Research ArticleA Probabilistic Physics of Failure Approach forStructure Corrosion Reliability Analysis
Chaoyang Xie12 and Hong-Zhong Huang1
1School of Mechatronics Engineering University of Electronic Science and Technology of China Chengdu 611731 China2Institute of Systems Engineering China Academy of Engineering Physics Mianyang 621000 China
Correspondence should be addressed to Chaoyang Xie xiezycaepcn
Received 14 January 2016 Revised 7 April 2016 Accepted 5 June 2016
Academic Editor Ramana M Pidaparti
Copyright copy 2016 C Xie and H-Z Huang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Corrosion is recognized as one of the most important degradation mechanisms that affect the long-term reliability and integrity ofmetallic structures Studying the structural reliability with pitting corrosion damage is useful for risk control and safety operationfor the corroded structure This paper proposed a structure corrosion reliability analysis approach based on the physics-basedfailure model of pitting corrosion where the states of pitting growth pit-to-crack and cracking propagation are included in failuremodel Then different probabilistic analysis methods such as Monte-Carlo Simulation (MCS) First-Order Reliability Method(FORM) Second-Order Reliability Method (SORM) and response surface method are employed to calculate the reliability Atlast an example is presented to demonstrate the capability of the proposed structural reliability model and calculating methods forstructural corrosion failure analysis
1 Introduction
Corrosion is recognized as one of the most importantdegradation mechanisms that affect the long-term reliabil-ity and integrity of metallic structures [1] Corrosion cancause several kinds of defect on structure with the stressload and different corrosion environment such as uniformcorrosion corrosion fatigue pitting corrosion and hydrogenembrittlement As the increase of service time for metallicstructure the corrosion damage will grow and the pit willchange to a crack especially with static or circle loads So thereliability and lifetime predictions of structure with pittingcorrosion damage are significant for structure safety and riskmitigation
The development of effective localized corrosion damagemodel is essential for reliability assessment A simple empiri-cal model was developed to describe the relationship betweencorrosion time and pitting depth based experimental dataIn recent years some stochastic method has been made inmodeling pitting corrosion through Markov chains Caleyoet al used a nonhomogenous Markov process to model pitdepth growth [2] Valor et al proposed a new stochastic
model in which pitting initiation is modeled as a Weibullprocess [3] Another type of corrosion damage model hasbeenmade based on electrochemical andmechanical processGoswami and Hoeppner identified a seven-stage conceptualmodel for corrosion fatigue The electrochemical effects inpit formation and the role of pitting in fatigue and corrosionfatigue crack nucleation behavior were considered in thatmodel Shi and Mahadevan represented a computationalimplementation approach based on the seven-stage concep-tual model for corrosion fatigue life prediction [4] HarlowandWei proposed a three-stage probabilisticmodel includingcrack initiation surface crack to grow into a through crackand crack fracture But all the models focus on structure withcircle loads and the stress effect was not included in the pitgrowth phase Stress corrosion crack is another importantfailure model for structure with static loadsWu [5] proposeda probabilistic-mechanistic approach focused on modelingSCC propagation of Alloy 600 SG tubes with uncertaintyBut the pit growth process was not contained in this modelA transition model for pitting to corrosion fatigue cracknucleationwas first proposed byKondo and further discussedby Harlow and Wei [6]
Hindawi Publishing CorporationInternational Journal of CorrosionVolume 2016 Article ID 1343587 7 pageshttpdxdoiorg10115520161343587
2 International Journal of Corrosion
This paper proposed a physics-based failure model forstructure with pitting corrosion damage The mechanicalstress effects for pitting growth are coupled in the integratedcorrosion process The three stages of pitting corrosion dam-age pitting growth pit-to-crack and cracking propagationare considered in the developed physics-based failure modelThen the time-dependent limit state function of corrosionstructure is defined by fracture theory The Monte-CarloSimulation FORM and SORM are employed to calculate thereliability
2 Reviews of Structure ReliabilityAnalysis Method
21 MCS Method Consider a performance model 119866(X)under the existence of uncertainties where the system fails if119866(X) lt 0 If the joint probability density function of randomvariables X is 119891X the statistical description of a probabilisticperformance that fails is then completely characterized by thecumulative density function (CDF) as
119875119891 = Pr (119866 lt 0) = int119866lt0
119891X (X) 119889X (1)
In practice the integration boundary 119866(X) = 0 and thehigh dimensionality make it difficult or even impossible to
obtain an analytical solution to the probability integration in(11)
The MCS method is showed as follows [7 8]
119875119891 =1
119873
119899
sum
119894=1
119868 (119866)
119868 (119866) =
1 119866 lt 0
0 119866 gt 0
(2)
where 119873 is the total number of simulations 119868(119866) is an indi-cator function However the MCS method needs expensivecomputational cost
22 FORM The approximation methods are therefore usedfor reliability analysis in order to reduce the computationalcost FORM [9 10] solves the probability integral by simpli-fying the performance function 119866(X) using the first-orderTaylor series expansion at the Most Probable Point (MPP)The flowchart of the FORM for reliability analysis is shownin Figure 1 FORM involves three steps to approximate theintegral [11 12]
Step 1 Transform original random variables X in X-spaceto standard normal random variables U in U-space and theperformance function is expressed as 119866(U)
Step 3 After calculating the reliability index 120573 is obtainedby an optimization problem The probability in (1) is thencomputed analytically by the following equation
Figure 1 The flowchart of the FORM for reliability analysis
where 120581119894 denotes the 119894th main curvature of the performancefunction 119892(U) The approximation of the performance func-tion in FORM and SORM is shown in Figure 2
3 Physics of Failure Modeling forPitting Corrosion
Thedegradation (damage) process for the crack developmentand propagation has been studied and modeled in manydifferent ways [16] In the proposed method the pittingcorrosion crack consists of four stages as shown in Figure 3
The first stage in the corrosion damage process is pittingnucleation It is related to the electrochemical process duringcorrosion and the pit initial size and nucleation time dependon factors such as materials corrosion environment loadsand electrolytesThis process is very complex and the physicsmodel of pit nucleation is not well understood yet In thispaper we therefore assume the initial size of pit as a randomvariable This distribution can be obtained by experimentaldata
A simplified model for pit growth proposed by HarlowandWei is used [6 17]Themodel assumes a pit of hemispher-ical shape growing at a constant volumetric rate in accordance
U2
U1
o
g gt 0g lt 0
g = 0
120573
FORM
SORM
MPP ulowast
Figure 2 Comparison of FORM and SORM
Pit nucleation Pit growth Pit-to-crackcaused by stress
Crack propagation
Figure 3 Pitting corrosion growth process with stress
with Faradayrsquos law from an initial radius size The rate of pitgrowth is given as
119889119881
119889119905=1198721198940
119911119865120588exp (minusΔ119867
119877119879) (11)
where 119881 is the hemispherical pit volume 119881 = 212058711988633 119886
is the pit size 119872 is the molecular weight of the material1198940 is the pitting current coefficient 119911 is the valence 119865 isFaradayrsquos constant 120588 is the material density Δ119867 is theactivation energy 119877 is the gas constant and 119879 is the absolutetemperature
Based on the electrochemical theory the electrical cur-rent on an electrode depends on the electrode potential Ageneral representation of the polarization of an electrode isdescribed in the Butler-Volmer equation
119869 = 1198950 exp(119911119865 (120593 minus 120593eq)
119877119879) (12)
where 119869 is electrode current density Am2 1198950 is exchangecurrent density Am2 120593 is electrode potential V and 120593eqis equilibrium potential V When the stress is applied onthe metal materials with elastic deformation the equilibriumpotential will be varied according to Gutmanrsquos theory [18 19]
Δ120593eq = minusΔ119875119881119898
119911119865 (13)
4 International Journal of Corrosion
where Δ119875 is the spherical part of macroscopic stress tensorexcess pressure (Pa) and119881119898 is the molar volume of the metalThe current density with the stress effects will be changed as
119869 = 1198950 exp(119911119865 (120593 minus 120593eq minus Δ120593eq)
119877119879)
= 1198950 exp(119911119865 (120593 minus 120593eq)
119877119879) exp(
Δ119875119881119898
119877119879)
(14)
The coefficient-exp(Δ119875119881119898119877119879) is used for depicting thevariance of corrosion current with stress effects comparedwithout stress situation
In this paper the effects of stress applied are consideredbased on (14) then the pit growing depth at the time 119905 can bechanged to take into account the coupled effects of stress loadand corrosion environment as
119886 (119905)
= (31198721198940
2120587119911119865120588exp (minusΔ119867
119877119879) exp(
119881119898Δ119875
119877119879) lowast 119905 + 119886
30)
13
(15)
where 1198680 is the pitting current 119881119898 is the molar volume of thematerial and Δ119875 is the bulk component of stress tensor Asthe pit depth grows the stress intensity factor at the pit tip willincrease correspondingly and when it reaches a value beyondthe threshold value of stress corrosion crack (SCC) the pitwill transform into a crack and the SCC will thus occur Thecriteria of a pit transforming into a crack can be obtainedbased on the thresholds of SCC and the fracture toughnessas [6 7]
119870pit ge 119870ISCC (16)
where119870ISCC is the threshold of stress corrosion cracking and119870pit is the stress intensity factor for the surface of the pit From
the fracture mechanics theory the critical size of a pit at thetransition to a crack can be calculated as
119886119888119894 =1
120587(
119870119868119862Φ
112119896119905Δ120590)
2
(1 minus 120578120588119911119865
120572119872120590119904
) (17)
where 119870ISCC is the threshold stress intensity factor of SCCwhich means the minimum stress required for SCC propaga-tion 119870119868119862 is the fracture toughness of the material 120578 is theanodic polarization potential 120572 is the coefficient constant120590119904 is the yield stress Φ is shape factor and 119896119905 is the stressconcentration factor of the pit hole
