Int J Fract (2015) 191:31–51DOI 10.1007/s10704-015-9994-4

SPECIAL INVITED ARTICLE CELEBRATING IJF AT 50

Creep and creep–fatigue crack growth

Ashok Saxena

Received: 28 November 2014 / Accepted: 24 January 2015 / Published online: 17 February 2015© The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract Creep and creep–fatigue considerations areimportant in predicting the remaining life and safeinspection intervals as part of maintenance programsfor components operating in harsh, high temperatureenvironments. Time-dependent deformation associatedwith creep alters the crack tip stress fields establishedas part of initial loading which must be addressed inany viable theory to account for creep in the vicin-ity of crack tips. This paper presents a critical assess-ment of the current state-of-the-art of time-dependentfracture mechanics (TDFM) concepts, test techniques,and applications and describes these important devel-opments that have occurred over the past three decades.It is concluded that while big advances have been madein TDFM, the capabilities to address some significantproblems still remain unresolved. These include (a) ele-vated temperature crack growth in creep-brittle materi-als used in gas turbines but now also finding increasinguse in advanced power-plant components (b) in pre-dicting crack growth in weldments that inherently havecracks or crack-like defects in regions with microstruc-tural gradients (c) in development of a better fundamen-tal understanding of creep–fatigue–environment inter-actions, and (d) in prognostics of high temperature

A. Saxena (B)College of Engineering, University of Arkansas, 130 AJohn A. White Hall, 790 W. Dickson Street, Fayetteville,AR 72701, USAe-mail: asaxena@uark.edu

component reliability. It is also argued that while theseproblems were considered intractable a few years ago,the advances in technology do make it possible to sys-tematically address them now and advance TDFM toits next level in addressing the more difficult but realengineering problems.

Keywords Cracks · Fatigue · Creep · Welds · Ni-basealloys · Ferritic steels · Ct parameter · C∗-integral ·(Ct )avg parameter

1 Introduction

Structural components that experience harsh environ-ments such as high temperatures and stresses are com-mon in land-based steam and gas turbines, in aircraftengines, and in power-plant components. These com-ponents are often cast, forged or welded making themprone to having crack-like manufacturing defects. Theconsiderations that are involved in determining the riskof fracture in these components are several that include:

• Transient and steady-state thermal stresses• Hold times, static and cyclic stresses due to external

loading• Environmental effects• Creep deformation (primary, secondary, tertiary

creep) and rupture• Creep–fatigue and environmental damage mecha-

nisms

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32 A. Saxena

Fig. 1 The estimated relationship between flaw size at inspec-tion and the remaining life in steam turbine casings in regionsexposed to different temperatures (Saxena et al. 1986)

• Varying material properties due to temperature gra-dients

• Complex crack geometries and variable amplitudeloading

• In-service degradation of material properties• Weldments with microstructural gradients• Directionally solidified and single crystal materi-

als with anisotropic creep and plastic deformationproperties

Such a set of complex conditions are difficult toaddress in a single comprehensive theory so researchershave endeavored to break-up the problem into smallerelements and address them one or two at a time. This hasled to a comprehensive set of analytical tools and mod-els that allow us to approach real problems using themost appropriate analytical tools for specific situations.Examples of such analyses date back to nineteen eight-ies in the form of remaining life and inspection intervalcalculations in steam turbine casings used in ships togenerate the auxiliary power as shown in Fig. 1 (Saxenaet al. 1986). For regions exposed to 538 ◦C (1,000 ◦F),and a crack depth of 6 mm (0.25 in.) in the steam

passage-ways of steam turbine casings, the estimatedremaining life was 800 cycles or at 50 cycles/year,about 16 years. In this example the crack growth ratesestimated from direct measurements of oxide thicknesson crack surfaces excavated from turbine casings vali-dated the analytical estimates thus, significantly raisingthe confidence levels in the models that were developedspecifically for this application.

1.1 Early developments in TDFM

Some of the earliest studies involving the use of fracturemechanics to predict crack growth rates at elevated tem-peratures are in the work of James (1972) who success-fully extended the idea of using the cyclic stress inten-sity parameter, �K , to correlate fatigue crack growthbehavior of 304 stainless steel at elevated temperaturesat various frequencies. About the same time, Sivernsand Price (1973) unsuccessfully attempted to correlatecreep crack growth rate to the stress intensity parame-ter, K . Under static loading, K uniquely characterizesthe crack tip stress field only upon loading at time,t = 0+. With elapse of time, stress relaxation at thecrack tip occurs due to creep deformation and there isno longer a uniqueness in the relationship between Kand the crack tip stress fields (Saxena 1980; Riedel andRice 1980). On the other hand under cyclic loading, thefatigue crack growth rate does continue to be uniquelycharacterized by �K but only if the time related para-meters such as the frequency of loading and the load-ing waveform are kept constant (Saxena 1980; Saxenaet al. 1981; Riedel 1983). The functional relationshipbetween fatigue crack growth rate and �K is given byEq. 1.

da

d N= f (�K , frequency, waveform, R, T ) (1)

where, da/dN = fatigue crack growth rate and R = loadratio of the loading waveform and T = test temperature.

Landes and Begley (1976) proposed the C∗-integralfor characterizing creep crack growth rates, basedon the mathematical analogy between C∗ and Rice’sJ -integral (Rice 1968), as a candidate parameter forcharacterizing creep crack growth rate. They alsorecognized its importance as a crack tip parametercapable of uniquely characterizing the crack tip stressfields (Eq. 2a) under the idealized conditions of wide-spread secondary-stage creep throughout the cracked

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Creep and creep–fatigue crack growth 33

body by making another analogy to the HRR fieldsproposed for J-Integral (Hutchinson 1968; Rice andRosengren 1968). C∗ was also related to the stress-power dissipation rate (U∗) in their paper as shownin Eq. (2b). They attempted to experimentally demon-strate the validity of C∗ by conducting experimentson alloy A286 using both dominantly bend-type, com-pact, C(T), specimens and middle crack tension, M(T),specimens. Their results only met with partial successbecause the condition of widespread secondary creepwas not met in their specimens for much of the dura-tion of the tests. Consequently, the creep crack growthrates measured from C(T) and M(T) specimens differedfrom each other. Attempts to find J-like parameters tocorrelate creep crack growth rates were also made inEurope (Nikbin et al. 1976) and in Japan (Taira et al.1979) but the association with crack tip stress fields wasnot recognized in these studies. Saxena (1980) showedthat creep crack growth rates were uniquely correlatedwith C∗-Integral for C(T) and M(T) specimens for 304stainless steel at 594 ◦C (1,100 ◦F) because the nec-essary conditions of widespread steady-state creep (aterm interchangeably used with secondary creep in theliterature and here) were established shortly after load-ing and the creep crack growth rate occurred primarilyunder this tailor-made conditions for C∗ to describethe crack tip conditions. In the same paper it was alsoshown how the value of C∗-Integral can be analyti-cally calculated from equations derived to estimate J -Integral for cracked bodies under fully plastic condi-tions such as in Kumar et al. (1981).

