PROBABILISTIC MODEL FOR PITTING CORROSION AND FATIGUE LIFE

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Conférence Internationale sur le Soudage, le CND et l’Industrie des Métaux, IC-WNDT-MI’10 Oran, 27 - 28 novembre 2010

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PROBABILISTIC MODEL FOR PITTING CORROSION AND FATIGUE LIFE ESTIMATION FOR TURBINE BLADES

Y. ASSOUL^a,b , V.SIJACKI^a,N.BACHA^b, D.SEMMAR^b

a Faculty of Mechanical Engineering, University of Belgrade, Serbia

b Faculty of Engineering Sciences, Université Saâd Dahleb Blida, Algéria.

[email protected]

Abstract:

Service conditions of turbine blades are very complicated because of the mechanical loadings and severe environmental conditions of the aggressive and hot fluid stream. The pressure and temperature variations provoke mechanical load variation at the start up and the shut down and could aggravated the blades dynamics proprieties in the rotating system.

Different corrosion mechanisms could be developed in turbine blades, in according to the operational environment, but it has been shown by several updated studies that dominant failure mechanisms in turbine blades is fatigue crack progression, initiated at pitting corrosion defect. This failure mechanism is a complex electrochemical and mechanical one.

To prevent the occurrence of an undesirable failure, the presented research, aims to find a procedure to estimate the life and the damage evolution in studied structural parts, and to schedule optimal time to reaper with convenient repairs actions. By considering the very large dispersion of several parameters and the not exactly well define models yet, the one that is developed in this study is the probabilistic damage tolerance model. The model is taking in account, that most parameters are likely random, and trying to found the best probabilistic evaluation of failure occurring. Several works in engineering sciences are concerned with the degree of knowledge on statistical distributions of the model parameters, to limit the randomness. Some of them are mentioned in this work.

The model studied is the conceptual model with seven stages, the pitting initiation, the crack growth, transition from pit to crack growth, short crack growth, transition from the short crack growth to long crack growth, long crack growth and finally the fracture. This complex model comport several not well defined physical concepts, and some better known entities. This mechanistic model is embedded to a probability philosophy, to estimate the probability of failure according to well known classical deterministic models.

1. INTRODUCTION

A heat engine is one that converts heat energy into mechanical energy. The steam turbine as gas engines are classified as heat engine. Steam and gas turbines are used in industry for several critical purposes: to generate electricity by driving an electric generator and to lead equipments such as compressors, fans, and pumps. They are often potential critical components over the industry fields. For the industry efficiency, a high reliability is needed. The turbine blades are the elements which enables this energy conversion. These elements are subjected to various loading conditions, when transforming the fluid kinetic energy, of gazes or steam, in mechanical energy by rotating the shaft. Thus, the blades in the turbine are exposed to pressure variations, temperature variations, corrosive environment, and may beehive weakly to dynamical conditions

The blades are subjected to complex mechanisms of damage, progressing during the service time leading to the loss of component life. The damage mechanisms responsible of

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blades ruptures are investigated in several works [1-5], regarding to the macro- and microscopic aspects of fractures surfaces, a fatigue mechanism [1-5] with corrosion pits initiation are observed. In order to enhance service conditions and to prevent failures, the research is looking for the ways to understand and to estimate the state of the element and its damage evolution [6, 7]. Several works are aimed to reach this knowledge, the most recent trends and according to the latest standards all over the word [8], are the probabilistic approaches. They take in account the vagueness, the uncertainties and the randomness of the values and parameters used in the estimating models. This probabilistic frame work uses two principal philosophical concepts: Frequentist approach and the Bayesian one [9]. Frequentist approach as objective probabilistic approach is the most used, for its capability of being mathematically implemented and widely understood and easily computed [9-10].

Safe lives were estimated in Naim’s article [10] for of blade life with taking in account functional uncertainties during the evolution of the operations of flight engine. He stated that the limit state according to variable distribution on loads and temperature effects high pressure, he estimates the safe lives of the blades to fatigue and creep damage. In The work done in [12] the same procedure was used to define the safe lives to turbo reactor high pressure blades by a deterministic way.

