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[Izuno, Hitoshi](https://orcid.org/0000-0003-0503-3621), [Demura, Masahiko](https://orcid.org/0000-0002-7308-3041), [Tabuchi, Masaaki](https://orcid.org/0000-0002-8781-559X), Mototake, Yoh-ichi, Okada, Masato

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[Data-based selection of creep constitutive models for high-Cr heat-resistant steel](https://mdr.nims.go.jp/datasets/6dd4bd5a-33ec-4eda-bfe9-93e9f80a176d)

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Data-based selection of creep constitutive models for high-Cr heat-resistant steelFull Terms & Conditions of access and use can be found athttps://www.tandfonline.com/action/journalInformation?journalCode=tsta20Science and Technology of Advanced MaterialsISSN: (Print) (Online) Journal homepage: https://www.tandfonline.com/loi/tsta20Data-based selection of creep constitutive modelsfor high-Cr heat-resistant steelHitoshi Izuno, Masahiko Demura, Masaaki Tabuchi, Yoh-ichi Mototake &Masato OkadaTo cite this article: Hitoshi Izuno, Masahiko Demura, Masaaki Tabuchi, Yoh-ichi Mototake& Masato Okada (2020) Data-based selection of creep constitutive models for high-Crheat-resistant steel, Science and Technology of Advanced Materials, 21:1, 219-228, DOI:10.1080/14686996.2020.1738268To link to this article:  https://doi.org/10.1080/14686996.2020.1738268© 2020 The Author(s). Published by NationalInstitute for Materials Science in partnershipwith Taylor & Francis Group.View supplementary material Published online: 27 Apr 2020. Submit your article to this journal Article views: 723 View related articles View Crossmark data Citing articles: 1 View citing articles https://www.tandfonline.com/action/journalInformation?journalCode=tsta20https://www.tandfonline.com/loi/tsta20https://www.tandfonline.com/action/showCitFormats?doi=10.1080/14686996.2020.1738268https://doi.org/10.1080/14686996.2020.1738268https://www.tandfonline.com/doi/suppl/10.1080/14686996.2020.1738268https://www.tandfonline.com/doi/suppl/10.1080/14686996.2020.1738268https://www.tandfonline.com/action/authorSubmission?journalCode=tsta20&show=instructionshttps://www.tandfonline.com/action/authorSubmission?journalCode=tsta20&show=instructionshttps://www.tandfonline.com/doi/mlt/10.1080/14686996.2020.1738268https://www.tandfonline.com/doi/mlt/10.1080/14686996.2020.1738268http://crossmark.crossref.org/dialog/?doi=10.1080/14686996.2020.1738268&domain=pdf&date_stamp=2020-04-27http://crossmark.crossref.org/dialog/?doi=10.1080/14686996.2020.1738268&domain=pdf&date_stamp=2020-04-27https://www.tandfonline.com/doi/citedby/10.1080/14686996.2020.1738268#tabModulehttps://www.tandfonline.com/doi/citedby/10.1080/14686996.2020.1738268#tabModuleData-based selection of creep constitutive models for high-Cr heat-resistantsteelHitoshi Izuno a, Masahiko Demura a, Masaaki Tabuchib, Yoh-ichi Mototakec and Masato Okadaa,daResearch and Services Division of Materials Data and Integrated System, National Institute for Materials Science, Ibaraki, Japan;bResearch Center for Structural Materials, National Institute for Materials Science, Ibaraki, Japan;cThe Institute of Statistical Mathematics, Tokyo, Japan;dGraduate School of Frontier Sciences, The University of Tokyo, Chiba, JapanABSTRACTThere are two types of creep constitutive equation, one with a steady-state term (steady-statetype) and the other with no steady-state term (non-steady-state type). We applied the Bayesianinference framework in order to examine which type is supported by experimental creepcurves for a Grade 91 (Gr.91) steel. The Bayesian free energy was significantly lower for thesteady-state type under all the test conditions in the ranges of 50–90 MPa at 923 K, 90–160 MPaat 873 K and 170–240 MPa at 823 K, leading to the conclusion that the posterior probability wasvirtually 1.0. These findings mean that the experimental data supported the steady-state-typeequation. The dependence of the evaluated steady-state creep rate on the applied stressindicates that there is a transition in the mechanism governing creep deformation around120 MPa.ARTICLE HISTORYReceived 9 August 2019Revised 25 December 2019Accepted 19 January 2020KEYWORDSCreep constitutive equation;grade 91 steel; bayesian freeenergy; model selectionmethod; theta method;steady-state creepCLASSIFICATION404 Materials informatics /Genomics; 106 MetallicmaterialsCONTACT Masahiko Demura demura.masahiko@nims.go.jp Research and Services Division of Materials Data and Integrated System, NationalInstitute for Materials Science, Namiki 1-1, Tsukuba, Ibaraki, JapanSupplemental data for this article can be accessed here.SCIENCE AND TECHNOLOGY OF ADVANCED MATERIALS2020, VOL. 21, NO. 1, 219–228https://doi.org/10.1080/14686996.2020.1738268© 2020 The Author(s). Published by National Institute for Materials Science in partnership with Taylor & Francis Group.