High temperature dislocation processes in precipitation hardened crystals investigated by a 3D discrete dislocation dynamics
Graphical abstract
Introduction
The discrete dislocation dynamics (DDD) proved to be a powerful modelling methodology that can contribute to a better understanding of plastic deformation on the scale of individual dislocations, when the mutual dislocation-dislocation interactions must be taken into account Bulatov and Cai, 2006, Pollock and LeSar, 2013. Therefore, recent approaches to a meso-scale dislocation modelling are often based on the DDD methodology where Peach-Koehler and osmotic forces Peach and Koehler, 1950, Hirth and Lothe, 1992, Caillard and Martin, 2003 jointly provide a driving force for displacements of dislocation segments. The Peach-Koehler force can involve all external and internal contributions to the local stress field acting at a segment position.
Since more than thirty years, the DDD investigations focused on dislocation ensembles in 2D. Here we mention only limited number of studies that either investigated dislocation structures consisting of infinite straight dislocations perpendicular to the simulation plane, e.g. Lepinoux and Kubin, 1987, Cleveringa et al., 1997, Dlouhý et al., 2001, Deshpande et al., 2003, Miller et al., 2004, Zhu et al., 2014 or had taken fully into account a flexibility of dislocation lines and addresses planar configurations of dislocations in one particular slip or climb plane, e.g. Mohles and Fruhstorfer, 2002, Hiratani et al., 2003, Gao et al., 2011. In spite of the 2D approximation, these models helped understanding of a wide variety of deformation mechanisms, e.g. (i) the dislocation patterning Lepinoux and Kubin (1987), (ii) the composite microstructure plasticity Cleveringa et al. (1997), (iii) the stability of dislocation structures in γ-channels of modern superalloys Dlouhý et al. (2001), (iv) the dislocation plasticity around fatigue cracks Deshpande et al. (2003), the dislocation structures generated under a circular indenter Miller et al. (2004) and (viii) the Hall-Petch relation Zhu et al. (2014). 2D planar models with flexible dislocation lines were employed to account for the dislocation glide in randomly distributed particle arrangements Mohles and Fruhstorfer (2002), (ii) the dislocation glide in a field of local obstacles Hiratani et al. (2003) and (iii) various in-plane climb processes controlled by pipe diffusion Gao et al. (2011). With respect to the high temperature dislocation mechanisms, some of the 2D models also incorporated dislocation climb to study (i) the formation of specific dislocation structures during creep of nickel-based superalloys Probst-Hein et al. (1999), (ii) the structure of low angle tilt boundaries and their interactions with circular particles Cheng et al., 2009, Holec and Dlouhý, 2005 and, more recently, also to rationalize stress exponents of minimum creep rate observed experimentally Keralavarma et al. (2012).
As a matter of principle, the 2D DDD models address either only straight dislocations and thus cannot account for effects related to the dislocation line curvature and corresponding self stresses or restrict the interactions between dislocation segments to a single plane. Therefore, an effort has been made to extend the 2D DDD methodology to complete 3D models, see e.g. Schwarz (1999). A power of the 3D DDD methodology was first fully demonstrated in a pioneering work by Devincre and Kubin (1997), where elementary processes like a 3D operation of Frank-Read sources and/or cross-slip were investigated. Since then, further numerous 3D DDD studies have been performed for a low temperature regime, where diffusion does not significantly contribute to the deformation process. Here, we reference some numerical studies which, from the perspective of the present work, represent important technical steps in DDD simulations. These investigations focused on (i) the plasticity of small confined volumes Arsenlis et al., 2007, Weygand et al., 2001, (ii) the interactions of dislocations with short cracks Depres et al. (2004), (iii) the size effect in metal matrix composites Groh et al. (2005), (iv) the formation of multi-junctions in dislocation networks Bulatov et al. (2006), (v) the interactions of dislocations with particles during the deformation of particle hardened alloys Bako et al. (2007), (vi) the plasticity of BCC crystals Tang et al., 1998, Chaussidon et al., 2008, Wang and Beyerlein, 2011, Srivastava et al., 2013, (vii) the influence of the initial dislocation distribution on work hardening in finite-sized samples Motz et al. (2009), (viii) the interactions between slip systems in α-iron Queyreau et al. (2009) and (ix) the influence of particle misfit stresses on dislocation glide Gao et al. (2015).