σi j =(

C∗

In Ar

) 1n+1

σi j (θ, n) (2a)

C∗ = − 1

B

dU∗

da(2b)

where, A and n are steady-state creep constants in aNorton’s power–law relationship, r =distance from thecrack tip, θ is the angular coordinate, σi j (θ, n) is anangular function tabulated with the HRR fields, B isthickness of the planar cracked body and a is the cracksize.

2 Stress analysis of cracks in the small-scale-creepregime

With establishment of C∗ as the crack tip parame-ter for characterizing creep crack growth rates under

widespread steady-state creep conditions, the attentionin early nineteen eighties turned to its limitations andsearch for parameters that could be used under small-scale creep conditions and would become identical toC∗ when widespread steady-state creep conditions areestablished. This body of analytical and numerical sim-ulations work can be divided in two parts (a) consid-eration of stationery cracks and (b) consideration ofmoving cracks.

2.1 Analysis of stationery cracks

Work of Riedel and Rice (1980), Ohji et al. (1979) andBassani and McClintock (1981) was key in unlockingthe mechanics of this problem. The Riedel and Rice andOhji et al. analytically derived the crack tip stress fieldsahead of the crack tip under small-scale creep condi-tions while Bassani and McClintock (1981) numeri-cally verified these analytical results and extended theanalysis by proposing a parameter, C(t), that charac-terized the amplitude of the crack tip stress singularityfrom small-scale creep all the way to widespread creepfor steady-state creep deformation, Eq. 3. For wide-spread creep, C(t) becomes identical to C∗ and forsmall scale creep it is related to a line integral that ispath dependent given in Eq. 4.

σi j =(

C(t)

In Ar

) 1n+1

σi j (θ, n) (3)

For small-scale creep,

C (t) =∫

Γ →0

[W ∗ sin θ − Ti

(∂ ul

∂x1

)]ds (4)

where, T is the traction vector, W ∗ is the stress-powerdensity and u is the displacement vector, � is a con-tour that begins on the lower crack surface and trav-els counter clock-wise ending on the upper crack sur-face. The contour must be defined in a region wherecreep deformation dominates near the crack tip and thusthe condition � → 0. Under widespread creep, C(t)becomes identical to C∗ by definition. Schemes weredeveloped (Bassani and McClintock 1981) to estimateC(t) analytically and the following equations were pro-posed.

For t � t1, C (t) ≈ K 2(1 − ν2)

E A (n + 1) t(5a)

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34 A. Saxena

For t � t1, C (t) = C∗ (5b)

For 0 <t

t1< ∞,

C (t) ≈ C∗[

K 2(1 − ν2)

E A (n + 1) t+ 1

](5c)

The transition time, t1, between small-scale creep andwidespread creep was obtained by equating Eqs. 5aand 5b for the two extreme conditions to derive Eq. 6:(Riedel and Rice 1980; Ohji et al. 1979).

t1 = K 2(1 − ν2)

E A (n + 1) C∗ (6)

Based on Eqs. 3 and 5a, b, c, C(t) is an attractiveparameter for characterizing creep crack growth ratebut its major disadvantage is that it cannot be mea-sured at the loading pins or it is not uniquely relatedto the stress-power dissipation rate in the small-scalecreep regime like C∗ is in the widespread creep regime.Recall that C∗ serves both functions. The former short-coming is particularly a problem because one is thencompletely dependent on accurate constitutive equa-tions for estimating C(t) which must include, in addi-tion to elastic deformation and steady-state creep, theeffects of instantaneous plasticity, primary creep, ter-tiary creep, 2-D versus 3-D stress fields in the cracktip region and the scale of creep deformation. All thesedeformation phenomena are operative in the crack tipregion so it is not sufficient to just account for elas-tic deformation and steady-state creep as is assumed inEqs. (5).

Addressing the above shortcoming, Saxena (1986)proposed the Ct parameter, measurable at the load-lineof the cracked body and thus including contributionsfrom all the different deformation mechanisms presentat the crack tip. Ct generalizes the stress-power dissi-pation rate definition of C∗ into the small-scale creepregime in the following way:

(Ct )ssc = − 1

B

∂U∗t (a, t, Vc)

∂a(7)

where, U∗t is the instantaneous stress-power at any

time, t , that is also a function of the crack size, a, andthe deflection rate due to creep, Vc. Expressions werederived to measure the value of (Ct )ssc in specimens(Saxena 1986), Eq. (8), and subsequently it was alsouniquely related to the rate of expansion of the creepzone at the crack tip (Bassani et al. 1989), Eq. (9).

(Ct )ssc = PVc

BW

F ′

F(8)

(Ct )ssc = 2K 2(1 − ν2)

EW

F ′

Fβrc (9)

where, W=width of the cracked body, E=elastic mod-ulus, β ≈ 1

3 , P=Load on the specimen, ν =Poisson’s

ratio, F = K -calibration factor= ( KP

)BW

12 , F ′ =

d F/d( aW ), rc = rate of expansion of the creep zone

size at the crack tip. Ct also then uniquely character-izes the rate of expansion of the creep zone size andis therefore an attractive candidate crack tip parameterand it is also by definition identical to C∗ under wide-spread creep conditions. However, under small-scalecreep conditions Ct = C(t) (Saxena 1998).

2.2 Growing crack considerations in creep-ductileand creep-brittle materials

The growing cracks perturb the stationery crack fieldsand if this perturbation occurs over a sizeable regionnear the crack tip compared to the region of dominanceof the stationery crack tip fields, crack parameters suchas K , C∗ and Ct may be unable to correlate uniquelywith creep crack growth rates. The material character-istics themselves play a big role in determining whichof the above parameters, if any, are appropriate. Thus, aclassification of the different material types is in order.

High temperature structural materials may be clas-sified as creep-ductile or creep-brittle materials. Creep-ductile materials are ones in which the crack growth isaccompanied by substantial amounts of time-dependentcreep strains and the region of influence of growingcrack fields is small in comparison to the creep zonesize. Similarly, in the case of widespread creep, theregion of influence of growing crack fields are smallin comparison to the zone in which C∗ dominatesthe crack tip fields. Examples of these materials arechromium (Cr), Molybdenum (Mo) and vanadium (V)containing ferritic or bainitic steels used in the fos-sil power-plant applications. Creep-brittle materials, onthe other hand, are those in which creep crack growthoccurs in the presence of small amounts of creep defor-mation so the creep zone size ahead of the movingcrack tip always remains constrained within a smallregion. This class of materials includes Nickel basesuperalloys used commonly in the land-based gas tur-bines and in aero-engines and are now finding usage

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Creep and creep–fatigue crack growth 35

in advanced power-plant components and in advancednuclear power stations.

