Bergische Universität WuppertalFachbereich Mathematik und NaturwissenschaftenInstitute of Mathematical Modelling, Analysis andComputational Mathematics (IMACM)Preprint BUW-IMACM 17/01L. Mäde, S. Schmitz, G. Rollmann, H. Gottschalk and T. BeckPROBABILISTIC LCF RISKEVALUATION OF A TURBINE VANE BYCOMBINED SIZE EFFECT AND NOTCHSUPPORT MODELINGFebruary 19, 2017

Preprint – Preprint – Preprint – Preprint – Preprint – PreprintPROBABILISTIC LCF RISK EVALUATION OF A TURBINE VANE BY COMBINEDSIZE EFFECT AND NOTCH SUPPORT MODELINGLucas MädeGas Turbine Department of Materials and TechnologySiemens AGBerlin, 10553GermanyEmail: [email protected] GottschalkFaculty of Mathematics and Natural ScienceBergische Universität WuppertalWuppertal, Nordrhein-Westfalen, 42119GermanyEmail: [email protected] SchmitzGas Turbine Department of Materials and TechnologySiemens AGBerlin, 10553GermanyEmail: [email protected] BeckInstitute of Materials Science and EngineeringTechnische Universität KaiserslauternKaiserslautern, Rheinland-Pfalz, 67653GermanyEmail: [email protected] RollmannGas Turbine Department of Materials and TechnologySiemens AGMülheim an der Ruhr, Nordrhein-Westfalen, 45473GermanyEmail: [email protected] probabilistic risk assessment for low cycle fatigue (LCF)based on the so-called size effect has been applied on gas-turbinedesign in recent years. In contrast, notch support modeling forLCF which intends to consider the change in stress below the surface of critical LCF regions is known and applied for decades.Turbomachinery components often show sharp stress gradientsand very localized critical regions for LCF crack initiations sothat a life prediction should also consider notch and size effects.The basic concept of a combined probabilistic model that includes both, size effect and notch support, is presented. In manycases it can improve LCF life predictions significantly, in particular compared to E-N curve predictions of standard specimenswhere no notch support and size effect is considered. Here, anapplication of such a combined model is shown for a turbinevane.NOMENCLATURELCF Low cycle fatigueE-N curve Curve of strain amplitude vs. crack initiation lifeCMB Coffin-Manson-Basquin (model)σ 0f Fatigue strength coefficientb Fatigue strength exponentε 0f Fatigue ductility coefficientc Fatigue ductility exponentE Cyclic Young’s modulusNi Load cycles until crack initiationm Weibull shape parameterη Weibull scale parameterχ Normalized gradient of equivalent elastic stressχT Normalized gradient of temperatureσe Equivalent elastic stressNSP Notch support parametersΩ 3D component domain1

Preprint – Preprint – Preprint – Preprint – Preprint – Preprint Ω 2D domain surfaceMLE Maximum likelihood estimationFEA Finite Element AnalysisTMF Thermo mechanical fatigueTBC Thermal barrier coatingBC Bond coatINTRODUCTIONIt is well known that the number of cycles until initiation ofa fatigue crack in Ni-based superalloys is subjected to considerable statistical scatter, see e.g. [1] for a discussion of staticalscatter in fatigue experiments. The design of gas turbines andtheir safe and reliable operation therefore requires a methodology that is capable to accurately quantify risk levels for low cyclefatigue (LCF) crack initiation, crack growth and ultimate failure. Traditional deterministic design approaches however predict absolute safety below a specified ’safe’ number of servicecycles and failure just above it. Such life prediction models giveclear answers on one hand but on the other hand they do notprovide an adequate description of the real world. At the sametime, such a binary description narrows the business options ofgas turbine power plant operators and service providers, wherein some instances taking a non safety relevant economic risk inexchange for an even bigger economic opportunity might be a rational way of decision making. Of course, such decisions need tobe underpinned by a proper risk assessment. Responding to thisneed, in [2, 3], a probabilistic model, based on a local Weibullhazard