International Journal of Engineering Science 49 (2011) 1424–1434

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International Journal of Engineering Science

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Continuum metrics for deformation and microrotation from atomisticsimulations: Application to grain boundaries

Garritt J. Tucker a, Jonathan A. Zimmerman c,⇑, David L. McDowell a,b

a School of Materials Science and Engineering, Georgia Institute of Technology, 771 Ferst Drive N.E., Atlanta, GA 30332-0245, USAb Woodruff School of Mechanical Engineering, Georgia Institute of Technology, 801 Ferst Drive, Atlanta, GA 30332-0405, USAc Sandia National Laboratories, Livermore, CA 94550, USA

a r t i c l e i n f o

Article history:Received 15 October 2010Accepted 24 March 2011Available online 23 April 2011

Keywords:Grain boundary deformationAtomistic simulationsMicrorotationContinuum metrics

0020-7225/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.ijengsci.2011.03.019

⇑ Corresponding author. Tel.: +1 (925) 294 2437.E-mail address: jzimmer@sandia.gov (J.A. Zimme

a b s t r a c t

Motivated by a desire to incorporate micro- and nanoscale deformation mechanisms intocontinuum mechanical models of material behavior, we apply recently developed volume-averaged metrics to the results of atomistic simulations to investigate deformation andmicrorotation in the vicinity of grain boundaries. Three-dimensional bicrystalline struc-tures are employed to study the inelastic deformation behavior under uniaxial tensionand simple shear at a temperature of 10 K. Each bicrystal is constructed by molecular stat-ics followed by thermal equilibration under NPT using an embedded atom method poten-tial for copper. Strain is imposed in each simulation cell at a constant 109 s�1 strain rateapplied perpendicular and parallel to the grain boundary plane for tension and shear,respectively. A variety of grain boundary deformation mechanisms arise and the resultingdeformation and microrotation fields are examined. We also include an analysis showinghow microrotation varies as a function of distance from the grain boundary with increasingstrain for different grain boundary deformation mechanisms. This work demonstrates thatcritical interface behavior can be extracted from atomistic simulations using volume-averaged metrics, offering a potential avenue for translating fundamental information tocontinuum theories of grain boundary deformation in polycrystalline materials.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Grain boundaries (GBs) and their deformation accommodation mechanisms significantly influence the mechanical behav-ior of polycrystalline (PC) materials (Sutton & Balluffi, 1995), especially as grain size is reduced to the nanometer scale(Gleiter, 1989; Krasilnikov & Raab, 1999; Kumar, Van Swygenhoven, & Suresh, 2003). Materials with an average grain sizeof less than 100 nm are considered nanocrystalline (NC) (Koch, 2003). In recent years, tremendous efforts have been under-taken to understand and define their structure/property relationships (Dao, Lu, Asaro, De Hosson, & Ma, 2007; Guduru,Murty, Youssef, Scattergood, & Koch, 2007; Kumar et al., 2003; Warner, Sansoz, & Molinari, 2006). In NC metals, a varietyof GB processes during annealing and deformation have been observed both experimentally (Chen, Ma, et al., 2003; Chen,Wang, Qian, & Lu, 2003; Li et al., 2008; Liao et al., 2004; Liao et al., 2003) and with atomistic simulations (Monk & Farkas,2007; Schiøtz, Vegge, Di Tolla, & Jacobsen, 1999; Van Swygenhoven, Derlet, & Frøseth, 2004, 1999; Van Swygenhoven &Weertman, 2006; Yamakov, Wolf, Phillpot, & Gleiter, 2002). Some commonly observed GB-mediated processes includeheterogeneous dislocation nucleation (Liao et al., 2003; Van Swygenhoven, Derlet, & Hasnaoui, 2002; Tschopp & McDowell,2008), GB sliding (Farkas, Mohanty, & Monk, 2008; Schiøtz, Di Tolla, & Jacobsen, 1998; Schiøtz & Jacobsen, 2003; Van

. All rights reserved.

rman).

