A statistical study of the relationship between plastic strain and lattice misorientation on the surface of a deformed Ni-based superalloy

doi:10.1016/j.actamat.2020.05.029

Acta Materialia

Volume 195, 15 August 2020, Pages 555-570
第 195 卷，2020 年 8 月 15 日，第 555-570 页

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Abstract 摘要

Misorientation data from Electron Backscatter Diffraction (EBSD) is often used to identify strain localisation and quantify plastic strain at the microstructural scale. However, the exact relationship between local plastic strain and misorientation and how it changes at the grain and sub-grain level has not been studied in detail. We have used high resolution digital image correlation (HRDIC) to measure plastic strain at the sub-micron scale on the surface of a nickel superalloy strained to 2%. The strain values have been correlated to different misorientation measures at the grain and subgrain scale, over several hundreds of grains. We show that although the grain mean plastic strain is positively correlated to the lattice misorientation, there is a large scatter in the correlation, which depends on the misorientation measure used. There is also essentially no correlation between the magnitude of grain strain and grain orientation derived parameters like the Schmid factor and the Taylor factor, largely due to deformation bands at the mesoscale that are not crystallographic. At these strain levels, the relationship between misorientation and plastic strain is affected by the differences in how slip (discontinuous) and lattice rotation (continuous) develop, by local grain interactions and the development of transgranular strain localisation. It is therefore effectively not possible to quantify plastic strain within individual grains using EBSD derived misorientation values alone, although some measures of misorientation are more appropriate than others if there is an understanding of the underlying local plastic phenomena. Whereas slip is localised in slip bands, the misorientation varies smoothly in a manner that is only weakly spatially correlated to the slip. These findings have implications for the modelling of the deformed state of polycrystalline metals at the microstructural scale using continuum mechanics.
电子背散射衍射（EBSD）的取向数据常用于识别应变局部化和在微观结构尺度上量化塑性应变。然而，局部塑性应变与取向之间的确切关系以及它在晶粒和亚晶粒尺度上的变化尚未得到详细研究。我们使用高分辨率数字图像相关（HRDIC）测量了应变至 2%的镍基高温合金表面的亚微米尺度塑性应变。应变值与数百个晶粒的晶粒和亚晶粒尺度的不同取向度量相关联。我们发现，尽管晶粒平均塑性应变与晶格取向呈正相关，但相关性存在很大散布，这取决于所使用的取向度量。此外，晶粒应变的大小与晶粒取向参数（如 Schmid 因子和 Taylor 因子）之间基本没有相关性，这主要是因为中尺度变形带并非晶体学特征。在这些应变水平下，取向差与塑性应变之间的关系受到滑移（不连续）和晶格旋转（连续）发展方式差异的影响，受到局部晶粒相互作用和穿晶应变局部化发展的影响。因此，仅使用 EBSD 衍生的取向差值无法有效量化单个晶粒内的塑性应变，尽管如果对潜在的局部塑性现象有所了解，某些取向差度量比其他度量更合适。滑移在滑移带中局部化，而取向差则以平滑的方式变化，这种变化与滑移的空间相关性较弱。这些发现对使用连续介质力学在微观结构尺度上对多晶金属变形状态进行建模具有启示意义。

Graphical abstract 图示摘要

Keywords 关键词

Plasticity

Deformation

Misorientation

Digital image correlation

Electron backscatter diffraction

塑性变形位错数字图像相关电子背散射衍射

1. Introduction 1. 引言

The deformation response of an advanced polycrystalline alloy is determined by the deformation behaviour and interactions of the constituent phases and grains that make up its microstructure [1]. Studying these interactions is a powerful way of understanding the strengthening and damage mechanisms that control their performance, which in turn will help develop new, improved alloys and tailor microstructures for a given application. One of the prevailing ideas is that damage develops through the localisation of stresses and plastic strains at the microstructural scale, which leads eventually to failure, e.g. [2]. The local plastic strain can create an accumulation of defects (dislocations and vacancies) that lead to damage but can also produce stress concentrations, e.g. ahead of blocked slip bands [3], that cause damage. However, not all plastic strain creates defects and local plastic strain is difficult to measure at the microstructural scale, particularly in the bulk and post-mortem. Instead, researchers often use misorientation data from electron backscatter diffraction (EBSD) as a proxy for local strain and to link local deformation to damage.
先进多晶合金的变形响应由构成其微观结构的各相和晶粒的变形行为及相互作用决定[1]。研究这些相互作用是理解控制其性能的强化和损伤机制的有效途径，进而有助于开发新型改进合金和为特定应用定制微观结构。一种普遍的观点是，损伤通过在微观结构尺度上应力与塑性应变的局部化而发展，最终导致失效，例如[2]。局部塑性应变可以产生缺陷（位错和空位）的累积，这些缺陷会导致损伤，但也可以产生应力集中，例如在受阻滑移带前方[3]，这些应力集中会导致损伤。然而，并非所有塑性应变都会产生缺陷，在微观结构尺度上尤其难以测量局部塑性应变，特别是在块体材料和死后分析中。相反，研究人员通常使用电子背散射衍射（EBSD）的取向数据作为局部应变的替代指标，并将局部变形与损伤联系起来。

