Luis A. VarelaDepartment of Mechanical Engineering,

The University of Texas at El Paso,

El Paso, TX 79968

e-mail: Lavarela@miners.utep.edu

Calvin M. StewartAssistant Professor

Department of Mechanical Engineering,

The University of Texas at El Paso,

El Paso, TX 79968

Modeling the Creep of HastelloyX and the Fatigue of 304Stainless Steel Using the Millerand Walker Unified ViscoplasticConstitutive ModelsHastelloy X (HX) and 304 stainless steel (304SS) are widely used in the pressure vesseland piping industries, specifically in nuclear and chemical reactors, pipe, and valveapplications. Both alloys are favored for their resistance to extreme environments,although the materials exhibit a rate-dependent mechanical behavior. Numerous unifiedviscoplastic models proposed in literature claim to have the ability to describe the inelas-tic behavior of these alloys subjected to a variety of boundary conditions; however, typi-cally limited experimental data are used to validate these claims. In this paper, twounified viscoplastic models (Miller and Walker) are experimentally validated for HX sub-jected to creep and 304SS subjected to strain-controlled low cycle fatigue (LCF). Bothconstitutive models are coded into ANSYS Mechanical as user-programmable features.Creep and fatigue behavior are simulated at a broad range of stress levels. The resultsare compared to an exhaustive database of experimental data to fully validate the capa-bilities and performance of these models. Material constants are calculated using therecently developed Material Constant Heuristic Optimizer (MACHO) software. This soft-ware uses the simulated annealing algorithm to determine the optimal material constantsthrough the comparison of simulations to a database of experimental data. A qualitativeand quantitative discussion is presented to determine the most suitable model to predictthe behavior of HX and 304SS. [DOI: 10.1115/1.4032319]

Keywords: unified viscoplastic model, Miller, Walker, inelastic behavior, Hastelloy X,304 stainless steel, constant optimizer, creep, low cycle fatigue

1 Introduction

Nuclear and chemical reactors can be used to generate electric-ity while minimizing greenhouse gas emissions [1]. The efficiencyof these reactors is related to the high temperature at which theyoperate. High-operating temperatures are desired for better effi-ciency. The reactors require cooling systems to keep them withinsafe operating temperature. The coolant system is the intermediateheat exchanger (IHX) which exchanges heat between the primaryheated coolant that comes directly from the reactor at a tempera-ture of around 950 �C and the secondary working fluid whichcools down the primary coolant [1–3]. IHX performance is crucialfor a safe operation and a high efficiency. Therefore, accurateIHX component design is pivot to achieve high-quality reactors. Itis essential to have a deep understanding of the mechanical behav-ior of the constituent material under servicelike conditions [4].

Since the IHX operates at elevated temperature and extremeconditions, a material resistant to high temperature, thermal andmechanical stresses, corrosion, and oxidation is required. Thenickel-base superalloy HX is an attractive candidate material. It isfavored for piping applications because of its high nickel content,which provides excellent mechanical properties at high tempera-ture, including high resistance to creep, oxidation, and corrosion[5–10]. A second candidate material for this piping application is304SS.

In order to optimize the design of IHX components, a detailedmodeling of the material’s behavior under any loading condition

is essential. On the other hand, a better understanding of the mate-rial behavior leads to less conservative designs that in returnreduce the cost of hardware and components [11]. There has beenconsiderable effort to develop unified constitutive models capableof describing the inelastic behavior of HX and 304SS. These“unified” models are designed to model the multiple deformationmechanisms present during various loading cases, such as stressrelaxation, monotonic tension, creep, and fatigue. Historically,numerous viscoplastic models have been proposed in literature,such as Chaboche, Bodner, Hart, Miller, Walker, Bodner–Partomamong many others [12–17]. However, few researchers have vali-dated these models for HX under creep and for 304SS under LCFconditions. The response of viscoplastic constitutive model isdriven by the material constants, where the constants are charac-teristic of each material type. Material constants are typically cal-culated using specific types of experimental data that activate thedeformation mechanisms of interest. The complexity of the con-stitutive equations and the considered temperature ranges dictatethe total number of material constants required to model the me-chanical behavior. The process to determine these material con-stants is not well documented for most viscoplastic constitutivemodels; therefore, there exist gaps in the calculation process thatlead to the “unsystematic” calculation of material constants. Thisunsystematic calculation of constants might result in improperusage of the viscoplastic models.

In the present work, Miller [18] and Walker [19] unified visco-plastic constitutive models are exercised and analyzed to deter-mine the most accurate model describing the inelasticdeformation of HX under creep and 304SS under fatigue. Toensure a systematic calculation of material constants, a numericaloptimization software is used for both constitutive models. HX

Contributed by the Materials Division of ASME for publication in the JOURNAL OF

ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received June 4, 2015; finalmanuscript received December 13, 2015; published online January 29, 2016. Assoc.Editor: Vadim V. Silberschmidt.

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and 304SS are used in the present work to test the model’s equa-tions and the optimization software capabilities to model morethan one material behavior. The numerical simulations of both themodels are presented and compared to an exhaustive database ofHX and 304SS experimental data, and the correspondencebetween them is estimated by the percentage error and coefficientof determination. Finally, a discussion of model performance andlimitations are provided.