density approach is being used for LCF crack initiationprediction. It inherently considers the statistical size effect andthe inhomogeneity of surface stress and has been applied to different gas turbine components, such as blades and compressordiscs [4]. Further probabilistic models for LCF have been proposed by other authors, e.g. [5–7]. Okeyoyin et al. used a probabilistic framework for computation of fatigue notch factors forhigh cycle fatigue (HCF) based on a random distribution of failure inducing defects in the volume of material [8]. While Hertel et al. [6] take into account notch support factors, this is notthe case for the other papers. The modes [6, 7] however arebased on the Paris-Erdogan law of crack growth and an initialflaw size distribution in the sense of strength-probability-time(SPT)-diagrams in ceramics. Also, the model [6] has been set upand validated for several steels and not for superalloys. So thesemodels considerably differ from the local probabilistic model forLCF proposed and validated in [2, 3]. However, notch supportfactors were not included beyond the statistical size effect. Thisgap is here closed for the first time where the notch support effect arising from stress gradients in the volume combined withthe size effect is included in the probabilistic framework and experimental validation work is provided. Here the crack initiationprediction prediction of a turbine vane is presented as a use caseof the probabilistic model with notch support implementation.Section 1 provides a brief repetition of the key steps to understanding the local probabilistic model with a focus at the notchsupport mechanism in Subsection 1.2. The following Subsection1.3 outlines methodology and results of the notch support modelvalidation for 850 C. A crack initiation prediction of a turbinevane is discussed in Section 2, emphasizing the differences of theresults when neglecting (Subsections 2.1) and enabling (Subsection 2.2) notch support in the model.1LOW CYCLE FATIGUEFailure due to strain driven LCF surface crack initiationplays an important role for highly loaded engineering parts, suchas turbine components made of Ni-based super alloys. Sincethese alloys feature high yield strength and relatively low ductility, the initiation of surface cracks of a critical length couldlead to rapid failure of the component under cyclic load.The empirical relationship between maximum load (strainamplitude) at the component and the number of load cycles tofailure Ni , the E-N curve (or Woehler-curve), is the basis for thelocal probabilistic model discussed in this article. Several deterministic models for Woehler-curves exist. A well known is theCoffin-Manson-Basquin (CMB) equation describing Woehlercurves in strain controlled fatigue [1, 9–11],εa σ 0fE(2Ni )b ε 0f (2Ni )c .(1)The parameters σ f , b (fatigue strength coefficient/exponent) andε f , c (fatigue ductility coefficient/exponent) are material parameters obtained by fitting test data while E is the cyclic Young’smodulus. The first summand at the r.h.s. of Eqn. (1) describes EN curves dominated by elastic strain, while the second summandat the r.h.s accounts for dominating plastic strain. In the conventional safe life approach of crack initiation prediction the CMBmodel is used in combination with safety factors to account fornatural scatter, size effects and additional effects influencing theLCF mechanism. The probabilistic approach to crack initiationprediction introduced in [2] is explicitly accounting for the firsttwo of these influences. For more reliable and realistic predictions, the notch support effect is additionally implemented in thismodel.1.1Local Probabilistic Model for LCFThe motivation of using a probabilistic model for crack initiation prediction is the need to account for the scatter which isgenerally observed in fatigue events and the statistical size effect.The latter accounts for the different probabilities of crack initiation in bodies of equal shape but different size when subjected toequal stress.