After the transition from a pit to a crack the corrosiondamage process will turn into the crack propagation stageUnlike corrosion fatigue crack which is usually caused bythe combination of cyclic load and a corrosive environmentthe stress corrosion cracking is generally induced by a statictensile or torsional load to open and sustain the crackThe stress intensity factor at the crack propagation stage isgenerally a function of the total stress and the crack lengthSimilarly the corrosion fatigue failure usually occurs once thestress intensity factor reaches a value beyond the thresholdvalue 119870119868119862 Following studies reported in the literature [12ndash14] in this study it is also assumed that the empirical modelas shown below can be used for stress corrosion crackpropagation which is similar to the fatigue crack propagationmodeled by Parisrsquo Law
119889119886
119889119905= 119862 (119870119868 minus 119870ISCC)
119898 (18)
where 119862 and 119898 are the model constants of crack propaga-tion Note that the developed pit growth model takes intoaccount the coupled effects of corrosion environment and themechanical stresses at the pit growth stage of the corrosiondamage process
Following the terminology of structure reliability meth-ods and stress-strength interference theory corrosion failurecan generally be defined as the stress intensity factor exceed-ing the fracture toughness based on fracture mechanicsAccordingly with the assumptions that (1) the corrosion pitof hemispherical shape grows at a constant volumetric rate(2) the corrosion current density with the stress effects willbe changed as (3) and (3) the pit will transform to crackwhen the stress effect dominates the corrosion pit growththe corresponding time-dependent limit state function forcorrosion reliability analysis can be written as
119866 (x 119905) = 119870119868119862 minus 120573120590radic120587119886 (119905) =
119870119868119862 minus 120573120590radic120587(31198721198940
2120587119911119865120588exp (minusΔ119867
119877119879) exp (
119881119898Δ119875
119877119879) lowast 119905 + 119886
30)
13
(Pit stage)
119870119868119862 minus 120573120590radic120587(119886119888119894 + int
119905
119905119888119894
119862 (119870119868 minus 119870ISCC)119898119889119905) (Crack stage)
(19)
where 120573 is the shape parameter for the crake 120590 is the staticstress load and 119886(119905) is the pit depth in pitting growth stage
or the crack size in crack propagation stage at a given time 119905Based on the time-dependent limit state function expressed
International Journal of Corrosion 5
Corrosion time (days)
Pit d
epth
cra
ck si
ze (m
)
times10minus3
5
45
4
35
3
25
2
15
1
05
00 50 100 150 200
Figure 4 Pit depth growth curve with different corrosion time
in (10) 119866(x 119905) gt 0 denotes the safe state whereas 119866(x 119905) lt 0
represents the corrosion failure state
4 Numerical Example
A case study is employed in this section to demonstratethe proposed physics of failure based corrosion model andreliability analysis approach The case study structure isidealized as an infinite plate with a pitting corrosion defectwhile the pit corrosion occurs on the surface of the structurematerial and the corrosive environment is assumed to beknown The structure material considered in this case studyis the aluminum alloy since it has been widely used inaerospace structures energy engineering and marine engi-neering applications The uncertainties involved in materialproperties as well as the corrosion model parameters areconsidered and modeled with Gaussian random variablesThe random and deterministic parameters used in this casestudy are listed in Tables 1 and 2 respectively [4 6] As shownin Table 1 seven different random variables are employed inthis case study as specified by the mean values and standarddeviations where the standard deviations for all randomvariables have been taken as 5 of the given mean valuesTo demonstrate the proposed corrosion model at the pitgrowth stage considering the coupled effects of corrosionenvironment and the mechanical stresses the corrosion timehas been assumed to begin right after the pit nucleation andwithout losing the generosity the random pit nucleation hasnot been considered in this example
Figure 4 provides 100 random realizations of the pitgrowth curve over time considering the random inputs asshown in Table 1 It is clear from the figure that the randominputs yield a large deviation for the pit growth curve overtime and a few curves rising obviously after the pit transformsto crack propagation Figure 5 shows that the 119866 value at thecorrosion time equals 200 days Note that the corrosion timedoes not include the pit nucleation time
The structure reliability of pitting corrosion damage is cal-culated using MCS and FORMTheMCS is used with a largenumber of samples (119873 = 100000) as the benchmark solution
Gva
lue d
istrib
utio
n
02
018
016
014
012
01
008
006
004
002
00 10 20 30 40 50 60 70 80 90 100
G value
Figure 5 119866 value distribution at 119905 = 200 days
Table 1 Random variables for case study
Random variables Unit Mean STD (meanlowast)1198860 m 198119890 minus 6 5101198680 cs 65119890 minus 6 510119870119868119862 MPam2 35 510120590119904 MPa 470 510Δ120590 MPa 90 510119862 mdash 28119890 minus 11 510119898 mdash 116 510
Table 2 Deterministic model parameters for the case study
to check the accuracy of the FORMmethodWith the randomvariables shown in Table 1 the corrosion reliability analysisis then carried out with different corrosion times using twodifferent approaches as mentioned earlier and the analysisresults are summarized in Tables 3 and 4 The comparisonof the reliability estimations and the absolute percentageerrors obtained using the FORM and SORM compared withthe MCS results are also shown in Figures 6 and 7 It is
6 International Journal of Corrosion
Table 3 The reliability results with different corrosion time
Corrosion time (days) ReliabilityMCS FORM FORM error () SORM SORM error ()
Figure 6 Reliability analysis results at different corrosion times
clear from the figures that the reliability estimation errorfor the FORM tends to increase with longer corrosion timemainly due to the linearization of the limit state the FORMused Besides the accuracy performance the efficiencies ofcorrosion reliability analysis using the FORM SORM andMCS are also compared as the results shown in Table 4 inwhich the number of sample points being evaluated for thelimit state function is employed as the accuracy measureAs shown in the table the FORM generally requires lesssample points to be evaluated to conduct reliability analysiscompared with the MCS method The results also show thatthe FORM is more efficient than SORM but its calculationaccuracy is lower than SORMMoreover due to the gradient-based searching process employed by the FORM to find theMPP the gradient information must be provided where inthis study the finite difference method has been employedfor the FORM to provide the required information As thegradient-basedmethod is used forMPP search the searchingprocess may not converge to the trueMPP efficiently in some
250 300 350 400 450 500
Corrosion time
Erro
r (
)
35
3
25
2
15
1
05
0
FORM errorSORM error
Figure 7 The computational error of FORM and SORM
scenarios as shown in the table for corrosion time of 250days and 300 days in which the total numbers of sampleevaluations for both have reached the upper limit of 800
5 Conclusion and Future Work
The pitting corrosion growth model considering the coupledeffects of stress and corrosion environment is proposed basedon mechanical theory And the time-dependent limit statefunction is discussed for stress corrosion crack failure basedon the corrosion growth model MCS and FORM are usedfor reliability assessment approachesThe example shows thatthe model is useful for structure reliability analysis but themodel parameters need verification by more experimentaldata Apply this mechanical model for structure corrosionfailure prognostics Integrate the Bayesian inference methodfor parameter estimation
International Journal of Corrosion 7
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the Foundation of ChinaAcademy Engineering Physics (CAEP) under Grant no2013B0203028 and the Technology Foundation Project underGrant no 2013ZK12
References
[1] R E Melchers ldquoThe effect of corrosion on the structuralreliability of steel offshore structuresrdquoCorrosion Science vol 47no 10 pp 2391ndash2410 2005
[2] F Caleyo J C Velazquez A Valor and J M Hallen ldquoMarkovchainmodelling of pitting corrosion in underground pipelinesrdquoCorrosion Science vol 51 no 9 pp 2197ndash2207 2009
[3] A Valor F Caleyo L Alfonso D Rivas and J M HallenldquoStochastic modeling of pitting corrosion a new model forinitiation and growth of multiple corrosion pitsrdquo CorrosionScience vol 49 no 2 pp 559ndash579 2007
[4] P Shi and SMahadevan ldquoDamage tolerance approach for prob-abilistic pitting corrosion fatigue life predictionrdquo EngineeringFracture Mechanics vol 68 no 13 pp 1493ndash1507 2001
[5] G Wu A probabilistic-mechanistic approach to modeling stresscorrosion cracking propagation in alloy 600 components withapplications [MS thesis] University of Maryland College ParkMd USA 2011
[6] D G Harlow and R PWei ldquoA probability model for the growthof corrosion pits in aluminum alloys induced by constituentparticlesrdquo Engineering Fracture Mechanics vol 59 no 3 pp305ndash325 1998
[7] A Haldar and S Mahadevan Probability Reliability and Statis-tical Methods in Engineering Design John Wiley amp Sons NewYork NY USA 2000
[8] H J Pradlwarter and G I Schueller ldquoLocal domain MonteCarlo simulationrdquo Structural Safety vol 32 no 5 pp 275ndash2802010
[9] G J Li Z Z Lu and C C Zhou ldquoDiscussion on local domainMonte Carlo simulation H J Pradlwarter and G I SchuellerStructural Safety 32 (2010) 275ndash280rdquo Structural Safety vol 40pp 78ndash80 2012
[10] M Di Sciuva and D Lomario ldquoA comparison between MonteCarlo and FORMs in calculating the reliability of a compositestructurerdquoComposite Structures vol 59 no 1 pp 155ndash162 2003
[11] Y-G Zhao and T Ono ldquoA general procedure for firstsecond-order reliabilitymethod (FORMSORM)rdquo Structural Safety vol21 no 2 pp 95ndash112 1999
[12] A Der Kiureghian ldquoThe geometry of random vibrations andsolutions by FORM and SORMrdquo Probabilistic EngineeringMechanics vol 15 no 1 pp 81ndash90 2000
[13] J Lim B Lee and I Lee ldquoSecond-order reliability method-based inverse reliability analysis using hessian update foraccurate and efficient reliability-based design optimizationrdquoInternational Journal for Numerical Methods in Engineering vol100 no 10 pp 773ndash792 2014
[14] M Hohenbichler and R Rackwitz ldquoImprovement of second-order reliability estimates by importance samplingrdquo Journal ofEngineering Mechanics vol 114 no 12 pp 2195ndash2199 1989
[15] J Zhang and X Du ldquoA second-order reliability method withfirst-order efficiencyrdquo Journal of Mechanical Design vol 132 no10 Article ID 101006 2010
[16] M C Alseyabi Structuring a probabilistic model for reliabilityevaluation of piping subject to corrosion-fatigue degradation[PhD thesis] University of Maryland College Park Md USA2009
[17] C Xie P Wang Z Wang and H Huang ldquoProbability ofcorrosion failure analysis using an adaptive sampling approachrdquoin Proceedings of the IEEE Conference on Prognostics and HealthManagement (PHM rsquo15) pp 1ndash8 Austin Tex USA June 2015
[18] E M Gutman Mechanochmistry of Solid Surfaces WorldScientific Publications Singapore 1994
