Concurrent atomistic–continuum simulations of dislocation–void interactions in fcc crystals
Introduction
Materials subjected to irradiation exhibit elevated yield strength and suffer from intrinsic softening and reduced ductility by virtue of localization of dislocation plasticity. Irradiation-induced defects act as obstacles to dislocation migration. Typical radiation-induced defects in fcc lattices include voids (Averback et al., 1977, Kluth et al., 2005, Kondo et al., 2008, Crocombette and Proville, 2011) and helium bubbles (Donnelly et al., 1983, Henriksson et al., 2005, Demkowicz et al., 2010, David et al., 2011), particularly for fusion applications (Zinkle, 2005). Transmission electron microscopy (TEM) studies indicate that these defects act as obstacles to dislocations and, when bypassed, lead to localization of deformation in regions where defect densities are reduced via dislocation interaction (Shilo and Zolotoabko, 2003, Shilo and Zolotoabko, 2007, Shingo et al., 2007, Wu et al., 2007). Unfortunately, in situ observations of dislocation–obstacle interactions are quite limited owing to elaborate sample preparation, and restricted spatial and temporal ranges of TEM.
Static analysis of void strengthening suggests that voids with diameter of ∼2 nm or larger generally act as ‘strong obstacles’, whereas voids with diameter less than ∼2 nm act as ‘weak obstacles’, respectively (Hull and Bacon, 2001). It was concluded that the stress required for dislocation depinning from these voids approaches the theoretical Orowan stress (Hull and Bacon, 2001, Hirth and Lothe, 1982, Shim et al., 2007). For predicting the in-service performance of metals in fusion energy facilities, however, understanding of the influence of dynamic deformation on void strengthening mechanisms is crucial and requires atomistic insight of unit processes of dislocation–void interactions. In the past ten years, molecular dynamics (MD) has been used extensively to investigate dislocation–obstacle interactions in irradiated metals (Wu et al., 2007, Shim et al., 2007, Harry and Bacon, 2002, Osetsky and Bacon, 2003, Bacon and Osetsky, 2005, Bacon et al., 2006, Terentyev et al., 2007, Cheng et al., 2010). Major limitations arise in using MD to simulate large systems, for example systems with dislocation line lengths on the order of microns and void spacing on the order of hundreds of nanometers. Continuum simulation tools such as dislocation dynamics (DD) (Amodeo and Ghoniem, 1990a, Amodeo and Ghoniem, 1990b, Kubin and Canova, 1992, Van der Giessen and Needleman, 1995, Zbib et al., 1998, Bulatov et al., 1998, Cai and Bulatov, 2004) provide approximate description of dislocations at these larger scales, based on elastic theory of lattice dislocations. However, DD typically suffers from the approximate nature of prescribed short-range interaction rules between mobile dislocations and obstacles, lacks the ability to model dislocation dissociation, and requires approximations for dislocation cross slip and other important local mechanisms.
Pioneering MD simulations of vibrating dislocations were conducted but limited to 2D Frenkel–Kontorowa models (Weiner et al., 1976). Those models neglected phonon drag effects. Recently, Gumbsch and co-workers (Cheng et al., 2010, Bitzek and Gumbsch, 2004, Bitzek and Gumbsch, 2005 ) investigated dislocation depinning through MD simulations considering dynamic/inertial effects. Their results suggest that dynamic inertial effects significantly lower the depinning stress. It was suggested that such inertial effects should not be ignored in computational models at higher scales. However, continuum models, such as DD, typically assume overdamped dislocation migration via a constitutive force–velocity relationship.
Due to the spatio-temporal complexity of dislocation dynamics, various multiscale modeling methods (McDowell, 2010) including sequential (Amodeo and Ghoniem, 1990a, Amodeo and Ghoniem, 1990b, Kubin and Canova, 1992, Van der Giessen and Needleman, 1995, Zbib et al., 1998, Bulatov et al., 1998, Cai and Bulatov, 2004, Shehadeh et al., 2006, Hu et al., 2007) and concurrent (Tadmor et al., 1996, Zhou and McDowell, 2002, Fago et al., 2004, Shilkrot et al., 2002a, Shilkrot et al., 2002b, Zamora et al., 2012) approaches have been developed to describe dislocation physics. In sequential, hierarchical multiscale modeling approaches, it is intended that the characteristics and understanding of dislocation–obstacle interactions obtained from MD are incorporated in higher length scale continuum DD or crystal plasticity simulations. These parameters include maximum obstacle force, critical cusp angle, and Peierls stress. Although particular long range dislocation–obstacle interactions can be represented within continuum treatments, it is difficult if not impossible to address complex short range interactions and processes (e.g., core interactions). Concurrent approaches such as the Quaiscontinuum method (QC) (Tadmor et al., 1996, Fago et al., 2004) or Coupled Atomistics Discrete Dislocation (CADD) (Shilkrot et al., 2002a, Shilkrot et al., 2002b, Zamora et al., 2012) seek to address the crucial question of how to reconcile a consistent treatment of dislocations that pass between atomic and continuum regions; heuristic numerical techniques and/or rules are invoked for passing dislocations across interfaces between atomistic and continuum domains or through coarse-grained continuum domains with adaptive mesh refinement. As a consequence of these specialized treatments, existing concurrent approaches are only suitable for 2D quasistatic simulations of dislocations. For 3D dynamic dislocation–obstacle interactions, the recently developed concurrent atomistic–continuum (CAC) method (Xiong et al., 2011, Xiong et al., 2012a, Xiong et al., 2012b) is suitable as a formal coarse-graining of MD and is pursued in this work.
Section snippets
Methodology
Fundamental to the CAC method is a unified formulation of atomistic and continuum representation of balance laws (Chen and Lee, 2005, Chen, 2006, Chen, 2009). The CAC formulation generalizes Kirkwood’s statistical mechanical theory of transport processes (Kirkwood, 1946, Irving and Kirkwood, 1950) to facilitate a two-level structural description of materials. It describes the structure of a crystalline material in terms of continuously-distributed lattice cells, but with a group of discrete
Effects of void spacings on the dislocation pinning–depinning
Fig. 1 shows the computational configuration of CAC models in this study. Single crystal Ni specimens (∼50 × 200 × 100 nm3) contain over 20 million atoms. The V-notch and four spherical voids with diameters of ∼5 nm are initially introduced into the models. The distance between the V-notch tip and the centers of the voids is ∼40 nm. Here the V-notch is employed to induce a highly localized stress concentration to drive the dislocations into the specimen to explore inertial effects of interactions with
Summary and discussion
In summary, dynamic processes of dislocation bypass of voids with diameters of ∼5 nm and spacing up to ∼155 nm in Ni subjected to high strain rate loading have been simulated using the CAC method. Such large length scales are accessible only to massive MD simulations. Critical depinning cusp angles and dislocation line configurations have been determined. Atomic scale mechanisms are identified, including drawing out of screw dipoles and nucleation and formation of Hirsch loops. This work has
Acknowledgements
The authors are grateful for the support of a collaborative National Science Foundation research grant (McDowell, CMMI-1232878 and Chen, CMMI-1233113) to further advance and apply the CAC method, which in its present form is a culmination of developments supported in part by NSF – United States (CMMI 1129976), DARPA – United States (N66001-10-1-4018), and Department of Energy – United States (DOE/DE-SC0006539) (Chen and Xiong). Any opinions, findings, and conclusions or recommendations
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