Slip transfer and plastic strain accumulation across grain boundaries in Hastelloy X

doi:10.1016/j.jmps.2012.02.001

Journal of the Mechanics and Physics of Solids
《固体力学与物理学杂志》

Volume 60, Issue 6, June 2012, Pages 1201-1220
第 60 卷，第 6 期，2012 年 6 月，第 1201-1220 页

https://doi.org/10.1016/j.jmps.2012.02.001 Get rights and content 获取权利和内容

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Abstract 摘要

In this study, high resolution ex situ digital image correlation (DIC) was used to measure plastic strain accumulation with sub-grain level spatial resolution in uniaxial tension of a nickel-based superalloy, Hastelloy X. In addition, the underlying microstructure was characterized with similar spatial resolution using electron backscatter diffraction (EBSD). With this combination of crystallographic orientation data and plastic strain measurements, the resolved shear strains on individual slip systems were spatially calculated across a substantial region of interest, i.e., we determined the local slip system activity in an aggregate of ∼600 grains and annealing twins. The full-field DIC measurements show a high level of heterogeneity in the plastic response with large variations in strain magnitudes within grains and across grain boundaries (GBs). We used the experimental results to study these variations in strain, focusing in particular on the role of slip transmission across GBs in the development of strain heterogeneities. For every GB in the polycrystalline aggregate, we have established the most likely dislocation reaction and used that information to calculate the residual Burgers vector and plastic strain magnitudes due to slip transmission across each interface. We have also used molecular dynamics simulations (MD) to establish the energy barriers to slip transmission for selected cases yielding different magnitudes of the residual Burgers vector. From our analysis, we show an inverse relation between the magnitudes of the residual Burgers vector and the plastic strains across GBs. Also, the MD simulations reveal a higher energy barrier for slip transmission at high magnitudes of the residual Burgers vector. We therefore emphasize the importance of considering the magnitude of the residual Burgers vector to obtain a better description of the GB resistance to slip transmission, which in turn influences the local plastic strains in the vicinity of grain boundaries.
在本研究中，采用高分辨率的离体数字图像相关（DIC）技术，测量了镍基高温合金 Hastelloy X 在单轴拉伸过程中的塑性应变累积，空间分辨率为亚晶粒级。此外，利用电子背散射衍射（EBSD）技术，以相似的空间分辨率表征了基体微观结构。通过结合晶体取向数据和塑性应变测量，在感兴趣区域（即包含约 600 个晶粒和退火孪晶的集合体）内，空间计算了各个滑移系统上的分切应变。全场 DIC 测量结果显示，塑性响应具有高度异质性，晶粒内部和晶界（GBs）之间的应变幅值存在显著差异。我们利用实验结果研究这些应变变化，特别关注了晶界滑移传递在应变异质性发展中的作用。对于多晶聚集体中的每一个晶界，我们已经确定了最可能的位错反应，并利用这些信息计算了由于滑移在各个界面传递所导致的残余伯格斯矢量和塑性应变的大小。我们还使用了分子动力学模拟（MD）来确定所选情况下滑移传递的能量势垒，这些情况产生了不同大小的残余伯格斯矢量。从我们的分析中，我们展示了残余伯格斯矢量的大小与晶界处塑性应变之间的反比关系。此外，分子动力学模拟揭示了在残余伯格斯矢量较大时滑移传递具有更高的能量势垒。因此，我们强调了考虑残余伯格斯矢量大小的重要性，以获得对晶界滑移传递阻力更好的描述，这反过来又影响了晶界附近的局部塑性应变。

Highlights 要点

► We experimentally study local deformation in the vicinity of grain boundaries identifying the resolved shear strains. ► We focus on the role of slip transmission across GBs in causing strain heterogeneities. ► High resolution DIC and EBSD are used to evaluate local slip system activity in the core and mantle domains. ► Energy barriers for slip transmission across grain boundaries are established from molecular dynamics. ► The magnitude of the residual Burgers vector influences the GB resistance.
► 我们通过实验研究晶界附近的局部变形，识别了分解剪切应变。 ► 我们关注滑移在晶界间的传递在导致应变异质性问题中的作用。 ► 采用高分辨率 DIC 和 EBSD 技术评估核心区和包覆区的局部滑移系统活动。 ► 通过分子动力学建立滑移在晶界间传递的能量势垒。 ► 剩余伯格斯矢量的幅度影响晶界的抗力。

Keywords 关键词

Grain boundaries

Microstructures

Dislocations

Polycrystalline material

Slip transmission

晶界微观结构位错多晶材料滑移传递

1. Introduction 1. 引言

During the deformation of polycrystalline metals, some grain boundaries (GBs) act as barriers that block slip and result in dislocation pile-ups and stress concentrations (Eshelby et al., 1951, Hall, 1951, Petch, 1953). Other boundaries allow either partial or full transmission of the incident dislocations across the GB. In the case of partial dislocation transmission, a residual dislocation is left in the GB plane, and in the case of full transmission, the cross-boundary dislocation reaction takes place with no residual Burgers vector, i.e., crossslip (Sutton and Balluffi, 2006). The magnitude of the residual Burgers vector at the GB plane has a predominant effect on the GB resistance against slip transmission (Lim, 1984, Lim and Raj, 1985, Lee et al., 1989). It is expected that those boundaries that are conducive to slip transmission result in low residual Burgers vectors in the GB plane and exhibit slip induced strains across both sides of the interface. In contrast, boundaries that block slip are expected to exhibit high strains in one of the grains across the interface, but relatively strain free zones in the adjacent grain. This correlation between GB resistance to slip transmission and the magnitude of plastic strains across GBs requires further quantitative investigation as it can improve our understanding of plasticity at the microstructural level and the buildup of strains that could be a precursor for damage initiation. In the present study, our aim is to develop not only a deeper understanding of strain accumulation in the vicinity of GBs within a polycrystalline aggregate, but also a quantitative evaluation of the resistance of GBs to slip transmission that can eventually be used in a predictive fashion. We focus on the role of slip transmission in the uniaxial plastic deformation response of the nickel-based superalloy, Hastelloy X.
在多晶金属的变形过程中，某些晶界（GBs）充当障碍物，阻碍滑移，导致位错堆积和应力集中（Eshelby 等人，1951 年，Hall，1951 年，Petch，1953 年）。其他晶界则允许入射位错部分或完全通过晶界。在部分位错传递的情况下，晶界平面上会留下残余位错；而在完全传递的情况下，晶界两侧的位错反应发生，没有残余伯格斯矢量，即交叉滑移（Sutton 和 Balluffi，2006 年）。晶界平面上的残余伯格斯矢量的幅度对晶界抵抗滑移传递具有主要影响（Lim，1984 年，Lim 和 Raj，1985 年，Lee 等人，1989 年）。预计那些有利于滑移传递的晶界会在晶界平面上产生较低的残余伯格斯矢量，并在界面两侧表现出滑移诱导应变。相反，阻碍滑移的晶界预计会在界面一侧的晶粒中表现出高应变，但在相邻晶粒中则相对应变较小。这种晶界对滑移传递的阻力与晶界处塑性变形的幅度之间的相关性需要进一步定量研究，因为它可以增进我们对微观结构水平塑性变形的理解，以及那些可能成为损伤起始前兆的应变积累。在本研究中，我们的目标不仅是更深入地理解多晶聚集体中晶界附近的应变积累，还定量评估晶界对滑移传递的阻力，最终可用于预测性应用。我们关注镍基超级合金 Hastelloy X 的单轴塑性变形响应中滑移传递的作用。

The fundamental understanding of slip transmission has been developed through studies of individual GBs (Livingston and Chalmers, 1957, Shen et al., 1986, Lee et al., 1989). In these carefully conducted experimental works, different parameters were utilized to help predict the experimentally observed slip transmission reactions. Livingston and Chalmers (1957) proposed a geometric criterion for predicting specific slip system activation across a boundary due to dislocation pile-up. A modified version of this criterion was introduced by Shen et al. (1986) in which they added an additional requirement that the resolved shear stress on the outgoing, activated system had to be maximized. Lim and Raj (1985), suggested that the residual dislocation in the grain boundary plane (i.e., in the case of partial transmission) plays an important role in the mechanism of slip transfer. Minimizing this residual dislocation in the boundary was later incorporated by Lee et al., 1989, Lee et al., 1990 as an additional criterion for slip transfer prediction (LRB criterion). An additional level of analysis was made possible through simulations at the atomic level, e.g., molecular dynamics (MD) simulations. This tool provides the means for quantitative studies of the details of dislocation/GB interactions for different dislocation types and GB structures. For example, Jin et al. (2008) have explained different interaction behaviors, in addition to slip transmission, using material dependent energy barriers to nucleate partial dislocations. Dewald and Curtin (2011) have used MD simulations to formulate a modified criterion for slip transmission which incorporates, in addition to the LRB criterion, the characteristics of the GB dislocations and GB steps, and the effect of non-Schmid stresses. These kinds of simulations in conjunction with experimental work have helped develop our fundamental understanding of dislocation/GB interaction.
对滑移传递的基本理解是通过研究单个晶界（Livingston 和 Chalmers，1957 年，Shen 等人，1986 年，Lee 等人，1989 年）而发展起来的。在这些精心设计的实验工作中，利用了不同的参数来帮助预测实验中观察到的滑移传递反应。Livingston 和 Chalmers（1957 年）提出了一个几何标准，用于预测由于位错堆积而在晶界处特定滑移系统的激活。Shen 等人（1986 年）引入了这个标准的改进版本，其中他们增加了一个额外的要求，即 outgoing、激活系统上的 resolved shear stress 必须最大化。Lim 和 Raj（1985 年）提出，晶界平面中的残余位错（即部分传递的情况）在滑移传递机制中起着重要作用。Lee 等人，1989 年，Lee 等人，1990 年后来将减少边界中的这种残余位错作为滑移传递预测的附加标准（LRB 标准）。通过原子级模拟（例如分子动力学模拟），实现了更深入的分析。这种工具为定量研究不同位错类型和晶界结构的位错/晶界相互作用细节提供了可能。例如，Jin 等人（2008）利用材料相关的能量势垒解释了除滑移传递外的不同相互作用行为，包括部分位错的形核。Dewald 和 Curtin（2011）利用分子动力学模拟提出了一个改进的滑移传递判据，该判据除了包含 LRB 判据外，还考虑了晶界位错和晶界台阶的特性，以及非 Schmid 应力的作用。这类模拟与实验研究相结合，有助于我们深化对位错/晶界相互作用的基本理解。

In the experimental studies on slip transmission, the residual Burgers vector (b_r) was established using transmission electron microscopy (TEM), while in MD it was predicted from simulations that consider single dislocation/GB interaction (Ezaz et al., 2010) or a pile-up (Dewald and Curtin, 2007a, Dewald and Curtin, 2007b, Dewald and Curtin, 2011). In both cases, a finite number of GBs have been considered. However, as the need to relate microstructural behavior to macroscale response through multiscale models and experiments is increasing, it will be of interest to examine such slip transmission or blockage over a much larger number of GBs than is possible in a TEM or through simulations. Therefore, for further insight into the role of b_r in slip transmission and plastic strain accumulation across multiple interfaces, the consideration of a large number of GBs in a polycrystalline aggregate will be pursued in this work.
在滑移传递的实验研究中，通过透射电子显微镜（TEM）建立了剩余伯格斯矢量（b _r ），而在分子动力学（MD）中，则根据考虑单个位错/晶界相互作用（Ezaz 等人，2010）或堆垛（Dewald 和 Curtin，2007a，Dewald 和 Curtin，2007b，Dewald 和 Curtin，2011）的模拟进行预测。在这两种情况下，都考虑了有限数量的晶界。然而，随着通过多尺度模型和实验将微观结构行为与宏观响应联系起来的需求日益增加，研究在 TEM 或通过模拟所能实现的远大于此数量的晶界上的滑移传递或阻塞将具有重要意义。因此，为了进一步深入了解 b _r 在多界面滑移传递和塑性应变累积中的作用，本研究将致力于在多晶聚集体中考虑大量晶界。

