

CONTENTS 

In TMF tests, a specimen is subjected to a desired temperature and mechanical strain with different phasings. Two baseline TMF tests are conducted in the laboratory with proportional phasings: inphase (IP), the maximum strain at the maximum temperature, and outofphase (OP), the maximum strain at the minimum temperature. The variation of thermal, mechanical, and total strain components 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 mechanical strain (_{mech}) is the sum of the elastic and inelastic strain components, while the total strain (_{tot} or _{t}) is the sum of thermal and mechanical strain components
_{tot}= _{th}+ _{mech}= (TT_{0})+ _{mech}  (1) 
where _{th} is the thermal strain, T_{0} is the reference temperature where the test was begun, T is the test temperature, and is the coefficient of thermal expansion.
A schematic of the stressstrain behavior corresponding to TMFOP and TMFIP cases is illustrated in Figure 2. In the TMFOP case, the material undergoes compression at high temperatures and tension at low temperatures. The inverse behavior is observed for the TMFIP case. The mean stress of the cycle is tensile in the TMFOP case, while it is compressive in the TMFIP case.
a  
b  
Figure 1. Strain versus time under (a) TMFOP and (b) TMFIP cases.^{3}  
a  b 
Figure 2. Stress versus strain under (a) TMFOP and (b) TMFIP cases.1 
Under TMF conditions, damage mechanisms prevalent in metals involve three major aspects: fatigue, environmental (oxidation), and creep damage. These damage mechanisms may act independently or in combination according to various materials and operating conditions, such as maximum and minimum temperatures, temperature range, mechanical strain range, strain rate, the phasing of temperature and strain, dwell time, or environmental factors.
Traditionally, fatigue damage is the cyclic plasticitydriven, time and temperatureindependent damage that exists whenever cyclic loading occurs. Creep refers to a material undergoing viscous deformation at a constant stress level. This type of deformation leads to intergranular creep cavity growth and rupture. Under TMF loading, however, creep deformation contributes to the formation and propagation of microcracks. Metals exposed to environments at high temperatures are subjected to corrosion by oxidation. This kind of corrosion is accelerated by a tensile stress. During TMF, brittle oxides can enhance the nucleation and propagation of fatigue microcracks and impede the rewelding of crack surfaces during unloading.
Due to complexity, a wellaccepted framework for the prediction of TMF life has been elusive. Various approaches have been taken, ostensibly nonisothermal generalizations of isothermally derived models.
(2) 
Assuming that linear damage is equal to one at failure, the equation may be rewritten in terms of the life, N_{f}, where damage is taken as equal to 1/N_{f},
(3) 
(4) 
where _{m} is the mechanical strain range, and C and d are material constants determined from lowtemperature isothermal tests.
(5) 
where h_{cr} is a critical crack length at which the environmental attack trails behind the cracktip advance, _{0} is the ductility of the environmentally affected material, B is the coefficient, is the exponent, and is the strainrate 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 the relative amount of oxidation damage for different thermal strain to mechanical strain ratios and is extracted from the experiments. K_{p}^{eff} is a parabolic oxidation constant and can be calculated by
(8) 
where T(t) is the temperature as a function of time, t_{c} is the cycle period, D_{0} is the diffusion coefficient for oxidation, Q is the activation energy for oxidation, and R is the gas constant.
(9) 
where is the effective stress, _{H} is the hydrostatic stress, K is the drag stress, _{1} and _{2} are scaling factors that represent the relative amount of damage occurring in tension and compression, H is the activation energy for the ratecontrolled 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 with an incremental evolution of damage for oxidation and creep terms within a TMF loading cycle. The model was applied to MARM247 and 1070 steel,^{1,5,6} and 80% of the predictions were within a factor of ± 2.5 of the experimentally measured lives.
