A novel approach to study dislocation density tensors and lattice rotation patterns in atomistic simulations
Highlights
► Dislocation networks are automatically identified in atomistic simulations. ► Dislocation density tensors are computed during nanoindentation in copper. ► Results are comparable to experiments on a larger length scale. ► Observed dislocation density patterns are explained by dislocation slip system. ► At low temperatures deformation twin boundaries accommodate large strain gradient.
Introduction
The distribution of lattice rotations and geometrically necessary dislocations (GNDs) under an indenter tip during nanoindentation has recently attracted attention in both experiments (Zaafarani et al., 2006, Zaafarani et al., 2008, Kysar et al., 2007, Larson et al., 2007, McLaughlin and Clegg, 2008, Demir et al., 2009) and simulations (Zaafarani et al., 2006, Zaafarani et al., 2008, Demir et al., 2009, Bouvier and Needleman, 2006, Hua and Hartmaier, 2010). Cross-sectional cuts through the indented material show patterns of rotations and counter-rotations, as well as an inhomogeneous distribution of GNDs in face centered cubic (FCC) metals. Although experimental results are a valuable source of information, high resolution observations are only available post-mortem and can therefore not capture the ongoing mechanisms of plasticity on very small scales; a gap that can be bridged by simulations. In the present work we study the lattice rotation patterns and dislocation densities during nanoindentation in copper single crystals on an atomistic scale using molecular dynamics (MD). In contrast to classical continuum mechanics simulations, MD does not rely on a model for material response. Instead the response in MD is a result of atomic interactions, making it a powerful tool to gain insight into material behavior at small scales, where dislocations are the most important lattice defect during plastic deformation.
Because of the extremely large number of dislocations even in a well annealed crystal, only densities of dislocations are used to characterize the dislocation microstructure in experiments on the microscale. On the nanoscale two definitions of dislocation densities are feasible and will be used to analyze the performed simulations. The density of dislocations can be either defined as a scalar value, computed by dividing the total length of all contained dislocations by a reference volume v, or as a dislocation density tensor (DDT)where the line direction and the Burgers vector of each dislocation is taken into account. The GND is considered to be the minimum amount of dislocations in a finite volume to preserve the continuity of a lattice in the presence of strain gradients and is related to the DDT. As opposed to a scalar dislocation density, the DDT does not consider dislocations of opposite Burgers vectors within the same volume element. Such dislocations, called statistically stored dislocations (SSDs), neither produce strain gradients nor do they contribute to the lattice rotation (Gao et al., 1999).
The DDT , also called the Nye-tensor, is directly related to the lattice curvature K as presented in the work of Nye (1953), bywhere I denotes the identity matrix. Therefore, by measuring the local curvature within a crystal, it is possible to derive the DDT. This approach to obtain local GND densities has been applied in experimental studies using X-ray microscopy by Larson et al. (2007) or in electron backscatter diffraction (EBSD) tomography by Demir et al. (2009).
By calculating the local curvature caused by lattice rotations on the atomistic scale, GNDs can be derived from atomistic data in a similar approach to experiments. A far more precise alternative is however the application of Eq. (1), but this approach requires the knowledge of the dislocation network. The task of obtaining dislocation networks from large scale atomistic simulations of plasticity, consisting of millions of atoms, is highly non-trivial. Neither is a priori information concerning dislocations and their Burgers vectors available, nor are there explicit dislocation entities present. The dislocation network has to be extracted purely from atomic coordinates. Recently Stukowski and Albe, 2010a, Stukowski and Albe, 2010b published two methods to acquire this kind of data, where Burgers vectors are identified by constructing minimal Burgers circuits around defect structures. Here we present another solution for this task, in which the Nye-tensor analysis method as described by Hartley and Mishin, 2005a, Hartley and Mishin, 2005b is extended and integrated into the method of dislocation line extraction (Begau et al., 2011). Combining both methods enables a fully automated process to obtain dislocation networks including Burgers vectors and to derive dislocation densities and dislocation density tensors.
Section snippets
Obtaining Burgers vectors from dislocation density tensors
This section introduces the modifications to the original Nye-tensor method, an approach to derive Burgers vectors without constructing Burgers circuits explicitly, and its coupling with the dislocation line extraction (skeletonization) method, to identify dislocation networks. For details concerning the implementation of the original Nye-tensor method we refer to the work of Hartley and Mishin, 2005a, Hartley and Mishin, 2005b.
The definition of an atomistic scale dislocation density tensor
MD model
In order to demonstrate the dislocation network analysis and to compare results with experiments, a large-scale MD simulation of nanoindentation in single-crystal copper has been performed, using an embedded atom method (EAM) potential developed by Mishin et al. (2001). This ensemble is consisting of 12.3 million atoms and has a size of 60 nm×60 nm×40 nm. The top free surface is the crystallographic (111) plane. Periodic boundary conditions are applied to the sides of the simulation box. The
Discussion and conclusion
The explanation for the observed DDT patterns in the simulation has been found to lie in the geometry of the active slip system of copper by a detailed analysis of the dislocation network and its evolution. GNDs are concentrated in certain domains due to the three-fold symmetry of the slip system. A comparison of the DDT patterns found in our simulations with experimental results shows remarkable similarities, although the studied length scales are at least two orders of magnitude
Acknowledgments
The authors acknowledge financial support through ThyssenKrupp AG, Bayer MaterialScience AG, Salzgitter Mannesmann Forschung GmbH, Robert Bosch GmbH, Benteler Stahl/Rohr GmbH, Bayer Technology Services GmbH and the state of North-Rhine Westphalia as well as the European Commission in the framework of the European Regional Development Fund (ERDF).
Role of the funding source: None of the above-mentioned financial supporters has been involved directly during study, data analysis and writing of the
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