[19] X G Jiang W Y Zhu and J M Xiao ldquoFractal analysis oforientation effect on KIC and KISCCrdquo Engineering FractureMechanics vol 51 no 5 pp 805ndash808 1995
This paper proposed a physics-based failure model forstructure with pitting corrosion damage The mechanicalstress effects for pitting growth are coupled in the integratedcorrosion process The three stages of pitting corrosion dam-age pitting growth pit-to-crack and cracking propagationare considered in the developed physics-based failure modelThen the time-dependent limit state function of corrosionstructure is defined by fracture theory The Monte-CarloSimulation FORM and SORM are employed to calculate thereliability
2 Reviews of Structure ReliabilityAnalysis Method
21 MCS Method Consider a performance model 119866(X)under the existence of uncertainties where the system fails if119866(X) lt 0 If the joint probability density function of randomvariables X is 119891X the statistical description of a probabilisticperformance that fails is then completely characterized by thecumulative density function (CDF) as
119875119891 = Pr (119866 lt 0) = int119866lt0
119891X (X) 119889X (1)
In practice the integration boundary 119866(X) = 0 and thehigh dimensionality make it difficult or even impossible to
obtain an analytical solution to the probability integration in(11)
The MCS method is showed as follows [7 8]
119875119891 =1
119873
119899
sum
119894=1
119868 (119866)
119868 (119866) =
1 119866 lt 0
0 119866 gt 0
(2)
where 119873 is the total number of simulations 119868(119866) is an indi-cator function However the MCS method needs expensivecomputational cost
22 FORM The approximation methods are therefore usedfor reliability analysis in order to reduce the computationalcost FORM [9 10] solves the probability integral by simpli-fying the performance function 119866(X) using the first-orderTaylor series expansion at the Most Probable Point (MPP)The flowchart of the FORM for reliability analysis is shownin Figure 1 FORM involves three steps to approximate theintegral [11 12]
Step 1 Transform original random variables X in X-spaceto standard normal random variables U in U-space and theperformance function is expressed as 119866(U)
Step 3 After calculating the reliability index 120573 is obtainedby an optimization problem The probability in (1) is thencomputed analytically by the following equation
Figure 1 The flowchart of the FORM for reliability analysis
where 120581119894 denotes the 119894th main curvature of the performancefunction 119892(U) The approximation of the performance func-tion in FORM and SORM is shown in Figure 2
3 Physics of Failure Modeling forPitting Corrosion
Thedegradation (damage) process for the crack developmentand propagation has been studied and modeled in manydifferent ways [16] In the proposed method the pittingcorrosion crack consists of four stages as shown in Figure 3
The first stage in the corrosion damage process is pittingnucleation It is related to the electrochemical process duringcorrosion and the pit initial size and nucleation time dependon factors such as materials corrosion environment loadsand electrolytesThis process is very complex and the physicsmodel of pit nucleation is not well understood yet In thispaper we therefore assume the initial size of pit as a randomvariable This distribution can be obtained by experimentaldata
A simplified model for pit growth proposed by HarlowandWei is used [6 17]Themodel assumes a pit of hemispher-ical shape growing at a constant volumetric rate in accordance
U2
U1
o
g gt 0g lt 0
g = 0
120573
FORM
SORM
MPP ulowast
Figure 2 Comparison of FORM and SORM
Pit nucleation Pit growth Pit-to-crackcaused by stress
Crack propagation
Figure 3 Pitting corrosion growth process with stress
with Faradayrsquos law from an initial radius size The rate of pitgrowth is given as
119889119881
119889119905=1198721198940
119911119865120588exp (minusΔ119867
119877119879) (11)
where 119881 is the hemispherical pit volume 119881 = 212058711988633 119886
is the pit size 119872 is the molecular weight of the material1198940 is the pitting current coefficient 119911 is the valence 119865 isFaradayrsquos constant 120588 is the material density Δ119867 is theactivation energy 119877 is the gas constant and 119879 is the absolutetemperature
Based on the electrochemical theory the electrical cur-rent on an electrode depends on the electrode potential Ageneral representation of the polarization of an electrode isdescribed in the Butler-Volmer equation
119869 = 1198950 exp(119911119865 (120593 minus 120593eq)
119877119879) (12)
where 119869 is electrode current density Am2 1198950 is exchangecurrent density Am2 120593 is electrode potential V and 120593eqis equilibrium potential V When the stress is applied onthe metal materials with elastic deformation the equilibriumpotential will be varied according to Gutmanrsquos theory [18 19]
Δ120593eq = minusΔ119875119881119898
119911119865 (13)
4 International Journal of Corrosion
where Δ119875 is the spherical part of macroscopic stress tensorexcess pressure (Pa) and119881119898 is the molar volume of the metalThe current density with the stress effects will be changed as
119869 = 1198950 exp(119911119865 (120593 minus 120593eq minus Δ120593eq)
119877119879)
= 1198950 exp(119911119865 (120593 minus 120593eq)
119877119879) exp(
Δ119875119881119898
119877119879)
(14)
The coefficient-exp(Δ119875119881119898119877119879) is used for depicting thevariance of corrosion current with stress effects comparedwithout stress situation
In this paper the effects of stress applied are consideredbased on (14) then the pit growing depth at the time 119905 can bechanged to take into account the coupled effects of stress loadand corrosion environment as
119886 (119905)
= (31198721198940
2120587119911119865120588exp (minusΔ119867
119877119879) exp(
119881119898Δ119875
119877119879) lowast 119905 + 119886
30)
13
(15)
where 1198680 is the pitting current 119881119898 is the molar volume of thematerial and Δ119875 is the bulk component of stress tensor Asthe pit depth grows the stress intensity factor at the pit tip willincrease correspondingly and when it reaches a value beyondthe threshold value of stress corrosion crack (SCC) the pitwill transform into a crack and the SCC will thus occur Thecriteria of a pit transforming into a crack can be obtainedbased on the thresholds of SCC and the fracture toughnessas [6 7]
119870pit ge 119870ISCC (16)
where119870ISCC is the threshold of stress corrosion cracking and119870pit is the stress intensity factor for the surface of the pit From
the fracture mechanics theory the critical size of a pit at thetransition to a crack can be calculated as
119886119888119894 =1
120587(
119870119868119862Φ
112119896119905Δ120590)
2
(1 minus 120578120588119911119865
120572119872120590119904
) (17)
where 119870ISCC is the threshold stress intensity factor of SCCwhich means the minimum stress required for SCC propaga-tion 119870119868119862 is the fracture toughness of the material 120578 is theanodic polarization potential 120572 is the coefficient constant120590119904 is the yield stress Φ is shape factor and 119896119905 is the stressconcentration factor of the pit hole
After the transition from a pit to a crack the corrosiondamage process will turn into the crack propagation stageUnlike corrosion fatigue crack which is usually caused bythe combination of cyclic load and a corrosive environmentthe stress corrosion cracking is generally induced by a statictensile or torsional load to open and sustain the crackThe stress intensity factor at the crack propagation stage isgenerally a function of the total stress and the crack lengthSimilarly the corrosion fatigue failure usually occurs once thestress intensity factor reaches a value beyond the thresholdvalue 119870119868119862 Following studies reported in the literature [12ndash14] in this study it is also assumed that the empirical modelas shown below can be used for stress corrosion crackpropagation which is similar to the fatigue crack propagationmodeled by Parisrsquo Law
119889119886
119889119905= 119862 (119870119868 minus 119870ISCC)
119898 (18)
where 119862 and 119898 are the model constants of crack propaga-tion Note that the developed pit growth model takes intoaccount the coupled effects of corrosion environment and themechanical stresses at the pit growth stage of the corrosiondamage process
Following the terminology of structure reliability meth-ods and stress-strength interference theory corrosion failurecan generally be defined as the stress intensity factor exceed-ing the fracture toughness based on fracture mechanicsAccordingly with the assumptions that (1) the corrosion pitof hemispherical shape grows at a constant volumetric rate(2) the corrosion current density with the stress effects willbe changed as (3) and (3) the pit will transform to crackwhen the stress effect dominates the corrosion pit growththe corresponding time-dependent limit state function forcorrosion reliability analysis can be written as
119866 (x 119905) = 119870119868119862 minus 120573120590radic120587119886 (119905) =
119870119868119862 minus 120573120590radic120587(31198721198940
2120587119911119865120588exp (minusΔ119867
119877119879) exp (
119881119898Δ119875
119877119879) lowast 119905 + 119886
30)
13
(Pit stage)
119870119868119862 minus 120573120590radic120587(119886119888119894 + int
119905
119905119888119894
119862 (119870119868 minus 119870ISCC)119898119889119905) (Crack stage)
(19)
where 120573 is the shape parameter for the crake 120590 is the staticstress load and 119886(119905) is the pit depth in pitting growth stage
or the crack size in crack propagation stage at a given time 119905Based on the time-dependent limit state function expressed
International Journal of Corrosion 5
Corrosion time (days)
Pit d
epth
cra
ck si
ze (m
)
times10minus3
5
45
4
35
3
25
2
15
1
05
00 50 100 150 200
Figure 4 Pit depth growth curve with different corrosion time
in (10) 119866(x 119905) gt 0 denotes the safe state whereas 119866(x 119905) lt 0
represents the corrosion failure state
4 Numerical Example
A case study is employed in this section to demonstratethe proposed physics of failure based corrosion model andreliability analysis approach The case study structure isidealized as an infinite plate with a pitting corrosion defectwhile the pit corrosion occurs on the surface of the structurematerial and the corrosive environment is assumed to beknown The structure material considered in this case studyis the aluminum alloy since it has been widely used inaerospace structures energy engineering and marine engi-neering applications The uncertainties involved in materialproperties as well as the corrosion model parameters areconsidered and modeled with Gaussian random variablesThe random and deterministic parameters used in this casestudy are listed in Tables 1 and 2 respectively [4 6] As shownin Table 1 seven different random variables are employed inthis case study as specified by the mean values and standarddeviations where the standard deviations for all randomvariables have been taken as 5 of the given mean valuesTo demonstrate the proposed corrosion model at the pitgrowth stage considering the coupled effects of corrosionenvironment and the mechanical stresses the corrosion timehas been assumed to begin right after the pit nucleation andwithout losing the generosity the random pit nucleation hasnot been considered in this example
