WNWWII“)WWll"lHllWllWWlHllWI“HI " Jim” fill 117' jjfillTllIffifl‘llfl'jfllln i§§_§__1 Lihfiud ‘r’ Michigan State University This is to certify that the thesis entitled CORROSION: STOCHATIC MODELLING OF THE INITIATION RATE AND DAMAGE ON CORROSION DAMAGE presented by Zhiyun Li has been accepted towards fulfillment of the requirements for M.S. dpgfimin Metallurgy Major professor Date 24%;, m I 957 0-7639 MS U i: an Affirmative Action/Equal Opportunity Institution \ V 7 I, , if MSU LIBRARIES -_ RETURNING MATERIALS: Piace in book drop to remove this checkout from your record. flfl§§_wiii be charged if book is returned after the date stamped beiow. “WT CORROSION: STOCHASTIC MODELLING OF THE INITIATION RATE AND EFFECT ON CORROSION DAMAGE by ZHIYUN LI A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Metallurgy, Mechanics, And Materials Science 1987 ABSTRACT CORROSION: STOCHASTIC MODELLING OF THE INITIATION RATE AND EFFECT ON CORROSION DAMAGE by ZHIYUN LI Corrosion is a two-step process: initiation and propagation. An initiation-time distribution model determined experimentally, was found to follow a Gamma function. Parameters for two corrosion processes were determined. Using the initiation distribution with a modified rate equation W a A ( t - r )b, "long-time" probability-damage-exposure time diagram were projected with 5%, 50%, 95% confidence, similar to a fatigue P-S-N diagram. ACKNOWLEDGEMENTS First and foremost, I would like to thank my advisor, Professor R. W. Summitt, for his patient guidance and beneficial discussions in making this thesis a success. Also, I would like to express my deepest gratitude to my parents for their boundless love, encouragement and full support. TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES I. II. III. IV. VI. INTRODUCTION THEORETICAL BACKGROUND AND LITERATURE REVIEW 1. kinetics of reactions 2. corrosion initiation 3. testing problems EXPERIMENTAL 1. experiment design 2. procesure RESULTS 1. data 2. statistical analysis DISCUSSION 1. initiation time model 2. projection to "long time" damage distribution 3. extension to a less severe environment CONCLUSION iii Page iv 13 16 21 23 30 32 40 44 52 63 Table 5. 6. 7. LIST OF TABLES Experimental and Calculated 10-year Weight Losses Larrabee-coburn Data from Kearny, N.J. Compared with that from South Bend, PA. Corrosion Initiation Time Data for Nails Corrosion Initiation Time Data for AA 2024-T3 Coupons Analysis of Steel Weight Losses Experimental and Calculated Weight Losses for Two Sites Calculated Parameters at Kure Beach & Bayonne Calculated Parameters of Four Steels at Kure Beach iv Page 18 3O 31 49 50 51 58 59 Figure 5. 6. 10. 11. 12. 13. 14. LIST OF FIGURES Relationship between short-term and long- term exposure periods, Kure Beach Relationship between short-term and long- term exposure periods, Block Island Relationship between short-term and long- term exposure periods, Bayonne Fatigue probability-stress-cycles, P-S-N, diagram Blue M electric vapor-temp humidity chamber AA 2024-T3 coupon surface before immertion AA 2024-T3 coupon surface after immersion (220 min) (a) corrosion initiated (b) corrosion coloured halo appeared (c) corrosion developed An edf of initiation corrosion time data for nails An edf of initiation corrosion time data for AA 2024-T3 coupon Histogram of initiation time data for nails Histogram of initiation time data for AA 2024-T3 coupon Histogram of initiation time data for AA 2024-T3 coupon Evalution of Pj(t) with time for Poisson process In illustration of probability-time curve Page 19 24 25 27 28 29 33 34 35 36 37 43 46 15. 16. 17. 18. 19. 20. Two-way distribution shown and P-D relationships Weight loss-time curve for at Kure Beach & bayonne Weight loss-time curve for at Kure Beach & Bayonne Weight loss-time curve for at Kure Beach & Bayonne Weight loss-time curve for steels at Kure Beach Weight loss-time curve for steels at Kure Beach schematic P-T No.32 steel No.54 steel No.11 steel No.63 & No.48 No.10 & No.13 48 55 56 57 61 62 I INTRODUCTION The results of corrosion experiments, which have been designed mainly to determine reaction kinetics, show wide variations, especially over short-time exposure. It is generally accepted that ,corrosion, like fatigue cracking and other cumulative damage processes, consists of two steps: (a) initiation, followed by (b) propagation. The initiation step, however, is generally ignored, hence its influence on experimental results is not known. Corrosion rates frequently are slow (mm/yr,etc), and if the kinetics are slow, it is possible that the initiation process also may be slow, thus the initiation times for three or five random samples may be widely separated and may contribute significantly to the observed dispersion of results. This study examined the initiation distribution experimentally for two simple systems in relatively severe environments, and the results were used to develop probability distribution models. These models were used to project corrosion damage for extended exposures, and then they were extrapolated to less severe environments, and