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A molecular dynamics (MD) simulation study to investigate the role of existing dislocations on the incipient plasticity under nanoindentation Ajith Ukwatta, Ajit Achuthan Department of Mechanical and Aeronautical Engineering, Clarkson University, Postsdam, NY 13699, USA a r t i c l ei n f o Article history: Received 3 January 2014 Received in revised form 7 April 2014 Accepted 1 May 2014 Available online 2 June 2014 Keywords: Dislocations Nanoindentation Molecular dynamics simulation Displacement burst a b s t r a c t Incipient plasticity is realized in the nanoindentation load-depth response as a load-drop under a dis- placement controlled loading condition or as a displacement-burst under a load controlled loading con- dition. Experimental results indicate that the characteristics of load-drop or displacement-burst such as the critical indentation load, critical indentation depth, magnitude of load-drop, or magnitude of dis- placement-burst, can vary substantially, depending on the microstructural conditions of the deformation volume. In this article, a molecular dynamic (MD) simulation study to investigate the role of existing dis- locations, particularly the role of mutual interaction of individual dislocations, on incipient plasticity under nanoindentation is reported. Dislocations are introduced into the perfect lattice structure of a cop- per sample by removing two adjacent layers of atoms. Subsequent stabilization of the this system results in extended edge dislocations (EEDs) consisting of two partial dislocations and a stacking fault. The nan- oindentation is simulated with three different tip radii on these samples consisting of various number of EEDs. Results show that, a substantial cross-slipping of atoms at constriction points formed by the motion of partial dislocations toward each other, driven by the indenter stress fi eld, is a potential deformation mechanism that yield load-drop (or displacement-burst). The effect of partial dislocations on critical indentation load, critical indentation depth, and magnitude of load-drop (or displacement-burst) is stud- ied. The mutual interaction of the local stress fi elds of neighboring partials have signifi cant infl uence on the deformation kinematics, and consequently, on the various characteristics of incipient plasticity. ? 2014 Elsevier B.V. All rights reserved. 1. Introduction Several experimental studies on understanding the mechanism of incipient plasticity observed in the form of displacement-burst in nanoindentation tests have been reported in the past decade. The displacement-burst is attributed to a spontaneous collective motion of dislocations on favorable slip planes. Experimental stud- ies have shown that the characteristics of displacement-burst changes signifi cantly with the sample conditions, such as variation in work-hardening or indentation near a grain boundary. Hence, the characterization of displacement-burst under various condi- tions may help in understanding the mechanism of deformation associated with incipient plasticity, especially the role of existing dislocations, and may aid in the design, development, and manu- facturing of materials with desirable properties. With the vast advancement in computational capabilities, molecular dynamic (MD) simulation has recently been developed as a reliable tool to study the nanoscale deformation. Early studies reported in the literature on the MD simulation of nanoindentation were by Landman et al. 1 and Hoover et al. 2 in the 1990s. Sev- eral studies that followed demonstrated the capability of MD sim- ulation to investigate various nanoscale deformation mechanisms under nanoindentation. For example, an MD simulation study by Fang et al. to understand the effect of temperature on nanoinden- tation showed that the elastic recovery decreased with increase in temperature 3. Similarly, studies investigating the anisotropy of metal at nanoscale by indenting in h100i; h110i and h111i direc- tions of Al and Cu demonstrated higher strength in h111i loading direction 4,5. The phase transformation of Si under nanoindenta- tion from Si-II phase to a mixture of Si-III and Si-XII phases has also been investigated using MD 6. Rate-dependent grain growth of nano-crystalline metal was studied by Tuker and Foiles 7 using MD based nanoindentation simulation as well. The effect of inden- ter geometry on the mechanical properties of titanium has also been studied using MD 8. MD simulation studies specifi cally focusing on investigating the mechanism of load-drop (or displacement-burst), and the associ- ated instability conditions, have made major contributions in understanding various aspects of this phenomena. Kelchner et al. /10.1016/matsci.2014.05.001 0927-0256/? 2014 Elsevier B.V. All rights reserved. Corresponding author. Tel.: +1 3152684429; fax: +1 3152686695. E-mail address: (A. Achuthan). Computational Materials Science 91 (2014) 329338 Contents lists available at ScienceDirect Computational Materials Science journal homepage: 9 studied the displacement-burst in perfect crystal, and attrib- uted the formation and looping of prismatic dislocation nucleation under the indenter tip as the fundamental mechanism that causes displacement-burst.Unlike nanoindentationexperiments,MD simulation enabled a detailed realtime monitoring of atomic scale deformation as the indentation progresses, showing nucleation of initial dislocations inside the bulk directly under the indenter. The centro-symmetry parameter, fi rst proposed in this study to visualize the dislocation, has been widely used by researchers. Effect of surface roughness on nanoindentation behavior and dis- placement burst of perfect crystal was studied by Zimmerman et al. 10. The presence of surface roughness was found to have signifi cant effect on the critical resolved shear stress (CRSS), with the CRSS decreasing with increase in surface