EBSD can provide a quantification of local lattice distortion, measured as lattice misorientation [4,5]. The usefulness of this technique for measuring deformation has been demonstrated in the literature. For example, average map misorientation, calculated by reference to the grain mean orientation or pixel neighbours in the deformed state, correlates well with applied plastic strain during both quasistatic loading in tension [6], [7], [8] and in compression [7], as well as during creep [7,9,10], and geometrically necessary dislocation map averages have been shown to correlate to the global plastic strain in polycrystalline copper under tension [11] but not in single crystal nickel [12]. Since grain misorientation uses a reference point in the grain it can be readily measured and mapped using EBSD, both post-mortem and in the bulk, via sectioning. Average map misorientation correlates well with the applied macroscopic strain, but local lattice misorientation should not correlate spatially with plastic deformation. This is because lattice misorientation is produced by gradients in plastic strain [13] and therefore it is not proportional to the magnitude of plastic strain but rather to its heterogeneity. For example, slip band termination at a grain boundary will result in local strain gradients and misorientation [14,15] and the extent of the misorientation will depend upon grain boundary character, dislocation slip type and the boundary condition associated with the free surface [16]. Therefore, lattice misorientation data is difficult to interpret without knowledge of the underlying local plastic phenomena. In metallic microstructures, deformation gradients and therefore lattice misorientation can be caused by the deformation incompatibility between different microstructural constituents, like different phases or grains of the same phase but with different crystallographic orientations, or by plastic instability within individual crystals [17]. This explains why mean grain misorientation correlates with macroscopic plastic strain. In a polycrystal experiencing strain hardening the strain gradients within individual grains should generally increase with increasing plastic strain. However, the correlation between plastic strain and local lattice rotation (as measured by misorientation) breaks down at the grain level, as demonstrated for copper deformed in tension to 5% [18]. Although covering hundreds of grains, in this copper study the EBSD misorientation measurements were performed at a very different spatial resolution to the strain measurements and therefore the origins of the scatter could not be studied in detail. Despite this shortcoming, EBSD measurements of lattice distortion are often used as a proxy of for locating and estimating plastic strain levels [19], [20], [21] and have been used to identify different fracture modes [22] and crack initiation modes [23]. Lattice distortion has been correlated to local stress corrosion crack (SCC) initiation sites [24] and shown as localised along crack paths [22], suggesting a relationship between local deformation and crack initiation and propagation, respectively. Lattice distortion has been shown as a variable when surrounding different types of SCCs [25] and it has been used to quantify the effect of the annealing temperature on residual deformation and the resulting effect of this localised deformation on rates of corrosion [26].
EBSD 可以提供局部晶格畸变的定量分析，测量方式为晶格取向差[4,5]。该技术在测量变形方面的有效性已在文献中得到证实。例如，通过参考变形状态下的晶粒平均取向或像素邻域计算出的平均图样取向差，与拉伸[6],[7],[8]和压缩[7]过程中的准静态加载以及蠕变[7,9,10]期间施加的塑性应变密切相关；几何必需位错图样平均已被证明与单晶铜在拉伸[11]下的全局塑性应变相关，但在单晶镍中则不相关。由于晶粒取向差利用晶粒内的参考点，因此可以通过 EBSD 方便地测量和绘制，包括死后分析和通过切片进行的整体分析。平均图样取向差与施加的宏观应变密切相关，但局部晶格取向差不应在空间上与塑性变形相关。这是因为晶格倾斜是由塑性应变梯度产生的[13]，因此它不与塑性应变的幅度成正比，而是与其不均匀性成正比。例如，滑移带在晶界处的终止会导致局部应变梯度和晶格倾斜[14,15]，而晶格倾斜的程度取决于晶界特性、位错滑移类型以及与自由表面相关的边界条件[16]。因此，如果没有对潜在局部塑性现象的了解，晶格倾斜数据很难解释。在金属微观结构中，变形梯度，因此晶格倾斜，可能是由不同微观结构组分之间的变形不兼容引起的，例如不同相或同一相但具有不同晶体学取向的晶粒，或由单个晶体的塑性不稳定性引起[17]。这解释了为什么平均晶粒倾斜与宏观塑性应变相关。在经历应变强化的多晶材料中，单个晶粒内的应变梯度通常应随着塑性应变的增加而增加。然而，在晶粒尺度上，塑性应变与局部晶格旋转（通过晶格旋转测量）之间的相关性会失效，正如对拉伸至 5%的铜所展示的那样[18]。尽管这项铜研究覆盖了数百个晶粒，但 EBSD 晶格旋转测量所采用的空间分辨率与应变测量非常不同，因此无法详细研究散布的原因。尽管存在这一缺陷，EBSD 晶格畸变测量通常被用作定位和估计塑性应变水平的替代方法[19]，[20]，[21]，并且已被用于识别不同的断裂模式[22]和裂纹萌生模式[23]。晶格畸变与局部应力腐蚀裂纹（SCC）的萌生位点相关联[24]，并且显示出沿着裂纹路径局部化[22]，这表明局部变形与裂纹萌生和扩展之间存在关系。晶格畸变已被证明是在周围不同类型的应力腐蚀裂纹（SCCs）时的一种变量[25]，并且已被用于量化退火温度对残余变形的影响以及这种局部变形对腐蚀速率的最终影响[26]。

