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Theoretical and Applied Fracture Mechanics

journal homepage: www.elsevier.com/locate/tafmec

The disparate data problem: The calibration of creep laws across test typeand stress, temperature, and time scales

Mohammad Shafinul Haque⁎, Calvin Maurice StewartUniversity of Texas El Paso, 500 West University Avenue, El Paso, TX 79902, USA

A R T I C L E I N F O

Keywords:Creep deformationCreep-damageStress-ruptureMinimum-creep-strain-rateExtrapolation, design mapsContinuum damage mechanics

A B S T R A C T

Creep data exists in a diaspora. It is not always available in the regime (short, intermediate-, and long-term) orform (minimum-creep-strain-rate, creep deformation, stress-rupture, etc.) of interest. This is particularly true fornew alloys where the time-to-qualification can take between 10 and 20 years. For new alloys, extrapolation isperformed using short-term creep data. When incorporating the uncertainty of the creep data, the probabilitydistribution in extrapolations expands with time; eventually leading to unrealistic predictions. Standards re-commend that extrapolation should not exceed three times the duration of the longest experiment. For example,a creep deformation model calibrated using short-term creep deformation data (e.g. 103 h) is highly unlikely togenerate realistic creep deformation predictions near 105 h. This study proposes integrating disparate creep datain the calibration process of continuum damage mechanics based constitutive laws that are capable of si-multaneously predicting minimum-creep-strain-rate, stress-rupture, deformation, and damage. It is hypothesizedthat cross-calibrating CDM constitutive models to any and every form of available creep data (across test type,stress range, and temperature range) will lead to more reliable extrapolations. This is demonstrated by in-tegrating minimum-creep-strain-rate, stress-rupture, and short-/intermediate-term creep deformation data intothe calibration process of two CDM constitutive models. Successful extrapolations are performed for minimum-creep-strain-rate, stress-rupture, and creep deformation. In addition, the approach facilitates the development ofcreep design maps that can be used for material selection and design and to compare the interpolation andextrapolation ability of the existing creep models.

1. Introduction

1.1. Motivation

Drives to extend the service life of existing power plants is leading tostructural components with lives beyond 106 h. In addition, the ongoingdevelopment of Advanced-Ultrasupercritical (A-USC) power plants hascreated material challenge where new materials are needed to with-stand operating pressures above 4000 psi and temperatures exceeding1400 °F. These conditions make it critically important to consider creepdeformation and damage in the design process of power plant compo-nents and systems. Knowledge concerning the long-term creep behaviorof existing and new materials is needed to make design decisions;however, often only limited creep data is available. The average time-to-qualification of new creep resistant alloys is between 10 and 20 years[1]. This development time is consumed by computational materialsdesign, thermomechanical processing optimization, and the physicalexperiments needed to qualify the long-term creep resistance of a

material [2].

1.2. Creep data

Usually, creep data is created by performing a variety of short- and/or long-term creep tests. The most common data collected areminimum-creep-strain-rate, stress-rupture, and deformation data asshown in Fig. 1. Minimum-creep-strain-rate data is easy to generateusing short-term creep tests but it also can be created in an acceleratedfashion using the stress relaxation tests (SRTs) where multiple iso-therms of minimum-creep-strain-rate data are generated from a singlespecimen [3]. Minimum-creep-strain-rate data is reported on a log-logscale with stress and per isotherm as depicted in Fig. 1a. Long-termstress-rupture data is generally accrued by national laboratories andbusiness entities who have a vested interest and suitable financialbacking to commit to long-term creep testing (continuous testing fordecades or more) [2,4]. An ideal stress-rupture curve on a log-log scaleexhibits a sigmoidal transition from high-stress to low-stress as

https://doi.org/10.1016/j.tafmec.2019.01.018Received 12 April 2018; Received in revised form 3 January 2019; Accepted 21 January 2019

⁎ Corresponding author.E-mail address: mhaque@miners.utep.edu (M.S. Haque).

Theoretical and Applied Fracture Mechanics 100 (2019) 251–268

Available online 24 January 20190167-8442/ © 2019 Elsevier Ltd. All rights reserved.

T

illustrated in Fig. 1b. At the ultimate tensile strength, the specimenruptures instantly while at very low stress, life is near infinite(> >106 h=114 years). For legacy materials such as 304 stainlesssteel, tests have been performed beyond 105 hours and there are on-going tests estimated to failure at ≫106 h [2]. Most previously per-formed and ongoing tests were initiated at a time when digital dataacquisition (DAQ) systems did not exist. Instead, a military log of theelongation and time was recorded, and tables of time-to-creep-straingenerated to profile the creep deformation responses. Most often the“full” creep deformation curve has not been digitized and the militarylogs have either been neglected and/or misplaced over the years oftesting. It is difficult to obtain full- long-term creep deformation curves.To bridge this gap, short/intermediate-term creep deformation tests areperformed with modern DAQs. The stress and temperature conditionsof these tests are carefully selected to induce rupture within 104 [1,5–7].A typical creep deformation curve is shown in Fig. 1c. Creep de-formation is composed of primary, secondary, and tertiary regimes withboth the appearance and dominance of these regimes dependent onstress and temperature. Minimum-creep-strain-rate (Fig. 1a) and stress-rupture (Fig. 1b) exist as points of interest on a creep deformation curveat a set isostress and isotherm. For legacy materials, the availability offull- long-term creep deformation curves are limited. For modern ma-terials, not enough time has passed for long-term curves to have beengenerated. A gap in long-term creep deformation data exists. To bridgethe gap, extrapolation is necessary.

1.3. Extrapolation

The extrapolation of creep behavior can be performed by fittingcreep models to short/intermediate creep data and then simulatingoutside of the available data points. These extrapolations assume thatthe existing trend in the available data persists and that the models’functional form accurately describes the creep behavior in the extra-polated region. In general, extrapolations are not reliable and are dif-ficult to assess. Empirical evidence shows that the 0% and 100% re-liability bands for stress-rupture expand as the applied stress is reduced

such that at low stress and elevated temperature the bands can existdecades apart [2]. The combined uncertainty of extrapolations andmaterial performance necessitate that generous safety factors be ap-plied in design [8]. A new material with 30,000 h of creep data, ex-trapolated to 100,000 h has no guarantee of reliability. Creep extra-polation methods have evolved with the evolution of creep models. Theearliest extrapolation approach uses Time-Temperature Parametric(TTP) models for stress-rupture prediction [9,10]. These models exist asa relationship between time and temperature expressed as a parametricfunction of stress. When calibrated to sufficient short duration stress-rupture data, extrapolations to long-life can be made. The TTP modelscan be further utilized to predict the time to percent-creep-strain atvarious increments (such as 0.1, 1, 5%, etc.) or to predict the time toonset of tertiary creep [11,12]. Standards recommended that TTP ex-trapolations should not exceed three times the duration of the longestexperiment [13]. Recent work has shown that in some cases this re-striction can be eased by performing rigorous statistical analysis[14,15]. The European Creep Collaborative Committee (ECCC) hasoutlined a standard procedure for this process [16]. The application ofTTP models cannot predict minimum-creep-strain-rate, deformationcurve, and damage and thus have limited application in design.

1.4. Constitutive laws

Over time, the focus has shifted from extrapolations using simpleTTP models to the use of constitutive models. First, the constitutivemodel is calibrated to short/intermediate-term creep data to determinethe material constants per isotherm within the given stress range.Second, the material constants are defined as a function of temperatureand/or stress. Third, interpolation and extrapolation are performed,either analytically or numerically (using the finite element method).

