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Research Papers

A Predictive Framework for Dislocation-Density Pile-Ups in Crystalline Systems With Coincident Site Lattice and Random Grain Boundaries OPEN ACCESS

[+] Author and Article Information
David M. Bond

North Carolina State University,
Raleigh, NC 27695-7190
e-mail: dmbond@ncsu.edu

Mohammed A. Zikry

College of Engineering,
North Carolina State University,
Campus Box 7910/3154 EBIII,
Centennial Campus,
Raleigh, NC 27695-7190
e-mail: zikry@ncsu.edu

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received June 16, 2016; final manuscript received September 9, 2016; published online February 13, 2017. Assoc. Editor: Xi Chen.

J. Eng. Mater. Technol 139(2), 021023 (Feb 13, 2017) (8 pages) Paper No: MATS-16-1184; doi: 10.1115/1.4035494 History: Received June 16, 2016; Revised September 09, 2016

Evolving dislocation-density pile-ups at grain-boundaries (GBs) spanning a wide range of coincident site lattice (CSL) and random GB misorientations in face-centered cubic (fcc) bicrystals and polycrystalline aggregates has been investigated. A dislocation-density GB interaction scheme coupled to a dislocation-density-based crystalline plasticity formulation was used in a nonlinear finite element (FE) framework to understand how different GB orientations and GB-dislocation-density interactions affect local and overall behavior. An effective Burger's vector of residual dislocations was obtained for fcc bicrystals and compared with molecular dynamics (MDs) predictions of static GB energy, as well as dislocation-density transmission at GB interfaces. Dislocation-density pile-ups and accumulations of residual dislocations at GBs and triple junctions (TJs) were analyzed for a polycrystalline copper aggregate with Σ1, Σ3, Σ7, Σ13, and Σ21 CSLs and random high-angle GBs to understand and predict the effects of GB misorientation on pile-up formation and evolution. The predictions indicate that dislocation-density pile-ups occur at GBs with significantly misoriented slip systems and large residual Burger's vectors, such as Σ7, Σ13, and Σ21 CSLs and random high-angle GBs, and this resulted in heterogeneous inelastic deformations across the GB and local stress accumulations. GBs with low misorientations of slip systems had high transmission, no dislocation-density pile-ups, and lower stresses than the high-angle GBs. This investigation provides a fundamental understanding of how different representative GB orientations affect GB behavior, slip transmission, and dislocation-density pile-ups at a relevant microstructural scale.

FIGURES IN THIS ARTICLE
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The structure and properties of GBs in a crystalline material strongly influence the material's mechanical response [1,2], and the interaction of lattice dislocations with GBs is a significant aspect in determining and characterizing behavior at different scales. Experimental observations have shown that specific CSL and low-angle GBs transmit dislocations more easily than high-angle GBs [36], which results in local stress relaxation [7] and increased strength and ductility [813]. Substantial barriers to dislocation transmission are associated with random and CSL high-angle GBs, and it can lead to dislocation pile-ups [1418] and localized stress concentrations [19,20] that result in crack nucleation and propagation [9,21,22]. As shown by these experimental observations, individual GB structure, properties, and orientations can have a substantial impact on material behavior. These transmission electron microscopy and scanning electron microscope experimental observations and measurements, however, are difficult to scale to the polycrystalline aggregate level due to the evolving behavior of dislocation densities and pile-ups.

Computational investigations have provided fundamental insights into local conditions to further understand the evolving anisotropic behavior of GBs to plastic deformation. Dislocation-GB interactions have been characterized using MDs [14,17,2326], discrete dislocation dynamics (DDDs) [2729], and finite element methods (FEMs) [22,3033]. The MD analysis of dislocations impinging on a GB by Koning et al. [17] and Sangid et al. [26] resulted in criteria for dislocation transmission. Although MD studies provide atomic-scale detail in the dislocation-GB interaction process, MD models are limited by physically unrealistic temporal and spatial physical scales and a low density of dislocations [25,34]. Experiments have shown that there are complex interactions from multiple active slip systems at the GB that evolve during plastic deformation [35,36].