Experimental techniques such as electron backscattering diffraction (EBSD) identify grain orientations and grain boundary types (Engler and Randle, 2010), and digital image correlation (DIC) provides full-field strain measurements (Sutton et al., 1983, Efstathiou et al., 2010). Utilization of both tools to study a large ensemble of grains allows for a better quantitative understanding of the influence of GBs on the development of local strain heterogeneities during plastic deformation. In this study, high-resolution DIC is used to obtain measurements of plastic strain accumulation across GBs. This information, along with crystallographic orientation measurements from EBSD is utilized to establish the residual dislocations due to slip transmission. Based on these quantitative results, we investigate the relationship between b_r and the magnitudes of strains across GBs due to slip transmission.
实验技术如背散射电子衍射（EBSD）可识别晶粒取向和晶界类型（Engler 和 Randle，2010），而数字图像相关（DIC）提供全场应变测量（Sutton 等人，1983；Efstathiou 等人，2010）。利用这两种工具研究大量晶粒，可以更定量地理解晶界（GBs）在塑性变形过程中对局部应变异质性的发展影响。在本研究中，采用高分辨率 DIC 来获取晶界处塑性应变累积的测量数据。结合来自 EBSD 的晶体学取向测量信息，用于确定因滑移传递而产生的残余位错。基于这些定量结果，我们研究了 b _r 与滑移传递导致的晶界处应变幅值之间的关系。

The material investigated in this study, Hastelloy X, is a nickel based super alloy which is designed for high temperature applications. In previous studies, researchers have investigated Hastelloy X under static (e.g., Rowley and Thornton, 1996), fatigue (e.g., Miner and Castelli, 1992), crack growth (e.g., Huang and Pelloux, 1980) and creep loading conditions (e.g., Kim et al., 2008). However, the local material response at a microstructural level has not been previously studied and will be pursued in the current work with emphasis on the slip transmission across GBs.
本研究调查的材料 Hastelloy X 是一种镍基超级合金，设计用于高温应用。在先前的研究中，研究人员已对 Hastelloy X 在静态（例如，Rowley 和 Thornton，1996）、疲劳（例如，Miner 和 Castelli，1992）、裂纹扩展（例如，Huang 和 Pelloux，1980）和蠕变载荷条件下进行了研究。然而，在微观结构水平上的局部材料响应尚未被研究过，并且将在当前工作中着重于晶界滑移传递的研究。

In summary, we seek to further investigate the role of the residual Burgers vector in slip transmission and plastic strain accumulation at the mesoscale. To accomplish this, we report the analysis of a systematic experiment (described in Section 2), in which DIC, in conjunction with EBSD, is used to interrogate deformation in the vicinity of GBs for an entire microstructure (Section 3). We establish the shear strains on crystallographic slip systems on both sides of every GB in the microstructure (i.e., determine local slip system activity). From this information, regarding activated slip systems, we compute estimates of the residual Burgers vector and magnitudes of strain accumulation in the mantle regions (as defined in Section 3.2) due to slip transmission. The results reported in this study provide a better understanding of the role of GBs in mediating slip and causing local deformation heterogeneities.
总之，我们旨在进一步研究残余伯格斯矢量在介观尺度滑移传递和塑性应变累积中的作用。为此，我们报告了一项系统实验的分析（详见第 2 节），其中结合数字图像相关法和电子背散射衍射技术，对整个微观结构中晶界附近的变形进行检测（第 3 节）。我们确定了微观结构中每个晶界两侧晶体学滑移系统上的剪切应变（即确定局部滑移系统活动性）。基于这些信息，关于激活的滑移系统，我们计算了由于滑移传递而在包覆区域（如第 3.2 节所定义）产生的残余伯格斯矢量和应变累积的估计值。本研究报告的结果有助于更好地理解晶界在调节滑移和导致局部变形异质性方面的作用。

2. Material and methods 2. 材料和方法

2.1. Material 2.1. 材料

Commercially available polycrystalline Hastelloy X, a nickel-based superalloy, was investigated in this study. The alloy was solution heat treated by the manufacturer at 1177 °C. Dog bone specimens with 4.0×3.2 mm cross sectional gage area were electric discharge machined from a 3.2 mm thick sheet in the as received condition. The overall sample size was selected based on the load frame loading capacity and the maximum sample size that can be investigated using the EBSD system. The surface of the specimen was mechanically polished using SiC paper (up to P1200) followed by finer polishing using alumina polishing powder (up to 0.3 μm) and vibro-polishing with colloidal silica (0.05 μm). The final surface finish was adequate for microstructural surface characterization using EBSD.
本研究研究了市售的多晶 Hastelloy X，一种镍基高温合金。该合金由制造商在 1177 °C 进行固溶热处理。从退火状态下的 3.2 mm 厚板材上用电火花加工成 4.0×3.2 mm 横截面的狗骨式试样。样品的总尺寸是根据载荷框架的加载能力和 EBSD 系统可研究的最大样品尺寸选择的。试样的表面使用 SiC 砂纸（最高 P1200）进行机械抛光，然后用氧化铝抛光粉（最高 0.3 μm）进行更精细的抛光，最后用胶体二氧化硅进行振动抛光（0.05 μm）。最终表面光洁度足以使用 EBSD 进行微观结构表面表征。

After sample preparation, a 1.0×0.6 mm region of interest was outlined on the specimen’s surface using Vickers indentation marks (fiducial markers as discussed in Carroll et al. (2010)). A Scanning Electron Microscope (SEM) equipped with an EBSD detector was used to characterize the microstructure of the specimen in the region of interest. Measurement spacing of 1 μm was selected, and a total of eight area scans were necessary to cover the region of interest at the selected magnification (300×). Fig. 1(a) shows a grain orientation map of the selected region of interest. No texture was observed in the aggregate consisting of approximately 600 grains. The percentage of annealing twin boundaries (Σ3 type GBs using the coincident site lattice, CSL, notation) was about 30% of the total number of GBs, as seen in the enlarged view of Fig. 1(b).

2.2. High resolution plastic strain measurements using DIC

In DIC, images of the deformed region of interest are correlated to a reference image (of the same region prior to deformation) to make full-field measurements of displacements. Afterwards, the in-plane strain fields are calculated through differentiation of the vertical and horizontal displacement fields (Sutton et al., 1983). To obtain accurate strain measurements with sub-grain level resolution using DIC, it is necessary to increase the magnification at which reference and deformed images are captured (Efstathiou et al., 2010). This reduces the field of view and thus imposes limitations on the area/number of grains that can be studied. The ex situ technique used in this study, described in detail in Carroll et al. (2010), addresses this problem and enables high resolution measurements over relatively large areas by capturing and stitching enough high magnification images to cover the required region of interest.

Following EBSD, a fine speckle pattern was applied to the sample’s surface for DIC measurements. Reference images were captured using an optical microscope at 31×magnification (0.14 μm/pixel). Fig. 2 shows an example of a reference image with the speckle pattern and subset size (101×101 pixels) used in the current work. A total of 316 overlapping images were required to cover the region of interest outlined by the indentation marks. These reference images were stitched together to generate one ultra-high resolution reference image that covered the complete region of interest. The specimen was then deformed in uniaxial tension using a servo-hydraulic load frame to 2.2% nominal strain (using strain control at 1.83×10⁻⁴ s⁻¹ strain rate) and unloaded (using load control). After unloading, the total residual strain was 2% nominal strain measured using a 12.7 mm (½″) gage length extensometer. Subsequently, 316 deformed images were captured and stitched, following the same procedure used for the reference images. In plane displacements were obtained from DIC and the results were differentiated to obtain the high resolution strain fields. The subset size used for DIC (14 μm) is smaller than the average grain size (50 μm) allowing for sub-grain level deformation measurement resolution (average number of DIC correlation points per grain=350). The fiducial markers visible in both the EBSD orientation map and DIC contour plots, allow for accurate alignment of the measured strain fields and the underlying microstructure in the region of interest (Carroll et al., 2010). The advantage of such a measurement procedure is that it enables quantitative analysis of the plastic strain fields in relation to the underlying microstructure of the polycrystalline specimen. Different aspects of the microstructure, such as GBs and grain orientation, coupled with their influence on plastic strain accumulation were investigated using this technique.
在 EBSD 之后，为进行 DIC 测量，在样品表面应用了细小颗粒图案。使用光学显微镜在 31 倍放大倍数下（0.14 μm/像素）拍摄参考图像。图 2 展示了一个参考图像示例，其中包含颗粒图案和本研究中使用的子集尺寸（101×101 像素）。需要 316 张重叠图像来覆盖由压痕标记界定的感兴趣区域。这些参考图像被拼接生成一张覆盖完整感兴趣区域的高分辨率参考图像。然后使用伺服液压加载框架对样品进行单轴拉伸至 2.2%名义应变（使用应变控制，应变速率为 1.83×10 ⁻⁴ s ⁻¹ ），随后卸载（使用载荷控制）。卸载后，使用 12.7 mm（½″）量程引伸计测量总残余应变为 2%名义应变。随后，按照与参考图像相同的程序拍摄并拼接了 316 张变形图像。通过 DIC 获得平面位移，并对结果进行微分以获得高分辨率应变场。用于 DIC 的子集尺寸（14 μm）小于平均晶粒尺寸（50 μm），允许进行亚晶粒级别的变形测量分辨率（每个晶粒的平均 DIC 相关点数=350）。在 EBSD 取向图和 DIC 等高线图中可见的基准标记，使得测量应变场与感兴趣区域中的基础微观结构能够精确对齐（Carroll 等人，2010）。这种测量程序的优势在于它能够定量分析多晶样品中塑性应变场与其基础微观结构的关系。利用该技术研究了微观结构的各个方面，如晶界和晶粒取向，以及它们对塑性应变积累的影响。

3. Results and analysis 3. 结果与分析

3.1. Local plastic strain
3.1. 局部塑性应变

Fig. 3(a)–(c) show the contour plots of the horizontal strain field ε_xx, the shear strain field ε_xy, and vertical strain field ε_yy (along the loading direction). These components of the strain tensor were acquired using DIC. By assuming plastic incompressibility, the component in the residual plastic normal strain along the third direction, ε_zz, was calculated using (1)

ε_{z z} = - 1 \times (ε_{x x} + ε_{y y}) .

The shear strain components in the third direction, ε_xz and ε_yz remain unknown in our analysis (as in any surface measurement technique). Using the measured and calculated components of the plastic strain tensor and by assuming the unknown components as zero, an estimate of the effective plastic strain, ε_eff^p, was calculated spatially using the following equation:
我们分析中的第三方向剪切应变分量ε _xz 和ε _yz 仍然未知（与任何表面测量技术一样）。通过使用测量的和计算的塑性应变张量分量，并假设未知分量为零，我们使用以下公式空间地计算了有效塑性应变ε _eff ^p 的估计值：(2)

ε_{eff}^{p} = \sqrt{\frac{2}{3} (ε_{i j} \times ε_{i j})} .