In this paper we aim to make a direct comparison between surface lattice misorientation and plastic strain at high spatial resolution and over a wide field of view containing a large number of grains in an attempt to achieve measurements that are statistically representative of the surface studied. We carried out these measurements on a precipitation-strengthened Ni superalloy deformed to a small plastic strain of around 2%, a level of plastic strain that is relevant to the small deformations and lattice rotations experienced in fatigue testing and in service and typically found in previous attempts to use EBSD to quantify plastic strain. We used automated data acquisition to cover an area 1 × 0.5 mm with a spatial resolution of ∼120 nm and developed automated data analysis routines for the manipulation of sub-grain scale data, making it possible to correlate directly HRDIC plastic strain information and EBSD lattice distortion information [27]. Because there are many different ways to manipulate EBSD data in order to obtain lattice distortion information [5,[28], [29], [30]], we calculated nine different measures of lattice misorientation from the same Hough-indexed EBSD data set. These values were compared to the local plastic strain, measured using HRDIC, both at the mean grain-to-grain variation scale and on the pixel-to-pixel sub-grain scale. Finally, we assessed the validity of global boundary condition orientation-based predictors of deformation, such as the Taylor factor and Schmid factor in a statistical manner by looking for correlations between these predictors and the actual plastic deformation and lattice misorientation measured on the grain scale. These calculations are easily performed in commercial EBSD post-processing software and often used to understand local plastic deformation at the microstructural scale [19], [20], [21]. However, we demonstrate that these predictors break down at the local level and cannot be used to predict the magnitude of local strain.
在这篇论文中，我们旨在对表面晶格取向差和塑性应变进行直接比较，以在较高的空间分辨率和包含大量晶粒的宽视场范围内进行测量，从而获得能够代表所研究表面的统计代表性测量结果。我们在这项测量中使用了经过沉淀强化的镍基高温合金，其塑性应变约为 2%，这一水平的塑性应变与疲劳试验和服务中经历的小变形和晶格旋转相关，并且通常出现在以往使用 EBSD 量化塑性应变的尝试中。我们采用自动化数据采集方法覆盖了 1 × 0.5 mm 的区域，空间分辨率为~120 nm，并开发了用于处理亚晶粒尺度数据的自动化数据分析程序，使得 HRDIC 塑性应变信息和 EBSD 晶格畸变信息能够直接关联[27]。由于存在多种不同的方法可以用来处理 EBSD 数据以获得晶格畸变信息[5,[28], [29], [30]]，我们从同一 Hough 索引的 EBSD 数据集中计算了九种不同的晶格取向差测量值。这些值与使用 HRDIC 测量的局部塑性应变进行了比较，比较尺度包括平均晶粒间变异尺度和像素到像素的亚晶粒尺度。最后，我们通过寻找这些预测因子与晶粒尺度上实际测量的塑性变形和晶格畸变之间的相关性，以统计方式评估了基于全局边界条件方向变形预测器的有效性，例如泰勒因子和施密特因子。这些计算可以在商业 EBSD 后处理软件中轻松完成，并常用于理解微观结构尺度的局部塑性变形[19]、[20]、[21]。然而，我们证明了这些预测因子在局部尺度上失效，不能用于预测局部应变的幅度。

