Materials Science & Engineering A 582 (2013) 47–54

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Materials Science & Engineering A

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Dislocation/precipitate interactions in IN100 at 650 1C

Kimberly Maciejewski a, Mustapha Jouiad b, Hamouda Ghonem a,n

a Department of Mechanical Engineering, University of Rhode Island, Kingston, RI 02881, USAb Division of Physical Science & Engineering, Mechanical Engineering Program, KAUST, Saudi Arabia

a r t i c l e i n f o

Article history:Received 26 April 2013Received in revised form28 May 2013Accepted 3 June 2013Available online 11 June 2013

Keywords:Nickel-based superalloyGamma prime precipitates particle shearingParticle loopingCritical resolved shear stress

93/$ - see front matter & 2013 Elsevier B.V. Ax.doi.org/10.1016/j.msea.2013.06.004

esponding author. Tel.: +1 401 874 2909.ail address: ghonem@egr.uri.edu (H. Ghonem)

a b s t r a c t

The influence of γ′ size on critical resolved shear stress in alloy IN100 at 650 1C has been examined byconsidering dislocation/precipitate interactions involving particle shearing and Orowan by-passingmechanisms. To achieve this, heat treatment procedures were carried out on smooth specimens toproduce materials with variations in secondary and tertiary γ′ size, while maintaining their respectivevolume fractions. These specimens were subjected to strain-controlled fully reversed cyclic loading at650 1C. Thin foils extracted from these specimens, post-testing, were examined by transmission electronmicroscopy to identify the nature of the precipitate/dislocation interactions during plastic deformation.Results indicated the presence of shearing and Orowan by-passing mechanisms. These observations havebeen used as a basis to calculate the critical resolved shear stress as a sum of components contributed bysolid solution and by γ′ particles being sheared and looped. In this analysis, a critical particle size definingthe shearing/looping transition has been determined and this has been used to calculate the relativevolume fraction and size of particles contributing to the critical resolved shear stress. These analyticalresults have been compared with those experimentally obtained at 650 1C using smooth specimens withdifferent precipitate sizes.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

Over the last few decades, many authors have examined therole of microstructure on deformation and damage mechanisms inprecipitation strengthened Nickel based superalloys, with a focuson the role of dislocation/precipitate interactions [1–8]. Precipitatesize and volume fraction have been shown to influence thehardening behavior and in particular, the yield strength. In thisregard, the γ′ precipitate/dislocation interactions are described interms of shearing and/or Orowan by-passing mechanisms. Thetransition between these mechanisms is controlled by the pre-cipitate size and volume fraction. The work of Shenoy et al. [9] onIN100 implies that all γ′ particle/dislocation interactions areshearing. The initial critical resolved shear stress (CRSS) isdescribed as being proportional to volume fraction and inverselyproportional to size for secondary γ′ (γ′s) and primary γ′ (γ′p)particles. These precipitates are incorporated through shearingby strongly coupled dislocations, while the smaller particles,tertiary γ′ (γ′t), are incorporated through shearing by weaklycoupled dislocations. For γ′t, the CRSS is proportional to volumefraction and size. These patterns of γ′ particle/dislocation interac-tions are consistent with how Milligan et al. [10] incorporates γ′s

ll rights reserved.

.

and γ′t into the yield stress for alloy IN100. However, their workshows that the γ′p are of comparable strength as that of the matrixmaterial and thus has no influence on the yield strength. Theseauthors have shown that for IN100 there is no evidence ofdislocation looping. Conversely, in the work by Heilmaier et al.[11] on IN100 at room temperature, CRSS is incorporated throughshearing by weakly and strongly coupled dislocations, as well as,looping by edge and screw dislocations. Their work has shownthat the transition between mechanisms is controlled by theparticle size and suggests that both shearing and looping mechan-isms exist simultaneously. Similar results are obtained by Reppichet al. [12] on Nimonic 105. For this alloy, large overaged particles ofa diameter greater than 120 nm (volume fraction 22%) and 450 nm(volume fraction 51%), show differences in the experimental andtheoretical data of yield stress, implying that both mechanismsoperate. In addition, they suggest that the transition betweenshearing and looping occurs over a range of particle sizes, ratherthan a distinct critical size. Thus, for a continuum with a givenmean particle size it is possible that both mechanisms operate.This is supported by transmission electron microscopy evidence ofNimonic 105 with particle sizes of 75 nm, 220 nm and 320 nm,showing the existence of shearing, shearing and looping, andlooping respectively. Sinha's work [13] on IN-738LC, shows thatduring long creep tests, γ′ precipitates ripen and coarsen, becomeirregularly shaped and develop rafted microstructures of plate orrod morphology. This has been addressed by using short-term