the results were compared with published test data. Finally, published literature data were analyzed, from the standpoint of sample variation, to infer initiation rates. II THEORETICAL BACKGROUND AND LITERATURE REVIEW 1. KINETICS OF REACTIONS The need to establish a relationship between corrosion weight losses (or pitting depth) and exposure time is obvious, in order to predict the performance of a material from a field test, as well as to assess the effect of environmental factors on materials. A general corrosion damage equation has been proposed by several authors, w .. Atb. ‘ (1) where M is corrosion loss ( either penetration depth or weight loss ), t is exposure time, and A and b are empirical constants and are functions of environmental conditions Miller and coworkers [1] pointed out that the exponential b theoretically takes on the value of approximately 1/2, when corrosion is limited by the diffusion rate of the relevant species through a semi- permeable film of reaction products. This would be the case for most aluminum alloys. When the corrosion products are flocculant or soluble and offer no protection, linear corrosion kinetics are observed and b a 1. The above equation may be expanded in the following manner to account for the presence of pollutants in the environment, for example, so and Cl-, 2 w a Atb(P1, 92). (2) where P1 and P2 are damage coefficients governed by the presence of pollutants in the atmosphere. The experimental results show that the maximum corrosion rate for the aluminum alloys were obtained in solutions which contained both salt and acids. Godard [2] developed an equation for determining pitting depth, 0 - c (t)1/3. (3) Knotkova-Cermakova et al. [3] performed corrosion tests at nineteen corrosion stations in the North-Bohemian modern region. On the basis of the results of three years of the described program, they analyzed the dependence of corrosion on time with the use of the power function ( W = atb ), the expanded power function ( W = atb+Ct ), and the power function passing into a linear and hyperbolic function [ W = a + bt -(ac/t +c)]. The best correlation was attained for the hyperbolic function. All the analyses above seems to give a fair representation of corrosion phanomena in many sites, but there are some problems predicting long-time damage, and extrapolating the damage to other environments. Bragard and Bonnarens [4] tried to use the power function law to predict long-time behavior of a steel from short-time experimental data. Their experimental data resulted from a long-term program carried out at three sites in Belgium: Liege (industrial site), Ostende I (marine site, considerable salt mist and fog), and Ostende II (marine site, protected from the sea by a dune). For each steel, values of A and b were calculated from the weight losses after 1, 2, 3, and 4 years exposure. The two parameters then were used to predict the weight loss after 10 years. The predicted and actual 10-year. values for weight losses are given in Table 1. For Liege and Ostende II, the calculated values are generally underestimated. For Liege, the predicted and actual values are very close, while for Ostende II the agreement is poorer. For Ostende I, the calculated values are nearly always misleading. The law M = Atb is valid in numerous cases only as far as deviations of :10 percent or even :30 percent are allowed. For the sake of generalization and interpretation of this law, the relationship in logarithmic form was mm. n.NN o.oN o... Yon n.nn no . ..o. 06. N 2.. v... v.cN MN. Won 93 .N . mN. ad. MN m... N.oN Nan No.. o.wN «:2 8.. .6. 90. NN oN. NAN ndn .N.. voN «In No. v.2 . no. .N .N.. n.Nv N.Nn 3.5 «do v.xN m... v..N TN 8 no. v.NN N.mN X... v.3. 9.3 we . v.2 .6. o. 8.. QMN o.vN 9... n.Nn n.Nn no . Nd. ow. w. No. N.NN ..mN oN. .6... adv Nod 06 m... N. 2... c.vN o.nN 3.. N60 mac 8.. n.» fix o. 2.. TN o.NN N... ..m. N.oN cod . N... wd. n. 0... Von no... 3... ad... «on Na. .6. N6. v. . . . 0.»... . . . . . . odwN . . . co.— n.NN NAN n. No. de odN 3.. .dN can No. o6. ad. N. v... NdN m.oN c... o.mN N? No. 90. no. .. N... v.N. ndN 3.. v.vN ..on no... o.N N.N o. 8.. ch m..N 3... mdv memo no.0 o.” ..w o NO... ON.” mom on... ..n.. a..v 3.. n.N. ma. m . . . 9.x . . . . . . non. . . . a... N.NN n.NN N 8.. NSN ..oN on... N.Nn ch 3.. mi. 9n. 0 8.. 9mm 93” Nod Nan «Na 3.. No. 9N. n N. c.NN v.wN NNd New N.vn 3.. N6. no. v 3.. «:3 RN“ m. .. Ron «AK 3.. 93 EN m . . . ode . . . . . . N..a. . . . No. 0.; N.Nn N N... o.wN 0.2 mod 9.2.. w...“ Ne. w... ”.2 . 3.23:0 33.330 .358 032330 cu.a.:u.~u .358 33.330 3.2.530 .ScuE .uufi icooxm .todxm -tucxm 3.58.316 . 