Figure 4 provides 100 random realizations of the pitgrowth curve over time considering the random inputs asshown in Table 1 It is clear from the figure that the randominputs yield a large deviation for the pit growth curve overtime and a few curves rising obviously after the pit transformsto crack propagation Figure 5 shows that the 119866 value at thecorrosion time equals 200 days Note that the corrosion timedoes not include the pit nucleation time
The structure reliability of pitting corrosion damage is cal-culated using MCS and FORMTheMCS is used with a largenumber of samples (119873 = 100000) as the benchmark solution
Gva
lue d
istrib
utio
n
02
018
016
014
012
01
008
006
004
002
00 10 20 30 40 50 60 70 80 90 100
G value
Figure 5 119866 value distribution at 119905 = 200 days
Table 1 Random variables for case study
Random variables Unit Mean STD (meanlowast)1198860 m 198119890 minus 6 5101198680 cs 65119890 minus 6 510119870119868119862 MPam2 35 510120590119904 MPa 470 510Δ120590 MPa 90 510119862 mdash 28119890 minus 11 510119898 mdash 116 510
Table 2 Deterministic model parameters for the case study
to check the accuracy of the FORMmethodWith the randomvariables shown in Table 1 the corrosion reliability analysisis then carried out with different corrosion times using twodifferent approaches as mentioned earlier and the analysisresults are summarized in Tables 3 and 4 The comparisonof the reliability estimations and the absolute percentageerrors obtained using the FORM and SORM compared withthe MCS results are also shown in Figures 6 and 7 It is
6 International Journal of Corrosion
Table 3 The reliability results with different corrosion time
Corrosion time (days) ReliabilityMCS FORM FORM error () SORM SORM error ()
Figure 6 Reliability analysis results at different corrosion times
clear from the figures that the reliability estimation errorfor the FORM tends to increase with longer corrosion timemainly due to the linearization of the limit state the FORMused Besides the accuracy performance the efficiencies ofcorrosion reliability analysis using the FORM SORM andMCS are also compared as the results shown in Table 4 inwhich the number of sample points being evaluated for thelimit state function is employed as the accuracy measureAs shown in the table the FORM generally requires lesssample points to be evaluated to conduct reliability analysiscompared with the MCS method The results also show thatthe FORM is more efficient than SORM but its calculationaccuracy is lower than SORMMoreover due to the gradient-based searching process employed by the FORM to find theMPP the gradient information must be provided where inthis study the finite difference method has been employedfor the FORM to provide the required information As thegradient-basedmethod is used forMPP search the searchingprocess may not converge to the trueMPP efficiently in some
250 300 350 400 450 500
Corrosion time
Erro
r (
)
35
3
25
2
15
1
05
0
FORM errorSORM error
Figure 7 The computational error of FORM and SORM
scenarios as shown in the table for corrosion time of 250days and 300 days in which the total numbers of sampleevaluations for both have reached the upper limit of 800
5 Conclusion and Future Work
The pitting corrosion growth model considering the coupledeffects of stress and corrosion environment is proposed basedon mechanical theory And the time-dependent limit statefunction is discussed for stress corrosion crack failure basedon the corrosion growth model MCS and FORM are usedfor reliability assessment approachesThe example shows thatthe model is useful for structure reliability analysis but themodel parameters need verification by more experimentaldata Apply this mechanical model for structure corrosionfailure prognostics Integrate the Bayesian inference methodfor parameter estimation
International Journal of Corrosion 7
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the Foundation of ChinaAcademy Engineering Physics (CAEP) under Grant no2013B0203028 and the Technology Foundation Project underGrant no 2013ZK12
References
[1] R E Melchers ldquoThe effect of corrosion on the structuralreliability of steel offshore structuresrdquoCorrosion Science vol 47no 10 pp 2391ndash2410 2005
[2] F Caleyo J C Velazquez A Valor and J M Hallen ldquoMarkovchainmodelling of pitting corrosion in underground pipelinesrdquoCorrosion Science vol 51 no 9 pp 2197ndash2207 2009
[3] A Valor F Caleyo L Alfonso D Rivas and J M HallenldquoStochastic modeling of pitting corrosion a new model forinitiation and growth of multiple corrosion pitsrdquo CorrosionScience vol 49 no 2 pp 559ndash579 2007
[4] P Shi and SMahadevan ldquoDamage tolerance approach for prob-abilistic pitting corrosion fatigue life predictionrdquo EngineeringFracture Mechanics vol 68 no 13 pp 1493ndash1507 2001
[5] G Wu A probabilistic-mechanistic approach to modeling stresscorrosion cracking propagation in alloy 600 components withapplications [MS thesis] University of Maryland College ParkMd USA 2011
[6] D G Harlow and R PWei ldquoA probability model for the growthof corrosion pits in aluminum alloys induced by constituentparticlesrdquo Engineering Fracture Mechanics vol 59 no 3 pp305ndash325 1998
[7] A Haldar and S Mahadevan Probability Reliability and Statis-tical Methods in Engineering Design John Wiley amp Sons NewYork NY USA 2000
[8] H J Pradlwarter and G I Schueller ldquoLocal domain MonteCarlo simulationrdquo Structural Safety vol 32 no 5 pp 275ndash2802010
[9] G J Li Z Z Lu and C C Zhou ldquoDiscussion on local domainMonte Carlo simulation H J Pradlwarter and G I SchuellerStructural Safety 32 (2010) 275ndash280rdquo Structural Safety vol 40pp 78ndash80 2012
[10] M Di Sciuva and D Lomario ldquoA comparison between MonteCarlo and FORMs in calculating the reliability of a compositestructurerdquoComposite Structures vol 59 no 1 pp 155ndash162 2003
[11] Y-G Zhao and T Ono ldquoA general procedure for firstsecond-order reliabilitymethod (FORMSORM)rdquo Structural Safety vol21 no 2 pp 95ndash112 1999
[12] A Der Kiureghian ldquoThe geometry of random vibrations andsolutions by FORM and SORMrdquo Probabilistic EngineeringMechanics vol 15 no 1 pp 81ndash90 2000
[13] J Lim B Lee and I Lee ldquoSecond-order reliability method-based inverse reliability analysis using hessian update foraccurate and efficient reliability-based design optimizationrdquoInternational Journal for Numerical Methods in Engineering vol100 no 10 pp 773ndash792 2014
[14] M Hohenbichler and R Rackwitz ldquoImprovement of second-order reliability estimates by importance samplingrdquo Journal ofEngineering Mechanics vol 114 no 12 pp 2195ndash2199 1989
[15] J Zhang and X Du ldquoA second-order reliability method withfirst-order efficiencyrdquo Journal of Mechanical Design vol 132 no10 Article ID 101006 2010
[16] M C Alseyabi Structuring a probabilistic model for reliabilityevaluation of piping subject to corrosion-fatigue degradation[PhD thesis] University of Maryland College Park Md USA2009
[17] C Xie P Wang Z Wang and H Huang ldquoProbability ofcorrosion failure analysis using an adaptive sampling approachrdquoin Proceedings of the IEEE Conference on Prognostics and HealthManagement (PHM rsquo15) pp 1ndash8 Austin Tex USA June 2015
[18] E M Gutman Mechanochmistry of Solid Surfaces WorldScientific Publications Singapore 1994
[19] X G Jiang W Y Zhu and J M Xiao ldquoFractal analysis oforientation effect on KIC and KISCCrdquo Engineering FractureMechanics vol 51 no 5 pp 805ndash808 1995
Figure 1 The flowchart of the FORM for reliability analysis
where 120581119894 denotes the 119894th main curvature of the performancefunction 119892(U) The approximation of the performance func-tion in FORM and SORM is shown in Figure 2
3 Physics of Failure Modeling forPitting Corrosion
Thedegradation (damage) process for the crack developmentand propagation has been studied and modeled in manydifferent ways [16] In the proposed method the pittingcorrosion crack consists of four stages as shown in Figure 3
The first stage in the corrosion damage process is pittingnucleation It is related to the electrochemical process duringcorrosion and the pit initial size and nucleation time dependon factors such as materials corrosion environment loadsand electrolytesThis process is very complex and the physicsmodel of pit nucleation is not well understood yet In thispaper we therefore assume the initial size of pit as a randomvariable This distribution can be obtained by experimentaldata
A simplified model for pit growth proposed by HarlowandWei is used [6 17]Themodel assumes a pit of hemispher-ical shape growing at a constant volumetric rate in accordance
U2
U1
o
g gt 0g lt 0
g = 0
120573
FORM
SORM
MPP ulowast
Figure 2 Comparison of FORM and SORM
Pit nucleation Pit growth Pit-to-crackcaused by stress
Crack propagation
Figure 3 Pitting corrosion growth process with stress
with Faradayrsquos law from an initial radius size The rate of pitgrowth is given as
119889119881
119889119905=1198721198940
119911119865120588exp (minusΔ119867
119877119879) (11)
where 119881 is the hemispherical pit volume 119881 = 212058711988633 119886
is the pit size 119872 is the molecular weight of the material1198940 is the pitting current coefficient 119911 is the valence 119865 isFaradayrsquos constant 120588 is the material density Δ119867 is theactivation energy 119877 is the gas constant and 119879 is the absolutetemperature
Based on the electrochemical theory the electrical cur-rent on an electrode depends on the electrode potential Ageneral representation of the polarization of an electrode isdescribed in the Butler-Volmer equation
119869 = 1198950 exp(119911119865 (120593 minus 120593eq)
119877119879) (12)
where 119869 is electrode current density Am2 1198950 is exchangecurrent density Am2 120593 is electrode potential V and 120593eqis equilibrium potential V When the stress is applied onthe metal materials with elastic deformation the equilibriumpotential will be varied according to Gutmanrsquos theory [18 19]
Δ120593eq = minusΔ119875119881119898
119911119865 (13)
4 International Journal of Corrosion
where Δ119875 is the spherical part of macroscopic stress tensorexcess pressure (Pa) and119881119898 is the molar volume of the metalThe current density with the stress effects will be changed as
119869 = 1198950 exp(119911119865 (120593 minus 120593eq minus Δ120593eq)
119877119879)
= 1198950 exp(119911119865 (120593 minus 120593eq)
119877119879) exp(
Δ119875119881119898
119877119879)
(14)
The coefficient-exp(Δ119875119881119898119877119879) is used for depicting thevariance of corrosion current with stress effects comparedwithout stress situation
In this paper the effects of stress applied are consideredbased on (14) then the pit growing depth at the time 119905 can bechanged to take into account the coupled effects of stress loadand corrosion environment as
119886 (119905)
= (31198721198940
2120587119911119865120588exp (minusΔ119867
119877119879) exp(
119881119898Δ119875
119877119879) lowast 119905 + 119886
30)
13
(15)
where 1198680 is the pitting current 119881119898 is the molar volume of thematerial and Δ119875 is the bulk component of stress tensor Asthe pit depth grows the stress intensity factor at the pit tip willincrease correspondingly and when it reaches a value beyondthe threshold value of stress corrosion crack (SCC) the pitwill transform into a crack and the SCC will thus occur Thecriteria of a pit transforming into a crack can be obtainedbased on the thresholds of SCC and the fracture toughnessas [6 7]
