Arising from long-term high temperature service, the microstructure of nickel-base (Ni-base) superalloy components undergoes thermally and deformation-induced aging characterized by isotropic coarsening and directional coarsening (rafting) of the $\gamma \u2032$ precipitates. The net result of the morphological evolutions of the $\gamma \u2032$ particles is a deviation of the mechanical behavior from that of the as-heat treated properties. To capture the influence of a rafted and isotropic aged microstructure states on the long-term constitutive behavior of a Ni-base superalloy undergoing thermomechanical fatigue (TMF), a temperature-dependent crystal viscoplasticity (CVP) constitutive model is extended to include the effects of aging. The influence of aging in the CVP framework is captured through the addition of internal state variables that measure the widening of the *γ* channels and in-turn update the material parameters of the CVP model. Through the coupling with analytical derived kinetic equations to the CVP model, the enhanced CVP model is shown to be in good agreement when compared to experimental behavior in describing the long-term aging effects on the cyclic response of a directionally solidified (DS) Ni-base superalloy used in hot section components of industrial gas turbines.

## Introduction

Airfoils used in gas turbine engines for aircraft and power generation are often made from single crystal (SX) and directionally solidified (DS) nickel-base (Ni-base) superalloys. Ni-base superalloys are the dominant material for this application due to their excellent mechanical properties and corrosion resistance at temperatures in excess of half their melting temperatures. In the case of modern cast Ni-base superalloys, their superior mechanical properties are derived from the combination of the face-centered cubic solid solution strengthened *γ* matrix and coherent face-centered cubic intermetallic $\gamma \u2032$ strengthening precipitates that comprise 50–70% of the alloys volume fraction. In the fully heat-treated form, the $\gamma \u2032$ precipitates take on a cubic geometry [1–3] and are the attributable reason for the anomalous yield behavior in Ni-base superalloys.

While the $\gamma \u2032$ is coherent with the *γ* matrix, coherency stresses between the two phases do exist, that with the application of an external stress at temperatures in excess of 850 °C causes directional coarsening or rafting of the $\gamma \u2032$ [4]. Further, the manner in which the $\gamma \u2032$ precipitates coarsen is dependent on both the sign of the applied stress and lattice mismatch $\delta =2(a\gamma \u2032\u2212a\gamma /a\gamma \u2032+a\gamma )$. In the case of negative lattice misfit alloys, of which modern Ni-base superalloys are, the application of a tensile stress results in the $\gamma \u2032$ directionally coarsening normal to the applied stress (N-rafts), and in the case of a compressive stress the $\gamma \u2032$ aligns parallel to the applied stress (P-rafts) [5]. Further, after long-term exposure the $\gamma \u2212\gamma \u2032$ phases undergo topological inversion, where the $\gamma \u2032$ phase entirely encases the former *γ* matrix. Overall, the morphological changes in the $\gamma \u2032$ have the net result of influencing the mechanical behavior of Ni-base superalloys and cause deviations from the idealized as-heat treated material.

To date, few attempts have been made at developing constitutive models that account for the effects of microstructure evolution on the constitutive response of Ni-base superalloys. Of the microstructure-sensitive models reported [6–8], all have shown microstructure-sensitivity on a limited basis. The advantage of constitutive models that are able to predict the evolution of the constitutive response with aging is that the entire service behavior can be simulated. This paper presents a set of viscoplastic constitutive equations within the crystal plasticity framework. This model accounts for aging of the $\gamma \u2032$ microstructure through linking an aging kinetics model to the crystal plasticity equations. To be shown is the applicability of the combined crystal plasticity model and microstructure sensitivity to predicting the thermomechanical fatigue (TMF) behavior of the alloy.