2 HX

HX is a face center cubic nickel–chromium–iron–molybdenumsolid-solution-strengthened Ni-base superalloy which possessesan outstanding combination of oxidation resistance and highstrength at temperatures of up to 1000 �C. A large number of stud-ies have been performed on this material characterizing the ten-sile, rupture, and creep deformation behavior [20]. The HXexperimental data used in the present study come from Ref. [8];the alloy used for the experiments was a commercial type hot-rolled plate with 19 mm of thickness. The creep specimen usedhad a cylindrical form of 30 mm in gauge length and 6 mm in di-ameter. The constant load creep test was conducted at differentstress levels 35 MPa, 30 MPa, 25 MPa, 20 MPa, 18 MPa, 16 MPa,and 14 MPa at 950 �C. At 950 �C, HX exhibited an averageYoung’s modulus of 144 GPa, a Poisson’s ratio of 0.29, and a0.2% yield strength of 121 MPa. The nominal chemical composi-tion of the HX is provided in Table 1.

3 304 Stainless Steel

304SS is an austenitic iron–nickel–chromium alloy that pos-sesses high strength and high resistance at elevated temperatures.Due to its repeated use in piping applications, this material is con-sidered in the present study. The 304SS experimental data used inthe present work were found in Ref. [21]. The specimen used dur-ing the fatigue testing was rod annealed and cold finished improv-ing strength and straightness. LCF tests were conducted at 600 �Cand at 0.5% and 0.7% strain amplitude (DeÞ. The tested materialwas prepared to meet A276 and A479 ASTM standards. The nom-inal chemical composition of 304SS is provided in Table 2.

4 Miller Model

Miller model is a unified viscoplastic model also called“MATMOD,” proposed in 1975 by Miller to model the viscoplas-tic behavior of materials subjected to high temperature and load[22]. This model was developed based on the physical mecha-nisms that take place within the material during deformation. Thisunified model has the ability of modeling transient (primary)

creep, steady-state (secondary) creep, monotonic short-time plas-tic deformation, cyclic hardening and softening, Bauschingereffect, rate, temperature, annealing, and the accumulation andinteraction of these effects with respect to time [23]. This modelconsists of a set of three rate-dependent constitutive equations anda pair of auxiliary equations which introduce the temperaturedependence modeling capability. The constitutive equationsrequire eight material constants that are characteristic of each ma-terial type. The analytical approach to find the constants involvesthe use of auxiliary equations and/or plotting of experimentaldata. Optimization of the constant values is performed until thedata reach a desired region or desired shape [18]. To calculatematerial constants, the required experimental data come fromshort-time monotonic, creep, and cyclic loading tests at elevatedtemperatures. The original MATMOD equations do not replicatethe nonlinear cyclic hysteresis loop of experimental data well. Tocorrect this issue, modern MATMOD equations are structured tomodel “normal” and “anomalous” Bauschinger’s effect; however,modern MATMOD equations are too complex for easy determina-tion of material parameters. The original MATMOD equations areconsider in the present work.

The inelastic strain rate equation (1) is based on Garofalo’ssteady-state creep equation [22]; it was modified to include dragand rest stress variables. This equation is dependent on the appliedstress r and the two state variables R (rest stress) and D (dragstress) [23]. The strain rate equation (1) requires three materialconstants B, h0, and n; the constant h0 carries the temperaturedependence, n represents the rate sensitivity of the stress, and B isa coefficient that carries the units

_e ¼ Bh0 sin hjr� Rj

D

� �1:5( )n

sgn r� Rð Þ (1)

The rest stress rate equation (2) describes the rate of kinematichardening with respect to time; rest/back stress represents thestress field within the material caused by stuck dislocations. Thefirst term in the rest stress rate equation represents the amount ofhardening of the material, it is produced by the pile up of disloca-tions against obstacles; the second term represents the recoverythat is produced by the climbing and cross-slip of dislocations andby thermal recovery. The constants H1, B, and A1 represent mate-rial constants; h0 is the temperature dependence term whichaccounts for thermal recovery. The constant H1 determines howrapidly the rest stress reaches a saturated value during cyclic load-ing. The signum function, sgn and absolute value enable reverseplastic flow [23]

_R ¼ H1 _e � H1Bh0½sin hðA1jRjÞ�nsgnðRÞ (2)

The drag stress rate equation (3) describes the rate of isotropichardening with respect to time; drag stresses impede dislocationmotion caused by grain boundaries, solute atoms, and precipitateparticles. Just as in the rest stress rate equation, in (Eq. (3)), thefirst term represents the hardening and the second term representsthe recovery of the material. During steady-state, the hardeningand recovery terms are equal resulting in _D¼ 0. The same effectis applicable to the rest stress equation, meaning that at steady-state the inelastic strain rate is dependent on the steady-state stressonly. It is important to note that the drag stress rate equation isdependent on rest stress R and drag stress D; this dependence wasestablished to give this equation the ability to consider cyclicloading by limiting the amount of isotropic hardening due to dragstress. The constants H2, C2, and A2 are material constants

_D ¼ H2j_ej½C2 þ jRj � ðA2=A1ÞD3� � H2C2Bh0½sin hðA2D3Þ�n (3)

The term h0 represents the temperature dependence of themodel and its value is given by (Eq. (4)) and/or (Eq. (5)) depend-ing on the initial temperature. This dependence is due to the

Table 1 HX chemical composition (wt. %) [8]