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 m_{f} and C_{f} 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 the cyclichardening exponent, Y is a geometric correction factor, and a is the crack length. The function f(1/n') is given by
(15) 
(16) 
where
if m_{f} 1. 
if m_{f} = 1  (17) 
= 2 Y^{2}C_{f}, a_{0} and a_{f} 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 powerreleaserate parameter, ,
(19) 
where C_{c} and m_{c} are the experimentally determined creep constant and exponent, respectively.
(20) 
where is the creep strain rate, t_{t} is the time within a cycle during which tensile strain accumulates, and t_{c} 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 under investigation.
The oxidation component of the microcrack propagation is also correlated using the J parameter with an additional time and temperature dependence
(22) 
where m_{0} and are experimentally determined constants, t is the cycle time, and the coefficient, C_{0}, is defined as
(23) 
where , B, and k are experimentally determined constants; Q_{ox} is the experimentally measured activation energy of the effective cracktip oxidation and growth process; the parameter = <_{Tmin}>; and _{Tmin} is the stress at the minimum temperature. T_{eff} is an effective temperature and is defined as
(24) 
where t_{min} and t_{max} are the times at which the minimum and maximum temperatures occur during a temperature cycle, respectively, and t = t_{max}  t_{min}.
Among the damagerate models is one proposed by Reucket and Remy, which accounts for the interactions 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 a dominant microcrack and that the initiation period can be neglected for practical purposes. Thus, the damage equations were derived assuming that the elementary crack growth results from an advance associated with crack opening under fatigue and from an additional contribution due to oxidation at the crack 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 a tensile stress
(26) 
where
(27) 
where _{in} is the inelastic strain range, _{max} is the maximum cyclic tensile stress, and _{u} is the ultimate tensile strength in monotonic tension.
The contribution to crack advance due to oxidation is derived assuming that fracture of the formed oxide spike occurs at each tensile stroke. The oxidation rate at the crack tip is measured by the depth of interdendritic oxide spikes formed before fracture, l_{ox},
(28) 
The interdendritic oxidation obeys a t^{1/4} 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, the depth 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 some metallographic 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 2528, the following equation can be derived for the crack depth, a, as a function of the number 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 fatiguedamage model applicable to TMF, which also assumes that fatigue life is spent in the growth of microcracks.^{10} The oxidation and fatigue interaction are considered in the following manner: exposure to high temperatures during the TMF cycle oxidizes the material at the crack tip; then, high stress ranges at medium temperatures give rise to fatigue damage in the material that has been embrittled by oxidation.
Consider a volume element of size, , ahead of the crack tip. This volume element will endure an incremental damage, dD, during dN cycles under oxidizing and cyclicloading conditions. This damage increment is given by
(34) 
with R = 1  _{yy}/_{yy} when _{yy} _{yy}, and R = 0 when _{yy} > _{yy}, where _{eq} is the Von Mises equivalent stress range averaged over the microstructural element at the crack tip; _{yy} is the maximum tensile value of the normal stress of the crack tip at a distance ; S_{0}, 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 the virgin 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 volume element will be given by the condition
(39) 
N() is, thus, computed through the set of Equations 3437 using an iterative procedure, cycle by cycle, until the condition of Equation 39 is fulfilled. The model gives good predictions of lives for an inphase cycle when using a diamondshaped cycle (Figure 4).
An engineering procedure for TMF crackgrowthrate predictions is presented by Dai and colleagues, which is based on isothermal data.^{11} The model uses a linear summation of the contributions to crack growth of the two dominant mechanisms, which are active at minimum and maximum temperatures of the cycle—namely, the mechanical fatigue and environmentally assisted crack growth
(40) 
where (da/dN)_{fat}, the mechanical fatigue contribution, is correlated by a fracture mechanics parameter, K_{eff}
(41) 
where the correlated function, f, is determined by isothermal or TMF tests. Assuming that the cracks were opened when the loads were tensile,
(42) 
where N_{max} is the peak value of the measured load during cycling, a is the crack length measured from the specimen edge, B is the thickness of the specimen, W is the width of the specimen, and H is the geometrical function that characterizes the singleedge notched (SEN) specimen geometry as well as the boundary conditions.