3.298.3me .255..qu .. occufiO . occufio owe; . > » - mcoumccom can cumwcum he .mmoq ucmeB umokloa woumaooamo com HmucoENMooxm H wanna expressed Log Wy - kLog Wx + n. (4) where Wy is the weight loss after y years, Wx is the weight loss after x years, and k and n coefficients depend upon the site and the time periods x and y A set of data were selected from the literature and calculated. All plots of Figures 1, 2, 3 show definitely that empirical relationships can be developed between the corrosion losses at different exposure times, but in a band, instead of a straight line. The width of the band for 3.5 or 5 years of data prediction is narrower than that for 1 year. Mikhailovsky [16], based on the mechanism of nonstationary atmospheric corrosion, proposed quite definite physical-chemical interpretations, expressing the metal corrosion rate: K=K 1-0). (5) o ( where K0 is the corrosion rate at the active surface, a is the fraction of the surface covered by the corrosion components having the protective properties. At same time 0 = b ( K - K'). (6) where b is a coefficient, and I K is the steady corrosion rate equal to the 60 /V./ 30 // f W U1 q < E Q mm o 9 D (p F 3,0“ 5 / b 3'5 V G V) 8 $,0 KURE BEACH , LB 54 srrus 2 b 5 , 2 3 4 S 6 7 8 9 IO IS WEIGHT LOSS (g/dm‘) . 1.5 YEARS . 0.15YEARS Figure 1. Relationship between short-term and long-term exposure periods, Kure Beach. by Bragard and Bonnarens 4o . X o Pg/dm“ ag/dm‘ BLOCK ISLAND“ / 30 %? / \ ‘g/dm" / / / v Ig/dm‘ x 5 2 3 a 5 6 7 8 9 10 IS WEIGHT LOSS (g/dm‘) e LIYEARS o SYEARS Figure 2. Relationship between short-term and long-term exposure periods, Block Island. by Bragard and Bonnarens 8 BAYONNE 32 S7EEL$ .\\ i i Ing‘ Sng‘ %: g WEIGHT LOSS (g/am‘) 17.! YEARS 0 A R o Vqti 4 s 5 WEIGHI L055 (9 7 8 910 IS /dm‘) .1 YEAR 0 5.: YEARS Figure 3. Relationship between short-term and long-term exposure periods, Bonnarens Bayonne. by Bragard and IO protective surface layer damage rate under the action of environment medium. Differentiation of Equations (5) 8 (6) with respect to time and solution of the differential equation system for K, yields K a Roe-‘6’ + K'. (7) w = (KO/a) (1 - e'fi' ) + x'. (8) where p - Rob, and W = corrosive loss in time. At the initiation time (7 = O), K 4 K (the maximum 0 corrosion rate ): when 7 4 a the corrosion rate is stabilized and K 4 K'. K, K0, and K. are functions of the temperature-moisture and aerochemical atmospheric complex. Metal corrosion in natural atmospheres should be considered as a combination of corrosive processes developed under adsorption and drop-liquid (phase) water film. Considering the effect of moisture and aerochemical air parameters on the rates of metal corrosion, the metal corrosion within the year ( KO ) can be represented by t2 :1 = + o K(pm 2, E '(ph) K 1 l where K(ph) and K(a) are the average metal corrosion rates under phase and adsorption water layers in g/mE/yr t n t n )3: E '(ph)’ Z: Z 7(a) are the total life time of 1 1 I] phase and of asorption water layers on metals in hr/year Assuming that the mean corrosion K(ph a) is proportional to I the active impurities, then, a (0) ch ch '5 “ Ko [ K(a.1:>h>” B 10-15 mcg/ma and [Cl'] > 1 mg/m8 showed that the 2 acceleration of steel corrosion 8:32; caused by sulfur dioxide also depends on temperature and may be represented by (ch) __ (ch) B(ph) - B(ph, t=0)+ 7 t (g/m? hr/l mg/m? 802) where ngfii t-O) is the corrosion acceleration at 0 °C. 1 is the temperature coefficient of acceleration [<3/III"‘(hr)/11|Irc, rc is the critical age, A is the nucleation frequency, and p is the probability of dying. Dallek & Foley [7] investigated the influence of anions on the initiation of pitting and kinetics of pit 16 growth on aluminum alloy type 7075. The rate equation is then (1/1) = k [A1]m [x']“. (12) where (1/7), the reciprocal of the induction time, is taken as the rate of pit initiation, k is the rate constant, [Al] the aluminum atom concentration, [x'] the halide ion concentration, and m and n are the respective orders of reaction. A deterministic treatment for the pitting of Fe-17Cr is presented by Macdonald [8], for the statistical nature of the breakdown treatment of films on metal surfaces in which it is assumed that breakdown sites on the surface are distributed normally in terms of the cation vacancy diffusivity. This analysis, which is based on the point defect model for the breakdown of passive films, predicts a near-normal distribution in the breakdown voltage, but a high asymmetric distribution in the induction time. The model is able to account for the experimental observed distribution in the breakdown voltage and induction time for the pitting of Fe-17Cr in 3.5% NaCl solution at 30 0C, as reported by T. Shibata (1983) and T. Shibata and T. Takeyama (1976-1981). 