recent developments in the thermomechanical fatigue life prediction of superalloys

CONTENTS

INTRODUCTIONGENERAL LINEARACCUMULATIONDAMAGE MODELS

Fatigue-Damage TermEnvironmental-Damage (Oxidation)TermCreep-Damage Term

DAMAGE-RATE MODELSTHE TMF/SRP METHOD

The TMF/TS-SRPMethodThe CumulativeCreep-FatigueDamage (TMF/SRP-Damage Coupling)Model

THE MODIFIEDEFFECTIVE J-INTEGRALFRACTURE MECHANICSMODELOTHER APPROACHESDISCUSSIONReferences

The following article is a component of the April 1999 (vol. 51, no. 4) JOM and ispresented as JOM-e. Such articles appear exclusively on the web and do not

have print equivalents.

Mechanical Behavior: Overview

Recent Developments in the ThermomechanicalFatigue Life Prediction of SuperalloysChangan Cai, Peter K. Liaw, Mingliang Ye, and Jie Yu

The models of thermomechanical fatigue life prediction used forsuperalloys can be classified into five types: general damagemodels, damage-rate models, thermomechanicalfatigue/strain-range partitioning methods, modified J-integralmodels, and empirical models. The formulation, which simulatesthe damage mechanisms and predicts the thermomechanicalfatigue lives in various models, has been specified. However,in-depth understanding and theoretical modeling ofthermomechanical fatigue damage mechanisms and interactionsare still lacking.

INTRODUCTION

Thermomechanical fatigue (TMF) with or without superimposedcreep is the primary life-limiting factor for engineering componentsin many high-temperature applications.1-2 In early work,isothermal fatigue (IF) tests performed at various temperatures,mechanical strain ranges, and strain rates were used to estimate thelife of members undergoing thermomechanical damage. A majorconcern has been the difficulty of simulating thermal stress cyclingin the laboratory. However, isothermal tests may not capture manyof the important damage micromechanisms under varyingtemperature conditions. Recently, considerable effort has beendevoted to developing TMF tests to simulate the behavior of thematerials undergoing thermomechanical fatigue. Some studies haveattempted to develop TMF life-prediction methods.

In TMF tests, a specimen is subjected to a desired temperature andmechanical strain with different phasings. Two baseline TMF testsare conducted in the laboratory with proportional phasings:in-phase (IP), the maximum strain at the maximum temperature,and out-of-phase (OP), the maximum strain at the minimumtemperature. The variation of thermal, mechanical, and total straincomponents with time in OP and IP cases is illustrated in Figure 1.These two types of phasings reproduce many of the mechanisms that develop under TMF. The mechanicalstrain ( mech) is the sum of the elastic and inelastic strain components, while the total strain ( tot or t) is thesum of thermal and mechanical strain components

Recent Developments in the Thermomechanical Fatigue Life Prediction ... http://www.tms.org/pubs/journals/jom/9904/cai/cai-9904.html

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a

b

Figure 1. Strain versus time under (a) TMF-OP and (b)TMF-IP cases.3

a b

Figure 2. Stress versus strain under (a) TMF-OP and(b) TMF-IP cases.1

tot= th+ mech= (T-T0)+ mech (1)

where th is the thermal strain, T0 is the reference temperature where the test was begun, T is the testtemperature, and is the coefficient of thermal expansion.

A schematic of the stress-strain behavior corresponding to TMF-OP and TMF-IP cases is illustrated inFigure 2. In the TMF-OP case, the material undergoes compression at high temperatures and tension atlow temperatures. The inverse behavior is observed for the TMF-IP case. The mean stress of the cycle istensile in the TMF-OP case, while it is compressive in the TMF-IP case.

Except in the baseline TMF tests, the phasing of strainand temperature can vary to a great extent. Realisticsimulation-type cycles are preferred, which often havea large hysteresis. One example is a diamond-shapedcycle (Figure 3), with the intermediate temperaturebeing reached at the largest tensile and compressivestrains, respectively, and the highest and lowesttemperatures at zero strain.

Under TMF conditions, damage mechanisms prevalentin metals involve three major aspects: fatigue,environmental (oxidation), and creep damage. Thesedamage mechanisms may act independently or incombination according to various materials andoperating conditions, such as maximum and minimumtemperatures, temperature range, mechanical strainrange, strain rate, the phasing of temperature and strain,dwell time, or environmental factors.