Preprint – Preprint – Preprint – Preprint – Preprint – PreprintIn a probabilistic framework for crack initiation prediction,the number of cycles to crack initiation Ni is regarded as a random variable whose statistics is characterized by the cumulative distribution function FNi (n) and probability density functionfNi (n). Here the local approach from [3] is taken up again. It assumes all members of {(Ni ) j } j 1.k for the subsets of an arbitrarypartition {A j } j 1.k of the component surface Ω, to be independent from each other because initial cracks only cover the rangeof few grains. The concept of the hazard rate h(n) was chosento quantify the risk for crack initiation in every surface patch A jindividually as its property of additivity for stochastically independent variables allows partitioning of the risk analysis of thebody’s surface. That is a requirement of the local approach forcrack initiation prediction at the entire surface of an arbitrarilyshaped body. The hazard rate is calculated byfNi (n)P(n Ni n n Ni n) . n 0 n1 FNi (n)h(n) lim(2)Thus h(n) · n is the probability of crack initiation within the cycle n n where n is the cycle increment [13]. The hazard ratefor the first crack initiation at the entire surface Ω is the sumof those for all A j , h kj 1 h j since one assumes the numberof load cycles until crack initiation (Ni ) j to be stochastically independent in every A j . The second assumption states that crackinitiation risk is a functional of only local strain and temperaturefields εa (x) and T (x) since no long range order phenomena occurin the continuum mechanics model for polycrystalline materials.That is why h(n) is the surface integral of ρ(n; εa , T ) the in thelimit of an infinitesimal fine partition {Ai }i with ρ(n; εa , T ) being a local functional of strain and temperature field at the surface Ω.h(n) Z Ωρ(n; εa (x), T (x)) dA(3)The cumulative hazard function H(n) is defined asH(n) Z n0h(t) dt.(4)where m is the Weibull shape parameter and η the Weibull scaleparameter. While η determines the position of the distributionin the domain, m influences the shape of the distribution (broador peaked) and thus the expected scatter of events. The wholeconcept of the local approach leads to an integration formula forη which adds up the risk for crack initiation along the examined surface. Using equations (5) for the Weibull distributionone findsZmρ(n; εa (x), T (x)) dA ·η ΩfNi (n),1 FNi (n)FNi (n) 1 e H(n) .(5)In [2] the number of cycles to crack initiation Ni is assumed tobe Weibull distributed with the cumulative distribution function m nFNi (n) 1 exp ,η(6)(7)The local approach effectively states that all distributions for(Ni ) j scale individually dependent on the load state. Consequently the integrand in Eqn. (3) ismρ(n; εa , T ) Nidet (εa , T ) nNidet (εa , T ) m 1.(8)There Nidet (εa (x), T (x)) is the deterministic number of life cyclesat every point x of the body’s surface Ω.The integrand Eqn. (8) allows independent integrations oversurface and time, which are necessary to receive the cumulativehazard function H(n) according to equations (3) and (4). Thecumulative hazard function is then found to have the formulaH(n) nm ·Z1m dA, Ω Nidet(9) mwhere the remaining integrand 1/Nidet (x) is defined as hazarddensity.In the following Section 2 plots of the hazard density fieldat the geometry are shown in order to visualize the risk at thecomponents surface. This is preferable compared to plots of Nidetas one can add up values of ρ(n; εa (x), T (x)) from arbitrary surface spots to receive the overall hazard density for the combinedsurface. This is a convenient way to directly assess the criticalityof different surface subsets. From equations (5), (6) and (9) onefurther derives a formula for the Weibull scale parameter:H(n) and h(n) fulfill the relationsh(n) m 1n.ηη Z1m dA Ω Nidet! 1/m.(10)The deterministic life Nidet (x) of one surface patch is determinedby numerically solvingεa (x) σ 0fE b c2Nidet (x) ε 0f 2Nidet (x) .(11)