Ni C Mn Si Cr Co Cu Mo

48.04 0.082 0.82 0.42 21.91 0.79 0.13 8.65

W Fe B Al Ti P N S

0.44 19.0 0.0020 0.17 0.007 0.013 0.015 0.0003

Table 2 304SS chemical composition (wt. %) [21]

Fe Cr Ni C Mn Cu

69 19 9.25 0.04 1 0.5

Mo Si S P Co N

0.5 0.5 0.015 0.023 0.1 0.05

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assumption that the apparent activation energy Q is temperaturedependent. The value of Q is constant at a temperature above0.6 Tm (where Tm represents the melting temperature) and variantat temperatures below. Constant Q represents the activationenergy of processes related to inelastic deformation that are ther-mally activated, k represents the universal gas constant, and T isthe working temperature

h0 ¼exp

�Q

kT

� �for T � 0:6Tm (4)

exp�Q

0:6kTmln

0:6Tm

T

� �þ 1

� �� �for T < 0:6Tm

8>>><>>>:

(5)

In the present work, the temperature dependence function hasbeen deactivated by setting the activation energy Q¼ 0, whichresults in a temperature factor of h0 ¼ 1. The temperature functionwas deactivated because only isothermal tests are considered inthis study; moreover, this assumption eliminates one material con-stant, simplifying the material constant determination process.

5 Walker Model

The Walker model is a unified viscoplastic model, developedby Walker [19]; originally, this model was called functionaltheory. The Walker model has the capability of modeling transient(primary) creep, steady-state (secondary) creep, monotonic short-time plastic deformation, cyclic hardening and softening, Bau-schinger effect, stress relaxation, strain rate, and temperatureeffects [19]. The model describes the inelastic behavior of materi-als using five equations, three constitutive rate equations, and twolinear equations. The constitutive equations require 14 materialconstants that are calculated using cyclic hardening and softeningexperimental data. It should be noted that according to Walker[19], only nine material constants are required to model creep andLCF behavior. The process to solve for the material constants hasnot been found in literature; thus, numerical optimization software(recently developed at The University of Texas at El Paso) hasbeen used in the present work. A distinguishing characteristic ofthis model is the temperature dependence. The temperaturedependence is present only in some explicit terms of the constitu-tive equations; therefore, temperature dependence is incorporatedby making the material constants a function of temperature. Thus,material constants must be determined at every temperature level.So the number of material constants increases dramatically undernonisothermal conditions. The strain rate equation (6) is based ona power law and represents the strain rate behavior with respect toapplied stress r, rest stress R, and drag stress D. Constants B andn are material constants, the sgn function and absolute value ena-ble reversed plastic flow

_e ¼ Bjr� Rj

D

� �n

sgn r� Rð Þ (6)

The rest stress rate equation (7) introduces kinematic hardeninginto the model and provides the ability to account for Bauschingereffect. The equation requires the material constants n1, n2, and n3,temperature T, temperature rate T_, recovery relation _G, inelasticstrain e, and the inelastic strain rate _e. The growth law for the reststress accounts for strain hardening with its first two terms, andfor recovery effects with the last two terms which are dependenton the recovery relation. The recovery relation is given by Eq. (8)that represents a relationship between dynamic and static recoveryof the material. The dynamic recovery term (first term of the equa-tion) governs kinematic recovery in the presence of inelastic strainrate. The static recovery term (second term of the equation) gov-erns kinematic recovery in the absence of an inelastic strain rate.

The equation requires material constants n3, n4, n5, n6, R, Rs, andm, accumulated inelastic strain ek, and the strain rate magnitude ek_

_R ¼ n1 þ n2ð Þ_e þ @n1

@T_Te� R� R0 � n1eð Þ _G � 1

n2

@n2

@T_T

� �(7)

_G ¼ n3 þ n4 exp �n5ek

� �_ek þ n6

jRjjRSj

m�1

(8)

The original model proposed by Walker [19] contains tempera-ture rate terms _T in the rest stress rate equation. These terms areresponsible for modeling rest stress changes due to temperaturechanges during nonisothermal tests. However, in the present workonly isothermal tests are considered; therefore, the temperaturerate term _T is set to zero. The material constants n1 and n2 deter-mine how fast the rest stress grows, until it reaches saturation.When the magnitude of the constants is large, the rest stress willsaturate so promptly that in theory it will saturate within the elas-tic region; in a hysteresis loop, this will cause a perfectly plastichysteresis loop. With intermediate values, it will take longer forthe rest stress to saturate. This will cause a rounded behavior ofthe hysteresis loop. With small values, the rest stress willslowly saturate and hysteresis loops will exhibit a further roundedbehavior [19].

The drag stress equation (9) introduces isotropic hardening intothe model and provides the ability to account for cyclic hardeningand softening of the material. Material variables n7 and k2 controlthe isotropic behavior of the model depending on the accumulatedinelastic strain ek ; therefore, the precision of these variables is ofextreme importance to accurately model isotropic hardening orsoftening of the material

D ¼ k1 � k2 expð�n7ekÞ (9)

Equation (10) sets the accumulated inelastic strain rate _ek equalto the magnitude of the inelastic strain rate _e. The accumulatedinelastic strain rate _ek in the Walker model is equivalent to theaccumulated rest stress, R in the Miller model. Walker [19] doesnot assign any microstructural meaning to the rest and drag stress;only their effects in terms of hardening and softening phenomenaare considered

_ek ¼ j_ej (10)

As mentioned earlier, only nine material constants are requiredfor this study. The constants K2, n1, n4, n5, and n7 are disabled.