## Experimental

The Ni-base superalloy utilized for in this work for calibration of the microstructure sensitive crystal viscoplasticity (CVP) model is a second-generation DS Ni-base superalloy of composition given in Table 1. In the standard heat-treated state, the alloy is comprised of cuboidal $\gamma \u2032,\u20090.7\mu m$ in length as shown in Fig. 1(a). The $\gamma \u2032$ accounts for approximately 70% of the volume fraction. Further, the lattice misfit, *δ*, is negative over the entire temperature range of interest (20–950 °C). The lattice misfit was calculated with JMatPro v8 (Sente Software Ltd., Surrey, UK). As a result, an N-raft microstructure will form under pure tensile loadings and a P-raft microstructure under pure compressive loads.

The effects of the artificial aging treatments upon the microstructure of the material used in the cyclic fatigue calibration experiments are summarized in Table 2. Stress-free exposure increased the $\gamma \u2032$ cube size to 0.85 *μ*m on average. The tensile precreep treatment resulted in a fully N-rafted structure, where the average width of the *γ* channels perpendicular to the applied load was increased from 0.18 to 0.3 *μ*m. The compressive precreep resulted in the *γ* channels parallel to the applied load having a width of 0.25 *μ*m as depicted in Fig. 1(d). In all cases, the microstructure measures were determined through two point correlations statistics as reported in Ref. [9]. While similar magnitudes of time and stress were applied to the material while creeping in both tension and compression, the differences in channel width suggest the existence of a tension/compression asymmetry as reported for other DS and SX Ni-base superalloys [10].

### Cyclic Fatigue.

Isothermal calibration experiments were performed on cylindrical dog-bone specimens having a gage section length of 13.2 mm (0.52 in) and diameter of 6.35 mm (0.25 in) made from a commercial DS Ni-base superalloy of chemical composition given in Table 1. The calibration experiments were performed on a 44.5 kN (10 kip) MTS closed-loop servo-hydraulic test frame controlled by a MTS FlexTest 40 controller. Strain feedback was provided by a 12.7 mm (0.5 in) gage length MTS high-temperature extensometer. Specimen heating was controlled by a closed-loop Ameritherm 2 kW induction heating system utilizing a 26 gage type-K thermocouples spot welded within the gage section of the specimen. A set of isothermal calibration fatigue experiments was conducted for each microstructure following the temperatures and material orientations given in Table 3.

The test matrix is based on previously reported protocols [11] and represents the minimum number of experiments required to fully calibrate the CVP model across the 20–1050 °C temperature range for each of the given microstructures. Contained within each of the calibration experiments are blocks of five cycles at strain rates 10^{−3} and $10\u22124\u2009(1/s)$ as illustrated in Fig. 2, respectively, which captures the fatigue regime material response, followed by a single compressive strain hold to capture the creep-like relaxation, and closing with a creep-fatigue (CF) interaction cycle at a strain rate of $10\u22125\u2009(1/s)$. Further details on the calibration experiments can be found in Ref. [11]. Within the strain rate-sensitive temperature regime (*T* ≥ 750 °C), most Ni-base superalloys become cyclically stable within a short period, as such five cycles are deemed sufficient to capture the stable response at a given strain rate [12]. For temperatures below 750 °C where the cast high $\gamma \u2032$ Ni-base superalloys are rate independent, conducting the experiment at a constant strain rate $(10\u22122\u2009(1/s))$ until a stabilized hysteresis is obtained is acceptable. With the given strain ranges, this typically occurs within the first few hundred cycles.

Isotropic coarsening of the microstructure was undertaken through placing specimen within a Lindberg Blue tube furnace at 1000 °C for a period of 720 h. A tensile precreep treatment at 950 °C for 300 h and 150 MPa was used for generating the N-raft microstructure state. Artificial aging of the material under compressive-creep conditions was done with a special purpose built frame discussed elsewhere [9]. In this case, the material was exposed to compressive loads of 150 MPa for 300 h periods.

### Kinetics Experiments.

To derive the information necessary for calibration of the isotropic coarsening component of the rafting model, material was thermally exposed in open air for periods ranging between 96 and 672 h at temperatures of 900, 950, and 1000 °C under stress free conditions. Below 900 °C, the rate of coarsening was too slow for there to be measurable effects on the microstructure for the given exposure times. To determine the stress-assisted aging kinetics, specimens of a tapering hour glass geometry (Fig. 3) were used under tensile creep according to the schedule in Table 4.

