International Journal of Fatigue · 2017-12-30 · plane stress mode according to standard ASTM E647 [28]. For measuring the crack size and crack opening displacement, COD, an EOS

International Journal of Fatigue 105 (2017) 191–207

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International Journal of Fatigue

journal homepage: www.elsevier .com/locate / i j fa t igue

Evolution of the texture and microstructure in a nickel based superalloyduring thermo-mechanical fatigue (TMF), using a modified integratedmodel and experimental results

http://dx.doi.org/10.1016/j.ijfatigue.2017.08.0280142-1123/� 2017 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: [email protected] (M. Esmaeilzadeh), qods@

semnan.ac.ir (F. Qods), [email protected] (H. Arabi), [email protected](B.M. Sadeghi).

M. Esmaeilzadeh a, F. Qods a,⇑, H. Arabi b, B.M. Sadeghi ba Faculty of Metallurgical and Materials Engineering, Semnan university, Semnan, Iranb School of Metallurgy and Materials Engineering, Iran University of Science & Technology, Tehran, Iran

a r t i c l e i n f o a b s t r a c t

Article history:Received 14 May 2017Received in revised form 29 August 2017Accepted 30 August 2017Available online 1 September 2017

Keywords:Thermo-mechanical fatigueTextureODFTaylor methodMonte Carlo modelPhase fieldHastelloy X

Microstructure & texture evolutions in Hastelloy X superalloy during thermo-mechanical fatigue (TMF)were investigated in this research. For this purpose TMF tests were performed in two different ways usinglinear elastic fracture mechanics (LEFM) and elastic plastic fracture mechanics (EPFM). These methodscan produce small and large scale yielding in plane stress mode, respectively. On the base of dataobtained in these tests a modified integrated model for prediction and simulation of texture evolutionhas been proposed. This model consists of a mechanical and a thermal part relating variations of stressand temperature occur during TMF tests. For the mechanical part of the model, a modified Taylor methodbased on strain partitioning in the grains orientations was used; and for the thermal part of the model, anintegrated phased field and Monte Carlo models with harmonic Fourier approach for orientations distri-bution function (ODF) was used. The effects of Twin boundaries and dispersed secondary phase particleswere also considered in the model. The validation tests show a good correlation between experimentalresults and the proposed model for texture development during TMF. To predict the initiation of unstablecrack growth up to failure of the specimen during TMF, a new parameter in the form of reduced criticalODF value has also been introduced in this research.

� 2017 Elsevier Ltd. All rights reserved.

1. Introduction

Thermo mechanical fatigue is a process dealing with simultane-ous changes occur in a sample subjected to mechanical load at hightemperatures [1]. Hastelloy X is a nickel based superalloy possess-ing a combination of high temperature strength, good oxidationresistance, suitable fabricability and resistant to stress-corrosioncracking [2]. This alloy is used mainly in the hot section parts ofgas turbines which are exposed to TMF. TMF behavior of nickelbased super alloys has been investigated in several researches[3–8]. Most of these researches used strain for damage assessment.Fracture mechanics for predicting fatigue life in the in-phase (IP)and out-of-phase (OP) states has been used in these researches.

Some studies [9–11] reported the evolution of crystallographicgrain texture, microstructure and formation of secondary phasesduring TMF and their effects on the fatigue crack initiation andgrowth. The effects of crystallographic orientation and stacking

fault energy (SFE) have also been studied during TMF [12]. Resultsof these researches demonstrated that low-angle, high-angle andTwin boundaries have the most, the least and medium effects onresisting fatigue cracking respectively.

In order to study the effects of crystallographic orientationsevolve during TMF on fatigue crack initiation, statistical data ofmany grains are required. These data can be obtained by orienta-tion distribution function (ODF) and misorientation distributionfunction (MODF). It has been stated [12] that for statistical studyof crystallographic orientation and its effect on fatigue, the com-monly used electron backscatter diffraction (EBSD) technique isnot suitable because of the difficulties exist for locating a smoothsurface for the post-fatigue samples.

For simulating and predicting the texture development duringgrain growth at high temperature, some researches used MonteCarlo method [13,14]. The effects of boundary properties and initialmicrostructure in texture evolution during grain growth have alsobeen investigated by the phase-field method [15,16]. In the phase-field method, misorientation angle of a grain boundary sets itsenergy and its mobility, so the evolution of its texture can be deter-mined by the time-dependent Ginzburg–Landau theory [17].

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192 M. Esmaeilzadeh et al. / International Journal of Fatigue 105 (2017) 191–207

In addition, many experimental studies have been accom-plished on texture evolution of various pure and alloyed materialsat high temperature including those performed for recrystalliza-tion and grain growth and their effect on TMF [18–22].

Texture development during plastic deformation is said to bedue to crystals rotation [23]. One of the most common modelsfor texture development has been presented by Taylor [24]. He sta-ted that the plastic strain of all crystallites within a polycrystallinematerial is equal to the macroscopic plastic strain. Therefore onecan use this model for predicting the texture development on thebase of plastic strain. In addition of the Taylor model, the use ofcrystal plasticity finite element method (CPFEM) and the advancedLamel method can provide models for predicting crystallographictexture based on the strain path [23]. Cyclic deformation at ele-vated temperature in a polycrystalline nickel based superalloyhas been modeled by using crystal-plasticity constitutive model[25] and the grain interaction (GIA) model. Advanced Taylor clustertype model has been used for predicting the deformation texture ofmaterials with low stacking fault energies when plastic deforma-tion occurs by crystallographic slip and twinning [26].