119870pit ge 119870ISCC (16)
where119870ISCC is the threshold of stress corrosion cracking and119870pit is the stress intensity factor for the surface of the pit From
the fracture mechanics theory the critical size of a pit at thetransition to a crack can be calculated as
119886119888119894 =1
120587(
119870119868119862Φ
112119896119905Δ120590)
2
(1 minus 120578120588119911119865
120572119872120590119904
) (17)
where 119870ISCC is the threshold stress intensity factor of SCCwhich means the minimum stress required for SCC propaga-tion 119870119868119862 is the fracture toughness of the material 120578 is theanodic polarization potential 120572 is the coefficient constant120590119904 is the yield stress Φ is shape factor and 119896119905 is the stressconcentration factor of the pit hole
After the transition from a pit to a crack the corrosiondamage process will turn into the crack propagation stageUnlike corrosion fatigue crack which is usually caused bythe combination of cyclic load and a corrosive environmentthe stress corrosion cracking is generally induced by a statictensile or torsional load to open and sustain the crackThe stress intensity factor at the crack propagation stage isgenerally a function of the total stress and the crack lengthSimilarly the corrosion fatigue failure usually occurs once thestress intensity factor reaches a value beyond the thresholdvalue 119870119868119862 Following studies reported in the literature [12ndash14] in this study it is also assumed that the empirical modelas shown below can be used for stress corrosion crackpropagation which is similar to the fatigue crack propagationmodeled by Parisrsquo Law
119889119886
119889119905= 119862 (119870119868 minus 119870ISCC)
119898 (18)
where 119862 and 119898 are the model constants of crack propaga-tion Note that the developed pit growth model takes intoaccount the coupled effects of corrosion environment and themechanical stresses at the pit growth stage of the corrosiondamage process
Following the terminology of structure reliability meth-ods and stress-strength interference theory corrosion failurecan generally be defined as the stress intensity factor exceed-ing the fracture toughness based on fracture mechanicsAccordingly with the assumptions that (1) the corrosion pitof hemispherical shape grows at a constant volumetric rate(2) the corrosion current density with the stress effects willbe changed as (3) and (3) the pit will transform to crackwhen the stress effect dominates the corrosion pit growththe corresponding time-dependent limit state function forcorrosion reliability analysis can be written as
119866 (x 119905) = 119870119868119862 minus 120573120590radic120587119886 (119905) =
119870119868119862 minus 120573120590radic120587(31198721198940
2120587119911119865120588exp (minusΔ119867
119877119879) exp (
119881119898Δ119875
119877119879) lowast 119905 + 119886
30)
13
(Pit stage)
119870119868119862 minus 120573120590radic120587(119886119888119894 + int
119905
119905119888119894
119862 (119870119868 minus 119870ISCC)119898119889119905) (Crack stage)
(19)
where 120573 is the shape parameter for the crake 120590 is the staticstress load and 119886(119905) is the pit depth in pitting growth stage
or the crack size in crack propagation stage at a given time 119905Based on the time-dependent limit state function expressed
International Journal of Corrosion 5
Corrosion time (days)
Pit d
epth
cra
ck si
ze (m
)
times10minus3
5
45
4
35
3
25
2
15
1
05
00 50 100 150 200
Figure 4 Pit depth growth curve with different corrosion time
in (10) 119866(x 119905) gt 0 denotes the safe state whereas 119866(x 119905) lt 0
represents the corrosion failure state
4 Numerical Example
A case study is employed in this section to demonstratethe proposed physics of failure based corrosion model andreliability analysis approach The case study structure isidealized as an infinite plate with a pitting corrosion defectwhile the pit corrosion occurs on the surface of the structurematerial and the corrosive environment is assumed to beknown The structure material considered in this case studyis the aluminum alloy since it has been widely used inaerospace structures energy engineering and marine engi-neering applications The uncertainties involved in materialproperties as well as the corrosion model parameters areconsidered and modeled with Gaussian random variablesThe random and deterministic parameters used in this casestudy are listed in Tables 1 and 2 respectively [4 6] As shownin Table 1 seven different random variables are employed inthis case study as specified by the mean values and standarddeviations where the standard deviations for all randomvariables have been taken as 5 of the given mean valuesTo demonstrate the proposed corrosion model at the pitgrowth stage considering the coupled effects of corrosionenvironment and the mechanical stresses the corrosion timehas been assumed to begin right after the pit nucleation andwithout losing the generosity the random pit nucleation hasnot been considered in this example
Figure 4 provides 100 random realizations of the pitgrowth curve over time considering the random inputs asshown in Table 1 It is clear from the figure that the randominputs yield a large deviation for the pit growth curve overtime and a few curves rising obviously after the pit transformsto crack propagation Figure 5 shows that the 119866 value at thecorrosion time equals 200 days Note that the corrosion timedoes not include the pit nucleation time
The structure reliability of pitting corrosion damage is cal-culated using MCS and FORMTheMCS is used with a largenumber of samples (119873 = 100000) as the benchmark solution
Gva
lue d
istrib
utio
n
02
018
016
014
012
01
008
006
004
002
00 10 20 30 40 50 60 70 80 90 100
G value
Figure 5 119866 value distribution at 119905 = 200 days
Table 1 Random variables for case study
Random variables Unit Mean STD (meanlowast)1198860 m 198119890 minus 6 5101198680 cs 65119890 minus 6 510119870119868119862 MPam2 35 510120590119904 MPa 470 510Δ120590 MPa 90 510119862 mdash 28119890 minus 11 510119898 mdash 116 510
Table 2 Deterministic model parameters for the case study
to check the accuracy of the FORMmethodWith the randomvariables shown in Table 1 the corrosion reliability analysisis then carried out with different corrosion times using twodifferent approaches as mentioned earlier and the analysisresults are summarized in Tables 3 and 4 The comparisonof the reliability estimations and the absolute percentageerrors obtained using the FORM and SORM compared withthe MCS results are also shown in Figures 6 and 7 It is
6 International Journal of Corrosion
Table 3 The reliability results with different corrosion time
Corrosion time (days) ReliabilityMCS FORM FORM error () SORM SORM error ()
Figure 6 Reliability analysis results at different corrosion times
clear from the figures that the reliability estimation errorfor the FORM tends to increase with longer corrosion timemainly due to the linearization of the limit state the FORMused Besides the accuracy performance the efficiencies ofcorrosion reliability analysis using the FORM SORM andMCS are also compared as the results shown in Table 4 inwhich the number of sample points being evaluated for thelimit state function is employed as the accuracy measureAs shown in the table the FORM generally requires lesssample points to be evaluated to conduct reliability analysiscompared with the MCS method The results also show thatthe FORM is more efficient than SORM but its calculationaccuracy is lower than SORMMoreover due to the gradient-based searching process employed by the FORM to find theMPP the gradient information must be provided where inthis study the finite difference method has been employedfor the FORM to provide the required information As thegradient-basedmethod is used forMPP search the searchingprocess may not converge to the trueMPP efficiently in some
250 300 350 400 450 500
Corrosion time
Erro
r (
)
35
3
25
2
15
1
05
0
FORM errorSORM error
Figure 7 The computational error of FORM and SORM
scenarios as shown in the table for corrosion time of 250days and 300 days in which the total numbers of sampleevaluations for both have reached the upper limit of 800
5 Conclusion and Future Work
The pitting corrosion growth model considering the coupledeffects of stress and corrosion environment is proposed basedon mechanical theory And the time-dependent limit statefunction is discussed for stress corrosion crack failure basedon the corrosion growth model MCS and FORM are usedfor reliability assessment approachesThe example shows thatthe model is useful for structure reliability analysis but themodel parameters need verification by more experimentaldata Apply this mechanical model for structure corrosionfailure prognostics Integrate the Bayesian inference methodfor parameter estimation
International Journal of Corrosion 7
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the Foundation of ChinaAcademy Engineering Physics (CAEP) under Grant no2013B0203028 and the Technology Foundation Project underGrant no 2013ZK12
References
[1] R E Melchers ldquoThe effect of corrosion on the structuralreliability of steel offshore structuresrdquoCorrosion Science vol 47no 10 pp 2391ndash2410 2005
[2] F Caleyo J C Velazquez A Valor and J M Hallen ldquoMarkovchainmodelling of pitting corrosion in underground pipelinesrdquoCorrosion Science vol 51 no 9 pp 2197ndash2207 2009
[3] A Valor F Caleyo L Alfonso D Rivas and J M HallenldquoStochastic modeling of pitting corrosion a new model forinitiation and growth of multiple corrosion pitsrdquo CorrosionScience vol 49 no 2 pp 559ndash579 2007
[4] P Shi and SMahadevan ldquoDamage tolerance approach for prob-abilistic pitting corrosion fatigue life predictionrdquo EngineeringFracture Mechanics vol 68 no 13 pp 1493ndash1507 2001
[5] G Wu A probabilistic-mechanistic approach to modeling stresscorrosion cracking propagation in alloy 600 components withapplications [MS thesis] University of Maryland College ParkMd USA 2011
[6] D G Harlow and R PWei ldquoA probability model for the growthof corrosion pits in aluminum alloys induced by constituentparticlesrdquo Engineering Fracture Mechanics vol 59 no 3 pp305ndash325 1998
[7] A Haldar and S Mahadevan Probability Reliability and Statis-tical Methods in Engineering Design John Wiley amp Sons NewYork NY USA 2000
[8] H J Pradlwarter and G I Schueller ldquoLocal domain MonteCarlo simulationrdquo Structural Safety vol 32 no 5 pp 275ndash2802010
[9] G J Li Z Z Lu and C C Zhou ldquoDiscussion on local domainMonte Carlo simulation H J Pradlwarter and G I SchuellerStructural Safety 32 (2010) 275ndash280rdquo Structural Safety vol 40pp 78ndash80 2012
[10] M Di Sciuva and D Lomario ldquoA comparison between MonteCarlo and FORMs in calculating the reliability of a compositestructurerdquoComposite Structures vol 59 no 1 pp 155ndash162 2003
[11] Y-G Zhao and T Ono ldquoA general procedure for firstsecond-order reliabilitymethod (FORMSORM)rdquo Structural Safety vol21 no 2 pp 95ndash112 1999
[12] A Der Kiureghian ldquoThe geometry of random vibrations andsolutions by FORM and SORMrdquo Probabilistic EngineeringMechanics vol 15 no 1 pp 81ndash90 2000