Hui and Riedel (1981), Reidel and Wagner (1981),Hui (1983, 1986) proposed that in order for steady-statecrack growth conditions characterized by K and C∗ toexist, both the crack growth rate and the magnitudeof crack tip parameters must change slowly with time.These conditions are expressed by Eqs. (10) and (11)for the conditions of small-scale creep dominated by Kand widespread creep dominated by C∗, respectively.The dots signify time derivatives of those quantities.

|a| ≤[

an−2

E AK n−1

] 2n−3

(10a)

∣∣K ∣∣ ≤[

an−1

E2 A2 K n+1

] 1n−3

(10b)

|a| ≤ A

(En+1C∗n−1

an−3

)1/2

(11a)

∣∣C∗∣∣ ≤ A

((EC∗)n+1

an−1

)1/2

(11b)

Conditions bound by Eqs. (10) and (11) under whichsteady-state creep crack growth uniquely characterizedeither by K or C∗ may not frequently be encounteredin practice. Therefore, it is also important to under-stand the crack tip mechanics during transient periodswhich may account for a significant portion of the crackgrowth during a laboratory test or in a component.

Transient regions are defined as (a) the region priorto the establishment of C∗ dominated crack tip con-ditions for cracks growing at speeds slow enough tobe considered stationery for practical purposes such asin creep-ductile materials or (b) the region in whichcracks grow at sufficient speeds in creep-resistant orcreep-brittle materials to preclude C∗ or Ct dominatedconditions but also steady-state conditions dominateduniquely by the stress intensity parameter, K, are notachieved. It may take substantial amounts of crackextension prior to the rates being uniquely correlatedto K .

Figure 2 shows the results of finite element simula-tions that were carried out by Hall et al. (1997) to under-stand the development of the crack tip creep defor-mation with time in real tests conducted on a creep-resistant aluminum alloy. These simulations were car-ried out on C(T ) specimens used in actual tests in which

the creep deformation properties, the specimen geom-etry and the load levels were identical in two separatesimulations. In one case the crack size versus time his-tory input into the simulations were the experimentallymeasured crack size with time. These conditions weretypical of creep-brittle materials. In the other case, thecrack growth rates were artificially decreased by a fac-tor of ten to allow more creep strains to accumulateat the crack tip. These conditions are characteristic ofcreep-ductile materials. The crack tip creep zone evo-lution is shown to be strongly influenced by the crackgrowth rate and the creep zone size and shape are quitedifferent for the two cases as evident from Fig. 2. Alsoin the creep-brittle case, the creep zone shape aheadof the moving crack tip changes with time and there-fore, steady-state conditions cannot develop during thetests and K is not expected to uniquely characterize thecreep crack growth rates that was in fact experienced asshown in Fig. 3 (Hamilton et al. 1997) where every testproduced a different correlation between creep crackgrowth rate and K .

In creep-ductile materials, the extent of the regionin which the growing crack field is dominant is negli-gible and the creep zone size evolves from small-scaleto widespread creep conditions giving rise to transientcrack tip conditions (Saxena 1986). The creep crackgrowth rates have been shown in such instances touniquely correlate with the Ct parameter (Saxena 1986and Saxena et al. 1994) as seen in Fig. 4. In these testsconducted on C(T) specimens that were 254 mm wide,it was shown that very significant amounts of crackextension occurred prior to the establishment of wide-spread creep conditions and in spite of that, all creepcrack growth rate data correlated well with Ct . Thus, weconclude that our understanding of creep crack growthin creep-ductile materials under a wide range of tran-sient and steady-state conditions is good but not so forcreep-brittle materials for which there does not exist aCt type parameter.

To distinguish between creep-ductile and creep-brittle materials, Saxena et al. (1983) proposed amethod based on partitioning the measured load-linedisplacement rate, V , during testing between that asso-ciated with creep deformation (Vc) for no crack growthand that due to change in elastic (Ve) and plastic (Vp)

compliance of the specimen due to crack extension. Themeasured load-line displacement rate (V ) is the sum ofall three. For steady-state creep crack growth to occurthat is correlated with either C∗ or Ct , Vc � (Ve+Vp).

123

36 A. Saxena

Fig. 2 Finite elementsimulations of creep crackgrowth under creep-ductileand creep-brittleconditions Hall et al. (1997).The specimens analyzedwere C(T) type with a widthof 50.8 mm and a startingcrack size of 20.3 mm

They estimated the value of Ve + Vp by Eq. 12:

Ve + Vp ≈ aB

p

[2k2

E+ (m + 1)Jp

](12)

where, Jp is the fully plastic component of the J -integral and m is the plastic exponent in the power–lawrelationship between plastic strain and applied stressfor the material. It is required that for creep crackgrowth rate to correlate with Ct , that Vc � (Ve + Vp).In practice, the condition is expressed as Vc ≥ 0.8V .

Standard test methods for characterizing creep crackgrowth rates in creep-ductile materials are available

via the American Society for Testing and Materials(ASTM) test standard E1457 [ASTM Standard E1457].This method currently only applies to creep-ductilematerials for which accepted crack tip parameters areavailable to correlate the creep crack growth behavior.In the future, it would be useful to develop these meth-ods to include creep-brittle materials also.

2.3 Consideration of primary creep

A creep constitutive law which combines elastic defor-mation, primary creep, and steady-state creep can bewritten as in Eq. 13.

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Creep and creep–fatigue crack growth 37

Fig. 3 Creep crack growth behavior of a creep-brittle Al alloy2519-T87 (Hamilton et al. 1997). The lack of uniqueness in thecorrelation between da/dt and K is apparent in data from variousspecimens that are identical except in the starting crack size andthe applied load

Fig. 4 Creep crack growth rate as a function of Ct parameter in1Cr–1Mo–0.25V steel at 538 ◦C under small-scale creep to wide-spread creep in large C(T) specimens (Saxena et al. 1994). VAH1and VAH 2 refer to two specimens tested at different load levelsand the arrows point to crack growth rates from the beginning ofthe test going forward in time

ε = σ

E+ A1ε

−pσ n1(1+p) + Aσ n (13)

In Eq. 13, the first and the third terms on the righthand side of the equation pertain to elastic and sec-ondary creep strains while the middle term pertainsto primary creep strains that contain three constants,A1, n1, and p that are obtained from regression of pri-

mary creep data. If primary creep condition dominatesthe deformation behavior, Eq. 13 can be simplified toEqs. 14a, b, c.