In the case of analytical extrapolation, the constitutive laws arerearranged/modified to predict points of interest such as the rupturelife, damage state, creep ductility, minimum-creep-strain-rate, etc.[17–19]. Hayhurst found that 30x stress-rupture extrapolations arepossible with ± 20% accuracy when using the theta-projection

Nomenclature

ε(%) creep strainε ε ε , , cr finalmin ( −%hr 1) creep strain rate, minimum-creep-strain-rate,

final creep strain rateω ω, cr (unit less) creep damage, critical damageω ( −hr 1) creep damage ratet t t, ,r T (hr) time, rupture life, target timeσ σ σ, ,max min (MPa) stress, maximum stress, minimum stress

T T T, ,max min (°C) temperature, maximum temperature, minimumtemperature

t σ TΔ , Δ , Δr (hr, MPa, °C) range of rupture, stress, and temperatureT σ t, ,s st s (°C, MPa, hr) temperature, stress, and time stepA n, ( − −%MPa hrn 1, unit less) Norton Power law constantsB σ, s ( −%hr , MPa1 ) McVetty Creep law constantsM θ ψ, , ( − −MPa hrψ 1, unit less) Kachanov-Rabotnov model constantsQ σ χ ϕ λ, , , ,t ( −hr 1, MPa, unit less) Sinh model constants

Fig. 1. Disparate types of creep data (a) minimum-creep-strain-rate exhibiting a sigmoidal behavior, (b) stress-rupture data that can be sub-divided into short-term,intermediate, and long-term data, (c) creep deformation including primary, secondary and tertiary regimes.

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constitutive model by assuming that rupture will occur if the strainreaches a certain multiple of the creep rate [20]. Maruyama and Brownestablished that theta-projection can further be used to extrapolate theminimum-creep-strain-rate, creep deformation, and rupture curves[21,22]. Abdallah and colleagues showed that extrapolated creep de-formation curves can be generated by modifying the Wilshire rupturemodel [23].

In the case of numerical extrapolation, the constitutive models areprogrammed into the finite element (FE) method and the service en-vironment and geometry of a structural component simulated. Thecalibrated material constants are programmed as a function of stressand/or temperature, which enables interpolation and extrapolation ofthe creep properties per time step. Using FE analysis, different load andtemperature intensities, rates, and distributions, as well as differentgeometry than those applied in experiments, can be simulated. Rarelydoes the available creep data match the conditions found in an FEAsimulation. For example, an ideal structural component is designed inthe elastic regime and yet the stress-rupture behavior of most alloysat< 40% of the yield strength is rarely empirically known. Continuumdamage mechanics (CDM) models are popular for extrapolation becausethe minimum-creep-strain-rate, stress-rupture, creep deformation, anddamage are furnished using a single set of coupled differential equa-tions [24–27]. The application of CDM in calculating damage and crackgrowth under creep, fatigue, creep-fatigue, and other detrimental en-vironments is well-posed [28–30]. Many uniaxial CDM models havebeen extended to a multiaxial form [31]. Rules concerning the designand life-extension of structural components using CDM models exist[32]. Models also exist for the creep of anisotropic materials; in-corporating a damage tensor that accommodates various degrees ofanisotropic damage [33,34,35]. The mathematical and numerical sta-bility of CDM models has also been investigated [36,37].

1.5. Creep design maps

Once models (TTP, constitutive, CDM, or otherwise) are calibratedand the material properties functionalized, the interpolation and ex-trapolation ability can be reported graphically by creating a CreepDesign Map. In these maps, creep properties (such as minimum-creep-strain-rate, rupture, and/or time-to-creep-strain) are presented in con-tour where stress and temperature form the axes. Three uses are im-mediately evident for creep design maps. They can be used for

(a) Design [22,38]. For a given material, creep design maps can be usedto determine which boundary conditions meet specific design re-quirements such as rupture life, elongation, damage, etc.;

(b) Material selection [39]. Once creep design maps have been gener-ated for several materials, the maps can be used to select the mostappropriate material for given boundary condition. Hypothetically,Ashby-like plots can also be created;

(c) Model comparison [40]. Creep design maps are specific to themodel used to create them. If creep design maps are generated formore than one model, the interpolation and extrapolation ability ofthe models can be compared.

A variety of creep design maps have been created. Brown and Evanscreated time-to-creep-strain and/or rupture design maps for the extra-polation of short-term creep data [22]. Hyde and colleagues suggestedthe use of such design maps to determine realistic test and operatingconditions [38]. Fujiyama developed Ashby-like maps to facilitatematerial selection under creep-fatigue conditions [39]. Stewart createdtime-to-creep-strain design maps to identify the dominant creep regime(primary, secondary, or tertiary) using CDM [40].

1.6. Problem statement

It is evident that gaps exist in the creep data available for mostmaterials. Creep data is not always available in the regime (short, in-termediate-, and long-term) or form (minimum-creep-strain-rate, creepdeformation, stress-rupture, etc.) of interest. It is hypothesized thatcross-calibrating a constitutive model to any and every form of creepdata will lead to more reliable extrapolations. This requires integratingdisparate creep data into the calibration process of a constitutive model.A CDM constitutive model simultaneously predicts minimum-creep-strain-rate, stress-rupture, deformation, and damage using a single setof differential equations. The coupled nature of the creep strain rate anddamage rate equations of CDM models enable calibration using dis-parate creep data. A comparison of traditional and disparate calibrationapproaches is presented in Table 1.

2. Objectives

The objectives of this study are to

A. provide a detailed procedure for calibrating CDM models usingdisparate creep data;

B. demonstrate the reliability of extrapolations via post-audit valida-tion, where the calibrated model is compared to additional post-audit creep data that was not used in the calibration process;

C. illustrate a variety of creep design maps and demonstrate that theinterpolation and extrapolation ability of CDM models can becompared graphically using creep design maps.

The objectives are achieved using the following approach. First, thecalibration procedure for disparate creep data is introduced. Next, theKachanov-Rabotnov (KR) and Sine-hyperbolic (Sinh) CDM models areintroduced. These models are calibrated per isotherm to minimum-creep-strain-rate, creep deformation, and stress-rupture data collectedfrom various sources. Afterward, the calibrated material constants aredefined as a temperature-dependent-function of temperature for inter-polation and extrapolation. Next, extrapolations are performed forminimum-creep-strain-rate, stress-rupture, and creep deformation.Next, post-audit validation is performed by collecting additional post-audit creep data and compare the data to the pre-calibrated model.Finally, minimum-creep-strain-rate, creep deformation, damage, andrupture creep design maps are generated and the performance of thetwo models compared.

Table 1Comparison of traditional and disparate calibration approaches.

Traditional Disparate Data

Data Type Short/intermediate-term creep deformation data(data within 104 hours is available)

Long-term creep rupture data. (data up to 106 hours is available)Short/intermediate-term creep deformation data. (data within 104 hours is available)Short/long-term minimum-creep-strain-rate data. (data over a wide range of stress andtemperature is available)

Result When post-audit validated, predictions are unrealistic and exhibit lowreliability

When post-audit validated, predictions are realistic and exhibit higher reliability

Reason Not enough data is provided to properly calibrate for long-term creepbehavior. Predictions are sensitive to specimen-to-specimen variations

Calibration incorporates data across a wide range of stress, temperature, and duration.Data collected across many specimens increases the reliability of the predictions

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3. Model calibration procedure

The first objective of this study is to provide a detailed procedure forcalibrating CDM models using disparate creep data. The procedure fordisparate database calibration consists of six steps illustrated in Fig. 2.The first step is to select a model of interest. The second step is tosegregate the model into equations related to each type of creep data(e.g. minimum-creep-strain-rate, stress-rupture, creep deformation etc.)such that they can be calibrated independently. In the steps three andfour, the material constants are calibrated to creep data and regressionanalysis is performed to convert the material constants into tempera-ture-dependent functions, f T( )i respectively. The next step is to post-audit validate the model using additional post-audit creep data not usedin the calibration process. In the sixth step interpolation and extra-polation of the creep behavior are performed to develop design maps. Inthe following sections, this procedure is demonstrated using two CDMmodels for 304 stainless steel.