In this research, an integrated model on the base of mechanicaland thermal contribution of TMF on texture evolution has beenproposed. This model was used for simulating and predicting thetexture evolution during TMF cycles. For the mechanical part, amodified Taylor method based on strain partitioning in the grainsorientations was used. Also for the thermal part of the proposedmodel, an integrated model consisting of the phase field and MonteCarlo methods with harmonic Fourier approach for ODF was used.Worth mentioning, some parameters of this integrated modelshould be measured by experimental tests and can’t be quoted orextracted from other models.

For using the thermal part of the proposed model, one shouldobtain the grain boundaries mobility and energy. In addition, theenergy of twin boundaries presented by the minimum location ofgrain boundary energy curve vs misorientation angle must be cal-culated and also the effects of the secondary phases on TMF shouldbe calculated via Zener equation. The validation tests for texture

Table 1Mean chemical composition of Hastelloy X in wt.%

Element Ni Cr Fe Mo

Wt% 47.2 22.8 18.5 8.2

Fig. 1. The compact tensio

have shown a good correlation between experimental results andthe texture predicted by model during TMF. In the model pre-sented in this research, a new parameter based on the reduced crit-ical ODF value was defined and used for predicting the start ofunstable crack growth which leads to sample failure in TMF.Finally, using the MODF and the coincidental site lattice (CSL)model [27], the effects of special high angle grain boundaries(e.g. R7, R13b and R17b) in TMF cracking were investigated.

2. Experimental methods

The material used in this study was Hastelloy X, a nickel basedsuperalloy, with a chemical composition shown in Table 1. It wasin the form of plate shape with 2 mm thickness. These plates wereheat treated at 1175 � C for 1 h and then rapidly cooled in air.

Tensile compaction test (CT) specimens were fabricated withthe dimensions given in Fig. 1 [8]. These specimens were precracked before they were subjected to tensile compaction tests inplane stress mode according to standard ASTM E647 [28].

For measuring the crack size and crack opening displacement,COD, an EOS 700D digital camera with 5184 ⁄ 3156 pixel resolu-tion and a crack gauge apparatus were used. For application ofinduction heating in TMF, an elliptic coil composed of 7 turns ofcopper tube was used. The elliptic shape of coil can provide a sym-metrical heating in two sides of the plate. The power used waswithin the range of 1.6–2 KW. Water circulation for cooling of cop-per coil was used. Also grips and fixtures for holding the specimenwere used. Fig. 2 shows set up of TMF equipments [8].

TMF tests were performed by utilizing the small and large scaleyielding stresses belonged to linear elastic fracture mechanics(LEFM) and elastic-plastic fracture mechanics (EPFM) respectively.To apply these two theories of fracture mechanics in the design ofTMF tests, the requirements of ASTM E647 [28] standard werefulfilled. For LEFM, the specimen should be pre-dominantlyelastic at all values of applied force, thus the length of CTspecimen’s uncracked ligament should be greater than value of

W Mn Co Si P,S

0.71 0.49 0.73 0.24

Table 3Induction heating conditions used to calculate various parameters in theproposed model.

Temperature K (�C) Holding time (min)

803 (5 3 0) 5803 (5 3 0) 15803 (5 3 0) 30903 (6 3 0) 5903 (6 3 0) 15903 (6 3 0) 301003 (7 3 0) 51003 (7 3 0) 151003 (7 3 0) 30

Fig. 2. Set up of TMF equipments.

M. Esmaeilzadeh et al. / International Journal of Fatigue 105 (2017) 191–207 193

ð4=pÞðKmax=rYSÞ2 when K is the stress intensity factor and rYS is the0.2% offset yield strength. For the larger applied forces, EPFMrequirements were provided. The conditions of performing thesetests at various temperatures and modes, i.e. in-phase (IP) andout of phase (OP), are presented in Table 2. For measuring and cal-culating various parameters in the proposed model during graingrowth, various conditions for induction heating parameters suchas annealing temperature and time were applied. These conditionsare shown in Table 3. The microstructures were fully examined bySEM (TESCAN VEGA/XMU system) and optical microscope. The tex-ture of the samples after TMF tests were evaluated by X-raydiffractions (RIGAKU ULTIMA IV system). The areas for each ofthe sample considered for texture examination was about 20⁄20mm2 and include at least 10 mm length of the cracked area. Fordetermining the ODFs from pole figures, a generalized sphericalharmonic functions, MTEX [29] tools in the MATLAB softwarewas used.

In order to use the MTEX tools for determining the ODFs, theexperimental results of (1 1 1), (2 0 0) and (2 2 0) X-ray diffractionsas the inputs should be imported to the software by three matrixesthat each matrix includes three columns which are polar angle, azi-muth angle and intensity. The ODF determination from a pole fig-ure was done by the function calcODF using an algorithmexplained in the next part in Eqs. (1)–(4). The ODF as the outputof MTEX can be plotted in two dimensional sections through theEuler angles space. By default the sections are at constant anglesphi2. The number of sections can be specified by the user. Onecan also use the MTEX tools in MATLAB for programming any codesin the field of texture. In this study, a code was developed to pre-dict the texture evolution during TMF using the proposed modelexplained in the next part in Eqs. (5)–(17). For the future readers

Table 2The conditions of TMF tests at various temperatures, loads, TMF phases.

a0 (crack length mm) Force (N) TMF phase Temperature

17 0–1500 IP 873–101317 0–1500 OP 1013–87319 0–2600 IP 643–77319 0–2600 OP 773–643

to reproduce the methods and results, the code of the proposedmodel written by the authors in MATLAB is given in the supple-mentary material 1.