## Material Model

### Crystal Viscoplasticity Constitutive Formulation.

where *β*(*T*) is the coefficient of thermal expansion of the material as a function of temperature, and $F\theta n$ and $F\theta n+1$ are the thermal strain deformation gradients and the beginning and end of a finite time–step, respectively. *T* is the temperature at the end of the cycle and *T _{o}* is the temperature at the beginning of the thermal cycle.

where $Lp$ is the material time derivative of $Fp$, and $Fp\u22121$ is the inverse of $Fp$ and $so\alpha $ is a unit vector in the slip direction, $mo\alpha $ is a unit vector in the direction normal to the slip plane, *α* is a slip system, and $\gamma \u02d9\alpha $ is the inelastic shear strain rate on slip system *α* and *N*_{slip} is the number of slip systems. In the case of Ni-base superalloys, a homogenized face-centered cubic structure of 12 octahedral ${111}\u27e8110\u27e9$ and six cubic ${100}\u27e8110\u27e9$ slip systems is assumed, with cubic slip systems only activated above 750 °C [13].

*χ*is the back stress and

^{α}*κ*is the threshold stress on the

^{α}*α*slip system. The drag stress,

*D*, is a measure of the flow potential of the material and effectively accounts for the component of flow resistance than can be overcome by thermal fluctuation. The reference inelastic shear strain rate, $\gamma \u02d9o$, is related to the maximum strain rate associated with the limit on dislocation velocity. The rate-insensitive regime is set by

^{α}*B*and

_{o}*n*

_{2}collectively, while the rate sensitive or creep regime is determined by

*n*

_{1}. The temperature-dependent diffusivity function, Θ(

*T*), typically of a Arrhenius form is defined later.

*μ*is the shear modulus at 0°K. The back stress evolution equation is a Frederick–Armstrong relation with modifications to account for temperature changes [18]

_{o}The back stress is associated with the evolution of the dislocation structure within the material and has a temperature rate term to allow for consistency in temperature-dependent loadings [20]. The first term represents hardening due to dislocations piling up on obstacles while the second captures the dynamic recovery, which is the reduction in dislocation density due to annihilation of dislocations of opposite sign. As a note, $R\chi =(h\chi /h\chi d)$ represents the saturated value of the back stress. The third term accounts for the evolution of the back stress due to changes in temperature and the last term accounts for the static thermal recovery of the back stress during creep like loadings, which is important for long-term dwells.

*κ*captures the temperature-dependent resistance to dislocation movement. The last three terms of Eq. (9) capture the non-Schmid effects that result in an increased resistance of dislocation movement through the $\gamma \u2032$ precipitates prior to dislocation looping, with the strongest effects occurring in the neighborhood of 750 °C [22]. Additionally, the functional form of Eq. (9) captures the anomalous increase in yield strength as temperature is increased [21,23,24]. The $\kappa e\alpha $ term in Eq. (8) evolves according to a competition between dislocation storage,

_{o}*h*, rearrangement,

_{κ}*h*, and annihilation mechanisms,

_{κd}*h*

_{κs}where *Q _{o}* is the activation energy for thermally activated dislocation bypass of obstacles having a value similar to the self-diffusion of nickel in the alloy,

*R*is the universal gas constant, and

*T*is the absolute melting temperature of the alloy.

_{m}To account for effect of intergranular interactions, the Taylor constraint whereby grains of different orientations are assumed to undergo identical deformation gradients is used. Further details on the implementation of the Taylor constraint in the CVP model framework can be found in the following Refs. [9] and [13].

$\gamma -\gamma \prime $ Microstructure Evolution.

*γ*matrix are inherently linked to one another such that dimensional changes in the $\gamma \u2032$ precipitates result in changes the

*γ*matrix. This has been experimental observed and quantified by Epishin et al. [24]. As a result, the $\gamma \u2212\gamma \u2032$ microstructure can be represented as a periodic unit cell as illustrated in Fig. 4, where the periodicity

*λ*in each of the primary crystallographic directions (

*i*= [001], [010], or [100]) is given by

where *w _{i}* is the

*γ*channel width, and $\gamma \u2032$ cube thickness is given by

*h*. Equation (12) holds true so long as topological inversion of the $\gamma \u2212\gamma \u2032$ structure has not occurred. Further, the [001] direction is assumed to be the principal frame.