4. Model selection and segregation

The classic Kachanov-Rabotnov (KR) and modern Sine-hyperbolic(Sinh) CDM-based creep constitutive models are selected to vet thedisparate creep data calibration approach. These models consist ofcoupled creep strain rate and damage rate equations that can be utilizedto predict minimum-creep-strain-rate, creep deformation, damage, andrupture. A brief description of these models is given below. A detailstep-by-step derivation of the KR and Sinh models with analyticalprocedures to calibrate the material constants are reported elsewhere[36,37,41].

Select the creep model of interest. (Kachanov-Rabotnov, Sinh, etc.)Model Selection

Segregate the model into equations related to each type of creep data. (Minimum-Creep-Strain-Rate, Stress-Rupture, Creep Deformation, etc.)Segregation

Calibrate the material constants of the equations using the creep data.Calibration

Use regression analysis to convert material constants into temperature-dependent functions. fi (T)Regression

Take the pre-calibrated model and compare it to additional data not used in the calibration process.Validation

Make interpolative and extrapolative prediction of creep behavior. Plot these predictions as Creep Design Maps.Design

Fig. 2. Calibration procedure for CDM models to disparate creep data.

Table 2Kachanov-Rabotnov CDM model.

Purpose Material constants Equations No.

Creep strain rate A n, =−

ε A ( )crσ

ωn

1(1)

Damage rate M ψ θ, , =−

ω Mσψ

ω θ(1 )

(2)

Minimum-Creep-Strain-Rate (Norton Power Law) A n, =ε Aσ nmin (3)Damage θ

= − ⎡⎣

− − + ⎤⎦

++

ω t ω( ) 1 [(1 ) 1] 1ttr r θ

θ1

1/( 1) (4)

Rupture prediction M ψ θ, , =+

trθ σψM

1( 1)

(5)

Analytical damage A n, = −ω ε ε A σ ε A( ) [( / ) ]/( / )cr cr n cr n1/ 1/ (6)

Table 3Sine-hyperbolic CDM model.

Purpose MaterialConstants

Equations No.

Creep strain rate B σ λ, ,s ==

ε B σ σ λωλ ε ε

sinh( / ) exp( );ln( / )

cr s

final

3/2

min

(7)

Damage rate Q ϕ σ χ, , ,t = − − ( )ω ϕω sinh exp( )Q ϕϕ

σσt

χ[1 exp( )] (8)

Minimum-creep-strain-rate(McVetty Law)

B σ, s =ε B σ σ sinh( / )smin (9)

Damage ϕ = − ⎡⎣

− − − ⎤⎦

ω t ϕ( ) ln 1 [1 exp( )]ϕ

ttr

1 (10)

Rupture prediction Q σ χ, ,t= ⎡

⎣⎤⎦

−( )t Q sinhrσσt

χ 1 (11)

Analytical damage B σ λ, ,s= ⎡

⎣⎤⎦{ }ω ε( ) ln

λεcr t

B σ σs1 ( )

sinh( / )

2/3 (12)

Table 4Statistics of disparate creep database of 304SS.

Test Type Isotherms Stress range No. ofTests

(°C) (MPa)

Minimum-creep-strain-rate[5,41,45–48]

566, 593, 649, 760,816

3–250 141

Stress-rupture [5,6,45,48–50] 593, 649, 732, 760,843, 899, 954

5–271 216

Creep deformation [6] 600, 650, 700 160–320 30

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4.1. Kachanov-Rabotnov (KR) model

The equations and material properties associated with theKachanov-Rabotnov (KR) model are provided in Table 2. The KR modelconsists of a coupled creep strain rate, εcr and damage rate, ω [Eqs. (1)and (2)] respectively [42,43]. The KR model can be segregated intominimum-creep-strain-rate, damage, and rupture predictions [Eqs.(3)–(5)] respectively. Analytical damage can be calculated from ex-periments by rearranging [Eq. (1)] as shown in [Eq. (6)]. A detail step-by-step derivation of these equations has been previously reported

[36,37].

4.2. Sine-hyperbolic (Sinh) model

The equations and material properties associated with the Sine-hyperbolic (Sinh) model are provided in Table 3 [41,44]. Similar to theKR model, the Sinh constitutive model [(7) and (8)] can be segmentedinto minimum-creep-strain-rate, damage, and rupture predictions [Eqs.(9)–(11)] respectively. The Sinh analytical damage is obtained by re-arranging [Eq. (7)] as shown in [Eq. (12)]. A detail step-by-step

Fig. 3. Minimum-creep-strain-rate versus stress using the KR (straight lines) and Sinh (curved lines) models [Eqs. (3)] and [(9)] respectively.

Fig. 4. Stress-rupture predictions using the KR (straight lines) and Sinh (curved lines) models [Eqs. (5)] and [(11)] respectively.

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derivation of these equations has been previously reported [36,37].

5. Calibration

The disparate creep data calibration procedure is demonstratedusing a disparate creep database of 304 stainless steel. In this sectioneach type of creep data is calibrated separately. Later, the results will becombined into the disparate data approach. Minimum-creep-strain-ratedata at five isotherms are collected from literature [5,41,45–48]. Stress-rupture data at eight isotherms is pulled from multiple sources[5,6,45,48–50]. Quintuplicate creep deformation tests at six stress le-vels across three isotherms are gathered from an article by Kim et al [6].The statistics of the disparate creep database are summarized inTable 4. It is important to note, that the metadata associated with theseexperiments (chemical compositions, form, thermomechanical proces-sing, etc.) was not restricted, thus this data represents a combination ofmany different variations of 304SS.

The performance of the KR and Sinh models in predicting theminimum-creep-strain-rate and stress-rupture are evaluated using thenormalized mean-squared error, NMSE

= ∑

= ∑

= ∑

=−

=

=

NMSE

X X

X X

¯

¯

n in X X

X X

sim n in

sim i

n in

i

11

( )¯ ¯

11 ,

exp1

1 exp,

sim i i

sim

, exp, 2

exp

(13)

where n is the total number of data points and Xsim and Xexp are the

simulated and experimental data points respectively. An NMSE value ofzero indicates that the experimental data and model are identical.

5.1. Minimum-creep-strain-rate

Minimum-creep-strain-rate predictions using KR and Sinh [Eqs. (3)]and [(9)] are plotted in Fig. 3. The calibrated material constants arelisted in Table 4. Examining the data, it is observed that the 593, 649,and 760 °C isotherms exhibit a sigmoidal behavior while the 566 and816 °C isotherms are linear. The KR predictions are linear on a log-logscale and are not able to model the sigmoidal behavior observed in thedata; under-predicting the minimum-creep-strain-rate at low-stresslevel. The KR model can be calibrated to fit high-stress or low-stressdata but the model cannot fit both regions simultaneously. A possiblesolution to this problem is to use a step function with two different setsof material constants to accommodate the sigmoidal behavior; how-ever, using a step function to switch between the material constantsintroduces numerical instability in the simulations. Hosseini andHoldsworth reported that the assumption of a sudden change in creepmechanics for minimum-creep-strain-rate is unrealistic [51]. The Sinhmodel can model the sigmoidal behavior. In the Sinh model, the con-stant σs controls the trajectory of the bend on a log-log scale. The NMSEof the KR and Sinh model are 4.63 and 3.75 respectively.

5.2. Stress-rupture

Stress-rupture predictions using KR and Sinh [Eqs. (5)] and [(11)]are plotted in Fig. 4. The calibrated material constants are listed inTables 6 and 7 for KR and Sinh respectively. It is observed that both KRand Sinh agree with the stress-rupture data. The KR prediction is linearon a log-log scale which is not ideal considering the sigmoidal bendobserved in the data. The Sinh model fits the sigmoidal bend in thedata. The superiority of the Sinh model over the KR is reflected in NMSEstatus. The NMSE of the KR model is 321.59 compared to 89.44 of theSinh model.