G.J. Tucker et al. / International Journal of Engineering Science 49 (2011) 1424–1434 1425

Swygenhoven & Derlet, 2001), and GB migration (Agnew, Elliott, Youngdahl, Hemker, & Weertman, 2000; Farkas, Frseth, &Van Swygenhoven, 2006; Kizuka, 2007). Furthermore, combinations of these mechanisms have been observed in NC fccmetals (Van Swygenhoven, Spaczer, Caro, & Farkas, 1999; Yamakov, Wolf, Phillpot, Mukherjee, & Gleiter, 2004; Zheng, Zhang,Chen, Lu, & Mai, 2009), and their interplay in constituting deformation accommodation in NC metals is still largely unknown.

Developing a theoretical understanding of GB deformation mechanisms in NC metals is no trivial task. While atomisticsimulation methods have provided insight into competing inelastic deformation mechanisms, few efforts have been madeto connect these observed mechanisms to continuum theory, which would be useful for engineering analysis of materialphenomena at length and time-scales beyond the limit inherent to atomistic simulations (i.e., molecular dynamics). Onestrategy for addressing this issue is to extract time and volume-averaged continuum measurables from atomistic simula-tions, which have direct implications for continuum-level theories. Accordingly, Zimmerman, Bammann, and Gao (2009) de-fined an atomic deformation gradient that is easily calculated within atomistic simulations. The present authors (Tucker,Zimmerman, & McDowell, 2010) built upon the work of Zimmerman et al. to develop formulations for microrotation andvorticity that can also be calculated from atomistic simulation data. Both of these efforts are useful in interpreting thekinematics of materials at the nanoscale in the context of continuum mechanics, and build upon key concepts developedby Eringen and coworkers (cf. Eringen, 1999) to illuminate microscale behavior of materials. Much of Eringen’s contributionsare especially relevant to atomistics, as they involve non-local continuum theories (Eringen, 2002) and the inter-atomicpotentials used are typically non-local in nature, extending beyond nearest-neighbor interactions.

The connection between Eringen’s concepts for micromorphic materials and the atomic deformation gradient developedby Zimmerman et al. (2009) was noted in the recent article by Zimmerman, Jones, and Templeton (2010), and warrants fur-ther exploration. The objective of the current work is to investigate the inelastic deformation of GBs utilizing the nano/microscale continuum metrics developed in Zimmerman et al. (2009), and Tucker et al. (2010). Simulations of symmetrictilt grain boundaries of copper using a bicrystal interface model are performed wherein the bicrystal configuration is subjectto uniaxial tension and simple shear loading conditions. Various GBs are employed under either tension or shear to activatedifferent deformation mechanisms (e.g., dislocation emission, GB sliding, and GB migration) and are subsequently analyzedwith respect to these metrics for deformation and microrotation. This analysis also includes a comparison of how the micro-rotation varies with distance from the GB and with increasing applied strain for the various GBs.

2. Computational methodology

Symmetric tilt grain boundaries (STGBs) were created as bicrystal interfaces. Two distinct crystal lattices, symmetricallyrotated around a common tilt axis (h001 i or h110 i), were separated by a planar STGB with its normal in the Y-direction, tiltaxis in the Z-direction, and GB period in the X-direction. Using molecular statics employing an embedded atom methodpotential for copper (Mishin, Mehl, Papaconstantopoulos, Voter, & Kress, 2001) and 3D periodic boundary conditions, initialminimum energy GB structures were determined. The initial GB structures were consistent with previous computationalstudies (Rittner & Seidman, 1996; Sansoz & Molinari, 2005; Spearot, Tschopp, Jacob, & McDowell, 2007; Tschopp,Tucker, & McDowell, 2008). Then molecular dynamics (MD) simulations were performed to achieve thermal equilibrationat 10 K under NPT for at least 10 ps. The resulting bicrystalline simulation cell dimensions were approximately16 nm � 32 nm � 16 nm, containing at least 700,000 atoms.