To address these challenges, we have developed a multiple-slip dislocation-density based crystalline plasticity constitutive framework that incorporates a physically based dislocation-density GB interaction scheme accounting for dislocation-density transmission and blockage with a specific focus on the microstructural scale relevant for GBs and dislocation-density interactions, which is essential for physically realistic predictions of evolving dislocation-density pile-ups. We incorporated the dislocation-density based crystalline plasticity constitutive and the dislocation-density GB interaction scheme into a nonlinear FE framework. With this approach, significant details of the local effects of dislocation-GB interactions that have mostly been captured through atomistic schemes can be elucidated at the microstructural scale and applied to physically realistic problems. We will use the nonlinear FE framework to investigate the heterogeneous mechanical response of GBs in fcc bicrystals and polycrystals with different CSL orientations, such that potential sites for dislocation-density pile-ups can be predicted. The specific CSL GBs investigated are Σ1, Σ3, Σ7, Σ13, and Σ21, which were chosen because they span low- and high-angle misorientations. This paper is organized as follows: the dislocation-density crystalline plasticity formulation and the dislocation-density GB interaction scheme are presented in Sec. 2, bicrystal and polycrystal results are presented and discussed in Sec. 3, and a summary of the salient results and conclusions are given in Sec. 4.

In this section, the multiple-slip crystal plasticity constitutive formulation, including the coupled mobile and immobile dislocation-density evolution equations and dislocation-GB interaction scheme, will be presented.

Multiple-Slip Crystal Plasticity Formulation.

The crystal plasticity constitutive framework used in this study is based on the formulation developed in Refs. [22,37,38], and only a brief outline will be presented here. It can be assumed that the velocity gradient is decomposed into a symmetric deformation rate tensor Dij and an antisymmetric spin tensor Wij [39]. The deformation rate and spin tensors can then be additively decomposed into elastic and plastic parts, which are defined in terms of the crystallographic slip rates on each slip system. A power law relation characterizes the rate-dependent constitutive description on each slip system, and the reference stress used is a modification of widely used classical forms [40] that relate reference stress to immobile dislocation-density,  ρim(α).

Following the approach of Zikry and Kao [41], it is assumed that, for a given deformed state of the material, the total dislocation-density,  ρ(α), can be additively decomposed into mobile, ρm(α), and immobile dislocation-density, ρim(α). Furthermore, the mobile and immobile dislocation-density rates can be coupled through the formation and the destruction of junctions as the stored immobile dislocations act as obstacles for the evolving mobile dislocations. This is the basis for taking the evolution of mobile and immobile dislocation-densities as Display Formula

(1)dρmαdt=|γ˙α|[gsourb2(ρimαρmα)gminter-ρmαgimmob-bρimα]
Display Formula
(2)dρimαdt=|γ˙α|[gminter+ρmα+gimmob+bαρimαgrecovρimα]

where gsour is a coefficient pertaining to an increase in the mobile dislocation-density due to dislocation sources, gminter are coefficients related to the trapping of mobile dislocations due to forest intersections, cross-slip around obstacles, or dislocation interactions, grecov is a coefficient related to the rearrangement and annihilation of immobile dislocations, and gimmob are coefficients related to the immobilization of mobile dislocations. These nondimensional coefficients are determined as functions of the crystallography and deformation mode of the material by considering the generation, interaction, and recovery of dislocation-densities as discussed by Shanthraj and Zikry [42].

Dislocation-Density Grain-Boundary Interaction Scheme.

The dislocation-density GB interaction scheme is outlined in this section. If an incoming slip system does not completely coincide with an outgoing slip system, dislocation-density transmission through the GB will leave residual dislocations in the boundary due to the conservation of lattice defect vectors [5,16,26,35]. The energy for the formation of a residual dislocation can be considered as the energy barrier for dislocation transmission [43,44]. Based on a line-tension model developed by Koning et al. [17], this energy barrier due to creation of GB residual dislocations for incoming and outgoing slip systems α and β can be postulated as Display Formula

(3)UGB(αβ)=κGΔbeff2Δ2

where κ is approximately equal to 0.5, G is the shear modulus, Δbeff is the effective residual Burger's vector, which is a function of the misorientation of slip planes and the magnitude of the true residual Burger's vector. Δ2 is the length of the residual dislocation, and it is a function of the resolved shear stress for outgoing slip system β.