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permitsunrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.http://orcid.org/0000-0003-0503-3621http://orcid.org/0000-0002-7308-3041https://doi.org/10.1080/14686996.2020.1738268http://www.tandfonline.comhttps://crossmark.crossref.org/dialog/?doi=10.1080/14686996.2020.1738268&domain=pdf&date_stamp=2020-04-201. IntroductionCreep is a time-dependent phenomenon and is con-trolled by various mechanisms depending on the stage.In general, it starts at a relatively high rate followed bya decrease in the rate in the primary stage, and then therate reaches a minimum and appears to become con-stant in the secondary stage. In the tertiary stage, thecreep deformation gradually accelerates with time. Thenature of the dominant mechanism in each stage can becharacterized by the time dependence of the creep ratein most cases. For example, a decrease or increase in thecreep rate is expected to indicate a change in the micro-structure and/or in the amount and the structure ofdefects such as dislocations and voids. A constant creeprate, however, does not necessarily mean that the con-trolling microstructure is stable since an apparentsteady state could occur when the decrease and increasein creep rate are balanced owing to the occurrence ofboth types of microstructural change. To examine thedominant mechanisms in the secondary stage, it iscrucial to distinguish between the intrinsic and extrinsicmechanisms when a constant creep rate is observed.In this study, we propose a Bayesian inference fra-mework to judge whether there is an intrinsic steadystate, on the basis of observed creep strain curves. Here,we reduce the problem to a model selection probleminvolving two types of creep constitutive equation, i.e.one with a steady-state term and the other with nosteady-state term. If there is no steady state, the pro-posed framework should choose the equation with nosteady-state term. For the non-steady-state model, weadopt the theta projection [1], which includes exponen-tial-type deceleration and acceleration terms corre-sponding to the primary and tertiary stages,respectively. For the steady-state model, we adda linear time-dependent term, which was introducedin the modified theta projection [2]. In summary, thetwo creep constitutive equations are written as follows:ε ¼ ε0 þ A 1� exp �αtð Þð Þ þ _εst þ B exp βtð Þ � 1ð Þmodified theta projection; steady-state modelð Þ(1)ε ¼ ε0 þ A 1� exp �αtð Þð Þ þ B exp βtð Þ � 1ð Þtheta projection; non-steady-state modelð Þ;(2)where ε and t are the creep strain and test time,respectively. α and β are the time constants for theprimary and tertiary stages, respectively, and ε0, A, andB are constants with the dimension of strain. Theconstant _εs in equation (1) corresponds to the steady-state creep rate in the secondary stage. For the modelselection, we use the Bayesian inference framework.As detailed in the next section, we calculate, for eachmodel, the Bayesian free energy which can be used asan indicator to evaluate which model is closer to theactual model that generated the observed data. To ourbest knowledge, there is no study to apply the Bayesianinference framework to the model selection problemin creep constituent equations.The target material is one of the 9Cr-1Mo-Nb-V(Gr.91) steels, which are employed for high-temperature piping at thermal power plants owing totheir excellent creep strength. Table 1 lists the previousstudies in which the constitutive equation (1) or (2)was applied to the Gr.91 steel [2–4]. Both equationshave been used and there is no consensus on whetherthe steady-state term is necessary.2. Bayesian inference framework for modelselectionIn the Bayesian inference framework [5], the posteriorprobability of each model based on given data can beevaluated as