Shi and Mahadevan [12] studied the complex probabilistic pitting corrosion and fatigue life estimate on trough the damage tolerance procedures. This model is qualified by complex by [13]. They used the seven stages probabilistic model for pitting fatigue, identified by Goswani and Hopper [14] using the basic Paris model for crack growth and the Faraday law for electrochemical interaction between the metal and its environment. This model was intensively studied and developed by Harlow and Wei [15 -18]. The mechanism involved in the study is the evolution of damage from the electrochemical corrosion pit initiation to brittle fracture. The damage progression goes from crack formation to short crack propagation and finally to the long crack propagation. The steps of the damage evolution and coexistence of two damage evolution at the same time that are the fatigue and the corrosion have their own variability. Both the chemical and the mechanical proprieties could not be gauge accurately in a deterministic sense. Probabilistic aspects combined with the mechanical degradation models shows reasonable estimates for damage process and life. In fatigue studies two aspects are considered generally, the safe lives which include the part of service where hypothetically no damage is initiated, the crack growth where there is a defect growing with the variation of loads. This work will present variable that could interact on the damage of the element and their variability trough the work found in the literature and along those to steps of the fatigue life of the component.

In this paper the probabilistic model addresses the randomness affecting the nucleation ad growth processes in the four stages of time to compute the life of the pitting corrosion fatigue. The Monte Carlo simulation is used for our probabilities estimation for its efficiency [9, 12, and 13].

2. CASE STUDIES OF BLADE FAILURE

In those last decades it has been demonstrated [1-4] that the turbine blades fail to fatigue mechanisms with pitting corrosion initiation in both steam turbines and gas turbines (figure 1-3). Added to fatigue, mechanism of damage in the blades is multiple, oxidation and creep in high pressure, corrosion in low pressure. But all case presents the pitting initiation as the originating point of the crack growth. The crack grows from a initiated pit with an random aspect to a crack level with a specified form and specified rate of growth in this level linear concepts of fracture mechanics are adopted (LEFM), when the crack reaches certain grater level cracking rate versus time: the crack changes from slow crack growth to a faster called

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the long crack growth. This is very dangerous state for any structure in service it leads very rapidly to static fracture.

3. LOAD DETERMINATION

To estimate the lives for the complex way of damage, it is necessary to find the load according to the thermo dynamical conditions, then the thermo mechanical varying stresses state, with the thermal and mechanical properties in function of temperatures. The thermal cycle is defined in this work by using classical formulations of thermodynamics rules; depend of the cycle of the engine. Adding centrifugal effect to pressure effect mechanical varying stresses are found by strength material rules. The temperature variations during normal operation induce a thermal deformation and thus a thermal varying stress in the blades material. Several studies take in account the dynamical proprieties that are the eigenvalues and the mode shapes.

Figure 1: THP turbine blade of Turbo propeller (S/N: ALLISON 501-D22) with 48000h of service without reapers [19]:

Figure 2: THP turbine blade of Turpropller (S/N: ALLISON 501-D22) with 39000 h of service without reapers [19]

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Figure 3 - THP creep crack defect on blade with 3 years of service without reapers [5].

Figure 4- Microscopically aspect of crack initiated from a corrosion pit [20]

Figure 5 – Procedure of the stress calculation in the turbine blades.

4. MODEL OF PITTING CORROSION FATIGUE LIFE

The total fatigue life may be represented by the summation of the four time phases in accordance with the model of Shi [12]

1. Time to defect initiation.

Uncertain Inputs:

 Thermal dilatation λ (T°)

 Young modulus E(T°) .

) (T

 Calculation of mechanical stress σmec( Xi)

Deduce from thermal cycle thermal stresses σ thmax (X_i) σ_thmin(X_i).

Total thermo mechanical stress cycle (TMC) with its alternatives and mean values Static equilibrium equation

Uncertain Inputs:

 Fluid’s Temperatures and pressure.

 Cooling temperature.

 Random variables

Corrosion pit

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2. The pit growth

3. The transition from pit crack to short crack growth 4. The slow crack growth

5. The second transition that represents the variation of crack growth from short to long crack.

6. The long crack growth 7. Fracture.

Figure 6- The seven steps of Mahadevan’s damage tolerant model

Initiation model:

In order to find the time to initiation is a problem of many scientific developments [20,21] the raison and the mechanisms of the pit initiation are still investigated to limit the randomness of this mechanism The pit initiation is considered actually as a complete random process but several studies tries to found the reasons which lead to the pit development. [1, 20] suppose that initiation of pit is according to presence of inclusions. Martin [22] concludes in ones of the most recent researches, thank to atomic force microscopy that the pit initiation is more likely (70%) to happen in the nano-plastically deformed locations.