∈ = A1ε−pσ n1(1+p) (14a)

or

∈ = [(1 + p)A1t]1/(1+p) σ n1 (14b)

and

∈ = A1[(1 + p) A1t] −pp+1 σ n1 (14c)

Riedel (1981) recognized that Eq. 14b for constanttime is analogous to power–law relationship betweenstress and strain, thus, a J -integral can be defined thatwill be path-independent for any given time t . He alsopointed to another path-independent integral, C∗

h whichis independent of time for primary creep conditions asin Eq. 15.

J = C∗h t

11+p (15)

C∗h can be obtained by replacing A by [(1 + p) A1t] 1

(1+p)

and n by n1 in equations for determining C∗. Further,from Eq. 14c, C∗-integral can also be derived that is afunction of time and related to C∗

h by Eq. 16.

C∗(t) = C∗h

(1 + p)tp

1+p(16)

These equations permit accounting for primary creep,if present, in estimating the magnitude of crack tip para-meters. This approach can also be extended for small-scale creep. For further details, the readers are referredto Saxena (1998).

This section shows that the crack tip mechanics forstationery cracks under a variety of crack tip deforma-tion conditions is well understood. For creep-ductilematerials where the rate of advance of the crack tip ismuch slower than the rate of creep deformation at thecrack tip, this understanding is sufficient for the pur-poses of analyzing components. In creep-brittle mate-rials, where there is competition between the evolutionof stress and strain fields due to time-dependent defor-mation and due to crack growth, there are no unifyingconcepts that are currently available to analyze thesesituations. This is a major gap in technology that cur-rently exists and prevents the use of time-dependentfracture mechanics approach to an important class of

123

38 A. Saxena

high temperature materials such as intermetallics andNi base alloys and high temperature ceramics. This canbe addressed with more research.

Fig. 5 Fatigue crack growth behavior at of a 1Cr–1Mo–0.25Valloy at 538 ◦C (Grover 1997; Grover and Saxena 1999). The dataare a collection of results from samples taken from base metal(D1 and D2) and from weldments (TC1, TC2, and TC-5)

3 Creep–fatigue crack growth

3.1 Crack tip parameters for creep–fatigue loading

For cyclic loading, the crack tip stress fields duringeach cycle will at least partially regenerate every cycleif small-scale creep and small-scale plastic deformationcan be maintained. Thus even in the presence of a holdtime that follows the rising load portion of the wave-form, crack growth rates can be uniquely characterizedby �K for that loading waveform and cycle time. Ifeither the cycle time or loading waveform changes, thefunctional relationship between the crack growth rateper cycle, da/dN, and �K will change. This was earlierexpressed in Eq. 1.

Some examples of such correlations are shown inFigs. 5 and 6 for creep-ductile and creep-brittle materi-als. These relationships are valid only for time periodsof cycles that are much less than the transition timedefined in Eq. 6. For longer times, the validity of thesmall-scale-creep conditions cannot be assured so the

Fig. 6 a Fatigue crackgrowth behavior of Inconel718 at various loadingfrequencies (Floreen andKane 1980) and b forvarious hold times forAstroloy (Pelloux andHuang 1980)

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Creep and creep–fatigue crack growth 39

Fig. 7 Fatigue crack growth rate in a Cr–Mo–V creep-ductilesteel at 5,380 ◦C for a waveform containing a hold time of 60 sat the maximum load (Grover and Saxena 1995)

validity of Eq. 1 also comes into question. In Fig. 5,results are presented from multiple tests on a creep-ductile 1 Cr–1Mo–0.25 V steel at 538 ◦C at a load-ing frequency of 1 Hz subjected to continuous cyclingusing a triangular waveform at different load ampli-tudes (Grover 1997; Grover and Saxena 1999). All testsyield identical crack growth rates when correlated with�K . In Fig. 6, the loading frequency is varied for tests

conducted on a creep-brittle Inconel 718 (Floreen andKane 1980) and Astroloy (Pelloux and Huang 1980)materials. We see a clear layering of the data with differ-ent loading frequencies or hold times validating Eq. 1.The cyclic crack growth rates show a definite trend withincreasing hold time or increasing cycle time but stillshow a good correlation with �K .

When hold time is added to the cyclic loadingwaveforms of creep-ductile materials, the fatigue crackgrowth rates tend to show a hook in the behavior whencorrelated to �K , Fig. 7 (Grover and Saxena 1995).This behavior is due to ratcheting of the minimum load-line deflection with each cycle because the specimenaccumulates creep strains near the crack tip that are notfully reversed during unloading. For such situations,the use of the (Ct )avg parameter defined by Saxenaand Gieseke (1987) is recommended.

Studies (Grover and Saxena 1999; Yoon et al. 1993)performed on creep-ductile power plant materials usingC(T) specimens have shown (Ct )avg to be the mosteffective crack tip parameter in correlating creep–fatigue crack growth rates under trapezoidal wave formwith a hold time, th . In such cases, the time dependentcrack growth, (da/dt)avg , during the hold period isexpressed as a function of the average magnitude ofthe Ct parameter,(Ct )avg , defined in Eq. 17.

(Ct )avg = 1

th

∫ th

0Ct dt (17)

(Ct )avg is estimated from Eq. 18 for test specimens inwhich both load and load-line deflection behavior withtime are measured:

(Ct )avg = ΔPΔVc

(B BN )1/2 W th(F ′/F) (18)

where, �P is the applied load range, �Vcis the dif-ference in load-line displacement between the end andstart of the hold time, th , during a cycle. F ′/F is givenby Eq. 19.

F ′

F=

[(1

2 + a/W )

)+

(3

2(1 − a/W )

)]

+[ (

4.64 − 26.64 (a/W ) + 44.16 (a/W )2 − 22.4 (a/W )3)0.886 + 4.64 (a/W ) − 13.32 (a/W )2 + 14.72 (a/W )3 − 5.6 (a/W )4

](19)

The value of (Ct )avg from Eq. 18 is appropriate forthe small scale creep regime. When hold times aretoo small for reliable measurement of load-line dis-placement range because of the resolution limitationsof the extensometer, (Ct )avg can be estimated usingEq. 20 (Yoon et al. 1992).

(Ct )avg = (Ct )ssc + C∗ (20)

(Ct )ssc can be estimated using Eq. 21 (Saxena andGieseke 1987):

(Ct )ssc = 2αβ(1 − ν2

)E

Fcr (θ, n)�K 4

W

(F ′/F

)(E A)

2n−1 t

− n−3n−1

h

(21)

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40 A. Saxena

Fig. 8 Fatigue crackgrowth rate behavior of9Cr-1Mo steel at 625 ◦C asa function of �K(Narasimhachary andSaxena 2013)

where

α = 1

2π

((n + 1)2

1.38n

) 2n−1

(22)

For θ = 90◦, the value of β ≈ 0.33 and Fcr (90◦, 8.24)

= 0.387 (Saxena and Gieseke 1987). C∗can be esti-mated using Eq. 23

C∗ = A (W − a) h1 (a/W, n)

(P

1.455η1 B (W − a)

)n+1

(23)

where, η1 for C(T) specimen is given by Eq. 24.