roughness. The irreg- ularities in steps such as kinks were found to act as strong anchor points for dislocations making them diffi cult to move along the edge 11. The surface irregularities also acted as heterogeneous dislocation nucleation cites those activated before the displace- ment-burst 12. These unstable dislocations, caused the indenta- tion load depth curve to deviate from Hertzian behavior before the transition to plastic region occurred. Based on studying a per- fect crystal sample, Begau et al. 13 reported that the initial dislo- cation was nucleated in the bulk material right below the indenter when the shear stress approached the theoretical strength. An algorithm to skeletonize the dislocation in order to form the dis- crete atomic arrangement into smooth continuous curves was used. All of these studies have considered perfect crystal condition for their simulation. However, the presence of statistically stored dis- locations (SSDs) is expected to have signifi cant infl uence on the incipient plasticity and its evolution. Even for well annealed metals with the dislocation density in the range of 106108cm=cm3, the average dislocation spacing is comparable to the size of the defor- mation volume of a low load nanoindentation experiment. For cold worked metals with a larger dislocation density in the range of 10101011cm=cm3 , the infl uence of dislocations on the nanoscale deformation mechanisms is even more critical. In order to investigate the effect of existing dislocations, the nanoindentation on a plastically deformed material was simulated by Loads et al. in a recent study 14. The plastic deformation was introduced into a CaF2perfect crystal with an initial indentation using a large spherical indenter of 12 nm radius. A second indenta- tion with a 4 nm radius indenter was simulated on this already deformed sample, and the deformation characteristic of this higher dislocation density region was studied. A gradual transitioning of the deformed region from elastic to plastic, without showing any signifi cant load relaxation, was reported in this study. In a similar study, the effect of nano-machining on Cu single crystal was inves- tigated 15. A scratch was introduced on the sample surface with a 4 nm spherical tip, followed by the nanoindentation with a 3 nm spherical tip. As expected, a decrease in the critical displace- ment-burst load for samples with larger scratch depth was observed. A study on the effect of pre-existing vacancies in nanoin- dentation also showed that the critical displacement-burst load decreased in the presence of vacancies 16. Effect of fi nite temper- ature and the presence of vacancies have also been studied, and the results demonstrated that the yielding is sensitive to the size of defects and loading conditions 17. Yield stress decreased signifi - cantly at the elevated temperature and in the presence of large defects. Effect of existing dislocations has also been studied in the context of understanding the role of grain boundaries on nano- scale deformation mechanisms 18. The initial dislocation nucle- ation was found to occur at twin boundaries, well before the nucleation of homogeneous dislocation in the bulk region, due to the lower fault energy of twin boundaries when compared to the unstable stacking-fault energy of the bulk. Similarly, a study on the effect of an atom island on nanoindentation has also been reported 19. For a smaller atom island the critical load was lower and the dislocations nucleated at the island boundaries, while for relatively larger island the effect was insignifi cant. In summary, despite many idealizations involved, MD simula- tions successfully demonstrated incipient plasticity under nanoin- dentation in the presence of a complex network of pre-existing dislocations introduced through a pre-deformation step. This sta- tistical approach to understand the collective infl uence of a group of pre-existing dislocations on deformation associated with nano- indentation provided great insight into the underlying mechanism. However, the mechanism of mutual interaction of individual dislo- cations, and its role on incipient plasticity, cannot be captured in these studies due to the complex nature of the introduced disloca- tions, making the isolation of individual effects extremely diffi cult. Understanding the mechanism of interaction of individual disloca- tions is critical to explain their role in the observed collective behavior as a function of dislocation density. Moreover, this under- standing is fundamental for developing physics based theories to describe incipient plasticity. It is also important to note that the pre-existing dislocation distribution introduced through a pre- indentation, or a pre-scratch, is very specifi c to those pre-loading conditions, and need not necessarily represents the nature and dis- tribution of the dislocation network in an annealed or uniformly work-hardened samples. The objective of the present study was to understand the role of existing dislocations on the mechanism of incipient plasticity, by investigating the interactions between the indentation stress fi eld and the local stress fi eld of individual dislocations. Partial disloca- tions were introduced into the sample by removing layers of atoms and then stabilizing the sample. The pair of straight partial disloca- tions and the associated stacking fault formed during the stabiliza- tion, though quite different geometrically from the randomly distributed complex network of dislocation loops expected in real samples, are ideal for studying the individual stress fi eld interac- tions, without having to deal with the complexities of the geome- try. Being partial dislocations, these dislocations do have both screw and edge characteristics initially. In addition, these pre- existing partial dislocations also bow-out or bow-in under the very initial part of nanoindentation loading, thereby gaining more real- istic characteristics of individual dislocations. Results from the present study clearly demonstrate the role of local stress fi eld of each dislocation on their mutual stability, and also on the incipient plasticity. Signifi cant cross-slipping of atoms was found as a major deformation mechanism that drives the incipient plasticity. 