To assess and analyze the effect of microstructure, local crystallographic orientations from EBSD were numerically overlaid on the DIC strain data, i.e., for each point, spatial strain and orientation data were consolidated. Eventually, each point in the field has four components of the strain tensor (ε_xx, ε_yy, ε_xy, and ε_zz), the effective plastic strain, and an associated local crystallographic orientation. This allowed for a superposition of the grain boundaries on all the strain contour plots as shown for example for the ε_yy in Fig. 3(c).
为了评估和分析微观结构的影响，从 EBSD 中获取的局部晶体学取向被数字叠加到 DIC 应变数据上，即对于每个点，空间应变和取向数据被整合。最终，该区域中的每个点都具有应变张量的四个分量（ε _xx ，ε _yy ，ε _xy ，和ε _zz ）、有效塑性应变以及相关的局部晶体学取向。这使得可以在所有应变等值线图上叠加晶界，例如图 3(c)中所示ε _yy 的情况。

A significant level of heterogeneity is observed from the strain contour plots. In Fig. 3(c), regions rendered in dark red have strains higher than 3% (ε_yy field average=2%), while regions rendered in dark blue have strains around 0%, and some are even negative, i.e., compressive, in certain areas. This variation in strain appears to be associated with the local microstructure. For example, high strains were visually detected in the vicinity of many grain boundaries (Fig. 3(d)). In the next section, we quantitatively assess the correlation between the presence of GBs and the local heterogeneities in the plastic strains.
从应变等值线图中可以观察到显著的异质性。在图 3(c)中，深红色渲染的区域应变超过 3%（ε场平均=2%），而深蓝色渲染的区域应变约为 0%，在某些区域甚至为负值，即压缩应变。这种应变变化似乎与局部微观结构有关。例如，在许多晶界附近可以观察到高应变（图 3(d)）。在下一节中，我们将定量评估晶界存在与塑性应变局部异质性之间的相关性。

3.2. Grain boundary mantles
3.2. 晶界包层

Full-field strain measurements at the grain level enable us to address the specific regions of grains, which facilitate accumulation of heterogeneous strains. This is accomplished through separating each grain into a core and mantle demarcation utilizing the combination of high resolution DIC and EBSD. The core refers to the interior of the grain and the mantle to the region in the vicinity of the neighboring grain, i.e., the region of the grain near the GB (Meyers and Ashworth, 1982). Hence, each grain has several mantles corresponding to the number of neighboring grains. Mantle regions, which exhibit high strains on both sides of the interface, can be associated with slip transmission across the GB, while boundaries that block slip, i.e., shielding, have high strains in one of the mantles and low strains in the other mantle across the interface.
在晶粒尺度上的全场应变测量使我们能够针对晶粒的特定区域进行分析，从而促进非均匀应变的积累。这是通过结合高分辨率 DIC 和 EBSD 将每个晶粒分离为核心和包膜区域来实现的。核心指晶粒内部，包膜指邻近晶粒附近的区域，即靠近晶界（GB）的区域（Meyers and Ashworth, 1982）。因此，每个晶粒都有若干个包膜，对应于其邻近晶粒的数量。在界面两侧表现出高应变的包膜区域可以与晶界滑移的传递相关联，而阻碍滑移的边界，即屏蔽边界，在界面一侧的包膜中具有高应变，而在另一侧的包膜中具有低应变。

In this section, we establish an experimental estimate of the GB mantle size from DIC strain measurements and grain boundary locations. To do so, we calculated the spatial distance from each DIC measurement point to the nearest grain boundary, as shown schematically in the inset of Fig. 4(a). This facilitates binning the strain data based on distance from the GBs. As the focus in this study is on high strain regions that can be associated with slip transmission across GBs, only points with high strains were considered (>ε_eff^p field average=2.08%). These points were then binned based on their distance from the nearest GB, and the average strain in each bin was computed. Fig. 4(a) shows a plot of the average effective plastic strain ε_eff^p versus distance from the closest boundary (only points>ε_eff^p field average=2.08% were considered in this figure). We observe that the magnitude of the elevated strains decrease as we move away from the GBs (mantle regions) and approach the grain interior regions (core).
在本节中，我们通过 DIC 应变测量和晶界位置建立了晶界包覆层尺寸的实验估计。为此，我们计算了每个 DIC 测量点到最近晶界的空间距离，如图 4(a)插图所示。这有助于根据与晶界的距离对应变数据进行分组。由于本研究关注的是可能与晶界滑移传递相关的高应变区域，因此仅考虑了高应变点（>ε _eff ^p 场平均=2.08%）。然后，我们根据这些点与最近晶界的距离进行分组，并计算了每个分组的平均应变。图 4(a)显示了平均有效塑性应变ε _eff ^p 与最近边界距离的曲线图（该图仅考虑了>ε _eff ^p 场平均=2.08%的点）。我们观察到，随着我们远离晶界（包覆层区域）并接近晶粒内部区域（核心），高应变的幅度逐渐减小。

At a distance of ∼10 μm from the boundary, there is an inflection point in the change of strain with respect to distance from the GB as observed from the change in slope of the black dashed lines in Fig. 4(a). This point was used as an experimentally measured estimate of the GB mantle size, since it marks a transition in slip response while moving away from the GB and approaching the core of the grain. Once the mantle size was determined, mantle points were selected for each specific boundary across the entire microstructure, i.e., select points within 10 μm from the GBs. Each GB has two mantles associated with it; one on each side of the GB across the interface. Fig. 4(b) shows an example of GB mantles defined in this work. Delineating the mantle regions helps in determining the core points for each specific grain by subtracting all the mantle points from the total number of points in each grain.
距离边界约 10 μm 处，从图 4(a)中黑色虚线的斜率变化可以看出，应变随距离边界的变化出现拐点。该点被用作实验测量的边界套层尺寸估计值，因为它标志着远离边界并接近晶粒核心时滑移响应的转变。一旦确定了套层尺寸，就在整个微观结构中为每个特定边界选择套层点，即选择距离边界在 10 μm 范围内的点。每个边界都有两个与之相关的套层，分别位于界面两侧的边界上。图 4(b)展示了本工作中定义的边界套层示例。通过勾画套层区域，可以通过从每个晶粒的总点数中减去所有套层点来确定每个特定晶粒的核心点。

Strain histograms of ε_eff^p in cores and mantles over the entire microstructure in the area of interest, approximately 600 grains, are shown in Fig. 5. The histogram of the points belonging to mantles, i.e., red histogram, shows a range of strains between 0 and 6%, while points belonging to cores, i.e., black histogram, show a smaller strain range between 0.1 and 4.2%. Less scatter around the mean was also observed for the core histogram (standard deviation=0.8) compared to mantle histogram (standard deviation=0.9). We also notice in Fig. 5 that the mean strain for mantle points (2.05%) is less than the mean strain for core points (2.18%). This is primarily attributed to the low strains in some of the mantle regions which lower the mean strain of the mantle points. We emphasize that the presence of mantle points with low strains does not contradict the results presented in Fig. 4(a) as only points with relatively high strains were used to construct that figure as opposed to considering all the points, regardless of their magnitude, in the results presented in Fig. 5.
在感兴趣区域（约 600 个晶粒）的整个微观结构中，芯区和壳区的ε _eff ^p 应变直方图如图 5 所示。壳区点（即红色直方图）的直方图显示应变为 0 至 6%的范围，而芯区点（即黑色直方图）的应变范围较小，为 0.1 至 4.2%。与壳区直方图（标准差=0.9）相比，芯区直方图的均值周围散布较少（标准差=0.8）。我们还注意到，在图 5 中，壳区点的平均应变（2.05%）低于芯区点的平均应变（2.18%）。这主要归因于壳区某些区域的低应变，这些低应变降低了壳区点的平均应变。我们强调，存在低应变的壳区点并不与图 4(a)中的结果相矛盾，因为构建该图时仅使用了相对高应变的点，而图 5 中的结果则考虑了所有点，无论其大小如何。

As seen from the inset image on the right side of Fig. 5, some of the highly strained regions extend across the boundary into the neighboring grain and show continuous slip traces across the GBs as shown, for example, in Fig. 6. This can be an indication of slip transmission across the boundary (transmitting mantles). Other regions (inset images on the left side of Fig. 5) show high strains on only one side of the GB and relatively low strains on the other side. This case can be associated with blockage or shielding (shielding mantle). All the GBs in the region of interest were individually interrogated and classified as either shielding or transmitting GBs based on the strain magnitudes measured in mantle regions across each interface. For the study of slip transmission, we focused our analysis only on the transmitting mantles, which exhibit high strains on both sides of the GB.
从图 5 右侧的插图可以看出，一些高应变区域延伸至边界另一侧的邻近晶粒，并在晶界上显示出连续的滑移痕迹，例如图 6 所示。这可能是滑移通过边界传递的标志（传递包层）。其他区域（图 5 左侧的插图）只在晶界的一侧显示高应变，而另一侧应变相对较低。这种情况可能与阻塞或屏蔽（屏蔽包层）有关。研究区域内的所有晶界都进行了单独分析，并根据每个界面包层区域的应变大小将其分类为屏蔽晶界或传递晶界。在研究滑移传递时，我们仅分析了表现出晶界两侧高应变的传递包层。

3.3. Local slip system activity
3.3. 局部滑移系统活动

In order to establish estimates of the residual Burgers vector due to slip transmission, additional information regarding the crystallographic slip systems involved in the transmission process is required. Traditionally, determining the active slip systems from experiments has been accomplished through a combination of slip trace analysis and Schmid factor calculations (Zhang and Tong, 2004, Zhao et al., 2008, Bartali et al., 2009, Bieler et al., 2009). A different method for determining the active slip systems utilizes the measured local plastic strains and crystal orientation to solve for the crystallographic shear strains (Tatschl and Kolednik, 2003). Results utilizing both of these approaches are presented in 3.3.1 Slip trace analysis, 3.3.2 Crystallographic shear strain increments, respectively.
为了建立由滑移传递引起的残余伯格斯矢量的估计，需要关于传递过程中涉及的晶体学滑移系统的额外信息。传统上，通过滑移迹分析和小施密特因子计算相结合的方法（Zhang and Tong, 2004, Zhao et al., 2008, Bartali et al., 2009, Bieler et al., 2009）从实验中确定活动滑移系统。确定活动滑移系统的另一种方法利用测量的局部塑性应变和晶体取向来求解晶体学剪切应变（Tatschl and Kolednik, 2003）。使用这两种方法的计算结果分别在第 3.3.1 节滑移迹分析和第 3.3.2 节晶体学剪切应变增量中给出。

3.3.1. Slip trace analysis
3.3.1. 滑移迹分析

In fcc materials, 12 possible slip systems exist; three {1 1 0} directions on each of four {1 1 1} planes. An activated slip system creates a slip trace on the sample’s surface (provided that the slip plane of that system is not parallel to the sample’s surface). Observing and identifying the slip systems associated with the slip traces is used to determine the activated slip systems. Slip traces are seen from SEM micrographs as shown in Fig. 7 (for the same region shown earlier in Fig. 3(d)). We used the crystal orientation measurements from EBSD to determine the slip planes, not systems, associated with the observed traces. The procedure involves rotating the slip plane normal vectors, n, from crystal frame to sample frame, and then finding the intersection of the slip planes (define by the resulting normal vectors in sample frame) with the sample’s surface (Engler and Randle, 2010). These intersections represent the possible traces of slip planes (colored lines in Fig. 7). Matching of a possible slip trace with an observed one confirms the activation of that slip plane. For example, the central grain in Fig. 7 shows clear activation of the (1̄ 1 1̄) slip plane (blue line) and the (1 1 1̄) plane (magenta line). Schmid factors were used to make further specification regarding the slip directions associated with these slip planes.

Table 1 lists the Schmid factors of the 12 slip systems for the central grain in Fig. 7 (uniaxial loading conditions in the vertical direction). Systems 6 and 10 have the highest Schmid factors and, thus, they are the most likely to be activated compared to other systems with lower Schmid factors. This is the most probable case for the grain under consideration since these systems are on the (1̄ 1 1) and the (1 1 1̄) slip planes, which are active from slip traces. Obviously, this approach does not provide quantitative information regarding the degree of slip activity in each system. Also, the spatial differences in slip system activation leading to the heterogeneities within grains, observed in Fig. 3(d), can not be detected or quantitatively measured. This calls for a better global approach capable of extracting slip system activation quantitatively across the entire region of interest. In the next section, an alternative method, which provides local information about slip system activation, e.g., in GB mantles, is presented.

Table 1. Schmid factors of the 12 slip systems for the central grain shown in Fig. 7 (see Appendix for details on Schmid factor calculation).