(da/dN)_{env} is the timedependent contribution related to the kinetics of oxygeninduced embrittlement. For TMF, it is computed using
(43) 
where is a material constant, Q is the apparent activation energy for oxygen transport, T is the temperature in ^{o}K, and t_{1} and t_{2} pertain to the starting and finishing times within one cycle where oxygeninduced embrittlement operates. Assuming that t_{1} and t_{2} are related to the opening and closure processes of the crack, the timedependent contributions (Equation 43) were estimated by integrating over the tensile load part of the cycle. Combining Equation 41 (mechanical fatigue) with Equation 43 (contribution of oxygeninduced embrittlement) allows the accurate prediction of the actual crack growth rates (da/dN)_{tot}.
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} 
For each of the inelastic deformation types, a life relation is established, typically given as
(44) 
where A_{ij} and a_{ij}, i,j = p, c (i.e., pp, cp, pc, and cc modes) are materialdependent parameters. When these modes are present in a complex cycle, their damaging effects may be combined by the use of the interaction damage rule (IDR), and a calculation for the life is performed. The IDR can be stated as
(45) 
where F_{pp}, F_{cp}, F_{pc}, and F_{cc} correspond to the relative amount of inelasticstrain type present (e.g., _{pp}/_{in}, _{cp}/_{in}, _{pc}/_{in}, and _{cc}/_{in}), and the terms N_{pp}, N_{cp}, N_{pc}, and N_{cc}, correspond to the life levels established from Equation 44 at the inelastic strain range, _{in}.
SRP has since been successfully formulated on a total strain basis (TSSRP), extending it into the isothermal lowstrain, longlife regime where the inelastic strains are small and difficult to determine. This extension, plus the introduction of the bithermal test to characterize TMF behavior, has permitted the SRP method to be applied to general TMF life prediction problems.^{14–16} (Bithermal experiments utilized a trapezoidalwave temperature versus time profile wherein mechanical straining was imposed only during the 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.)
Fractographic and metallographic investigations were conducted on the specimens of some materials, such as Haynes 188, that were fatigued to failure under IP and OP bithermal and thermomechanical loading conditions. These investigations identified the prevailing damaging mechanisms in both bithermal and thermomechanical fatigue tests. For these identified materials, bithermal fatigue tests could be used as the basis for TMF life prediction.
Total Strain RangeLife Relations
The total strain range is the sum of two terms, _{e} and _{in}. Each term is a power law function of N_{f0} (the subscript 0 on N_{f} denotes the fatigue life for an equivalent zero mean stress condition) and appears as one of a family of straight lines on the loglog coordinates of Figure 6. The solid lines represent upper and lower bounds on the inelastic and elastic strainrange terms. The equation for the general life relation is shown in Figure 6 and is written as
(46) 
or
(47) 
Figure 6. A schematic of strain rangelife relations for the total strain version of strainrange partitioning.^{14} 
The intercepts, B and C', are firstorder functions of many of the variables (e.g., time per cycle, strain rate, environment, wave shape, and temperature) that are known to influence TMF lives. For example, both B and C' typically decrease as the time per cycle increases (lower frequency, lower strain rate, or increased holdtime), since this introduces the phenomena of creep and environmental attack. Creep and environmental attack are usually regarded as detrimental damage mechanisms. As temperature increases, B usually decreases (primarily due to creepweakening effects, but also due to environmental attack), and C' may do either.
Inelastic Strain RangeLife Relations
The intercept, C', is evaluated as
(48) 
where ij = pp, pc, or cp, and F_{ij} is the fractional value of each inelastic strain range present in any given TMF cycle (i.e., the partitioning of _{in}).
(49) 
(50) 
(i.e., F_{ij }= 1.0). F_{ij} can be written as an exponential relation of the time per cycle, normalized by a function 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 tests would be utilized to measure F_{ij} and determine the constants, A', , and m.
In addition to F_{ij}, the other factors influencing the value of C' are C_{ij} and c. Bithermal fatigue test results provide the data base to determine the constants, C_{ij} and c.