2. Material and experimental methodology
2. 材料与实验方法

2.1. Material and sample 2.1. 材料和样品

The material studied was polycrystalline Ni-based superalloy RR1000 following uniaxial tension. RR1000 is a γ/γ’ alloy that was developed at Rolls Royce plc. for use as turbine disks in the rotational hot sections, has high tensile strength and retains good fatigue properties over its operational lifetime [31]. Its development is detailed in a comprehensive review elsewhere [32]. Here, we have applied heat treatments to obtain a unimodal distribution of γ’ size [33]. This was achieved through first applying a γ’ super-solvus solution heat treatment followed by an oil quench and a γ’ sub-solvus heat treatment with subsequent very slow cooling to promote growth of existing γ’ rather than nucleation of new one [33]. All heat treatments were performed in an Ar atmosphere. This procedure resulted in a unimodal size distribution of γ’ precipitates with diameter ∼ 250 nm and a volume fraction ∼ 40%. In this microstructure slip predominantly occurs by cutting of the γ’ particles [34] and therefore deformation heterogeneity between the two phases can be ignored, i.e. it can be treated as a single phase polycrystalline material for the purposes of this study. A characterisation of the grains after heat treatment is provided in Fig. 1, in which we show the average grain orientations and the grain size distribution (both post-deformation) in parts a) and b), respectively. The grain size was determined as the diameter of the equivalent circle of the grain area, and the mean grain size is ∼20 μm. The inverse pole figure (IPF) scatter-plot in part a) is shown from the loading direction and demonstrates that the texture is random. Almost half (0.47) the grain boundaries in this microstructure are Σ3 annealing twin boundaries.
所研究的材料是单轴拉伸下的多晶 Ni 基高温合金 RR1000。RR1000 是一种γ/γ'合金，由 Rolls Royce plc 公司开发，用于旋转热部件中的涡轮盘，具有高抗拉强度，并在其工作寿命中保持良好的疲劳性能[31]。其开发细节在另一篇综合综述中有详细描述[32]。在此，我们通过热处理获得了γ'尺寸的单峰分布[33]。这是通过首先进行γ'过饱和固溶热处理，然后进行油淬，接着进行γ'亚饱和热处理，并随后进行非常缓慢的冷却来实现的，以促进现有γ'的生长而不是新γ'的形核[33]。所有热处理均在氩气气氛中进行。该工艺产生了直径约为 250 nm、体积分数约为 40%的γ'析出物的单峰尺寸分布。在这种微观结构中，滑移主要通过切割γ'颗粒发生[34]，因此可以忽略两相之间的变形不均匀性，即在本研究中可以将其视为单相多晶材料。热处理后的晶粒表征如图 1 所示，其中分别展示了 a)部分的平均晶粒取向和 b)部分的晶粒尺寸分布（均为变形后）。晶粒尺寸被定义为晶粒面积的等效圆直径，平均晶粒尺寸约为 20 μm。a)部分的逆极图（IPF）散点图显示加载方向，表明织构是随机的。这种微观结构中近半数（0.47）的晶界是Σ3 退火孪晶界。

The tensile specimens were machined by electric discharge machining (EDM) to the geometry detailed in Fig. 1 part c). The EDM recast layer was removed by grinding and polishing with a final step of polishing with colloidal silica (0.06 μm) for ∼10 min before pre-deformation orientation mapping. Then, a gold speckle pattern was applied to the sample surface to act as fiducial markers for the image correlation process. Images of the speckle pattern were obtained before and after uniaxial tensile deformation and the resulting displacement field was calculated. The gold particles were removed by polishing with colloidal silica before post-deformation EBSD. The following contains details relating to this procedure.
拉伸试样通过电火花加工（EDM）加工成图 1c)中详细描述的几何形状。通过研磨和抛光去除 EDM 再铸层，并在预变形取向映射前用胶体二氧化硅（0.06 μm）进行最后一步抛光，持续约 10 分钟。然后在样品表面施加金点状图案，作为图像相关过程的基准标记。在单轴拉伸变形前后获取点状图案的图像，并计算所得位移场。在变形后 EBSD 前，通过胶体二氧化硅抛光去除金颗粒。以下包含与此程序相关的详细信息。

All displacement data, orientation data and in-house data manipulation procedures in python have been made open access and are available to the community [27,35].
所有位移数据、取向数据以及 Python 中的内部数据处理程序均已公开，并可供社区使用[27,35]。