3.3.2. Crystallographic shear strain increments

The local change in the plastic strain, due to crystallographic slip, is achieved through increments of shear, dγ^α, in the activated slip systems (Kocks and Chandra, 1982). The individual components of the plastic strain tensor are given by the following equation: (3)

d ε_{i j}^{p} = \frac{1}{2} \sum_{α = 1}^{s} (n_{i}^{α} l_{j}^{α} + n_{j}^{α} l_{i}^{α}) d γ^{α} = \sum_{α = 1}^{s} (m_{i j}^{α}) d γ^{α},

where α is the slip system number (see Table 1 for planes and directions), s is the number of slip systems (12 for fcc), n^α is the vector defining the normal to slip plane for system α, and l^α is the vector defining the slip direction. Three components of the plastic strain tensor were measured from DIC (ε_xx, ε_yy, ε_xy). The fourth component, ε_zz, was calculated by assuming plastic incompressibility (Eq. (1)). Also, n^α and l^α are known for fcc crystals (as shown in Table 1). Solving for the scalar quantities dγ^α at each spatial point provides local information about slip system activation across the entire region of interest.

The problem that arises when attempting to solve Eq. (3) is that the number of activated slip systems is not generally known. Also, if the number of activated systems is assumed, five has been proposed as sufficient number of systems necessary to satisfy compatibility (Taylor, 1938), the problem of which combination to select from the twelve possible systems arises. Taylor proposed a model to solve this problem. In his formulation, the combination which minimizes the sum of the absolute values of the shear increments is considered to be the combination that is actually operative (Taylor, 1938). Since then, different models and constitutive formulations, that are more physically based, have been proposed to solve this problem (Roters et al., 2010). In the current work, a visco-plastic constitutive model was used to solve for the shear strain increments spatially across the entire microstructure. In the formulation used, which is standard in many crystal plasticity frameworks, the shear strain rate is written as a function of the resolved shear stress on each slip system (Hutchinson, 1976):
在尝试求解公式(3)时出现的问题是，激活的滑移系统的数量通常未知。此外，如果假设激活系统的数量，有人提出五个系统是满足兼容性所需的最小数量（Taylor，1938），这就产生了从十二个可能的系统中选择哪一组合的问题。Taylor 提出了一个模型来解决这个问题。在他的公式中，被认为实际起作用的是使剪切增量绝对值之和最小的组合（Taylor，1938）。此后，人们提出了不同的模型和更基于物理的本构公式来解决这个问题（Roters 等人，2010）。在当前工作中，使用粘塑性本构模型来求解整个微观结构中剪切应变增量的空间分布。在所使用的公式中，该公式在许多晶体塑性框架中是标准的，剪切应变率被写为每个滑移系统上解离剪切应力的函数（Hutchinson，1976）：(4)

\frac{{\dot{γ}}^{α}}{{\dot{γ}}_{0}} = {| \frac{τ^{α}}{\bar{τ}} |}^{n} s g n (τ^{α})

where

{\dot{γ}}^{α}

is the shear rate on slip system α, τ^α is the resolved shear stress,

\bar{τ}

is a reference stress state, and

{\dot{γ}}_{0}

and n are material parameters that describe the reference strain rate and the slip rate sensitivity, respectively. The term “sgn(τ^α)” in Eq. (4) is present in order to ensure that

{\dot{γ}}^{α}

and τ^α have the same sign (i.e., positive work is being done). Eq. (4) can be rewritten as follows: (5)

{\dot{γ}}^{α} = \frac{{\dot{γ}}_{0}}{\bar{τ}} {| \frac{τ^{α}}{\bar{τ}} |}^{n - 1} τ^{α} .

As the emphasis in our analysis is to obtain the shear strain increments, dγ^α, and not solving for the exact kinetics, i.e., the shear stress τ^α, we take

{\dot{γ}}_{0} = \bar{τ} = 1

and n=20, i.e., rate insensitive.

If we make the simplification that

d γ^{α} = {\dot{γ}}^{α}

, we can substitute for the shear increments from Eq. (5) into the net strain Eq. (3), producing a relation between the total plastic strain and resolved shear stresses (6)

{\dot{ε}}_{i j}^{p} = \sum_{α = 1}^{s} (\frac{{\dot{γ}}_{0}}{\bar{τ}} {| \frac{τ^{α}}{\bar{τ}} |}^{n - 1} τ^{α}) (m_{i j}^{α}) .

The resolved shear stress on each slip system (τ^α) is related to the stress tensor (σ_ij) through the following equation: (7)

τ^{α} = m_{pq}^{α} σ_{pq} .

Substituting Eq. (7) into Eq. (6): (8)

{\dot{ε}}_{i j}^{p} = [\sum_{α = 1}^{s} (\frac{{\dot{γ}}_{0}}{\bar{τ}} {| \frac{τ^{α}}{\bar{τ}} |}^{n - 1} m_{i j}^{α} m_{pq}^{α})] σ_{pq} .

Given the local plastic strain rate (here we make the simplification that

{\dot{ε}}_{i j}^{p} = d ε_{i j}^{p}

) and the orientation data (to be able to transform from sample frame to crystal frame) we solved Eq. (8) – the non-linear system of equations was solved using an iterative solver – for the stresses σ_pq. Subsequently, the resolved shear stresses τ^α were calculated using Eq. (7). Finally, Eq. (5) was used to back-substitute for the shear strain rates. Following this procedure, the shear strain increments can be specified on all the 12 slip systems (i.e., no question of choosing the active slip systems arises; all systems with nonvanishing resolved shear stress are potentially active). By performing this calculation for each spatial point across the region of interest, we acquired spatial information about slip system activation across the entire microstructure.

An example of the results obtained using this procedure is shown in Fig. 8(a), which depicts a contour surface plot of the crystallographic shear strain increment on system 10 across the entire region of interest, i.e.,

| d γ^{10} | on (11 ¯ 1) [1 ¯ 10]

slip system. Note that the direction of slip system 10 in sample coordinates is different for each grain depending on its orientation. In Fig. 8, the dark blue color indicates no activity of that particular slip system, while the red color indicates slip system activation. For better visualization, an enlarged view of a particular sub-region is shown in Fig. 8(b). Compared to the slip trace analysis, this approach offers a quantitative description of activation on the slip system level with spatial information capable of capturing differences within a single grain or cluster of grains across the entire aggregate. This is important for our slip transmission analysis, since we wish to focus on slip activity in mantle regions.

Contour plots similar to Fig. 8(a) were generated for the other eleven slip systems. For the sake of brevity, in Fig. 9, we only show the ones with the highest shear increments in the same region shown earlier in Figs. 3(d) and 7. Different slip systems were activated in various spatial regions of the central grain, while certain slip systems, such as 5 and 8, show activity only in the vicinity of some grain boundaries, i.e., mantle regions. This spatial information, regarding activated slip systems on both sides of every grain boundary in the region of interest (approximately 1600 GBs), along with crystallographic grain orientation was used to provide insight into the local deformation behavior in grain boundary regions. In the next section, this information is used to study the role of the residual Burgers vector in slip transmission and strain accumulation across grain boundaries.

3.4. Residual Burgers vector

As previously mentioned, one of the possible outcomes of dislocation-grain boundary interaction is partial slip transmission of the incoming slip across the GB, which leaves behind a residual dislocation in the GB plane. The following dislocation reaction equation is used to define b_r based on the Burgers vectors of the dislocations on both sides of the GB (Sutton and Balluffi, 2006): (9)

{\vec{b}}_{r} = {\vec{b}}_{1} - {\vec{b}}_{2},

where b₁ and b₂ are the Burgers vectors of the incident and transmitted dislocations across the GB, respectively (see schematic in Fig. 10). When information about these quantities is lacking, i.e., we do not know the exact types of the incident and transmitted dislocations, the difference between the slip directions (l₁ and l₂) can be used as an approximation of the magnitude of b_r (10)

| {\vec{b}}_{r} | = | {\vec{l}}_{1} - {\vec{l}}_{2} | .

In the most general case, 144 possible interactions exist for slip transmission at each GB. This number of possible interactions is obtained by considering the 12 slip systems (fcc material) within grain 1 (incident) and 12 slip systems within grain 2 (transmitted). Note that the same reaction is obtained by reversing the order of the incoming and outgoing dislocations (considering the magnitude of b_r). For example, if slip system 1 is the incident slip system in the first grain and slip system 2 is the transmitted slip system in the other grain across the GB, then the magnitude of the residual Burgers vector resulting from this interaction (Eq. (10)) will be similar to the reaction obtained if the order of incident and transmitted slip is reversed, i.e., slip system 2 in the second grain becomes incident and system 1 becomes transmitted. Notice that this order reversal of the incident and transmitted slip systems implies reversing the slip directions of both systems involved in the transmission process, e.g., if the slip direction l₁ is incident slip for a GB, then −l₁ is transmitted slip for the same boundary. To specify whether a slip system is incident or transmitted relative to a specific GB, we check the sign of the inner product between the slip direction and the vector defining the normal to the GB plane (n_GB). If n_GB is defined as shown in Fig. 10, then a positive inner product between the slip direction of the slip system in Grain 1 and n_GB indicates that the slip direction is incident relative to that GB and a negative inner product indicates that it is transmitted. We therefore check and adjust the signs of the slip directions (slip directions are shown in Table 1 for fcc crystals) for each of the possible slip transmission interactions to guarantee that one of the systems is incident and the other transmitted relative to the GB under consideration.

In our analysis, we have individually selected each GB and interrogated the 144 possible slip transmission interactions by checking if both of the slip systems associated with each interaction were activated in the mantle regions of that GB. In the case of confirmed activation of both slip systems, through the DIC strain measurements, an experimental estimate of b_r was calculated for that specific GB. If multiple slip systems were activated in the mantles, several estimates of b_r were calculated. Fig. 11 shows two of these possible interactions for a transmitting mantle case. In this figure, the magnitude of the residual Burgers vector and the grain boundary Schmid factor parameter are shown for each of the two possible interactions. The plotted parameters are defined using the following equations: (11)

S c H^{(α, β)} = S c H_{i ncident}^{α} + S c H_{transmitted}^{β}

(12)

| b_{r}^{(α, β)} | = | l_{incident}^{α} - l_{transmitted}^{β} |,

where ScH^(α,β) is the GB Schmid factor parameter,

S c H_{incident}^{α}

is the Schmid factor of slip system α (of incident dislocation),

S c H_{transmitted}^{β}

is the Schmid factor of slip system β (of transmitted dislocation), and

| b_{r}^{(α, β)} |

is the magnitude of the residual Burgers vector from the interaction that involves incident dislocations on system α and transmitted dislocations on slip system β.

One of the cases plotted in Fig. 11 has a high GB Schmid factor parameter and a relatively low |b_r| (the point on the left side of the figure). This interaction represents slip system 6 being activated in both grains across the GB (notice that although both systems are 6, they have different orientations in the sample frame). The contour plots of the shear strains show activation of systems 6 in mantle regions across the GB. The activation of both slip systems associated with this interaction point is considered an indication of slip transmission across the GB. The point to the right in Fig. 11 represents the interaction between systems 6 and 7. Even though the value of the Schmid factor parameter is high, no activation of system 7 in grain 2 was observed, i.e., no transmission from system 6 in grain 1 to system 7 in grain 2. Notice the high magnitude of the residual Burgers vector associated with this possible interaction. Also, the geometric condition, θ, which describes the angle between the lines of intersection between slip planes of the incident and transmitted dislocations and the GB plane (shown in Fig. 10), for systems 6 and 7 is larger than the geometric condition for systems 6 and 6, where transmission is observed (θ^6,7=24.5°>θ^6,6=11.5°). This makes transmission less favorable due to larger misalignment between slip planes. The active slip systems, experimentally determined from DIC strain measurements, are in concurrence with the established criterion for predicting slip transmission (Lee et al., 1989).

