A Lattice-misfit-dependent Micromechanical Approach in Ni-based Single Crystal Superalloys
Graphical abstract
Introduction
Ni-based single-crystal superalloys are the materials of choice for a wide range of temperature/stress creep regimes [1,2]. It is due to the precipitation of a high volume fraction (close to 70%) of the long-range ordered L12 γ’ phase that comes as cubes coherently embedded in a face-centered cubic (fcc) solid solution γ matrix. For temperatures higher than 800°C in the low-stress regimes, the γ’ precipitates display a directional coarsening, also known as γ’ rafting [3,4]. If the natural lattice misfit, i.e., the lattice misfit before deformation, is negative/positive, the rafts will be perpendicular/parallel to the load axis. This microstructure evolution happens with an increase in the γ channel width and a modification of the lattice misfit defined as δ = 2(aγ′ − aγ)/(aγ′ + aγ) where aγ′ and aγ are the lattice parameter in the deformed state for the γ’ and γ phases, respectively. The natural lattice misfit increases in absolute value with the temperature due to a difference in the thermal coefficient between γ and γ’ and also because some alloying elements from the γ’ phase, such as Al, Ti, and Ta, contribute to the γ phase when γ’ dissolves which increases its lattice parameter [5,6]. The constrained lattice misfit, viz. the misfit during deformation, was shown to evolve during constant-stress [7,8], variable-stress [9], [10], [11], and non-isothermal [12], [13], [14] creep. Diologent et al. [7] tracked in-situ the evolution of the constrained lattice misfit during isothermal creep. They found that it increases in absolute value during the primary creep stage, which corresponds to the increase of γ/γ’ interfacial dislocations, then decreases in absolute value during the secondary creep stage due to relaxation. So, even if a fundamental understanding of the lattice misfit evolution during creep seems to have been reached, most of the modeling efforts consider either a constant lattice misfit [15], [16], [17], [18] or an incomplete description of its evolution [19]. Unfortunately, the way the lattice misfit evolves during cyclic loading and strain-controlled experiments is not available due to a lack of experimental data.
Nonetheless, it was shown, as for creep [20], [21], [22], [23]; low cycle fatigue (LCF) [24,25]; and thermo-mechanical loading [26,27], that rafting and, therefore, lattice misfit affect monotonic and cyclic loading [28], [29], [30]. Mackay et al. [28] showed on a single-crystal Ni-Al-Mo-Ta alloy that rafted microstructures have smaller yield (at 0.2%) and rupture stresses in tension for temperatures ranging from 930 to 1040°C. The same qualitative results were obtained by Pessah-Simonetti et al. on the first generation Ni-based single crystal superalloy MC2 for a 650-950°C range [29]. The same temperature range was also investigated by Espié [30] to determine the effect of the microstructure on the monotonic and cyclic behavior for the AM1 alloy. If 650°C revealed no effect because of the shearing of the γ’ particles independently of their shape, 950°C showed modifications. Espié tested the mechanical response of creep-obtained and cyclic-obtained (Δε=2%) rafted microstructures to monotonic strain-controlled tensile tests at 9. 10−4 s−1. Both rafted microstructures revealed smaller hardening at 2% than the cuboidal one. Contrary to Espié, who found no difference between creep-obtained and cyclic-obtained rafts in terms of softening, Gaubert [31] noticed that rafts obtained by cycling hardening lead to a more considerable softening compared to rafts obtained by creep. The difference might be because Espié used samples aged at 950°C while Gaubert's were aged at 1050°C leading to a significant difference in the γ’ volume fraction (70 vs. 60%) and γ channel widths. Therefore, Is the widening of the γ channels or, the modification of the lattice misfit, or perhaps a mix of these two factors responsible for the above-mentioned discrepancy in the softening behavior? The softening observed in the monotonic and cyclic mechanical responses for creep-rafted and cyclic-rafted microstructures is often interpreted as a consequence of the decrease in the Orowan stress, i.e., an increase of the γ channel widths. However, relaxation tests at 1050°C/εt = 1.2% performed on an initially undeformed and on a creep-rafted (1050°C/150 MPa for 72h and 0.65% deformation) microstructure showed that the asymptotic value, i.e. the remaining internal stress, is 1.6 times higher for the initially undeformed specimen (120 MPa vs. 75 MPa) whereas the two tests started at the same stress magnitude [31]. One should expect that a deformed specimen would lead to a higher value of the internal stresses. If the difference in the γ channel widths (110 nm for undeformed vs 300 nm for rafted) is used to explain the smaller internal stress value of the rafted specimen, then it is important to note that the decrease in the Orowan stress should only be around 35 MPa at 1050°C. It is clear from these experiments that something else is at stake and that the difference in the lattice misfit is something that should be considered in the numerical calculations. Moreover, interpreting the changes in mechanical behavior with just the Orowan stress might be erroneous and deserves more detailed studies. Such studies would provide a better understanding on how internal stresses evolve and, therefore, on the evolution of the hardening variables employed in constitutive modeling, which could, for instance, allow for better predictions of size effects [32] that was attributed to surface relaxation effects on the lattice misfit during experiments on micro-pillars [33].