follows: The posterior probability ofa model l ¼ l wlð Þ (wl: parameter vector of model l)under given data D is considered. The posterior prob-ability P ljDð Þ for model l is given byp ljDð Þ ¼ pðDjlÞp lð Þp Dð Þ (3)from Bayes’ theorem. On the other hand, let Λ be theset of all the considered models. P ljDð Þ is evaluated asTable 1. Some previous studies of creep curves in which the theta or modified theta method was applied to Gr.91 steel.PapersSteady(modifiedtheta)Non-steady(theta or revisedtheta)Othermodels Test conditions Minimum creep rateSecondary(steady- state)creepFactors influencing creep model equationselectionHoldsworth et al. (2007) [2]Applied Not applied – – – –Long-term creep characterization ofGr.91 steel by modified creepconstitutive equationsKim et al. (2011) [3]NotappliedApplied Omega 873 K 130–200MPa10�1 � 10�4/h YesCreep-rupture life prediction for 9Cr-1Mo-Nb-V weld metalMiyakita et al. (2015) [4]NotappliedApplied Omega 883 K 150 MPa 3:85� 10�5/h Yes898 K 110–170MPa3:25� 10�3–1:15� 10�5/h923 K 90–150 MPa 6:10� 10�3–2:16� 10�5/hSci. Tech. Advan. Mater. 21 (2020) 220 H. IZUNO et al.p ljDð Þ ¼ pðljDÞPλ2Λ pðλjDÞ¼pðDjlÞp lð Þp Dð ÞPλ2ΛpðDjλÞp λð Þp Dð Þ¼ pðDjlÞPλ2Λ PðDjλÞ(4)under the assumption that p lð Þ is the same uniformdistribution among all the considered models of Λ.pðDjlÞ is the model evidence and given bypðDjlÞ ¼ � dwlpðDjwl; lÞφðwljlÞ, where φðwljlÞ is theprior distribution of parameters wl of model l. LetFl ¼ �ln pðDjlÞð Þ. Fl is called the Bayesian free energyof l [5,6]. Finally, p ljDð Þ can be evaluated asp ljDð Þ ¼ exp �Flð ÞPλ2Λ exp �Fλð Þ : (5)Although the direct calculation of the Bayesian freeenergy (equation (4)) is difficult in general, it is possiblefor a linear model with variables having normal distribu-tions. That is, let model l be a c-dimensional linearmodel : y ¼Pci¼1 wixi wl ¼ wif gð Þ, and the numberof data be N. If both pðDjwl; lÞ and φðwljlÞ are multi-variate normal distributions of dimensions N and c withvariances σ2 and σ21, respectively, the Bayesian free energyis given bywhere X ¼ xif gjh iand y ¼ yj� �meaning given dataare a c� N matrix of the explanatory variables and anN-dimensional vector of the objective variable, respec-tively, and the superscript T means the transpose ofvector.This integration can be performed directly byGaussian integration. The first exponential term corre-sponds to the bias of quadratic errors between the modeland data, and minimizing this term may cause the over-fitting problem. However, there is also a second expo-nential term for evaluating the complexity of the model.Thus, minimizing the Bayesian free energy is a robustway of optimizing the model parameters since it isequipped with a mechanism to avoid the overfittingproblem [5].The question is how to reduce the nonlinear creepconstitutive equations into linear problems. Here, wetreat the exponential terms as constant and hyperpara-meterize the time constants α and β. That is, by lettingR α; tð Þ ¼ 1� exp �αtð Þ and S β; tð Þ ¼ exp βtð Þ � 1, andapplying R and S to equations (1) and (2), they arereduced to the following linear functions:ε ¼ ε0 þ ARþ _εst þ BS (1)ε ¼ ε0 þ ARþ BS: (2)For these linear functions, we can calculate theBayesian free energy by applying equation (6). Then,we optimize the hyperparameters α and β so that theBayesian free energy is minimized.Figure 1(a) shows the algorithm for optimizing αand β to minimize the Bayesian free energy based ona grid search method. Firstly, we performeda conventional nonlinear fitting to the experimentaldata to obtain initial guesses of α and β, respectively.Around the initial guesses, we then defined the firstsearch area within the range from double to half ofthem, setting the number of grids to 100 for eachparameter. We calculated the Bayesian free energy ateach grid point and determined the point that exhib-ited the lowest Bayesian free energy within the searcharea. When the lowest free energy point was at theedge of the search area, we redefined a new search areaaround the point in the same way as we defined thefirst search area, that is, we set it within the range fromdouble to half of the value of the point. This is becausethere might be a point with a lower value outside theprevious search area. We continued the same searchprocess until we found the lowest free energy pointinside the search