6 Numerical Optimization Software

Previous efforts to develop methods of calculating the materialconstants of viscoplastic models have been documented [16,24].These techniques and regression software involve the analyticalmanipulation of equations and experimental data. This results ininconsistent calculations of material constants, since the manipu-lation of experimental data can differ from user to user. Thenumerical optimization software used in the present work is theMACHO software. This new FORTRAN-based software was developedto optimize material constants of complex constitutive models.This software allows the optimization of material constants bycomparing experimental data to multiple iterations of numericalsimulated mechanical tests. The provided experimental data andboundary conditions play a fundamental role in the optimizationprocess. In the case of comparing viscoplasticity models, MACHO isextremely helpful because it warranties the consistent systematiccalculation of material constants. By using the same optimizationparameters for the considered constitutive models, a fair compari-son between both models can be performed since the materialconstants for both models were calculated/optimized under thesame conditions. MACHO is based on the simulated annealing

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multimodal optimization algorithm that explores the entire func-tion surface and can perform both uphill and downhill moves; thisallows the exploration of not only a local minima but also thesearch of a global minima [25,26].

The optimization process is illustrated in Fig. 1. It is as follows:initial guess constants, boundary conditions, and experimentaldata are provided by the user. To avoid algorithm failure, initialguess constants should be values that will converge to a solution;to determine initial guess values, both constitutive models wereprogramed in spreadsheets, where experimental data were plottedand different material constants values from literature [13] weremodified until the initial guess constants converged to a reasona-ble noninfinite solution. Next, finite element (FE) simulations ofthe desired mechanical test are performed. The approach used toperform the FE simulation is executed based on the test type: loadcontrolled or displacement controlled. For load control, the pro-cess is simple where the applied stress is directly applied in theconstitutive equations. For displacement controlled, the radialreturn mapping technique is used where Newton–Raphson itera-tions provide an updated stress and solve the constitutive equa-tions. Next, the calculation of the objective value occurs. Theobjective function indicates the goodness-of-fit of the FE simula-tion with respect to the experimental data. The objective functionis calculated based on the type of test; for load controlled a terti-ary creep function is used, while a LCF function is used for dis-placement controlled. The objective value of each data set issummed into one total objective value. Afterward, the total objec-tive value is compared to convergence criterion. If convergencehas not been reached, then the simulated annealing optimizationalgorithm is executed. The simulated annealing algorithm willproduce a new set of material constants and restart the processuntil convergence is achieved [21].

The simulated annealing algorithm has optimization parametersthat controlled the efficiency of optimization. The optimizationparameters are the number of constants to optimize (N), initialtemperature (T), temperature reduction factor (RT), number ofsteps at each temperature level before cutting temperature (NT),maximum number of iterations (MAXEVL), and set upper (UB)and/or lower (LB) boundaries for the optimized constants. Thefine-tuning of these parameters contributes to a faster andmore accurate optimization of constants. For creep simulations,the following optimization parameters where fixed: MAXEVL¼ 500,000, T¼ 100 �C, RT¼ 0.5, NT¼ 5*N, UB¼ 1� 1020, andLB¼�3.4, respectively. On the other hand, for LCF simulationsthe following optimization parameters were fixed: MAXEVL¼ 500,000, T¼ 125 �C, RT¼ 0.25, NT¼ 20*N, UB¼ 1� 1020,and LB¼ 0. Numerical optimization using HX creep data wasperformed. The total of optimized constants is NMiller¼ 7 andNWalker¼ 9. The total objective function values with respect toiteration are shown in Fig. 2. It is observed that during the early

iterations, the objective function values exhibit large oscillationsfor both constitutive models; these oscillations are the result ofsimulated annealing encountering sets of material constants thatdo not converge. Similarly, it can be observed how after severalthousand iterations both models stabilize and converge to a con-stant objective function value (observed as a constant horizontalline in the figure). The Miller model reaches convergence fasterthan Walker. The convergence difference is attributed to the num-ber of material constants required, where seven material constantsare needed for Miller model and nine for Walker model. The ini-tial and final objective function values for Miller and Walker HXcreep are shown in Tables 3 and 4. The initial objective functionvalue difference between Miller and Walker model is 3.04%,whereas the final objective function value difference is only 0.4%where Walker model provides the better fit. The goodness-of-fit orobjective function value is reasonably similar for both models.

Numerical optimization using 304SS LCF data was performed.The number of optimized constants is NMiller¼ 7 and NWalker¼ 9.The initial and optimized material constants for Miller and Walker304SS LCF are shown in Tables 5 and 6. The total objective func-tion values with respect to iteration are shown in Fig. 3. It isobserved that the Walker model exhibited a larger variation of theobjective function value than the Miller model. The Miller modelreaches convergence faster than Walker. The initial and finalobjective function values for Miller and Walker 304SS LCF areshown in Tables 5 and 6, respectively. The initial objective func-tion value difference between Miller and Walker model is14.32%, whereas the final objective function value difference is

Fig. 1 MACHO material constant optimization process

Fig. 2 Objective function evolution during optimization ofMiller and Walker for HX creep

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46.40% where Walker provides the better fit. Unlike the creepoptimization, the LCF optimization shows that the final objectivefunction values diverge.

