Modelling plastic deformation in a single-crystal nickel-based superalloy using discrete dislocation dynamics
- B. Lin^{1},
- M. S. Huang^{2},
- F. Farukh^{3},
- A. Roy^{1},
- V. V. Silberschmidt^{1} and
- L. G. Zhao^{1}Email author
https://doi.org/10.1186/s40759-016-0012-y
© The Author(s). 2016
Received: 7 April 2016
Accepted: 11 October 2016
Published: 15 November 2016
Abstract
Background
Nickel-based superalloys are usually exposed to high static or cyclic loads in non-ambient environment, so a reliable prediction of their mechanical properties, especially plastic deformation, at elevated temperature is essential for improved damage-tolerance assessment of components.
Methods
In this paper, plastic deformation in a single-crystal nickel-based superalloy CMSX4 at elevated temperature was modelled using discrete dislocation dynamics (DDD). The DDD approach was implemented using a representative volume element with explicitly-introduced precipitate and periodic boundary condition. The DDD model was calibrated using stress–strain response predicted by a crystal plasticity model, validated against tensile and cyclic tests at 850 °C for <001 > and <111 > crystallographic orientations, at a strain rate of 1/s.
Results
The DDD model was capable to capture the global stress–strain response of the material under both monotonic and cyclic loading conditions. Considerably higher dislocation density was obtained for the <111 > orientation, indicating more plastic deformation and much lower flow stress in the material, when compared to that for <001 > orientation. Dislocation lines looped around the precipitate, and most dislocations were deposited on the surface of precipitate, forming a network of dislocation lines. Simple unloading resulted in a reduction of dislocation density.
Conclusions
Plastic deformation in metallic materials is closely related to dynamics of dislocations, and the DDD approach can provide a more fundamental understanding of crystal plasticity and the evolution of heterogeneous dislocation networks, which is useful when considering such issues as the onset of damage in the material during plastic deformation.
Keywords
Discrete dislocation dynamics Representative volume element Crystal plasticity Monotonic loading Cyclic deformationBackground
Nickel-based superalloys are primarily used for rotating turbine blades and discs in the hot section of gas turbine engines thanks to their exceptional high-temperature mechanical properties, which are attributed to their characteristic two-phase microstructure: a ductile γ-matrix phase and a coherent L1_{2}-ordered γ'-precipitate phase. Most nickel-based superalloys have a volume content of the precipitate phase that ranges between 40% and 70%. Nickel-based superalloys are usually exposed to high static or cyclic loads in non-ambient environments, so a reliable prediction of their mechanical properties at elevated temperature is essential for adequate damage-tolerance assessment of components.
Mechanical behaviour of engineering alloys can be studied at different length scales, ranging from atomic to macroscopic scale (McDowell 2010). At the macro/continuum scale, a physically-based crystal-plasticity theory was employed successfully for description of mechanical deformation of single crystals and polycrystals, including body-centered-cubic (Kothari and Anand 1998), face-centered-cubic (Balasubramanian and Anand 2002) and hexagonal-close-packed lattice structures (Hasija et al. 2003). Combined with a finite-element method, the crystal plasticity theory is capable to predict both global and local stress–strain responses (Kothari and Anand 1998; Balasubramanian and Anand 2002; Hasija et al. 2003), evolution of a crystallographic grain texture (Kothari and Anand 1998) and micro-structural crack nucleation (Dunne et al. 2007) in metallic materials under monotonic, creep and fatigue loading conditions. It is also well known that inelastic deformation in metallic alloys is caused by the motion of a large number of dislocations. In the past decade, discrete dislocation dynamics (DDD) (Zbib and de la Rubia 2001) was developed to compute plastic deformation directly from the evolution of collective dislocation segments, particularly three-dimensional (3D) DDD models (Kubin et al. 1992). The DDD method can explicitly model interactions between dislocations on different slip systems and internal microstructure, as well as the formation of heterogeneous dislocation networks such as slip bands, under both monotonic and cyclic loading conditions.