area. Then, we performed the finaloptimization step, for which we set the search areawithin the range from α� 0:002 to αþ 0:002 and β�0:0001 to βþ 0:0001 throughout all the creep testconditions. The final optimization step enabled us toobtain the optimized values with uniform accuracy forall creep test conditions. Figure 1(b) shows how ouralgorithm worked in the case of applying the thetamethod to the creep strain curve for 873 K/160 MPa.In the search, we assumed that α and β were positivefrom the characteristics of the experimental creepcurves. The computational calculations were per-formed on Python 3.5.2/NumPy 1.13.1/SciPy 0.19.0/scikit-learn 0.18.1 on Ubuntu 16.04 with a PrecisionTower 7910 (Dell, USA).3. Experimental proceduresPart of the creep results used in the study was pre-viously reported [7], namely, most of the rupture timedata except for those at 823 K/170 MPa, 873 K/90 MPa,F σ2; σ21� � ¼ �ln � dwl12πσ2� �N2exp � 12σ2yT � wlTX� �yT � wlTX� �T� �12πσ21� �c2exp � 12σ21wlTwl� � !(6)Sci. Tech. Advan. Mater. 21 (2020) 221 H. IZUNO et al.and 923 K/50 MPa, and three creep strain curves at873 K/160, 140, and 130 MPa. The procedures for thecreep tests and the sample preparation were also givenin the previous work [7]. The chemical composition ofthe Gr.91 steel used in this study is summarized inTable 2. We conducted creep tests under several con-stant loads at temperatures of 823 K, 873 K, and 923 K.Specimens with a gauge of 6 mm diameter and 30 mmlength were cut from a plate. The nominal strain wasobtained from the measured displacement normalizedby the initial gauge length. For the analysis, the periodfrom the beginning of the creep test up to 98% of thetotal rupture time was used to exclude the extrinsicincrease in the creep rate attributable to the significantFigure 1. (a) Schematic diagram of model selection method with grid search. (b) shows how our algorithm worked in the case ofapplying the theta method to the creep strain curve for 873 K/160 MPa. The four heat maps show the distribution of Bayesian freeenergy by greyscale, where the point of the lowest free energy in each search area is indicated by the yellow cross mark. Thegreen, red, and purple rectangles inserted in the top three maps show the next search area, respectively. The right-bottom plotshows the transition from the first search area to the final search area.Table 2. Chemical composition of the 9Cr-1Mo-Nb-V (Gr.91) steel.C Si Mn P S Cu Ni Cr Mo V Nb Al N(mass %) 0.10 0.25 0.43 0.006 0.002 0.012 0.06 8.87 0.93 0.19 0.07 0.014 0.06Sci. Tech. Advan. Mater. 21 (2020) 222 H. IZUNO et al.increase in true stress with the reduction of the crosssection.4. Results and discussionTable 3 lists the creep test conditions, observed rup-ture time, and minimum creep rate. The creep rupturetimes and minimum creep rates obtained in this studywere within the same ranges as those obtained inpreviously reported studies [2–4], and the consistencyin the range of minimum creep rates can be partlyconfirmed by comparing Tables 1 and 3. The mini-mum creep rate and the creep rupture time in eachcreep test have an inversely proportional relationship.Figure 2 shows the creep strain curve at 873 K/160MPa. The shape of the curve was upper convex in theprimary stage from the start of the test to 20 h, showinga monotonic decrease in the creep rate. Then, it becamealmost a straight line from 20 to 300 h in the secondarystage, indicating that the creep rate was constant. In thetertiary stage after 300 h, the curve increased gradually,showing the increase in creep rate, and finally, thespecimen ruptured at 501.4 h. All the creep strain curvesobtained in this study had these three stages. The mini-mum creep rates listed in Table 3 were obtained fromthe gradient of the linear part of the creep strain curvesin the second stage.Using the obtained creep strain curves, we calculatedthe Bayesian free energies for the two types of constitutiveequation, i.e. themodified theta projection (equation (1))and the theta projection (equation (2)). Table 3 sum-marizes the results. Under all the test conditions, themodified theta projection, i.e. the steady-state model,exhibited lower Bayesian free energies than the thetaprojection, i.e. the non-steady-state model. This meansthat the experimental data supported the steady-statemodel rather than the