting corrosion of steel, Prob. Engineering Mech., 23: 482-488.   Melchers RE (2014). Bi-modal trend in the long-term corrosion of aluminium alloys, Corros. Sci. 82: 239-247.   Provan, J.W. and Rodriguez, E.S. (1989). Development of a Markov description of pitting corrosion, Corrosion, 45(3) 178-192. Stineman, R.W. (1980). A consistently well-behaved method of interpolation, Creative Comp., 6(7) 54-57.   Sun S, Q. Zheng, D. Li and J. Wen (2009). Long-term atmospheric corrosion behaviour of aluminium alloys 2024 and 7075 in urban, coastal and industrial environments, Corros. Sci., 51: 719-727. Turnbull, A. (1993). Review of modelling of pit propagation kinetics, Br. Corros. J., 28(4) 297-308.  Vargel, C. (2004). Corrosion of Aluminium, London: Elsevier.  


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