3. The proposed model for texture evolution during TMF

For texture evolution during TMF, an integrated model based oncontribution of mechanical and thermal parts of TMF was devel-oped. The main approach of the proposed model for texture evolu-tion based on the changes in the amplitudes of ODF, f ðu1;U;u2Þduring TMF due to stress and temperature variations.

The Euler angles u1, U, and u2 refer to three rotations that,when performed in the correct sequence, transform the specimencoordinate system onto the crystal coordinate system [30]. Thefirst two angles, u1 and U, show the position of the [0 0 1] crystaldirection relative to the specimen axes. In other words, the crystal

K Fatigue fracture mechanic Interval time for half cycle (min)

LEFM 0.46LEFM 0.46EPFM 0.43EPFM 0.43


rotate about the normal direction ND (1st angle, u1); then rotatethe crystal out of the plane (about the [1 0 0] axis, U); finally the3rd angle (u2) tells you how much to rotate the crystal about[0 0 1].

The ODF has been defined as a probability for density functionof orientations f (g), in the form of the Euler angles and g is givenby multiplication of three rotation matrices as g = gu2. gU. gu1. SoODF is given by the following relationship [30]:

f ðgÞ ¼ VgVdg

;when : dg ¼ 18p2

sinUdu1dUdu2 ð1Þ

where V is the sample volume and Vg is the volume of all crystal-lites with the orientation g in the angular element dg.

ODFs can also be given as a linear combination of harmonicfunctions on two-dimensional sphere S2 and on the rotation groupSO(3), [31]. Moreover, any ODF e SO(3) has an associated Fourierexpansion as the following relationship. An implementation ofthe algorithm is freely available as part of the texture analysis toolMTEX [29] in the MATLAB software.

f ðu1;U;u2Þ ¼X1l¼0

Xþlk;k0¼�l

ðlþ 12Þ½1=2�

2pf̂ ðl;k;k0ÞDkk0l ð2Þ

with Fourier coefficients f̂ ðl;k;k0Þ; l 2 N;k;k0 ¼ �l; . . . ;þl: thegeneralized spherical harmonics functions, Dkk

0l defines as:

Dkk0

l ðu1;U;u2Þ ¼ exp ð�iku1Þdkk0

l ð cos ðUÞÞ exp ð�ik0u2Þ ð3Þwhere

dkk0

l ðtÞ ¼ Skk0ð�1Þl�k0

2lðlþ k0Þ!

ðl� k0Þ!ðlþ kÞ!ðl� kÞ!

� �12

� ð1� tÞk�k0

ð1þ tÞkþk0" #1

2

� dl�k0

dtl�k0 ð1� tÞl�kð1þ tÞlþk ð4Þ

Skk0 is the correction factor for the normalization used for the spher-

ical harmonics; S0kk¼1 if k;k0P0;ð�1Þk if k0P0;k


where ui is velocity of the finite element in i direction, _cðaÞ is slip

rate from a -th slip system with slip plane nðaÞi , slip direction bðaÞi

and mðaÞij ¼ bðaÞi nðaÞj , Wpij is plastic spin and Xij is the lattice spinmeans the crystal rotation.

As it has been mentioned before in the Taylor model, the strainsfor all of grains are assumed to be the same and equal to macro-scopic plastic strain. Regarding to some studies [9,10,33] aboutlocalization and accumulation of strain in fatigue crack growth,e.g. in Hastelloy X superalloy, and considering the same amountof strain for all the grains may generate erroneous results. To han-dle somehow this error in this study, partitioning of the plasticstrains in grains according to their orientations distribution wasconsidered in the proposed model. For this purpose the Taylor fac-tor M [34] was defined as the following equation.

M ¼Xsa¼1

dcðaÞ

deijð10Þ

where dcðaÞ and deij are increments of shear strain in the slip sys-tems and the plastic strain increment applied externally respec-

Fig. 3. ODF evolution (algorithm 1&2 and experimental results) for a given orientationcycle interval for algorithm 1, b) until failure with 100 cycles interval for algorithm 1.

tively, a is the slip system number and s is the number ofactivated slip systems (i.e. 12 for fcc).

Birosca et al. [11] said that M is a function of the lattice orien-tation of the grain; the large values of M represents a large amountof shear strain in the slip system required for specific plastic strainapplied externally while for grains with lower values of M, moreefficient slip systems are available for plastic deformation.