[13] J Lim B Lee and I Lee ldquoSecond-order reliability method-based inverse reliability analysis using hessian update foraccurate and efficient reliability-based design optimizationrdquoInternational Journal for Numerical Methods in Engineering vol100 no 10 pp 773ndash792 2014
[14] M Hohenbichler and R Rackwitz ldquoImprovement of second-order reliability estimates by importance samplingrdquo Journal ofEngineering Mechanics vol 114 no 12 pp 2195ndash2199 1989
[15] J Zhang and X Du ldquoA second-order reliability method withfirst-order efficiencyrdquo Journal of Mechanical Design vol 132 no10 Article ID 101006 2010
[16] M C Alseyabi Structuring a probabilistic model for reliabilityevaluation of piping subject to corrosion-fatigue degradation[PhD thesis] University of Maryland College Park Md USA2009
[17] C Xie P Wang Z Wang and H Huang ldquoProbability ofcorrosion failure analysis using an adaptive sampling approachrdquoin Proceedings of the IEEE Conference on Prognostics and HealthManagement (PHM rsquo15) pp 1ndash8 Austin Tex USA June 2015
[18] E M Gutman Mechanochmistry of Solid Surfaces WorldScientific Publications Singapore 1994
[19] X G Jiang W Y Zhu and J M Xiao ldquoFractal analysis oforientation effect on KIC and KISCCrdquo Engineering FractureMechanics vol 51 no 5 pp 805ndash808 1995
where Δ119875 is the spherical part of macroscopic stress tensorexcess pressure (Pa) and119881119898 is the molar volume of the metalThe current density with the stress effects will be changed as
119869 = 1198950 exp(119911119865 (120593 minus 120593eq minus Δ120593eq)
119877119879)
= 1198950 exp(119911119865 (120593 minus 120593eq)
119877119879) exp(
Δ119875119881119898
119877119879)
(14)
The coefficient-exp(Δ119875119881119898119877119879) is used for depicting thevariance of corrosion current with stress effects comparedwithout stress situation
In this paper the effects of stress applied are consideredbased on (14) then the pit growing depth at the time 119905 can bechanged to take into account the coupled effects of stress loadand corrosion environment as
119886 (119905)
= (31198721198940
2120587119911119865120588exp (minusΔ119867
119877119879) exp(
119881119898Δ119875
119877119879) lowast 119905 + 119886
30)
13
(15)
where 1198680 is the pitting current 119881119898 is the molar volume of thematerial and Δ119875 is the bulk component of stress tensor Asthe pit depth grows the stress intensity factor at the pit tip willincrease correspondingly and when it reaches a value beyondthe threshold value of stress corrosion crack (SCC) the pitwill transform into a crack and the SCC will thus occur Thecriteria of a pit transforming into a crack can be obtainedbased on the thresholds of SCC and the fracture toughnessas [6 7]
119870pit ge 119870ISCC (16)
where119870ISCC is the threshold of stress corrosion cracking and119870pit is the stress intensity factor for the surface of the pit From
the fracture mechanics theory the critical size of a pit at thetransition to a crack can be calculated as
119886119888119894 =1
120587(
119870119868119862Φ
112119896119905Δ120590)
2
(1 minus 120578120588119911119865
120572119872120590119904
) (17)
where 119870ISCC is the threshold stress intensity factor of SCCwhich means the minimum stress required for SCC propaga-tion 119870119868119862 is the fracture toughness of the material 120578 is theanodic polarization potential 120572 is the coefficient constant120590119904 is the yield stress Φ is shape factor and 119896119905 is the stressconcentration factor of the pit hole
After the transition from a pit to a crack the corrosiondamage process will turn into the crack propagation stageUnlike corrosion fatigue crack which is usually caused bythe combination of cyclic load and a corrosive environmentthe stress corrosion cracking is generally induced by a statictensile or torsional load to open and sustain the crackThe stress intensity factor at the crack propagation stage isgenerally a function of the total stress and the crack lengthSimilarly the corrosion fatigue failure usually occurs once thestress intensity factor reaches a value beyond the thresholdvalue 119870119868119862 Following studies reported in the literature [12ndash14] in this study it is also assumed that the empirical modelas shown below can be used for stress corrosion crackpropagation which is similar to the fatigue crack propagationmodeled by Parisrsquo Law
119889119886
119889119905= 119862 (119870119868 minus 119870ISCC)
119898 (18)
where 119862 and 119898 are the model constants of crack propaga-tion Note that the developed pit growth model takes intoaccount the coupled effects of corrosion environment and themechanical stresses at the pit growth stage of the corrosiondamage process
Following the terminology of structure reliability meth-ods and stress-strength interference theory corrosion failurecan generally be defined as the stress intensity factor exceed-ing the fracture toughness based on fracture mechanicsAccordingly with the assumptions that (1) the corrosion pitof hemispherical shape grows at a constant volumetric rate(2) the corrosion current density with the stress effects willbe changed as (3) and (3) the pit will transform to crackwhen the stress effect dominates the corrosion pit growththe corresponding time-dependent limit state function forcorrosion reliability analysis can be written as
119866 (x 119905) = 119870119868119862 minus 120573120590radic120587119886 (119905) =
119870119868119862 minus 120573120590radic120587(31198721198940
2120587119911119865120588exp (minusΔ119867
119877119879) exp (
119881119898Δ119875
119877119879) lowast 119905 + 119886
30)
13
(Pit stage)
119870119868119862 minus 120573120590radic120587(119886119888119894 + int
119905
119905119888119894
119862 (119870119868 minus 119870ISCC)119898119889119905) (Crack stage)
(19)
where 120573 is the shape parameter for the crake 120590 is the staticstress load and 119886(119905) is the pit depth in pitting growth stage
or the crack size in crack propagation stage at a given time 119905Based on the time-dependent limit state function expressed
International Journal of Corrosion 5
Corrosion time (days)
Pit d
epth
cra
ck si
ze (m
)
times10minus3
5
45
4
35
3
25
2
15
1
05
00 50 100 150 200
Figure 4 Pit depth growth curve with different corrosion time
in (10) 119866(x 119905) gt 0 denotes the safe state whereas 119866(x 119905) lt 0
represents the corrosion failure state
4 Numerical Example
A case study is employed in this section to demonstratethe proposed physics of failure based corrosion model andreliability analysis approach The case study structure isidealized as an infinite plate with a pitting corrosion defectwhile the pit corrosion occurs on the surface of the structurematerial and the corrosive environment is assumed to beknown The structure material considered in this case studyis the aluminum alloy since it has been widely used inaerospace structures energy engineering and marine engi-neering applications The uncertainties involved in materialproperties as well as the corrosion model parameters areconsidered and modeled with Gaussian random variablesThe random and deterministic parameters used in this casestudy are listed in Tables 1 and 2 respectively [4 6] As shownin Table 1 seven different random variables are employed inthis case study as specified by the mean values and standarddeviations where the standard deviations for all randomvariables have been taken as 5 of the given mean valuesTo demonstrate the proposed corrosion model at the pitgrowth stage considering the coupled effects of corrosionenvironment and the mechanical stresses the corrosion timehas been assumed to begin right after the pit nucleation andwithout losing the generosity the random pit nucleation hasnot been considered in this example
Figure 4 provides 100 random realizations of the pitgrowth curve over time considering the random inputs asshown in Table 1 It is clear from the figure that the randominputs yield a large deviation for the pit growth curve overtime and a few curves rising obviously after the pit transformsto crack propagation Figure 5 shows that the 119866 value at thecorrosion time equals 200 days Note that the corrosion timedoes not include the pit nucleation time
The structure reliability of pitting corrosion damage is cal-culated using MCS and FORMTheMCS is used with a largenumber of samples (119873 = 100000) as the benchmark solution
Gva
lue d
istrib
utio
n
02
018
016
014
012
01
008
006
004
002
00 10 20 30 40 50 60 70 80 90 100
G value
Figure 5 119866 value distribution at 119905 = 200 days
Table 1 Random variables for case study
Random variables Unit Mean STD (meanlowast)1198860 m 198119890 minus 6 5101198680 cs 65119890 minus 6 510119870119868119862 MPam2 35 510120590119904 MPa 470 510Δ120590 MPa 90 510119862 mdash 28119890 minus 11 510119898 mdash 116 510
Table 2 Deterministic model parameters for the case study
to check the accuracy of the FORMmethodWith the randomvariables shown in Table 1 the corrosion reliability analysisis then carried out with different corrosion times using twodifferent approaches as mentioned earlier and the analysisresults are summarized in Tables 3 and 4 The comparisonof the reliability estimations and the absolute percentageerrors obtained using the FORM and SORM compared withthe MCS results are also shown in Figures 6 and 7 It is
6 International Journal of Corrosion
Table 3 The reliability results with different corrosion time
Corrosion time (days) ReliabilityMCS FORM FORM error () SORM SORM error ()
Figure 6 Reliability analysis results at different corrosion times
clear from the figures that the reliability estimation errorfor the FORM tends to increase with longer corrosion timemainly due to the linearization of the limit state the FORMused Besides the accuracy performance the efficiencies ofcorrosion reliability analysis using the FORM SORM andMCS are also compared as the results shown in Table 4 inwhich the number of sample points being evaluated for thelimit state function is employed as the accuracy measureAs shown in the table the FORM generally requires lesssample points to be evaluated to conduct reliability analysiscompared with the MCS method The results also show thatthe FORM is more efficient than SORM but its calculationaccuracy is lower than SORMMoreover due to the gradient-based searching process employed by the FORM to find theMPP the gradient information must be provided where inthis study the finite difference method has been employedfor the FORM to provide the required information As thegradient-basedmethod is used forMPP search the searchingprocess may not converge to the trueMPP efficiently in some
250 300 350 400 450 500
Corrosion time
Erro
r (
)
35
3
25
2
15
1
05
0
FORM errorSORM error
Figure 7 The computational error of FORM and SORM
scenarios as shown in the table for corrosion time of 250days and 300 days in which the total numbers of sampleevaluations for both have reached the upper limit of 800
5 Conclusion and Future Work