5.3. Creep deformation and damage

In this section, the traditional calibration approach is demonstrated,while at the same time the final material constants (ω λ ϕ, ,cr ) needed forthe disparate calibration are obtained. The traditional calibration ap-proach is similar to the disparate calibration approach in that we gatherminimum-creep-strain-rate, stress-rupture, and deformation data. Thedifference is that in the traditional approach, the data comes from asingle source: available creep deformation curves. Since, creep de-formation data is more expensive to gather than just minimum-creep-strain-rate, time-to-strain, or stress-rupture data, a majority of creepdeformation data is only available at short-term (less than 104 h).Calibration using short-term creep deformation data exclusively leadsto inaccurate extrapolation in the long-term. The traditional calibrationapproach operates as follows

• Creep deformation curves are obtained as the only source for cali-bration.

• The minimum-creep-strain-rate data is extracted from each curveusing finite difference and the KR (A n, ) and Sinh (B σ, s) materialconstants are determined numerically per isotherm.

• The stress-rupture data of each curve is extracted and the KR(M θ ψ, , ) and Sinh (Q σ χ, ,t ) material constants are determined nu-merically per curve. Note: For KR, these material constants are de-termined concurrent with the next step.

• Using the previously determined material constants, the final set ofKR ω( )cr and Sinh λ ϕ( , ) material constants are determined numeri-cally by fitting the partially-calibrated models to the creep de-formation data.

Table 5The minimum-creep-strain-rate material constants of KR and Sinh models for304SS.

Kachanov-Rabotnov Sinh

Temp. A n B σs(°C) (%MPa−n hr−1) (%hr−1) (MPa)

566 3.82E−35 13.50 3.72E−12 11.21593 1.10E−29 11.95 1.22E−09 11.98649 2.75E−22 9.15 5.10E−07 13.43760 4.81E−13 6.10 5.11E−04 16.52816 4.54E−10 5.05 4.37E−03 18.20

Table 6KR stress-rupture material constants [Eq. (5)].

Temp. M θ ψ(°C) ( − −MPa hrψ 1)

593 1.5E−22 12.00 7.90649 3.9E−20 10.00 7.50732 1.1E−15 7.00 6.00760 2.3E−14 6.00 5.70843 1.0E−11 5.00 5.00899 8.0E−10 4.00 4.45954 9.0E−08 3.50 3.60

Table 7Sinh stress-rupture material constants [Eq. (11)].

Temp. Q σt χ(°C) (hr−1) (MPa)

593 2.6E−5 118.0 5.20649 3.9E−5 80.0 4.60732 9.5E−5 56.0 3.80760 2.2E−4 45.0 3.60843 5.5E−4 28.0 3.10899 9.9E−4 19.0 2.90954 1.7E−3 12.8 2.70

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Creep deformation and damage predictions of the KR and Sinhmodels using traditional calibration are plotted in Figs. 5 and 6 re-spectively. The calibrated material constants are listed in Tables 8 and 9for KR and Sinh respectively. The material constants were obtained byfitting the models to the creep deformation curve nearest the middle ofthe scatter band. Both models provide reasonable predictions of creepdeformation and damage. In the KR model critical damage, ωcr variesbetween 0.2 and 0.3 while in the Sinh model critical damage is alwaysunity. Extrapolation using the traditional calibration approach is de-monstrated in Section 7 and found to be highly inaccurate. In the dis-parate calibration approach, the creep deformation data provides the(ω λ ϕ, ,cr ) material constants only. These constants are relatively in-sensitive to stress and temperature.

6. Regression

In order to make creep predictions at any temperature using thedisparate data approach, the tables of independent minimum-creep-strain-rate, stress-rupture, and creep deformation constants (Tables5–9) must be converted into temperature dependent functions. Pre-ference is given to linear, exponential, and/or power functions to avoidinflection points [10].

The temperature-dependent functions for the minimum-creep-strainrate (Table 5) and stress-rupture material constants (Tables 6 and 7) ofKR (A n M θ ψ, , , , ) and Sinh (B σ Q σ χ, , , ,s t ) are listed in Table 10 andplotted versus temperature in Fig. 7. The smoothness of regressionarises from the use of both numerical optimization and manual iteration

Fig. 5. Creep deformation simulations at 600, 650, and 700 °C, (a–c) Kachanov-Rabotnov and (d–f) Sinh model [Eqs. (1)] and [(7)] respectively.

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Fig. 6. Analytic damage evolution at 600, 650, and 700 °C, (a–c) Kachanov-Rabotnov and (d–f) Sinh model [Eq (4)] and [Eqs. (10)] respectively.

Table 8KR material constants for 304SS.

Temp, T Stress, σ A n ωcr M θ ψ(°C) (MPa) (%MPa−n hr−1) (MPa−ψ hr−1)

700 160 6.53E−31 12.78 0.29 9.35E−11 12.5 3180 0.28 1.07E−10 18.5

650 240 4.26E−33 12.98 0.17 2.69E−11 20.0260 0.27 1.31E−10 12.4

600 300 1.56E−35 13.36 0.25 9.49E−12 38.0320 0.18 2.37E−11 27.0

Table 9Sinh material constants for 304SS.

Temp, T Stress, σ B σs λ ϕ Q σt χ(°C) (MPa) (%hr−1) (MPa) (hr−1) (MPa)

700 160 1.34E−4 27.99 4.52 5.5 1.7E−4 86.01 3180 4.24 3.8

650 240 9.64E−10 13.5 3.02 3.5 1.9E−9 41.88260 2.40 1.8

600 300 2.31E−7 24.76 3.72 5.99 4.6E−7 76.00320 1.93 2.62

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to balance minimizing-calibration-error with temperature-dependence-trends. The scatter bands in the disparate datasets provided an oppor-tunity for some limited tuning.

Temperature-dependent functions for the creep-deformation mate-rial constants of KR (ωcr in Table 8) and Sinh (λ ϕ, in Table 9) cannot becreated. The material constants do not exhibit a distinct trend with

temperature. The material constants appear to trend with stress; how-ever, not enough creep deformation data is available to perform re-gression analysis. When compared to the creep ductility of the availableexperiments, the coefficient of variations are similar suggesting thatthese material constants carry a majority of material and test-relateduncertainties. Similar to what is often done for creep ductility, the

Table 10Material constants as a function of temperature (°C).

Test Type KR R2 Sinh R2

Minimum-creep-strain-rate = − −A Tlog( ) 1.84E11· 3.534 0.99 = − −B Tlog( ) 5.0E12· 4.231 0.99

= −n T125.92·exp( 3.94E - 3· ) 0.99 =σ T3.84·exp(1.9E - 3· )s 0.99

Stress-rupture = −M Tln( ) 71.61·ln( ) 507.44 0.99 = −Q Tln( ) 9.26·ln( ) 69.942 0.98= −θ T87.61·exp( 3.37E - 3· ) 0.98 = −σ T4295·exp( 6.1E - 3· )t 0.99= − +ψ T0.0118· 14.905 0.98 = −χ T39293· 1.4 0.98

Creep deformation =ω 0.29cr NA = = =ω λ ϕ1, 4.5, 3.0cr NA

Fig. 7. Temperature dependence of the KR (a–c) and Sinh (d–f) material constants.

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materials constants are fixed. In KR, the critical damage is fixed to=ω 0.29cr the maximum observed to predict a conservative creep

ductility. In Sinh, the material constants =λ 4.5 is fixed to the max-imum to predict a conservative creep ductility, while the materialconstant =ϕ 3.0 set to the average to prevent an over calculation ofdamage evolution.

Using the above temperature-dependent functions and fixed mate-rial constants of the disparate calibration approach, the constitutivemodels can predict creep over a wide range of stress and temperatureconditions.