_{i}*γ*channels width is given by sum of the individual kinetics processes occurring

where $w\u02d9i$ is the cumulative rate of channel widening, $w\u02d9icoar$ is the rate of *γ* channel widening associated with isotropic coarsening, and $w\u02d9iraft$ is the rate of *γ* channel widening associated with rafting. The directional coarsening mechanism occurs as a result of cross-diffusion of *γ* and $\gamma \u2032$-forming atoms between the horizontal (vertical) *γ*-channels, which become wider, and vertical (horizontal) *γ*-channels, which shrink and finally disappear under tensile (compressive) loads. Isotropic coarsening in Ni-base superalloys during rafting occurs as a result of Ostwald ripening was shown to occur [2]. Neglecting the isotropic coarsening component of Eq. (13) has been shown to result in a 20% under prediction in the amount of coarsening that occurs in the $\gamma \u2212\gamma \u2032$ microstructure [24,26–28].

#### Isotropic Coarsening.

*K*is the isotopic coarsening constant,

_{o}*Q*

_{coar}is the activation energy for isotropic coarsening,

*R*is the ideal gas constant, and

*T*is the temperature. Further, during isotropic coarsening a proportional growth relationship within the unit-cell can be assumed maintained [26]

#### Directional Coarsening.

*γ*channel widening when under uniaxial tensile loading was the relation

where *σ*_{VM} is the von Mises yield stress which is used as a quantification of the rate of change of the precipitate dimensions and $\sigma idev$ is the diagonal components of the deviatoric stress tensor, which when normalized by the von Mises stress defines the relative growth of the $\gamma \u2032$ precipitates in the noted dimension *i*, *Q*_{raft} is the activation energy required for rafting, and *U*(*T*) is the activation volume at a temperature *T*.

where *T _{o}* is the cutoff temperature below which rafting does not occur, and n and

*U*are model parameters. Equation (19) is based on rafting experiments conducted on the Ni-base superalloy CMSX-4 in the temperature range 850–1050 °C over the stress range 50–350 MPa [25]. In replacing the uniaxial stress used by Epishin et al. [31] in their derivation, Tinga et al. [8] utilized experimental findings showing strong correlation between the equivalent stress and the degree of rafting and used the von Mises stress as a means to quantify the rate of change of the precipitate dimensions. The deviatoric stress serves the purpose of defining the relative $\gamma \u2032$ precipitate growth in each direction.

_{T}Additionally, Epishin et al. [25] observed rafting kinetics operated until the *γ* channels parallel to the applied uniaxial load had widened to a value roughly three times the initial channel width (3*w _{o}*) independent of the applied stress. After attaining a channel thickness of 3

*w*, coarsening of the $\gamma \u2032$ was once again controlled by isotropic coarsening. Shown in Fig. 5 are the fits to experimental data obtained for the rate of gamma channel widening of the alloy used in this study.

_{o}### Microstructure-Sensitive Parameter Formulation.

Experimentally, the impacts of directional and isotropic coarsening of the $\gamma \u2032$ precipitates on the mechanical behavior are most pronounced on the yield strength of the material (Fig. 6) and subsequently the hardening behavior once plastic deformation has occurred [8,9,32]. Typical is the yield strength of the as-heat treated material achieves the greatest yield strength across the 20–950 °C temperature range, with the directionally coarsened microstructures being bound by the as-heat treated and overaged cubodial $\gamma \u2032$ microstructures as illustrated in Fig. 6. Interestingly, once stabilized under LCF conditions, the stress state of the aged microstructures have been reported to converge to that of the as-heat treated material [32,33]. However, as reported in these instances, the life of the aged material is inferior to the as-heat treated material [32,33]. As a result of the above-noted experimental observations, modifications to the functional form of the material parameters that control the threshold stress (*κ _{o}*), and the evolution of the backstress (

*R*and

_{χ}*h*) were carried out.