Traditionally, fatigue damage is the cyclic plasticity-driven, time- and temperature-independent damage thatexists whenever cyclic loading occurs. Creep refers to amaterial undergoing viscous deformation at a constantstress level. This type of deformation leads tointergranular creep cavity growth and rupture. UnderTMF loading, however, creep deformation contributesto the formation and propagation of microcracks.Metals exposed to environments at high temperaturesare subjected to corrosion by oxidation. This kind ofcorrosion is accelerated by a tensile stress. During TMF,brittle oxides can enhance the nucleation andpropagation of fatigue microcracks and impede therewelding of crack surfaces during unloading.

Due to complexity, a well-accepted framework for the prediction of TMF life has been elusive. Variousapproaches have been taken, ostensibly nonisothermal generalizations of isothermally derived models.

GENERAL LINEAR ACCUMULATION DAMAGE MODELS

Neu and Sehitoglu1,3–5 have developed a general model for high-temperature fatigue, including thermalmechanical fatigue. This model incorporates damage accumulation due to fatigue, environment(oxidation), and creep processes. Damages per cycle from fatigue, environmental attack (oxidation), andcreep are summed to obtain a total damage per cycle


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(2)

Assuming that linear damage is equal to one at failure, the equation may be rewritten in terms of the life,Nf, where damage is taken as equal to 1/Nf,

(3)

Fatigue-Damage Term

Fatigue damage is represented by fatigue mechanisms, which nominally occur at ambient or lowtemperatures. The fatigue life term, Nf

fat, is represented by the strain-life relation

(4)

where m is the mechanical strain range, and C and d are material constants determined fromlow-temperature isothermal tests.

Environmental-Damage (Oxidation) Term

The oxidation damage is based on crack nucleation and growth through an oxide layer. It can beexpressed by

(5)

where hcr is a critical crack length at which the environmental attack trails behind the crack-tip advance, 0 is the ductility of the environmentally affected material, B is the coefficient, is the exponent, and isthe strain-rate sensitivity constant. The values of all above constants are determined by experiments. ox

is a phasing factor for environmental damage and is defined as

(6)

(7)

where is the ratio of thermal to mechanical strain rates, and ox is a constant as a measure of therelative amount of oxidation damage for different thermal strain to mechanical strain ratios and isextracted from the experiments. Kp

eff is a parabolic oxidation constant and can be calculated by

(8)

where T(t) is the temperature as a function of time, tc is the cycle period, D0 is the diffusion coefficientfor oxidation, Q is the activation energy for oxidation, and R is the gas constant.

Creep-Damage Term


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The creep-damage term is a function of temperature, effective stress, and hydrostatic stress components,and can be expressed by

(9)

where is the effective stress, H is the hydrostatic stress, K is the drag stress, 1 and 2 are scalingfactors that represent the relative amount of damage occurring in tension and compression, H is theactivation energy for the rate-controlled creep mechanism, and A and m are material constants.

(10)

(11)

the constant, creep, defines the sensitivity of the phasing to the creep damage.

This model couples a viscoplastic, nonisothermal constitutive model of material under investigation withan incremental evolution of damage for oxidation and creep terms within a TMF loading cycle. The modelwas applied to MAR-M247 and 1070 steel,1,5,6 and 80% of the predictions were within a factor of ± 2.5of the experimentally measured lives.

DAMAGE-RATE MODELS

Among these models is the formulation proposed by Miller and colleagues, who use a physicallymeasurable quantity, such as crack length, as specific definitions of damage.7 The TMF life-predictionmodel is based on the concept of microcrack propagation and explicitly accounts for damage due tofatigue, creep, and oxidation.

The general form of the equation is

(12)

where a is the crack length, and N is the cycle number.

The fatigue component of microcrack propagation is correlated using the J parameter

(13)

where mf and Cf are constants. If J is taken as

(14)

where is the stress range and e and p are the elastic and plastic strain ranges, respectively, n' is thecyclic-hardening exponent, Y is a geometric correction factor, and a is the crack length. The functionf(1/n') is given by


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(15)

If Equation 13 is integrated between appropriate initial and final crack sizes, the integrated form of fatiguecomponents is obtained

(16)

where

if mf 1.

if mf = 1 (17)

= 2 Y2Cf, a0 and af are the initial and final crack sizes, respectively, and is defined as

(18)

The creep component of the microcrack propagation is correlated using the stress power-release-rate

parameter, ,

(19)

where Cc and mc are the experimentally determined creep constant and exponent, respectively.

(20)

where is the creep strain rate, tt is the time within a cycle during which tensile strain accumulates, andtc is the time within a cycle during which compressive strain accumulates. The Macaulay brackets, < >,are defined as

(21)

The creep strain rate, , is determined using a viscoplastic constitutive law for the material underinvestigation.