In this study we assume that the life to initiation is the sale life for a component calculated by Masson Coffin’s law (equation 1). Those blades could suffer even creep and corrosion. Creep is considered by Larson Miller formulation, and corrosion is took in account by dividing by 3 [7], the life found.

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To consider the interaction between fatigue and creep damages, several rules of damage accumulation could be used. This accumulation can be considered as linear or non

c f f b f

f N N

E (2 ) (2 )

2  



 

Pit inition safe life

Pit initiation

Pit growth

Slow crack growth

Long crack growth

Fractu

re Static rupture Fast crack growth Slow crack growth Fraday’s Law

 Pit curent random.

 Pit geometry random .

 Geometry variation random

 Manson Coffin formulation

Pit growth

Change in the rate in of crack growth

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linear. For this work linear Miner rule is used. The damage is total when it reaches the unity.

Then the safe number of cycle, of start up and shut down, are found.

Pit growth stage

This stage concerns pit growth by constituent particles and involves electrochemical process.

The parameters of the Faraday’s law formulation are highly random especially the pit current, and the geometrical pit shape. Turnbull and all [21] are performing published researches to understand how the passivity breaks downs creating interactively the pit growing.

Rajansankar [13] studied trough empirical investigations and probabilistic approaches the rate and the shape of the pit growth. In this model, the pit is assumed to grow at a constant volumetric rate according to Faraday’s law. This law gives tin, time to pit growth. This time is function of Molecular weight, valence of the material, Faraday constant, activation energy, universal gas constant, absolute temperature and the pitting current. The pit is assumed to be a half prolate spheroid in infinite plate.



 



 

 RT

H nF

MI dt

dv po

 exp (1)

where M is molecular mass of the materiel, n is the valence, F is the Faraday constant, ϕ is the density, ΔH is the activation energy, R is the perfect gas constant, T absolute temperature.

The volume of the pit is prolate spheroid:

(2) Where a and c are half lengths of the major and minor axes respectively

Determination of the first transition stage

This third stage considers the transition from the pit growth to short fatigue crack growth where the effect of stress concentration factor is taken in account the transition is assumed to occur when:

Kpit 



K



_crack

(3) Applying Faraday law to define the time of propagation and resolving the equation of transition to define the critical pit size gives the transition.

Short crack growth:

The fourth stage involves chemical and micro structural factors and their interactions a Paris law with specific parameter is used in this stage to define the tscg, which is function of the crack transition to the classical long crack growth. Several studies [12,18] demonstrated the important effect of this step that the short crack rate exceeds the long crack rate, but the stresses are below those in long crack growth.

(4) Determination of transition 2

In analytical approach the transition size from short crack growth to long crack growth is equating the rates of growth of the short crack and long crack. This is treated, in this work as a deterministic way but it is a random variable.

Long crack growth: The classical Paris law is used in this stage, dependant on material constant for long crack growth and the stress intensity factor. The stress concentration factor is took a crack in infinite plate in regard. The time tlcg is calculated to the final critical size for long crack growth. This value defines the limit to consider for repair, took 1mm.

2 2

V 3ca

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(5) Fracture: It considers a crack size which is unacceptable or in need having repair, lees then the crack size when real fracture occurs.

The time is evaluated for fatigue corrosion life as:

(6) Where Tf is the time of service, tin is the initiation time, tpg is the pit growth, tscg and the t_lcgis the long crack growth.

5. RANDOM VARIABLE IN THIS MODEL

The model above includes several uncertain data and some up to now yet totally random variables. The model consider with specific application, specific variables according the found and investigated data. This considers an empirical research and a statistical analysis.

And the actual main hurdle is defining the statistical distribution information about the random variables. The model along the seven stage presents many random variables, the stress range is one of them, and it evaluation have an amount of random variables. In the damage tolerant model, the most significant random variables are:

1. The pit initiation step: Time to initiation, and it is taken as lognormal distribution with the determinist value as the safe life calculated in procedure above mentioned.

2. The pit growth step : The random variables involves in this step are the pit current constant, the number of clustered particles on which depends the pit current, the pit shape geometry and seems to obey to lognormal distribution, material microstructures [20-23] .

3. The transition step: The transition step includes the variables that affect the pit growth and the short crack growth.

4. The short crack growth: The random variables that affect the short growth are the stresses range, the stress concentration factors, rate of growth, the distribution adopted are generally took as lognormal, and some studies investigated the Weibull distribution.