The 3D DDD models also represent an important link in the multiscale modelling chain of plasticity phenomena. As typical examples, these models provide input for continuum-based crystal plasticity calculations Cui et al. (2015a) and can be combined with numerical techniques based on finite element analysis Zbib et al., 1998, Zbib and de la Rubia, 2002. A recent review on the applications of the 3D DDD techniques has been published by Po et al. (2014). Furthermore, the aforementioned 3D DDD model (Zbib et al., 1998, Zbib and de la Rubia, 2002) has been extended to address also dislocation climb and applied to a numerical study of Ni-based superalloys (Huang et al., 2012, Yang et al., 2015). Combined models which bridge atomistic and continuum scale dislocation plasticity were further used to account for the nucleation and rapid multiplication of dislocations under the nano-indenter tip Engels et al. (2012), interactions of dislocations with Guinier-Preston zones during fast deformations Yanilkin et al. (2014), irradiation hardening effects in alpha Fe single crystals Li et al. (2014) and effect of image forces during plastic deformation in heteroepitaxial films Cui et al. (2015b).
The present study employs the 3D DDD technique in order to investigate collective dislocation phenomena in precipitation hardened crystals subjected to loadings at high temperatures. In particular, we focus on challenging issues associated with threshold stresses observed in dispersion strengthened alloys, e.g. Lund and Nix, 1976, Hausselt and Nix, 1977a, Hausselt and Nix, 1977b. These stresses are generally lower than an Orowan stress (that would result from a given size and distribution of dispersoids) but are much higher than thresholds expected for diffusion controlled bypassing of particles Lagneborg, 1973, Brown and Ham, 1971. In order to rationalize the observed threshold stresses, a mechanism different from the Orowan process, namely a dislocation detachment from dispersoids, was suggested as a source of the threshold-like behaviour Nardone and Tien, 1983, Schröder and Arzt, 1985, Arzt and Wilkinson, 1986. The concept of the localized dislocation climb stabilized by an attractive dislocation-particle interaction and completed by the thermally activated detachment of the dislocation segment at a particle departure side demonstrated a considerable potential in explaining various aspects of the experimental data Rösler and Arzt (1990). However, in the number of cases, the detachment model has failed to rationalize the high-temperature strength of precipitation-hardened alloys, see e.g. Čadek et al. (1997). In this respect, it should be pointed out that either the Orowan process or the detachment models are based on one “typical” dislocation which interacts with particles under an assistance of the applied stress and an internal stress that is averaged out over a “typical” volume of the microstructure. The averaged out internal stress cannot capture correctly individual long-range dislocation-dislocation interactions and, in particular, effects associated with a formation and migration of low angle dislocation boundaries Exell and Warrington, 1972, Caillard and Martin, 1982, Vogler and Blum, 1990, Blum, 1993, Heilmaier and Reppich, 1997. In view of the indicated shortcomings, Holec and Dlouhý (2005) performed a 2D DDD numerical study focused on migration of low angle tilt boundaries in a two dimensional array of particles. Indeed, their results showed that, depending on the distribution and size of the particles and also on the density of the boundary dislocations (boundary tilting angle), the migration of the boundary can completely cease provided that the externally applied stress is lower than a certain threshold value. Results presented in Holec and Dlouhý (2005) were encouraging but, even though the 2D DDD model incorporated diffusion assisted climb processes, it could not capture effects associated with the curvature of dislocation lines. This represents a serious limitation since the development of the line curvature should be considered as a key process responsible for the wrapping of the boundary dislocations round the particles. Therefore, we have extended the model to a full 3D version in this study. The model builds upon similar recent developments, which incorporated the diffusion assisted dislocation climb as an inherent part of the dislocation segment dynamics Mordehai et al., 2008, Liu et al., 2011, Bako et al., 2011, Haghighat et al., 2013, Liu et al., 2014, Huang et al., 2014, Po and Ghoniem, 2014, Yang et al., 2015, Gu et al., 2015. However, we emphasize that, except the already mentioned 2D investigation by Holec and Dlouhý (2005), none of the modelling studies referenced here has dealt with the migration of low angle dislocation boundaries in the array of particles. Moreover, our model introduces some additional features. First, it considers dislocation segments of an arbitrary character, including segments of the mixed type. This extends the modelling methodology developed by Mordehai et al. (2008), who approximated smooth dislocation lines by step-like chains consisting exclusively of edge and screw segments. Second, we also introduce new, physically based, numerical techniques how to deal with the collisions of dislocation segments with particles and how to treat their motion along the particle-matrix interface. From these reasons, we first describe our 3D DDD model in section 2 and its numerical implementation in section 3. The results of some benchmark tests are presented in section 4 while section 5 is dedicated to the numerical study of the low angle boundary migration in the matrix with an array of particles. Results of the numerical study are discussed in section 6 and general conclusions are summarized in section 7.