Both crystal plasticity and DDD methods were employed to investigate mechanical behaviour of nickel-based superalloys. The crystal plasticity was applied to study creep (Zhao et al. 2006), fatigue (Kiyak et al. 2008), thermal-mechanical fatigue (Shenoy et al. 2005), indentation (Zambaldi et al. 2007) and gradient-dependent deformation (Meissonnier et al. 2001) of single crystal nickel superalloys. Application of crystal plasticity was also extended to model stress–strain response and fatigue crack nucleation for polycrystalline nickel superalloy (Dunne et al. 2007), with microstructure considered as one of the major factors affecting its fatigue and creep properties. The DDD method was applied for simulation of cutting of dislocation pairs into a precipitate phase (Huang et al. 2012), prediction of a critical resolved shear stress (CRSS) (Vattré et al. 2009), simulation of orientation-dependent mechanical response (Vattré et al. 2010), and study of the role of dislocation dissociation in plastic behaviour (Huang and Li 2013) of single-crystal nickel-based superalloys.
In this paper, the DDD model was presented to study plastic deformation in a single-crystal nickel-based superalloy CMSX4 under monotonic and cyclic loading conditions at 850 °C. The material’s mechanical response predicted by the validated CP model at 1/s strain rate in both <001 > and <111 > orientations was used to calibrate the DDD model. The work aims to provide a micromechanics-based understanding of plastic deformation behaviour of the material.
Methods
Crystal plasticity model
where F ^{e} represents elastic stretching and rigid-body rotation of the crystal and F ^{ p } describes crystallographic slip along slip planes due to dislocation motion.
where C _{11}, C _{12} and C _{44} are material elastic constants.
where \( {\overset{.}{\gamma}}^{\alpha } \) is the shear strain rate on the slip system α, n ^{ α } is the slip-plane normal and m ^{ α } is the slip direction vector for a generic slip system α.
where κ is the Boltzmann constant, τ^{α} is the resolved shear stress on the slip system α, θ is the absolute temperature, μ and μ_{0} are the shear moduli at θ and 0 K, respectively, and F_{0}, \( {\widehat{\tau}}_0 \), p, q and \( {\overset{.}{\gamma}}_0 \) are material constants. The Macaulay brackets imply that 〈x〉 = x for x > 0 and 〈x〉 = 0 for x ≤ 0.
where \( {S}_0^{\alpha } \) is considered as the initial value of the slip resistance for a slip system. h _{ s } and d _{ D } are the material constants linked to static and dynamic recovery terms, respectively.
where (r _{ D }) is a dynamic recovery function which introduces the inherent dependency between the slip resistance and the back stress.
The crystal plasticity model was developed within the large strain framework. When applying this model to small strain problems, the computation shows high efficiency and easy convergence as the nonlinear behaviour is not well developed yet due to fairly limited plastic deformation. Also the flow rule given by Eq. (5) is dependent on temperature θ.
Finally, the calibrated CP model was used to predict the monotonic response and the first loop of a cyclic stress–strain diagram of the studied material at a high strain rate 1/s (see Figs. 1 and 2), which were used to calibrate the DDD model (see the following sections). The reason for choosing a high strain rate for the DDD model lies in the fact that the computation cost of this model is extremely high and a high strain rate can lead to quicker convergence.
Discrete dislocation dynamics
RVE model
Peach-Koehler force, equations of motion and short-range interactions
where b _{ i } and ξ _{ i } are the Burgers vector and the line sense vector of segment i, respectively.
where \( {\boldsymbol{F}}_i^{glide} \) is the glide component of the Peach-Koehler force F _{ i } on the slip plane, τ _{ APB } is an internal or back stress introduced by the antiphase boundary in the γ′ phase, τ _{ F } is the constant friction stress, B is the viscous drag coefficient and abs(x) represents the absolute value of x.
where κ is the Boltzmann constant, f _{ D } is the Debye frequency factor, l _{ s } is the screw segment length, T is the absolute temperature and ΔH _{ 0 } is the activation enthalpy for KW locks. In this work, the KW unlocking stress τ _{ KW } was calcukated as 600 MPa from Eq. (10), based on the parameter values suggested in the paper of Demura et al. (2007). Other thermal-related effects such as dislocation climb are not considered in the current DDD code yet.