The oxidation component of the microcrack propagation is also correlated using the J parameter with anadditional time and temperature dependence

(22)

where m0 and are experimentally determined constants, t is the cycle time, and the coefficient, C0, isdefined as


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(23)

where , B, and k are experimentally determined constants; Qox is the experimentally measured

activation energy of the effective crack-tip oxidation and growth process; the parameter = < Tmin>;and Tmin is the stress at the minimum temperature. Teff is an effective temperature and is defined as

(24)

where tmin and tmax are the times at which the minimum and maximum temperatures occur during atemperature cycle, respectively, and t = tmax - tmin.

Figure 3. Temperature-strain cycles for differentTMF tests (Tu—upper temperature, Tl—lowtemperature).2

Figure 4. A comparison betweencalculated lives to a 0.1 mm deepcrack in TMF of bare IN-100specimens and experimentaldata.10

This model was applied for MAR-M247, and in general, the correlation was within a scatter of ±2 of themedian.

Among the damage-rate models is one proposed by Reucket and Remy, which accounts for theinteractions between fatigue and oxidation by superimposing both kinds of damages.8–10

The model considers that TMF damage in conventionally cast superalloys is mostly the growth of adominant microcrack and that the initiation period can be neglected for practical purposes. Thus, thedamage equations were derived assuming that the elementary crack growth results from an advanceassociated with crack opening under fatigue and from an additional contribution due to oxidation at thecrack tip. The damage equation can be written as

(25)

The fatigue contribution to the crack advance is estimated assuming that the crack is opened only by atensile stress

(26)

where


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(27)

where in is the inelastic strain range, max is the maximum cyclic tensile stress, and u is the ultimatetensile strength in monotonic tension.

The contribution to crack advance due to oxidation is derived assuming that fracture of the formed oxidespike occurs at each tensile stroke. The oxidation rate at the crack tip is measured by the depth ofinterdendritic oxide spikes formed before fracture, lox,

(28)

The interdendritic oxidation obeys a t1/4 kinetics, where t is time,

(29)

where t is the cycle period, and

(30)

where 0 is the oxidation constant at a given temperature, 0 is a threshold (above this threshold, thedepth of oxide formed at every cycle increases with the maximum stress according to a power law,Equation 30), and n is the exponent. These parameters are identified from tensile tests and somemetallographic measurements of crack depth for the specimens.

Under TMF, as temperature varies, the right side of Equation 29 has to be averaged over the cycle, and an

average oxidation constant, , is computed as

(31)

By Equations 25-28, the following equation can be derived for the crack depth, a, as a function of thenumber of cycles, N:

(32)

or the number of cycles, N, corresponding to a given crack depth, a,

(33)

Equation 32 or 33 can be used for TMF life prediction with good results.

Remy and colleagues have proposed another fatigue-damage model applicable to TMF, which alsoassumes that fatigue life is spent in the growth of microcracks.10 The oxidation and fatigue interaction are


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considered in the following manner: exposure to high temperatures during the TMF cycle oxidizes thematerial at the crack tip; then, high stress ranges at medium temperatures give rise to fatigue damage inthe material that has been embrittled by oxidation.

Consider a volume element of size, , ahead of the crack tip. This volume element will endure anincremental damage, dD, during dN cycles under oxidizing and cyclic-loading conditions. This damageincrement is given by

(34)

with R = 1 - yy/ yy when yy yy, and R = 0 when yy > yy, where eq is the Von Mises equivalentstress range averaged over the microstructural element at the crack tip; yy is the maximum tensile valueof the normal stress of the crack tip at a distance ; S0, M, and are constants at a given temperature; and

(35)

where is the critical fracture stress (i.e., the critical value of yy when N approaches infinity) of thevirgin alloy, and c

ox is that of the alloy embrittled by oxidation, which is defined as

(36)

where is a constant at a given temperature; the function, f, is determined from the experiments; and

(37)

where t is the exposure time at a given temperature, and ox is an oxidation constant.

When the temperature varies, the right side of Equation 37 has to be averaged over the cycle, and an

average oxidation constant, ox, is computed as

(38)

where t is the cycle period.

Using Equations 35 and 36, c is computed at every cycle, and the number of cycles to break the volumeelement will be given by the condition

(39)

N( ) is, thus, computed through the set of Equations 34-37 using an iterative procedure, cycle by cycle,until the condition of Equation 39 is fulfilled. The model gives good predictions of lives for an in-phasecycle when using a diamond-shaped cycle (Figure 4).