The experimental estimates of |b_r| that we established for each GB are similar to the transmission case shown in Fig. 11. As indicated earlier, several values of |b_r| were calculated if multiple slip systems were activated in the mantle regions. Fig. 12(a) shows a histogram of the minimum |b_r| for each of the transmitting GBs in the region of interest (∼1000 GBs). Three distinct peaks in the number of GBs having similar |b_r| are observed in Fig. 12(a). Most of these boundaries were characterized as Σ3 type GBs, i.e., twins. The first peak is at |b_r|=0. This magnitude of the residual Burgers vector is associated with cross slip leaving no residual in the GB plane (Hirth and Lothe, 1992), as shown schematically in Fig. 12(b). The second peak is at

| b_{r} | = | (a / 6) < 211 > | = a / \sqrt{6}

; this type of reaction leaves a partial dislocation step at the GB as shown in Fig. 12(c). The third peak is at

| b_{r} | = | (a / 2) < 101 > | = a / \sqrt{2}

, which leaves a full dislocation step at the GB as shown in Fig. 12(d). Examples of these three different cases, from our experimental results plotted in Fig. 12(a) are shown in Eqs. (13), (14), (15) below: (13)

\frac{a}{2} {[0 1 \bar{1}]}_{1} \to \frac{a}{2} {[\bar{1} \bar{1} 0]}_{2} + b_{r} \Rightarrow b_{r} = 0 \Rightarrow | b_{r} | = 0

(14)

\frac{a}{2} {[1 0 1]}_{1} \to \frac{a}{2} {[\bar{1} 0 1]}_{2} + b_{r} \Rightarrow b_{r} = \frac{a}{6} {[\bar{1} 2 2]}_{1} \Rightarrow | b_{r} | = a / \sqrt{6}

(15)

\frac{a}{2} {[1 1 0]}_{1} \to \frac{a}{2} {[0 1 1]}_{2} + b_{r} \Rightarrow b_{r} = \frac{a}{2} {[1 0 \bar{1}]}_{1} \Rightarrow | b_{r} | = a / \sqrt{2} .

To investigate the relation between |b_r| and the accumulation of strain due to slip transmission, we used the experimentally determined estimates of |b_r| and the strain measurements across GBs (mantle regions). First, we calculated the strain across each transmitting mantle in the region of interest, and then binned the GBs based on the minimum estimate of |b_r| and finally calculated the average strain across the GBs for each bin. Fig. 13 shows a plot of strain across GBs versus |b_r|. Higher strains were measured across those GBs that have low |b_r| indicating a lower GB resistance to slip transmission.

4. Molecular dynamics simulations
4. 分子动力学模拟

By the use of atomistic simulations, GBs are reconstructed from distinct orientations of crystal lattices (i.e., Σ7 GB – 38.2° twist about [1 1 1] and Σ9 GB – 38.9° tilt about [1 1 0]), which allows for direct calculation of the GB resistance to plastic deformation. The simulations consisted of Ni atoms using the Foiles-Hoyt Embedded Atom Method (EAM) potential (Foiles and Hoyt, 2006). This EAM potential was chosen to match the intrinsic, γ_SF=127 mJ/m², and unstable, γ_US=255 mJ/m², stacking fault energies of the material, which compare well with experimental values of 125–128 mJ/m² and ab initio calculations of 273 mJ/m² for the γ_SF and γ_US energies, respectively (Siegel, 2005). It is critical to obtain reasonable values of the unstable stacking fault energy as this parameter has been tied to the mechanics and nucleation of dislocations (Rice, 1992). We model the GB for this Ni-based superalloy as pure Ni. This is indeed a simplification, although a necessary one, given the difficulties in obtaining suitable cross pair-potentials between atoms.

Molecular Dynamics (MD) simulations are employed in the form of a Sandia National Laboratories code called LAMMPs (Plimpton, 1995, Plimpton, 2007). A single GB is constructed in each simulation, which is oriented horizontally across the simulation box. Periodic boundary conditions are enforced to allow for calculations of bulk material properties; hence the simulations are not restricted in length scale to nanocrystalline material. The simulation box is deformed using an NPT ensemble, where the number of atoms in the simulation box, N, the pressure in the three directions (stress free boundaries), P, and the system temperature, T (10 K), are held constant throughout the simulation. Uniaxial tension is applied perpendicular to the GB via a strain controlled test with a strain-rate of 10¹⁰ s⁻¹. This high strain-rate is indicative of MD simulations, although the results were verified with smaller strain-rates to ensure the same dislocation mechanics.

A void is introduced into the system to facilitate dislocation nucleation leading to slip-GB interaction, as shown in a schematic and atomistic snapshot view in Fig. 14(a) and (b), respectively. To grasp the role of the GBs on the energetics of each system, the potential energy of each atom was measured during the simulation. A control volume was placed at the intersection of the dislocation and GB along the atoms which play a role in the interaction (selected via the centro-symmetry parameter (CSP) (Kelchner et al., 1998); hence it is not a simple cubic box. Extreme care was taken to select the position of only the relevant defect atoms (as indicated by the CSP). By subtracting the energy of the atoms in the initial relaxed position from the total calculated energy (upon loading) and normalizing by the control volume, we arrive at the interaction energy. In order to verify these MD calculations of the energy barrier, a system was constructed without a GB to mimic slip in an fcc lattice. The result of our MD control volume method was in concurrence with the generalized stacking fault energy and produced a modest 6% difference (Sangid et al., 2011), thus validating this procedure. Details concerning the methodology of these MD simulations can be found in Sangid et al. (2011).

This procedure was repeated for the CSL Σ7 and Σ9 GBs, as the slip-GB interaction and resulting dislocation reactions are shown in Fig. 15(a) and (b), respectively, using Visual Molecular Dynamics (Humphrey et al., 1996), an atomistic configuration viewer program. The CSP (Kelchner et al., 1998) is utilized to locate and color the defects within the material based on its position with respect to its nearest neighbors (red indicates a partial dislocation, while the golden color denotes a stacking fault). For clarity of presentation, defect-free atoms that do not participate in the interaction are deleted from the MD simulation snapshots. As expected, the incident dislocation is on a closed pack (1 1 1) plane, in the form of a leading partial dislocation and in the shape of a loop containing mixed edge and screw components. After interaction with the GB, a partial dislocation is transmitted into the adjacent grain on a glissile plane. Given the dislocation reactions, the residual Burgers vector of the slip transmission process is calculated according to Eq. (9). The residual Burgers vector is a/6[1 1̄ 2] in the case of slip transmission through a Σ7 GB and 2a/27[2̄ 2̄ 1] for the Σ9 GB. Therefore the magnitude of the residual Burgers vector is larger in the Σ7 GB, 0.41a, compared with the Σ9 GB, 0.22a, where a is the lattice constant of the material.

The resulting energy barriers for slip to penetrate the Σ7 and Σ9 GBs are shown in Fig. 15(c) and (d), respectively. The elastic strain energy of the system is removed from the energy calculations. The local fluctuations in energy due to temperature induced vibrations are small compared to the value of the energy barrier. Additionally, a local drop in energy is seen as the system nucleates a dislocation from the void. The energy of the system dramatically rises when the dislocation impinges upon the GB. The dislocation is temporarily impeded as the local energy builds at the site of slip-GB interaction. Once the dislocation traverses the GB, the energy of the system reduces as can be seen in the energy barrier profiles in Fig. 15(c) and (d). It can be seen that the Σ7 GB provides a greater energy barrier to slip than the Σ9 GB. In the cases shown, slip in the form of a leading partial dislocation interacts with a Σ7 or Σ9 GB, thus providing a consistent means for comparison. For the Σ9 case, once the first partial has transmitted into the adjacent grain, a second partial travels towards the GB but we only consider the interaction of the first leading partial. Thus, the rise in energy associated with the slip – Σ9 GB interaction in Fig. 15(b) is only based on the first dislocation interaction with the GB. The second leading partial is sufficiently far from the GB during slip transmission of the first partial, and does not affect the calculated barrier.

5. Discussion 5. 讨论

The experimental and analysis procedure adopted in this study provides point-wise comparisons between strain fields (from DIC) and microstructure (from EBSD). The significance of this approach is that it enables quantitative analysis of local deformation in the vicinity of each GB within a polycrystalline aggregate. For each GB, we have identified the GB type (Σ value), crystallographic orientation, plastic strains, and local slip system activity in mantle regions across the interface. These experimental results were used to study slip transmission across GBs and how it relates to local strain heterogeneities in the plastic response. Although more details can be attained from higher resolution techniques, e.g., TEM, the limitations associated with small viewing areas makes considering a large number of grain boundaries, at comparatively lower resolutions, valuable for an improved understanding of the deformation response in the polycrystalline aggregate.

The full field DIC measurements show a high level of heterogeneity in the plastic response. The buildup of strains in some cases and the shielding in others was associated with deformation in mantle regions. In contrast, less scatter around the nominal residual strain was generally observed in grain interiors, i.e., cores. As the focus in this study is on deformation in the vicinity of grain boundaries, a distinction between core and mantle regions for each grain was made based on the experimental results. This kind of discretization of the measured strain fields was made possible through the utilization of high resolution DIC and EBSD. The strain contour plots (Fig. 3) and the strain histograms (Fig. 5) of mantle regions show a wide distribution of strain magnitudes in the vicinity of grain boundaries indicating that, at least at the applied strain levels employed here, not all mantles are in an advanced stage of hardening as we usually expect. Similar results from experiments (Tschopp et al., 2009) and simulations (Rollett et al., 2009) have been reported in the literature. Based on our experimental results we proposed a classification of each mantle as one of two types, high strain mantle or low strain mantle. We further associated these two types of mantles with shielding or slip transmission across GBs. In the case of shielding, the GBs have mantles with small and large strain combinations (Shielding mantles in Fig. 5); and in the case of transmission, both mantles exhibit high strains (Transmitting mantles in Fig. 5). This demarcation of mantles based on strain magnitudes allows a better characterization of the local plastic deformation and how it relates to the microstructure of the material. For example, as we show in this paper, some of the variations in plastic strain accumulation across GBs are attributed to different magnitudes of b_r due to slip transmission.

In the current work, we have considered an entire aggregate and determined the residual Burgers vector and strain magnitudes across every GB due to slip transmission. The residual Burgers vector, b_r, was calculated using the slip directions of the incoming and outgoing active slip systems across each GB (Eq. (10)) and the strain magnitudes across GBs were computed by adding the average

ε_{eff}^{p}

in both mantles across each interface. Since a large number of GBs were investigated, we were able to establish the relation between |b_r| and the magnitudes of strain across GBs (Fig. 13). The higher strains across certain boundaries, at low |b_r|, are attributed to lower GB resistance against slip transmission, while lower strains across GBs, at high |b_r|, are attributed to higher GB resistance against slip transmission. This result confirms the importance of the residual Burgers vector in slip transmission (Lim, 1984, Lim and Raj, 1985, Lee et al., 1989) and also indicates that it is essential for describing the local strain magnitudes in the vicinity of GBs for a polycrystalline aggregate.

An additional level of understanding of the role of |b_r| in slip transmission was obtained from Molecular Dynamics (MD) simulations by calculating the energy barriers against slip transmission. As our aim was to further elucidate the relation between |b_r| and GB resistance to slip transmission, two simulation cases that result in different magnitudes of b_r were considered here (shown in Fig. 15(a) and (b)). In the first case, we observe transmission through a Σ7 GB. This interaction left a higher residual Burgers vector compared to the other simulated case, where transmission through a Σ9 GB was considered (0.41a>0.22a, where a is the lattice spacing). The corresponding energy barriers presented in Fig. 15(c) and (d) show a higher energy barrier for the Σ7 case (higher |b_r|) compared to the Σ9 case (lower |b_r|). This difference in energy barriers, analogous to the GB resistance to slip transmission, leads to varying degrees of strain magnitudes across the GBs as shown in Fig. 13, i.e., higher slip induced strains across the GBs with lower resistance to slip transmission. This utilization of MD for energy barriers calculations is valuable since we can more accurately interpret the role of each GB in impeding slip deformation by understanding the physics of the material. We emphasize that the use of MD in conjunction with EBSD and DIC provides insights that cannot be gleaned by consideration of only one of these methods.

It should be pointed out that slip transmission depends not only on the magnitude of b_r but also on other parameters such as the type of the boundary, loading conditions, i.e., resolved shear stress (RSS), and the geometric condition, θ, (Fig. 10). Lee et al. (1989) have indicated that there is a competition between b_r and the RSS to determine the final outcome of the slip transmission process. The systems that produce the absolute minimum residual might not be active due to lower RSS compared to other systems. Also, an activated system with the maximum RSS may cease operation if it generates residual dislocations with large magnitudes. Therefore, within a polycrystalline aggregate, where each grain is under a different state of stress, consideration of both of these factors is required for a better description of the GB resistance to slip transmission.