A back-force model (Yashiro et al. 2006) was introduced to simulate shearing of the γ′ precipitate by a series of superdislocations. The leading and trailing dislocations form a superdislocation, separated by an antiphase boundary (APB). To determine whether the dislocation segment, entering into the γ′ precipitate, was a leading or a trailing dislocation, the following method was applied in the DDD model. If F _{ app } ⋅ F _{ int } < 0 and abs(F _{ int }) > 0.25χ _{ APB }, it is the trailing dislocation that entered the γ′ precipitate. If F _{ app } ⋅ F _{ int } ≥ 0 and abs(F _{ int }) > 0.25χ _{ APB }, it is the leading dislocation that entered the γ′ precipitate. Here, F _{ app } and F _{ int } are the glide forces induced by the external loads and the stress at the centre of dislocation segment i introduced by an interacting dislocation, respectively. The inherent APB energy per unit area is represented by χ _{ APB }.
Besides glide under the PK force, dislocations interacted with each other directly under mechanical loading. Direct interactions considered in the DDD model included annihilation and formation of jogs and junctions. A complete list of the interaction rules for dislocation dynamics can be found in Rhee et al. (1998).
Dislocation-induced plastic strain and computation of external stress
where dA is the incremental area swept out by the segment, n _{ k } or n _{ l } is the component of the normal vector of the glide plane, b _{ k } or b _{ l } is the component of the Burger’s vector, V is the RVE volume, and A _{ slip } is the collection of surfaces active in deformation.
where E is the Young’s modulus.
Results and discussions
Parameter calibration
Material parameters for DDD model at 850 °C
Orientation | E (GPa) | v | b (nm) | B (Pa s) | τ _{ F } (MPa) | χ _{ APB } (mJ/m^{2}) | Initial dislocation density (m^{−2}) |
---|---|---|---|---|---|---|---|
<001> | 91.92 | 0.379 | 0.25 | 8.3e-6 | 200 | 162.5 | 2.5e + 13 |
<111> | 244.55 | 0.179 |
It should be noted, only isothermal behaviour was simulated in this work and the model parameters were calibrated against the test data at a temperature of 850 °C. Consequently, the calibrated model parameters already reflected the mechanical behaviour at 850 °C. If changed to other temperature, model parameter (including the friction stress for dislocations in the γ phase) would need to be re-calibrated. Further developments are required in order to capture the thermal effect automatically, especially under thermal fatigue.
Monotonic tensile response
According to Taylor’s model, the hardening can be naturally captured by the formation of junctions, jogs and polarized dislocation structures in DDD simulations (Huang et al. 2012). Generally, higher dislocation density induces higher hardening. But in nickel superalloys, the Taylor hardening mechanism might not be dominant, since no significant junctions formed in the γ-matrix channels, as most dislocations are deposited at the matrix-precipitate interface instead of the channels. As a result, higher dislocation density does not necessarily mean higher fractions of junctions, jogs and polarized dislocation structures.
Cyclic response
The DDD model simulates dislocation junctions, jogs and tangles directly, relating hardening of materials directly to these dislocation interactions. The model can also capture the evolution of dislocation structure/configuration and dislocation density, supporting our understanding of intrinsic mechanisms of plastic deformation. However, temporal and spatial scales of DDD simulations are still limited. This is why almost all DDD work in literature is limited to simulations at relatively high strain rates (up to 20/s). Still, the DDD model can help elucidation of plastic deformation at the scale of discrete dislocations, which can assist the development of an appropriate continuum constitutive theory.