7. Validation

The second objective of this study is to demonstrate the reliability ofextrapolations using post-audit validation. Post-audit validation iswhere the pre-calibrated model is compared to additional post-auditcreep data that was not used during calibration. Additional creep datafor 304SS is gathered from the Japan National Institute of MaterialsScience (NIMS) material database [2]. It is important to note that themetadata of the calibration database is not identical to the metadata ofthe NIMS database. Minimum-creep-strain rate data at three isotherms,stress-rupture data at five isotherms, and long-term creep deformationcurves at 650 °C are collected. The statistics of the disparate databaseare summarized in Table 11.

The NIMS minimum-creep-strain-rate data is compared to KR andSinh simulations plotted in Fig. 8. It is observed that the KR and Sinh

simulations at 600 and 650 °C reasonably agree with the NIMS data;however, the predictions at 700 °C do not agree with the NIMS data.This is because the 649 °C isotherm (applied in calibration) overlapswith the 700 °C isotherms from NIMS as shown in Fig. 8. The cause ofthis overlap could be due to (a) the limited number of tests performed at700 °C not capturing the scatter band expected at 700 °C, (b) differencesin the chemical composition and thermomechanical processing andform of the material between the two data sources, (c) differences in theequipment, test procedure, and geometry between the two data sources.

The NIMS stress-rupture data is compared to KR and Sinh simula-tions shown in Fig. 9. Both the KR and Sinh model reasonably agreewith the NIMS data. The KR and Sinh predictions are not identical. Theydiverge in the extrapolated regions at creep life below 102 and above105 h. The Sinh predictions are more conservative than KR. This is dueto the sigmoidal bend in the Sinh model. The stress-rupture predictionsare superior to the minimum-creep-strain-rate predictions.

Extrapolation using the traditional calibration approach at 61MPaand 650 °C are performed following the steps described in Section 5.3.The creep ductility is found to be less than 0.0078% for both modelswhen extrapolated to 105 hours. The predicted ductility is very lowcompared to the NIMS database showing a creep ductility of 18%.Improved extrapolation can be achieved by following the disparatecalibration approach.

The KR and Sinh simulations using the disparate calibration ap-proach are compared to the NIMS creep data shown in Fig. 10. In themodels, creep ductility and deformation are dependent on the predictedrupture life, with ductility increasing with rupture life. Stress-rupturedata exhibits uncertainty, where at low stress, rupture life can spanmultiple decades. This is observed in both the calibration and NIMSstress-rupture data in Fig. 10a. It would be unwise to extrapolatewithout considering the uncertainty of stress-rupture and its effect oncreep deformation. In this study, creep deformation predictions aremade at four rupture points of interest. The four rupture points are:

• model predicted rupture. The calibrated rupture predictions [Eqs.(5)] and [(11)] are applied. These predictions take an average paththrough the stress-rupture data and can be considered 50% reliable.

• 100% reliability rupture. The model predictions are shifted

Table 11Statistics of the post-audit validation data from NIMS [2].

Test Type Isotherms Stressrange

No. ofTests

(°C) (MPa)

Minimum-creep-strain-rate 600, 650, 700 29 to 216 20Stress-rupture 450, 475, 500, 550, and

65061 to 412 52

Creep deformation 650 61, 98 2

Fig. 8. Post-audit validation: minimum-creep-strain-rate prediction of the KR (straight lines) and Sinh (curved lines) models against NIMS data at 600, 650, and700 °C.

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downwards to pass through the shortest experimental rupture-life.These are considered 100% reliable predictions because the actualrupture life is longer than the predicted life. These predictions areextremely conservative.

• 0% reliability rupture. The model predictions are shifted upwards topass through the longest experimental rupture-life. These are con-sidered 0% reliable predictions because the actual rupture life isshorter than the predicted life. These predictions are extremely non-conservative and are not ideal for design.

• actual rupture. The actual rupture life of each creep deformationcurve is input into the CDM models. This gives the opportunity toeliminate the specimen-specific uncertainty of rupture.

Long-term creep deformation curves conducted at 61 and 98MPa at650 °C with rupture lives of 100,491 and 9064 h and creep ductility of23% and 18% respectively are collected. Creep deformation is predictedfor these two tests using the four rupture points as shown in Fig. 10b.Predictions are categorized as reasonable (those which can be con-sidered physically realistic) and unreasonable (those which cannot beconsidered physically realistic) considering the uncertainty of creepbehavior. An analysis of the KR and Sinh predictions at the four rupturepoints follows

• model predicted rupture: The KR model produces a reasonableprediction of creep ductility and rupture at 61MPa; however, KR isextremely conservative at 98MPa, overpredicting creep ductility,and underpredicting stress-rupture. The Sinh model, at both 61 and98MPa are reasonable, underpredicting creep ductility and stress-rupture.

• 100% and 0% reliability rupture: The KR and Sinh models aremodified with the rupture predictions [Eqs. (5)] and [(11)] re-spectively replaced with 100% and 0% reliability rupture. Simulatedrupture is defined as when damage [Eqs. (4)] and [(10)] is equal tothe critical damage; therefore, in some cases, a simulated ruptureoccurred shorter or longer than the designated 100% and 0% re-liability rupture. The statistics of the 100% and 0% reliability

simulations are provided in Table 12. Overall, the creep ductilityrange of the KR model is unreasonable while the creep ductilityrange for Sinh model is reasonable.

• Actual rupture: The KR model produces a reasonable prediction ofcreep ductility and rupture at 61MPa; however, at 98MPa, it isextremely conservative, over predicting the creep ductility (95%instead of 23%) and underpredicting rupture. The Sinh model pro-duces excellent predictions of creep ductility and rupture at bothstress levels. Overall, the Sinh model produces more reasonablepredictions than the KR model.

These results are remarkable. The models were calibrated usingcreep deformation curves with rupture near 102 h. It is found for Sinh,using actual rupture data, that the model can accurately predict creepdeformation up to 105 h when given the actual rupture data. The meansthe Sinh model is accurate across five decades. Using the model pre-dicted rupture and reliability bands, a design engineer can select thedesired reliability in a design taking into account the uncertainty ofcreep behavior.

At this stage, the first and second research objectives have beenachieved. The calibration procedure has been successfully applied totwo CDM models for 304SS. In post-audit validation, the minimum-creep-strain-rate predictions are reasonable with an exception at 700 °C(due to overlapping isotherms between the calibration and NIMS da-tabase). The stress-rupture predictions are reasonable but the predic-tions of the two models diverge in extrapolation. The creep deformationpredictions are excellent for the Sinh model and unreasonable for theKR model. The Sinh model, when calibrated using the disparate creepdata, is remarkably accurate across five decades of creep deformation,damage, and life.

8. Creep design maps

The third objective of this study is to illustrate a variety of creepdesign maps and compare the interpolation and extrapolation ability ofCDM models. An algorithm is developed to generate creep design maps

Fig. 9. Post-audit validation: Stress-Rupture prediction of the KR (Straight lines) and Sinh (curved lines) models against NIMS data at 450, 475, 500, 550, and 650 °Cdata.

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for the minimum-creep-strain-rate, stress-rupture, creep deformation,and damage. Using these maps, the KR and Sinh models are compared.

8.1. Algorithm

To generate these creep design maps, a mathematical algorithm isdeveloped in MATLAB to perform interpolation and extrapolation at atarget time for a given range of stress and temperature as outlined inFig. 11. The algorithm is divided into two steps: (Step 1) to perform theinterpolation and extrapolation and (Step 2) to plot the creep designmaps. In the first step, the temperature-range TΔ , stress-range σΔ ,target time tT , and the temperature step, stress step, and time step

increment (T σ t, ,s s s) are the inputs. The temperature-dependent mate-rial constant functions are applied to determine the material constantsat any given temperature. The material constants are plugged into theconstitutive model to determine the variables of interest (minimum-creep-stain-rate, rupture, deformation, and/or damage) at the targettime over the given stress and temperature range. In the second step,the variables of interest are plotted to create creep design maps byfollowing the steps shown in Fig. 11 (step 2). Using the algorithm, thefollowing creep design maps are generated.