non-steady-state model. We alsocalculated the posterior probability from the Bayesianfree energy. Here, we explain the actual calculation inthe case of 873 K/90 MPa, where the difference inBayesian free energy is the smallest among all the testconditions. The Bayesian free energies were −9612 and−9585 for the modified theta and theta projection meth-ods, respectively. The posterior probability for eachmethod can be calculated from equation (5) with the setof models Λ ¼ modified-theta; thetaf g as follows:P modified thetaj873K=90MPað Þ¼ exp �9612ð Þexp �9612ð Þ þ exp �9585ð Þ ¼ 0:999999999998(7)andP thetaj873K=90MPað Þ ¼ exp �9585ð Þexp �9612ð Þ þ exp �9585ð Þ¼ 1:87952881654� 10�12:(8)Although the difference in the Bayesian free energy isonly 27, the posterior probability is clearly higher forthe modified theta projection and is virtually one. Asshown in Table 3, we confirmed that the posteriorprobabilities of the modified theta projection andtheta projection are virtually one and zero under allthe test conditions, respectively. Consequently,a model assuming a steady-state term was explicitlysupported by the creep data obtained from the Gr.91steel.Figure 3(a,b) show the experimental creep straincurves and those calculated with the parameters at theminima of the Bayesian free energies for the modifiedtheta and theta projections, respectively, for 823 K/240MPa. The experimental curves (black dots) were wellTable 3. Observed rupture time and creep rate, Bayesian free energy, posterior probability of the two models, ratio of fittingparameters α for the modified theta method (αModθ) and theta method (αθ), and the ratio of the steady-state region to the rupturetime for the modified theta method in the present ASME Gr.91 steel creep tests. Values for the Bayesian free energy are rounded tointegers. Values for the posterior probability, the ratio of fitting parameters α, and the ratio of the steady-state region are roundedto three significant figures.Test conditions Free energyPosteriorprobabilityTemperature(K) Stress (MPa)Rupture(h)Minimum creep rate(1/h)ModifiedTheta ThetaModifiedtheta ThetaαθαModθSteady-state regionratio823 240 290.7 1.76� 10�4 −2322 −2002 1 0 0.0359 0.563220 2392.5 1.05� 10�5 −2980 −2473 1 0 0.0646 0.709200 10,432.9 3.58� 10�6 −3883 −3370 1 0 0.0546 0.585190 18,514.7 1.44� 10�6 −11,016 −9453 1 0 0.0651 0.554170 47,104.5 3.46� 10�7 −24,936 −22,427 1 0 0.00813 0.500873 160 501.4 9.65� 10�5 −2625 −2270 1 0 0.0167 0.579140 2550.2 1.48� 10�5 −3992 −3432 1 0 0.0271 0.579130 6036.8 5.33� 10�6 −2826 −2449 1 0 0.0232 0.483110 19,907.0 9.60� 10�7 −11,258 −10,693 1 0 0.00107 0.419100 40,307.4 5.48� 10�7 −12,410 −11,586 1 0 0.00107 0.503923 90 73,960.9 2.83� 10�7 −9612 −9585 1 0 0.00124 0.34290 928.0 3.50� 10�5 −3141 −3026 1 0 0.000102 0.46980 2726.1 1.18� 10�5 −2757 −2688 1 0 0.0000820 0.46070 8385.6 4.75� 10�6 −3537 −3428 1 0 0.000129 0.38250 60,181.2 6.19� 10�7 −16,654 −15,000 1 0 0.0159 0.618Sci. Tech. Advan. Mater. 21 (2020) 223 H. IZUNO et al.reproduced by the calculated ones (gray lines) for bothprojections. In detail, the theta projection methodexhibited some discrepancy, especially in the primaryregion up to 50 h (see the inset of Figure 3(b)). Incontrast, themodified theta projectionmethod followedthe experimental curves very well even in the primarystage, as shown by the inset of Figure 3(a).The difference in the reproducibility of the creepstrain curve is more clearly depicted in the creep strainrate curves shown in Figure 3(c, d), which are plots ofthe log of the creep strain rate against time. For moredetailed analysis, we decompose the terms in eachequation. The derivatives of equations (1) and (2)with respect to time t are_ε ¼ Aα exp �αtð Þ þ _εs þ Bβ exp βtð Þ (9)_ε ¼ Aα exp �αtð Þ þ Bβ exp βtð Þ: (10)In the plot of log _ε versus t, the primary, steady-state,and tertiary creep terms are asymptotic to the linesln _εprimary� � ¼ ln Aα exp �αtð Þð Þ ¼ ln αAð Þ � αt (11)ln _εsecondary� � ¼ ln _εsð Þ (12)ln _εtertiary� � ¼ ln Bβ exp βtð Þð Þ ¼ ln βBð Þ þ βt; (13)respectively. In Figure 3(c, d), we drew the lines givenby equations (11) – (13) for the modified theta pro-jection method (Figure 3(c)) and the lines given byequations (11) and (13) for the theta projectionmethod (Figure 3(d)). The modified theta projectionmethod traced the changes in the creep strain