K. Maciejewski et al. / Materials Science & Engineering A 582 (2013) 47–5448

strain relaxation (creep) and strain recovery tests. Sinha's work hasshown that fitting of experimental data to a Norton power lawform, has a strong dependence of the exponent on stress andmicrostructure. Sinha suggests that the change in value of thisexponent is from a transition in which dislocations climb overgamma prime particles to that in which they cut the particles.Furthermore, the work of Del Valle et al. [14] on Inconel X-750alloy have shown a critical γ′ particle radius exists, below whichparticle shearing occurs and above which the Orowan loopingmechanism operates.

The studies mentioned above, among others [see for example[15,16]], express different views on secondary γ′ precipitate/dis-location interactions. The goal of this paper is to examine thesemechanisms on the basis of the particle size and their relativeinfluence on the hardening behavior in IN100. To achieve this, thefirst section of the paper describes heat treatment proceduresaiming at producing microstructures with variations in theirprecipitate sizes while maintaining the respective volume frac-tions. In order to identify the nature of the precipitate/dislocationinteractions during plastic deformation, strain-controlled fullyreversed cyclic loading tests were carried out on specimens havingas-received, as well as modified microstructures, at 650 1C. Thinfoils were extracted from these specimens post-testing and weresubjected to transmission electron microscope examination toidentify the shearing and Orowan by-passing mechanisms inrelation to the γ′ particle size. These results were then used as abasis to calculate the components of critical resolved shear stresswith contributions from secondary and tertiary γ′ particles takinginto consideration their relative dislocation/particle interactions.

2. Microstructure control

The role of secondary γ′ particle size on the critical resolvedshear strength is the focus of this work, thus, variation in this sizeis achieved by varying the heat treatment sequence of the as-received alloy. The model material in this work is the powdermetallurgy Inconel 100 (IN100) with the chemical composition (inweight percent): 4.85 Al, 4.24 Ti, 18.23 Co, 12.13 Cr, 3.22 Mo, 0.71V,0.071 Zr, 0.02 B, 0.072C and balance Ni [17]. The as-receivedcondition, shown in Table 1, has seen the typical three stage heattreatment sequence; subsolvus solutioning, stabilization andaging. This treatment has resulted in the microstructure shownin Fig. 1a–c, which has an average grain size of 5 μm, and trimodaldistribution of γ′ particles, denoted primary, secondary and tertiary(γ′p, γ′s and γ′t). The volume fraction and size of the γ′p are 24.2% and0.9 μm, γ′s are 26.7% and 72.1 nm and γ′t are 7.5% and 4.7 nm;respectively. The particle size is reported as mean equivalentparticle radius. Two additional heat treatments were performedon the as-received material in order to vary their γ′ statistics. Theseheat treatments, denoted overaged and water quenched, are listedin Table 1. Both heat treatments are performed at subsolvusconditions, thus maintaining the same grain size, as well as, theγ′p volume fraction and size as that of the as-received material (seeFig. 1d and f). In addition, the volume fractions of the secondaryand tertiary γ′ show little variation, thus, an average value of 26.7%

Table 1Heat treatment conditions of IN100 material and the corresponding γ′t and γ′s sizes (rep

HT condition Solutioning Stabilization

As-received 1149 1C/2 h/oil quench 982 1C/1 h/fan coolOveraged / 815 1C/75 min/cool aWater quenched 1135 1C/2 h/water quench 815 1C/2 h/ air cool

for γ′s and 7.5% for γ′t was considered for all microstructureconditions. The overaged heat treatment condition, shown inFig. 1d and e, where an additional stabilization and aging sequenceis applied, allows coarsening of both the γ′s and γ′t; whereas, thewater quenched condition (see Fig. 1f and g) shows a decrease inthe γ′s particle size. In this case, the fast cooling rate from thesolutioning temperature allows for precipitation of the γ′s particlesbut does not provide sufficient kinetics for coarsening. The agingcondition, which is similar to that of the as-received material,maintains the size of the γ′t. The mean tertiary and secondary γ′sizes for the three microstructures are listed in Table 1. The meanparticle radius of γ′t for the three microstructure conditions werein the range of 2–16 nm, while γ′s particles were in the range of20–140 nm. Thus, a threshold particle size of 20 nm has been usedhere as the transition limit between γ′t and γ′s particle size.