In the proposed model in the current study, M considered to bea function of orientation distribution;Mðu1;U;u2Þ. Strain (e) wasconsidered to be approximately CTODth in the vicinity of the crack tipin the plane stress mode [35] when CTOD is crack tip openingdisplacement and th is thickness of CT sample. Macroscopicplastic strain around the crack was defined by the followingequation.

epmacro ¼Igðu1;U;u2Þ

XadcðaÞg

Mðu1;U;u2Þ ¼CTODth

ð11Þ

where gðu1;U;u2Þ indicates all possible grain orientations withinthe sample, dcðaÞg is increment of shear strain in the slip systems

(u1 ¼ 150�;U ¼ 2:5�;u2 ¼ 165�Þ during TMF-LEFM for a) until 30 cycles with one

Fig. 4. Illustration of crystal rotation in grains due to slip deformation with different orientations (different M factors).


due to grain orientation g and Mðu1;U;u2Þ is the related Taylorfactor.

3.2. Thermal part: Integrated model for texture evolution duringgrain growth

Based on the conventional Monte Carlo simulations about theeffect of interfacial anisotropy on misorientation texture develop-ment during grain growth (see i.e. [14]), the total system energycan be calculated by summation of grain boundary energies cðhijÞhaving misorientation angle hij [13].Using definition of orientationdistribution function in Eqs. (1)–(4), the total system energy can berewritten for this model as the following relationship.

Esys ¼ E0 þXmj¼i

Xmi¼1

ViV

VjV

NgbcðhijÞ

¼ E0 þI I

f ðgiÞdgif ðgjÞdgjNgbcðhijÞ ð12Þ

where ViV �VjV is the probability for the presence of grain boundary

density between grains with orientations gi, Ngb is the number ofgrain boundaries in the sample and m the number of grain orienta-tions in the sample. The grain boundary energy for low angleboundaries used here (i:e: hm � 15�) is based on the Read-Shocklyfunction [36] as presented in Eq. (12).

cðhijÞ ¼cmax

hhm

ð1� ln hhmÞ; h < hm

cmax ; h P hm

�ð13Þ

For conventional high angle boundaries, cmax was used. Also forespecial high angle boundaries (e.g. twinning), theoretical andexperimental results for nickel [37] and MODF of a cyclicallydeformed nickel specimen in Euler space [30] were used, in orderto see the effects of twin boundaries by this model.

The main approach of this model for texture evolution duringgrain growth is by application of a relationship similar to themultiple orientation phase-field method [15,16]. The differenceis that the ODF is used instead of non-conserved long-rangeorder (lro) parameters. The texture evolution in the system canbe described by the equation presented in this model, Eq. (13),this equation is based on time dependent Ginzburg-Landau the-ory [17].

@f ðgiÞ@t

¼ �Li @Esys@f

ð14Þ

where Li is the grain boundary mobility and t is the time. By substi-tuting Eqs. (11) and (12) into Eq. (13) the following equation wasextracted for texture evolution.

@f ðu1;U;u2; t; TÞ@t

¼ �Lðu1;U;u2; TÞBðu1;U;u2ÞNgbðT; tÞcmaxð15Þ

where T is absolute temperature and B as a function of initial ODFcan be written in a harmonic Fourier series by using the followingrelationship.

@Esys@f ðgiÞ

¼ BðgiÞNgbcmax ð16Þ

It should be noticed that dependency of parameters B and Ngb toorientations (u1;U;u2Þ, temperature and time should be mea-sured from experimental results and calculated by Eqs. (12), (13)and (16). Worth mentioning the proposed model (i.e. integrationof Monte Carlo and phased field models) enable us to obtain thegrain boundaries mobility and energy by an inverse method and

using experimental results (e.g. @f ðu1;U;u2Þ@t ). For this purpose the fol-

lowing relationship based on the probability of the grain sites flip-ping [38] and Eq. (13) were used to measure the grain boundariesenergy including random grain boundary energy (cmax Þ.

Pi!j ¼MðhijÞcðhijÞMmax cmax

; DEsys 6 0ðDf j P 0ÞMðhijÞcðhijÞMmax cmax

exp �DERT� �

; DEsys > 0ðDf j < 0Þ

8<: ð17Þ

where MðhijÞ ¼ Lj � Li.The random grain boundary energy (cmax Þ can be calculated by

solvingPm

j¼1Pi!j ¼ � DPiPi :Consequently, by measuring and calculating various parameters

in Eq. (14), f ðu1;U;u2; t; TÞ as the thermal part of texture evolu-tion during TMF can be extracted.

4. Results and discussion

To extract the above mentioned parameters in the proposedmodel, the microstructures and ODF of as received sample andthe heat treated samples one should know the heating conditionapplied on the samples. These conditionswere presented in Table 3.Fig. 5 shows the initial microstructure, the related pole figures andthe ODF of the as received material after solution aging heat treat-ment. The average measured grain size was 45 lm and the meansize of large precipitates aggregated within the grains and those

Fig. 5. The microstructure, related pole figures and ODF of the as received material.


distributed finely within grains were 4.5 lm and 1.6 lm respec-tively. The most of precipitates seem to be meu (l) phase as theywere rich in Mo, Ni, Cr, Fe andW elements. 27% of the grain bound-aries were due to twins. The related ODF of the as received sampleshows slightly the formation of copper texture ((1 1 2) h1 1 1i,g = (90, 35, 45)).

The microstructure evolution of samples under conditions pre-sented in Table 3 is shown in Fig. 6(a). The average grain sizes atvarious times and temperatures were measured and used for

extraction of the related Arrhenius equations for evaluation ofgrain growth. It was found by considering the total area of the sam-ples (Atotal) and the average number of grain neighbors (or edges, e)in the tested plane that it is possible to calculate the Ngb as thenumber of grain boundaries for the proposed model (Eq. (14)) bythe following relationship.