A dislocation transmits through the GB when a Frank-Read source is activated in the presence of a GB obstacle using the line tension model. Activation occurs when critical conditions are reached, defined as Display Formula

(4)ψ=Δ1Δ2=cosνsinνcos B, and tanB=Δb22b(α)2

where ν is the angle of intersection between incoming and outgoing slip planes, Δ1 is the length of the dislocation-GB intersection line, Δ2 is the length of the residual dislocation, Δb is the residual Burger's vector, and b(α) is the Burgers vector of dislocations on the incoming slip system. The case of ν0 with a residual Burgers vector Δb is energetically equivalent to a case of ν=0 with an increased effective residual Burger's vector Δbeff [17]. Therefore, an effective residual Burger's vector is calculated that increases with the increasing angle of intersection as Display Formula

(5)(ΔbeffΔb)2=1+(ψ22ψcosν+1)1/2

Dislocation-density transmission is considered on the most energetically favorable outgoing slip system by selecting the lowest value of the transmission energy barrier from incoming slip system α to all possible outgoing slip systems β as Display Formula

(6)UGB(α)=minβUGB(αβ)

The power law equation for slip-rate at the GB has been modified [16] to include a GB transmission factor (GBTF) based on the energy barrier to account for the interactions of dislocation-densities with GBs as Display Formula

(7)γ˙(α)=γ˙ref(α)[τ(α)τref(α)][|τ(α)|τref(α)]1m1GBTF(α)

where k is the Boltzmann constant, T is the absolute temperature, and GBTF(α)=e(UGB(α)/kT). It can range from 0 to 1, where 0 corresponds to full blockage of dislocation-densities and 1 corresponds to complete transmission.

The rate at which dislocations impinge upon the GB is derived from dividing the dislocation velocity, obtained from the Orowan relation as |γ˙(α)|/b(α)ρm(α), by the mean distance traveled by the dislocation 1/ ρm(α). Assuming a square root relation and normalizing by Burger's vector, we obtain the following expression for the rate of accumulation of residual GB dislocations: Display Formula

(8)ρ˙GB(α)=Δbb(α)|γ˙(α)|ρm(α)b(α)

Effective Burger's Vector of Residual Dislocations in Copper Bicrystals.

The influence of the effective Burger's vector of residual dislocations, Δbeff, on dislocation transmission at GBs, as well as the relation of Δbeff to static GB energy, has been investigated in this section for fcc copper bicrystals. The effective Burger's vector, Δbeff, was calculated for two different GB rotation axes, [001] and [111], which were chosen to represent a wide range of GB misorientations. To elucidate the effect of Δbeff on dislocation-density transmission, a convergent mesh of 2500 elements for a 1 mm × 1 mm bicrystal of fcc copper with CSL GBs spanning a range of Δbeff was deformed quasi-statically in tension to 5% nominal strain. The CSL GBS that were investigated were Σ3, Σ13, and Σ7 because they included low- and high-angle misorientations. The rotation angles about the [111] axis for the CSL GBs chosen were as follows: the Σ13 CSL had a 27.8 deg misorientation, the Σ7 CSL had a 38.2 deg misorientation, and the Σ3 CSL had a 60 deg misorientation. The material properties for pure copper are listed in Table 1.

Effective Burger's Vector and Grain Boundary Energy.

We analyzed how the effective Burger's vector of fcc bicrystals varied with GB misorientation and we compared our predictive trends with static MD results from Ref. [26]. An effective Burger's vector was obtained for each possible slip system interaction for the 12 fcc slip systems in a bicrystal (Eq. (5)), and the average magnitude of the effective Burger's vector, |Δbeff|, was obtained at the GB for the three slip systems with the smallest |Δbeff| for each rotation angle (Fig. 1). This average |Δbeff| was compared with static MD predictions [26] for the same misorientations. As the results indicate, the static GB energy and average |Δbeff| for GBs with [001] and [111] rotation axes (Fig. 1) have the same trends as those predicted in Ref. [26]. The maximum and minimum GB energies occur at the same rotation angles, but |Δbeff| were generally lower than the MD predictions by a factor of three. The quantitative differences between the atomistic and the microstructural predictions are due to differences in physical scales and computational approaches, but, most importantly, the trends predicted for GB energy at the microscale are consistent with those predicted using MD. The correlation between the maximum and minimum values of both methods is an indication that the effective Burger's vector, |Δbeff|, can be used to ascertain, on a relevant microstructural scale, GB energies for fcc GBs with these rotation axes.

Grain Boundary Dislocation-Density Transmission.