The creep and LCF objective function values are not compara-ble due to the area under the experimental creep (strain versustime) and fatigue curves (stress versus time) being of dissimilarmagnitude. The magnitude of the final material constants foundfor creep and LCF of both models are dramatically different. Themechanical behavior of HX and 304SS behavior is not so dramati-cally different to suggest such large swings in magnitude. Thissuggests that MACHO is database-dependent, physics agnostic, andonly concerned with function minimization and not the realism ofthe material constants. An attempt was made to optimize the full(14 constant) Walker model for HX creep and 304SS LCF. It wasfound that the additional constants increased the magnitude of thefinal objective function value (when the maximum number of iter-ations was fixed to 500,000) in both cases. This suggests that thefull Walker model is over defined. The nine constants suggestedin literature and used in this study are ideal [19].

7 Results

The simulated creep curves of HX using Miller are shown inFig. 4. It is observed that even though this model is designed forprimary and secondary creep only, the physics agnostic MACHO

code found optimized material constants that enable the ability toaccurately model tertiary creep at high stress levels. On the otherhand, at low stress levels the model is able to predict only primary

Table 3 Miller initial guess and optimized constants for HXcreep

Material constant Initial guess value Optimized value Units

B 0.4511� 1014 0.4123� 1013 s�1

n 3.2003 3.2608 —H1 0.3299� 10� 4 0.1438� 10�3 MPaH2 0.2184� 106 0.3392� 107 —A1 11.3510 7.0297 MPa�1

A2 0.1242� 10�13 0.1772� 10�14 MPa�1

C2 0.1203� 10�27 0.2573� 10�15 MPaObj. Funct. 1290 1106 —

Table 4 Walker initial guess and optimized constants for HXcreep

Material constant Initial guess value Optimized value Units

B 1.46� 10�7 0.1354� 10�6 —K1 4.5870 4.5183 MPaK2 0.0 0.0 —Ro 15 15 MPaRs 7.3055 8.3276 —m 6.3459 6.3480 —n 4.8584 4.8582 —n1 0.0 0.0 —n2 �3.4 �3.4 MPan3 �2.0 �2.0 —n4 0.0 0.0 —n5 0.0 0.0 —n6 1� 10�8 �0.2746 s�1

n7 0.0 0.0 —Obj. Funct. 1251 1101 —

Table 5 Miller initial guess and optimized constants for 304SSLCF

Material constant Initial guess value Optimized value Units

B 0.10� 10�14 0.5686� 10�4 s�1

n 5.8 1.2661 —H1 100.0 0.4139� 10�3 MPaH2 280.0 9705.50 —A1 0.7420� 10�4 0.2407� 10�5 MPa�1

A2 0.800 0.4139� 10�1 MPa�1

C2 0.100 0.1599� 109 MPaObj. Funct. 129,831 3292 —

Table 6 Walker initial guess and optimized constants for304SS LCF

Material constant Initial guess value Optimized value Units

B 0.7719� 10�5 0.41691� 10�3 —K1 27.386 169.16 MPaK2 0.0 0.0 MPaRo 2.22 14.696 MPaRs 0.8181� 1020 0.1813� 1020 —m 0.5250� 1020 0.7803� 1019 —n 22.0 7.8255 —n1 0.0 0.0 MPan2 6954.1 24.252 MPan3 182.89 279.18 —n4 0.0 0.0 —n5 0.0 0.0 —n6 0.9457� 1010 0.1388� 1020 s�1

n7 0.0 0.0 —Obj. Funct. 151,525 1764 —

Fig. 3 Objective function evolution during optimization ofMiller and Walker for 304SS LCF

Fig. 4 Miller creep deformation of HX creep at 950 �C [8]

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and secondary creep. As stress decreases, the Miller modelbecomes less accurate. A good correspondence between experi-mental data and simulations is observed for r � 25 MPa, wheremultistage creep is predicted. Scatter is present between experi-mental data and simulations at r < 25 MPa.

The drag stress/isotropic hardening behavior of the Millermodel for HX creep is shown in Fig. 5. Simulations show that thedrag stress linearly decreases with time. At high stresses, the dragstress reduction will be larger and faster, meaning that the higherthe applied stress the less time it will take to have a drag stress orisotropic hardening reduction. On the contrary, at low stress, thereduction of drag stress is smaller and slower.

The rest stress/kinematic hardening behavior of Miller modelfor HX creep is shown in Fig. 6. The rest stress represents the

amount of stress that stuck dislocations are causing within the lat-tice due to the applied stress. Simulations show how as timeincreases so do rest stress. At high stress, the rest stress is high. Atlow stress, the rest stress is low.

The simulated creep curves of HX using Walker are shown inFig. 7. Walker model exhibits high precision predictions of creepat high stress. However, at low stress it only describes the primaryand secondary creep regime. As stress decreases, the model accu-racy decreases. A good correspondence between experimentaldata and simulation is observed at r � 20 MPa, where multistagecreep is predicted. Scatter is present between experimental dataand simulation at r < 20 MPa.