Micro-mechanical modeling is a way to answer the overarching question asked in the previous paragraph. However, except the work by the authors [34], most micromechanical finite element simulations were performed with only one γ’ particle or idealized γ’ precipitate shapes [35], [36], [37], [38]. This does not provide full confidence in the results knowing that the shape of the γ’ precipitates depends on the value of the natural lattice misfit [39] and that precipitates will have different shapes between dendrite and interdendrite regions due to micro-segregations [40]. Therefore, it seems a dead end to study the effect of lattice misfit for certain microstructural states without having faithful representations of intricate γ/γ’ microstructures in all their details to account for realistic internal stress and strain distributions. The best approach to address this issue would be coupled phase-field crystal-plasticity models [41], [42], [43]. However, if such models can well predict rafting and to simulate macroscopic responses in certain conditions, they consider γ’ as either elastic or deforming elastically due to a very large critical resolved shear stress, which seems in contradiction with what was experimentally observed [44]. In addition, these studies do not control equilibrium concentrations in the presence of plasticity, which is essential in the computation of the chemical free energy density, and consider constant misfit strains.
A much more reliable and straight forward approach for short thermo-mechanical loadings, i.e, loading conditions that do not trigger microstructure evolutions, would be to improve the formulation of the recent multi-scale approach employed by the authors in [34], namely perform macro-scale finite-element calculations on realistic statistically volume elements (SVEs) obtained by phase-field simulations. On one hand, this approach leverages the advantages of both phase-field and finite-element crystal plasticity models and, on the other hand, gives the flexibility to build and calibrate a robust micromechanical model connecting the two. Furthermore, averaging or homogenization-based approaches [15,45,46] are sensitive to the volume fraction of the precipitates, but not to the shape differences in γ’ precipitates owing to different lattice misfits considered in this study. It is worth pointing out that the approach of utilizing phase-field SVEs in the finite-element crystal plasticity model has one major drawback: only short duration tests can be simulated while the microstructure is essentially non-evolving. However, this drawback is not a limitation to the aim of the present study, which is to bring forth the importance of considering the effect of misfit strains to better understand the effect of microstructural state on the macro-scale performance of Ni-based single-crystal superalloys. The study will, therefore, use realistic 3D phase-field microstructures (cuboidal and rafted) with the experimentally-consistent γ’ volume fractions as well as shapes and distributions of the γ’ precipitates at the envisioned temperature/stress conditions to simulate strain-controlled tensile tests up to 2% total strain and strain-controlled cyclic tests with Δεt = 2% both at a strain rate of .
It is important to hereby point out that a large range of natural lattice misfits, viz. from -0.2% to -0.5%, is investigated to be relevant with the large modifications of the lattice misfit during heat treatments, as shown in [47], and help the design of materials with improved performance. Thus, the changes in the natural lattice misfits are neither due to different chemical compositions nor temperature variations [48], which would, otherwise, make the set of parameters discussed in section 2.2.2 invalid. The authors want to point out that unlike the previous study [34], the present phase-field informed micro-mechanical model built into a crystal plasticity framework is sensitive to lattice misfit strain, the details of which are explained in detail in section 2.2.1. Furthermore, the authors also want to emphasize that the lattice misfit values (natural and constrained) will be considered constant during the calibration and the FE simulations, even though it is probably incorrect when such simulations last for a long time (>1300s), like in Fig. 2b, and Fig. 3b and d. It is the reason why the FE simulations performed in this study are for short tests: 20s and 200s for the monotonic and cyclic loadings, respectively. Considering that the lattice misfit does not evolve during short tests is twofold: 1) there is not enough time to trigger microstructural evolutions as well as to allow the γ/γ’ interfacial dislocations rearranging to form a network and 2) the simulated thermo-mechanical loadings do not include softening. It was shown that γ/γ’ interfacial dislocation networks are rapidly formed when softening occurs [49] and that hexagonal networks are responsible for relieving misfit strains at {111} interfaces when square networks are for {100} interfaces [50,51]. It is also consistent with what Carroll et al. [52] found, which is that the interfacial dislocation networks were not formed during creep deformation in an alloy with a natural lattice misfit that was close to zero. Therefore, for fast thermo-mechanical loading, there is not enough time for the γ/γ’ interfacial dislocations to rearrange into dislocation networks to accommodate the misfit.
Section snippets
Phase-field microstructures
Phase-field microstructures are utilized for the micromechanical approach. The phase-field framework was already described in the two previous articles by the authors [53, 54]. For the sake of completeness, the model is revisited here. This section provides the basic formulation and information regarding the parameters to generate the bi-phased γ/γ' systems similar to the microstructural states commonly encountered in Ni-based single crystal superalloys: cuboidal microstructural states
Results and Discussion
The phase-field SVEs, imported into the FE crystal plasticity framework, are enforced with boundary conditions that are prescribed on individual volume element V in a similar fashion as what can be found in [83,84].
- –
kinematic uniform boundary conditions: the displacement is imposed at point that belongs to the boundary ∂V such that:
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uniform traction boundary condition is applied, i.e.
Conclusion
Ni-based single crystal superalloys are defined by their complex bi-phased microstructures that take different shapes depending on the heat-treatment and thermo-mechanical conditions. In this study, a novel multi-scale multi-model numerical strategy is devised to understand the effect of a microstructural state of material on their macroscale performance by utilizing:
- 1
Thermodynamically consistent and experimentally verified microstructural configurations derived from phase-field SVE realizations
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The simulations were performed using the computing resources from Laboratory for Molecular Simulation (LMS) and High Performance Research Computing (HPRC) at Texas A&M University. The authors are grateful to Mr. James Fillerup and the financial support from AFOSR through Award No.: FA9550-17-1-0233 to carry out this study. We also acknowledge Dr. Adrian Loghin and GE Global Research for their support and interest in our research.
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