η1 =[(

2a

W − a

)2

+ 2

(2a

W − a

)+ 2

]1/2

−[(

2a

W − a

)+ 1

](24)

The first term on the right hand side of the Eq. (21) rep-resents the small-scale creep contribution and secondterm represents the extensive creep contribution to thevalue of (Ct )avg . The function h1 (a/W, n) for variousa/W and n are available in Kumar et al. (1981).

The fatigue crack growth rates, da/d N , are plottedas a function of�K , as shown in Fig. 8 (Narasimhacharyand Saxena 2013). The data with hold times of 60 and600 s do not uniquely correlate with �K even between

tests performed at the same hold times, during the ini-tial portions of the tests. There is also a significant con-tribution of time dependent crack growth under bothhold time conditions signaling creep–fatigue interac-tions being present. These data were next plotted withthe (Ct )avg parameter.

The average time-dependent crack growth rates,(da/dt)avg , were calculated using both the linear dam-age summation model and the dominant damage model.The governing equation for the linear damage summa-tion model under creep–fatigue condition is given byEq. 25.

da

d N=

(da

d N

)cycle

+(

da

d N

)time

(25)

where (da/d N )time is related to (da/dt)avg by:

(da

dt

)avg

= 1

th

(da

d N

)time

(26)

The first term on the right hand side of the Eq. 25accounts for cycle dependent crack growth rate and thesecond term represents the time-dependent part. In thisapproach, the cycle-dependent and the time-dependentcrack growth rates during creep–fatigue conditionsare assumed to be independently additive. The cycle-dependent crack growth rate (da/d N )cycle was deter-mined from fatigue tests at 625 ◦C without hold time.

123

Creep and creep–fatigue crack growth 41

Fig. 9 Average time rate ofcrack growth duringhold-time as a function(Ct )avg of for hold times of60 and 600 s for 9Cr–1Mosteel at 625 ◦C(Narasimhachary andSaxena 2013)

By re-arranging the equation, (da/dt)avg , can be cal-culated by Eq. 27.

(da

dt

)avg

= 1

th

[(da

d N

)−

(da

d N

)cycle

](27)

The alternate approach for representing creep–fatiguecrack growth is the dominant damage hypothesis. Thegoverning equations for estimating the crack growthrate from this model is given by Eq. 28.

da

d N= max

[(da

d N

)cycle

,

(da

d N

)time

](28)

where (da/dt)avg is expressed by Eq. 29.

(da

dt

)avg

= 1

th

(da

d N

)time

(29)

In this approach, the cycle-dependent and the time-dependent crack growth rates during creep–fatigueconditions are assumed to be competing mechanismsand the crack growth rate is determined by the greaterof the two. Both approaches, in other words, the dom-inant damage hypothesis and the damage summationhypothesis, are used to plot the data so that the differ-ences between the two approaches can be assessed.

Figure 9 shows the plot between the average crackgrowth rates estimated from Eqs. (27) and (29) corre-lated with (Ct )avg (Narasimhachary and Saxena 2013).The differences in the crack growth behavior for both60 and 600 s hold times from the linear damage sum-mation and the dominant damage hypothesis are small.Neither of the two approaches addresses true inter-action between creep and fatigue but they bound theextent of the interactions between two extremes whichis useful in engineering applications. The first approachsupposes no interaction while the other supposes thatall the crack growth is associated with only one of thetwo mechanisms. Thus, a conservative way to accountfor creep–fatigue interactions is to use Eq. 28 for ana-lyzing creep–fatigue crack growth data because it isconservative and then use Eq. 25 in applications whencreep–fatigue crack growth rates are being predicted incomponents.

Another observation from Fig. 9 is the difference inthe creep–fatigue crack growth rates between 60 and600 s hold time in the 9Cr ferritic/martensitic steels. Inprevious studies involving lower Cr content steels, thedata for hold times ranging from 10 s to 24 h follows thesame trend when correlated with (Ct )avg (Yoon et al.1993; Saxena and Gieseke 1987) and all fall withinthe scatter-band for the creep crack growth behavior.It should be expected that for long hold times, evenfor the 9Cr material tested in this study, the time-rateof crack growth must converge with the creep crack

123

42 A. Saxena

growth rates. When creep–fatigue crack growth ratesbecome identical to creep crack growth rates and howthat varies from material to material, deserves a morein-depth investigation.

3.2 Creep–fatigue–environment interactions

Several models for creep–fatigue interactions (Sax-ena et al. 1981; Saxena and Bassani 1984) and forcreep–fatigue–environment interactions (Saxena 1983)to account for hold time effects have been proposed.Predictions from these earlier phenomenological mod-els are shown in Fig. 6b in which the data from 0and 2 min hold-time tests are used to derive the modelconstants and then the resulting equation is used topredict the expected creep–fatigue–environment domi-nated crack growth rates for 15 min hold time tests. Thefigure shows the good agreement between the modelprediction and the test results.

A new crack growth model for creep–fatigue envi-ronment effects was developed by Evans and Saxena(2009) incorporating a thermally activated dislocationmodel of Yokobori et al. (1975) and an environmentallyassisted time-dependent crack growth model to predictthe creep–fatigue–environment interactions in creep-brittle materials. This new model addresses the issuesmore comprehensively by including effects of temper-ature. The cycle dependent component, (da/d N )cycle,can be divided into the athermal and the thermally acti-vated dislocation components and the time rate of crackgrowth in this model is given by Eq. 30.da

dt= α A exp

(− Q

RG T

)

C0

⎛⎝4πγ 3

3kT

[k2

tr

] 1n+1

β(σxx (θ, n) + σyy(θ, n)

)⎞⎠

(30)

Equation 30 can be written as a simplified Eq. 31.da

dt= A′ exp

(Ψ K

21+n

), (31)

where,

Ψ = Ψ ′(

1

T

) (1

t

) 1n+1

, (32)

A′ = A′′ exp

(− Q

RG T

)(33)

and A′ and Ψ are determined from the regression analy-sis, Ψ ′ and A′′ are constants, Q is the activation energy

of the reaction, and t is the time at any point along thecycle. A comprehensive model can be obtained by com-bining the time and cycle dependent terms and usingthe definition that Kmax = �K/(1 − R) to get(

da

d N

)tot

= qΔK n1 + c0 · ΔK m0+ c1RT · exp

(− Q0

RG T

)