We investigated how the effective Burger's vector of residual dislocations varied for copper bicrystals with Σ3, Σ13, and Σ7 CSL GBs and how it was related to GB transmission. These were chosen since they span a wide range of misorientation angles. For the [111] axis, the three relative maximums for average |Δbeff| occurred at rotation angles of 38.2 deg (Σ7), 89 deg (Σ13), and 158 deg (Σ7) with CSL orientations based on Brandon's criterion [46]. A specific GB orientation is not unique to one rotation angle, so multiple rotation angles would have the same average |Δbeff|. The predictions indicate that a GB's dislocation transmission is inversely related to its average |Δbeff| (Fig. 2). For example, a bicrystal with minimum average |Δbeff| (Σ3) had a GBTF of almost 1 on its most active slip system, where very little slip was obstructed. The Σ7 CSL bicrystal, on the other hand, substantially impeded dislocation transmission with a GBTF of 0.67 on its most active slip system. The Σ13 CSL bicrystal had similar transmission behavior with a GBTF of 0.64. These predictions are consistent with experimental observations by Abuzaid et al. who reported an inverse relationship between the magnitude of the residual Burger's vector and GB transmission [47].

Grain Boundary Effects in Polycrystalline Aggregates.

The influence of GBs and triple junctions on the microstructural response of a polycrystalline fcc aggregate has been investigated in this section to further understand how bicrystal behavior scales to polycrystalline behavior. A 1 mm × 1 mm model for fcc copper with 16 grains, with material properties as listed in Table 1, was used with a plane strain convergent FE mesh of 5900 elements. Euler angles for rotations about the [111] axis were applied to each grain in the aggregate. A distribution of GBs (Table 2) with the [111] rotation axis was chosen because it includes GBs with both low and high dislocation-density transmissions. Misorientations between grains were determined using the rotation matrices of each crystal in the system (Fig. 3). The methods detailed in Ref. [48] were used to compare these misorientations with tabulated data [49] to accurately determine CSL relationships within an allowable deviation based on the Brandon criterion, Δθ15 Σ1/2 [46]. The distribution of GBs in the aggregate is as follows: 12.1% Σ1, 33.3% Σ3, 21.2% Σ7, 6.1% Σ13, 18.2% Σ21, and 9.1% random high-angle GBs. A displacement load was applied to the top surface of the model at a quasi-static strain rate, and the bottom surface was fixed (Fig. 4).

Dislocation-Density Pile-Ups and Stress Accumulation.

To demonstrate how pile-ups of residual dislocation-densities at grain boundaries led to stress accumulations, the normalized (by the yield stress) normal stress and residual GB dislocation-density at 3% nominal strain are shown in Fig. 5. The maximum normal stress was 22 and occurred in three locations indicated region 1, region 2, and region 3 (Fig. 5(a)). Region 1 was a Σ7 CSL boundary, and the two other maximum stress locations, region 2 and region 3, were near triple junctions. Most of the highly stressed GBs in the aggregate were Σ7 CSL boundaries, where the largest dislocation-density pile-ups occurred. The lower GB stresses, corresponding to the Σ1 and Σ3 CSL GBs, did not have any dislocation-density pile-ups. This is consistent with the bicrystal results (Fig. 2), as well as experimental observations that GBs with aligned slip planes can easily transmit dislocations, which would inhibit pile-up formation and stress accumulation at GBs [4,7,50,51].

Dislocation-Density Pile-Ups at Grain Boundaries.

Dislocation-density pile-ups on active slip systems occurred at the Σ7 CSL GB, as indicated region 1 (Fig. 5), when the transmission of dislocation-densities was impeded at the boundary. The immobile dislocation-density and GBTF of the most active slip system for each grain in region 1 are shown in Fig. 6. In grain A, the most active slip system was (11¯1)[011]. The maximum normalized immobile dislocation-density was 3.37 × 107, and the GBTF was 0.49. In grain B, the most active slip system was (1¯1¯1)[011] with a maximum normalized immobile dislocation-density of 2.7 × 107 and a GBTF of 0.57. The low transmission from both sides of the GB was due to the large slip plane misalignment of the Σ7 CSL GB, which has the largest average |Δbeff| for GBs with the [111] rotation axis, as seen in Fig. 2. Similar transmission blockage behavior at a Σ7 GB was predicted by MD calculations [14]. Furthermore, the large |Δbeff| of active slip systems at the interface led to the accumulation of residual dislocation-densities in the GB. Hence, the normal stress at the GB increased as the pile-up of dislocation-densities both on active slip systems and within the GB increased and evolved (Fig. 7).

Dislocation-Density Pile-Ups at Triple Junctions.