3. Dislocation/precipitate interactions at 650 1C

In order to identify the nature of the precipitate/dislocationinteractions during plastic deformation, a series of strain-controlled fully reversed cyclic loading tests were carried out, at650 1C, on low cycle fatigue specimens that have been heat treatedin the manner described in Table 1. The mechanical testing wascarried out using a servo hydraulic test machine, equipped with ahigh temperature furnace and quartz rod extensometer. Fullyreversed cyclic stress–strain tests (R¼−1) are performed, at astrain rate of 1�10−5 s−1, with strain ranges varying from 70.6to 70.9%. Cyclic stress–strain curves for the various microstruc-ture conditions are shown in Fig. 2.

In order to analyze the deformation modes in term of disloca-tion mechanisms, transmission electron microscopy (TEM) inves-tigations were carried out by means of Titan FEI TEM operating at300 kV. TEM discs were prepared using a twin jet polisher, in anelectrolyte of 10% perchloric acid in methanol, of the as-receivedand water quenched microstructure (as described in Table 1).These samples were cut from low cycle fatigue specimens, whichhave been cyclically loaded at a strain rate of 10−5 s−1 and strainrange of 70.08% at 650 1C, perpendicular to the loading axis. Thedeformation substructure of the as-received material is shown inFig. 3 and the water quenched microstructure is shown in Fig. 4.

The as-received substructure shows dislocations which arecurved and often blocked against secondary γ′ precipitates whichindicates high activation of the Orowan by-passing mechanism.While, there is some evidence of isolated shearing of particles,designated by S in Fig. 3, the Orowan by-passing mechanism is theprevailing one in the as-received material and is linked to the γ′particles size and their volume fraction which control the γ channelwidth. Similar deformation mechanisms were observed in failedspecimens after dwell fatigue on U-720 and was related to theaccumulated plastic deformation [18] and after creep tests in NR3[19]. On the other hand, in the water quenched microstructure, theγ′ precipitates seem to be sheared by supershockley dislocations a/3(112) type (designated by Sh in Fig. 4a) leading to the stacking faultformation appearing as contrasted fringes between dislocations.This mechanism is described in References [20–22], as two super

orted as mean equivalent particle radius).

Aging rγ0t (nm) rγ0s (nm)

732 1C/8 h/air cool 4.7 72.1t 1 1C/min 732 1C/30 h/air cool 10.6 79.3

732 1C/8 h/air cool 4.7 37.9

10 m1 m 1 m

10 m 10 m1 m

50 nm

m

Fig. 1. Secondary electron micrographs (SEM) of the as-received IN100 microstructure, swabbed with AG-21 etchant for 15 s, showing (a) γ′p located along the grainboundaries and (b) γ′s in the matrix. (c) Transmission electron micrograph of the as-received IN100 showing details of the γ′t. Similarly, (d and e) are the overaged materialand (f and g) are the water quenched material, corresponding to those listed in Table 1.

Strain (mm/mm)

Stre

ss (M

Pa)

-1500

-1000

-500

0

500

1000

1500

Strain (mm/mm)

Stre

ss (M

Pa)

-1500

-1000

-500

0

500

1000

1500

Strain (mm/mm)-0.010 -0.005 0.000 0.005 0.010 -0.010 -0.005 0.000 0.005 0.010 -0.010 -0.005 0.000 0.005 0.010

Stre

ss (M

Pa)

-1500

-1000

-500

0

500

1000

1500

Fig. 2. Experimental cyclic stress-strain loops obtained a strain rate of 1�105 s−1 at 650 1C, with strain ranges varying from 70.6 to70.9% for the three conditions,(a) as-received, (b) overaged and (c) water quenched IN100 microstructures, see Table 1.

200 nm 200 nm

S

S

S

S

Fig. 3. Post mortem observations of the as-received material showing the activity of Orowan by-passing mechanism: (a) bright field image at low magnification, (b) darkfield image showing curved dislocations surrounding γ′ particles and some isolated γ′ particles shearing (S).

K. Maciejewski et al. / Materials Science & Engineering A 582 (2013) 47–54 49

200 nm 100 nm

Fig. 4. Post mortem observations of the water quenched material showing the activity of particles shearing mechanism: (a) two different systems are activated (principal SP)and secondary system (SS), supershockley dislocations a/3(112) type (Sh) and some isolated Orowan by-passing (Or), (b) Zoom in showing the secondary γ′ shearing (SII) andthe tertiary γ′ shearing (SIII).

Table 2Experimentally obtained yield stress (0.2% offset) and CRSS values for microstruc-tures corresponding to those listed in Table 1.