An engineering procedure for TMF crack-growth-rate predictions is presented by Dai and colleagues,which is based on isothermal data.11 The model uses a linear summation of the contributions to crackgrowth of the two dominant mechanisms, which are active at minimum and maximum temperatures of thecycle—namely, the mechanical fatigue and environmentally assisted crack growth


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(40)

where (da/dN)fat, the mechanical fatigue contribution, is correlated by a fracture mechanics parameter, Keff

(41)

where the correlated function, f, is determined by isothermal or TMF tests. Assuming that the cracks wereopened when the loads were tensile,

(42)

where Nmax is the peak value of the measured load during cycling, a is the crack length measured from thespecimen edge, B is the thickness of the specimen, W is the width of the specimen, and H is thegeometrical function that characterizes the single-edge notched (SEN) specimen geometry as well as theboundary conditions.

(da/dN)env is the time-dependent contribution related to the kinetics of oxygen-induced embrittlement.For TMF, it is computed using

(43)

where is a material constant, Q is the apparent activation energy for oxygen transport, T is thetemperature in oK, and t1 and t2 pertain to the starting and finishing times within one cycle where oxygen-induced embrittlement operates. Assuming that t1 and t2 are related to the opening and closure processesof the crack, the time-dependent contributions (Equation 43) were estimated by integrating over thetensile load part of the cycle. Combining Equation 41 (mechanical fatigue) with Equation 43 (contributionof oxygen-induced embrittlement) allows the accurate prediction of the actual crack growth rates(da/dN)tot.

THE TMF/SRP METHOD

Among the principal alternative approaches that treat the creep-fatigue interaction synergistically is thestrain-range partitioning (SRP) approach of Manson, Halford, and Hirschberg.12,13

a b c d

Figure 5. SRP cycle types showing (a) pp type, (b) cp type, (c) pc type, and (d) cc type cycles.18

SRP is based on the observation that the inelastic strain range is the controlling variable on fatigue life.


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Specifically, SRP recognizes four modes of cyclic inelastic straining, each of which may be accompaniedby a distinguished deformation/damage micromechanism characteristic. These modes are: pp—theinelastic strain range is considered to be composed of plasticity (p) in both tension and compression,cp—creep strain in tension and plastic strain in compression, pc—plastic strain in tension and creep strainin compression, and cc—creep strain in both tension and compression (Figure 5). The four modes have acertain fundamental significance in that any creep-fatigue cycle may be decomposed into theseconstituent components.

For each of the inelastic deformation types, a life relation is established, typically given as

(44)

where Aij and aij, i,j = p, c (i.e., pp, cp, pc, and cc modes) are material-dependent parameters. When thesemodes are present in a complex cycle, their damaging effects may be combined by the use of theinteraction damage rule (IDR), and a calculation for the life is performed. The IDR can be stated as

(45)

where Fpp, Fcp, Fpc, and Fcc correspond to the relative amount of inelastic-strain type present (e.g., pp/in, cp/ in, pc/ in, and cc/ in), and the terms Npp, Ncp, Npc, and Ncc, correspond to the life levelsestablished from Equation 44 at the inelastic strain range, in.

SRP has since been successfully formulated on a total strain basis (TS-SRP), extending it into theisothermal low-strain, long-life regime where the inelastic strains are small and difficult to determine. Thisextension, plus the introduction of the bithermal test to characterize TMF behavior, has permitted the SRPmethod to be applied to general TMF life prediction problems.14–16 (Bithermal experiments utilized atrapezoidal-wave temperature versus time profile wherein mechanical straining was imposed only duringthe isothermal portions at the maximum and minimum temperatures. Moreover, during the bithermal tests,the temperature was changed while the specimen was held at zero load.)

The TMF/TS-SRP Method

The method proposed by Halford and colleagues, referred to as TMF/TS-SRP, is based upon threerelatively recent developments: the total strain version of the method of SRP, the bithermal testingtechnique for measuring TMF behavior, and advanced viscoplastic constitutive models.

Fractographic and metallographic investigations were conducted on the specimens of some materials, suchas Haynes 188, that were fatigued to failure under IP and OP bithermal and thermomechanical loadingconditions. These investigations identified the prevailing damaging mechanisms in both bithermal andthermomechanical fatigue tests. For these identified materials, bithermal fatigue tests could be used as thebasis for TMF life prediction.

Total Strain Range-Life Relations

The total strain range is the sum of two terms, e and in. Each term is a power law function of Nf0 (thesubscript 0 on Nf denotes the fatigue life for an equivalent zero mean stress condition) and appears as oneof a family of straight lines on the log-log coordinates of Figure 6. The solid lines represent upper andlower bounds on the inelastic and elastic strain-range terms. The equation for the general life relation isshown in Figure 6 and is written as

(46)


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Figure 6. A schematic of strain range-liferelations for the total strain version ofstrain-range partitioning.14

or

(47)

In general, the slopes, b and c, and intercepts, B and C', inEquation 47 would be functions of the specific thermal-cyclingconditions. Observations on the behavior of many materialsunder a wide range of fatigue conditions suggest that b and care fixed. Thus, the families of elastic-life relations are parallelto one another, as are the inelastic-life relations.