We infer three different types of reactions involving twin boundaries (Σ3 GBs) as shown in Fig. 12. The same types of reactions were experimentally observed through TEM and reported in the literature for selected transmission cases (fcc materials), e.g., |b_r|=0, i.e., cross slip (Lee et al., 1989) and

| b_{r} | = a / \sqrt{6}

, i.e., leaving partial dislocation step in the GB plane (Lee et al., 1990). Also, some of these reactions have been reported and discussed in much more detail in studies using MD simulations (Dewald and Curtin, 2011). The current study reports similar magnitudes of b_r by investigating a large number of grain boundaries in a polycrystalline aggregate. This approach provides further quantitative insight of the relative importance of each of the observed reactions, involving Σ3 GBs, in polycrystalline deformation. For example, we observe that the number of transmitting Σ3 GBs with |b_r|=0 is considerably larger than the number of Σ3 GBs with

| b_{r} | = a / \sqrt{6}

| b_{r} | = a / \sqrt{2}

(Fig. 12). This in turn results in higher strain magnitudes across these interfaces, with |b_r|=0, compared to other GBs.

The experimental results presented in this paper highlight the importance of the magnitude of the residual Burgers vector due to slip transmission on the local plastic strains in the vicinity of grain boundaries. It should be pointed out that, although some of the heterogeneities in the local plastic strains are attributed to slip transmission, slip transfer is not the only contributing mechanism to the development of such heterogeneities, e.g., slip nucleation and elastic anisotropy have a contribution which we do not account for in the current study. Therefore, for a deeper perspective of the reasons why strain accumulation occurs at certain boundaries and not at others, and to discern between slip transmission and pure nucleation that might also take place at the GBs, further analysis of the other contributing mechanisms would be required. In situ experiments can be advantageous in that regard as deformation can be monitored in real time compared to ex situ experiments (i.e., after deformation) as performed in this study. However, there are many challenges related to the experimental setup and procedure which have to be overcome before such in situ experiments are possible (Carroll et al., 2010). In addition, ex situ experiments afford a much higher resolution of DIC measurements than in situ ones.

Another limitation that is worth pointing out is the fact that all the analyses have been performed on the surface of the material. No insight into subsurface effects is possible through the utilization of the DIC and EBSD techniques as employed in this study. This particular issue can be addressed with a combination of high energy X-ray diffraction for three dimensional orientation mapping (Lienert et al., 2011) and crystal plasticity simulations (Rollett et al., 2010). Internal strain measurements could be performed using Digital Volume Correlation – a 3D extension of DIC – but this is a method still under development (Gates et al., 2011). Therefore, high resolution DIC measurements, as performed in this study, are only possible at the surface. Careful experimental and simulation work which utilizes all of the previously mentioned techniques would help further explorations of subsurface effects and how they influence, for example, slip transmission across GBs.

Despite the previously mentioned limitations, which might affect the results obtained at individual GBs, we believe that the impact on the general observations made in the current work would be minimum as the average response of a large number of GBs was considered to make the final conclusions presented in this paper.
尽管之前提到的局限性可能会影响单个晶界的实验结果，但我们认为，由于最终结论是基于大量晶界的平均响应得出的，因此这些局限性对当前工作所做的一般性观察的影响将是最小的。

6. Conclusions 6. 结论

High resolution DIC and EBSD were used to study the uniaxial plastic deformation response of a polycrystalline sample in relation to the underlying microstructure. The aim of the work was to develop a deeper perspective of strain accumulation in the vicinity of grain boundaries within a polycrystalline aggregate. The conclusions of this study are summarized as follows:
采用高分辨率 DIC 和 EBSD 研究了多晶样品的单轴塑性变形响应与其微观结构的关系。这项工作的目的是更深入地了解多晶聚集体中晶界附近的应变积累情况。本研究的结论总结如下：

1.
We present an experimental and analysis procedure, that provides point-wise comparisons between strain fields (from DIC) and microstructure (from EBSD). The significance of this approach is that it enables quantitative analysis of local deformation in the vicinity of every grain boundary within a polycrystalline aggregate. These experimental tools were utilized to provide further insight into the role of the residual Burgers vector in slip transmission and plastic strain accumulation in the vicinity of GBs. This correlation between plastic strain magnitudes across GBs and the residual Burgers vector has not been investigated before. A better quantitative understanding of the local plastic strain magnitudes is of significant importance since the development of such deformation heterogeneities is a precursor to crack initiation.
我们提出了一种实验和分析程序，该程序可在应变场（来自 DIC）和微观结构（来自 EBSD）之间进行逐点比较。这种方法的重大意义在于它能够对多晶聚集体中每个晶界附近的局部变形进行定量分析。这些实验工具被用于进一步深入了解残余伯格斯矢量在晶界附近滑移传递和塑性应变累积中的作用。这种晶界处塑性应变幅度与残余伯格斯矢量之间的相关性之前尚未被研究过。对局部塑性应变幅度的更好定量理解非常重要，因为这种变形异质性的发展是裂纹萌生的前兆。
2.
For an entire aggregate, we determined the residual Burgers vector and strain magnitudes across every GB due to slip transmission. Since a large number of GBs was investigated, we were able to establish an inverse relation between |b_r| and the magnitudes of strain across GBs. To the best of our knowledge, no similar results have been presented in the literature where such a large number of GBs was considered with sufficient details to derive general conclusion concerning the impact of slip transmission on the development of local deformation heterogeneity.
对于整个聚集体，我们确定了由于滑移传递而在每个晶界处产生的残余伯格斯矢量和应变大小。由于研究了大量晶界，我们能够建立|b _r |与晶界处应变大小之间的逆关系。据我们所知，在文献中尚未提出过类似结果，其中考虑了如此大量的晶界，并提供了足够的细节来得出关于滑移传递对局部变形非均匀性发展影响的普遍结论。
3.
The MD simulations revealed a higher energy barrier to slip transmission at high |b_r|. These energy barriers, analogous to the GB resistance to slip transmission, have an influence on the strain magnitudes across GBs.
分子动力学模拟揭示在高|b _r |时滑移传递的能量势垒更高。这些能量势垒类似于晶界对滑移传递的阻力，对晶界处的应变大小有影响。
4.
The higher strains across certain boundaries, at low |b_r|, were associated with lower GB resistance against slip transmission while lower strains across GBs, at high |b_r|, were attributed to higher resistance against slip transmission.
在低|b _r |时，某些晶界处的较高应变与晶界对滑移传递的较低阻力有关，而在高|b _r |时，晶界处的较低应变则归因于对滑移传递的较高阻力。
5.
The reactions we inferred for slip transmission across Σ3 GBs revealed a larger number of boundaries with |b_r|=0, i.e., cross slip, compared to other types of reaction resulting in higher magnitudes of the residual Burgers vector. This in turn results in higher strain magnitudes across these interfaces, with |b_r|=0, compared to other GBs.
我们推断的Σ3 晶界滑移传递反应显示，与其他导致残余伯格斯矢量幅值更高的反应类型相比，存在更多|b _r |=0 的边界，即交叉滑移。这反过来导致在这些界面上的应变幅值更高，与|b _r |=0 的其他晶界相比。
6.
We made a distinction between core and mantle regions for each grain through the utilization of high resolution DIC and EBSD. We also proposed a classification of each mantle as a high or low strain mantle and associated these two types of mantles with shielding or slip transmission across GBs. This demarcation of mantles based on strain magnitudes allows a better characterization of the local plastic deformation and how it relates to the microstructure of the material.
我们通过高分辨率 DIC 和 EBSD 技术区分了每个晶粒的核心区和幔区。我们还提出了将每个幔区分类为高应变幔区或低应变幔区的分类方法，并将这两种类型的幔区与晶界上的屏蔽或滑移传递相关联。基于应变幅值的幔区划分，可以更准确地描述局部塑性变形及其与材料微观结构的关系。

Acknowledgments 致谢

This work was supported by the Midwest Structural Sciences Center (MSSC), which is supported by the Air Vehicles Directorate of the U.S. Air Force Research Laboratory under contract number FA8650-06-2-3620. The work is also partially supported by NSF grants CMMI-09-26813 and partly by DMR-08-03270. The authors thank Prof. Armand J. Beaudoin for enlightening discussions. EBSD results were obtained with the assistance of Dr. Jim Mabon at the Frederick Seitz Materials Research Laboratory Central Facilities, University of Illinois, which are partially supported by the U.S. Department of Energy under grants DE-FG02-07ER46453 and DE-FG02-07ER46471.
这项工作得到了中西部结构科学中心（MSSC）的支持，该中心由美国空军研究实验室的航空器部根据合同号 FA8650-06-2-3620 提供支持。这项工作还部分得到了 NSF 资助项目 CMMI-09-26813 和 DMR-08-03270 的支持。作者感谢 Armand J. Beaudoin 教授的启发性讨论。EBSD 结果是在伊利诺伊大学弗雷德里克·塞兹材料研究实验室中央设施助理 Dr. Jim Mabon 的帮助下获得的，该设施部分得到了美国能源部根据资助项目 DE-FG02-07ER46453 和 DE-FG02-07ER46471 的支持。

Appendix 附录

This appendix provides details of the coordinate transformation between crystal and sample frames. We also report the procedure followed to calculate the Schmid factors, shown for example in Table 1, from EBSD crystal orientation measurement (i.e., Euler angles).
本附录提供了晶体坐标系和样品坐标系之间坐标变换的详细信息。我们还报告了计算 Schmid 因子（例如表 1 所示）的步骤，这些因子来自 EBSD 晶体取向测量（即欧拉角）。

1.
Using the Euler angles (φ₁, Φ, φ₂), the rotation matrix g is determined using the following equation (Bunge definition).
使用欧拉角（φ ₁ ， Φ，φ ₂ ），根据以下方程（Bunge 定义）确定旋转矩阵 g。(A1) $g = [\begin{matrix} \cos φ_{1} \cos φ_{2} - \sin φ_{1} \sin φ_{2} \cos Φ & \sin φ_{1} \cos φ_{2} + \cos φ_{1} \sin φ_{2} \cos Φ & \sin φ_{2} \sin Φ \\ - \cos φ_{1} \sin φ_{2} - \sin φ_{1} \cos φ_{2} \cos Φ & - \sin φ_{1} \sin φ_{2} + \cos φ_{1} \cos φ_{2} \cos Φ & \cos φ_{2} \sin Φ \\ \sin φ_{1} \sin Φ & - \cos φ_{2} \sin Φ & \cos Φ \end{matrix}]$
2.
To transform from sample frame to crystal frame, the following equations are used
要从样品坐标系转换为晶体坐标系，使用以下方程(A2) $C_{crystal} = g C_{sample} (where C is a vector)$ (A3) $A_{crystal} = g A_{sample} g^{- 1} (where A is a second order tensor, e . g ., strain tensor ε_{ij})$
3.
The Schmid factor for a particular slip system α, that is defined by slip plane normal n^α and slip direction l^α, can be found using the following equation:
特定滑移系统α的 Schmid 因子，由滑移面法向 n ^α 和滑移方向 l ^α 定义，可以使用以下公式计算：(A4) $Schmid {Factor}^{α} = | (L_{crystal} \cdot n^{α}) (L_{crystal} \cdot l^{α}) |$ where L_crystal is the loading direction written in crystal frame (found using Eq. (A2)). All vectors in Eq. (A4) are unit vectors. n^α and l^α are known for fcc crystals (listed in Table 1).
其中 L _crystal 是晶体坐标系中的加载方向（通过公式(A2)确定）。公式(A4)中的所有向量都是单位向量。对于面心立方晶体，n ^α 和 l ^α 是已知的（见表 1）。
4.
From DIC we establish the strain tensor in sample frame (ε_sample). To write the strain tensor in crystal frame, Eq. (A3) is used.
通过数字图像相关法（DIC）我们建立了样品坐标系中的应变张量ε _sample 。要写出晶体坐标系中的应变张量，使用公式(A3)。(A5) $ε_{crystal} = g ε_{sample} g^{- 1}$