8.2. Reading a design map and application

In this study, four different design maps (minimum-creep-strain-rate, rupture life, strain, and damage) are introduced. An illustration ofa generic design map is provided in Fig. 12. The design maps arecontours plots. Temperature is on the x-axis and stress is on the y-axis.The variable of interest (minimum-creep-strain-rate, rupture life, strain,damage, etc.) is plotted as a contour value, z (indicated by color orisolines). The user can select the target temperature and stress for theirdesign and identify the contour value. The contour value represents thepredicted variable of interest. The tensile properties of the material are

Fig. 10. Post-audit-validation: Creep deformation prediction of the KR and Sinh models against NIMS data at 650 °C. Time is on a log-scale.

Table 12100% and 0% reliability predictions for KR and Sinh.

Stress, σ Rupture range, tΔ r (hrs) Creep ductility (%)

(MPa) KR Sinh KR Sinh

61 17872–558490 9374–275710 2.2–69 1.2–3798 578–18087 588–17287 6–189 1.3–38

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plotted as lines on the design map so that the user can identify the onsetof rate-independent plasticity and ductile failure. The user can alsoselect and plot a line representing a critical value, Xcr of the variable ofinterest. By doing so, the user can establish a design envelope of stressesand temperatures within which a design is safe.

A design and maintenance engineer can apply design maps in twoways. In the deterministic approach, the design engineer uses the de-sign map as-is to predict behavior and establish the design envelope.The engineer may choose to apply a safety factor to the critical valueline to accommodate both the uncertainty of design and materials. Inthe probabilistic approach, the engineer would need to modify the al-gorithm and add probabilistic functions (such as Monte-Carlo methods,etc.) for a more quantitative assessment of uncertainty. The outcome ofthe modified algorithm would be a probability distribution of thevariable of interest. To plot the results, the engineer needs to establishwhat the acceptable probability is and plot it or develop live plotswhere the acceptable probability can be varied, and the design mapsupdate to reflect the engineers’ selection. For brevity, only the de-terministic approach is investigated herein.

8.3. Minimum-creep-strain-rate

In this section, the design maps developed for minimum-creep-

strain-rate are discussed. A parametric simulation is performed from550 to 850 °C and 0 to 400MPa as shown in Fig. 13. It is observed thatthe Sinh model is more conservative (predicts higher minimum-creep-strain-rate) than the KR model in the low-stress region (below 30MPa)and high-stress region (above 250MPa) due to the sigmoidal bend ofthe Sinh model. The KR model is more conservative than the Sinhmodel from 50 to 250MPa. The KR model exhibits an inflection at hightemperature (> 750 °C) and high stress ( > >σ220 170 MPa). This isdue to the evolution of the minimum-creep-strain-rate constants A andn. The constant A, increases with temperature, while n decreases. Theinteraction of these constants in [Eq. (3)] produces a minimum-creep-strain-rate inflection within the given stress and temperature range.This is not observed in the Sinh model for the given stress and tem-perature range.

Minimum-creep-strain-rate design maps can be can used to estimatethe rupture life (when the rupture design map is not available) using theMonkman-Grant model [52]. Furthermore, the minimum-creep-strain-rate design maps can be used to determine the onset of tertiary creepregime using the Theta projection model [53]. The onset of tertiarycreep represents the time at which appreciable creep damage begins toaccumulate and grow within the material. The onset of tertiary creep isan ideal metric for safe-life design. In the safe-life approach, no de-tectable damage is allowed in design; rather, the component is retired

Fig. 11. Algorithm to generate creep design maps.

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after a designated duration of service.

8.4. Stress-rupture

In this section, the creep design maps developed for stress-ruptureare discussed. Parametric simulations are performed to produce stress-rupture design maps as shown in Fig. 14. It is observed that the KR andSinh rupture predictions are similar for life> 102 h and notably dif-ferent for life< 102 hrs. This deviation is due to the sigmoidal bend inthe Sinh model previously observed in Figs. 4 and 9.

Both models exhibit creep-life above the UTS. It is hypothesized thatis phenomenon is related to the creep activation temperature. The creepactivation temperature is a temperature above which creep begins toappreciably impact the mechanical behavior. The creep activationtemperature is typically found between < <T T T0.3 0.5m m where Tm isthe melting temperature of the material.

Herein, the models spontaneously produce the creep activation

temperature. The predicted creep activation temperatures for the KRand Sinh are 656 °C and 575 °C respectively; that is T0.47 m and T0.41 m ofthe melting temperature of 1400 °C. Both numbers are reasonable.Overall, stress-rupture design maps can be used as a tool in safe-lifedesign. An engineer can apply a safety-factor to the stress-rupture de-sign maps and proceed with the design.

8.5. Creep deformation and damage

Parametric simulations are performed to produce creep deformationand damage design maps for the KR and Sinh as shown in Figs. 15 and16 respectively. The design envelope is consider anything below thecritical strain and/or damage. For this study, the critical strain is set tounity and the critical damage for KR and Sinh are set to 0.29 and unityrespectively. The engineer can select the target time at which to gen-erate design maps. In this study, maps are plotted at two target times: 1and 100,000 h. The former was selected because it shows the stratifi-cation of the contour bands and allows a discussion of creep-dominantversus ductile failure. The later was selected because it represents theuseful design life of hot path components in both industrial gas (IGT)and steam turbines (ST). In equivalent base operating hours, 100,000 hcoincides with the second major inspection in many IGT and ST fleets.

In the 1 h design maps, the contour bands are easy to perceive. Therelevant creep deformation and damage maps are Figs. 15a, c and 16a, crespectively. The contour bands appear similar to exponential decayfunctions. As the operating temperature increases, the contour bandscollapse to zero allowable stress at some critical temperature. The KRand Sinh predictions are distinct; with the bands in the Sinh maps beingmore grouped together and appearing to collapse to zero allowablestress at a lower critical temperature. The onset of ductile failure isplotted using the ultimate tensile strength (a bold dashed line). In bothKR and Sinh, there are regions where creep failure will occur beforeductile failure. The shape and intercepts of these regions are distinct.Ductile failure becomes dominant at a lower temperature as indicatedby the UTS line passing though contours where the critical values arenot exceeded.

In the 100,000 h design maps, the design envelope collapses, withthe creep failure region increasing dramatically in size while becomingmore similar in shape across the various maps. The relevant creep de-formation and damage maps are Fig. 15b, d and Fig. 16b, d respec-tively. The engineer has a much smaller design envelope if a design-lifeof 100,000 h is to be achieved. It is observed that the critical

Fig. 12. Illustration of design map. The map includes lines indicating the tensileproperties of the material. Contour colors/lines indicate the value of the vari-able of interest. The light grey shaded region represents the design envelopewhere stress and temperature are below the user selected critical value. (Forinterpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.)

Fig. 13. Minimum-creep-strain-rate (%/hr) design maps for the (a) KR [Eq. (3)] and (b) Sinh [Eq. (9)] models respectively. The white lines indicate the 0.2% yieldstrength and the bold dashed line represents the ultimate tensile strength (UTS).

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Fig. 14. Stress-rupture (hr) design maps for the (a) KR [Eq. (5)] and (b) Sinh [Eq. (11)] models respectively. The white lines indicate the 0.2% yield strength and thebold dashed line represents the ultimate tensile strength (UTS).

Fig. 15. Creep deformation (strain, unitless) design maps for the (a, b) KR [Eq. (1)] and (c, d) Sinh [Eq. (7)] models respectively. The white lines indicate the 0.2%yield strength and the bold dashed line represents the ultimate tensile strength.