2.2. Orientation mapping 2.2. 取向映射

Orientation mapping was performed in the region of interest both before and after deformation using a field emission gun (Zeiss Sigma) SEM equipped with an Oxford Instruments electron backscatter diffraction (EBSD) system, consisting of a NordlyNano detector and Aztec software version 3. In the non-deformed state the EBSD scans were obtained before gold remodelling and in the deformed state the gold layer was removed by ∼60 s of polishing with colloidal silica. Scans were performed at an operating voltage of 30 kV and a probe current of ∼7.8 nA using the 120 μm aperture in high current mode. An area of ∼1 × 0.5 mm was scanned with a step size of 0.5 μm at an acquisition rate of ∼35 Hz. The Oxford Instruments Aztec software was used for Hough-based indexing of diffraction patterns by comparison to theoretical patterns.
取向映射在变形前后对感兴趣区域进行了检测，使用配备牛津仪器电子背散射衍射（EBSD）系统的场发射枪（蔡司 Sigma）SEM，该系统由 NordlyNano 探测器及 Aztec 软件版本 3 组成。在非变形状态下，EBSD 扫描在金重塑前获得，而在变形状态下，通过用胶体二氧化硅抛光约 60 秒去除金层。扫描在 30 kV 的工作电压和约 7.8 nA 的探针电流下进行，使用 120 μm 孔径在高电流模式下进行。扫描区域约为 1 × 0.5 mm，步长为 0.5 μm，采集频率约为 35 Hz。通过将衍射图与理论图进行比较，使用牛津仪器 Aztec 软件进行基于 Hough 的标定。

2.3. High resolution digital image correlation
2.3. 高分辨率数字图像相关

2.3.1. Gold remodelling 2.3.1. 金重塑

The EBSD-induced carbon contamination was removed by a ∼30 s polish with colloidal silica and a gold speckle was then applied to the sample surface using the gold remodelling technique [36]. A 25–40 nm thick gold layer was deposited onto the sample surface using an Edwards S150B sputter coater at a rate of 5–8 nm min⁻¹. Following this, the remodelling was performed in a water vapour environment for 3 h. This was achieved by placing the sample on a hotplate at 300 °C together with a beaker of water and a larger, inverted beaker to cover both the small beaker and the sample and topping up the water as necessary.
EBSD 引起的碳污染通过使用胶体二氧化硅进行约 30 秒的抛光去除，然后使用金重塑技术[36]在样品表面施加金点。使用爱德华兹 S150B 溅射涂层机以 5-8 纳米/分钟的速度将 25-40 纳米厚的金层沉积到样品表面。之后，在含水蒸气环境中进行重塑 3 小时。这是通过将样品放置在 300°C 的热板上，同时放置一个水杯和一个更大的倒置水杯来覆盖小水杯和样品，并根据需要添加水来实现的。

2.3.2. Image acquisition 2.3.2. 图像采集

Backscattered electron images of the gold speckle pattern were obtained before and after the deformation step using a FEI Magellan HR 400 L FE-SEM. To maximise the spatial resolution, the microscope was operated at a voltage of 5 kV with a + 2 kV stage bias and a probe current of 0.8 nA. A working distance of 4 mm was chosen to maximise the signal-to-noise ratio. A mosaic of 40 columns × 20 rows was used to collect 800 images with a 20% overlap, corresponding to a field of view of ∼ 1 × 0.5 mm². Each image was of resolution 2048 × 1768 pixels and a pixel size of 14.6 nm. After the deformation step, great care was taken to ensure that the contrast and brightness of the post-deformation images closely represented those in the non-deformed state in order to minimise systematic errors.
背散射电子图像在变形步骤前后通过 FEI Magellan HR 400 L FE-SEM 获取。为最大化空间分辨率，显微镜在 5 kV 电压下操作，带有+2 kV 的样品台偏置和 0.8 nA 的探针电流。选择 4 mm 的工作距离以最大化信噪比。使用 40 列×20 行的马赛克来收集 800 张图像，具有 20%的重叠，对应视野约为 1 × 0.5 mm²。每张图像分辨率为 2048 × 1768 像素，像素大小为 14.6 nm。变形步骤后，特别注意确保变形后图像的对比度和亮度与非变形状态下的情况尽可能接近，以最小化系统误差。