Triple junctions (TJs) have a significant influence on a material's mechanical response and can be favored sites for damage nucleation [23,52,53], and this is due to multiple slip and dislocation density interactions at the junction. In this section, we investigated the behavior of the TJ indicated region 2 (Fig. 5), which consisted of Σ1, Σ7, and Σ13 CSL GBs. The most active slip system in the TJ region was (11¯1)[011] with a maximum normalized immobile dislocation-density of 6.21 × 107 (Fig. 8). The slip planes of the neighboring grains were significantly misaligned due to the Σ7 and Σ13 CSL GBs, which inhibited dislocation-density transmission in the TJ. This resulted in dislocation-density pile-ups and high localized stresses as indicated by the low dislocation-density transmission values. In contrast to the Σ7 and Σ13 CSL GBs, the Σ1 CSL GB had high transmission values, no dislocation-density pile-ups, and lower normal stresses.

In the TJ region, the most active slip directions in grains A and B were nearly aligned (Fig. 9). The slip direction in grain C was misaligned from the other two grains' slip directions, and the slip system was less active than the others by an order of magnitude. The slip directions in grains B and C had the highest misalignment, and the GB between them had the highest normal stress.

The misalignment of slip planes and directions in the TJ led to local differences in shear slip behavior that were largest in boundaries with large |Δbeff|. Shear slip across the Σ7 and Σ13 CSL GBs had large changes, whereas shear slip across the Σ1 CSL GB was almost homogeneous across the boundary (Fig. 10). The shear slip heterogeneities in the TJ were a result of each GB's dislocation-density transmission behavior—the Σ1 CSL had high transmission, but the other CSLs had significant dislocation-density impedance. These predictions are consistent with the observations of Patriarca et al., who observed a correlation between the magnitude of residual Burger's vector, |Δb|, and local strain differences across GBs [54]. They observed large variations in strain across boundaries where |Δb| was high, but in GBs where |Δb| approached zero, both sides of the boundary had similar amounts of strain.

A dislocation-density GB interaction scheme was coupled to a dislocation-density crystalline plasticity formulation to investigate the pile-up of dislocation densities at GBs spanning a wide range of CSL and random GB misorientations in fcc bicrystals and polycrystalline aggregates. An effective Burger's vector of residual dislocations was obtained for fcc bicrystals and compared to MD predictions of static GB energy for the same GB misorientation angles and axes. Predicted maximum and minimum had the same trends as those predicted by MD simulations. This indicates that the effective Burger's vector can be used to predict GB energy at a microstructural level for fcc GBs with the specified rotation axes.

The effects of the effective Burger's vector on dislocation-density transmission through fcc bicrystals with Σ3, Σ13, and Σ7 CSL GBs, which were chosen since they span low- and high-angle misorientations, were also investigated. The transmission in the bicrystals corresponded to the degree of slip system alignment—the Σ7 CSL had the largest misorientation of slip systems and the Σ3 CSL had the least. The Σ7 CSL bicrystal had the highest resistance to dislocation transmission for GBs with the [111] rotation axis, the Σ7 CSL bicrystal had the lowest, and the Σ13 CSL bicrystal was between the Σ7 and the Σ3.

To understand how bicrystal behavior scales to polycrystalline behavior, an fcc copper polycrystal with GBs with Σ1, Σ3, Σ7, Σ13, Σ21 and random high-angle GBs was quasi-statically deformed in tension. The bicrystal and polycrystalline aggregate predictions for GB transmission were consistent. Differences in shear slip behavior were larger across high-angle GBs, such as random high-angle GBs and Σ7, Σ13, Σ21 CSLs, than low-angle GBs, such as Σ1 and Σ3 CSLs, due to high dislocation-density impedance of the high-angle GBs. Dislocation-density pile-ups and accumulations of residual dislocation-densities at GBs with large slip system misalignments, such as Σ7 CSLs, resulted in localized stress accumulations in those GB regions. GBs with small slip system misorientations, such as Σ1 and Σ3 CSLs, had high transmission and did not accumulate dislocation-density pile-ups or high stresses, which was consistent with experimental observations. This investigation provides a fundamental understanding of how different representative GB orientations affect GB behavior, slip transmission, and dislocation-density pile-ups at a relevant microstructural scale.

This material is supported by the U.S. Office of Naval Research as a Multi-Disciplinary University Research Initiative under Grant No. N00014-10-1-0958.