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J. Mol. Graphics, 14 (1996), p. 33
View PDF View article Google Scholar
Hutchinson, 1976
J.W. Hutchinson
Bounds and Self-consistent estimates for creep of polycrystalline materials. Proceedings of the Royal Society of London Series A
NATO ASI Ser., Ser. C, 348 (1976), p. 101
Google Scholar
Jin et al., 2008
Z.H. Jin, P. Gumbsch, K. Albe, E. Ma, K. Lu, H. Gleiter, H. Hahn
Interactions between non-screw lattice dislocations and coherent twin boundaries in face-centered cubic metals
Acta Mater., 56 (2008), p. 1126
View PDF View article View in Scopus Google Scholar
Kelchner et al., 1998
C.L. Kelchner, S.J. Plimpton, J.C. Hamilton
Dislocation nucleation and defect structure during surface indentation
Phys. Rev. B, 58 (1998), p. 11085
View in Scopus Google Scholar
Kim et al., 2008
W.-G. Kim, S.-N. Yin, Y.-W. Kim, J.-H. Chang
Creep characterization of a Ni-based Hastelloy-X alloy by using theta projection method
Eng. Fract. Mech., 75 (2008), p. 4985
View PDF View article View in Scopus Google Scholar
Kocks and Chandra, 1982
U.F. Kocks, H. Chandra
Slip geometry in partially constrained deformation
Acta Metall., 30 (1982), p. 695
View PDF View article View in Scopus Google Scholar
Lee et al., 1989
T.C. Lee, I.M. Robertson, H.K. Birnbaum
Prediction of slip transfer mechanisms across grain boundaries
Scr. Metall., 23 (1989), p. 799
View PDF View article View in Scopus Google Scholar
Lee et al., 1990
T.C. Lee, I.M. Robertson, H.K. Birnbaum
An In Situ transmission electron microscope deformation study of the slip transfer mechanisms in metals
Metall. Trans. A, 21 (1990), p. 2437
View in Scopus Google Scholar
Lienert et al., 2011
U. Lienert, S. Li, C. Hefferan, J. Lind, R. Suter, J. Bernier, N. Barton, M. Brandes, M. Mills, M. Miller, B. Jakobsen, W. Pantleon
High-energy diffraction microscopy at the advanced photon source
JOM Journal of the Minerals, Metals and Materials Society, 63 (2011), p. 70
Crossref View in Scopus Google Scholar
Lim, 1984
L.C. Lim
Slip-twin interactions in nickel at 573 K at large strains
Scr. Metall., 18 (1984), p. 1139
View PDF View article View in Scopus Google Scholar
Lim and Raj, 1985
L.C. Lim, R. Raj
Continuity of slip screw and mixed crystal dislocations across bicrystals of nickel at 573 K
Acta Metallurgica, 33 (1985), p. 1577
View PDF View article View in Scopus Google Scholar
Livingston and Chalmers, 1957
J.D. Livingston, B. Chalmers
Multiple slip in bicrystal deformation
Acta Metall., 5 (1957), p. 322
View PDF View article View in Scopus Google Scholar
Meyers and Ashworth, 1982
M.A. Meyers, E. Ashworth
Model for the effect of grain size on the yield stress of metals
Philos. Mag. A:, 46 (1982), p. 737
Google Scholar
Miner and Castelli, 1992
R. Miner, M. Castelli
Hardening mechanisms in a dynamic strain aging alloy, HASTELLOY X, during isothermal and thermomechanical cyclic deformation
Metall. Mater. Trans. A, 23 (1992), p. 551
View in Scopus Google Scholar
Petch, 1953
N.J. Petch
The cleavage strength of polycrystals
J. Iron. Steel Inst., 174 (1953), pp. 25-28
Google Scholar
Plimpton, 1995
S. Plimpton
Fast parallel algorithms for short-range molecular dynamics
J. Comput. Phys., 117 (1995), pp. 1-19
View PDF View article View in Scopus Google Scholar
Plimpton, 2007
S. Plimpton
Large−scale Atomic/Molecular Massively Parallel Simulator
Sandia National Laboratories (2007)
Google Scholar
Rice, 1992
J.R. Rice
Dislocation nucleation from a crack tip: an analysis based on the Peierls concept
J. Mech. Phys. Solids, 40 (1992), p. 239
View PDF View article View in Scopus Google Scholar
Rollett et al., 2009
A.D. Rollett, R.A. Lebensohn, M. Groeber, Y. Choi, J. Li, G.S. Rohrer
Stress hot spots in viscoplastic deformation of polycrystals
Modell. Simul. Mater. Sci. Eng. (2009), p. 18
Google Scholar
Rollett et al., 2010
A.D. Rollett, R.A. Lebensohn, M. Groeber, Y. Choi, J. Li, G.S. Rohrer
Stress hot spots in viscoplastic deformation of polycrystals
Modell. Simul. Mater. Sci. Eng. (2010), p. 18
Google Scholar
Roters et al., 2010
F. Roters, P. Eisenlohr, L. Hantcherli, D.D. Tjahjanto, T.R. Bieler, D. Raabe
Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: theory, experiments, applications
Acta Mater., 58 (2010), p. 1152
View PDF View article View in Scopus Google Scholar
Rowley and Thornton, 1996
M.A. Rowley, E.A. Thornton
Constitutive modeling of the visco-plastic response of Hastelloy-X and aluminum alloy 8009
J. Eng. Mater. Technol., 118 (1996), p. 19
Crossref View in Scopus Google Scholar
Sangid et al., 2011
M.D. Sangid, T. Ezaz, H. Sehitoglu, I.M. Robertson
Energy of slip transmission and nucleation at grain boundaries
Acta Mater., 59 (2011), p. 283
View PDF View article View in Scopus Google Scholar
Shen et al., 1986
Z. Shen, R.H. Wagoner, W.A.T. Clark
Dislocation pile-up and grain boundary interactions in 304 stainless steel
Scr. Metall., 20 (1986), p. 921
View PDF View article View in Scopus Google Scholar
Siegel, 2005
D.J. Siegel
Generalized stacking fault energies, ductilities, and twinnabilities of Ni and selected Ni alloys
Appl. Phys. Lett., 87 (2005), p. 121901
Google Scholar
Sutton and Balluffi, 2006
A.P. Sutton, R.W. Balluffi
Interfaces in Crystalline Materials
Oxford Classical Texts, Oxford (2006)
Google Scholar
Sutton et al., 1983
M.A. Sutton, W.J. Wolters, W.H. Peters, W.F. Ranson, S.R. McNeill
Determination of displacements using an improved digital correlation method
Image Vision Comput., 1 (1983), p. 133
View PDF View article View in Scopus Google Scholar
Tatschl and Kolednik, 2003
A. Tatschl, O. Kolednik
On the experimental characterization of crystal plasticity in polycrystals
Mater. Sci. Eng., A, 356 (2003), p. 447
View PDF View article Google Scholar
Taylor, 1938
G.I. Taylor
Plastic strain in metals
J. Inst. Met., 62 (1938), pp. 307-325
View in Scopus Google Scholar
Tschopp et al., 2009
M.A. Tschopp, B.B. Bartha, W.J. Porter, P.T. Murray, S.B. Fairchild
Microstructure-dependent local strain behavior in polycrystals through <i>In-Situ</i> Scanning Electron Microscope Tensile Experiments
Metall. Mater. Trans. A, 40 (2009), p. 2363
Crossref View in Scopus Google Scholar
Zhang and Tong, 2004
N. Zhang, W. Tong
An experimental study on grain deformation and interactions in an Al-0.5%Mg multicrystal
Int. J. Plast., 20 (2004), p. 523
View PDF View article View in Scopus Google Scholar
Zhao et al., 2008
Z. Zhao, M. Ramesh, D. Raabe, A.M. Cuitiño, R. Radovitzky
Investigation of three-dimensional aspects of grain-scale plastic surface deformation of an aluminum oligocrystal
Int. J. Plast., 24 (2008), p. 2278
View PDF View article View in Scopus Google Scholar

Cited by (244)

Grain-boundary kinetics: A unified approach
晶界动力学：一种统一方法
2018, Progress in Materials Science
2018, 材料科学进展
Grain boundaries (GBs) are central defects for describing polycrystalline materials, and playing major role in a wide-range of physical properties of polycrystals. Control over GB kinetics provides effective means to tailor polycrystal properties through material processing. While many approaches describe different GB kinetic phenomena, this review provides a unifying concept for a wide range of GB kinetic behavior. Our approach rests on a disconnection description of GB kinetics. Disconnections are topological line defects constrained to crystalline interfaces with both step and dislocation character. These characteristics can be completely specified by GB bicrystallography and the macroscopic degrees of freedom of GBs. GB thermal fluctuations, GB migration and the ability of GBs to absorb/emit other defects from/into the delimiting grains can be modeled via the nucleation, propagation and reaction of disconnections in the GB. We review the fundamentals of bicrystallography and its relationship to disconnections and ultimately to the kinetic behavior of GBs. We then relate disconnection dynamics and GB kinetics to microstructural evolution. While this review of the GB kinetics literature is not exhaustive, we review much of the foundational literature and draw comparisons from a wide swath of the extant experimental, simulation, and theoretical GB kinetics literature.
Slip band-grain boundary interactions in commercial-purity titanium
2014, Acta Materialia
The interaction between slip bands and grain boundaries in commercial-purity titanium was examined using cross-correlation-based electron backscatter diffraction. At a low strain level, three types of interactions were observed: blocked slip band with stress concentration; slip transfer; and blocked slip band with no stress concentration. The stress concentration induced by the blocked slip band was fitted with Eshelby’s theoretical model, from which a Hall–Petch coefficient was deduced. It was found that the Hall–Petch coefficient varies with the individual grain boundary. We investigated the geometric alignment between the slip band and various slip systems to the neighbouring grain. Stress concentration can be induced by the blocked slip band if the slip system is poorly aligned with 〈a〉 prismatic, pyramidal or basal slip systems in the neighbouring grain. Transfer of slip across the boundary occurs when there is good alignment on 〈a〉 prismatic or 〈a〉 pyramidal slip systems. Other stress-relieving mechanisms are possible when the best alignment is not with the slip system that has the lower critical resolved shear stress.
Grain boundary segregation engineering in metallic alloys: A pathway to the design of interfaces
2014, Current Opinion in Solid State and Materials Science
Grain boundaries influence mechanical, functional, and kinetic properties of metallic alloys. They can be manipulated via solute decoration enabling changes in energy, mobility, structure, and cohesion or even promoting local phase transformation. In the approach which we refer here to as ‘segregation engineering’ solute decoration is not regarded as an undesired phenomenon but is instead utilized to manipulate specific grain boundary structures, compositions and properties that enable useful material behavior. The underlying thermodynamics follow the adsorption isotherm. Hence, matrix-solute combinations suited for designing interfaces in metallic alloys can be identified by considering four main aspects, namely, the segregation coefficient of the decorating element; its effects on interface cohesion, energy, structure and mobility; its diffusion coefficient; and the free energies of competing bulk phases, precipitate phases or complexions. From a practical perspective, segregation engineering in alloys can be usually realized by a modest diffusion heat treatment, hence, making it available in large scale manufacturing.
Dislocation interactions with grain boundaries
2014, Current Opinion in Solid State and Materials Science
Hydrogen-induced intergranular failure of iron
氢引起的沿晶断裂
2014, Acta Materialia
The hydrogen embrittlement of a commercial-grade pure iron was examined by using repeated stress-relaxation tests under simultaneous cathodic hydrogen charging. The hydrogen-charged iron, containing an estimated 25.8 appm H, fractured after repeated transients, with a total strain of ∼5%. The fracture mode was intergranular. Thermal activation measurements show a decrease in activation volume and free energy, which is consistent with hydrogen enhancing the dislocation velocity. The microstructure beneath the intergranular facets displays a dislocation cell structure more complex than expected for intergranular fracture and this strain-to-failure. It is proposed that hydrogen accelerates the evolution of the dislocation microstructure through the hydrogen-enhanced plasticity mechanism and this work-hardening of the matrix along with the attendant hydrogen concentration at the grain boundaries are crucial steps in causing the observed hydrogen-induced intergranular failure.
The physics of fatigue crack initiation
2013, International Journal of Fatigue
2013, 《国际疲劳杂志》
The fatigue life of a component can be expressed as the sum of two segments of life: (a) the number of loading cycles required to initiate a crack and (b) the number of cycles it takes that crack to propagate to failure. In this review, the primary emphasis is relating the fatigue crack initiation to the microstructure of the material. Many studies have focused on this phenomenon over the years and the goal of this paper is to put this work in perspective and encourage future work of fatigue in polycrystals based on the material’s microstructure. In order to address fatigue, it is necessary to understand the mechanisms that facilitate crack initiation. Slip irreversibilities exist in a material and accumulate during fatigue loading. At the defect level, irreversibilities are a result of dislocations: annihilating, cross-slipping, penetrating precipitates, transmitting through grain boundaries, and piling-up. These slip irreversibilities are the early signs of damage during cyclic loading. The dislocations subsequently form low-energy, stable structures as a means to accommodate the irreversible slip processes and increasing dislocation density during cyclic forward and reverse loading. The result is strain localizing in a small region within the materials, i.e. persistent slip bands and dislocation cells/bundles. Strain localization is a precursor to crack initiation. This review paper will focus on experimental observations of strain localization and the theory and numerical analysis of both slip irreversibilities and low energy configuration defect structures. This fundamental understanding is necessary to study persistent slip bands in FCC metals and alloys including the appropriate characterization, theory, and modeling. From this fundamental knowledge both micromechanical and crystal plasticity models can be used to predict crack initiation, which are also reviewed. Finally, this review ends with a discussion of the future of fatigue modeling and experiments.