5. The transition step from short crack growth to long crack growth. This step is depending on the two adjacent stages.

6. The long crack growth: This step depends on stresses range, stress concentration factors and propagation rate.

7. Fracture: This step depends on the residual resistance and the stress range.

6. METHODS FOR PROBABILISTIC ANALYSIS

The limit state or this analysis is performed to compute the cumulative distribution function of the corrosion fatigue life. The random variables used in this model are the time of the four steps considering the variables in the time calculations as deterministic values.

The distribution of the time is taken as the lognormal distribution with a mean value calculated trough deterministic calculations.

The limit state is defined by:

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Where Xi, the random variables are tin, tpg, tscg, tlcg and tf. The tf represent the service time. The limit state evaluated is.

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It is recognized that the first reliability order method and second order reliability method, deals with continuous and normal distributions, it is not the case generally. Often the probabilistic problem are non linear and not continuous.

For accuracy, robustness, and validation purposes the Monte Carlo simulation may be used [12, 13] .

The Monte-Carlo simulation is used to estimate the probability of rupture for a great number of samples , equation 10. The required sample size for a desired percentage of error ε% is found using the formula 11.

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Where P^Tf is the true probability of failure and N the sample required.

7. APPLICATION

The application done is the investigation on the probability of failure, of the Allison turbo reactor blade, made of super alloy figure 3.It presented failure before the first constructor scheduled remount. The lives found for this application calculated in [5,11] are given in table 1. This application attempts to simulate the probability of failure of this studied blade for a set of service times. The limit state considered is G (Xi) where the random variables are the times. Two distributions were evaluated to perform the estimation of failure probability, the normal distribution figure 7 (a) and the lognormal distribution in (b).

The very high probabilities of failures found even at small time service demonstrated the design problem on those types of blades. Table 2; Shows the efficiency of the Monte Carlo simulation, by increasing the number of tries, and stabilize to .01% at million tries.

Table 1 Data for the time the four stages and the deterministic

Stage Pit

initiation

Pit growth Short crack

growth

Long crack growth

Cycle Number 10666 3848 35479 35479

Defect length µm 1,00E-03 1 1000

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(a) (b)

Figure 7: Failure probability for normal distribution (a) and lognormal distribution (b) on service time.

Figure 8: Distribution fitting for three different cases

Table 2 Errors in probability estimations.

µ for lognormal distribution 9,6 9,7 9,8 9,9 10

% Error for 10e3 tries 1,974 1,755 1,5748 1,3416 1,0198

% Error for 10e6 tries 0,0019 0,0017 0,0015 0,0011 0,0004

8. CONCLUSION

This study evaluated the most important variables and parameters which contribute to the progress of fatigue damage in a blade. It adopted a probabilistic frame work around the four time of damage progress by adopting a limit state. It has been noted that the two first steps are the ones where the damages initiating, are very important and they affect greatly the reliability of the element. The pit initiation step is yet not well known and is totally random.

The pit growth is also not well recognized yet. In the third step is the step that take the maximum time, this time could support the maintenance actions, by scheduling optimal time to reaper with convenient repairs actions. A Matlab7 probabilistic frame work were realised with a Monte Carlo simulation.

The high pressure Allison’s gas turbine blade application studied presented a serious conception problem and must be redesigned by the constructor.

References:

1- V.Sijacki.Zeravcic, G.Bakic, M.Dukic, B.Andjelkic, D.Milankovic: “Malfunctioning During service life” FromFracture Mechanics to structural integrity assessment.

Monographie 8^th Intfracture mechanics School , DIVK TMF Belgrade, Serbia 2004.

2- V.Sijacki and all “ Studija O Stanju i Upotrebljivosti Lopatica Turbine Niskog Pritiska Tipa 290N i 290 R Termoelaktrane Nikola Tesla B” Masinski Faculte,Beograd Universit, 1994, Beograd.

3- V.Sijacki and all “ Studija O Ispitivanju Uzroka Loma Lopatica Drugog Fluxsa Turbine Turbonaponjne Pumpe na Bloku B” Masinski Faculte,Beograd Universit, 1995, Beograd. Izvastaj 12 – 16 12.04./1995

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4- V.Sijacki and all “ Studija O Ispitivanju Uzroka Loma Lopatica Drugog Fluxsa Turbine Turbonaponjne Pumpe na Bloku B2 TENT B” Masinski Faculte,Beograd Universit, 1995, Beograd. 1995.

5- Assoul.Y, M.Kouchrane and all “ Damage tolerant model for calculation the life of HP gas turbine to fatigue and corrosion damage”, JIP2007

6- R.I. Stephens, A. Fatemi, Metal Fatigue in Engineering, John Willey & Son New York, 2001.