_{χ}where $h[001]o$ and $h[100]o$ are the initial widths of the *γ* channels in the [001] and [100] crystallographic reference frame. And *h*_{[001]} and *h*_{[100]} are the widths of the *γ* channels in the [001] and [100] crystallographic directions at some time *t* after thermomechanical exposure. Internal state variables similar to Eqs. (23) and (24) and have been used previously by Fedelich et al. [34] for identifying the microstructure state in a single dimension. Due to the relationship the dimensions $\gamma \u2032$ precipitate share, only two of the three dimensions are needed to establish the state of the $\gamma \u2032$ morphology. In the case of N-rafts, the expectation is that the *γ* channel increases in width in the [001] direction and contract in the [100] direction, hence *η* > *ζ*. Likewise, for P-rafts the *γ* channel to increase in width in the [001] direction, *ζ* > *η* as shown in Fig. 4. For stress-free coarsening *η* = *ζ* > 1. In the initial as-heat treated configuration, the widths of all *γ* channels are assumed equivalent, i.e., $w[001]o=w[100]o=w[010]o$.

*κ*,

_{o}*R*, and

_{χ}*h*are formulated as multiplicative polynomials that are dependent on temperature and each of the microstructure variables

_{χ}*c*(

_{ζ}*T*) and

*c*(

_{η}*T*) are polynomials of the form $c\zeta (T)=a0+a1T+a2T2$, where

*a*

_{0},

*a*

_{1}, and

*a*

_{2}are constants. In the case of the transverse orientation, a linear interpolation is utilized.

## Implementation and Calibration

The CVP model utilized in this work is an adapted version of the CVP model implemented as an abaqus user material (UMAT) subroutine in the fortran programming language by McGinty [35] and modified by Shenoy [13,36] for the DS Ni-base superalloy GTD-111. Further details on the implementation are discussed elsewhere [11,13,35].

Following the experimental and calibration protocols reported by Kirka et al. [11], the temperature and microstructure dependent parameters are developed by fitting the CVP model to isothermal data sets discussed in the experimental methods section. The stabilized hysteresis behavior at each strain rate was utilized for fitting the CVP model parameters. The resultant material parameters for the microstructure sensitive CVP model are listed in Tables 5–11 with the corresponding *η* and *ζ* values for each microstructure given previously. The calibration fits for polynomials associated with Eqs. (25)–(27) are given in Table 9.

Reflected in the calibrated material parameters are several trends. The elastic constants are independent of microstructure. This is largely consistent with the basis that the elastic modulus of Ni-base superalloys depends primarily on the $\gamma \u2032$ volume fraction and overall alloy chemistry. At temperatures below the anomalous yield regime (*T* ≤ 750 °C), the material parameters are less sensitive to the aged microstructures; this can be attributed to the $\gamma \u2032$ precipitates being easily sheared by dislocation pairs. In addition, the material parameters are insensitive to microstructure at 1050 °C; this can be attributed to the isotropic flow behavior of Ni-base superalloys at very high temperatures.

The CVP model fits of the experimental isothermal hysteresis curves are shown in Figs. 7–10. Generally, good agreement is obtained between each of the experimental calibration curves and the CVP model for each of the microstructures and different temperatures. In the case of the isotropically coarsened, N-raft, and P-raft microstructures, all exhibit deviations at 750 and 850 °C from the experimentally measured behavior. Within the 750–850 °C temperature window, Ni-base superalloys experience a change in deformation mechanism associated with anomalous yielding which is not directly captured by the CVP model framework [21]. Discussed elsewhere is the calibration of the model to the off-axis constitutive behavior of the material, as the focus here is placed on the construct of microstructure sensitivity in the CVP framework [9].