Preprint – Preprint – Preprint – Preprint – Preprint – PreprintThe computational realization of this model is a tool that usesFinite Element Models (FEA) as input and computes Nidet at allintegration points. The parameters σ 0f , ε 0f , b, c for Eqn. (11)are now valid for one small surface patch and are thus independent of the investigated component geometry, given that the meshof the FEA input is sufficiently small. Hence one can also interpret them as material parameters. They are simultaneouslyderived with m from maximum likelihood fits of specimen testdata. Shape m and scale η entirely define the distribution function in Eqn. (6) from which the 50 %-quantile is regarded as theprobabilistic average life until crack initiation. By integrating1/Nimdet (x) over the entire surface in Eqn. (10), the presented local probabilistic model inherently incorporates the statistical sizeeffect and accounts for material scatter through the shape parameter m.Notch Support EffectComponents with notches or other inhomogeneous geometry features exhibit domains of concentrated stress at the respective location when subjected to a load. Geometry inducedstress concentration leads to inhomogeneous stress fields in theaffected domain while the highest values are usually occurringat the surface. Whereas domains near the surface quickly reachyield strength and are therefore plastically strained, domains further inside the body still support the structure since they experience smaller stresses and therefore impede failure. That is whythe crack initiation life of parts exhibiting spatially inhomogeneous stress fields under cyclic load is higher than predicted bythe CMB equation for the maximum occurring strain εa . Siebel etal. have approached a quantification of this phenomenon, knownas notch support effect, with a support number nNSdomain Ω so that the gradient is well defined at the surface Ω.Since nχ can be seen as the strain equivalent to Eqn. (12), itis combined with the CMB equation to Eqn. (16) instead ofEqn. (11) for Nidet (x) computation. This shifts the E-N curveto higher life because nχ 1. Then, Weibull scale η, computedby integrating 1/Nimdet (x) over the surface, and shape m define adistribution FNi (n) for load cycles n until crack initiation that accounts simultaneously for size effect and notch support effect.Thus, the probabilistic model for LCF with combined size effectand notch support is given by the following Weibull approach: m n,FNi (n) 1 exp η! 1/mZ1,η m dA Ω Nidet1.2nNS σnotched/obsobserved fatigue strength .expected fatigue strength σhomogeneous(12)They considered nNS to be directly proportional to the stress gradient in the loaded component [12]. This is a well justified approach since quickly abating loads (high concentration) implylarger low stress domains to support the structure. Hence, a stressgradient based support factor nχ is also used to consider the notchsupport effect in the use case described in this paper, where nχ isdependent on χ(x) and material dependent notch support parameters1 A and k. They are simultaneously derived with the CMBparameters from LCF test data as described in Section 1.3.χ(x) 1 σe (x) with x Ωσe (x)(13)is the derivative of the elastic von Mises stress σe normalizedwith its surface value. Note that σe is a scalar field in the whole1 SeeFig. 10.36 on page 378 in [11] for relationship between nχ and χ(14)(15)σ 0f b cεa (x) 2Nidet (x) ε 0f 2Nidet (x) .nχ (x)E(16)Note that besides of the notch support effect, the statistical sizeeffect is also playing an important role in the LCF life of irregularly shaped components because critical stresses usually occurin confined domains which are small compared to the entire component.1.3Calibration of Notch Support Parameters andModel ValidationAs mentioned in the previous subsection, the CMB-, notchsupport- and shape parameters for the combined local probabilistic model are estimated from material test data. The principleof the model calibration and validation procedure is shortly described here.In order to calibrate the notch support model, LCF-test dataof a specimen with homogeneous, cylindrical gauge section (redsquares in Fig. 1 [2]) is simultaneously fitted with test data of aspecimen with a circumferential notch of radius 2.4 mm (greencircles in Fig. 1 [14]). Both sample types are made of polycrystalline cast RENE 80 and tested in strain controlled LCF at850 C. The local probabilistic model extended with notch support in Subsection 1.2 requires the parameter setθ σ f , b, ε f , c, A, k, m T(17)which is determined by maximum likelihood estimation (MLE).Apart from parameters A and k in θ , the fitting procedure follows [2] from this point on. Determining the minimum of thenegative log-likelihood function is achieved with Nelder-Meadoptimization. The parameter estimate θ̂ is then used to compute the median E-N curve of another notched specimen with