To investigate deformation mechanisms, two different methodologies were implemented to impose uniaxial tension andsimple shear. For uniaxial tension (Fig. 1(a)), a constant strain rate (109 s�1) deformation was applied in the Y-direction (per-pendicular to the GB plane) under NPT at 10 K. Three-dimensional periodic boundary conditions were maintained, while

Fig. 1. 3D schematic showing the computational setup and conditions for (a) uniaxial tension and (b) simple shear.

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simulation cell stresses along the lateral bounds were allowed to relax to zero normal stress. Simulations continued until 1–2% tensile strain beyond maximum tensile stress and after significant GB deformation (i.e., leading partial dislocation nucle-ation) occurred.

In shear (Fig. 1(b)), periodic boundary conditions were maintained in the X- and Z-directions (parallel to GB plane), butconstrained surfaces were imposed normal to the GB plane (Y-direction). Atoms located within three times the potential cut-off distance from either the top or bottom surface (normal to the Y-direction) were held fixed in their face-centered cubic(fcc) lattice positions and confined to move as rigid groups. All atoms located between these two rigid atomic groups weremobile in the MD simulations, while atoms belonging to either rigid group were not. This constraint imposed interatomicforces on neighboring mobile atoms, and suppressed the influence of free surfaces on mobile atom behavior. Next, a constantvelocity in the X-direction was applied to the top rigid group, while the bottom rigid group was held fixed in all directions.However, due to the small time-step of MD (1 fs) used and large value of velocity applied to the top group (corresponding toa 109 s�1 constant strain rate), a linearly ramped velocity field was also applied to all mobile atoms to reduce the possibilityof shock wave generation (Delogu, 2007; Horstemeyer, Baskes, Prantil, Philliber, & Vonderheide, 2003).

To evaluate GB deformation kinematics, microscale continuum metrics formulated for use within an atomistic framework(Tucker et al., 2010; Zimmerman et al., 2009) were computed in addition to an estimate of atomic strain. In particular, thedeformation gradient (F), lattice microrotation (/), and vorticity (w) were computed. These metrics offer different insightinto GB deformation behavior as compared to traditionally employed atomic measures of energy, centrosymmetry, slip vec-tor, and common neighbor analysis. A sense of path or deformation history was uncovered due to the use of reference neigh-bor lists in computing F and /; in contrast, instantaneous deformation behavior was uncovered by w (Tucker et al., 2010).

We next summarize each microscale metric formulation used for estimating atomic properties; the reader is encouragedto review the works by Zimmerman et al. (2009) and Tucker et al. (2010) for a more detailed formulation of F, /, and w. Theatomic deformation gradient, Fa

iI , is defined as

Xn

b¼1

ðxabÞiðXabÞM

!iM

¼ FaiI

Xn

b¼1

ðXabÞIðXabÞM

!IM

ð1Þ

where lowercase symbols represent current configuration quantities, uppercase symbols are reference configuration quan-tities, a identifies the atom of interest, b is a neighbor of atom a, and n is the number of neighbors belonging to atom a. Thisformulation provides flexibility in the non-local domain description by using different neighbor sets. After F is calculated,multiplicative decomposition (as shown in Tucker et al. (2010)) can be utilized to estimate both the rotation tensor, R,and the right stretch tensor, U, from

F ¼ RU ð2Þ

It was shown by Zimmerman et al. (2009) that this atomic-scale deformation gradient, F, satisfies the compatibility relationcurl(F) = 0 in the interior of a deformed crystal, even when defects such as dislocations and stacking faults have been created.