The intercepts, B and C', are first-order functions of many ofthe variables (e.g., time per cycle, strain rate, environment,wave shape, and temperature) that are known to influenceTMF lives. For example, both B and C' typically decrease asthe time per cycle increases (lower frequency, lower strain rate,or increased hold-time), since this introduces the phenomena ofcreep and environmental attack. Creep and environmentalattack are usually regarded as detrimental damage mechanisms.As temperature increases, B usually decreases (primarily due tocreep-weakening effects, but also due to environmental attack), and C' may do either.

Inelastic Strain Range-Life Relations

The intercept, C', is evaluated as

(48)

where ij = pp, pc, or cp, and Fij is the fractional value of each inelastic strain range present in any givenTMF cycle (i.e., the partitioning of in).

(49)

(50)

(i.e., Fij = 1.0). Fij can be written as an exponential relation of the time per cycle, normalized by afunction of the total strain range

(51)

The constants, A', ', and m, are determined from multiple regression analyses of analytical data.

If a viscoplastic model for a material has not been evaluated, the IP and OP bithermal pp, cp, and pc testswould be utilized to measure Fij and determine the constants, A', , and m.

In addition to Fij, the other factors influencing the value of C' are Cij and c. Bithermal fatigue test resultsprovide the data base to determine the constants, Cij and c.

Elastic Strain Range-Life Relations

In the elastic strain-range versue life relations, the exponent, b, was determined from bithermal


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experiments. The intercept, B, is expressed as

(52)

where n is the cyclic strain-hardening exponent, and n is related to b and c by the expression n = b/c. Kijis a time-dependent, elastic-modulus-normalized cyclic strength coefficient defined by

(53)

The relation between Kij and cycle time can be expressed as

(54)

Normally, the values of the constants, A1' and m1, are ascertained using a viscoplastic model or the least-squares curve fit of the bithermal test data.

For the specific conditions of the cycle, the intercepts, B and C', and the exponents, b and c, will beevaluated. Then the total strain range versus cyclic life failure curve will be established, and the TMF lifeprediction can be made easily.

A sample TMF/TS-SRP life prediction curve is shown in Figure 7a. The assessment of the TMFlife-prediction capability of the TMF/TS-SRP method is shown in Figure 7b. It is seen that the agreementis generally within the conventional factor of two associated with high-temperature fatigue life predictionin the nominally low-cycle fatigue regime.

a b

Figure 7. (a) A sample TMF/TS-SRP life-prediction curve (zero mean stress condition) constructed

for OP B1900+Hf. 483 871°C, 4 min./cycle, t = 0.72%, observed life = 546 cycles, andpredicted life = 491 cycles. (b) An assessment of TMF life-prediction capability of theTMF/TS-SRP method for two high-temperature aerospace superalloys.14

The Cumulative Creep-Fatigue Damage (TMF/SRP-DamageCoupling) Model

Most applications that involve TMF loading also include complex multilevel cyclic loading patterns, sothat cumulative damage effects are also a concern. The model of cumulative creep-fatigue damage,proposed by McGaw, rests on the notion that certain basic damaging modes of cyclic deformationcharacterize the failure behavior of a material and utilize the SRP method as its basis.17,18

A set of damage-curve relations is proposed for the modeling of damage accumulation according to each


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of the four fundamental deformation and damage modes given in SRP

(55)

where ij, i,j = p,c may be considered as material-dependent parameters, and the damage variable, D,possesses a range of 0 (undamaged condition) to 1 (failure) over the domain of the life fraction n/Nij of 0to 1.

When multiple damage modes are operative, an additional set of damage coupling relations is required,

(56)

where the functions, gij, i,j = p,c [where gpp(Dpp) Dpp] must be determined by experiments.

The coupling relations recognize that what is considered as damage in one context cannot be necessarilyregarded as damage in another (e.g., pp-type cycling is associated with transgranular, classical fatigue-typedamage, while cp-type cycling is associated with intergranular, creep-type damage). However, thecoupling relations do imply that the damage state is relatable. The coupling relations provide a mapping orcorrespondence between damage modes.

To model the more general problem wherein multiple damage modes may be present within a single cycle,one additional relationship must be established—a description of how the various damaging contributionsmay be synthesized to provide a means of assessing the cumulative damage contribution

(57)

This equation describes the cumulative damage behavior of a complex cycle (with life Nf) as life fraction,expressed in terms of the four damage variables, Dpp, Dcp, Dpc, and Dcc. Note that in general, thecoupling relations must be invoked to recast Equation 57 in terms of one effective damage variable, D.