Preprint – Preprint – Preprint – Preprint – Preprint – Preprintnotch radius 0.6 mm (solid blue line). All test data points andE-N curves are shown in Fig. 1.2PROBABILISTIC LIFE PREDICTION FOR A TURBINE VANE UNDER THERMOMECHANICAL LOADSIn this section the probabilistic model is applied to a turbinevane made of polycrystalline cast RENE 80. All probabilisticanalyses of its LCF crack initiation life are based on FEA simulating a thermomechanical load as in the operating state. Themodel consists of tetrahedral elements and heat transfer analysis is performed assuming an undamaged system of thermal barrier coating (TBC) and bond coat (BC). Note that the present examination neglects the complex mechanical interaction betweencoating and substrate material. The structural analysis deliversthe elastic strain tensor and temperature field at all nodes. Fromstress tensor data the elastic-plastic strain field ε(x) (Fig. 2 (a))is calculated which is then further utilized for computing the deterministic life field Nidet (x) according to Eqn. (11) together withthe temperature data.highFIGURE 1. E-N CURVES AND TEST DATA OF HOMOGENOUSAND NOTHCED SPECIMEN: FIT AND PREDICTIONThe error bars in Fig. 1 indicate the 92.5 %-confidence interval of the respective median life. Uncertainties in parameter estimation are computed by parametric bootstrapping. 2000bootstrap sample2 sets are generated from the original crack initiation life distributions and fitted with the mentioned MLE procedure. 200 of the resulting median E-N curves are plotted ingrey for each specimen geometry. The confidence intervals hererepresent the 92.5 % percentile range of the uncertainty in curveprediction. Hence they do not cover the observed residual scatterwhich is e.g. in this case of superalloys also dependent on thegrain orientation and the location of the initial cracks comparedto the probing tips of the extensometer in the LCF experiment.The calibration curves (dashed) show that the observed test datais well described by the current notch support model. Additionally, the solid blue E-N curve, a pure prediction, shows a goodvalidation for another notch specimen data set (blue triangles) notused for calibration. These findings verify the appropriability ofthe χ-approach for the available test data. This motivates the application of the notch support model to a turbine vane discussedin Subsection 2.2.2 The minimum number of bootstrap samples according to [13] is chosen forfeasible computation times.low(a) von Mises stress(b) TemperatureFIGURE 2. NODAL FEA-VALUES OF TEMPERATURE ANDVON MISES STRESS AT PRESSURE SIDE OF TURBINE VANENote that instead of nodal values all results of the FEA postprocessor, explicitly the surface integration in Eq. (10), are evaluated and plotted at the coordinates of integration points of thefinite elements3 at the domain surface Ω by applying the corresponding shape functions for interpolation.2.1Probabilistic Lifing Without Notch SupportIn this subsection the same crack initiation prediction modelas presented in [3] is applied. Having calculated Nidet (x), one canproject the hazard density (1/Nidet (x))m onto the vane as shownin Fig. 3.It indicates areas of increased risk at the transitions fromairfoil to inner shroud and outer shroud which occur only at thetrailing edge. This strongly correlates with the locations of highest stress observed in Fig. 2 (a). However, since the material parameters are also temperature dependent, the vane temperature3 Quadratures of higher order than two are chosen to rule out numerical nonlinearities, compare [3]