Fig. 5 Miller creep drag stress of HX at 950 �C

Fig. 6 Miller creep rest stress of HX at 950 �C

Fig. 7 Walker creep deformation of HX creep at 950 �C [8]

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The drag stress behavior of Walker model for HX creep isshown in Fig. 8. The drag stress behavior is a straight line for allstress levels. This is expected for Walker’s model because thedrag stress equation is defined as the nonevolutionary k1 materialconstant. The remaining terms of the drag stress equation weredeactivated because they were not required to describe creep andcreep-fatigue behavior according to literature. This proves that theWalker model drag stress has no physical meaning.

The rest stress behavior of Walker model for HX creep isshown in Fig. 9. As time increases, the rest stress decreases. How-ever, the rest stress reduction is dependent on the stress. At lowstress, the rest stress reduction is slow and short, whereas at highstress levels the drag stress reduction is fast and larger. The reststress value decreases because of the deactivation of constant n1

in the rest stress rate equation. This behavior proves that theWalker model rest stress has no physical meaning.

The drag stress evolution of the two models is completely dis-similar due to the nature of the respective equations, whereWalker uses a nonevolutionary material constant and Millermodel uses an evolutionary drag stress equation that softens. Therest stress evolution of the two models is also dissimilar, whereWalker exhibits linear to power law softening and Miller modelexhibits linear to power law function hardening as stressincreases. These phenomena are attributed to the different mathe-matical functions present in the inelastic strain rate equations andthe physics agnostic nature of MACHO. No literature exists thatdefines the acceptable magnitude of material constants for either

model. The phenomenological nature of these constitutive modelsmakes the determination of material constants through experi-ments a fictitious effort at best. Microstructural length scale visco-plastic models would enable experimental validation of materialconstants; however, the mathematical formulation of these modelsis highly dependent on the microstructural characteristics of theselected material system. A more alloy insensitive model wasdesired in this study.

The results of 304SS LCF simulation at 600 �C using Miller areshown in Figs. 10 and 11 for De ¼ 0.005 and De ¼ 0.007, respec-tively. The hysteresis loops De¼ 0.005 in Fig. 10 show that thesimulated LCF curve of Miller underpredicts the hardening. In thenegative stress region, the simulations display a much softerbehavior than experimental data. At this strain amplitude, Millerunderpredicts LCF since the predicted hardening and softeningbehavior are smaller than the actual behavior. The hysteresis loopsDe¼ 0.007 in Fig. 11 show that the hardening portion of the initialcycle is underpredicted by the simulated curve. Overall, there isgood correspondence between the simulated and the experimentalcurve; however at the simulated maximum softening region(lower stress point) and the middle hardening region (positivestress, negative strain area) there is a underprediction of the actualbehavior.

The results of 304SS LCF simulation at 600 �C using Walkerare shown in Figs. 12 and 13 for De ¼ 0.005 and De ¼ 0.007,respectively. The hysteresis loops De¼ 0.005 in Fig. 12 show that

Fig. 8 Walker creep drag stress of HX at 950 �C

Fig. 9 Walker creep rest stress of HX at 950 �C

Fig. 10 Miller hysteresis loops of 304SS LCF at 600 �C,De 5 0.005 [21]

Fig. 11 Miller hysteresis loops of 304SS LCF at 600 �C,De 5 0.007 [21]

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at the maximum softening region (lower stress point), a good cor-respondence can be observed for the first cycle while during sub-sequent cycles there are some discrepancies leading tounderprediction. At the middle hardening region (positive stress,negative strain area), there is a small underprediction of thebehavior; however, at the maximum hardening region (higheststress point) there is a good hardening prediction. The hysteresisloops De¼ 0.007 in Fig. 13 show that an overall close correspon-dence between the simulated and experimental curves can beobserved; however, there is a slight overprediction of the behaviorat the maximum hardening region. No comparison figures of dragand rest stress for LCF simulations are presented due to the dis-similar nature of their treatment by Miller and Walker as demon-strated in the HX creep simulations.

8 Analysis

In order to determine which constitutive model is the best indescribing HX creep and 304SS LCF, it is required to consider theaccuracy and user friendliness of each model. To facilitate thedecision process, a quantitative and qualitative discussion is pre-sented. To establish a numerical or quantitative evaluation of thebehavior of each model with respect to the experimental data, thetime step size of each model was set equal to the time step size ofthe experimental data. To do so, for HX creep, ten experimentaldata points were selected as a base time step size and by using

linear interpolation, ten numerical simulation data points (strainand time) were obtained for each constitutive model at exactly thesame time. With these new data points, the ten experimental datapoints and the 20 numerical simulation data points (ten from eachmodel) were compared. This procedure was not necessary for theLCF data of 304SS since the time step size is the same betweenthe simulations and the experimental data. Two quantitative meas-ures were used in this study. The first quantity used in this study isthe mean percentage error (MPE) (Eq. (11))

MPE ¼ 100

n

X���� #SimData � #ExpData

#ExpData

���� (11)

where n represents the total number of data points considered ateach stress level and strain amplitude. The MPE represents theaverage percentage error by which the simulated data differ fromthe experimental data. Therefore, the smaller the MPE, the moreaccurate the constitutive model. The second quantity used in thisstudy is the coefficient of determination (R2) (Eq. (12))

R2 ¼ 1� SSres

SStot

; SStot ¼X

i

yi � �yð Þ2; SSres ¼X

i

yi � fið Þ2

(12)

where i is the number of data points; yi and fi are the experimentaland simulated data, respectively; and �y is the mean value of theexperimental data. The coefficient of determination is a numberthat specifies how well the simulated data fit the experimentaldata at each stress level. When R2 ¼ 1, the simulation perfectlymatches the experimental data; thus, the higher the coefficient ofdetermination, the more accurate the constitutive model is at aspecific stress level or strain amplitude.