HT condition s0:2%y (MPa) τ0:2%CRSS (MPa)

As-received 1115 364Overaged 1080 353Water quenched 1180 386

K. Maciejewski et al. / Materials Science & Engineering A 582 (2013) 47–5450

partial dislocations combined in the γ′ precipitate to form a perfect a(011) type dislocation for L12 ordered structure. It is also worthnoticing that γ′ shearing by a/3(112) dislocations concerns bothsecondary γ′ precipitates (designated by SII in Fig. 4b) and tertiary γ′particles (designated by SIII in Fig. 4b). This complex γ′ shearing wasobserved in References [19,23] and was associated to γ′ hardeningby Tian et al. [24]. In addition, the existence of other slip systems,where particle shearing is still a preferred deformation mode(designated by SP for primary slip system and SS for secondary slipsystem in Fig. 4a), shows that this mechanism is the dominantmode in the water quenched material. Some isolated Orowan by-passing may occur (designated by Or in Fig. 4a) but as describedabove, γ′ particle shearing seems to be the prevailing deformationmechanism due to particles size and the width of the γ channels.Similar complex shearing of both γ′ particles was observed in nickelbase superalloys for disks [18,19,23].

Results of this TEM investigation show that for a constantvolume fraction, the γ′ size and as a result the γ channel width,control the active deformation modes in the IN100 alloy at 650 1C.Large γ′s size, as in the as-received microstructure, is dominated bythe Orowan by-passing mechanism, while the smaller γ′s size, as inthe water quenched material, is dominated by particle shearing.In both materials, however, the γ′t particles are sheared. Theseobserved dislocation interaction mechanisms with respect to theparticle size, will be considered in the following sections to assessthe relative contribution of the secondary and tertiary γ′ particlepopulation to the total critical resolved shear stress.

4. Critical resolved shear stress

The experimental values of yield strength of the IN100 mate-rial, were determined from the strain-controlled tests carried outat 650 1C on low cycle fatigue specimens that have been heattreated in the manner discussed in Table 1. From these tests,shown in Fig. 2, the yield stress was determined as the 0.2% offset.This stress can be written in terms of critical resolved shear stress(CRSS) as

τ0:2%CRSS ¼ s0:2%y =M ð1Þ

where the Taylor factor, M, is 3.06 [11,14]. These two stress valueswere obtained for the three microstructure conditions and arelisted in Table 2. The CRSS can be written as the sum of differentstrengthening components as [9,10]

τCRSS ¼ τss þ τγ0 ð2Þwhere τss and τγ are the contributions of solid solution and γ′

precipitates, respectively, to τCRSS . Each of these contributions arediscussed below.

4.1. Intragranular γ′ particle contribution to CRSS

The second term in Eq. (2), τγ′, is decomposed into particlescontributing to shearing and looping, expressed as

τγ0 ¼ τshearγ0 þ τloopγ0 ð3Þ

The general trend in literature shows that tertiary γ′ precipi-tates' are mainly sheared by dislocations, thus they do notcontribute to the τloopγ0 . On the other hand, secondary γ′ precipitates'can be sheared and/or looped by dislocations, depending on theirparticle size relative to a critical size. The two terms in Eq. (3)referring to CRSS due to shearing and looping can then beexpressed as

τshearγ0 ¼ τshearγ0t þ τshearγ0s ð4Þ

τloopγ0 ¼ τloopγ0s ð5Þ

where τγ′t and τγ′s are the contributions of tertiary and secondary γ′precipitates' to the CRSS, respectively. These contributions arediscussed below.

4.1.1. ShearingThe shearing mechanism considered here is shearing by weakly

coupled dislocations, as this provides a simpler mathematicaltreatment when identifying the shearing/looping transition. Inthis mechanism, the slip in the particle is accompanied by theformation of antiphase boundary and the increase in strengtharises from the energy increase. CRSS due to shearing by weaklycoupled dislocations is proportional to r and f and is given byCourtney [25] as

τshear ¼ 0:7μAPBμb

� �3=2 f rb

� �1=2

ð6Þ

Particle Radius (nm)

P(u

)

0.0

0.5

1.0

1.5

2.0

2.5

Particle Radius (nm)0 20 40 60 80 100 120 1400 20 40 60 80 100 120 140

P(u

)

0.0

0.5

1.0

1.5

2.0

2.5

Particle Radius (nm)0 10 20 30 40 50 60 70

P(u

)

0.0

0.5

1.0

1.5

2.0

2.5

Fig. 5. γ′s particle distributions, P, of (a) as-received (b) overaged and (c) water quenched IN100 microstructures, corresponding to those listed in Table 1. The particledistribution (solid line) given by Eq. (10) is superimposed over the experimentally obtained particle distributions (bars) in each figure.