The model can be readily extended to treat the case of TMF through the use of the bithermal fatigueapproximation to TMF. To address TMF, the life relations, damage-curve equations, and damage-couplingrelations can be directly replaced by bithermal counterparts. Experiments consisting of two-level loadingsof TMF (both IP and OP) followed by isothermal fatigue to failure were conducted on 316 stainless steel.It was found that the model gave good predictions for the OP two-level tests and provided reasonablebounds for the IP two-level tests.

THE MODIFIED EFFECTIVE J-INTEGRAL FRACTUREMECHANICS MODEL

A model by Nisseley was developed to predict TMF crack initiation and estimate Mode I crack growth ingas turbine hot-section gas-path superalloys.19 The model is based on a strain-energy-density fracturemechanics approach modified to account for thermal exposure and single-crystal anisotropy.

An effective J-integral was developed for TMF crack-life prediction. The effective J-integral is anempirical strain-energy-density crack growth parameter and is not a rigorous fracture mechanicsparameter.

The developed effective J-integral fracture mechanics TMF model is summarized as


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(58)

where a is the crack depth, da/dN is the crack-growth rate, Jeff is the effective J-integral (an empiricalstrain-energy-density fracture mechanics parameter), A and c are material constants, and

(59)

where Q0 and T0 are material constants, T is the absolute temperature, and dt is the time increment.

(60)

where a0 is the initial material defect size (material constant), b is a material constant, is the crack-boundary-correction factor, and Jth is the threshold J-integral that may be constant or temperature-dependent. For the latter, a good empirical temperature-dependent formula of Jth is

(61)

where J0 is a constant and TJ is the temperature at which time-dependent deformation (creep) becomessignificant in monotonic tensile tests.

Where f is a crystallographic strain-energy-correction factor,

(62)

where f<111> is a factor that resolves max into the maximum normal octahedral slip plane stress, Emax isthe elastic modulus in the max direction and at the max temperature, E<111> is elastic modulus in the<111> crystal direction and at the max temperature, and is the effective total strain-energy density.

(63)

where is the material constant, and We is the elastic strain-energy density

(64)

where m = 1 if min > 0 and m = 0 if min 0.

Wp is the plastic strain energy density:

(65)

where > 0, is the stress, max is the maximum stress, min is the minimum stress of a cycle in which max occurs, and d in is the inelastic strain increment.

This TMF model's material constants are determined from a combination of uniaxial TMF and oxidationtests and by applying a nonlinear least-squares-regression analysis. Stresses were obtained from a


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Figure 8. Hysteretic energy lifecurves for a 316 stainless steelsubjected to TMF tests in thetemperature range of 399-621°C.21

nonlinear finite-element analysis of a TMF specimen strain-temperature history. Nonlinear stress-strainbehavior was predicted using unified viscoplastic constitutive models.

The TMF life-model formulation is limited to transgranular cracking. The model can capture many TMFcracking effects, such as coating thickness, single-crystal anisotropy, cycle wave shape, dwell, andthermal exposure. TMF life predictions based on the modified effective J-integral model were in goodagreement with observed uniaxial TMF specimen lives.

OTHER APPROACHES

Other TMF life-prediction approaches are mostly empirical models or engineering models.

Bernstein and colleagues developed a model to predict the TMF life ofindustrial gas-turbine blades.20 The model is a semi-empirical approach,similar to most engineering models that are actually used to predictlow-cycle fatigue. The formulation does not attempt to separately modelthe different mechanisms of damages (i.e., fatigue, creep, andenvironmental attack).

The model developed for gas-turbine blades in electric power generationis based upon the parameters of strain range, strain amplitude ratio, anddwell time. This combination of parameters is intuitively reasonable,

since the strain-amplitude ratio can account for mean strain effects, and the dwell time can considerenvironmental and mean stress effects caused by steady-state stress relaxation, both of which can affectthe fatigue life.

The general form of the model is

(66)

where Nf is the fatigue life; A is the strain ratio, A = amp/ mean ( amp = , mean = , where all strainsare mechanical strains); is the total strain range; th is the dwell time; and Ci (i = 0, 1, 2, and 3) is theempirical constant, determined from multiple regression of the TMF test data. In a test, the modeladequately described the behavior of IN-738LC superalloy for the TMF-OP test conditions in thetemperature range of 427-871°C.

When the TMF model was used to predict the fatigue life of the in-service gas turbine blades, which mayexperience the different cycle types, a linear damage rule was used to account for the different cycletypes

(67)

where the damage ratio, D', was assumed to be one at 100% life consumed. Ni is the number of cycles ofstraining-type i, and Nif is the number of cycles to failure of type i.