7- Jaap.Schijve, Fatigue of structures and materials, KluwerAcademic PublishersNetherlands, 2003.

8- R.Patel, « High Temperature Component Analysis Overview of essment.and Disignprocedures”, ECCC recomadations Volume 9 Part II [Issue 1] Volume 9 ,2005 9- B.M.Ayyub, R.HMcCuen “Probability,Statistics and Relaibility of Engineers and

Scientist” Second Edition CHAPMAN&HALL/CRC NewYork 2003.

10- M. Naeem, R. Singh, D. Probert, International Journal Of Fatigue 22 2000, 147 -160 . 11- S.Benbelaid , Y.Assoul « Modèle pour la prediction de la vie pour la fatigue

thermomécanique et le fluage », PFE département d’aéronautique, Blida, 2004.

12- P. Shi, S.Mahadevan, “Damage tolerance approach for probabilistic pitting corrosion fatigue life prediction” , Engineering Fracture Mechanics, 68 2001 , 1493-1507.

13- J.Rajansakar,N.NageshIYer, “ A probability based model for growth of corrosion pit in aluminium alloys” Engineering Fracture mechanics 73 (2006) 553-570.

Elsevier.com.

14- T.K. Goswani, D.W. Hopner, “Pitting Corrosion Fatigue on Structural Materials", Chang CI, Sun CT,Structural intergrity, in aging aircraft, New York ASME, pp 45 1995.

15- D.G.Harlow, R.P.Wei , “Probability approach for corrosion and corrosion fatigue life”, AIAA J ,32(10) 2073-9.

16- D.G.Harlow, R.P.Wei, “A probabilistic model for growth of corrosion pits in Aluminium alloys induced by constituent particles”, Engineering Fracture Mechanics 1998, 59(3),305-25.

17- D.G.Harlow, “Probability Versus Statistical Modeling Examples From Fatigue Life Prediction”, International Journal of Reliability, Quality and Safety Engineering Vol. 12, No.6 (2005) 535–550, World Scientific Publishing Company.

18- R.P.Wei, D.G.Harlow, “Mechanistically based probability modeling, life prediction and reliability assessment”, Modeling Simulation Material Sciences Eng. 13 (2005) R33–R51.

19- B.Weber,H.J.Robert,P.Paul, “Industrial application of gas turbines comity” , Paper No: 05-IAGT-2.2, Canada - 12-14 October, 2005.

20- K.M. Perkins, M.R. Bache, “Corrosion fatigue of a 12%Cr low pressure turbine blade steel in simulated service environments” International Journal of Fatigue 27 (2005) 1499–1508, www.elsevier.com/locate/ijfatigue

21- A.Turnbull,L.N.McCartney,S.Zhou, “ A model To predict the evolution of pitting corrosion and the pit-ti-crack transition incorporating statistically distributed input parameters”, Corrsion Science 48(2006) 2084-2105. www.elsevier.com/locate/corsci.

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22- F.A.Martin,C.Bataillon,J.Cousty, “ In Situ AFM detection on pit onset localization on a 304L Stainless steel. Corrosion Science 50(2008) 84.

www.elsevier.com/locate/corsci.

23- Y.Zhang,D.Macdonald, M.Urquedi.MacDonald,G.R.Englhart,R.B.Dooley “Passivity breakdowns On AISI Type 405 stainless steel in chloride-containing borte buffer solution”Corrosion science 48(2006),3812-3823. www.elsevier.com/locate/corsci 24- A.P. Tschiptschin a, C.R.F. Azevedo,“Failure analysis of turbo-blower blades”,

Engineering Failure Analysis 12 (2005) 49–59, www.elsevier.com/locate/engfailanal 25- A.Bagaviev, A.Ulbrich , “Life assessment of turbine components Based on

deterministic and probabilistic procedures”, International Journal of Pressure Vessels and Piping 81 (2004) 55–859.

26- A.Grandt, “Fundamentam Of Structural Integrity Damage tolerant design and Nondestructive Evaluation”,

27- V.S.Sastri, Edwaed Ghali, Mimoun Elboujdaini, “Corrosion Prevention and Protections”, Jhon Wiley and SonsLtd, England, 2007.

28- Y.Kondo, “ Prediction of Fatigue Crack Initiation Life Based on Pit Growth”, Corrosion , 1998 , 45(1),7 11.