K. Maciejewski et al. / Materials Science & Engineering A 582 (2013) 47–54 51

where the Taylor factor M is 3.06 [11,14], the shear modulus μ is69469 MPa, the antiphase boundary energy APB is 0.14 J/m2

[11,26], Burger's vector b is 0.253 nm [11], f is volume fractionand r is the mean particle radius.

4.1.2. LoopingThe second type of hardening occurs when a particle size is

above a critical radius and the favorable mechanism becomesdislocation looping. The CRSS due to dislocation looping isexpressed as a function inversely proportional to the channelwidth, l, which can be formulated in the following manner [27]:

τloop ¼ μb

L−2r¼ μb

lð7Þ

where L is the mean spacing between particles (measured fromcenter to center), and l is the mean channel size. The measuredmean values of l are found to be both a function of r and f whichcan be fitted into an expression written as: l¼ αrð1−f Þ=f , where αis a fitting parameter taken as 2.775. The fitting parameter isdetermined by comparing the theoretical value of l with experi-mental measurements. This expression of l is similar to othersreported in literature [27]. Substituting this expression into Eq. (7)we obtain

τloop ¼ αμbf

rð1−f Þ ð8Þ

4.1.3. Shearing/looping transitionFor a constant volume fraction of particles, Eqs. (6) and (8)

show that τshear increases with particle size and τloop decreaseswith particle size. A peak stress exists at a critical size where thetransition between particle shearing and looping occurs. Thiscritical size is obtained by equating Eqs. (6) and (8) and solvingfor size

rc ¼α

0:7

� �2=3 μb2

APBf

1−fð Þ2

!1=3

ð9Þ

This equation shows that the critical radius, rc, increases withvolume fraction, f. Using the total volume fraction's for tertiary andsecondary γ′ particles, rc is calculated to be 35.3 nm and 63 nm,respectively. This critical size will be used to determine thefractions of particles with r less and greater than rc. In order todefine these volume fractions, the secondary γ′ particles areassumed to follow the classical Lifshitz–Sloyozov–Wagner (LSW)type of distribution, that assumes the particles which have beencoarsened during the heat treatment have reached a steady state

distribution, P given by Ardell as [28,29]

P r; rð Þ ¼ P uð Þ ¼4u2

93

3þu

� �7=3 −3=2u−3=2

� �11=3exp u

u−3=2

� �for uo1:5

0 for u≥1:5

8<:

ð10Þwhere r is the radius random variable, r is the mean radius and u isthe normalized; i.e., r/r. The mean particle size, as determined as:

r¼ ∑N

i ¼ 1ri=N ð11Þ

where N is the total number of particles in the distribution P:

N¼Z 1:5

0PðuÞ du¼

Z 1:5r

0Pðr; rÞ dr ¼ 1 ð12Þ

and the summation of r is represented by

∑N

i ¼ 1ri ¼

Z 1:5r

0rPðr; rÞ dr ð13Þ

The maximum limit of the distribution is at u¼1.5 whichcorresponds to r¼1.5r. The γ′s particle distributions, P, for thethree microstructure conditions are shown in Fig. 5. Details ofconstructing these particle distribution plots are given in Refer-ence [30]. In the current analysis, the secondary and tertiary γ′particle populations are treated as two separate distributions, thusallowing for a simpler mathematical calculation of the distributionparameters. In this case, when considering the critical particleradius, rc, which falls within the limits of this distribution(0orco1:5r) and identifies the transition between shearing andlooping, then the volume fraction of particles which are sheared isf shear (0ororc ) and the volume fraction of particles which arelooped is f loop (rcoro1:5r). These fractions can be calculated as,see [31]

f shear ¼ f

R rc0 r3Pðr; rÞ drR 1:5r0 r3Pðr; rÞ dr

ð14Þ

f loop ¼ f

R 1:5rrc

r3Pðr; rÞ drR 1:5r0 r3Pðr; rÞ dr

ð15Þ

The mean size of particles which belong to the portions f shear

and f loop are defined as

rshear ¼R rc0 rPðr; rÞ drR rc0 Pðr; rÞ dr ð16Þ

rloop ¼R 1:5rrc

rPðr; rÞdrR 1:5rrc

Pðr; rÞdrð17Þ

K. Maciejewski et al. / Materials Science & Engineering A 582 (2013) 47–5452

If rc41.5r, then the entire particle population is sheared andthe corresponding rloop and f loop are zero and therefore, τloopγ0t ¼ 0.Thus, the first term in Eq. (4) is represented by