ratewell in the whole region, i.e. the extensive drop in theprimary stage, the steady state in the secondary stage,and the gradual increase in the tertiary stage. On theother hand, the theta projection method failed tofollow the extensive drop in the primary stage andshowed a very gradual decrease in this stage owing tothe trend in both the primary and secondary regionsbeing represented by a single average line. It turnsout that the modified theta model well captured thegradient of the extensive drop in the primary stage(blue broken line) and that of the gradual increase inthe tertiary stage (green dotted line). In addition,note that most of the secondary stage is governedby the steady-state term (equation (12)) as shownby the red dot-dashed line. That is, the contributionof the other time-dependent terms (equations (11)and (13)) can be regarded as negligible in the second-ary stage. This implies that there is a steady state fora certain period of time in the creep deformation. Onthe other hand, for the theta projection method, nosteady state appears, so one can expect an apparentsteady state by compensating for the decrease in theprimary stage and the increase in the tertiary stage. Inpractice, as shown in Figure 3(d), the decelerationrate was underestimated in the theta projectionmethod since it tended to represent both the primaryand secondary stages by a single deceleration term(equation (11)). Table 3 lists the ratio of the timeconstant α, indicating the deceleration rate, for themodified theta and theta projection models. The ratioclearly shows that the theta projection method sig-nificantly underestimated the rate of decrease com-pared with the modified theta projection method.We evaluated the time of the steady-state region asa fraction of the entire creep rupture time, asFigure 2. Creep strain curve of Gr.91 steel at 873 K/160 MPa.Sci. Tech. Advan. Mater. 21 (2020) 224 H. IZUNO et al.schematically shown in Figure 4(a). Here, the steady-state region was defined as the region where the mag-nitude of the steady-state term is at least 10 timeslarger than that of the primary or tertiary term withrespect to the creep rate. The steady-state region at823 K/240 MPa is identified as being between twosolid gray vertical lines in Figure 4(a). In the sameway, the steady-state regions were identified for theother test conditions (see Figure S1 in the supplementmaterial). Figure 4(b) shows the time fraction of thesteady-state region versus the creep life under varioustesting conditions. The average fraction of all testresults is 0.516 and the standard deviation is 0.0923,indicating that almost 50% of the creep deformation isgoverned by a mechanism leading to the steady statefor the Gr.91 steel.The present conclusion supports a widely usedapproximation named Norton’s law in the numericalsimulation of creep deformation behavior [8]. TheNorton’s law assumes a constant creep rate obeyingthe power law of the applied stress [9,10]. To our bestknowledge, there was so far no report confirming thevalidity of the assumption of constant creep rate usinga rational method based on creep curves. The presentBayesian inference method revealed that it is reason-able to assume a constant creep rate at least up to 50%of creep life for the Gr.91 steel.The power factor in the Norton’s law is related tothe dominant mechanism of the creep deformation andit is represented by the gradient of a linear curve in thedouble logarithmic plot of the creep rate in the steadystate, _εs versus the applied stress. Figure 5 plots thedouble logarithmic plot of the steady-state creep rateversus the applied stress. The standard deviation of theobtained posterior probability of each rate is shown bythe error bar, though the magnitude is so small,10−7 h−1 order at most, that the error bars are notvisible at all. As shown in Figure 5, there was a bentof the curve at a stress of 120 MPa at 873 K, indicatingthe change in the mechanism with respect to theapplied stress. Let us divide the stress region belowand above 120 MPa; thus, each data points at 923 Kand 823 K belongs to the lower and higher stressregion, respectively. It turned out that all the datapoints at 923 K can be fitted to a linear curve withthe same gradient as that for the fitting line in the lowerstress region at 873 K. Similarly, the fitting curve at823 K