3. TESTING PROBLEMS 17 The American Society of Testing and Materials Committee G-1 have generated several series of field tests for a variety of materials at five locations since 1907. Tests have triplicate exposed panels for removal periods of 1, 2, 3, 5, 7, 10, 20 years. Evaluation of these materials includes weight losses, pitting depth, and changes in mechanical properties. Results have been reported widely, [4], [10], [11], [12], [13], [14] as averages. These data will be discussed later. The purposes of these tests, among other, was to estimate corrosion resistance for different metallic systems and their protection (including commercial products) under different working and storage conditions. Even though only three or five samples of each material were removed at various time intervals, the total number of panels was still large. The dispersion of result also was large, Table 2, often several times the mean value. Commonly, one specimen had corroded severely, while an adjacent identical specimen was uncorroded. Considering this dispersion, the life prediction only can be made within a probability limit as in the case of fatigue. Figure 4 shows the schematic diagram of fatigue P-S-N curve. If a corrosion initiation time distribution can be determined and combined with suitable kinetics and 18 Table 2 Larrabee-coburn Data from Kearny, N.J., Compared with that from South Bend, PA. Kearny, South Bend, Data Factor New Jersey Pennsylvania No. of observations 270 270 Mean corrosion rate(mils) 4.500 4.164 Standard deviation 3.732 1.989 Highest observation 41.7 16.5 Lowest observation - 1.6 1.3 Range 40.1 15.2 from "The Statistic Analysis of Atmospheric Corrosion Data" by R. A. Legault & J. G. Dalal 19 7S Rotating bending SCMLBS ieupered at 600°C (01) 1 I 70 \ 3 . m. g 65 a x *‘6693 1 so ,__., —Q.QQ>'° 1 Probdaility at failure, PPM *‘ _ 1 11 l _1 1 1 1 l 1 1 1 1 1 1 10’ 1o“ 10’ Number oi cycles,N 750 7C!) 650 550 Figure 4. Fatigue probability-stress-cycles, P-S-N, diagram. S (Nlmm?) 20 statistical treatment, a predicted damage level at certain exposure time or failure life can be computed. I I I EXPERIMENTAL 1 . EXPERIMENT DESIGN Suppose that the observed variance in corrosion test data results largely from dispersion of a stochastic initiation process and random sampling followed by uniform corrosion kinetics, which would be the same for specimens of the same materials in a uniform corrosive environment. Then it would follow that at any time after initial exposure, these specimens would exhibit different amounts of corrosion damage in some proportion to their different time of initiation. In a real environment, of course, the corrosivity might change and such changes would accentuate or diminish differences caused by the initiation rate. For example, environmental severity may worsen between the time of initiation of sample i and sample j, thus the difference between the two samples at a time following the initiation of i would be larger than otherwise expected. Likewise, if the conditions became milder, the difference would be narrowed. There appear to be no published efforts to determine such initiation distributions experimentally, although some stochastic models have been suggested. The difficulties are that the corrosion initiation is hard to detect and evaluate, and a large number of specimens are 21 22 needed. We have designed deliberately simple-minded experiments with a simple criterion to detect initiation. Since true initiation occurs on a microscopic scale, it is not detectable, consequently, first visible observation of corrosion was used as the criterion, assuming that the short time interval between initiation and observation does not distort the results significantly. This is not unlike the use of 0.02% offset in tensile testing to determine the yield stress of a metal. Immersion of zinc coupons into aqueous 1.0 N hydro- chloric acid would result in near-instantaneous initiation in all samples, hence the initiation times could not be measured. At the other extreme, atmospheric exposure of aluminum in desert conditions would exhibit corrosion initiations spread over several years. All specimens should be placed simultaneously in the same environment with the corrosivity held constant. Using a statistically valid number of specimens, these tests' should provide sufficient data for analysis. The purpose of the experiments is to construct a model for corrosion initiation, two systems were selected: plain- carbon steel in a humid atmosphere and an aluminum alloy immersed in aqueous sodium chloride. These environments were selected to be severe so that initiation would occur over a short time period. 23 2 . PROCEDURE Experiment I was performed in a Blue M Electric Vapor- Temp Cnamber Model VP-100, Figure 5, with controlled humidity and temperature. The 25 plain-carbon steel carpenters nails were cleaned by conventional procedures, i.e. with 600# grit abrasive paper removal of rust, and rinsing with acetone and petroleum ether to remove oils. Following cleaning, no rust or stain was visible at 100x magnification. The samples were hung from a rack in the chamber with nynon filament and at relative humidity: 90:5% temperature: 85 i50F. The above conditions were calibrated before testing. The initiation time of individual smples was recorded when the first visible orange stain appeared on the surface. Experiment II used immersion of AA 2024-T3 in 0.01N aqueous NaCl without stirring. The specimens were cut into 3/4" x 1 1/2" x 1/16" coupons, with a hole near the top for hanging on glass hooks. The samples were wet polished with 400# grit and 600# grit abrasive paper, rinsed, and degreased with acetone. No evidence of corrosion was visible at 400x magnification, see Figure 6. The second 24 .3": ..