τshearγ0t ¼ 0:7μAPBμb

� �3=2 f γ0trγ0tb

� �1=2

ð18Þ

where rγ0t and fγ′t are the mean particle size and volume fractionfor the γ′t precipitates. On the other hand, when shearing andlooping exist simultaneously (0orco1.5r), Eqs. (6) and (8) areredefined as

τshear ¼ 0:7μAPBμb

� �3=2 f shearrshear

b

!1=2

ð19Þ

τloop ¼ αμbf loop

rloop 1−f loop� � ð20Þ

This is the case for the γ′s precipitates, thus the terms inEqs. (4) and (5) are then defined as

τshearγ0s ¼ 0:7μAPBμb

� �3=2 f shearγ0s rshearγ0s

b

!1=2

ð21Þ

τloopγ0s ¼ αμbf loopγ0s

rloopγ0s ð1−f loopγ0s Þð22Þ

The relative size and volume fraction of γ′s particles which aresheared and looped by dislocations are calculated using Eqs. (14)–(17) relative to rc¼63 nm calculated by Eq. (9). These values arereported in Table 3 for the three microstructures. The criticalparticle size and the particle distribution for the γ′s particlepopulation for the three microstructure conditions are shown inFig. 6. Fig. 6a and b shows that rc falls within the distribution, thusshearing and looping exist simultaneously for the as-received andoveraged microstructures. On the other hand, Fig. 6c shows that allγ′s precipitates for the water quenched microstructure are shearedby dislocations. Substituting the size and volume fraction of theγ′t particles into Eq. (18) and the relative size and volume fractionof γ′s particles into Eqs. (21) and (22) yields the CRSS due to γ′t andγ′s respectively. The summation of these terms, expressed byEqs. (3), (4) and (5), represent the total contribution of tertiaryand secondary γ′ particles to the CRSS, as shown in Table 4 for thethree microstructure conditions. Using the total volume fraction ofγ′t and γ′s particles, the CRSS components due to particle shearingand looping are calculated using Eqs. (6) and (8) for a range ofparticle radii between 0 and 200 nm. These are plotted againstparticle radius in Fig. 7, where the dashed (γ′t) and solid (γ′s) linesare for volume fractions of 7.5% and 26.7%, respectively. Inaddition, the values for τγ′t and τγ′s, as defined in Table 4, are alsoplotted in Fig. 7. This figure indicates that for a volume fraction of26.7% (corresponding to γ′s precipitates), particles with ro63 nmare sheared by dislocations, while particles with r463 nm are

Table 3The mean size (rγ0s) and volume fraction (f γ0s), as well as, the average secondary γ′

particle size and volume fraction contribution to both shearing and looping as

defined by rshearγ0s , rloopγ0s , f shearγ0s , and f loopγ0s for the as-received, overaged and waterquenched microstructures, corresponding to those listed in Table 1.

HT Condition rγ0s(nm)

f γ0s(%)

rshearγ0s

(nm)f shearγ0s

%rloopγ0s

(nm)f loopγ0s

(%)

As-received 72.1 26.7 50.1 2.2 79.2 24.6Overaged 79.3 26.7 49.8 1.1 85.2 25.7Waterquenched

37.9 26.7 37.9 26.7 0 0

looped by dislocations. Similarly, for a volume fraction of 7.5%(corresponding to γ′t precipitates), the critical radius is calculatedas 35.3 nm. For the three microstructure conditions, all γ′t pre-cipitates are sheared by dislocations. This observation is consistentwith the deformation mechanisms described in Section 3 throughTEM analysis of two LCF tested microstructures. Fig. 6c shows thatthe water quenched microstructure has rorc for the entire γ′spopulation, thus, the CRSS due to γ′s arises solely due to shearing,as shown in Fig. 7. This is consistent with the TEM observations inSection 3 for the water quenched microstructure, showing that thedeformation mechanism concerning the γ′s particles was domi-nated by particle shearing. Fig. 6a and b show that the as-receivedand overaged microstructures have rc that falls within the dis-tribution P, thus shearing and looping exist simultaneously. For themean particle sizes 72.1 nm and 79.3 nm, the value of CRSS due toparticle looping calculated by Eq. (8) is less then τγ′s calculated byEqs. (21) and (22), represented by the solid line and symbol inFig. 7, respectively, indicating the presence of dual deformationmechanisms operating. This is consistent with the TEM observa-tions for the as-received material revealed that the dominantdeformation mode is Orowan by-passing, but there was alsoevidence of isolated shearing of γ′s particles. In addition, Table 3shows the floop is much greater than fshear for these two micro-structures, thus supporting that both mechanisms exist simulta-neously, but looping is dominant.