Elastic Strain RangeLife Relations
In the elastic strainrange versue life relations, the exponent, b, was determined from bithermal experiments. The intercept, B, is expressed as
(52) 
where n is the cyclic strainhardening exponent, and n is related to b and c by the expression n = b/c. K_{ij} is a timedependent, elasticmodulusnormalized cyclic strength coefficient defined by
(53) 
The relation between K_{ij} and cycle time can be expressed as
(54) 
Normally, the values of the constants, A_{1}' and m_{1}, are_{ }ascertained using a viscoplastic model or the leastsquares 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 be evaluated. Then the total strain range versus cyclic life failure curve will be established, and the TMF life prediction can be made easily.
A sample TMF/TSSRP life prediction curve is shown in Figure 7a. The assessment of the TMF lifeprediction capability of the TMF/TSSRP method is shown in Figure 7b. It is seen that the agreement is generally within the conventional factor of two associated with hightemperature fatigue life prediction in the nominally lowcycle fatigue regime.
A set of damagecurve relations is proposed for the modeling of damage accumulation according to each of the four fundamental deformation and damage modes given in SRP
(55) 
where _{ij}, i,j = p,c may be considered as materialdependent parameters, and the damage variable, D, possesses a range of 0 (undamaged condition) to 1 (failure) over the domain of the life fraction n/N_{ij} of 0 to 1.
When multiple damage modes are operative, an additional set of damage coupling relations is required,
(56) 
where the functions, g_{ij}, i,j = p,c [where g_{pp}(D_{pp}) D_{pp}] must be determined by experiments.
The coupling relations recognize that what is considered as damage in one context cannot be necessarily regarded as damage in another (e.g., pptype cycling is associated with transgranular, classical fatiguetype damage, while cptype cycling is associated with intergranular, creeptype damage). However, the coupling relations do imply that the damage state is relatable. The coupling relations provide a mapping or correspondence 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 contributions may 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 N_{f}) as life fraction, expressed in terms of the four damage variables, D_{pp}, D_{cp}, D_{pc}, and D_{cc}. Note that in general, the coupling 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 fatigue approximation to TMF. To address TMF, the life relations, damagecurve equations, and damagecoupling relations can be directly replaced by bithermal counterparts. Experiments consisting of twolevel loadings of 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 twolevel tests and provided reasonable bounds for the IP twolevel tests.
An effective Jintegral was developed for TMF cracklife prediction. The effective Jintegral is an empirical strainenergydensity crack growth parameter and is not a rigorous fracture mechanics parameter.
The developed effective Jintegral fracture mechanics TMF model is summarized as
(58) 
where a is the crack depth, da/dN is the crackgrowth rate, J_{eff} is the effective Jintegral (an empirical strainenergydensity fracture mechanics parameter), A and c are material constants, and
(59) 
where Q_{0} and T_{0} are material constants, T is the absolute temperature, and dt is the time increment.
(60) 
where a_{0} is the initial material defect size (material constant), b is a material constant, is the crackboundarycorrection factor, and J_{th} is the threshold Jintegral that may be constant or temperaturedependent. For the latter, a good empirical temperaturedependent formula of J_{th} is
(61) 
where J_{0} is a constant and T_{J} is the temperature at which timedependent deformation (creep) becomes significant in monotonic tensile tests.
Where f is a crystallographic strainenergycorrection factor,
(62) 
where f_{<111>} is a factor that resolves _{max} into the maximum normal octahedral slip plane stress, E_{max} is the 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 strainenergy density.
(63) 
where is the material constant, and W_{e} is the elastic strainenergy density
(64) 
where m = 1 if _{min} > 0 and m = 0 if _{min} 0.
W_{p} 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 oxidation tests and by applying a nonlinear leastsquaresregression analysis. Stresses were obtained from a nonlinear finiteelement analysis of a TMF specimen straintemperature history. Nonlinear stressstrain behavior was predicted using unified viscoplastic constitutive models.