Zamrik and colleagues have used the hysteretic-energy method to characterize the TMF behavior of type316 stainless steel.21 From the stabilized mid-life hysteresis loop, the energy per volume cycle or densitycan be calculated and fit to the energy-life relation by

(68)


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where U is the energy per volume cycle, W is the work or area of the hysteresis loop, A is the specimencross-sectional area, L is the specimen gage length, is the stress, d is the strain increment, A0 and a areconstants, and Nf is the number of cycles to failure. The hysteretic energy versus life approach isillustrated in Figure 8. It can be observed that the TMF-IP hysteretic energy life line is lower than theTMF-OP line, which suggests more damage.

Kanasaki and colleagues have carried out axial-strain-controlled low-cycle fatigue tests of a carbon steelin oxygenated high-temperature water under TMF-IP and TMF-OP conditions.22 Based on the assumptionthat the fatigue damage increased in a linear proportion to the increment of strain during cycling, afatigue-life prediction method was proposed.

(69)

where Tmax and Tmin are the maximum and minimum temperatures during TMF cycling, N(T) is thefatigue life in high-temperature water under isothermal conditions, and N' is fatigue life inhigh-temperature water under a TMF condition. For a carbon steel, STS410, under the conditions of DO =1 ppm (in pure water containing 1 ppm oxygen) and = 0.002% 1/s, N(T) was determined by theexperiment

(70)

A good relationship between the predicted N'pred and the experimental N'test was observed.

DISCUSSION

The models described here can be divided into two types:

General models of TMF life prediction based on fundamental physical mechanisms of TMF crackinitiation and propagation. Such models may be used to capture and simulate damage mechanismsand their interactions under TMF loading conditions. These models must be based on a number oftest data sets of various materials, include microstructural observations and analyses, and generallycouple viscoplastic constitutive models. But the forms or the sets of equations of these models arecomplex and not convenient in engineering applications.Engineering and empirical models, the objective of which is directed toward engineering practicesand direct applications. It is for a special material that an empirical model of TMF life prediction isestablished. The forms of this model are a little simpler and convenient for applications, but they arenot in common use among various materials.

Damage modeling under thermomechanical cyclic loading is still at an early stage. TMF cycling isexpected to introduce a multitude of cyclic deformation and damage mechanisms in superalloys, and theinfluence of these damage mechanisms on the material's fatigue-crack initiation and propagation behavioris not well understood at present. No generally accepted models of TMF fatigue-life prediction arecurrently available.

It is generally recognized that three dominant damage mechanisms (fatigue, oxidation, and creep damage)may occur during TMF loading conditions. Most proposed models of TMF fatigue-life prediction attemptto capture the effects of these damage mechanisms and their interactions. It is not certain if thesedominant damage mechanisms operate simultaneously, or if some of them run and others becomeinoperative during a special TMF loading condition.

There are many factors affecting TMF lives, such as materials, maximum and minimum temperatures,


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temperature range, strain rate, strain range, strain-temperature phase, and dwell and cycle times. There isa difficulty in that the models quantitatively simulate the interaction of damage mechanisms; at present,views on how to deal with this problem differ. Based on this and the complexity of alloys systems,TMF-life-prediction models are generally time-consuming and expensive.

To develop effective TMF life-prediction approaches for superalloys, more experimental work and datacollection are necessary on various materials placed under TMF loading conditions.