+∫ th

0A′′ exp

(− Q

RG T

)

× exp

(Ψ ′

(1

T

) (1

t

) 1n+1 ΔK

1 − R

2n+1

)dt

(34)

The total crack growth rate, (da/d N )tot , can thus bedetermined by obtaining a curve fit for the elevated tem-perature fatigue crack growth rate (FCGR) data. The(da/d N )ath crack growth rate can be determined fromregression of room temperature data and (da/d N )th isobtained from time and environment independent dataacquired at elevated temperature by testing in vacuum.The constants for this model were determined usingFCGR data obtained in air at 25 ◦C and in a vacuumand in air at 704 ◦C using a 0 and 10 s hold time for theNi-base superalloy ME3 (Evans and Saxena 2009). Forthe thermally activated dislocation based crack growthrate model, the material constant m was assumed tobe independent of temperature (Evans 2008). The acti-vation energy, Q, was estimated to be 33,455 J/mol.The agreement between the model and the experimen-tal results for 0 and 10 s is shown in Fig. 10 showinga good fit with the test data. Also, the prediction for 5and 30 s hold times fall within the expected behavior.

Several gaps exist in understanding and modelingthe creep–fatigue–environment interactions in creep-ductile and creep-brittle materials. The necessary ana-lytical and experimental tools for such work are avail-able and by focusing on that can raise the understand-ing and the related technology to the next higher levelneeded to design structural components that will oper-ate under extreme environment conditions.

4 Creep and creep–fatigue behavior in weldscontaining mis-matched creep properties

Weld integrity is often an issue with high tempera-ture components. However, welds present special chal-lenges because of the complexity of microstructuralgradients in the weld region where the defects are

123

Creep and creep–fatigue crack growth 43

Fig. 10 A comparisonbetween predicted andexperimental results for Nibase alloy ME3 (Evans2008)

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

10 100ΔK (MPa m1/2)

da/d

N (m

m/c

yc)

704C, Air, Fast Cool, 10 sec Hold704C, Air, Fast Cool, 0 sec HoldModel 5 sec hold timeModel 10 sec hold timeModel 0 sec hold timeModel 30 sec hold time

30 sec hold time

10 sec hold time

5 sec hold time

0 sec hold time

present and the environment in which they grow. Themicrostructure consists of the base metal (BM), weldmetal (WM) and a heat affected zone (HAZ). Typically,the boundaries between these regions are not sharpand the surfaces not planar so crack tips can meanderfrom one region to another as the cracks grow. Thecreep deformation properties may vary between thethree major regions of the weld and therefore it cannotbe considered homogeneous for analytical purposes,except for the elastic properties. Even while recog-nizing the complexities, the analytical models assumesharp interfaces and the crack to be located at the inter-face with different creep deformation properties oneither side of the interface. For these idealized situa-tions, it is shown that strains can preferentially accu-mulate in the weaker region and cause the cracks togrow in that region (Cretegny and Saxena 1998). Fur-ther, if we assume power-law creep to be operative oneither side of the interface, C∗ can be shown to be pathindependent making it measurable using the load-linedisplacement rate. Even for small-scale creep, the creepzone expansion will be uneven on the two sides of theinterface but its expansion rate will be measurable atthe loading points. This makes Ct to be a valid parame-ter for characterizing the creep crack growth rate and(Ct )avg to be valid for creep–fatigue conditions.

Creep and creep–fatigue crack growth tests wereconducted on weldments (Grover 1997) using differ-ent hold times, starting crack sizes, different mater-ial conditions such as welds with over-matched andunder-matched weld materials. Fatigue crack growthtests without hold time were conducted using speci-mens D1 and D2 from the base material, and TC1,

TC2, and TC5 from welds at a cyclic frequency of1 Hz, Fig. 5, showing virtually identical behavior fromall specimens. This shows that if the conditions remaindominantly elastic, the fatigue crack growth behaviorbetween the weld metal and the base metal remain sim-ilar.

The data from tests with different hold times of10, 60, and 600 s are plotted in Fig. 11 (Grover 1997;Grover and Saxena 1999; Saxena 2007). Included inthis plot are data from tests conducted on the base metaland also over-matched and under-matched weldmentsand the data from creep crack growth tests conductedon the base material. The (Ct )avg values in Fig. 11 wereobtained from load-line displacements measured dur-ing the hold time, thus, it accounts for the compositecreep deformation rates for the weld metals and therespective base metal. There seems to be some differ-ence between the CCG (creep crack growth rate) andthe CFCG behavior; the overmatched weld conditionseems to perform slightly better under creep–fatigueconditions.

It was also noted from these studies that the differ-ences between the (da/dt)avg versus (Ct )avg behav-ior between the various material conditions is small (afactor of four) compared to much larger differences inthe creep deformation behavior (almost two orders ofmagnitude depending on the applied stress). This, how-ever, does not imply that the different weld conditionswill result in creep–fatigue lives that are within a factorof four of each other. On the contrary, the differencescan be quite large because the higher creep rates inwelds that are less resistant to creep deformation willyield higher (Ct )avg compared to the ones that are more

123

44 A. Saxena

Fig. 11 Creep–fatigue crack growth data from base metal,under-matched welds and over-matched welds and creep crackgrowth data from the base material in weldments from a Cr–Mo–V steel. All tests were conducted at 565 ◦C (Grover 1997; Groverand Saxena 1999; Saxena 2007)

resistant to creep deformation at equal load levels andwill result in much shorter lives. Calculating (Ct )avg

from the measured load-line displacements normalizesthe effects of higher creep deformation rates and tendsto consolidate the CCG and CFCG data into a muchcloser band.

Figures 12 and 13 show the crack path of variousCFCG specimens from welds. Figure 12 shows thecrack path followed by a CFCG specimen from anunder-matched weld TC1 and the crack appears to firstfollow the fusion line and then it turns into the WM,as would be expected because the WM creep rates aresignificantly higher (68 times at 207 MPa stress level)than the BM creep rates. The (da/dt)avg versus (Ct )avg

behavior was higher for this condition than for the oth-ers. Figure 13 shows the crack path of a CFCG spec-

imen from the overmatched weldment TC5 in whichthe crack path meanders from the fusion line into theBM which is the less creep resistant constituent in thisweldment.