\ “Max... m... REEANW®w§ Zfl/(NW\ ///¢/1W\\ Figure 5. Blue M Electric Vapor-Temp Humidity Chamber 25 400x Figure 6. AA 2024-T3 coupon surface before immersion 26 phase particles or impurities appear grey or white surrounded by phase boundaries. Following cleaning, the samples were allowed to stand for at least 24 hours in a desiccator. The 100 specimens were mounted with glass hooks, immersed in 0.01 N NaCl (PH = 7.0) at room temperature ( 70° F ) and removed when the first stain appeared. The beginning and end times were recorded for each sample. The samples were cleaned with distilled water to wash the precipitates off, and then dried. After that, the samples were examined carefully under the microscope, to be sure that the determination of initiation time was made at the same damage level. The second phase particles appeared dark and were surrounded by a coloured halo. Figure 7 shows the different corrosion results of these samples immersed in 0.01 N NaCl for same time 220 minutes: (a) corrosion initiated (b) corrosion coloured halo appeared (c) corrosion developed I The criterion was set at Figure 7 (b). 27 400x (a) Figure 7. AA 2024-T3 coupon surface after 220 minutes immersion. (a) corrosion initiated 28 400x (b) Figure 7. AA 2024-T3 coupon surface after 220 minutes immersion. (b) corrosion coloured halo appeared 29 400x (C) Figure 7. AA 2024-T3 coupon surface after 220 minutes immersion. (c) corrosion developed IV. RESULTS DATA Plain carbon steel nails: Table 3 Corrosion Initiation Time For Nails Initiation Time (hours) 2.2 2.9 3.3 3.7 3.9 4.3 4.5 4.8 5.1 5.4 5.8 6.2 6.6 7.1 7.6 8.1 8.7 9.5 10.1 10.8 1.8 12.8 13.9 15.7 20.0 1 25 sample mean a ——-2 X1 = 7.7918 n 1 1 25 2 sample variance =-——-X (Xi-i) = 18.7556 n 1 standard deviation = 4.33 30 31 AA 2024-T3 coupon salt immertion Table 4 Corrosion Initiation Time For AA 2024-T3 coupons Initiation Time (minutes) 218 222 223 225 226 227 228 229 230 231 231 232 233 233 234 234 235 236 236 237 237 237 238 238 239 239 239 240 241 242 242 243 243 243 243 244 244 ‘244 245 245 246 246 246 246 247 247 247 248 249 249 250 250 250 251 251 252 253 253 253 254 254 256 257 258 258 259 259 ' 260 261 262 263 263 264 264 266 267 267 269 269 270 271 272 274 274 277 277 278 278 279 280 281 282 283 285 287 289 291 292 293 307 1 100 sample mean =:-— 2 xi = 253.13 n 1 1 100 sample variance ==-— 2 (xi-X)2= 356.17 n 1 1 100 third moment =-—- (Xi-X)3= 3646.55 n 1 standard deviation = 18.87 32 2. STATISTICAL ANALYSIS An empirical distribution function (edf) of corrosion initiation time data for nails is shown in Figure 8. The abscissa is time measured in hours, and the ordinate is fraction initiated. The estimated mean and variance are 7.79 and 18.76, respectively. Curve 2, a cumulative distribution function of Gamma distribution based on the above mean and variance accurately describing this edf, incorporates these features in a quantitative manner. The signal-to-noise ratio, m /a is 1.80. An empirical distribution function (edf) of corrosion initiation time data for AA 2024-T3 coupons in salt immersion is shown in Figure 9. The abscissa is time measured in minutes, and the ordinate is fraction initiated. The estimated mean and variance are 253.1 and 356.1, respectively. The signal-to-noise ratio, m / a is 13.41. Curve 2, generated cdf of Gamma distribution based on the above mean and variance, incorporates the edf very well. Figure 10 shows the histogram of corrosion initiation time for nails. The abscissa is time t measured in hours and the ordinate is frequency, indicates during the time FRACTION OF SAMPLES INITIATED 1.00-1 0.90-1 0.80- 0.70 4 0.60 - 0.50 -- 0.40-1 0.30 - 0.20 - 0.10--I .1 0.00 33 -- EDF -— GAMMA DISTRIBUTION TTrI frITITTIFITTTI1 I 0 5 10 15 20 TIME (HOURS) FIGURE 8. AN EDF OF INITIATION CORROSION TIME DATA FOR NAILS FRACTION OF SAMPLES INITIATED 34 —- EDF -- mum mmaunou I T r T T F 1 ‘IO 220 111* T1 280 290 300 310 TI T r T 240 250 260 270 TIME (MIN) T 230 FIGURE 9. AN EDF OF INITIATION CORROSION TIME DATA FOR AA 2024—T3 COUPON FREQUENCY 35 55-0-1 - GAMMA DISTRIBUTION 1::1 HISTOGRAM I I ‘ I I I r 1 \ 2 04 T1 \ m I \ I \ ’ \ I \ '7 I ‘1 I \ I \ I \ I \ —I __ \ 1.0 I N} r— V II \ I \\ I \ 4 I \ ‘1. I \ I \\ I \ I \b‘N‘ 0'0 f 1 T r T r T T l 0 1o 15 20 25 TIME (HOURS) FIGURE 10. HISTOGRAM OF INITIATION TIME DATA FOR NAILS FREQUENCY 36 12.0— I \ — GAMMA DISTRIBUTION / \ :3 HISTOGRAM 'I / \ I \ I \ 10.04 D]— F'_] \ I \ -I I \ I \ so + \\ I E‘ I \ ‘ \ II \ 6.0-d I—f I-- #T; -I I ‘ I \ ’1 \ I \ . I \ I \ I \ / \ ‘K -I’ \ \N “x- 0.0 g I ZII5 225 235 245 255 265 275 285 295 305 315 TIME (MIN) FIGURE 11. HISTOGRAM 0F INITIATION TIME DATA FOR AA 2024-T3 COUPON FREQUENCY 37 ] —- GAMMA DISTRIBUTION / \ 1::1 HISTOGRAM ~e T T I T r I T I ' I 1 215 225 235 245 255 265 275 285 295 505 515 TIME (MIN) FIGURE 12. HISTOGRAM OF INITIATION TIME DATA FOR AA 2024—T3 COUPON 38 interval, 1 hour, how many nails exhibited corrosion initiation. The Gamma distribution function (pdf) added on this histogram, fitted the histogram. Figures 11 and 12 show the histograms of initiation time for AA 2024-T3 coupons salt immersion, the