The quantitative measures for HX creep and 304SS LCF arefound in Tables 7 and 8, respectively. According to Table 7, athigh stress, where r � 18 MPa, Walker model exhibits a smallerMPE and the higher R2 values; however, at 30 MPa the MPE dif-ference is not significant. At r � 16 MPa, the Miller model exhib-ited a higher accuracy than the Walker model; however, at thesestress levels both models only predict primary and secondarycreeps. According to Table 8, at a strain amplitude of De ¼ 0:005,the Walker model exhibits a 43.60% smaller MPE than Millermodel and produces a higher R2. At a strain amplitude of

Fig. 12 Walker hysteresis loops of 304SS LCF at 600 �C,De 5 0.005 [21]

Fig. 13 Walker hysteresis loops of 304SS LCF at 600 �C,De 5 0.007 [21]

Table 7 MPE and R2 for Miller and Walker simulations for HXcreep

Miller model Walker model

Stress level (MPa) MPE R2 MPE R2

35 15.150 0.9831 10.520 0.983430 15.916 0.6220 15.622 0.798825 14.651 0.7001 10.910 0.926820 16.178 0.5811 15.096 0.698318 35.693 0.0700 33.298 0.099616 47.523 �0.177 48.502 �0.26814 54.405 �0.355 56.694 �0.497

Table 8 MPE and R2 for Miller and Walker simulations for304SS LCF

Miller model Walker model

Strain amplitude MPE R2 MPE R2

0.005 47.547 0.9860 26.814 0.99620.007 23.139 0.9946 22.688 0.9962

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De ¼ 0:007, the Walker model exhibits a 0.1591% smaller MPEthan Miller model and produces a slightly higher R2.

A final decision on the models can now be made. In terms ofuser friendliness, Walker requires nine material constants whileMiller model only requires seven; therefore, Miller is more userfriendly. The procedure to calculate Walker constants is not welldocumented, while the procedure for Miller has been explained[23]; thus again Miller is more user friendly. The Walker model ismore complex requiring five constitutive equations, while Millermodel only requires three; again Miller is more user friendly. Anexamination of the rest and drag stress evolution shows that Millermodel is more physically representative of hardening and soften-ing phenomenon than Walker. Finally, the MPE and R2 quantitiesshow that the Walker model more accurately predicts the experi-mental data. It is concluded that the Miller model is more userfriendly and exhibits a more physically germane evolution. TheWalker model more accurately predicts the experimental data.Both models represent a continuum level phenomenologicalapproach to viscoplasticity and thus do not represent true modelsof the physics of deformation in the subject materials. If numericaloptimization software is not available, then the Miller model isthe more appropriate model due to ease of implementation andconstant determination. If a software, such as MACHO, is available,then Walker model is more suitable due to the level of accuracythat can be achieved.

9 Conclusion

The following conclusions can be formulated from the exerciseof Miller and Walker constitutive models:

� Both phenomenological viscoplastic models can model morethan one alloy behavior.

� MACHO can be used to find the optimal material constants formultiple alloys; however, the software is physics agnosticand requires further development.

� Numerical optimization can give the Miller and Walker con-stitutive models the ability to model multistage creep; acapability that neither was designed to have.

� Both constitutive models can be used in nonisothermal con-ditions; however, the total number of material constantsrequired will increase, especially for the Walker model. Nu-merical optimization using software like MACHO could mini-mize the time and effort needed to determine these constants.

� The full (14 constant) Walker model is not required to modelcreep and LCF independently.

� If numerical optimization software is not available, then theMiller is the more appropriate model due to ease of imple-mentation and constant determination.

� If a software, such as MACHO, is available, then Walker modelis more suitable due to the level of accuracy that can beachieved.

References[1] Totemeier, T., and Tian, H., 2006, “Creep-Fatigue-Environment Interactions in

INCONEL 617,” J. Mater. Sci. Eng., 468–470, pp. 81–87.[2] Cabet, C., Carroll, L., and Wright, R., 2013, “Low Cycle Fatigue and Creep-

Fatigue Behavior of Alloy 617 at High Temperature,” ASME J. Pressure VesselTechnol., 135(2), p. 061401.

[3] Tachibana, Y., and Iyoku, T., 2004, “Structural Design of High TemperatureMetallic Components,” J. Nucl. Eng. Des., 233(1), pp. 261–272.

[4] Swaminathan, B., Abuzaid, W., Sehitoglu, H., and Lambros, J., 2014,“Investigation Using Digital Image Correlation of Portevin-Le Chatelier in Has-telloy X Under Thermo-Mechanical Loading,” Int. J. Plast., 64, pp. 172–192.

[5] Sakthivel, T., Laha, K., Nandagopal, M., Chandravathi, K., Parameswaran, P.,Penner Selvi, S., Mathew, M., and Mannan, S., 2012, “Effect of Temperatureand Strain Rate on Serrated Flow Behaviour of Hastelloy X,” J. Mater. Sci.Eng., 534(A), pp. 580–587.