Ngb ffi ðe� 2Þ4AtotalpD2

¼ ðe� 2Þ4Atotalp

1

ðk0 exp �QRT� �

tnÞ2ð18Þ

Fig. 6. (a) The microstructure and (b) ODF evolution of samples in Table 3.


where t, T, k0 and Q are time, absolute temperature, material con-stant and activation energy for boundary mobility, respectively.Also, the dependency of the random grain boundary energy forHastelloy X on temperature was calculated by simulations at vari-

ous temperatures based on Eq. (16) and Eqs. (19.a) and (19.b) wasobtained.

cmaxj

m2

� �ffi 3 � 10�9T2 þ 3 � 10�5T ð19:aÞ


cmaxj

mol

� �ffi 0:88 � T2 þ 8821:88 � T ð19:bÞ

Ebarrier ¼ 2:8 � 1013 � c�0:6GB ð19:cÞ

The calculated results of cmax is compatible to general idea thatsays the grain boundary free energy of a pure material (e.g. Ni) isexpected to decrease with increasing temperature, but in an alloy-ing system composed of different alloying elements the situationcan be slightly different [39]. In most cases, the solutes in the grainboundaries reduce the grain boundary energy. For Hastelloy Xsuperalloy, the results of SEM-back scatter showed that distribu-tion of Cr atoms in grain boundaries is 5%weight more than withingrains whereas for Mo atoms, it was in an inverse state. The radiusof Ni, Cr and Mo atoms are 124, 128 and 139 pm; the little differ-ence between Ni and Cr radius causes to atomic absorb to disloca-tions within grain boundaries; therefore the grain boundary energywas decreased. As Using Eq. (18), the random grain boundaryenergy at T = 803 K (T/Tm = 0.5) was 0.022 J/m2 which is much lessthan that of pure nickel (0.8 J/m2 at T/Tm = 0.5 acc to [40]). This lowgrain boundary energy for Hastelloy X superalloy was expected fora creep resistance alloy, since as the temperature increased thebulk solubility of the solutes atoms in the grain boundariesincreased resulted to an increasing in the grain boundary freeenergy according to Eq. (18).

It should be noticed for calculating the random grain boundaryenergy from Eq. (16); the experimental results of ODF derived fromX-ray diffraction were used for each orientation; so the values ofgrain boundary energies can be acceptable. Nevertheless, themolecular dynamic (MD) method was also accomplished to vali-date the employed Eq. (16). For this purpose, the results of MDsimulation on HastelloyX [9], which is a same material used in thisresearch, were employed. The related grain boundary energy wasalso calculated using the methodology of energy barriers duringslip presented in [41]. The values of energy barrier fordislocation-grain boundary interaction in the twin boundaries(R3) and higher energy grain boundaries R7, R13b and R17b(which were detected in this alloy; see Table 4) are 2⁄1012,1.2⁄1012, 8⁄1011 and 6⁄1011mj/m3 respectively [41]. The totalamount of energy barriers for the mentioned grain boundaries is4.6⁄1012mj/m3. Using MD simulation (Eq. (19.c) [41]), the random

Table 4All the possible fcc slip systems in Matrix and changing in Schmid factor calculated for bo

CSL R value Rotation angle (�) Boundary plane Matrix sli

3 60 {1 1 1} ð111Þ½0 �1ð�111Þ½11ð1 �11Þ½11ð11 �1Þ½�11

7 38.2 {1 1 1} ð111Þ½0 �1ð�111Þ½11ð1 �11Þ½11ð11 �1Þ½�11

13b 27.8 {1 1 1} ð111Þ½0 �1ð�111Þ½11ð1 �11Þ½11ð11 �1Þ½�11

17b 61.9 {2 2 1} ð111Þ½0 �1ð�111Þ½11ð1 �11Þ½11ð11 �1Þ½�11

When Dm < 0, the dislocation density will be piled up at the interface.* Maximum piled up dislocation density at the grain boundaries.** No pile up.

grain boundary energy at T = 803 K was calculated equal to0.0198 J/m2 which has a good correlation with that calculated bythe proposed model in Eq. (16) (0.022 J/m2).

Using Eqs. (15)–(17) for the conditions presented in Table 3, theparameters of the proposed model were calculated. Simulationswere performed by MTEX tools in the MATLAB software for 4958orientations. Fig. 6(b) shows the results of ODF evolutions for testconditions presented in Table 3. These results were predicted bythe thermal part of the model. It is clear that similar ODFs at highertemperatures can be obtained in less time. The Fourier coefficientpowers of ODFs for harmonic degree of l were calculated by Eq.(20) and the related amplitudes in Fig. 6(b) are shown in Fig. 7(a).

powerðlÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXþl

k;k0¼�l f̂2ðl;k;k0Þ

rð20Þ

Changes in the Fourier coefficient powers of ODFs by changingthe temperature and heating time can be measured experimentallyand be used directly in thermal part of the model, but by doing thatgrain size and grain boundary energy cannot be controlled. Fur-thermore, the Fourier coefficients allow for a complete characteri-zation of the ODF, are suitable instrument for calculating theevolution of mean macroscopic properties, in the specimens duringTMF (e.g. elastic properties by fourth order Fourier coefficients)[29]. Variation of the fourth order Fourier coefficients against timefor the samples presented in Table 3 are shown in Fig. 7(b).