In addition to the microscale continuum metrics outlined in both Zimmerman et al. (2009) and Tucker et al. (2010), anatomistic estimation of strain is calculated utilizing F. We estimate atomic strain, e (referred to as dilatation for small strain),from the first invariant of the Green strain tensor, E, i.e.,

e ¼ trðEÞ ð3Þ

where E is defined as

E ¼ 12½FT F� I� ð4Þ

and I is the identity tensor. From R, and its skew-symmetric part, Rskew,

Rskew ¼12ðR � RTÞ ð5Þ

the microrotation vector, /, is defined as

/k ¼ �12�ijkðRskewÞij; ð6Þ

where �ijk is the permutation tensor.The velocity gradient, L, is calculated using an analogous method to that used for calculating F, i.e.,

Xn

b¼1

ðvabÞiðxabÞm

!im

¼ Laik

Xn

b¼1

ðxabÞkðxabÞm

!km

ð7Þ

This description is based on the instantaneous atomic velocities (v) and neighbor list in the current (or spatial) configuration.From the additive decomposition of L into both the rate of deformation tensor, D, and the vorticity tensor, W,

L ¼ DþW ð8Þ

G.J. Tucker et al. / International Journal of Engineering Science 49 (2011) 1424–1434 1427

W is calculated from both L and its transpose as

W ¼ 12

L � LT� �

ð9Þ

Once W is calculated, the vorticity vector, w, is defined by

wk ¼ �12�ijkðWÞij ð10Þ

For each metric employed in this work, only the influence of first nearest neighbors are considered. However, as explainedby Tucker et al. (2010), the extent of non-locality can be increased with additional neighbor sets. For example, second andthird nearest neighbors can be included in each calculation, providing a greater volume-averaged estimation of each metric.Including additional neighbor sets must be accompanied, however, by appropriate weight functions to properly account fortheir influence on each computed quantity. The additional neighbor sets will smooth out metric spatial fluctuations andthermal noise with the proper weight function, as discussed by Tucker et al. (2010). Furthermore, for higher order gradients,additional neighbor sets can also be used as Zimmerman et al. (2009) has discussed.

3. Results and discussion

Commonly observed deformation mechanisms in NC metals (e.g., dislocation nucleation, GB sliding and migration) areexamined in this work using microscale continuum metrics. Different copper STGBs are deformed at a constant 109 s�1 strainrate, either perpendicular (tension) or parallel (shear) to the GB plane. Both uniaxial tension and simple shear deformationare employed in this work to activate different deformation mechanisms: dislocation nucleation, GB sliding, and GBmigration.

3.1. Dislocation nucleation

To investigate dislocation nucleation under uniaxial tension, the R5 (310)h001i h = 36.9� STGB is used, where (310) des-ignates the GB plane normal, h001i is the tilt axis, and the symmetric misorientation angle is 36.9�. At approximately 8.7%tensile strain, partial dislocations nucleate from the GBs into both lattices. Shown in Fig. 2 is a series of snapshots of contoursbased on (a) centrosymmetry, (b) microrotation, /, (c) vorticity vector, w, and (d) the first invariant of the strain tensor (E), e,viewing along h110i just after maximum tensile stress (8.8%). For uniaxial tension, the simulation setup shown in Fig. 1(a) isfollowed.

Fig. 2(a) shows partial dislocations nucleating from the GB into the lattice during uniaxial tension. Boundary regions adja-cent to the nucleation sites also undergo deformation, but do not nucleate/emit dislocations. Each microscale continuummetric offers additional information concerning this deformation. In Fig. 2(b), rendering of microrotation shows that the ef-fect of dislocation slip extends outside the partial dislocation and trailing stacking fault. Distinct microrotation fields areapparent for the stacking fault and nearby lattice regions, while the GB shows no clear microrotation fields. The averageatomic microrotation (considering first nearest neighbors only) for the stacking fault is approximately 0.07, while in latticeregions adjacent to the emitted partial dislocation and stacking fault, atomic microrotation ranges between 0.003 and 0.03.However, no atomic slip occurs in these regions. Additionally, GB regions near the nucleation site display no comparable lev-els of microrotation as that calculated along the stacking fault.