The justification for using time-dependent fracturemechanics for addressing crack growth in weldmentsutilizes a basic representation of weldments consist-ing of a bimaterial with a sharp crack located alongthe fusion line subjected to a load normal to the fusionline. This simple picture results from the fact that the J -Integral, and the C∗-Integral are both path-independentfor such a configuration under elastic-plastic and dom-inantly secondary creep conditions, respectively (Shihand Asaro 1988; Asaro and Shih 1989; Asaro et al.1993; Shih 1991; Nakagaki et al. 1991). This assuresthe uniqueness between the crack tip stress fields andthe magnitude of the crack tip parameter(s) and theability to measure their magnitudes at the loading pins.This model predicts that there is strain intensificationin the material that has lower resistance to deforma-tion with the implication that there is greater potentialfor cracks to grow on that side of the interface. Thisgeneral statement has been proven to be right by thedata that is presented above (Saxena 2007; Cretegnyand Saxena 1998). Thus, fracture mechanics has a def-inite role in representing crack growth in weldments. Itis thus necessary to also understand the limitations ofthis approach that will require the following additionalwork.

Finite element studies should be carried out toexplore a number of factors that influence the growthof cracks by creep and fatigue in weldments subjectedto high temperatures. An understanding of the evolu-tion of the crack tip fields and the balance of driving

Fig. 12 Fracture path of aCFCG specimen from theunder-matched weldmentTC1 showing a crack pathalong the fusion line and inthe weld metal region(Grover 1997; Saxena 2007)

123

Creep and creep–fatigue crack growth 45

Fig. 13 Fracture path in aCFCG specimen from anovermatched weldment TC5showing crack path alongthe BM/HAZ interfaceregion and then meanderinginto the base metal (Grover1997; Saxena 2007)

and resisting forces for cracks is fundamental to anyfracture analysis. Nonlinear and time-dependent finiteelement analyses must be performed to investigate thecrack tip stress fields as a function of time when a mis-match of creep properties exists at the interface. Analy-sis must initially focus on stationary cracks located atvarious distances from the weld fusion-line subjectedto static and cyclic loading of the type encounteredduring service. Creep conditions spanning small-scalecreep to extensive creep regimes must be examined.Consideration must be given to constitutive laws whichaccount for combinations of elastic, plastic, primarycreep, and secondary creep deformation, as appropri-ate for the system being modelled. The results of thesimulations must be used to develop simple expressionsfor estimating the magnitudes of crack tip parametersfor engineering components.

The analyses must treat the material on both sidesof the interface as inelastic and deformable. Crackedconfigurations with identical elastic properties but dif-ferent creep and plastic properties with due considera-tion to the microstructural gradients across the fusionline must be examined. Comparison of crack tip fieldswith those resulting in homogeneous bodies will revealthe trends that develop as the mismatch increases andthe property gradients intensify. Solutions must beobtained for both stationary and growing cracks. Thethickness of the interface layer and the property gradi-ents must be systematically varied in some parametricform, along with crack position in the interface layer,crack length and crack path.

The effect of graded interfaces on the transition fromsmall-scale creep to extensive creep must be examined.

The relationship between global stress-power releaserate with crack size to near tip values including theirrelationships to crack tip creep zone evolution must beexamined to assess the limitations of global parametersfor correlating creep and creep–fatigue crack growthrates. The analysis must also attempt to model cyclicloading conditions to understand the development ofcyclic plasticity at the crack tip and its subsequent effecton the evolution of the creep fields during hold time.

5 Creep and creep–fatigue crack growth testmethods

5.1 Creep crack growth test methods

The experimental methods for measuring creep crackgrowth rates have been standardized in an AmericanSociety for Testing and Materials (ASTM) standard(ASTM Standard E1457-07 2007). Thus, for long-termsustained loading conditions, the fracture mechanicsmethods for characterizing creep crack growth ratesare reasonably well established. Areas that need fur-ther development are discussed briefly.

• In creep-brittle materials, considerable amount ofcrack extension both in specimens and in com-ponents can occur under transient conditions. Theapproaches for characterizing the crack growth rateunder such transient conditions are not adequatelyestablished.

• The specimens recommended for testing in thestandard E1457 represent a variety with different

123

46 A. Saxena

Fig. 14 Creep crack growthbehavior of DS-GTD 111 atdifferent temperatures forspecimens machined suchthat the loading directionwas in the direction of the ofthe grains (LT) andtransverse to the direction ofthe grains (TL) (Ibanezet al. 2006a). The data alsoincludes tests conducted atdifferent temperatures

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02Ct (KN/m-hr)

da/d

t (m

m/h

r)

TL5-2TL5-1ATL9-2ALT5-2ALT9-2TL9-4

Filled: 871 CHollow:760 C

levels of constraint. The effects of crack tip con-straint on the creep crack growth behavior has notbeen systematically investigated so there is concernabout the equivalency of creep crack growth behav-ior obtained from various geometries with high andlow constraints.

• The deformation based global fracture mechan-ics parameters are only valid when creep cavita-tion damage is limited to a small region in thevicinity of a crack. If the damage is widespread,other approaches based on damage mechanics aremore appropriate. The limits on the use of fracturemechanics have not been systematically studied.

• Several high temperature materials in gas turbineapplications are single crystal or directionally solid-ified materials with strong directional character-istics. This problem has barely been addressed.The correlation between da/dt and Ct for direction-ally solidified GTD-111 Ni-based alloy is shown inFig. 14 (Ibanez et al. 2006a, b) for tests conductedon specimens machined from different orientations.Limited finite element simulations of such testshave also been conducted (Gardner et al. 2001; Maet al. 2014). More experiments as well as simula-tions should be conducted to enhance our under-standing of DS and single crystal materials.

• There are some guidelines in ASTM E1457 for test-ing weldments but there are still major gaps in theunderstanding of how to conduct the tests that are

more meaningful in replicating service conditions.Considering the importance of welds in high tem-perature components, this topic has not received theattention it deserves.

5.2 Creep–fatigue crack growth test methods

ASTM Standard E2760-10 (2010) addresses the stan-dard method for characterizing the creep–fatigue crackgrowth behavior. Areas of advancement of these meth-ods include the following.

• The Standard E2760 restricts creep–fatigue crackgrowth tests to only one specimen geometry andonly under conditions of load-control. The amountof crack extension in the specimens that is consid-ered valid is limited by ratcheting in the specimenswith cycles. Additional specimen geometries anddisplacement-controlled tests should be introducedin the standard method so the range of valid crackgrowth data can be extended to deeper crack sizesmaking better use of the specimens’ capability tomeasure crack growth.

• Advancements are needed to more accurately mea-sure changes in load-line displacement during thehold time and also the increases in crack size dur-ing the hold period to enhance our understandingof the creep–fatigue interactions.

123

Creep and creep–fatigue crack growth 47

• The interrelationships between the various dam-age mechanisms that operate under the severe ser-vice conditions are complex. Thus, the develop-ment of physics-based models for predicting designor remaining life must be guided by experimentalstudies that are specifically aimed at fundamentalunderstanding of these mechanisms. These exper-iments are challenging and demand precise envi-ronmental control capability, extremely high reso-lution in the measurement of displacements asso-ciated with creep strains and also in measurementsof crack increments at high temperatures such asduring hold time. Recent advancements in analyt-ical, experimental and computational techniquescan be used in concert to enable very significantadvances in our understanding of these componentlife-limiting considerations that are currently notfully understood.