Preprint – Preprint – Preprint – Preprint – Preprint – PreprinthighlowhighFIGURE 3.HAZARD DENSITY PLOT OF VANEfield (Fig. 2 (b)) influences the probability of crack initiation aswell.From the hazard density and Weibull shape m one can derivethe probability distribution function for crack initiation events using Eqn. (14) which is shown in Fig. 5 The ratio of probabilisticaverage life for LCF crack initiation and the deterministic life ofa certain smooth specimen subjected to equal maximum strain εais defined as the size effect factor. This turbine vane examinationresults in a size effect factor of 3.25. Although the entire vanesurface is much larger than the gauge section area of those LCFspecimens, the regions with critical hazard density at the vane areconfined to very narrow spots. Thereby the overall hazard rate forthe critical surface at the vane is smaller than the hazard rate fora standard LCF specimen and thus leads to higher probabilisticaverage life which corresponds to the statistical size effect.2.2 Probabilistic Lifing With Notch SupportIn order to consider the notch support effect for the probabilistic crack initiation life, the lifing algorithm calculates theanalytical derivatives of the FEA shape functions to obtain thevon Mises stress gradient at the domain surface as in Eqn. (13).The resulting χ field at the pressure side of the airfoil is shownin Fig. 4.High values of the related stress gradient occur at sharpshape transitions of the geometry, for example at the crosspiecesof the cooling channel outlets and the fillet radii. Negative χvalues (strain decreasing towards surface) are set to zero.If the notch support effect is considered in the calculation,the scale value η changes according to Eqn. (10) and thereforeshifts the probability distribution for crack initiation. The difference in the distribution functions for crack initiation for thewhole vane, computed considering and neglecting notch support,can be seen in Fig. 5. When enabling the notch support in the local probabilistic model by using Eqn. (16) instead of Eqn. (11),a larger Weibull scale parameter for the distribution is received.lowFIGURE 4. χ VALUES AT THE PRESSURE SIDE OF AIRFOIL,LINERARILY SCALEDFIGURE 5. PROBABILITIES OF CRACK INITIATION AT VANENEGLECTING AND CONSIDERING NOTCH SUPPORTThis reduces the slope of the distribution function and results in36 % higher life.Comparing the hazard densities in Fig. 6, one can notice thesimilar shape of the risk patches but at the same time decreasedvalues in the results of the notch support examination. Closerexamination of the upper section of the airfoil’s trailing edge isgiven in Fig. 7.Fig. 7 (a) shows two spots of distinctively visible strain concentrations originating mostly from inhomogeneous thermal expansion. This is illustrated in Fig. 7 (b) and Fig. 8 (c) whichshow the normalized temperature gradient χT . The local strains,seen in Fig. 7 (b), cause the stresses in the respective locations,as shown in Fig. 8 (a).The related stress gradient χ is mapped onto the geometryof the examined section in Fig. 8 (b). However the distinctive

Preprint – Preprint – Preprint – Preprint – Preprint – Preprintlarger spot of high stress at the trailing edge near the outer shroud(spot 1) is not recognizable in the χ-field. Since the shape in thatarea is relatively smooth compared to the spot in the edge of thetop cooling channel, the local χ-values are not significantly elevated compared to the surrounding material. In contrast, higherχ-values occur in spot 2 since it features a sharp geometry transition causing a very inhomogeneous stress distribution. The notchsupport effect should therefore have a higher impact on the probabilistic crack initiation life in spot 2 compared to spot 1.highlowspot 2spot 2spot 1spot 1(a) no notch support(b) notch supportFIGURE 6. HAZARD DENSITIES AT AIRFOIL FROM PREDICTIONS NEGLECTING AND CONSIDERING NOTCH SUPPORTlowspot 2highhigh(a) no notch supportspot 1(b) notch supportFIGURE 9. HAZARD DENSITY COMPARISON IN TRANSITIONFROM TRAILING EDGE OF AIRFOIL TO OUTER SHROUDlow(a) Strain(b) TemperatureFIGURE 7. ELASTIC STRAIN AND TEMPERATURE FIELD INTRANSITION FROM TRAILING EDGE OF AIRFOIL TO OUTERSHROUDIndeed, the hazard densities in spot 2 are lower than in spot1 of Fig. 9 (b). This is opposite to the situation in Fig. 9 (a)where higher hazard densities than in spot 1 are observed in spot2. Computing the probabilities of crack initiation only from selected integration points in the respective spots confirms this observation. Fig. 10 (b) shows a significantly larger decrease in riskdue to notch support for spot 2 compared to spot 1 in Fig. 10 (a).spot 2spot 1(b) χ(a) von Mises Stresshigh(a) Spot 1(b) Spot 2FIGURE 10. COMPARISON OF RISK DECREASE IN CRITICALSPOTS DUE TO NOTCH SUPPORTlow(c) χTFIGURE 8. VON MISES STRESS, χ AND χT FIELD IN TRANSITION FROM TRAILING EDGE OF AIRFOIL TO OUTER SHROUDThe size effect factor in this crack initiation prediction,which is approximately 4.43, is now a combined size effect fac-