2.3.3. Mechanical testing
2.3.3. 力学测试

The dogbone sample was deformed in uniaxial tension using a Zeiss–Kammrath 5 kN tension-compression microtester. The sample was deformed at a rate of 0.3 mm min⁻¹ to a global macroscopic uniaxial engineering strain of ∼ 0.02. The engineering stress-strain curve is shown in Fig. 1 part d). After testing, the sample was removed from the tester and post-deformation images were obtained in the unloaded state according to Section 2.3.2.
狗骨样条在 Zeiss–Kammrath 5 kN 拉伸-压缩微测试机上进行了单轴拉伸变形。样条以 0.3 mm/min 的速率变形至全局宏观单轴工程应变约为 0.02。工程应力-应变曲线如图 1(d)所示。测试后，样条从测试机上取下，根据 2.3.2 节，在无载荷状态下获取了变形后图像。

2.3.4. Image correlation 2.3.4. 图像相关

The image mosaics were stitched together using the Grind/Collection stitching plugin for FIJI [37] and cropped into 8 separate areas for image correlation due to maximum image size limitations for the DIC calculations. The stitched images from the deformed and non-deformed state were then correlated using LaVision's commercially available DIC software, DaVis 8 (LaVision, 2018) [38]. The systematic error was calculated by comparing two images in the non-deformed state. These images were from the same region but the sample was removed from the microscope and replaced before taking the second image. The two images were then correlated using decreasing interrogation window sizes to obtain the optimal interrogation window size [36]. We found that using a sub-window size of 8 × 8 pixels and no overlap resulted in the best compromise between spatial resolution, at around 120 nm, and the mean systematic error at approximately 0.01 effective strain (ɛ_eff) over approximately 32 million data points. The image correlation results in full-field u₁, u₂ displacements on the x₁, x₂ plane with normal x₃. The individual components of the displacement gradient tensor can then be found by the gradient of the displacement with respect to the macroscopic directions using Eq. (1).
图像拼贴使用 FIJI 的 Grind/Collection 拼接插件[37]拼接在一起，并根据 DIC 计算的最大图像尺寸限制裁剪成 8 个独立区域进行图像相关分析。随后，使用 LaVision 的商业 DIC 软件 DaVis 8（LaVision，2018）[38]对变形和非变形状态下的拼接图像进行相关分析。通过比较非变形状态下的两张图像来计算系统误差。这两张图像来自同一区域，但在拍摄第二张图像之前将样品从显微镜中移除并更换。然后使用逐渐减小的查询窗口大小对这两张图像进行相关分析，以获得最佳查询窗口大小[36]。我们发现，使用 8×8 像素的子窗口大小且无重叠，在空间分辨率约为 120 纳米和平均系统误差约为 0.01 有效应变（ɛ _eff ）之间取得了最佳平衡，覆盖了约 3200 万数据点。图像相关分析结果得到 x ₁ ，x ₂ 平面上的全场 u ₁ ，u ₂ 位移，法线为 x ₃ 。然后可以通过使用公式(1)对位移相对于宏观方向进行梯度运算，找到位移梯度张量的各个分量。(1)

\frac{\partial u_{i}}{\partial x_{j}} = [\begin{matrix} \frac{\partial u_{1}}{\partial x_{1}} & \frac{\partial u_{2}}{\partial x_{1}} \\ \frac{\partial u_{1}}{\partial x_{2}} & \frac{\partial u_{2}}{\partial x_{2}} \end{matrix}]

It is convenient to visualise the local strain by combining the four components of the displacement gradient tensor into an effective strain, ɛ_eff [1]. This is a shear-dominated term, which reflects the characteristic slip, and can be calculated using Eq. (2).
通过将位移梯度张量的四个分量组合成一个有效应变ɛ _eff [1]，可以方便地可视化局部应变。这是一个以剪切为主的项，反映了特征滑移，可以使用公式(2)进行计算。(2)

ε_{e f f} = \sqrt{\frac{1}{2} {(\frac{\partial u_{1}}{\partial x_{1}} - \frac{\partial u_{2}}{\partial x_{2}})}^{2} + \frac{1}{2} {(\frac{\partial u_{1}}{\partial x_{2}} + \frac{\partial u_{2}}{\partial x_{1}})}^{2}}