Fig. 2(c) and (d) shows the vorticity vector, w, and the first invariant of strain (e) calculated at 8.8% uniaxial strain. InFig. 2(c), it is apparent that w captures the current deformation at the leading partial dislocations, while no fields emergein other lattice or GB regions. This is primarily due to the dependence of w on instantaneous atomic velocities and neighborlists in the current (spatial) configuration. Fig. 2(d) displays an estimate for atomic dilatation, e, for dislocation nucleation inthe R5 boundary. This metric shows that atoms involved with dislocation nucleation and slip constitute the majority of cal-culated strain. However, scattered interfacial regions also exhibit high values of e, meaning that GB reordering and shufflingduring dislocation nucleation is captured by e and is relevant to understanding strain accommodation in GBs. GB atomicstructures (i.e., structural units) with greater free volume promote reordering and shuffling events. Therefore, during dislo-cation nucleation, higher free volume GB regions not directly involved in nucleation also demonstrate noticable strain.

3.2. Grain boundary sliding

To investigate GB sliding, the R9 (221)h110 i h = 141.1� copper boundary is deformed under simple shear. At approxi-mately 4.5% shear strain, the peak shear stress (2.1 GPa) is reached activating GB sliding. Once this critical shear stress isreached, the GB continues to deform via sliding with further imposed shear. Fig. 3 shows (a) microrotation and (b) strain(e) for the R9 (221) boundary under shear at approximately 6.0% strain. At this strain, the R9 boundary deforms in a nearperfectly plastic manner via GB sliding. The shear deformation of this boundary was investigated in the authors’ previouswork (Tucker et al., 2010). In that work, the periodic simulation cell length along the tilt axis (Z-direction) was about0.5 nm, but in the current investigation it has been lengthened to about 16 nm. Both dislocation nucleation and GB slidingwere observed during shear in Tucker et al. (2010); however, only GB sliding is observed in the current investigation.Therefore, the observed deformation mechanism(s) in this boundary depends on the simulation cell dimensions.

Fig. 2. Contour plots showing (a) centrosymmetry, (b) microrotation, (c) vorticity vector, and (d) e viewing along h110 i for the R5 (310) boundary underuniaxial tension at approximately 8.8% strain. At this strain, partial dislocations have been nucleated from the GB into both the upper and lower lattices,connected to the interface by a stacking fault. Notice that each metric provides different information regarding dislocation nucleation and deformationaccommodation in both the lattice and GB.

Fig. 3. Contour plots showing (a) microrotation and (b) e viewing along the h110 i tilt axis for the R9 (221) boundary at approximately 6.0% shear strain.Each plot shows that GB sliding has been activated at this imposed strain and deformation is confined to the GB region.

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A possible reason for this disparity in the observed deformation mechanism is that certain atomic structural units have a3D character. This is important when configurations with small periodic lengths in the tilt axis are considered, because the

G.J. Tucker et al. / International Journal of Engineering Science 49 (2011) 1424–1434 1429

full character of the structural units might not be sufficiently sampled. Also, for dislocation nucleation to occur, a critical re-solved shear stress value must be reached on the activated {111} slip plane in the slip direction. Previous results (Spearotet al., 2007; Tschopp et al., 2008) show that the E structural unit does indeed influence the dislocation nucleation stress un-der uniaxial tension from the R9 boundary due to atomic structure and interfacial free volume. However, under shear, thehigher free volume E structural units might induce GB sliding at lower resolved shear stress in larger periodic structuresprior to the activation of dislocation nucleation from the interface. For this boundary, images showing contours of additionalmetrics (e.g., w) do not present any substantial or interesting behavior; therefore, they are not included in Fig. 3. Shear defor-mation is imposed using the setup shown in Fig. 1(b).