4.2. Solid solution contribution to CRSS

The first term in Eq. (2), τss, represents the collective effect ofsolid solution, carbides, γ′p particles, as well as, the stress con-centration that builds up at boundaries. In this work, no changeshave affected the grain size or γ′p precipitate statistics in the threemicrostructure conditions. Thus, their relative hardening contri-butions are assumed to be constant. In addition, Milligan et al. [10]have shown that the γ′p particles are of comparable yield strengthto the matrix. The solid solution hardening of the matrix is basedon an approach given by Gypen and Deruyttere [32,33] whichassumes the hardening stress increases as a function of soluteconcentration [34,35]

τss ¼ 1M

∑i

ds

dffiffiffiffiffiCi

p ffiffiffiffiffiCi

pð23Þ

where C is concentration of solute i and ds=dffiffiffiffiffiCi

pis the strength-

ening coefficient of solute i. Roth et al. [35] determined thestrengthening coefficients for various elements based on the workof Mishima et al. [36] in which they studied binary Nickel alloyswith varying atomic percents of solutes. Table 5 shows thestrengthening coefficients for the elements which contribute tostrengthening of the gamma matrix, along with the chemicalcomposition of the gamma matrix. This value was measured at0.2% flow stress at 77K (−196.15 1C) by Mishima and coworkers[36]. This coefficient describes the strain induced in the lattice dueto an added impurity. Using the chemical composition andstrengthening coefficients given in Table 5 in Eq. (23) yields avalue of solid solution strengthening of 179 MPa which representsτss at −196.15 1C (77 K). Substituting the experimentally deter-mined CRSS and the values of τγ′ (see Table 4) into Eq. (2) andsolving for τss, results in an average value of 73 MPa. This valuewould represent the solid solution strengthening at 650 1C(923.15 K). This variation of τss with temperature can be fit to theFeltham equation [37]

Tτssμ

� �1=2

¼ K1−K2τss ð24Þ

where T is absolute temperature, μ is shear modulus, K1 and K2 areconstants. K2 is taken as 0.26 K/MPa from the work of Roth et al.

Particle Radius (nm)

P(u)

0.0

0.5

1.0

1.5

2.0

2.5r < rcr > rc

Particle Radius (nm)

P(u)

0.0

0.5

1.0

1.5

2.0

2.5r < rcr > rc

Particle Radius (nm)0 20 40 60 80 100 120 0 20 40 60 80 100 120 140 0 10 20 30 40 50 60

P(u)

0.0

0.5

1.0

1.5

2.0

2.5

r > rc

r < rc

Fig. 6. Secondary γ′ particle distributions, P, for the (a) as-received, (b) overaged and (c) water quenched IN100 microstructures, showing the relative portion of particles lessthan and greater than the critical size, rc.

Table 4CRSS contributions due to tertiary and secondary γ′ particles for microstructurescorresponding to those listed in Table 1.

HT condition τγ′t (MPa) τγ′s (MPa) τγ′ (MPa)

As-received 41 273 314Overaged 61 249 310Water quenched 41 219 259

' Size, r (nm)0 50 100 150 200

0

50

100

150

200

250

300

's

's

't

't

' (MP

a)

Fig. 7. CRSS of γ′ particles as a function of particle size. Experimental values(symbols) of CRSS for γ′t and γ′s, as defined in Table 4, compared with thosecalculated using Eqs. (6) and (8) for volume fractions of 7.5% and 26.7% correspond-ing to the γ′t (dashed line) and γ′s (solid line) particle populations, respectively.