Conclusions
The DDD model was used to simulate macroscopic responses of a single-crystal nickel-based superalloy CMSX4 subjected to monotonic and cyclic loadings at elevated temperature. The model can simulate orientation-dependent stress–strain behaviour of this alloy. Plastic deformation in metallic materials is closely related to dynamics of dislocations, and the DDD approach can provide a more fundamental understanding of crystal plasticity and corresponding heterogeneous strain fields, which is critical when considering such issues as the onset of damage in the material during plastic deformation.
Declarations
Acknowledgements
The work was funded by the EPSRC (Grants EP/K026844/1 and EP/M000966/1) of the UK. The crystal plasticity UMAT was originally developed and calibrated against the experimental data by Professor Esteban Busso, Professor Noel O’Dowd and their associates while they were with the Imperial College, London. The research leading to these results also received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement No. PIRSES-GA-2013-610547 TAMER. Research data for this paper is available on request from the project principal investigator Dr Liguo Zhao at Loughborough University (email: L.Zhao@Lboro.ac.uk).
Authors’ contributions
BL and MSH developed and carried out the DDD modelling work; FF performed the CP modelling work; AR, VVS and LZ conceived and supervised the work. All authors contributed to the writing and editing of the paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Asaro RJ (1983) Micromechanics of crystals and polycrystals. Adv Appl Mech 23:1–115View ArticleGoogle Scholar
- Balasubramanian S, Anand L (2002) Elasto-viscoplastic constitutive equations for polycrystalline fcc materials at low homologous temperatures. J Mech Phys Solids 50:101–126View ArticleMATHGoogle Scholar
- Busso EP (1990) Cyclic deformation of monocrystalline nickel alluminide and high temperature coatings. PhD dissertation, Massachusetts Institute of Technology, USAGoogle Scholar
- Demura M, Golberg D, Hirano T (2007) An athermal deformation model of the yield stress anomaly in Ni3Al. Intermetallics 15:1322–1331View ArticleGoogle Scholar
- Dennis RJ (2000) Mechanistic modelling of deformation and void growth behaviour in super-alloy single crystals. PhD dissertation, Imperial College London, UKGoogle Scholar
- Dunne FPE, Wilkinson AJ, Allen R (2007) Experimental and computational studies of low cycle fatigue crack nucleation in a polycrystal. Int J Plast 23:273–295View ArticleMATHGoogle Scholar
- Gao S, Fivel M, Ma A, Hartmaier A (2015) Influence of misfit stresses on dislocation glide in single crystal superalloys: A three-dimensional discrete dislocation dynamics study. J Mech Phys Solids 76:276–290MathSciNetView ArticleGoogle Scholar
- Giacomo P, Mohamed MS, Crosby T, Erel C, El-Azab A, Ghoniem N (2014) Recent progress in discrete dislocation dynamics and its applications to micro plasticity. JOM 66:2108–2120View ArticleGoogle Scholar
- Hasija V, Ghosh S, Mills MJ, Joseph DS (2003) Deformation and creep modeling in polycrystalline Ti-6Al alloys. Acta Mater 51:4533–4549View ArticleGoogle Scholar
- Hrutkay K, Kaoumi D (2014) Tensile deformation behavior of a nickel based superalloy at different temperatures. Mater Sci Eng A 599:196–203View ArticleGoogle Scholar
- Huang EW, Barabash RI, Wang Y, Clausen B, Li L, Liaw PK, Ice GE, Ren Y, Choo H, Pike LM, Klarstrom DL (2008) Plastic behavior of a nickel-based alloy under monotonic-tension and low-cycle-fatigue loading. Int J Plast 24:1440–1456View ArticleMATHGoogle Scholar