In the proposed model, ODF is a function of initial texture, tem-perature and time, f ðu1;U;u2; t; TÞ. Therefore because of temper-ature variation in TMF; ODF can be calculated by the followingrelationship.

Df ðu1;U;u2;Dt;DTÞ in TMF ¼Z t2t1

_f ðu1;U;u2; t;TÞ:dt ð21Þ

where _f ¼ dfdt. Relation between temperature and time in TMF pro-cess should be considered in Eq. (21). In this study, a simple Fourierequations (T ¼ aþ bcosðxtÞ þ csinðxtÞ) was used for establishingthe relationship between temperatures and time during TMF.

The results of texture evolution of the samples undergone TMFtests under conditions shown in Table 2 are illustrated in Fig. 8.

The contribution of thermal and mechanical parts on ODF of thesamples undergone TMF tests, were found to be according to thefollowing equations.

th sides of the grain boundaries and Twins.

p systems (for fcc materials) Dm ¼ mtwin �mmatrix1�; ½�101�; ½�110� 0.36 ** �0.18 �0.540�; ½0 �11�; ½101� �0.18 0.18 ** �00�; ½�101�; ½011� �0.36 �0.18 �0.540�; ½011�; ½101� �0.81 * �0.81 * �0

1�; ½�101�; ½�110� 0.24 ** �0.37 �0.610�; ½0 �11�; ½101� �0.31 0.04 ** �0.270�; ½�101�; ½011� �0.39 �0.07 �0.470�; ½011�; ½101� �0.69 �0.87 * �0.17

1�; ½�101�; ½�110� 0.16 ** �0.42 �0.590�; ½0 �11�; ½101� �0.36 �0 �0.3670�; ½�101�; ½011� �0.39 �0.19 �0.580�; ½011�; ½101� �0.62 �0.87 * �0.25

1�; ½�101�; ½�110� 0.34 ** �0.32 �0.670�; ½0 �11�; ½101� �0.36 0.24 ** �0.120�; ½�101�; ½011� �0.40 �0.14 �0.540�; ½011�; ½101� �0.72 * �0.77 * �0.05


Df thermalðu1;U;u2Þ ¼ aðu1;U;u2Þ � Nbðu1;U;u2Þ ð22:aÞ

Df mechanicalðu1;U;u2Þ ¼ cðu1;U;u2Þ � N þ dðu1;U;u2Þ ð22:bÞwhere N is the number of TMF cycles and a, b, c and d are the con-stants which are functions of orientations. Eqs. (22.a) and (22.b) aredue to curve fitting of the simulation results achieved via the pro-posed model only on Hastelloy X superalloy. The results of the men-tioned equations have also a good correlation with experimentalresults.

Fig. 8 shows that in TMF samples using LEFM, the initial coppertexture was strengthened by increasing the number of the cyclesand also a brass texture was developed. This later texture quanti-tatively was about 0.66 times the density of copper texture. InTMF samples using EPFM, rotated cubic texture was developedand strengthened after 100 cycles, and the initial copper texture

Fig. 7. (a) The Fourier coefficient powers of ODFs for Fig. 6(a) and

was weakened until 30 cycles and then it strengthened afterward.Worth mentioning texture evolution during stable crack growthuntil the start of unstable crack growth can be a criterion for pre-diction of the TMF behavior and life. The TMF life (i.e. number ofcycles require for starting the un-stable crack growth; Nf) at tem-perature range of 873–1013 K and force range of 0–1500 N forLEFM samples and temperature range of 643–773 K and forcerange of 0–2600 N for EPFM samples are 2752 and 1335 cycles,respectively. The texture developed in various samples indicatedabove before the beginning of unstable crack growth is shown inFig. 9(a, b).

It is possible to correlate the increments in J-integral and incre-ments in ODF amplitudes of known textures developed duringTMF. Fig. 9(c) shows by increasing the J-integral values until thestart of unstable crack growth marked by arrows. J-integral usedin this study and previous research [8], is due to macro scale and

(b) evolution of the related fourth order Fourier coefficients.

Fig. 9. The texture of TMF samples at (a) LEFM, (b) EPFM; as soon as starting of unstable crack growth. (c) Evolution of the related J-integrals and (d) maximum ODFamplitudes of known textures. The arrows in (c) show start of the un-stable crack growth and Apl is the area under load–displacement, BN is the net thickness, b0 = uncrackedligament and g = 2 + 0.522 b0/W.

Fig. 8. Texture evolution during TMF (LEFM & EPFM).


defined according to ASTM E 1820 [42]; its relationships with crackdimension and applied force on CT specimens were also shown inFig. 9(c). To predict the start of unstable crack growth, the measur-able parameters are required in this study; therefore the macro

scale j-integral and ODF were employed. J-integral could be pathdependent in reality especially in EPFM. It was investigatednumerically via FEM method in [43] for Mode I. its results showedthat the J-integral value in a contour near the crack tip is lower


than a far field contour including the crack tip. In this research, thedifference between values of j-integral in the near field and far fieldcontours calculated equal to 5.7% using the relationships inFig. 9(c).