All significant deformation in the R9 (221) boundary, shown in Fig. 3, exists entirely within the GB region during GB slid-ing. However, certain boundary regions show higher microrotation, while other regions show greater e. Sansoz and Molinari(2004, 2005) have previously shown that atomic shuffling is prevalent during shear for this R9 boundary, and point to theimportance of particular interface structural units when interpreting shear deformation behavior. For the R9 (221) boundary,atoms showing the greatest microrotation in Fig. 3(a) are those located within these high free volume structural units. Suchunits may be locations where atomic shuffling processes initiate, eventually resulting in GB sliding. Calculated e values forthese same GB regions do not show a similar trend (Fig. 3(b)). The calculated e for each atom estimates the local volume expan-sion of the lattice without regard to shape change of the local neighborhood. Although atoms with the greatest dilatation arelocated within high free volume interfacial regions, they do not necessarily possess high microrotation as well. This indicatesthat consideration of multiple metrics is necessary to gain a complete picture of the kinematics that occur at and near the GB.

3.3. Grain boundary migration

The R3 (111)h110i h = 109.5� coherent twin boundary (TB) and the R129 (881)h110i h = 169.9� boundary are investi-gated to explore the deformation response during GB migration under shear. These two boundaries are studied because theirmigration mechanisms differ. The R3 (111) TB migrates via the glide of twinning disconnections (Anderoglu et al., 2010;Hirth & Pond, 1996) along the twin boundary plane, while the R129 (881) behavior is dictated by the migration of theGB dislocation cores (with translation as well as migration components parallel and normal to the boundary plane, respec-tively, as discussed by Cahn, Mishin, & Suzuki (2006)). Twinning disconnections contain both dislocation and step character(Hirth & Pond, 1996). They glide along the twin boundary, resulting in twin migration normal to the plane. These differencesin the boundary migration behavior are captured using microscale continuum metrics in post-processing, averaged over anon-local atomic neighborhood. By employing reference neighbor lists, microrotation and e contours shown in Section 3.3.2provide a detailed rendering of the GB dislocation core paths (both translation and migration) during shear deformation,while F12 shows the atomic deformation within the lattice region traversed by the migrating boundary. To impose shearon both the R3 and R129 boundaries, the methodology shown in Fig. 1(b) is used.

3.3.1. R3 (111) coherent twin boundaryUnder shear, the R3 (111) boundary migrates in a direction perpendicular to the GB plane as shown in Fig. 4(a). The peak

shear stress occurs at approximately 9.5% shear strain coinciding with the onset of GB migration normal to the GB plane.During GB migration, regions traversed by the migrating GB undergo rotation, so that the new lattice orientation vectorsof this region correspond to those in the upper lattice. The resulting microrotation estimate is provided in Fig. 4(b), F12 inFig. 4(c), and e in Fig. 4(d). Each contour plot in Fig. 4 corresponds to 11% shear strain, but migration continues at higherstrains as well.

For each metric calculated in Fig. 4(b)–(d), the lattice region traversed by the migrating boundary is visible. The calculatedvalues for each metric are relatively constant within this lattice region. The deformation of each atom within this region rel-ative to its neighbors is similar and is not a function of position, as shown in Fig. 4(b)–(d). In other words, the deformation ofeach atom relative to its nearest neighbors is similar for all atoms traversed by the migrating boundary, indicating a rela-tively uniform atomic deformation mechanism underlying GB migration in this boundary. For the R3 (111) boundary,migration is due to the glide of twinning disconnections. However, this same mechanism is not always active for migratingGBs. For example, the R129 (881) boundary is a low-angle boundary (disorientation angle is 10.1�), and the dislocation corestructures have definite paths through the traversed lattice region.

3.3.2. R129 (881) grain boundaryGB migration is activated in the R129 (881) boundary at approximately 1.8% shear strain. Fig. 5(a) displays both the ori-

ginal (dotted white line) and current GB location. The metrics used to produce the images shown in Fig. 5(b)-(d) correspondto those in Fig. 4(b)–(d). However, in Fig. 5(b) and (d), the migration path of each dislocation core is evident, while no suchpattern is seen in the migration of the R3 coherent TB in Fig. 4.