- Huang MS, Li ZH (2013) The key role of dislocation dissociation in the plastic behaviour of single crystal nickel-based superalloy with low stacking fault energy: Three-dimensional discrete dislocation dynamics modelling. J Mech Phys Solids 61:2454–2472View ArticleGoogle Scholar
- Huang MS, Zhao LG, Tong J (2012) Discrete dislocation dynamics modelling of mechanical deformation of nickel-based single crystal superalloys. Int J Plast 28:141–158View ArticleGoogle Scholar
- Kear BH, Wilsdorf HGF (1962) Dislocation configurations in plastically deformed polycrystalline Cu3Au alloys. Tran Metall Soc AIME 224:382–386Google Scholar
- Kim GS, Fivel MC, Lee HJ, Shin C, Han HN, Chang HJ, Oh KH (2010) A discrete dislocation dynamics modeling for thermal fatigue of preferred oriented copper via pat-terns. Scripta Mater 63:788–791View ArticleGoogle Scholar
- Kiyak Y, Fedelich B, May T, Pfennig A (2008) Simulation of crack growth under low cycle fatigue at high temperature in a single crystal superalloy. Eng Fract Mech 75:2418–2443View ArticleGoogle Scholar
- Kothari M, Anand L (1998) Elasto-viscoplastic constitutive equations for polycrystalline metals: application to tantalum. J Mech Phys Solids 46:51–83View ArticleMATHGoogle Scholar
- Kubin L, Canova G, Condat M, Devincre B, Pontikis V, Brechet Y (1992) Dislocation structures and plastic flow: a 3D simulation. Solid State Phenom 23–24:455–472View ArticleGoogle Scholar
- Lukáš P, Kunz L (2002) Cyclic plasticity and substructure of metals. Mater Sci Eng A 322:217–227View ArticleGoogle Scholar
- McDowell DL (2010) A perspective on trends in multiscale plasticity. Int J Plast 26:1280–1309View ArticleMATHGoogle Scholar
- Meissonnier FT, Busso EP, O’Dowd NP (2001) Finite element implementation of a genera-lised non-local rate-dependent crystallographic formulation for finite strains. Int J Plast 17:601–640View ArticleMATHGoogle Scholar
- Rhee M, Zbib HM, Hirth JP, Huang H, de La Rubia TD (1998) Models for long-/short-range interactions and cross slip in 3D dislocation simulation of BCC single crystals. Model Simul Mater Sci Eng 6:467–492View ArticleGoogle Scholar
- Rice JR (1970) On the structure of stress–strain relations for time-dependent plastic deformation in metals. J Appl Mech 37:728–737View ArticleGoogle Scholar
- Shenoy MM, Gordon AP, McDowell DL, Neu RW (2005) Thermomechanical fatigue behavior of a directionally solidified Ni-base superalloy. J Eng Mater Tech 127:325–336View ArticleGoogle Scholar
- Vattré A, Devincre B, Roos A (2009) Dislocation dynamics simulations of precipitation hardening in Ni-based superalloys with high γ′ volume fraction. Intermetallics 17:988–994View ArticleGoogle Scholar
- Vattré A, Devincre B, Roos A (2010) Orientation dependence of plastic deformation in nickel-based single crystal superalloys: discrete-continuous model simulations. Acta Mater 58:1938–1951View ArticleMATHGoogle Scholar
- Yashiro K, Kurose F, Nakashima Y, Kubo K, Tomita Y, Zbib HM (2006) Discrete dis-location dynamics simulation of cutting of γ′ precipitate and interfacial dislocation net-work in Ni-based superalloys. Int J Plast 22:713–723View ArticleMATHGoogle Scholar
- Zambaldi C, Roters F, Raabe D, Glatzel U (2007) Modeling and experiments on the inden-tation deformation and recrystallization of a single-crystal nickel-base superalloy. Mater Sci Eng A 454–455:433–440View ArticleGoogle Scholar
- Zbib HM, de la Rubia TD (2001) A multiscale model of plasticity. Int J Plast 18:1133–1163View ArticleMATHGoogle Scholar
- Zhao LG, O’Dowd NP, Busso EP (2006) A coupled kinetic-constitutive approach to the study of high temperature crack initiation in single crystal nickel-base superalloys. J Mech Phys Solids 54:288–309View ArticleMATHGoogle Scholar