Preprint – Preprint – Preprint – Preprint – Preprint – Preprinttor because the notch support effect is incorporated in the hazard density and Weibull scale computation. Solving Eqn. (16)for Nidet (x) to use in the surface integration in Eq. (15) leads tohigher values, i.e. less hazard density and higher probabilistic average life, compared to using Eqn. (11). The algorithm does notonly consider stresses at the surface, but also the stress gradientwhich links to the stress field in the volume below the surface.The combined size effect is therefore increasing.DISCUSSION AND CONCLUSIONIn Section 1 the local probabilistic approach to LCF crackinitiation prediction that was previously presented in [2] and [3]is reviewed. As specified in Subsection 1.1, the assumption of locally confined, independent LCF crack initiation events at engineering parts of polycrystalline metal allows a local hazard density approach. Using a Weibull distribution for the number ofload cycles until crack initiation, the approach leads to a surfaceintegral over the hazard density for the scale parameter whichpays regard to the statistical size effect. Subsection 1.2 showshow the notch support effect is incorporated in this model in away that it combines with the statistical size effect. Calibrationand validation of the presented notch support approach is exemplary outlined in Subsection 1.3. Both versions, the old and theextended, are applied for crack initiation life prediction of a gasturbine vane in Section 2. Areas of high hazard density on thevane are confined to small regions at the transitions of the trailing edge to inner and outer shroud and to edges in the coolingchannel outlets.As expected, the model taking notch support into accountpredicts higher probabilistic life for the vane. By evaluating thecritical spot at the trailing edge and the spot in the cooling channel edge separately, it is shown that significantly more probabilistic life is predicted for the second spot. That correlates withthe philosophy of the implemented notch support model whichstates that larger stress gradients in a body lead to higher crackinitiation life compared to a model that only considers homogeneous stress fields. Also, as shown in Fig. 1, the model is in goodagreement with experimental evidence.However, the true LCF life of turbine components in operation is influenced by more factors such as TBC thickness andspallation, BC-substrate interaction behavior, thermomechanicalfatigue (TMF) effects, grain size distribution, creep, manufacturing tolerances, variations in the operating conditions and theuncertainties of the parameter estimate θ̂ . The model appliedfor the work presented here is not specifically considering those.A probabilistic framework that combines the TBC/BC systemlife with the structural base material life and extends the presentLCF-based model to TMF and even connects to fracture mechanics [15] yet poses considerable future tasks for probabilistic gasturbine life prediction.ACKNOWLEDGMENTWe wish to thank the gas turbine technology department ofthe Siemens AG for stimulating discussions and many helpfulsuggestions.REFERENCES[1] D. Radaj and M. Vormwald, 2007. “Ermüdungsfestigkeit:Grundlagen für Ingenieure”, 3rd edition, Springer BerlinHeidelberg.[2] S. Schmitz, T. Seibel, T. Beck, R. Rollmann, R. Krauseand H. Gottschalk, 2013. “A Probabilistic Model For LCF”,Computational Materials Science, 79, pp. 584–590.[3] S. Schmitz, H. Gottschalk, R. Rollmann and R. Krause,2013. “Risk Estimation for LCF Crack Initiation”, ASMEPaper GT2013-94899.[4] S. Schmitz, R. Rollmann, H. Gottschalk and R. Krause,2013. “Probabilistic Analysis of LCF Crack Initiation Lifeof a Turbine Blade under Thermomechanical Loading”,Proc. Int. Conf LCF 7.[5] B. Fedelich, 1998. “A stochastic theory for the problem ofmultiple surface crack coalescence”, International Journalof Fracture, 91, pp. 23–45.[6] O. Hertel and M. Vormwald, 2012. “Statistical and geometrical size effects in notched members based on weakestlink and short-crack modelling”, Engineering Fracture Mechanics, 95, pp. 72–83.

Email: [email protected] Sebastian Schmitz Gas Turbine Department of Materials and Technology Siemens AG Berlin, 10553 Germany Email: [email protected] Georg Rollmann Gas Turbine Department of Materials and Technology Siemens AG M ulheim an der Ruhr, Nordrhein-Westfalen, 45473 Ger