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Google Scholar

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Z.H. Jin, P. Gumbsch, K. Albe, E. Ma, K. Lu, H. Gleiter, H. Hahn
Interactions between non-screw lattice dislocations and coherent twin boundaries in face-centered cubic metals
Acta Mater., 56 (2008), p. 1126
View PDF View article View in Scopus Google Scholar

[19] Kelchner et al., 1998
C.L. Kelchner, S.J. Plimpton, J.C. Hamilton
Dislocation nucleation and defect structure during surface indentation
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View in Scopus Google Scholar

[20] Kim et al., 2008
W.-G. Kim, S.-N. Yin, Y.-W. Kim, J.-H. Chang
Creep characterization of a Ni-based Hastelloy-X alloy by using theta projection method
Eng. Fract. Mech., 75 (2008), p. 4985
View PDF View article View in Scopus Google Scholar

[21] Kocks and Chandra, 1982
U.F. Kocks, H. Chandra
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Acta Metall., 30 (1982), p. 695
View PDF View article View in Scopus Google Scholar

[22] Lee et al., 1989
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View PDF View article View in Scopus Google Scholar

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View in Scopus Google Scholar

[24] Lienert et al., 2011
U. Lienert, S. Li, C. Hefferan, J. Lind, R. Suter, J. Bernier, N. Barton, M. Brandes, M. Mills, M. Miller, B. Jakobsen, W. Pantleon
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Crossref View in Scopus Google Scholar

[25] Lim, 1984
L.C. Lim
Slip-twin interactions in nickel at 573 K at large strains
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View PDF View article View in Scopus Google Scholar

[26] Lim and Raj, 1985
L.C. Lim, R. Raj
Continuity of slip screw and mixed crystal dislocations across bicrystals of nickel at 573 K
Acta Metallurgica, 33 (1985), p. 1577
View PDF View article View in Scopus Google Scholar

[27] Livingston and Chalmers, 1957
J.D. Livingston, B. Chalmers
Multiple slip in bicrystal deformation
Acta Metall., 5 (1957), p. 322
View PDF View article View in Scopus Google Scholar

[28] Meyers and Ashworth, 1982
M.A. Meyers, E. Ashworth
Model for the effect of grain size on the yield stress of metals
Philos. Mag. A:, 46 (1982), p. 737
Google Scholar

[29] Miner and Castelli, 1992
R. Miner, M. Castelli
Hardening mechanisms in a dynamic strain aging alloy, HASTELLOY X, during isothermal and thermomechanical cyclic deformation
Metall. Mater. Trans. A, 23 (1992), p. 551
View in Scopus Google Scholar

[30] Petch, 1953
N.J. Petch
The cleavage strength of polycrystals
J. Iron. Steel Inst., 174 (1953), pp. 25-28
Google Scholar

[31] Plimpton, 1995
S. Plimpton
Fast parallel algorithms for short-range molecular dynamics
J. Comput. Phys., 117 (1995), pp. 1-19
View PDF View article View in Scopus Google Scholar

[32] Plimpton, 2007
S. Plimpton
Large−scale Atomic/Molecular Massively Parallel Simulator
Sandia National Laboratories (2007)
Google Scholar

[33] Rice, 1992
J.R. Rice
Dislocation nucleation from a crack tip: an analysis based on the Peierls concept
J. Mech. Phys. Solids, 40 (1992), p. 239
View PDF View article View in Scopus Google Scholar

[34] Rollett et al., 2009
A.D. Rollett, R.A. Lebensohn, M. Groeber, Y. Choi, J. Li, G.S. Rohrer
Stress hot spots in viscoplastic deformation of polycrystals
Modell. Simul. Mater. Sci. Eng. (2009), p. 18
Google Scholar

[35] Rollett et al., 2010
A.D. Rollett, R.A. Lebensohn, M. Groeber, Y. Choi, J. Li, G.S. Rohrer
Stress hot spots in viscoplastic deformation of polycrystals
Modell. Simul. Mater. Sci. Eng. (2010), p. 18
Google Scholar

[36] Roters et al., 2010
F. Roters, P. Eisenlohr, L. Hantcherli, D.D. Tjahjanto, T.R. Bieler, D. Raabe
Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: theory, experiments, applications
Acta Mater., 58 (2010), p. 1152
View PDF View article View in Scopus Google Scholar

[37] Rowley and Thornton, 1996
M.A. Rowley, E.A. Thornton
Constitutive modeling of the visco-plastic response of Hastelloy-X and aluminum alloy 8009
J. Eng. Mater. Technol., 118 (1996), p. 19
Crossref View in Scopus Google Scholar

[38] Sangid et al., 2011
M.D. Sangid, T. Ezaz, H. Sehitoglu, I.M. Robertson
Energy of slip transmission and nucleation at grain boundaries
Acta Mater., 59 (2011), p. 283
View PDF View article View in Scopus Google Scholar

[39] Shen et al., 1986
Z. Shen, R.H. Wagoner, W.A.T. Clark
Dislocation pile-up and grain boundary interactions in 304 stainless steel
Scr. Metall., 20 (1986), p. 921
View PDF View article View in Scopus Google Scholar

[40] Siegel, 2005
D.J. Siegel
Generalized stacking fault energies, ductilities, and twinnabilities of Ni and selected Ni alloys
Appl. Phys. Lett., 87 (2005), p. 121901
Google Scholar

[41] Sutton and Balluffi, 2006
A.P. Sutton, R.W. Balluffi
Interfaces in Crystalline Materials
Oxford Classical Texts, Oxford (2006)
Google Scholar

[42] Sutton et al., 1983
M.A. Sutton, W.J. Wolters, W.H. Peters, W.F. Ranson, S.R. McNeill
Determination of displacements using an improved digital correlation method
Image Vision Comput., 1 (1983), p. 133
View PDF View article View in Scopus Google Scholar

[43] Tatschl and Kolednik, 2003
A. Tatschl, O. Kolednik
On the experimental characterization of crystal plasticity in polycrystals
Mater. Sci. Eng., A, 356 (2003), p. 447
View PDF View article Google Scholar

[44] Taylor, 1938
G.I. Taylor
Plastic strain in metals
J. Inst. Met., 62 (1938), pp. 307-325
View in Scopus Google Scholar

[45] Tschopp et al., 2009
M.A. Tschopp, B.B. Bartha, W.J. Porter, P.T. Murray, S.B. Fairchild
Microstructure-dependent local strain behavior in polycrystals through <i>In-Situ</i> Scanning Electron Microscope Tensile Experiments
Metall. Mater. Trans. A, 40 (2009), p. 2363
Crossref View in Scopus Google Scholar

[46] Zhang and Tong, 2004
N. Zhang, W. Tong
An experimental study on grain deformation and interactions in an Al-0.5%Mg multicrystal
Int. J. Plast., 20 (2004), p. 523
View PDF View article View in Scopus Google Scholar

[47] Zhao et al., 2008
Z. Zhao, M. Ramesh, D. Raabe, A.M. Cuitiño, R. Radovitzky
Investigation of three-dimensional aspects of grain-scale plastic surface deformation of an aluminum oligocrystal
Int. J. Plast., 24 (2008), p. 2278
View PDF View article View in Scopus Google Scholar

Outline 概要

Cited by (244) 被引用次数 (244)

Figures (15) 图 (15)

Tables (1) 表格 (1)

Journal of the Mechanics and Physics of Solids
《固体力学与物理学杂志》

Abstract 摘要

Highlights 要点

Keywords 关键词

1. Introduction 1. 引言

2. Material and methods 2. 材料和方法

2.1. Material 2.1. 材料

2.2. High resolution plastic strain measurements using DIC

3. Results and analysis 3. 结果与分析

3.1. Local plastic strain
3.1. 局部塑性应变

3.2. Grain boundary mantles
3.2. 晶界包层

3.3. Local slip system activity
3.3. 局部滑移系统活动

3.3.1. Slip trace analysis
3.3.1. 滑移迹分析

3.3.2. Crystallographic shear strain increments

3.4. Residual Burgers vector

4. Molecular dynamics simulations
4. 分子动力学模拟

5. Discussion 5. 讨论

6. Conclusions 6. 结论

Acknowledgments 致谢

Appendix 附录

References

Cited by (244)

Grain-boundary kinetics: A unified approach
晶界动力学：一种统一方法

Slip band-grain boundary interactions in commercial-purity titanium

Grain boundary segregation engineering in metallic alloys: A pathway to the design of interfaces

Dislocation interactions with grain boundaries

Hydrogen-induced intergranular failure of iron
氢引起的沿晶断裂

The physics of fatigue crack initiation

Slip transfer and plastic strain accumulation across grain boundaries in Hastelloy X滑移转移和 Hastelloy X 中的晶界塑性应变累积

Abstract 摘要

Highlights 要点

Keywords 关键词

1. Introduction 1. 引言

2. Material and methods 2. 材料和方法

2.1. Material 2.1. 材料

2.2. High resolution plastic strain measurements using DIC

3. Results and analysis 3. 结果与分析

3.1. Local plastic strain3.1. 局部塑性应变

3.2. Grain boundary mantles3.2. 晶界包层

3.3. Local slip system activity3.3. 局部滑移系统活动

3.3.1. Slip trace analysis3.3.1. 滑移迹分析

3.3.2. Crystallographic shear strain increments

3.4. Residual Burgers vector

4. Molecular dynamics simulations4. 分子动力学模拟

5. Discussion 5. 讨论

6. Conclusions 6. 结论

Acknowledgments 致谢

Appendix 附录

References

Slip transfer and plastic strain accumulation across grain boundaries in Hastelloy X
滑移转移和 Hastelloy X 中的晶界塑性应变累积

3.1. Local plastic strain
3.1. 局部塑性应变

3.2. Grain boundary mantles
3.2. 晶界包层

3.3. Local slip system activity
3.3. 局部滑移系统活动

3.3.1. Slip trace analysis
3.3.1. 滑移迹分析

4. Molecular dynamics simulations
4. 分子动力学模拟