2.4. Data analysis routines
2.4. 数据分析程序

The EBSD orientation data and HRDIC displacement data was analysed using our own Python routines with NumPy for numerical computation [39] and Matplotlib for visualisation [40]. The routines are contained within the open source DefDAP library (Deformation Data Analysis in Python) [27]. The manipulation of orientations was performed with quaternion geometry, the details for which are well described elsewhere [41], [42], [43], and the grain-level misorientation was calculated in one of nine ways, as detailed in Table 1. All of these measures of misorientation calculate the pixel misorientation in the deformed state. One of the misorientation calculations considered is the kernel average misorientation (KAM). The KAM considers the local misorientation by the mean orientation difference to the pixels that surround it. The KAM calculation here was performed for a 3 × 3 array that surrounds the central pixel of interest and the misorientation was not allowed to exceed 5° so as to exclude high angle grain boundaries. The CMA measure of misorientation [30] can be thought of as a non-correlated KAM where the misorientation is calculated between each pixel and every other pixel in a grain and then the cumulative distribution of all the grain misorientations is plotted as a Weibull plot in order to determine the misorientation for which the cumulative distribution is equal to 0.632.
使用我们自研的 Python 程序，结合 NumPy 进行数值计算[39]和 Matplotlib 进行可视化[40]，对 EBSD 取向数据和 HRDIC 位移数据进行了分析。这些程序包含在开源的 DefDAP 库（Python 中的变形数据分析）[27]中。取向的操作采用四元数几何方法，其详细内容在其他地方有详细描述[41][42][43]，晶粒级别的取向偏差通过九种方法之一进行计算，具体见表 1。所有这些取向偏差的测量方法都计算了变形状态下的像素取向偏差。其中一种取向偏差的计算方法是核平均取向偏差（KAM）。KAM 通过计算其周围像素的平均取向差异来考虑局部取向偏差。在此处的 KAM 计算中，以目标中心像素周围 3×3 阵列进行计算，且取向偏差不允许超过 5°，以排除高角度晶界。 CMA 测量晶格倾斜的方法[30]可以看作是一种非相关 KAM 方法，其中计算每个像素与晶粒中每个其他像素之间的晶格倾斜，然后将所有晶粒的晶格倾斜的累积分布绘制成威布尔图，以确定累积分布等于 0.632 时的晶格倾斜。

Table 1. Lattice misorientation and GROD variant definitions.
表 1. 晶格倾斜和 GROD 变体定义。

Lattice misorientation 晶格倾斜	Description 描述
KAM	3 × 3 nearest neighbour kernel 3×3 最近邻核
CMA	Non-correlated KAM 非相关 KAM
GROD variants GROD 变体	Reference orientation 参考取向
Mean 平均值	Average orientation 平均取向
PQ_Max	Best Kikuchi pattern quality 最佳 Kikuchi 图样质量
Min KAM 最小 KAM	Lowest kernel misorientation 最低晶粒取向差
Mean Min 平均最小值	Lowest GROD Mean 最低 GROD 平均值
Mean Max 平均最大值	Highest GROD Mean 最高 GROD 平均值
Original orientation 原始取向	Pre-deformation orientation 变形前取向
Centroid 质心	Euclidean grain centre 欧几里得晶粒中心

Seven of the measures of misorientation in Table 1 are variants of the grain reference orientation distribution (GROD) [29]. For the GROD the pixel misorientation is calculated for each pixel within a grain relative to a reference orientation. Those reference orientations may be in the deformed or the non-deformed state.
表 1 中的七项取向偏差测量是晶粒参考取向分布（GROD）[29]的变体。对于 GROD，每个晶粒内的每个像素的像素取向偏差是相对于参考取向计算的。这些参考取向可能处于变形或未变形状态。

•
For the GROD Mean the reference orientation is the mean orientation in the deformed state.
对于 GROD 平均值，参考取向是变形状态下的平均取向。
•
For the GROD PQ_Max the reference orientation is that of the pixel in the deformed state with the best Kikuchi pattern quality.
对于 GROD PQ _Max ，参考取向是变形状态下 Kikuchi 图案质量最好的像素的取向。
•
For the GROD Min KAM the reference orientation is the pixel in the deformed state with the lowest 3 × 3 KAM value.
对于 GROD Min KAM，参考方向是变形状态下具有最低 3×3 KAM 值的像素。
•
For the GROD Mean Min and Max the reference orientation is the pixel in the deformed state with the minimum or maximum GROD Mean value, respectively (note that no pixel in the GROD Mean distribution will have a GROD Mean value of exactly zero).
对于 GROD Mean Min 和 Max，参考方向分别是变形状态下具有最小或最大 GROD Mean 值的像素（注意：GROD Mean 分布中没有任何像素的 GROD Mean 值恰好为零）。
•
For the GROD Original the misorientation is calculated as the pixel orientation relative to the grain mean orientation in the non-deformed state.
对于 GROD Original，晶格畸变角计算为像素方向相对于非变形状态下晶粒平均方向的角度。
•
For the GROD Centroid the reference orientation is pixel in the deformed state with the greatest Euclidean distance from the grain boundary.
对于 GROD Centroid，参考方向是变形状态下与晶界具有最大欧几里得距离的像素。