had the same gradient as that in the higher stressregion at 873 K. Thus, there were at least two regionshaving each mechanism in terms of the applied stress,irrespective of temperature. The Gr.91 steel exhibitslong-term stability of the martensitic structure owingto fine precipitates introduced by tempering [11].According to previous works [12–14], however, themartensitic structure is not necessarily stable and thelath boundary of the martensitic structure migratesunder an applied stress in creep deformation. Forexample, Endo et al. [14] claimed that the dislocationsFigure 3. Fitting of Gr.91 steel creep curve at 823 K/240 MPa with (a) modified theta method and (b) theta method. Log of thecreep strain rate against time for 823 K/240 MPa fitted with (c) modified theta method and (d) theta method. The insets in (a) and(b) are enlargements of the primary region. The formulae in (c) and (d) are the derivatives of each term of equations (1) and (2).Sci. Tech. Advan. Mater. 21 (2020) 225 H. IZUNO et al.composing the subgrain boundary like the lath bound-ary, which are pinned by the fine precipitates, canmigrate by unpinning under an applied stress abovea certain level and estimated that the stress necessaryfor unpinning is approximately 100 MPa. As men-tioned in the former paragraph, our analysis suggestedthat the mechanism leading to the steady state is shiftedaround almost the same stress level. The agreement ofthe critical stress level indicates that the mechanismgoverning the steady state is strongly related to thestability of the martensitic structure.Then, we separately discuss the possible mechanismin the two regions in terms of the applied stress belowand above ~120 MPa. In the lower stress region, thework by Endo et al. [14] suggests that the coarseningmechanism by the applied stress was not activated andwe thus consider that the martensitic structure shouldbe stabilized by the precipitates, leading to the steadystate of creep deformation. While the coarsening of thefine precipitates progressively occurs, there is a certainduration of the steady state; this is because there shouldbe a sufficient amount of fine precipitates in the initialmicrostructure. In fact, Armaki et al. [15] experimen-tally showed that there is an incubation time to initiatethe subgrain growth in high-Cr ferritic steels and thatunder a relatively low applied stress the incubation timeis reasonably long, compatible to that observed at staticrecovery condition without applied stress. Thus, thesteady state is supposed to end when the stabilizationeffect by the precipitates is lost by their coarsening. Onthe other hand, the martensitic structure is not stable inthe higher stress region [14]. A possible mechanismFigure 4. (a) Schematic figure of log of the creep strain rate against time with its fitting by modified theta method. Blue broken,red dot-dashed, and green dotted lines are the asymptotes of the corresponding terms of the equation (similar to FIG. 3.) The grayhorizontal line shows the region where the steady-state term (equation (12)) is more than 10 times larger than the other terms(equations (11) and (13)), (b) Steady-state length ratio of Gr.91 steel creep curve estimated by modified theta method. Each lengthis normalized by the rupture time of the creep test.Figure 5. Double logarithmic plot of the creep rate in the steady state, _εs versus applied stress. The standard deviation of theobtained posterior probability of each creep rate is given by the error bars.Sci. Tech. Advan. Mater. 21 (2020) 226 H. IZUNO et al.here is that the migration of the subgrain boundary byunpinning from the precipitates dominates the creepdeformation. Considering a high density of the sub-grain boundaries in the Gr.91 steel, it is likely that theirmigration can yield most of the creep deformation,though the quantitative confirmation is out of thescope of the present work. Consequently, it is assumedthat the steady state is controlled by the unpinning ofthe subgrain boundary from the precipitates.Lastly, we would like to discuss about a limitation inthe present model selection. The model selection resultdepends not only on the data but also on the set oftargeted models. That is, it is impossible to concludethat the modified theta projection is the best among allthe possible creep constitutive equations. There are