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Koning, M. , Kurtz, R. J. , Bulatov, V. V. , Henager, C. H. , Hoagland, R. G. , Cai, W. , and Nomura, M. , 2003, “ Modeling of Dislocation–Grain Boundary Interactions in FCC Metals,” J. Nucl. Mater., 323(2–3), pp. 281–289. [CrossRef]
Lee, T. C. , Robertson, I. M. , and Birnbaum, H. K. , 1989, “ Prediction of Slip Transfer Mechanisms Across Grain Boundaries,” Scr. Metall., 23(5), pp. 799–803. [CrossRef]
Liu, G. S. , House, S. D. , Kacher, J. , Tanaka, M. , Higashida, K. , and Robertson, I. M. , 2014, “ Electron Tomography of Dislocation Structures,” Mater. Charact., 87, pp. 1–11. [CrossRef]
Zikry, M. A. , 1994, “ An Accurate and Stable Algorithm for High Strain-Rate Finite Strain Plasticity,” Comput. Struct., 50(3), pp. 337–350. [CrossRef]
Ziaei, S. , and Zikry, M. A. , 2015, “ Modeling the Effects of Dislocation-Density Interaction, Generation, and Recovery on the Behavior of H.C.P. Materials,” Metall. Mater. Trans. A, 46(10), pp. 4478–4490. [CrossRef]
Asaro, R. J. , and Rice, J. R. , 1977, “ Strain Localization in Ductile Single Crystals,” J. Mech. Phys. Solids, 25(5), pp. 309–338. [CrossRef]
Franciosi, P. , Berveiller, M. , and Zaoui, A. , 1980, “ Latent Hardening in Copper and Aluminium Single Crystals,” Acta Metall., 28(3), pp. 273–283. [CrossRef]
Zikry, M. A. , and Kao, M. , 1996, “ Dislocation Based Multiple-Slip Crystalline Constitutive Formulation for Finite-Strain Plasticity,” Scr. Mater., 34(7), pp. 1115–1121. [CrossRef]
Shanthraj, P. , and Zikry, M. A. , 2011, “ Dislocation Density Evolution and Interactions in Crystalline Materials,” Acta Mater., 59(20), pp. 7695–7702. [CrossRef]
Ma, A. , Roters, F. , and Raabe, D. , 2006, “ Studying the Effect of Grain Boundaries in Dislocation Density Based Crystal-Plasticity Finite Element Simulations,” Int. J. Solids Struct., 43(24), pp. 7287–7303. [CrossRef]
Roters, F. , Eisenlohr, P. , Hantcherli, L. , Tjahjanto, D. D. , Bieler, T. R. , and Raabe, D. , 2010, “ Overview of Constitutive Laws, Kinematics, Homogenization and Multiscale Methods in Crystal Plasticity Finite-Element Modeling: Theory, Experiments, Applications,” Acta Mater., 58(4), pp. 1152–1211. [CrossRef]
Rezvanian, O. , Zikry, M. A. , and Rajendran, A. M. , 2008, “ Microstructural Modeling in fcc Crystalline Materials in a Unified Dislocation-Density Framework,” Mater. Sci. Eng.: A, 494(1–2), pp. 80–85. [CrossRef]
Brandon, D. G. , 1966, “ The Structure of High-Angle Grain Boundaries,” Acta Metall., 14(11), pp. 1479–1484. [CrossRef]
Abuzaid, W. Z. , Sangid, M. D. , Carroll, J. D. , Sehitoglu, H. , and Lambros, J. , 2012, “ Slip Transfer and Plastic Strain Accumulation Across Grain Boundaries in Hastelloy X,” J. Mech. Phys. Solids, 60(6), pp. 1201–1220. [CrossRef]
Sangid, M. D. , Ezaz, T. , and Sehitoglu, H. , 2012, “ Energetics of Residual Dislocations Associated With Slip–Twin and Slip–GBs Interactions,” Mater. Sci. Eng.: A, 542, pp. 21–30. [CrossRef]
Warrington, D. H. , and Bufalini, P. , 1971, “ The Coincidence Site Lattice and Grain Boundaries,” Scr. Metall., 5(9), pp. 771–776. [CrossRef]
Sutton, A. P. , and Balluffi, R. W. , 1995, Interfaces in Crystalline Materials, Clarendon Press, Oxford, UK.
Britton, T. B. , Randman, D. , and Wilkinson, A. J. , 2009, “ Nanoindentation Study of Slip Transfer Phenomenon at Grain Boundaries,” J. Mater. Res., 24(03), pp. 607–615. [CrossRef]
Kumar, K. S. , Van Swygenhoven, H. , and Suresh, S. , 2003, “ Mechanical Behavior of Nanocrystalline Metals and Alloys,” Acta Mater., 51(19), pp. 5743–5774. [CrossRef]
Ovid'ko, I. A. , and Sheinerman, A. G. , 2009, “ Enhanced Ductility of Nanomaterials Through Optimization of Grain Boundary Sliding and Diffusion Processes,” Acta Mater., 57(7), pp. 2217–2228. [CrossRef]
Patriarca, L. , Abuzaid, W. , Sehitoglu, H. , and Maier, H. J. , 2013, “ Slip Transmission in bcc FeCr Polycrystal,” Mater. Sci. Eng.: A, 588, pp. 308–317. [CrossRef]
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Yang, Y. , Wang, L. , Bieler, T. R. , Eisenlohr, P. , and Crimp, M. A. , 2011, “ Quantitative Atomic Force Microscopy Characterization and Crystal Plasticity Finite Element Modeling of Heterogeneous Deformation in Commercial Purity Titanium,” Metall. Mater. Trans. A, 42(3), pp. 636–644. [CrossRef]
Lim, H. , Carroll, J. D. , Battaile, C. C. , Buchheit, T. E. , Boyce, B. L. , and Weinberger, C. R. , 2014, “ Grain-Scale Experimental Validation of Crystal Plasticity Finite Element Simulations of Tantalum Oligocrystals,” Int. J. Plast., 60, pp. 1–18. [CrossRef]