frequencies in different time interval 5 minutes and 10 minutes. A pdf of Gamma distribution function attached on the histogram, which fitted the histogram well. A test of the null hypothesis shows that the initiation time distribution follows a Gamma distribution function for nails and AA 2024-T3 coupons, respectively, hence hypothesis test cannot be rejected. Hence, the corrosion initiation time distribution model is: 1 f(a; 0,R,a) = -—-—1{ (O'GIR-leXPI' F(R)0 0'0 J. (13) where f(r: 0,r,a), the probability density function, indicates the corresponding probability of initiation time 1, r is the initiation time of corrosion different from the expOsure time, 0, R, a are parameters, depending on the individual materials, environments and geometry. In the case of nails: 39 R = 3.237, 0 = 2.4071, a S 0 In the case of AA 2024-T3 coupon salt immersion: R = 13.5918, 0 - 5.1191, a 8 183.55. V DISCUSSION 1. INITIATION TIME MODEL It is presented by some authors, Summitt [5], et a1. theoretically that the corrosion initiation process, like most cumulative damage processes, fulfills the stochastic process-- Poisson process. Let us start at this point, i.e. the specimen can only be corroded or not corroded at any point in time, which might be a specific discrete state. Let N(t) denote the number of samples corroded in (Opt) pk - P { N(t) = k I k - O, 1, 2, ... thus Pk (t) is the probability that k specimens have corroded in the time interval (0,t]. Then the random variable N(t+At) - N(t) is the number of specimens corroded in the interval at. Suppose the N(t), N(t+At)-N(t) have same distribution and are independent of what happens at or before t. P { N (at) a 0 } = PC (at) = 1 - Aat +a(At) P{N(t+At-N(t) =0}=P(N(t) =0} = 1 - Aat + 0(at) Po (t+At) p { N(t+at) = O I p I N(t) = o, N(t+At) - N(t) = O I 40 41 = P I N(t) = O } P {N(t-I-Tt) - N(t) = 0} - Po(t) ( 1 - Aat + 0(At) I where it is assumed that each specimen corroded independently or Po'(t) - - A P0(t). Solve the differential equation, Po (t) =- e'“: t z 0. Continuing in this manner, we find that (“’1‘ -1t k! We see that at any time t, N(t) the number of specimens corroded has the Poisson distribution with parameter At. Figure 13 shows the Pj (t) as a function of j and t. We obtain from Equation (14) P{N(t)= 0)= e"At Let the random variable T denote the time from 0 to the first corroded (initiation time). Then P(T>t}=P(N(t)=O}=-e"‘t. FT(t)-P{th}=1-P{T>t}=l-e "At. Let rk denote the time to the kth corrOsion. It follows that rk = T1 + T2 + T3 + ... + T Tk denote the time from ( k-l ) k k= 1, 2, (15) th specimens corroded to the kth rk and N(t) are related through their distributions: 42 N (t) a k means k or less corrosions have taken place in ( 0, t ] So FN (k;t) = 1 - Ff ( t: k+1 ) (16) h -At (At)j Since FN ( k:t) = Z . 0 j! We obtain the pdf of rk+1 is A(At)k e "t f (t; k+1) 8 t>0 ' k I (17) = 0 ‘ elsewhere 1 Let A=— R=k+1 t=r-a 0 1 r-a ff( 7 - a; R ) = -——--§ (MAR lexp[- 1 (R-l)!0 1 P1: = -——§— (r-GIR 1 epr- I (18) P(R)0 0 Equation (14) is as the same as the pdf (probability density function ) of Gamma distribution drawn from the experimental data (9). So far, from the experimental hypothesis tests of the corrosion initiation time or from the analysis of the stOchastic process, we get the same function. All confirm 43 /. /,// / 7' r T I I I 0 P1 P2 P3 P4 P5 Figure 13. Evalution of Pj (t) with time for Poisson process. 44 that the corrosion initiation time distribution model is a Gamma distribution, which has physical meaning, and fitted the experimental data very well. Further, by the central limit theorem, as R a w, the r are asymptotically normally distributed. 2. PROJECTION TO "LONG TIME" DAMAGE DISTRIBUTION Like fatigue, one can build a P-D-T diagram to predict corrosion damage. The Probability-Damage-Exposure time diagram, a family of D-T curves each corresponding to a particular value of probability, can adequately express the two concepts of scatter, i.e., damage scatter-distribution, (P-D) and life (exposure time) scatter-distribution (P-T). An ordinary way of establishing such a P-D-T diagram for a given material in a given environment is to determine experimentally the P-T relations at different corrosion damage values and draw a P-D curve using t values expected for the P prescibed. Obviously, a hundred tests are needed to get t values at P - 1 percent for only a single value of D. It is not ratical to run hundreds of corrosion field tests. The aim of this paper is to project corrosion. So we attempt to give a more practical method of producing P-D-T diagrams with an accessible number of specimens. From the 45 previous discussion, it is obvious that the exposure time should be divided by two periods: one for corrosion initiation, one for corrosion development, so that the kinetical equation should be modified as follows: W=A(t-r)b (19) where t is the total exposure time 7 is the initiation time, depending on probability prescribed, t - r is the actual corrosion reaction time, and A, b parameters, depend on the materials and environment. One can use this formula- and