The related ODF values of the textures during TMF can be calcu-lated for some number of cycles, as shown in Fig. 9(d). Also one candefine a new parameter as the critical ODF value based on therelated critical J-integral. So when the value of J-integral and ODFvalue of a known texture reaches the critical values, crack growthtransit from stable growth to un-stable growth. In fact for TMFsamples using LEFM, critical brass textures resulted to failure,while for TMF samples using EPFM, critical rotated cube texturewas the cause of failure.

The effects of TMF (i.e. using in-phase and out-of-phase) on tex-ture evolution showed that in LEFM-TMF, no clear differencebetween IP and OP can be observed. This is due to small variationon stresses in plastic zone around the crack tip. However, in TMF-EPFM, the texture evolution in OP mode was different to that of IPmode as shown in Fig. 10. This figure shows importance of OP and

Fig. 11. The cube and rotated cube textures in Hastelloy X aft

Fig. 10. Evolution of the cube and rotated cube textures in the TMF-EPFM (IP&OP).

IP mode proposed in TMF-EPFM in establishment of the criticalODF value. The critical ODF values of copper and rotated cube tex-tures in TMF-EPFM are 14 and 10 respectively. These values wereachieved after 1335 cycles in IP-EPFM and after 1457 cycles inOP-EPFM.

The proposed model for development of some specific textureswas used for the samples subjected to TMF. Using mechanical partof the model for the samples undergone cold rolling, so that theirthicknesses reduced from 4 mm to 2 mm, showed that cube androtated cube textures were developed as initial textures as shownin Fig. 11. The yield strength, ultimate tensile strength and elonga-tion of the rolled sample before TMF were measured 1179 Mpa,1379 Mpa and 7%, respectively.

For the above sample with cube and rotated cube texture, theevolution of texture was simulated by the model for TMF-IP attemperature range of 643–773 K, force range of 0–2600 N andcrack length of 19 mm. It should be noted that due to the strength-ening of the sample during cold rolling, TMF sample using EPFMconditions in Table 2 classified as LEFM after 50% cold rolling. ItsODF value reaches to a critical value after 7383 cycles as shownin Fig. 12. Experimental validation tests showed that the failureof the specimen in TMF was occurred after 7566 cycles.

The results of ODF values in Fig. 10 and Fig. 12(b) demonstratethat the values of critical ODF depend upon initial texture; there-fore one can define a reduced critical ODF as Critical ODF valueyield stress which

is approximately constant for various initial textures before TMF.The reduced critical ODF value can indicate the start of un-stablecrack growth.

To validate the proposed model for texture development duringTMF, ODF values measured from X-ray diffraction and comparedwith simulated results of samples under conditions presented inTable 2. Comparison between these results is presented inFig. 13. These results show a good correlation between experimen-tal and simulated textures during TMF.

The effects of dispersed secondary phase particles were investi-gated in this study. A distribution of secondary phase particlesexerted the pinning pressure on the grain boundaries; thereforethe grain size can be controlled [44]. The famous Zener equation

er cold rolling from 4 mm to 2 mm in thickness direction.

Fig. 12. Texture evolution of Hastelloy X with initial texture of Fig. 11 after TMF-IP test with force of 0–2600 N, temperature of 643–773 K and 19 mm crack length. (a) ODFsafter 7383 cycles (failure). (b) Rotated cube texture evolution.


[45] as the following relationship shows how the grain size is con-trolled when r is the secondary phase particle radius, fv is the vol-ume fraction of the particles and Rlim is the limiting mean grainsize.

Rlim ¼ 4r3f v ð23Þ

So, the secondary phase particles can control NgbðT; tÞ and therate of ODF values presented in Eq. (17) and Eq. (14) respectively.Fig. 14 shows how changes in ODF for an orientation i.e.u1 ¼ 150�;U ¼ 2:5�;u2 ¼ 165�) was controlled by the secondphase particles in TMF samples using LEFM at temperature rangeof 873–1013 K, force range of 0–1500 N and the crack length of17 mm.

As studied in some literatures, a consistent conclusion has beendrawn on the grain size effect in nickel based superalloys, e.g. [46],as fatigue crack growth rate is higher in fine grains than coarsegrains. In the experiments of our research, it was observed a phe-nomenon related to effect of initial grain size on the fatigue crackgrowth rate as the following description.

One of the as-received specimens was heated for 15 min at1003 K. The average measured grain size was increased from45 lm to 69 lm. The room temperature fatigue test for both CTspecimens (as-received and heat treated samples) were carriedout at maximum force 6000 N that caused to a very large scaleyielding stress. The results showed that unstable crack growth inthe specimens with grain size of 45 lm and 69 lm were startedafter 15 and 57 cycles respectively. As it is known, the initial grainsize has a significant effect on the growth rate of the crack. This

Fig. 14. The effects of finely dispersed second phase particles on ODF evolution for a given orientation (u1 ¼ 150�;U ¼ 2:5� ;u2 ¼ 165�) in TMF–LEFMwith force of 0–1500 N,temperature of 873–1013 K and 17 mm crack length.

Fig. 13. Simulation ODF values vs experimental ODF values for TMF-LEFM&EPFM.


effect was considered in the proposed model for Hastelloy X by Eq.(17) and also the relationships in Fig. 6(a).