Bieler, T. R. , Eisenlohr, P. , Roters, F. , Kumar, D. , Mason, D. E. , Crimp, M. A. , and Raabe, D. , 2009, “ The Role of Heterogeneous Deformation on Damage Nucleation at Grain Boundaries in Single Phase Metals,” Int. J. Plast., 25(9), pp. 1655–1683. [CrossRef]
Koning, M. , Kurtz, R. J. , Bulatov, V. V. , Henager, C. H. , Hoagland, R. G. , Cai, W. , and Nomura, M. , 2003, “ Modeling of Dislocation–Grain Boundary Interactions in FCC Metals,” J. Nucl. Mater., 323(2–3), pp. 281–289. [CrossRef]
Lee, T. C. , Robertson, I. M. , and Birnbaum, H. K. , 1989, “ Prediction of Slip Transfer Mechanisms Across Grain Boundaries,” Scr. Metall., 23(5), pp. 799–803. [CrossRef]
Liu, G. S. , House, S. D. , Kacher, J. , Tanaka, M. , Higashida, K. , and Robertson, I. M. , 2014, “ Electron Tomography of Dislocation Structures,” Mater. Charact., 87, pp. 1–11. [CrossRef]
Zikry, M. A. , 1994, “ An Accurate and Stable Algorithm for High Strain-Rate Finite Strain Plasticity,” Comput. Struct., 50(3), pp. 337–350. [CrossRef]
Ziaei, S. , and Zikry, M. A. , 2015, “ Modeling the Effects of Dislocation-Density Interaction, Generation, and Recovery on the Behavior of H.C.P. Materials,” Metall. Mater. Trans. A, 46(10), pp. 4478–4490. [CrossRef]
Asaro, R. J. , and Rice, J. R. , 1977, “ Strain Localization in Ductile Single Crystals,” J. Mech. Phys. Solids, 25(5), pp. 309–338. [CrossRef]
Franciosi, P. , Berveiller, M. , and Zaoui, A. , 1980, “ Latent Hardening in Copper and Aluminium Single Crystals,” Acta Metall., 28(3), pp. 273–283. [CrossRef]
Zikry, M. A. , and Kao, M. , 1996, “ Dislocation Based Multiple-Slip Crystalline Constitutive Formulation for Finite-Strain Plasticity,” Scr. Mater., 34(7), pp. 1115–1121. [CrossRef]
Shanthraj, P. , and Zikry, M. A. , 2011, “ Dislocation Density Evolution and Interactions in Crystalline Materials,” Acta Mater., 59(20), pp. 7695–7702. [CrossRef]
Ma, A. , Roters, F. , and Raabe, D. , 2006, “ Studying the Effect of Grain Boundaries in Dislocation Density Based Crystal-Plasticity Finite Element Simulations,” Int. J. Solids Struct., 43(24), pp. 7287–7303. [CrossRef]
Roters, F. , Eisenlohr, P. , Hantcherli, L. , Tjahjanto, D. D. , Bieler, T. R. , and Raabe, D. , 2010, “ Overview of Constitutive Laws, Kinematics, Homogenization and Multiscale Methods in Crystal Plasticity Finite-Element Modeling: Theory, Experiments, Applications,” Acta Mater., 58(4), pp. 1152–1211. [CrossRef]
Rezvanian, O. , Zikry, M. A. , and Rajendran, A. M. , 2008, “ Microstructural Modeling in fcc Crystalline Materials in a Unified Dislocation-Density Framework,” Mater. Sci. Eng.: A, 494(1–2), pp. 80–85. [CrossRef]
Brandon, D. G. , 1966, “ The Structure of High-Angle Grain Boundaries,” Acta Metall., 14(11), pp. 1479–1484. [CrossRef]
Abuzaid, W. Z. , Sangid, M. D. , Carroll, J. D. , Sehitoglu, H. , and Lambros, J. , 2012, “ Slip Transfer and Plastic Strain Accumulation Across Grain Boundaries in Hastelloy X,” J. Mech. Phys. Solids, 60(6), pp. 1201–1220. [CrossRef]
Sangid, M. D. , Ezaz, T. , and Sehitoglu, H. , 2012, “ Energetics of Residual Dislocations Associated With Slip–Twin and Slip–GBs Interactions,” Mater. Sci. Eng.: A, 542, pp. 21–30. [CrossRef]
Warrington, D. H. , and Bufalini, P. , 1971, “ The Coincidence Site Lattice and Grain Boundaries,” Scr. Metall., 5(9), pp. 771–776. [CrossRef]
Sutton, A. P. , and Balluffi, R. W. , 1995, Interfaces in Crystalline Materials, Clarendon Press, Oxford, UK.
Britton, T. B. , Randman, D. , and Wilkinson, A. J. , 2009, “ Nanoindentation Study of Slip Transfer Phenomenon at Grain Boundaries,” J. Mater. Res., 24(03), pp. 607–615. [CrossRef]
Kumar, K. S. , Van Swygenhoven, H. , and Suresh, S. , 2003, “ Mechanical Behavior of Nanocrystalline Metals and Alloys,” Acta Mater., 51(19), pp. 5743–5774. [CrossRef]
Ovid'ko, I. A. , and Sheinerman, A. G. , 2009, “ Enhanced Ductility of Nanomaterials Through Optimization of Grain Boundary Sliding and Diffusion Processes,” Acta Mater., 57(7), pp. 2217–2228. [CrossRef]
Patriarca, L. , Abuzaid, W. , Sehitoglu, H. , and Maier, H. J. , 2013, “ Slip Transmission in bcc FeCr Polycrystal,” Mater. Sci. Eng.: A, 588, pp. 308–317. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Comparison of static MD calculated GB energy from Ref. [26] and average effective Burger's vector calculated for the [111] and [001] rotation axes for an fcc bicrystal