corrosion initiation distribution model ‘ to predict long ' time damage distribution. Theoretically, for each damage level, there is an exposure time distribution function with its own parameters, simular to the initiation time distribution- Gamma function respectively. The individual D-T curve might be made by calculating the exposure time corresponding to a given probability, shown in Figure 15. Figure 14 and equation (20) illustrate the method to decide the exposure time, i.e. Probability that corroded within time interval (o,t] = the area under the curve until tr. The more convenient way is to calculate the initiation time corresponding to the given probability and shift the 46 f(t) tupper t Exposure time PROB [ T < tupper 1 = AREA tu er = PP f(t)dt O t 1 t-a = J upper R ( t _ a )R-lexp[- ]dt 0 F(R) o = ‘Y (20) Figure 14. In illustration of probability-time curve 47 D-T curve along the time axis for calculated initiation time, as in Figure 15. Here, assume that the sample which initiates early corresponds to larger damage. In this way, one can get the damage band for a specific material and predict the damage which exist at any arbitrary exposure time with some confidence. Meanwhile, one can recognize that the distribution is a two-way distribution shown schematically in Figure 15, i.e. in exposure time and in damage. The shape of the exposure time distribution is strongly dependent on the damage level, but the damage distribution appears to show similarities at different exposure time. Since the Gamma distributions are skewed negatively as shown, the scatter will tend to high value. This phenominon is expected to explain the large dispersion of the data. As a test of Equation (15), several sets of data for low alloy steels were selected from the literature [10]. Their chemical compositions are shown in Table 5, weight losses in Table 6, and the compared experimental and calculated weight losses in Table 7. From Table 7, the calculated values are generally underestimated, although the predicted and actual values are very close. This assumption shows that the rate law W = A(t - r)b is valid as far as deviation less than :10%. In addition, the above actual values are the average of two or four experimental data points, so the predicted values 48 mdfinmcowumamu mum com Hum owumamcom asonm moofiusnfiuumfip amaloze .mH gunman u mafia whamoexm n . 8 distribution ANvm .mmmamu mo sufiaenmnoum —— cofiusnfiuumfiu was» musmodxm ssoI nqflren 49 Table 5. Analysis of Steel Analysis per cent Steel C Mn 81 S P Ni Cu Cr Mo No.48 0.17 0.89 0.05 0.03 0.075 0.16 0.47 -- 0.28 No.63 0.02 0.023 0.002 0.03 0.005 0.05 0.053 -- -- No.13 0.072 0.27 0.83 0.02 0.14 0.03 0.46 1.19 -- No.32 0.11 0.69 0.14 0.02 0.011 0.20 0.038 -- -- No.11 0.04 0.49 0.003 0.04 0.074 0.01 0.05 0.10 0.003 Table 6. 50 Weight Loss, 9 Kure Beach N. C. Bayonne, N. J. Steel years of exposure 0.5 1.5 3.5 7.5 15.5 1.0 3.0 5.1 7.1 9.1 No.32 2.1 3.9 7.4 13.1 21.5 4.7 7.2 8.8 10.5 12.2 No.54 2.5 4.8 8.9 15.0 26.5 5.1 8.3 10.3 12.8 13.5 No.11 2.1 4.2 8.1 13.5 20.9 4.9 7.9 9.7 12.0 12.6 No.48 108 305 409 709 1108 300 -- 501 -- 6.5 No.63 2.3 4.8 10.1 19.0 43.0 5.8 10.1 11.3 -- 16.1 No.13 1.6 3.5 4.9 7.9 11.8 1.8 -- 2.6 3.2 3.5 No.10 2.6 5.0 10.3 19.7 -- 5.6 9.3 11.3 14.5 17.7 51 Table 7. Experimental and Calculated Weight Losses for Two Sites Kure Beach (15.5 yr) Bayonne (9.1 yr) Steel Experiment Experiment Experi- Calcu- Experi- Calcu- mental lated mental lated Calculated Calculated No.32 21.5 20.86 1.03 12.2 11.44 1.066 No.54 26.5 25.39 1.043 13.5 13.13 1.028 No.11 20.9 21.06 0.992 12.6 12.374 1.018 52 are for 50 % confidence ( not actually for the average values are suspensive as population mean in so small sample space ). Considering the number of the samples and experimental error, this result is good. Most literature data are reported as means and standard deviations, hence no original data were available for doing a thorough statistical treatment. 3. EXTENSION TO A LESS SEVERE ENVIRONMENT Corrosion process is a complex process, which can be affected by many factors, e.g., water 802, chloride etc., and synergistic effects are possible. The overall process can be separated into several subprocesses, each with its own transition probability function. When the environment and materials are set, the probability space is set by a stochastic matrix at certain time, FP11(t,s), P12(-t,s) , . . . , P1n(t,s)' P :2: : S 23 ff co co .0 l". ._J LPn1(t,s), Pn2(t,s),..., P Pij represents the one step transition probability from state i to state j, s, are various factors which influence the probability, e.g., temperature 53 In a Markoff process, each Pij depends only on the state i-1, and is independent of other states. So from initiation state to final state, the probability P{ij} at the kth step is P { ij } =- 2...2 ajopjoj1°°°ij1jk , Using matrix notation pk - p0 Pk . (17) Different combinations of environment and material have a different probability matrix, for example, if the environment includes Cl“, S04“, and moisture, the matrix will be more complicated, and the