References

1. H. Sehitoglu, "Thermo-Mechanical Fatigue Life Prediction Methods," Advances in Fatigue LifetimePredictive Techniques, ASTM STP 1122 (1992), pp. 47–76.2. S.A. Kraft and H. Mughrabi, "Thermo-Mechanical Fatigue of the Monocrystalline Nickel-BaseSuperalloy CMSX-6," Thermomechanical Fatigue Behavior of Materials, ASTM STP 1263 (1996), pp.27–40.3. R. Neu and H. Sehitoglu, "Thermo-Mechanical Fatigue, Oxidation and Creep: Part 1—Experiments,"Met. Trans. A, 20A (1989), pp. 1755–1767.4. R. Neu and H. Sehitoglu, "Thermo-Mechanical Fatigue, Oxidation and Creep: Part 2-Life Prediction,"Met. Trans. A, 20A (1989), pp. 1769–1783.5. H. Sehitoglu and D.A. Boismier, "Thermo-Mechanical Fatigue of Mar-M247: Part 2—Life Prediction,"J. Eng. Mat. and Tech., 112 (1990), pp. 80–89.6. Y. Kadioglu and H. Sehitoglu, "Modeling of Thermomechanical Fatigue Damage in Coated Alloys,"Thermomechanical Fatigue Behavior of Materials, ASTM STP 1186 (1993), pp. 17–34.7. M.P. Miller et al., "A Life Prediction Model for Thermomechanical Fatigue Based on MicrocrackPropagation," ASTM STP 1186 (1993), pp. 35–49.8. J. Reuchet and L. Remy, "Fatigue Oxidation Interaction in a Superalloy-Application to Life Predictionin High Temperature Low Cycle Fatigue," Met. Trans. A, 14A (1983), pp. 141–149.9. E. Chataigner and L. Remy, "Thermomechanical Fatigue Behavior of Coated and Bare Nickel-BasedSuperalloy Single Crystals," Thermomechanical Fatigue Behavior of Materials, ASTM STP 1263 (1996),pp. 3–25.10. L. Remy et al., "Fatigue Life Prediction under Thermo-Mechanical Loading in a Nickel-BaseSuperalloy," Thermomechanical Fatigue Behavior of Materials, ASTM STP 1186 (1993), pp. 3–16.11. J. Dai, N.J. Marchand, and H. Hongoh, "Thermal Mechanical Fatigue Crack Growth in Titanium Alloy:Experiments and Modeling," Thermomechanical Fatigue Behavior of Materials, ASTM STP 1263(1996), pp. 187–209.12. S.S. Manson, G.R. Halford, and M.H. Hirschberg, "Creep Fatigue Analysis by Strain RangePartitioning," First Symposium on Design for Elevated Temperature Environment, (1971), pp. 12–24.13. S.S. Manson, "The Challenge to Unify Treatment of High Temperature Fatigue—A Partisan ProposalBased on Strain Range Partitioning," Fatigue at Elevated Temperatures, ASTM STP 520 (1973), pp.744–782.14. G.R. Halford et al., "Application of a Thermal Fatigue Life Prediction Model to High-TemperatureAerospace Alloys B1900+HF and Haynes 188," Advances in Fatigue Lifetime Predictive Techniques,ASTM STP 1122 (1992), pp. 107–119.15. G.R. Halford et al., "Thermomechanical and Bithermal Fatigue Behavior of Cast B1900+Hf andWrought Haynes 188," Advances in Fatigue Lifetime Predictive Techniques, ASTM STP 1122 (1992), pp.120–142.16. S. Kalluri and G.R. Halford, "Damage Mechanisms in Bithermal and Thermo-mechanical Fatigue ofHaynes 188," Thermomechanical Fatigue Behavior of Materials, ASTM STP 1186 (1993), pp. 126–143.17. M.A. McGaw, "Cumulative Damage Concepts in Thermomechanical Fatigue," ThermomechanicalFatigue Behavior of Materials, ASTM STP 1186 (1993), pp. 144–156.18. M.A. McGaw, "Cumulative Creep-Fatigue Damage Evolution in an Austenitic Stainless Steel,"Advances in Fatigue Lifetime Predictive Techniques, ASTM STP 1122 (1992), pp. 84–106.19. D.M. Nissley, "Thermomechanical Fatigue Life Prediction in Gas Turbine Superalloys: A FractureMechanics Approach," AIAA Journal, 33 (6) (1995), pp. 1114–1120.


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20. H.L. Bernstein et al., "Prediction of Thermal-Mechanical Fatigue Life for Gas Turbine Blades inElectric Power Generation," Thermomechanical Fatigue Behavior of Materials, ASTM STP 1186 (1993),pp. 212–238.21. S.Y. Zamrik, D.C. Davis, and L.C. Firth, "Isothermal and Thermomechanical Fatigue of Type 316Stainless Steel," Thermomechanical Fatigue Behavior of Materials, ASTM STP 1263 (1996), pp. 96–116.22. H. Kanasaki et al., "Corrosion Fatigue Behavior and Life Prediction Method under ChangingTemperature Condition," Effects of the Environment on the Initiation of Crack Growth, ASTM STP 1298(1997), pp. 267–281.

ABOUT THE AUTHORS

Changan Cai is an associate professor of engineering mechanics at GuiZhou University of Technology.

Peter K. Liaw is the Ivan Racheff Chair of Excellence and professor at the University of Tennessee.

Mingliang Ye is an associate professor of mining engineering with GuiZhou University of Technology.

Jie Yu is a professor of materials science and engineering with GuiZhou University of Technology.

For more information, contact P.K. Liaw, University of Tennessee at Knoxville, Department ofMaterials Science and Engineering, 427-B Dougherty Engineering Building, Knoxville, Tennessee37996-2200; (423) 974-6356; fax (423) 974-4115; e-mail [email protected].

Copyright held by The Minerals, Metals & Materials Society, 1999

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recent developments in the thermomechanical fatigue life prediction of superalloys

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