[6] Aghaie-Khafri, M., and Golarzi, N., 2008, “Forming Behavior and Workabilityof Hastelloy X Superalloy During Hot Deformation,” J. Mater. Sci. Eng.,486(1–2), pp. 641–647.

[7] Kim, W., Yin, S., Ryu, W., Chang, J., and Kim, S., 2006, “Tension and CreepDesign Stresses of the ‘Hastelloy-X’ Alloy for High-Temperature Gas CooledReactors,” J. Mater. Sci. Eng., 483–484, pp. 495–497.

[8] Kim, W., Yin, S., Kim, Y., and Chang, J., 2008, “Creep Characterization of aNi-Based Hastelloy-X Alloy by Using Theta Projection Method,” J. Eng. Fract.Mech., 75(17), pp. 4985–4995.

[9] Udoguchi, T., and Nakanishi, T., 1981, “Structural Behaviour of a WeldedSuperalloy Cylinder With Internal Pressure in a High Temperature Environ-ment,” Int. J. Pressure Vessels Piping, 9(2), pp. 107–123.

[10] Miner, R. V., and Castelli, M. G., 1992, “Hardening Mechanisms in a DynamicStrain Aging Alloy, Hastelloy X, During Isothermal and ThermomechanicalCyclic Deformation,” Int. J. Fatigue, 14(6), pp. 551–561.

[11] Krempl, E., 1974, “Cyclic Creep—An Interpretive Literature Survey,” Weld.Res. Counc. Bull., 195, pp. 63–123.

[12] Chang, T. Y., and Thompson, R. L., 1994, “A Computer Program for PredictingNonlinear Uniaxial Material Responses Using Viscoplastic Models,” NASALewis Research Center, Cleveland, OH, NASA Technical Memorandum No.83675.

[13] Hartmann, G., and Kollmann, F. G., 1987, “A Computational Comparison ofthe Inelastic Constitutive Models of Hart and Miller,” Acta Mech., 69(1), pp.139–165.

[14] Dombrovsky, L. A., 1992, “Incremental Constitutive Equations for Miller andBodner-Partom Viscoplastic Models,” Comput. Struct., 44(5), pp. 1065–1072.

[15] Lindholm, U. S., Chan, K. S., Bodner, S. R., Weber, R. M., Walker, K. P., andCassenti, B. N., 1984, “Constitutive Modeling for Isotropic Materials,” NasaLewis Research Center, Cleveland, OH, NASA Report No. 174718.

[16] James, G. H., Imbrie, P. K., Hill, P. S., Allen, D. H., and Haisler, W. E., 1987,“An Experimental Comparison of Several Current Viscoplastic ConstitutiveModels at Elevated Temperature,” ASME J. Eng. Mater. Technol., 109(2), pp.130–139.

[17] Chaboche, J. L., 2008, “A Review of Some Plasticity and Viscoplasticity Con-stitutive Theories,” Int. J. Plast., 24(10), pp. 1642–1693.

[18] Miller, A. K., 1976, “An Inelastic Constitutive Model for Monotonic, Cyclic,and Creep Deformation: Part I—Equations Development and AnalyticalProcedures,” ASME J. Eng. Mater. Technol., 98(2), pp. 97–105.

[19] Walker, K. P., 1981, “Research and Development Program for Nonlinear Struc-tural Modeling With Advanced Time-Temperature Dependent ConstitutiveRelationships,” NASA Lewis Research Center, Cleveland, OH, NASA ReportNo. 165533.

[20] Haynes International, “High Performance Alloys Technical Information: Has-telloy X Alloy,” Haynes International Inc., Kokomo, IN, Report No. H-3009C,accessed June 4, 2015, http://www.haynesintl.com/pdf/h3009.pdf

[21] Stewart, C. M., 2013, “Mechanical Model of a Gas Turbine Superalloy Subjectto Creep-Fatigue,” Ph.D. dissertation, Department of Mechanical and Aero-space Engineering, University of Central Florida, Orlando, FL.

[22] Miller, A. K., 1975, “A Unified Phenomenological Model for the Monotonic,Cyclic, and Creep Deformation of Strongly Work-Hardening Materials,” Ph.D.dissertation, Department of Materials Science and Engineering, Stanford Uni-versity, Stanford, CA.

[23] Miller, A. K., 1976, “An Inelastic Constitutive Model for Monotonic, Cyclic,and Creep Deformation: Part II—Application to Type 304 Stainless Steel,”ASME J. Eng. Mater. Technol., 98(2), pp. 106–113.

[24] Abdel-Kader, M. S., El-Hefnawy, N. N., and Eleiche, A. M., 1991, “A Theoreti-cal Comparison of Three Unified Viscoplasticity Theories, and Application tothe Uniaxial Behavior of Inconel 718 at 1100�F,” Nucl. Eng. Des., 128(3), pp.369–381.

[25] Corana, A., Marchesi, M., Martini, A., and Ridella, S., 1987, “Minimizing Mul-timodal Functions of Continuous Variables With the ‘Simulated Annealing’Algorithm,” ACM Trans. Math. Software, 13(3), pp. 262–280.

[26] Goffe, W. L., Ferrier, G., and Roger, J., 1994, “Global Optimization of Statisti-cal Functions With Simulated Annealing,” J. Econ., 60(1–2), pp. 65–100.

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