6 Applications

The performance of high temperature components suchas natural gas-fired turbines, aircraft turbines and steamturbines has steadily improved with the continuousdevelopment of advanced materials and design con-cepts. Since these materials are being pushed to thelimits of their capability, accurate mathematical mod-els such as ones described in the earlier sections of thispaper are needed to predict the life of high tempera-ture components to prevent unscheduled outages dueto sudden failures. There is also a need to develop real-istic inspection intervals and criteria to avoid suddenfailures but they must not be too conservative becauseof operational cost considerations.

Figure 15 shows a schematic diagram of all elementsof a methodology for prognostics of high temperaturecomponent reliability that includes the use of constitu-tive equations, crack formation and crack growth mod-els under high temperature conditions (Saxena 2013).The models must be able to account for complex con-ditions consisting of low frequency cyclic stress asso-ciated with thermal and mechanical loads generatedduring startup and shutdown, sustained stress resultingfrom centrifugal loading and aerodynamic loads dur-ing steady-state operation leading to creep conditions,and damage phenomena in high temperature materi-als under conditions of creep, oxidation, fatigue, andthe synergistic effects of creep–fatigue–oxidation. The

goal for technologies used for assessing reliability ofhot gas path parts in industrial and aircraft turbines andsteam turbines must be to realistically estimate:

• Frequency and extent of required inspections dur-ing outages and the criteria for run, repair, retire,decisions,

• The level of online monitoring needed to updatethe prognostics of reliability in the face of changingoperating conditions from those used to design thecomponents,

• Establishing better and more realistic process con-trol and inspection criteria for acceptance of criticalcomponents,

• Avoiding unexpected outages and reducing main-tenance costs and,

• Deriving the full useful life from capital intensiveequipment and components.

Due to advances in online sensors, better serviceand maintenance records and the availability of oper-ating data during service, including results from theprior non-destructive inspections, more reliable esti-mates of remaining life of high temperature compo-nents are possible that are individualized to specificcomponents based on their specific loading history.These predictions can replace the need to make gen-eralized recommendations that apply to whole fleet ofcomponents that by nature must be conservative. In thatcontext it is essential that all this service data be usedto improve the prognostics of high temperature com-ponents to reduce the uncertainty in lives as called forin Fig. 15. Use of expert systems for such applicationsmust be developed.

7 Summary, conclusions and recommendations forfuture work

This paper presents the progress made in the field oftime-dependent fracture mechanics to account for creepdeformation at the crack tip at elevated temperatures.The following bullets outline the progress that has beenmade and the future work that must be undertaken todevelop the technology further in this age where infor-mation gathering and retrieval has become a lot easierand it can be used to customize the prediction of life ofspecific components.

Analytical frame-work exists for predicting creepand creep–fatigue crack growth in creep-ductile mate-rials and to lesser degree for creep-brittle materials

123

48 A. Saxena

Fig. 15 A methodology forprognostics of hightemperature componentreliability

Prognostics of High Temperature Component

Reliability

Stress-strain-temperature history of fracture critical components

Loading history

-transient loading

Constitutive equations

-Cyclic loading

Telemetry data from remote sensors

Creep deformation

Elastic and plastic deformation

Creep-fatigue interactions and TMF Conditions

Crack Formation Models

-Static and cyclic loading

-HCF and LCF loading

-Effects of temperature

-Oxidation damage

-Microstructural degradation

-Thermal-mechanical loading

Crack Growth Models

-Fracture toughness behavior

-Effects of temperature

-Creep crack growth

-Creep-fatigue loading

-Creep-fatigue-oxidation effects

-Thermal-mechanical loading

Remaining Life Estimation

-On-line monitoring

-Off-line inspection intervals during dead

Risk Assessment and Field Validation

Use of field data to correct

life/inspection interval predictions from Physics-based

models

and DS materials. Additional work is needed to under-stand the transients due to the effects of growing cracksand the development of creep deformations in frontof a moving crack tip. For DS materials, we need toaccount for grain boundaries and difference in crys-tallographic orientations on the creep deformation atthe crack tip as the crack transitions from one grain tothe next

• Advancements are needed in efficiently collectingcrack growth data during creep–fatigue tests; in par-

ticular displacement controlled test methods shouldbe explored.

• Models for predicting the effects of creep and envi-ronment are presented in the paper. However, morefundamental studies are needed to better understandcreep–fatigue–environment interactions.

• Significant effort should be devoted to the appli-cation of the currently available models in aframe-work of risk assessment in conjunctionwith on-line and off-line damage monitoring andinspection data. Field validation of creep–fatigue

123

Creep and creep–fatigue crack growth 49

model predictions should be given a very highpriority.

• Some progress has been made in the assessment ofcrack growth in welds under creep–fatigue. Futurestudies must consider;

• Microstructural gradients and transition layersbetween the weld metal and the base metal,

• Sharp interfaces, though useful for qualitativejudgments, are not always realistic,

• Plastic and creep deformation zones and theirevolution in the welds, including their interac-tion with the transition layer must be studied indetail to understand the limitations of nonlinearfracture mechanics to problems of mismatchedwelds,

• Analytical crack growth studies performedshould model cracks that meander from oneregion of the weld to another to develop robustmodels.

Acknowledgments The author is deeply indebted to severalof his present and past collaborators for their insightful discus-sions and their work that have directly and indirectly contributedto the research described in this paper. Particularly noteworthyare the contributions of Drs. R. Viswanathan, David L. McDow-ell, Bilal Dogan, Stephen D. Antolovich, Hugo Ernst, John D.Landes, James A. Begley, John Bassani, Herbert Hui, HermanRiedel, and K. Yagi. His former PhD students and Post-docswhose contributions are invaluable include C.P. Leung, BrianGieseke, Kee Bong Yoon, Negussie Adefris, Richard H. Norris,Yancy Gill, Parmeet Grover, David Hall, B. Carter Hamilton,Jeffery Evans, Kip Findley, Heather Major, Alejandro Ibanez,Jidong Kang, Bruce Antolovich, Sau-wee Koh, V. Kalyanasun-daram, Santosh Narasimhachary, and V. Srinivasan. The finan-cial support of this work was provided by the Electric PowerResearch Institute, the National Science Foundation, and by sev-eral companies. The support from the George and Boyce Billings-ley Endowed Chair at the University of Arkansas is also gratefullyacknowledged.

Open Access This article is distributed under the terms of theCreative Commons Attribution License which permits any use,distribution, and reproduction in any medium, provided the orig-inal author(s) and the source are credited.

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