Section snippets
Fundamentals
The 3D DDD model presented in this study approximates a smooth dislocation line by a continuous polygonal chain composed of straight dislocation segments of an arbitrary character. This simplifies considerably analytical formulas describing the stress field of curved dislocations Hirth and Lothe (1992) at a cost of high number of line segments that are required in order to adequately describe curved dislocation lines. Linear theory of elasticity allows summing up contributions from an arbitrary
Dislocation line connectivity and operations on the segment chain
Nodes in Fig. 4 delimit three inner and two outer dislocation segments (full blue lines) and represent a starting configuration of the line at the beginning of the time integration cycle i. Midpoints of the inner segments are situated in positions and . In the cycle i, the three individual segments are displaced by , , and and are then represented by red dashed abscissas in their new positions. Thus after a parallel translation in a
Benchmark processes
In this section, we demonstrate the applicability of our 3D DDD model to various planar and spatial dislocation processes some of which operate at high temperatures and thus involve dislocation climb. We start with 2D benchmark simulations in order to justify the adopted modelling and numerical techniques.
Interactions of low angle dislocation boundaries with precipitates
A migration of low angle dislocation boundaries (LADBs) in crystals hardened by particles is a process that may govern plasticity of many materials subjected to loadings at high temperatures Hausselt and Nix, 1977a, Hausselt and Nix, 1977b, Exell and Warrington, 1972, Caillard and Martin, 1982, Čadek, 1988. In this study, we simulate the LADB migration in order to demonstrate a full potential of the 3D DDD model. The calculations involve all the new features including dislocation segments of
Approximation by segmented chains
In our study, the continuous dislocation lines are represented by a chain consisting of a finite number of straight dislocation segments. This approximation may distort driving forces that control glide and climb of smooth and curved dislocations. In order to estimate differences between continuous and segmented representations, we have investigated the behaviour of the glissile dislocation loop in subsection 4.1. Comparisons of our numerical results and earlier analytical solutions Hirth and
Summary and conclusions
Linear elasticity analytical solutions for a stress field of straight dislocation segments have been incorporated into a 3D discrete dislocation dynamics (DDD) model in order to investigate evolution of dislocation systems subjected to loadings at elevated temperatures. A sufficient flexibility of the dislocation lines is guaranteed due to a chain-like approximation of the smooth curved lines by straight dislocation segments with typical lengths between 3 and . The model advances similar
Acknowledgement
Financial support has been provided by the Czech Science Foundation under projects Nos. 14-22834S, 202/09/2073 and 106/09/H035 and by the Ministry of Education, Youth and Sports, projects Nos. CZ.1.07/2.3.00/20.0214 and COST P19–OC 162. Additional support has been obtained from the Institute of Physics of Materials, Academy of Sciences of the Czech Republic due to the development program No. RVO: 68081723.
J.S. and A.D. would like to thank the European Commission for a financial support from the
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