The TMF lifemodel formulation is limited to transgranular cracking. The model can capture many TMF cracking effects, such as coating thickness, singlecrystal anisotropy, cycle wave shape, dwell, and thermal exposure. TMF life predictions based on the modified effective Jintegral model were in good agreement with observed uniaxial TMF specimen lives.
Figure 8. Hysteretic energy life curves for a 316 stainless steel subjected to TMF tests in the temperature range of 399621°C.^{21} 
The model developed for gasturbine blades in electric power generation is based upon the parameters of strain range, strain amplitude ratio, and dwell time. This combination of parameters is intuitively reasonable, since the strainamplitude ratio can account for mean strain effects, and the dwell time can consider environmental and mean stress effects caused by steadystate stress relaxation, both of which can affect the fatigue life.
The general form of the model is
(66) 
where N_{f} is the fatigue life; A is the strain ratio, A = _{amp}/_{mean} (_{amp} = , _{mean} = , where all strains are mechanical strains); is the total strain range; t_{h} is the dwell time; and C_{i} (i = 0, 1, 2, and 3) is the empirical constant, determined from multiple regression of the TMF test data. In a test, the model adequately described the behavior of IN738LC superalloy for the TMFOP test conditions in the temperature range of 427871°C.
When the TMF model was used to predict the fatigue life of the inservice gas turbine blades, which may experience the different cycle types, a linear damage rule was used to account for the different cycle types
(67) 
where the damage ratio, D', was assumed to be one at 100% life consumed. N_{i} is the number of cycles of strainingtype i, and N_{if} is the number of cycles to failure of type i.
Zamrik and colleagues have used the hystereticenergy method to characterize the TMF behavior of type 316 stainless steel.^{21} From the stabilized midlife hysteresis loop, the energy per volume cycle or density can be calculated and fit to the energylife relation by
(68) 
where U is the energy per volume cycle, W is the work or area of the hysteresis loop, A is the specimen crosssectional area, L is the specimen gage length, is the stress, d is the strain increment, A_{0} and a are constants, and N_{f} is the number of cycles to failure. The hysteretic energy versus life approach is illustrated in Figure 8. It can be observed that the TMFIP hysteretic energy life line is lower than the TMFOP line, which suggests more damage.
Kanasaki and colleagues have carried out axialstraincontrolled lowcycle fatigue tests of a carbon steel in oxygenated hightemperature water under TMFIP and TMFOP conditions.^{22} Based on the assumption that the fatigue damage increased in a linear proportion to the increment of strain during cycling, a fatiguelife prediction method was proposed.
(69) 
where T_{max} and T_{min} are the maximum and minimum temperatures during TMF cycling, N(T) is the fatigue life in hightemperature water under isothermal conditions, and N' is fatigue life in hightemperature 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 the experiment
(70) 
A good relationship between the predicted N'_{pred} and the experimental N'_{test} was observed.
Damage modeling under thermomechanical cyclic loading is still at an early stage. TMF cycling is expected to introduce a multitude of cyclic deformation and damage mechanisms in superalloys, and the influence of these damage mechanisms on the material's fatiguecrack initiation and propagation behavior is not well understood at present. No generally accepted models of TMF fatiguelife prediction are currently 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 fatiguelife prediction attempt to capture the effects of these damage mechanisms and their interactions. It is not certain if these dominant damage mechanisms operate simultaneously, or if some of them run and others become inoperative during a special TMF loading condition.
There are many factors affecting TMF lives, such as materials, maximum and minimum temperatures, temperature range, strain rate, strain range, straintemperature phase, and dwell and cycle times. There is a 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, TMFlifeprediction models are generally timeconsuming and expensive.
To develop effective TMF lifeprediction approaches for superalloys, more experimental work and data collection are necessary on various materials placed under TMF loading conditions.
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 of Materials Science and Engineering, 427B Dougherty Engineering Building, Knoxville, Tennessee 379962200; (423) 9746356; fax (423) 9744115; email pliaw@utk.edu.
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