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temperature has decreased dramatically. The lowest critical tempera-ture appears in the creep deformation map of Sinh at 750 °C. The mapsuggests a design-life of 100,000 h is not viable for 304SS at tempera-tures greater than 750 °C. Overall, the Sinh model is more conservativewhen compared to KR.

These creep deformation and damage maps can be used in the da-mage-tolerant design. The damage-tolerant design is where damage isallowed to develop within a component up to a critical value. Simulateddamage can be correlated to physical damage using several damagequantification techniques [54]. Physical damage can be measured usingdestructive (small sample incised from in-service component) or non-destructive evaluation methods (in-service component is examinedduring major inspection). During the design phase, high-fidelity simu-lations can be employed in lieu of physical data.

9. Conclusion

The first objective of this study was to provide a detailed procedurefor calibrating CDM models using disparate creep data. The Kachanov-Rabotnov and Sine-Hyperbolic CDM models were calibrated to dis-parate minimum-creep-strain-rate, stress-rupture, and creep-deforma-tion data. It was observed that the models fit the disparate creep dataacross the given stress and temperature range. Overall, creep CDM

models can be successfully calibrated using disparate creep data.The second objective was to demonstrate the reliability of extra-

polation using post-audit validation of the calibrated models. This ob-jective was completed by comparing the pre-calibrated KR and Sinhmodels with additional post-audit creep data that was not used in thecalibration process. The minimum-creep-strain rate predictions arepartially reasonable. The stress-rupture predictions are reasonable. Thecreep deformation predictions of the Sinh model are excellent whilethose of the KR model are not. The Sinh model produces excellentpredictions of a 100,000 h creep deformation curve when the modelwas calibrated using creep deformation curves of less than 200 h.Overall, the Sinh model, when calibrated using the disparate creepdata, is remarkably accurate across five decades of creep deformation,damage, and life.

The third objective was to illustrate a variety of creep design mapsand demonstrate that the interpolation and extrapolation ability ofCDM models can be compared graphically using the creep design maps.This objective was completed by developing a MATLAB algorithm togenerate design maps for minimum-creep-strain-rate, stress-rupture,creep deformation, and damage. It is observed that the Sinh model issuperior in the prediction of creep behavior when compared to the KRmodel. The creep activation temperature spontaneously arises in thestress-rupture design map. Overall, creep design maps are an excellent

Fig. 16. Damage design maps for the (a, b) KR [Eq. (4)] and (c, d) Sinh [Eq. (10)] models respectively. The white lines indicate the 0.2% yield strength and the bolddashed line represents the ultimate tensile strength.

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tool for design, material selection, and model comparison.

Future work

In the future, the integration of probabilistic methods in the de-velopment of the design maps must be explored. By incorporatingprobabilities, the design engineer has more flexibility in selecting thedesired designed reliability of a component. In addition, the currentdesign maps are applicable for new components subject to static op-erating conditions only. The maps are not appropriate for componentswith prior service history whose operating conditions are altered (i.e.when operating conditions are altered between base-load, load-fol-lowing, and/or peak-load modes). Finally, the design maps are for staticconditions only. There is a need to develop algorithms that allow fortime-dependent non-isothermal and -isostress conditions (i.e creep-fa-tigue and/or thermomechanical fatigue”).

Acknowledgments

This material is based upon work supported by the Department ofEnergy, National Energy Technology Laboratory under Award Number(s) DE-FE0027581.

Appendix A. Supplementary material

Supplementary data to this article can be found online at https://doi.org/10.1016/j.tafmec.2019.01.018.

References

[1] W. Ren, C. Luttrell, Gen IV Materials Handbook Beta Release for Structural andFunctional Evaluation, US Department of Energy, 2006, Report No. ORNL/GEN4/LTR-06–027.

[2] Materials Creep Database of NIMS – National Institutes for Materials Science,Japan.<http://mits.nims.go.jp/index_en.html > .

[3] D.A. Woodford, Test methods for accelerated development, design and life assess-ment of high-temperature materials, Mater. Des. 14 (4) (1993) 231–242.

[4] J. He, R.R. Sandström, Basic modelling of creep rupture in austenitic stainless steels,Theor. Appl. Fract. Mech. 89 (2017) 139–146.

[5] H.E. McCoy, R.D. Waddel, Mechanical properties of several products from a singleheat of Type 304 stainless steel, J. Eng. Mater. Technol. 97 (4) (1975) 343–350.

[6] S.J. Kim, Y.S. Kong, Y.J. Roh, W.G. Kim, Statistical properties of creep rupture datadistribution for STS304 stainless steels, Mater. Sci. Eng., A 483 (2008) 529–532,https://doi.org/10.1016/j.msea.2006.12.153.

[7] S.G. Pantelakis, N.I. Vassilas, Assessment of creep damage in austenitic steel X-8-CrNiMoNb-16-16 at elevated temperature. Theor. Appl. Fract. Mech. 19 (2) (1993)113–118.

[8] G.A. Webster, R.A. Ainsworth, High-Temperature Component Life Assessment,Springer Science and Business Media, 2013.

[9] R. Sandström, Extrapolation of creep strain data for pure copper, J. Test. Eval. 27(1) (1999) 31–35, https://doi.org/10.1520/JTE12037J.

[10] M. S. Haque, C. Ramirez, C. M. Stewart, A novel metamodeling approach for time-temperature parameter models, in: ASME 2017 Pressure Vessels and PipingConference, American Society of Mechanical Engineers, pp.V005T11A021–V005T11A021.

[11] W.M. Cummings, R.H. King, Extrapolation of creep strain and rupture properties of1/2 per Cent Cr, 1/2 per Cent Mo, 1/4 per Cent V pipe steel, Procee. InstitutionMech. Eng. 185 (1) (1970) 285–299, https://doi.org/10.1243/PIME_PROC_1970_185_036_02.

[12] M.K. Booker, C.R. Brinkman, V.K. Sikka, Correlation and extrapolation of creepductility data for four elevated temperature structural materials (No. ORNL-TM–4992), Oak Ridge National Lab., Tenn. (USA), 1975.

[13] Method of Extrapolation Used in the Analysis of Creep Rupture Data, Parts 1 to 6:Derivation of Long-Term Stress-Rupture Properties, Annex to InternationalStandard ISO 6303, 1981.

[14] R.R. Sandström, A procedure for extended extrapolation of creep rupture data, J.Test. Eval. 31 (1) (2003) 58–64.

[15] R. Sandström, L. Lindé, Precision in the extrapolation of creep rupture data, J. Test.Eval. 27 (3) (1999) 203–210, https://doi.org/10.1520/JTE12063J.

[16] S. Holdsworth, The European Creep Collaborative Committee (ECCC) approach tocreep data assessment, J. Pressure Vessel Technol. 130 (2) (2008) 024001.

[17] C.M. Stewart, A.P. Gordon, Strain and damage-based analytical methods to de-termine the kachanov-rabotnov tertiary creep damage constants, Int. J. DamageMech. 21 (8) (2011) 1186–1201, https://doi.org/10.1177/1056789511430519.

[18] M.S. Haque, C.M. Stewart, Modeling the creep deformation, damage, and rupture ofhastelloy X using MPC Omega, Theta, and Sin-hyperbolic models, ASME 2016

Pressure Vessels and Piping Conference, Vancouver, Canada, 2016, https://doi.org/10.1115/PVP2016-63029 V06AT06A050-V06AT06A050.

[19] M.S. Haque, C.M. Stewart, Exploiting functional relationships between MPCOmega, Theta, and Sinh-hyperbolic continuum damage mechanics model,V06AT06A052-V06AT06A052, ASME 2016 Pressure Vessels and PipingConference, Vancouver, Canada, 2016, https://doi.org/10.1115/PVP2016-63089.

[20] D.R. Hayhurst, D.A. Lavender, N.G. Worley, A. Salim, An assessment of the θ-pro-jection method for the representation and extrapolation of creep data for a 1% Cr,12% Mo, 14% V steel tested at 565° C, Int. J. Press. Vessels Pip. 20 (4) (1985)289–317.