Grahic Jump Location
Fig. 2

Average effective Burger's vector for the [111] rotation axis with CSL bicrystals showing the grain boundary transmission factor of the most active slip system at a nominal strain of 5%

Grahic Jump Location
Fig. 3

Grain boundary misorientations determined using rotation matrices of each grain. The rotation matrices, R, were obtained by combining three successive rotations described by the Euler angles for each grain.

Grahic Jump Location
Fig. 4

Microstructural polycrystal model with GB orientations indicated

Grahic Jump Location
Fig. 5

(a) Normal stress and (b) GB residual dislocation-density at 3% nominal strain with maximum normal stress regions marked. Region 1 is a Σ7 CSL GB, and region 2 and region 3 are near triple junctions.

Grahic Jump Location
Fig. 6

Behavior for the most active slip systems in grains A and B at 3% nominal strain: (a) immobile dislocation-density for (11¯1)[011], (b) GBTF for (11¯1)[011], (c) immobile dislocation-density for (1¯1¯1)[011], and (d) GBTF for (1¯1¯1)[011]

Grahic Jump Location
Fig. 7

Normalized normal stress and grain boundary residual dislocation-density at the Σ7 CSL GB

Grahic Jump Location
Fig. 8

Behavior at triple junction at 3% nominal strain: (a) GBTF for most active slip system (11¯1)[011], (b) grain boundary residual dislocation-density, (c) immobile dislocation-density for slip system (11¯1)[011], and (d) normal stress

Grahic Jump Location
Fig. 9

Slip direction and immobile dislocation-density of the most active slip system of grains A, B, and C at the triple junction

Grahic Jump Location
Fig. 10

Shear slip behavior: (a) near the TJ with sampling lines and CSL GBs indicated, (b) shear slip across Σ1 CSL GB, (c) shear slip across Σ13 CSL GB, and (d) shear slip across the Σ7 CSL GB

Tables

Table Grahic Jump Location
Table 1 Material properties for fcc copper bicrystals [45]
Table Grahic Jump Location
Table 2 CSL GB distribution of microstructural model
Table Footer NoteSee table online for color.

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