The pitting corrosion growth model considering the coupledeffects of stress and corrosion environment is proposed basedon mechanical theory And the time-dependent limit statefunction is discussed for stress corrosion crack failure basedon the corrosion growth model MCS and FORM are usedfor reliability assessment approachesThe example shows thatthe model is useful for structure reliability analysis but themodel parameters need verification by more experimentaldata Apply this mechanical model for structure corrosionfailure prognostics Integrate the Bayesian inference methodfor parameter estimation
International Journal of Corrosion 7
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the Foundation of ChinaAcademy Engineering Physics (CAEP) under Grant no2013B0203028 and the Technology Foundation Project underGrant no 2013ZK12
References
[1] R E Melchers ldquoThe effect of corrosion on the structuralreliability of steel offshore structuresrdquoCorrosion Science vol 47no 10 pp 2391ndash2410 2005
[2] F Caleyo J C Velazquez A Valor and J M Hallen ldquoMarkovchainmodelling of pitting corrosion in underground pipelinesrdquoCorrosion Science vol 51 no 9 pp 2197ndash2207 2009
[3] A Valor F Caleyo L Alfonso D Rivas and J M HallenldquoStochastic modeling of pitting corrosion a new model forinitiation and growth of multiple corrosion pitsrdquo CorrosionScience vol 49 no 2 pp 559ndash579 2007
[4] P Shi and SMahadevan ldquoDamage tolerance approach for prob-abilistic pitting corrosion fatigue life predictionrdquo EngineeringFracture Mechanics vol 68 no 13 pp 1493ndash1507 2001
[5] G Wu A probabilistic-mechanistic approach to modeling stresscorrosion cracking propagation in alloy 600 components withapplications [MS thesis] University of Maryland College ParkMd USA 2011
[6] D G Harlow and R PWei ldquoA probability model for the growthof corrosion pits in aluminum alloys induced by constituentparticlesrdquo Engineering Fracture Mechanics vol 59 no 3 pp305ndash325 1998
[7] A Haldar and S Mahadevan Probability Reliability and Statis-tical Methods in Engineering Design John Wiley amp Sons NewYork NY USA 2000
[8] H J Pradlwarter and G I Schueller ldquoLocal domain MonteCarlo simulationrdquo Structural Safety vol 32 no 5 pp 275ndash2802010
[9] G J Li Z Z Lu and C C Zhou ldquoDiscussion on local domainMonte Carlo simulation H J Pradlwarter and G I SchuellerStructural Safety 32 (2010) 275ndash280rdquo Structural Safety vol 40pp 78ndash80 2012
[10] M Di Sciuva and D Lomario ldquoA comparison between MonteCarlo and FORMs in calculating the reliability of a compositestructurerdquoComposite Structures vol 59 no 1 pp 155ndash162 2003
[11] Y-G Zhao and T Ono ldquoA general procedure for firstsecond-order reliabilitymethod (FORMSORM)rdquo Structural Safety vol21 no 2 pp 95ndash112 1999
[12] A Der Kiureghian ldquoThe geometry of random vibrations andsolutions by FORM and SORMrdquo Probabilistic EngineeringMechanics vol 15 no 1 pp 81ndash90 2000
[13] J Lim B Lee and I Lee ldquoSecond-order reliability method-based inverse reliability analysis using hessian update foraccurate and efficient reliability-based design optimizationrdquoInternational Journal for Numerical Methods in Engineering vol100 no 10 pp 773ndash792 2014
[14] M Hohenbichler and R Rackwitz ldquoImprovement of second-order reliability estimates by importance samplingrdquo Journal ofEngineering Mechanics vol 114 no 12 pp 2195ndash2199 1989
[15] J Zhang and X Du ldquoA second-order reliability method withfirst-order efficiencyrdquo Journal of Mechanical Design vol 132 no10 Article ID 101006 2010
[16] M C Alseyabi Structuring a probabilistic model for reliabilityevaluation of piping subject to corrosion-fatigue degradation[PhD thesis] University of Maryland College Park Md USA2009
[17] C Xie P Wang Z Wang and H Huang ldquoProbability ofcorrosion failure analysis using an adaptive sampling approachrdquoin Proceedings of the IEEE Conference on Prognostics and HealthManagement (PHM rsquo15) pp 1ndash8 Austin Tex USA June 2015
[18] E M Gutman Mechanochmistry of Solid Surfaces WorldScientific Publications Singapore 1994
[19] X G Jiang W Y Zhu and J M Xiao ldquoFractal analysis oforientation effect on KIC and KISCCrdquo Engineering FractureMechanics vol 51 no 5 pp 805ndash808 1995
Figure 4 Pit depth growth curve with different corrosion time
in (10) 119866(x 119905) gt 0 denotes the safe state whereas 119866(x 119905) lt 0
represents the corrosion failure state
4 Numerical Example
A case study is employed in this section to demonstratethe proposed physics of failure based corrosion model andreliability analysis approach The case study structure isidealized as an infinite plate with a pitting corrosion defectwhile the pit corrosion occurs on the surface of the structurematerial and the corrosive environment is assumed to beknown The structure material considered in this case studyis the aluminum alloy since it has been widely used inaerospace structures energy engineering and marine engi-neering applications The uncertainties involved in materialproperties as well as the corrosion model parameters areconsidered and modeled with Gaussian random variablesThe random and deterministic parameters used in this casestudy are listed in Tables 1 and 2 respectively [4 6] As shownin Table 1 seven different random variables are employed inthis case study as specified by the mean values and standarddeviations where the standard deviations for all randomvariables have been taken as 5 of the given mean valuesTo demonstrate the proposed corrosion model at the pitgrowth stage considering the coupled effects of corrosionenvironment and the mechanical stresses the corrosion timehas been assumed to begin right after the pit nucleation andwithout losing the generosity the random pit nucleation hasnot been considered in this example
Figure 4 provides 100 random realizations of the pitgrowth curve over time considering the random inputs asshown in Table 1 It is clear from the figure that the randominputs yield a large deviation for the pit growth curve overtime and a few curves rising obviously after the pit transformsto crack propagation Figure 5 shows that the 119866 value at thecorrosion time equals 200 days Note that the corrosion timedoes not include the pit nucleation time
The structure reliability of pitting corrosion damage is cal-culated using MCS and FORMTheMCS is used with a largenumber of samples (119873 = 100000) as the benchmark solution
Gva
lue d
istrib
utio
n
02
018
016
014
012
01
008
006
004
002
00 10 20 30 40 50 60 70 80 90 100
G value
Figure 5 119866 value distribution at 119905 = 200 days
Table 1 Random variables for case study
Random variables Unit Mean STD (meanlowast)1198860 m 198119890 minus 6 5101198680 cs 65119890 minus 6 510119870119868119862 MPam2 35 510120590119904 MPa 470 510Δ120590 MPa 90 510119862 mdash 28119890 minus 11 510119898 mdash 116 510
Table 2 Deterministic model parameters for the case study
to check the accuracy of the FORMmethodWith the randomvariables shown in Table 1 the corrosion reliability analysisis then carried out with different corrosion times using twodifferent approaches as mentioned earlier and the analysisresults are summarized in Tables 3 and 4 The comparisonof the reliability estimations and the absolute percentageerrors obtained using the FORM and SORM compared withthe MCS results are also shown in Figures 6 and 7 It is
6 International Journal of Corrosion
Table 3 The reliability results with different corrosion time
Corrosion time (days) ReliabilityMCS FORM FORM error () SORM SORM error ()
Figure 6 Reliability analysis results at different corrosion times
clear from the figures that the reliability estimation errorfor the FORM tends to increase with longer corrosion timemainly due to the linearization of the limit state the FORMused Besides the accuracy performance the efficiencies ofcorrosion reliability analysis using the FORM SORM andMCS are also compared as the results shown in Table 4 inwhich the number of sample points being evaluated for thelimit state function is employed as the accuracy measureAs shown in the table the FORM generally requires lesssample points to be evaluated to conduct reliability analysiscompared with the MCS method The results also show thatthe FORM is more efficient than SORM but its calculationaccuracy is lower than SORMMoreover due to the gradient-based searching process employed by the FORM to find theMPP the gradient information must be provided where inthis study the finite difference method has been employedfor the FORM to provide the required information As thegradient-basedmethod is used forMPP search the searchingprocess may not converge to the trueMPP efficiently in some
250 300 350 400 450 500
Corrosion time
Erro
r (
)
35
3
25
2
15
1
05
0
FORM errorSORM error
Figure 7 The computational error of FORM and SORM
scenarios as shown in the table for corrosion time of 250days and 300 days in which the total numbers of sampleevaluations for both have reached the upper limit of 800
5 Conclusion and Future Work
The pitting corrosion growth model considering the coupledeffects of stress and corrosion environment is proposed basedon mechanical theory And the time-dependent limit statefunction is discussed for stress corrosion crack failure basedon the corrosion growth model MCS and FORM are usedfor reliability assessment approachesThe example shows thatthe model is useful for structure reliability analysis but themodel parameters need verification by more experimentaldata Apply this mechanical model for structure corrosionfailure prognostics Integrate the Bayesian inference methodfor parameter estimation
International Journal of Corrosion 7
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the Foundation of ChinaAcademy Engineering Physics (CAEP) under Grant no2013B0203028 and the Technology Foundation Project underGrant no 2013ZK12
References
[1] R E Melchers ldquoThe effect of corrosion on the structuralreliability of steel offshore structuresrdquoCorrosion Science vol 47no 10 pp 2391ndash2410 2005
[2] F Caleyo J C Velazquez A Valor and J M Hallen ldquoMarkovchainmodelling of pitting corrosion in underground pipelinesrdquoCorrosion Science vol 51 no 9 pp 2197ndash2207 2009
[3] A Valor F Caleyo L Alfonso D Rivas and J M HallenldquoStochastic modeling of pitting corrosion a new model forinitiation and growth of multiple corrosion pitsrdquo CorrosionScience vol 49 no 2 pp 559ndash579 2007
[4] P Shi and SMahadevan ldquoDamage tolerance approach for prob-abilistic pitting corrosion fatigue life predictionrdquo EngineeringFracture Mechanics vol 68 no 13 pp 1493ndash1507 2001
[5] G Wu A probabilistic-mechanistic approach to modeling stresscorrosion cracking propagation in alloy 600 components withapplications [MS thesis] University of Maryland College ParkMd USA 2011
[6] D G Harlow and R PWei ldquoA probability model for the growthof corrosion pits in aluminum alloys induced by constituentparticlesrdquo Engineering Fracture Mechanics vol 59 no 3 pp305ndash325 1998
[7] A Haldar and S Mahadevan Probability Reliability and Statis-tical Methods in Engineering Design John Wiley amp Sons NewYork NY USA 2000
[8] H J Pradlwarter and G I Schueller ldquoLocal domain MonteCarlo simulationrdquo Structural Safety vol 32 no 5 pp 275ndash2802010
[9] G J Li Z Z Lu and C C Zhou ldquoDiscussion on local domainMonte Carlo simulation H J Pradlwarter and G I SchuellerStructural Safety 32 (2010) 275ndash280rdquo Structural Safety vol 40pp 78ndash80 2012
[10] M Di Sciuva and D Lomario ldquoA comparison between MonteCarlo and FORMs in calculating the reliability of a compositestructurerdquoComposite Structures vol 59 no 1 pp 155ndash162 2003