Fig. 5(b) shows microrotation for the R129 (881) GB, and the paths of the dislocation core structures display lower micro-rotation than the majority of atoms not directly traversed by the migrating dislocation core structures. This behaviour indi-cates that the deformation of the neighboring fields of these atoms varies with respect to other nearby atoms. Fig. 5(c) showsF12, and once again, a relatively uniform deformation field is observed as was seen for the R3 (111) boundary. However, ealso shows the migration path of the dislocation cores, and atoms lying between these migration paths show no calculatedstrain (e). This is again in contrast to the results of the R3 (111) boundary (Fig. 4(d)), where no spatial variation of e emerged.

Fig. 4. Contour plots showing (a) centrosymmetry, (b) microrotation, (c) F12, and (d) e viewing along the h110 i tilt axis for the R3 (111) coherent TB atapproximately 11.0% shear strain. In (a), the original GB location is shown by the dashed white line. During shear deformation the GB migratesperpendicular to its plane to the current location shown in (a).

1430 G.J. Tucker et al. / International Journal of Engineering Science 49 (2011) 1424–1434

3.4. Microrotation comparison

Although atomic-scale deformation is complex, the microscale continuum metrics (e.g., deformation gradient,microrotation, vorticity, and strain) utilized in this work offer additional and sometimes complementary insight intoatomic deformation behavior. Figs. 2–5 show that microrotation provides distinct information regarding the defor-mation history of each atom. Moreover, as clearly highlighted in Section 3.3 on GB migration, microrotation rendersclear deformation paths based on current atomic quantities relative to reference configuration neighbor lists (Tuckeret al., 2010). To further evaluate microrotation, Fig. 6 shows the distribution of average atomic microrotation as afunction of distance from the original GB location (GBo) for increasing imposed strain, where blue1 strain labels inFig. 6(a)–(d) mark the corresponding strain for contours shown in Figs. 2–5, respectively. To determine average micro-rotation in the vicinity of the GB, atoms within 20 Å on either side of GBo are considered during straining. Averagemicrorotation is then calculated per atomic layer (parallel to the GB plane), where each layer is approximately 2 Åthick. This thickness value provided accurate estimations of average microrotation while preserving computational effi-ciency. It is important to remember that lattice regions on both sides of GBo are included in the average microrotationcalculation.

Fig. 6(a) shows microrotation evolution through dislocation nucleation in the R5 (310) boundary, Fig. 6(b) represents GBsliding in the R9 (221) boundary, and Fig. 6(c) and Fig. 6(d) represent GB migration of the R3 (111) and R129 (881)boundaries, respectively. Each image shows that average atomic microrotation increases as imposed strain increases, inde-pendent of distance from the GB. As each bicrystalline computational cell is strained, the lattices undergo elastic distortions

1 For interpretation of color in Fig. 6, the reader is referred to the web version of this article.

Fig. 5. Contour plots showing (a) centrosymmetry, (b) microrotation, (c) F12, and (d) e viewing along the h110 i tilt axis for the R129 (881) boundary atapproximately 3.5% shear strain. In (a), the original GB location is shown by the dashed white line. During shear deformation the GB migrates perpendicularto its plane to the current location shown in (a).

G.J. Tucker et al. / International Journal of Engineering Science 49 (2011) 1424–1434 1431

to accommodate the imposed strain until activating the corresponding deformation mechanism. Once the stress required foractivating inelastic deformation is achieved within the computation cell, an increase in the average microrotation is ob-served in a manner which reflects the strain accommodating mechanism. This behavior continues with further imposedstrain during both dislocation nucleation (Fig. 6(a)) and GB sliding (Fig. 6(b)).