[21] K. Maruyama, H. Oikawa, An extrapolation procedure of creep data for StDetermination: with special reference to Cr-Mo-V steel, J. Pressure Vessel Technol.109 (1) (1987) 142–146.

[22] S.R. Brown, R.W. Evans, B. Wilshire, A comparison of extrapolation techniques forlong-term creep strain and creep life prediction based on equations designed torepresent creep curve shape, Int. J. Press. Vessels Pip. 24 (3) (1986) 251–268.

[23] Z. Abdallah, K. Perkins, S. Williams, Advances in the Wilshire extrapolation tech-nique—full creep curve representation for the aerospace alloy Titanium 834, Mater.Sci. Eng., A 550 (2012) 176–182.

[24] J.J. Lemaitre, How to use damage mechanics. Nucl. Eng. Des. 80 (2) (1984)233–245.

[25] D. Wu, H. Jing, L. Xu, L. Zhao, Y. Han, Theoretical and numerical analysis of creepcrack initiation combined with primary and secondary stresses, Theor. Appl. Fract.Mech. 95 (2018) 143–154.

[26] M.S. Haque, C.M. Stewart, Comparison of a new Sinh-hyperbolic creep damageconstitutive model with the classic Kachanov-Rabotnov model using theoretical andnumerical analysis TMS, 144th Annual Meeting and Exhibition, Florida, USA, 2015,pp. 15–19, , https://doi.org/10.1007/978-3-319-48127-2_114.

[27] M.S. Haque, C.M. Stewart, A novel sin-hyperbolic creep damage model to overcomethe mesh dependency of classic local approach Kachanov-Rabotnov model,V009T12A023V009T12A023-, ASME 2015 International Mechanical EngineeringCongress and Exposition, (2015).

[28] J.L. Chaboche, Continuum damage mechanics: part II—damage growth, crack in-itiation, and crack growth, J. Appl. Mech. 55 (1) (1988) 65–72.

[29] M. Moattari, I.I. Sattari-Far, N. Bonora, The effect of subcritical ductile crackgrowth on cleavage fracture probability in the transition regime using continuumdamage mechanics simulation, Theor. Appl. Fract. Mech. 82 (2016) 125–135.

[30] S. Murakami, Progress of continuum damage mechanics, JSME Int. J. 30 (263)(1987) 701–710.

[31] Q. Xu, Creep damage constitutive equations for multi-axial states of stress for 0.5Cr0. 5Mo0. 25V ferritic steel at 590 C. Theor. Appl. Fract. Mech. 36 (2) (2001)99–107.

[32] D.R. Hayhurst, The use of continuum damage mechanics in creep analysis for de-sign, J. Strain Anal. Eng. Design 29 (3) (1994) 233–241.

[33] C.M. Stewart, A.P. Gordon, Y.W. Ma, R.W. Neu, An anisotropic tertiary creep da-mage constitutive model for anisotropic materials, Int. J. Press. Vessels Pip. 88 (8)(2011) 356–364.

[34] S. Murakami, Y. Liu, Mesh-dependence in local approach to creep fracture, Int. J.Damage Mech. 4 (3) (1995) 230–250, https://doi.org/10.1177/105678959500400303.

[35] C.M. Stewart, A.P. Gordon, Constitutive modeling of multistage creep damage inisotropic and transversely isotropic alloys with elastic damage, J. Pressure VesselTechnol. 134 (4) (2012) 041401.

[36] M.S. Haque, C.M. Stewart, Finite-element analysis of waspaloy using sinh creep-damage constitutive model under triaxial stress state, J. Pressure Vessel Technol.138 (3) (2016) 031408, , https://doi.org/10.1115/1.4032704.

[37] M.S. Haque, C.M. Stewart, The stress-sensitivity, mesh-dependence, and con-vergence of continuum damage mechanics models for creep, J. Pressure VesselTechnol. 139 (4) (2017) 041403, https://doi.org/10.1115/1.4036142.

[38] T.H. Hyde, W. Sun, J.A. Williams, Creep analysis of pressurized circumferential pipeweldments—a review, J. Strain Anal. Eng. Design 38 (1) (2003) 1–27.

[39] K. Fujiyama, Risk-based engineering for design, material selection and maintenanceof power plants, Mater. High Temp. 28 (3) (2011) 225–233.

[40] C.M. Stewart, A.P. Gordon, Modeling the temperature dependence of tertiary creepdamage of a Ni-based alloy, J. Pressure Vessel Technol. 131 (5) (2009) 051406.

[41] C. M. Stewart, A Hybrid Constitutive Model For Creep, Fatigue, And Creep-FatigueDamage, Ph.D. Dissertation, University of Central Florida, Orlando, Fl, 2013.

[42] L. M. Kachanov, The Theory of Creep, National Lending Library for Science andTechnology, Boston Spa, England, Chaps. IX, X, 1967.

[43] Y.N. Rabotnov, Creep Problems in Structural Members, Amsterdam, Weinheim,North Holland, WILEY-VCH Verlag GmbH and Co KGaA, 1969.

[44] P.G. McVetty, Creep of metals at elevated temperatures – hyperbolic-sine relationbetween stress and creep rate, Trans. ASME 65 (7) (1943) 761–767.

[45] M.K. Booker, Use of Generalized Regression Models for the Analysis of StressRupture Data, United States Department of Energy by Union Carbide Corporation,Contract W7405-eng-26, CONF-780609-5, Oak Ridge, Tn, 1978.

[46] M. Gold, W.E. Leyda, R.H. Zeisloft, The effect of varying degrees of cold work on thestress-rupture properties of Type 304 stainless steel, J. Eng. Mater. Technol. 97 (4)(1975) 305–312.

[47] T. Sritharan, H. Jones, The creep of Type 304 stainless steel at low stresses, ActaMetall. 28 (12) (1980) 1633–1639.

[48] H.K. Kim, F.A. Mohamed, J.C. Earthman, High-temperature rupture of micro-structurally unstable 304 stainless steel under uniaxial and triaxial stress states,Metall. Trans. A 22A (1991) 2629–2636.

[49] D.J. Wilson, Extrapolation of Rupture Data for Type 304 (18Cr-10Ni), Grade 22 (2-1/4Cr-1Mo) and Grade 11(1-1/4Cr-1/2Mo-3/4Si) Steels, Progress Report to: The

M.S. Haque, C.M. Stewart Theoretical and Applied Fracture Mechanics 100 (2019) 251–268

267

Metal Properties Council, Report 34006-2P, New York, NY, 1971.[50] J.E. Bynum, F.V. Ellis, B.W. Roberts, D.A. Canonico, High temperature creep of type

304 stainless steel, stress classification, robust methods, and elevated temperaturedesign; presented at the 1992 Pressure Vessels and Piping Conference, New Orleans,La, 1992, pp. 67–83.

[51] E. Hosseini, S.R. Holdsworth, E. Mazza, Stress regime-dependent creep constitutivemodel considerations in finite element continuum damage mechanics, Int. J.Damage Mech. 22 (8) (2013) 1186–1205, https://doi.org/10.1177/1056789513479810.

[52] F.C. Monkman, N.J. Grant, Proceedings of ASTM 56 (1956) 593–620.[53] R.W. Evans, J.D. Parker, B. Wilshire, The θ projection concept—a model-based

approach to design and life extension of engineering plant, Int. J. Press. Vessels Pip.50 (1–3) (1992) 147–160.

[54] L. Yang, A. Fatemi, Cumulative fatigue damage mechanisms and quantifyingparameters: a literature review, J. Test. Eval. 26 (2) (1998) 89–100.

M.S. Haque, C.M. Stewart Theoretical and Applied Fracture Mechanics 100 (2019) 251–268

268