corrosivity more severe. For such a Markoff process, if time is a continuous variable, this stochastic process is still a Poisson process with the time distribution corresponding to Gamma, but the parameters are changed, depending on the materials and environments. Analyses of the data from ASTM tests, Tables 6,7,8, were made for comparisons on the basis of environment and of materials. At Kure Beach, N.C. a marine site 240 m from the ocean, the main corrosive factor is Cl-. Bayonne, N.J. is an industrial location not far from a center of oil refining activity. Comparing the parameters of two groups, Table 8, a noticeable feature is that all the initiation times of three steels at Bayonne decreased, the exponent b decreased approximately 3/5 of those in Kure Beach, but A for steels 54 at Bayonne increased as 1 1/3 times those in Kure Beach. This feature is fitted to the steel corrosion mechanism that, during the very early stages of exposure, the weight loss depends on the environment, and later depends on the water which can reach the surface through the corrosion products. In Bayonne, the main pollutant is sulfur dioxide, so the steel corroded very fast; at Kure Beach, the main environmental factor is chloride., its high solubility would be expected to make it more difficult to plug pores in the rust coating. Corrosion rates have been decreasing more slowly at Kure Beach than Bayonne, as shown in Figures 16, 17, 18. From the above examples, one suggestion might be made that all parameters are functions of corrosivity factors or proper coefficients. In turn, however, one can predict the damage to a material in an untested environment. Of course, it is obvious that the kinetic equation cannot be extrapolated to an other environment using only P P 1' 2 damage coefficients from the literature [1]. Comparing the calculated parameters of two groups of steels at Kure Beach, it is found that changing some alloy elements will change the values of parameters considerably. No.48 steel has more manganese (0.87%) and copper (0.42%) than No.63, and its A value decreases 0.8 and the exponential b 0.323. No.13 steel has more silicon (0.83%), copper (0.44%) and chromium (1.128%) than No.10, and its A g drn'2 WEIGHT L088 55 + H BAYONNE 0 0 H KURE BEACH O I f I T T r I 1 T I T l I I I I 0.0 4.0 8.0 12.0 16.0 TIME (YEARS) FIGURE 16. WEIGHT LOSS—TIME CURVE FOR No.32 STEEL AT KURE BEACH 8c BAYONNE g drn'2 WEIGHT L088 56 H BAYONNE 0 0 H KURE BEACH 0 T 7* T T r T r l' I r r I 1 fit j 0.0 4.0 8.0 12.0 16.0 mm (YEARS) FIGURE 17. WEIGHT LOSS—TIME CURVE FOR No.54 STEEL AT KURE BEACH & BAYONNE 9 dm'2 WEIGHT LOSS 57 1 _ H BAYONNE 0 0- e-e KURE BEACH 9 I I I r r I I r F I I l I I T —l 0.0 4.0 8.0 1 2.0 1 6.0 TIME (YEARS) FIGURE 18. WEIGHT LOSS—TIME CURVE FOR No.11 STEEL AT KURE BEACH 8c BAYONNE 58 Table 8. Calculated Parameters at Kure Beach & Bayonne Steel parameters Kure Beach Bayonne A 3.44 4.85 No.32 0.666 0.39 r 0.264 0.072 A 4.2 5.61 No.54 b 0.66 0.39 7 0.2275 0.217 A 4.145 5.53 No.11 0.6 0.37 r 0.478 0.283 59 Table 9. Calculated Parameters of Four Steels at Kure Beach Steel A b r No.48 3.055 0.477 0.17 No.63 3.866 0.8 0.182 No.10 3.564 0.849 0.01 No.13 2.196 0.389 0.098 60 value decreases 1.37, b value decreases 0.46: the decreasing in b is significant, only as half as those of No.10. So corrodibility of the material also should be considered. In extending to a less severe environment, one formula might be suggested: W a A (s,m) [ t - r(8,m) ]b(s,m)' (18) where s is a corrosivity factor depending on environments, m is a material factor depending on corrodibility of the materials, and r(m,s) mainly depends on environment through the initiation time function parameters a, R, 0. Since no actual experimental data are available in the literature, it is not possible to check further the agreement between our formula and experimental results. Further long term exposure tests 'will be performed to produce data. It is possible apparantly to predict damage in a less severe environment using the above model. It is too early to draw general conclusions from these tests of limited nature. Further investigation will produce original data to find the relationship. g drn‘2 WEIGHT LOSS 6] H No.48 STEEL H No.63 STEEL T I T 12.0 16.0 I r I j ' I 0.0 4.0 8.0 I T r TIME (YEARS) FIGURE 19. WEIGHT Loss—TIME CURVE FOR No.53 a. No.43 STEELS AT KURE BEACH 9 dm'2 WEIGHT LOSS 62 TIME (YEARS) FIGURE 20. WEIGHT LOSS—TIME CURVE FOR No.10 8c No.13 STEELS AT KURE BEACH VI CONCLUSION: 1. When apparently identical specimens are exposed simutaneously to a corrosive environment, corrosion does not begin simutaneously. Corrosion initiations are distributed at random, and follow a Gamma function as shown by our experimental results. The parameters are fitted to each specific combination of alloy and environment. 2. Two sets of parameters were determined from the experimental results: nails in humid air: 0 = 2.4071 R = 3.237 a = 0. AA 2024-T3 coupons in aqueous salt immersion: 0 = 5.1191 R = 13.5918 a = 183.55. 3. 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