Partial dislocation nucleation occurs at approximately 8.7% uniaxial tensile strain in the R5 GB, and GB sliding is activatednear 4.5% shear strain in the R9 boundary. For the R5 GB, as additional dislocations are emitted from the boundary, averagemicrorotation in the vicinity of the GB increases. In Fig. 6(a), the initial set of partial dislocations have traveled more than20 Å from GBo by a strain of 0.088, which leads to the relatively large increase in microrotation at 20 Å from GBo. GB slidingin the R9 boundary exhibits a marked concentration of microrotation near the boundary compared to tensile dislocationnucleation. As shear strain increases, the average microrotation within about 5 Å of GBo increases substantially comparedto lattice regions greater than 5 Å from GBo, indicating the role of this GB structure during shear deformation and the local-ization of atomic microrotation during GB sliding.

However, the evolution of microrotation during GB migration differs from both partial dislocation nucleation andGB sliding. Once migration is activated, significant microrotation occurs as the migrating boundary traverses latticeregions. As evident in Fig. 6(c) and (d), the maximum level of microrotation within the bicrystal does not increaseproportionately with imposed strain. Rather, microrotation indicates the propagation of the GB as a moving frontthrough the lattice and the migration distance from the original GB location. For example, in Fig. 6(c), at approxi-mately 10% shear strain (just after peak shear stress), the R3 boundary has migrated a distance of about 13 Å result-ing in an average microrotation of about 0.095–0.097 for atoms traversed by the migrating GB. For the R129 GB, at2% shear strain (just after peak shear stress), the maximum average microrotation of about 0.045 is achieved andthe GB has traversed approximately 10 Å from GBo. Also displayed in Fig. 6(c) and (d), both the R3 and R129boundaries have propagated more than the 20 Å from GBo at the strain corresponding to the contours shown inFigs. 4 and 5.

Fig. 6. Average atomic microrotation as a function of distance from the original GB for (a) dislocation nucleation in the R5 (310) boundary, (b) GB sliding inthe R9 (221) boundary, (c) GB migration in the R3 (111) boundary, and (d) GB migration in the R129 (881) boundary. In each plot, the strain evolution ofmicrorotation is provided, and strains which correspond to nucleation and activation events for each mechanism are noted. The included arrows indicateplots of increasing strain.

1432 G.J. Tucker et al. / International Journal of Engineering Science 49 (2011) 1424–1434

4. Summary

Different GB deformation mechanisms were investigated using atomistic simulations under uniaxial tension and simpleshear at 10 K. Certain kinematic metrics from continuum mechanics were utilized in the analysis, and it was shown that eachmetric offered insight into aspects of each mechanism. The use of neighbor lists (reference and current) in this analysis pro-vides both history dependent and instantaneous atomic behavior. During partial dislocation nucleation, microrotation ishighest for stacking fault atoms, but nearby lattice regions also show microrotation. High free volume boundary regions pro-mote microrotation during sliding, and differences in GB migration are uncovered by both microrotation and strain (e). Themicrorotation in lattice regions near the GB for each deformation mechanism show characteristic behaviors. During both dis-location nucleation and GB sliding, the peak average microrotation increased with increasing strain, but was concentratednear the GB for sliding. During GB migration, the microrotation metric identifies the moving GB and is useful for determiningdifferences in migration behavior. It was shown that one continuum metric alone was not sufficient to uncover the defor-mation behavior of all mechanisms. Therefore, in more complex structures, such as NC materials, it would be necessaryto combine information provided from different metrics. In future work, we wish to investigate the effect of considering mul-tiple metrics to facilitate enhanced identification and characterization of competing mechanisms in the deformation of NCmaterials.

Acknowledgements

G.J. Tucker and D.L. McDowell are grateful for the support of the US National Science Foundation, NSF Grant CMMI-0758265 on Multiresolution, Coarse-Grained Modeling of 3D Dislocation Nucleation and Migration. This research was sup-ported in part by the National Science Foundation through TeraGrid resources provided by the TeraGrid Science Gateways

G.J. Tucker et al. / International Journal of Engineering Science 49 (2011) 1424–1434 1433

program. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a whollyowned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Adminis-tration under contract DE-AC04-94AL85000.

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