K. Maciejewski et al. / Materials Science & Engineering A 582 (2013) 47–54 53

for Nickel alloys [35], assuming that the arc length of a disloca-tion segment does not vary with solute concentration. Theconstant, K1, is dependent on solute concentration and interac-tion energy between a dislocation and solute atom. In this work,the composition of IN100 is constant, thus for the three micro-structure conditions, K1 is constant. K1 was determined bysubstituting the values of K2, μ, and τss at both 77 K and923.15 K into Eq. (24). This yields an average value ofK1¼49.7 K. Using these constants, the τss can be plotted as afunction of temperature, as shown in Fig. 8. From this analysis, τsscomputed using Eq. (24) provides values of 176 MPa and74.7 MPa at 77 K (−196.15 1C) and 923.15 K (650 1C), respectively.These values are comparable to those obtained previously.Substituting the values for τss and τγ′, into Eq. (2) yields the totaltheoretical CRSS, of 389, 385 and 334 MPa for the as-received,overaged and water quenched microstructures, respectively. Thevalidity of this sum is compared with that obtained

experimentally, with maximum error of 15.6%, which is consid-ered reasonable for the wide range of γ′s.

5. Conclusions

The influence of γ′ size on CRSS in alloy IN100 has beenexamined at 650 1C by considering dislocation/precipitate interac-tion mechanisms. A series of heat treatments were carried out onsmooth specimens of the IN100 alloy in order to vary thesecondary and tertiary γ′ sizes, while maintaining their respectivevolume fractions. These specimens were subjected to strain-controlled fully reversed cyclic loading at 650 1C followed byTEM examinations to identify the nature of the precipitate/dislocation interactions during plastic deformation. Results indi-cated the presence of both shearing and Orowan by-passing, ofwhich the dominance of each depends on the γ′ size and γ channelwidth in relation to a critical particle size. These observations havebeen used as a basis to calculate the CRSS as a sum of componentscontributed by solid solution and by particles being sheared andlooped. The main conclusions of this study concerning the P/MIN100 are summarized as follows:

-

Selected heat treatments are carried out to achieve changes inthe mean particle size of γ′s resulting in variations ranging from37 nm to 79 nm. The resulting particle size distributions inthese heat treated microstructure are fit to the classical LSWtype of distribution which indicates that the particles whichhave been coarsened during the heat treatment have reached asteady state condition.

-

TEM examination of IN100 specimens, with a constant γ′volume fraction, subjected to cyclic deformation at 650 1C hasshown that dislocations/γ′ interactions occur by both theshearing and Orowan by-passing and that these two mechan-isms could exist simultaneously. The occurrence of either ofthese two mechanisms depends on the precipitate size withrespect to a critical particle size. The as-received substructure,with large γ′s size, showed high activation of the Orowan by-passing mechanism with some evidence of isolated shearing.The water quenched microstructure with a smaller γ′s sizerelative to the as-received, showed that γ′ shearing by a/3(112) dislocations concerns both secondary γ′ and tertiary γ′particles. This analysis has also shown that in both materials,the γ′t particles are sheared.

-

The secondary and tertiary precipitate populations are treatedas two separate distributions with a specified volume fractionfor which an average particle size and critical shear/loopingtransition size are calculated. For a volume fraction of 26.7% forγ′s and 7.5% for γ′t, a critical particle size of 63 nm and 35.3 nmhas been determined, respectively. The critical particle size

Temperature (K)0 200 400 600 800 1000

τ ss (

MP

a)

50

75

100

125

150

175

200

Fig. 8. Solid solution strengthening component as a function of temperature,calculated using Eq. (24) (solid line) compared with experimental values (symbols)of τss, at 77 K (179 MPa) and 923.15 K (73 MPa).

Table 5Strengthening coefficient [35] and chemical composition [34] of the gamma matrix in IN100.

Ni Al Ti Co Cr Mo V

Strengthening coefficient MPa√frac 225 775 39 337 1015 408Chemical composition of γ wt% Bal 2.25 0.93 27.8 24.5 5.73 0.05

K. Maciejewski et al. / Materials Science & Engineering A 582 (2013) 47–5454

defines the portions of the secondary γ′ volume fractions thatare sheared or looped. In this work all tertiary γ′ particles areconsidered to be sheared since their sizes are less than thecorresponding critical one. This is supported by the fact that nolooping of γ′t has been observed in the TEM examinations.

-

The CRSS of IN100 at 650 1C has been calculated as the sum ofhardening components associated with solid solution and γ′precipitates. The solid solution contribution is calculated as afunction of solute concentration and temperature for the γmatrix compares well with that obtained experimentally. The γ′CRSS components are contributed by both γ′s and γ′t statistics.This is formulated in terms of both the shearing and loopingmechanisms. The CRSS being the sum of contributions fromsolid solution, γ′s and γ′t compared well with the CRSS experi-mentally obtained at 650 1C.

Acknowledgments

The authors acknowledge support provided by Dr. AgnieszkaWusatowska-Sarnek of Pratt & Whitney, East Hartford, Connecti-cut, during the heat treatment work in this study.

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