To study the effects of crystallographic orientations evolve dur-ing TMF on fatigue crack initiation, MODF of many grains withinthe TMF samples should be obtained. The misorientation betweentwo orientations g1 and g2 is defined as g1�1 ⁄ g2. MODF can be dis-played in Euler space. It was computed by kernel density estima-tion via Fourier series. It has been stated that the mainadvantages of the representation of misorientation distributionsin Euler space are: 1- the computation is relatively easy and 2-the MODF and its derivatives can be obtained [30]. Figs. 15 and16 illustrate the MODF in TMF samples at the start of unstablecrack growth for constant values of u2 and U respectively. Theobtained results showed that density of twin boundaries (R3)and higher CSL values boundaries (i.e. R7, R13b and R17b) inTMF samples using EPFM is approximately twice of samples usingLEFM at the start of unstable crack growth.

The above observations indicated that twin boundaries in TMFsamples using LEFM was less effective in fatigue cracks initiation,and cracks were mostly nucleated at Slip Bands; while fatiguecracking at twin boundaries was observed in TMF samples using

EPFM having large scale of internal stresses due to dislocations pileup at the interface of matrix and Twins. In order to quantify theeffects of the twin boundaries (R3) and higher CSL values bound-aries (i.e. R7, R13b and R17b) on TB cracking, the following rela-tionship based on the influence of crystallographic orientations[12] can be used.

nDm ¼ � Lðr� r0ÞKGb Dm ð24Þ

where nDm is the number of piled up dislocations at the interface oftwo slip planes in two adjacent grains with different orientation. mis the Schmid factor for each grain and Dm ¼ m2 �m1. G, b and L arethe shear modulus, dislocation Burgers vector and the averagewidth of the twin lamellas respectively. K is a constant and r isthe applied flow stress on the crack tip (plastic zone) and r0 isthe threshold stress for dislocations to move. So is approximatelyequals to the yield stress. When Dm < 0, the dislocation density willbe piled up at the interface.

Using the grain boundaries properties including rotation angle,rotation axis and boundary planes which were listed in [27], all thepossible fcc slip systems in Matrix and changing in Schmid factor

Fig. 16. MODF of the TMF- LEFM&EPFM samples as soon as start of unstable crack growth (failure) with U ¼ constant.

Fig. 15. MODF of the TMF- LEFM&EPFM samples as soon as start of unstable crack growth (failure) with u2 ¼ constant.


calculated for both sides of the grain boundaries and Twins aregiven in Table 4. The results presented in Table 4 showed thatCSL boundaries (i.e. twin boundaries (R3) and higher energy grainboundaries R7, R13b and R17b), in some of the slip systems of thematrix f111gh�110i and f111gh011i have maximum piled up dis-

location density and they had the most effect on fatigue cracksnucleation.

As a brief review on the nobility of the proposed modelobtained in this research are the Eqs. (5), (6), (11), (12), (14),(15), (16), (18), (19), (21) and (22) presented for calculation and


simulation of various parameters and compared for the first timewith the results obtained by experimental.

5. Conclusions

Based on the results obtained via the proposed model and theexperimental results, the following conclusions were drawn.

1- The new proposed model which based on linear superposi-tion law for integration of the mechanical and thermal partscan be used for predicting ODF evolution during TMF.

2- In regard to localized and accumulated strain in the vicinityof fatigue crack grown during TMF in HastelloyX superalloy,evolutions of texture in two mechanisms by LEFM and EPFMfor IP and OP modes were proposed.

3- Mechanical part of the proposed model in this study parti-tioned the plastic strains in grains according to their orienta-tions distribution in order to minimize the error in samestrain assumption.

4- An integrated method consisting of the phased field andMonte Carlo models with harmonic Fourier approach forODF was design in order to simulate the thermal part of tex-ture evolution during TMF.

5- A reduced critical ODF value was defined as Critical ODF valueyield stress in

this research. This is a suitable parameter which is approxi-mately constant for various initial textures and its valueindicates the start of un-stable crack growth in TMF.

6- Twin Boundaries in TMF-LEFM was less effective in fatiguecracks initiation and cracks were mostly nucleated at SlipBands; while fatigue cracking at twin boundaries wasobserved in TMF-EPFM having large scale of internal stressesdue to dislocations pile up at the interface of matrix andTwins.

7- The results showed that CSL boundaries (i.e. twin boundaries(R3) and higher energy grain boundaries (R7, R13b andR17b)), in some of slip systems of the matrixf111gh�110i and f111gh011i have maximum piled up dis-location density and they had the most effect on fatiguecracks nucleation.

Acknowledgements

We would like to acknowledge Mr A.Shirzad from OTEC Co -material Lab. for his cooperation in material characterization. Theauthors would also like to thank Mr. Ghorban Esmaeilzadeh asfor design & manufacturing of the induction heating systems andalso Mrs. N.Zakeri from Semnan university- mechanical propertiesLab For her technical assistance.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.ijfatigue.2017.08.028.

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Evolution of the texture and microstructure in a nickel based superalloy during thermo-mechanical fatigue (TMF), using a modified integrated model and experimental results1 Introduction2 Experimental methods3 The proposed model for texture evolution during TMF3.1 Mechanical part: modified Taylor model for texture evolution3.2 Thermal part: Integrated model for texture evolution during grain growth

4 Results and discussion5 ConclusionsAcknowledgementsAppendix A Supplementary materialReferences

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International Journal of Fatigue · 2017-12-30 · plane stress mode according to standard ASTM E647 [28]. For measuring the crack size and crack opening displacement, COD, an EOS