sev-eral sophisticated constitutive equations including moreterms and thus it would be helpful to apply this frame-work to these equations. Since the present model selec-tion framework used the hyperparameterization toreduce the non-linear equations into linear problem, itwould be necessary to modify the framework if applyingit to more complicated constitutive equations includingtoo many non-linear terms. This modification will bea future work. In spite of the limitation of the presentmodel selection, we still consider that our analyses totallydemonstrated the appropriateness of assuming the steadystate for the present Gr.91 steel. In fact, the period of thesteady state was significantly long, over 50% of the creepdeformation in average for all the test conditions (Figure4(b)), when the modified theta projection was assumed.Furthermore, the creep rate in the steady state was well fitby a simple power low of the applied stress, the Nortonlaw, assuming two regions (Figure 5). These facts meanthat more than 50% of creep deformation can be repre-sented as a steady state in the wide range of applied stress.5. ConclusionWe calculated the Bayesian free energy for the modifiedtheta projection with a steady-state term and the thetaprojection for aGr.91 steel under various testing conditionsin the ranges of 50–90MPa at 923K, 90–160MPa at 873 Kand 170–240 MPa at 823 K. The Bayesian free energy wasclearly lower for themodified theta projection under all theconditions and the calculated posterior probability wasvirtually 1.0. We thus concluded that the experimentaldata support the modified theta projection. Detailed ana-lysis of the strain rate versus time plot showed that morethan 50% of the creep deformation can be regarded asoccurring in the steady state. The double logarithmic plotof the steady-state creep rate versus applied stress indicatedthat there are two stress regions in terms of themechanismgoverning the steady state below and above ~120 MPa.The present study demonstrated that the Bayesianinference framework is useful to rationally judge whattype of the constitutive equation should be applied increep deformation. We consider that this frameworkmay lead to a useful insight not only for the creepdeformation but also for any other fields in materialsscience. For applying the framework, each model ina targeted controversial issue should be formulated. Inaddition, it would be necessary to modify the frameworkitself so as to handle all the model parameters as prob-abilistic ones, that is, a fully Bayesian inference frame-work would be crucial in some non-linear formulations.AcknowledgmentsThe authors are grateful to Dr. Masayoshi Yamazaki,Dr. Kazuhiro Kimura (National Institute for Materials Science,Japan), and Dr. Kenji Nagata (National Institute of AdvancedIndustrial Science and Technology, Japan) for their valuableadvice. This work was supported by Council for Science,Technology and Innovation (CSTI), Cross-Ministerial StrategicInnovation Promotion Program (SIP), “Structural Materials forInnovation” and “Materials Integration for RevolutionaryDesignSystem of Structural Materials” (Funding agency: JST).Disclosure StatementNo potential conflict of interest was reported by the authors.FundingThis workwas supported by the Council for Science, Technologyand Innovation (CSTI), Cross-Ministerial Strategic InnovationPromotion Program (SIP), ‘Structural Materials for Innovation’;‘Materials Integration for Revolutionary Design System ofStructural Materials’ (Funding agency: JST).ORCIDHitoshi Izuno http://orcid.org/0000-0003-0503-3621Masahiko Demura http://orcid.org/0000-0002-7308-3041References[1] Evans RW, Parker JD, Wilshire B. An extrapolationprocedure for long term creep strain and creep life pre-diction. In: Wilshire B, Owen DRJ, editors. 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Mater Trans. 2003;44:239–246.[15] Armaki HG, Chen R, Maruyama K, et al. Prematurecreep failure in strength enhanced high Cr ferritic steelscaused by static recovery of tempered martensite lathstructures. Mater Sci Eng A. 2010;527:6581–6588.Sci. Tech. Advan. Mater. 21 (2020) 228 H. IZUNO et al. Abstract 1. Introduction 2. Bayesian inference framework for model selection 3. Experimental procedures 4. Results and discussion 5. Conclusion Acknowledgments Disclosure Statement Funding ORCID References