2. Simulation method A copper sample with the dimension 25 ? 24 ? 24 nm along x; y and z axes, respectively, was considered for the study. The sample consisted of 1,250,000 atoms. The EAM potential Cu u3 was used to model the inter-atomic forces of the system 20. Indentation was performed in the ?111 plane, such that the FCC crystallographic structure of copper has one of the slip planes aligned perpendicular to the loading direction (Fig. 1). Shrink-warp boundary condition was used in the indentation direction (z axis), while the periodic boundary condition was used in x and y directions. The shrink-wrap boundary conditions do not let the atoms move out of the simulation box or enter into it. The atoms are free to move with the simulation box as the material deform. The periodic boundary condition, on the other hand, allows atoms to move across the simulation box; i:e. an atom which moves out from one side of the box will re-enter from the opposite side of the box. Simulations were conducted for room temperature condition, with the average temperature maintaining 330A. Ukwatta, A. Achuthan/Computational Materials Science 91 (2014) 329338 at 300 K. The average temperature was maintained throughout the simulation by re-scaling velocities of individual atoms at every 100 time steps. Before the indentation was performed, the total internal energy of the sample was minimized using the conjugate gradient (CG) method. Indentation load was then applied at the center of the top surface. The indentation was performed under displacement controlled loading condition, ramping the indentation-depth to a predetermined maximum value at a predetermined constant rate. Accordingly, the application of indentation load was simulated by deforming the top surface of the sample in such a way that the deformed surface followed geometry of the penetrated part of the rigid indenter tip for the given indentation depth. A loading rate of 0:1 per time step, with the time step size of 0.001 ps were used for the simulation. A maximum indentation depth of 1 nm was used in this study. The energy of the system was minimized using CG method at the end of each time step. The interaction between the indenter tip and the sample is governed by the relation Fr ?Kr ? R21 where F is the repulsive force between the indenter and an atom located at a distance of r from the center of the indenter tip, K is the force constant and R is the indenter radius. Stresses reported in this study are based on virial stress defi ni- tion calculated per atom basis. Virial stress tensor for a given atom (I) is given by Sab ? 1 2 X Np n1 r1aF1b r2aF2b ? 1 2 X Nb n1 r1aF1b r2aF2b ? 1 3 X Na n1 r1aF1b r2aF2b r3aF3b ? 1 4 X Nd n1 r1aF1b r2aF2b r3aF3b r4aF4b ? 1 4 X Ni n1 r1aF1b r2aF2b r3aF3b r4aF4b ?# 2 where the pairwise energy over the Npneighbors of atom I is given by the fi rst term. The r1and r2are position vectors of atoms in the pair, and F1and F2 are the corresponding forces due to the pair interaction. The second, third, fourth, and fi fth terms correspond to the energy contribution of Nbbond, Naangle, Nddihedral, and Niimproper interactions. Centro-Symmetry Parameter (CSP), as defi ned below, was used to visualize the dislocation nucleation and motion 9. CSP X N=2 i1 jR ! i R ! iN=2j 2 3 where N is the number of nearest neighbors of each atom and R ! i; R ! iN=2are vectors from the central atom to the two atoms in a given pair. An atom with the CSP value between 0.5 and 4 is con- sidered to be displaced to form a partial dislocation, while an atom with the CSP value in the range of 48 is considered to be in a stak- ing fault. Samples with existing dislocations were formed as follows. A perfect crystal atomic structure was fi rst formed and the system was stabilized using the CG method. Dislocations were then intro- duced by removing one or more sets of two half layers of atoms representing a full edge dislocation in (110) plane (Fig. 2a, c and e) 21. The system is again stabilized. For the case shown in Fig. 2a, the stabilization process decomposed the full edge disloca- tion into two partial edge dislocations, creating a stacking fault rib- bon. The two partial edge dislocations along with the associated stacking fault ribbon bounded between these two partials will be collectively referred to as an extended edge dislocation (EED). Three different sample conditions, namely, a single EED (1-EED), two EEDs (2-EEDs) and three EEDs (3-EEDs), were studied. For the 2-EEDs case, EEDs were formed by removing two half layers of atom from the middle of the sample as shown in Fig. 2c, fol- lowed by stabilization leading to 2-EEDs structure as shown in Fig. 2d. Similarly, 3-EEDs was introduced by removing atom layers as shown in Fig. 2e, and the resulting structure after stabilization is as shown in Fig. 2f. In this particular confi guration the stacking faults are in ?111 plane, normal to the indenter, making it more sensitive to the indentation stress fi eld. For convenience, the following sign convention is introduced to defi ne the orientation of two partials associated with an EED. A pair of partial dislocations associated with an EED is defi ned nega- tive if the layers of atoms above the stacking fault of those partials, and positive in the opposite case. Accordingly, for 1-EED case with the partials missing the atomic layer above the stacking fault, the partials are negative. For the 2-EEDs case, the two partials associ- ated with the top EED can be considered as having two additional layers of atom present above the stacking fault (or missing atom layers below the