[11] Y-G Zhao and T Ono ldquoA general procedure for firstsecond-order reliabilitymethod (FORMSORM)rdquo Structural Safety vol21 no 2 pp 95ndash112 1999
[12] A Der Kiureghian ldquoThe geometry of random vibrations andsolutions by FORM and SORMrdquo Probabilistic EngineeringMechanics vol 15 no 1 pp 81ndash90 2000
[13] J Lim B Lee and I Lee ldquoSecond-order reliability method-based inverse reliability analysis using hessian update foraccurate and efficient reliability-based design optimizationrdquoInternational Journal for Numerical Methods in Engineering vol100 no 10 pp 773ndash792 2014
[14] M Hohenbichler and R Rackwitz ldquoImprovement of second-order reliability estimates by importance samplingrdquo Journal ofEngineering Mechanics vol 114 no 12 pp 2195ndash2199 1989
[15] J Zhang and X Du ldquoA second-order reliability method withfirst-order efficiencyrdquo Journal of Mechanical Design vol 132 no10 Article ID 101006 2010
[16] M C Alseyabi Structuring a probabilistic model for reliabilityevaluation of piping subject to corrosion-fatigue degradation[PhD thesis] University of Maryland College Park Md USA2009
[17] C Xie P Wang Z Wang and H Huang ldquoProbability ofcorrosion failure analysis using an adaptive sampling approachrdquoin Proceedings of the IEEE Conference on Prognostics and HealthManagement (PHM rsquo15) pp 1ndash8 Austin Tex USA June 2015
[18] E M Gutman Mechanochmistry of Solid Surfaces WorldScientific Publications Singapore 1994
[19] X G Jiang W Y Zhu and J M Xiao ldquoFractal analysis oforientation effect on KIC and KISCCrdquo Engineering FractureMechanics vol 51 no 5 pp 805ndash808 1995
Figure 6 Reliability analysis results at different corrosion times
clear from the figures that the reliability estimation errorfor the FORM tends to increase with longer corrosion timemainly due to the linearization of the limit state the FORMused Besides the accuracy performance the efficiencies ofcorrosion reliability analysis using the FORM SORM andMCS are also compared as the results shown in Table 4 inwhich the number of sample points being evaluated for thelimit state function is employed as the accuracy measureAs shown in the table the FORM generally requires lesssample points to be evaluated to conduct reliability analysiscompared with the MCS method The results also show thatthe FORM is more efficient than SORM but its calculationaccuracy is lower than SORMMoreover due to the gradient-based searching process employed by the FORM to find theMPP the gradient information must be provided where inthis study the finite difference method has been employedfor the FORM to provide the required information As thegradient-basedmethod is used forMPP search the searchingprocess may not converge to the trueMPP efficiently in some
250 300 350 400 450 500
Corrosion time
Erro
r (
)
35
3
25
2
15
1
05
0
FORM errorSORM error
Figure 7 The computational error of FORM and SORM
scenarios as shown in the table for corrosion time of 250days and 300 days in which the total numbers of sampleevaluations for both have reached the upper limit of 800
5 Conclusion and Future Work
The pitting corrosion growth model considering the coupledeffects of stress and corrosion environment is proposed basedon mechanical theory And the time-dependent limit statefunction is discussed for stress corrosion crack failure basedon the corrosion growth model MCS and FORM are usedfor reliability assessment approachesThe example shows thatthe model is useful for structure reliability analysis but themodel parameters need verification by more experimentaldata Apply this mechanical model for structure corrosionfailure prognostics Integrate the Bayesian inference methodfor parameter estimation
International Journal of Corrosion 7
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the Foundation of ChinaAcademy Engineering Physics (CAEP) under Grant no2013B0203028 and the Technology Foundation Project underGrant no 2013ZK12
References
[1] R E Melchers ldquoThe effect of corrosion on the structuralreliability of steel offshore structuresrdquoCorrosion Science vol 47no 10 pp 2391ndash2410 2005
[2] F Caleyo J C Velazquez A Valor and J M Hallen ldquoMarkovchainmodelling of pitting corrosion in underground pipelinesrdquoCorrosion Science vol 51 no 9 pp 2197ndash2207 2009
[3] A Valor F Caleyo L Alfonso D Rivas and J M HallenldquoStochastic modeling of pitting corrosion a new model forinitiation and growth of multiple corrosion pitsrdquo CorrosionScience vol 49 no 2 pp 559ndash579 2007
[4] P Shi and SMahadevan ldquoDamage tolerance approach for prob-abilistic pitting corrosion fatigue life predictionrdquo EngineeringFracture Mechanics vol 68 no 13 pp 1493ndash1507 2001
[5] G Wu A probabilistic-mechanistic approach to modeling stresscorrosion cracking propagation in alloy 600 components withapplications [MS thesis] University of Maryland College ParkMd USA 2011
[6] D G Harlow and R PWei ldquoA probability model for the growthof corrosion pits in aluminum alloys induced by constituentparticlesrdquo Engineering Fracture Mechanics vol 59 no 3 pp305ndash325 1998
[7] A Haldar and S Mahadevan Probability Reliability and Statis-tical Methods in Engineering Design John Wiley amp Sons NewYork NY USA 2000
[8] H J Pradlwarter and G I Schueller ldquoLocal domain MonteCarlo simulationrdquo Structural Safety vol 32 no 5 pp 275ndash2802010
[9] G J Li Z Z Lu and C C Zhou ldquoDiscussion on local domainMonte Carlo simulation H J Pradlwarter and G I SchuellerStructural Safety 32 (2010) 275ndash280rdquo Structural Safety vol 40pp 78ndash80 2012
[10] M Di Sciuva and D Lomario ldquoA comparison between MonteCarlo and FORMs in calculating the reliability of a compositestructurerdquoComposite Structures vol 59 no 1 pp 155ndash162 2003
[11] Y-G Zhao and T Ono ldquoA general procedure for firstsecond-order reliabilitymethod (FORMSORM)rdquo Structural Safety vol21 no 2 pp 95ndash112 1999
[12] A Der Kiureghian ldquoThe geometry of random vibrations andsolutions by FORM and SORMrdquo Probabilistic EngineeringMechanics vol 15 no 1 pp 81ndash90 2000
[13] J Lim B Lee and I Lee ldquoSecond-order reliability method-based inverse reliability analysis using hessian update foraccurate and efficient reliability-based design optimizationrdquoInternational Journal for Numerical Methods in Engineering vol100 no 10 pp 773ndash792 2014
[14] M Hohenbichler and R Rackwitz ldquoImprovement of second-order reliability estimates by importance samplingrdquo Journal ofEngineering Mechanics vol 114 no 12 pp 2195ndash2199 1989
[15] J Zhang and X Du ldquoA second-order reliability method withfirst-order efficiencyrdquo Journal of Mechanical Design vol 132 no10 Article ID 101006 2010
[16] M C Alseyabi Structuring a probabilistic model for reliabilityevaluation of piping subject to corrosion-fatigue degradation[PhD thesis] University of Maryland College Park Md USA2009
[17] C Xie P Wang Z Wang and H Huang ldquoProbability ofcorrosion failure analysis using an adaptive sampling approachrdquoin Proceedings of the IEEE Conference on Prognostics and HealthManagement (PHM rsquo15) pp 1ndash8 Austin Tex USA June 2015
[18] E M Gutman Mechanochmistry of Solid Surfaces WorldScientific Publications Singapore 1994
[19] X G Jiang W Y Zhu and J M Xiao ldquoFractal analysis oforientation effect on KIC and KISCCrdquo Engineering FractureMechanics vol 51 no 5 pp 805ndash808 1995
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the Foundation of ChinaAcademy Engineering Physics (CAEP) under Grant no2013B0203028 and the Technology Foundation Project underGrant no 2013ZK12
References
[1] R E Melchers ldquoThe effect of corrosion on the structuralreliability of steel offshore structuresrdquoCorrosion Science vol 47no 10 pp 2391ndash2410 2005
[2] F Caleyo J C Velazquez A Valor and J M Hallen ldquoMarkovchainmodelling of pitting corrosion in underground pipelinesrdquoCorrosion Science vol 51 no 9 pp 2197ndash2207 2009
[3] A Valor F Caleyo L Alfonso D Rivas and J M HallenldquoStochastic modeling of pitting corrosion a new model forinitiation and growth of multiple corrosion pitsrdquo CorrosionScience vol 49 no 2 pp 559ndash579 2007
[4] P Shi and SMahadevan ldquoDamage tolerance approach for prob-abilistic pitting corrosion fatigue life predictionrdquo EngineeringFracture Mechanics vol 68 no 13 pp 1493ndash1507 2001
[5] G Wu A probabilistic-mechanistic approach to modeling stresscorrosion cracking propagation in alloy 600 components withapplications [MS thesis] University of Maryland College ParkMd USA 2011
[6] D G Harlow and R PWei ldquoA probability model for the growthof corrosion pits in aluminum alloys induced by constituentparticlesrdquo Engineering Fracture Mechanics vol 59 no 3 pp305ndash325 1998
[7] A Haldar and S Mahadevan Probability Reliability and Statis-tical Methods in Engineering Design John Wiley amp Sons NewYork NY USA 2000
[8] H J Pradlwarter and G I Schueller ldquoLocal domain MonteCarlo simulationrdquo Structural Safety vol 32 no 5 pp 275ndash2802010
[9] G J Li Z Z Lu and C C Zhou ldquoDiscussion on local domainMonte Carlo simulation H J Pradlwarter and G I SchuellerStructural Safety 32 (2010) 275ndash280rdquo Structural Safety vol 40pp 78ndash80 2012
[10] M Di Sciuva and D Lomario ldquoA comparison between MonteCarlo and FORMs in calculating the reliability of a compositestructurerdquoComposite Structures vol 59 no 1 pp 155ndash162 2003
[11] Y-G Zhao and T Ono ldquoA general procedure for firstsecond-order reliabilitymethod (FORMSORM)rdquo Structural Safety vol21 no 2 pp 95ndash112 1999
[12] A Der Kiureghian ldquoThe geometry of random vibrations andsolutions by FORM and SORMrdquo Probabilistic EngineeringMechanics vol 15 no 1 pp 81ndash90 2000
[13] J Lim B Lee and I Lee ldquoSecond-order reliability method-based inverse reliability analysis using hessian update foraccurate and efficient reliability-based design optimizationrdquoInternational Journal for Numerical Methods in Engineering vol100 no 10 pp 773ndash792 2014
[14] M Hohenbichler and R Rackwitz ldquoImprovement of second-order reliability estimates by importance samplingrdquo Journal ofEngineering Mechanics vol 114 no 12 pp 2195ndash2199 1989
[15] J Zhang and X Du ldquoA second-order reliability method withfirst-order efficiencyrdquo Journal of Mechanical Design vol 132 no10 Article ID 101006 2010
[16] M C Alseyabi Structuring a probabilistic model for reliabilityevaluation of piping subject to corrosion-fatigue degradation[PhD thesis] University of Maryland College Park Md USA2009
[17] C Xie P Wang Z Wang and H Huang ldquoProbability ofcorrosion failure analysis using an adaptive sampling approachrdquoin Proceedings of the IEEE Conference on Prognostics and HealthManagement (PHM rsquo15) pp 1ndash8 Austin Tex USA June 2015
[18] E M Gutman Mechanochmistry of Solid Surfaces WorldScientific Publications Singapore 1994
[19] X G Jiang W Y Zhu and J M Xiao ldquoFractal analysis oforientation effect on KIC and KISCCrdquo Engineering FractureMechanics vol 51 no 5 pp 805ndash808 1995
![Probabilistic Upscaling of Material Failure Using Random ... · spectral stochastic finite element method to plasticity and failure [4]. To solve multiscale failure problems such](https://static.documents.pub/doc/80x56/5edc0184ad6a402d66667c35/probabilistic-upscaling-of-material-failure-using-random-spectral-stochastic.jpg)