## Discussion

Utilizing the microstructure-sensitive CVP model framework, CF TMF conditions were simulated for temperature ranges associated with the evolution of the $\gamma \u2032$ morphology for stress-free coarsened, P-Raft, and N-Raft microstructures to validate the model calibration. The simulated behaviors are compared to the experimentally measured half-life stress–strain behaviors for CM247 LC as illustrated in Figs. 11–13 [32,37]. Creep-fatigue TMF conditions were used for validation of the microstructure sensitive CVP framework due to CF TMF being most similar to the conditions materials within industrial gas turbine engines experience in service, and therefore most relevant. In all cases, the CVP simulations were run to the experimentally reported half-life values and compared [9].

In validating the microstructure-sensitive CVP model on ability to predict CF TMF conditions, the model was able to accurately capture the relaxation behavior of the material and accumulated cyclic inelastic strain. Overall, the CVP model validation cases can be observed to be in agreement with capturing the stress-state response of the material with an evolving $\gamma \u2032$ morphology. Further, the model captures the peak stresses within 10% (50–80 MPa) of the experimental measured response of the TMF behavior for which the validation was carried out. However, when accounting for statistical material behavior in the cyclic response, this can be considered within the realm of reasonable variation for Ni-base superalloys [37].

Not fully captured by the microstructure-sensitive CVP framework is the experimentally observed reverse yielding that occurs during the low temperature component of the TMF cycle. The inability to capture the reverse yielding is potentially attributive to the reprecipitation of the $\gamma \u2032$ upon cool-down as the volume fraction of the precipitates is predicted to go from ≈55% to 70% when cycling between 950 and 600 °C. The temperature dependence of the $\gamma \u2032$ volume fraction is calculated through JMatPro and illustrated in Fig. 14. While, the $\gamma \u2032$ volume fraction is inherently captured in the isothermal parameter fits at each calibration temperature, the parameters themselves nor the model framework accounts for the transient behavior of the volume fraction of the $\gamma \u2032$ during thermal cycling.

In calibrating the model for the aged microstructure states, the greatest impact of the state of the microstructure on the model parameters was observed in the 750–950 °C temperature range, specifically for the hardening parameters *h _{χ}* and

*R*. This can be rationalized through the change in deformation mechanism starting in the 700–800 °C range from $\gamma \u2032$ shearing to $\gamma \u2032$ bypass for Ni-base superalloys [21].

_{χ}Experimentally, the morphological state and size of the $\gamma \u2032$ precipitates have been shown to have a great effect on the overall low cycle fatigue life under both TMF and isothermal conditions [4,32,37–39]. So much so that the presence of an aged microstructure can result in a ±2X difference in LCF life from that of the as-heat treated condition undergoing identical TMF conditions [32]. As such, the full utility of this microstructure-sensitive CVP model framework would be as an input for life models as an aid in designing gas turbine engine components [40–43].

## Conclusions

In this work, a microstructure-sensitive CVP model has been developed that captures the aging effects of the $\gamma \u2032$ precipitates on the cyclic response of Ni-base superalloys undergoing TMF. Through calibration of the model and comparison to experimental TMF data from a number of microstructure states, the following conclusions can be drawn:

- (1)
Microstructure sensitivity was successfully introduced into the CVP model framework through coupling the temperature dependent material parameters with the sensitivity of the $\gamma \u2032$ aspect ratio to temperature, stress, and time via two internal state variables.

- (2)
The microstructure-sensitive CVP model was successfully calibrated to the isothermal cyclic response of a DS Ni-base superalloy for three aged microstructure states in the 20–1050 °C temperature range.

- (3)
The microstructure-sensitive CVP model was validated through its ability to predict the experimentally observed stress state of the material at half-life while undergoing creep-fatigue TMF conditions.

## Funding Data

U.S. Department of Energy (DOE) (DE-FC26-05NT42644).

Computational Materials Design (CCMD), a joint National Science Foundation (NSF) Industry/University Cooperative Research Center at Penn State (IIP-1034965) and Georgia Tech (IIP-1034968).

Siemens Energy Inc., Orlando, FL through subcontract to DOE Award DE-FC26-05NT42644 on “Advanced Hydrogen Turbine Development”.