Note that while most misorientation functions explored here can be mapped at the sub-grain scale, the CMA cannot. The EBSD grain boundaries were defined as neighbouring pixels with a misorientation greater than 6° which successfully delineated individual grains and twins. The boundaries were then used to mask grains for both the EBSD and the HRDIC data. The EBSD grain boundaries were mapped onto the HRDIC data using manually selected homologous points (usually triple junctions that are obvious in both data sets) and an affine transform (resize, shear, rotation) via software implemented in Python. Single grain analysis of the HRDIC plastic strain data then becomes possible for hundreds of grains, as does its correlation to lattice misorientation data from EBSD. The sub-grain comparisons of strain to misorientation required smoothing of the strain data to obtain more of a continuum of strain, which was performed by applying a Gaussian kernel to the strain data using the convolution package within the Astropy library [44]. The sub-grain pixel-to-pixel correlations were performed by binning the grain strain data to the EBSD spatial resolution and performing a Spearman's rank correlation test on the data using the stats package within the Scipy library [45]. The statistical significance of the Spearman's rank was judged based on the value of the resulting coefficient in accordance with common practice [46], as significance tests produced unreasonably small p values due to the large number of data points in the correlations.
请注意，尽管此处探索的大多数晶格取向函数可以在亚晶粒尺度上进行映射，但 CMA 不能。EBSD 晶界被定义为相邻像素间晶格取向大于 6°的边界，这成功地划分了单个晶粒和孪晶。然后使用这些边界对 EBSD 和 HRDIC 数据中的晶粒进行掩膜。通过 Python 软件，使用手动选择的同源点（通常是两组数据中都很明显的三叉结点）和仿射变换（调整大小、剪切、旋转），将 EBSD 晶界映射到 HRDIC 数据上。这使得对 HRDIC 塑性应变数据进行单个晶粒分析成为可能，同时也可以将其与来自 EBSD 的晶格取向数据相关联。为了获得更连续的应变，对应变数据进行平滑处理以进行应变与取向的比较，这是通过使用 Astropy 库中的卷积包[44]将高斯核应用于应变数据来完成的。子晶粒像素间相关性通过将晶粒应变数据分箱至 EBSD 空间分辨率，并使用 Scipy 库中的 stats 包对数据进行 Spearman 等级相关检验来执行[45]。根据常规做法[46]，根据所得系数的值判断 Spearman 等级的相关性统计显著性，因为相关性中的数据点数量很大，导致显著性检验产生了不合理的较小 p 值。

3. Results 3. 结果

3.1. The spatial relationship between strain and misorientation at the mesoscale and mean grain scale
3.1. 中尺度及平均晶粒尺度下应变与取向的时空关系

The effective strain (ɛ_eff) map at ɛ_xx ∼ 0.02 is shown in Fig. 2 on a log scale. In this map the tensile axis is horizontal, the field of view is ∼1 mm by 0.5 mm and the spatial resolution is ∼120 nm. The supplementary material contains a full resolution image and a link to a video overview of this data set. Five grains are highlighted in the figure to demonstrate different slip characteristics, G1-G5. The ɛ_eff data and the EBSD lattice misorientation data, ϕ, based on GROD Mean (G Mean) are shown together for each grain G1-G5. We observe discrete, localised slip bands that are contained in individual grains, sometimes of a single slip plane (G1) and sometimes of two or three slip planes (G2, G3, respectively), but always on {111} (confirmed by slip trace analysis utilising grain orientation data from EBSD). We also observe cross slip (G4) and more diffuse strain within grains (G5), the latter possibly discrete but if so then beyond the resolution of the strain data. Importantly, we observe diffuse strain bands that span many grains at ∼ ±45 ° to the tensile axis that cannot be attributed to single grain behaviour. Grains with complicated slip geometries (G4) often show correspondingly complex distributions of lattice misorientation and grains with diffuse slip (G5) can result in unexpected misorientation distributions, such as misorientation banding that does not correspond to plastic slip bands (G5).
有效应变（ɛ _eff ）图在ɛ _xx ∼ 0.02 时显示于图 2，采用对数刻度。在该图中，拉伸轴为水平方向，视场范围为∼1 mm × 0.5 mm，空间分辨率为∼120 nm。补充材料中包含全分辨率图像以及该数据集的视频概览链接。图中突出了五个晶粒（G1-G5）以展示不同的滑移特征。对于每个晶粒 G1-G5，均显示了ɛ _eff 数据和基于 GROD 均值（G Mean）的 EBSD 晶格取向数据ϕ。我们观察到离散的、局部的滑移带，这些滑移带存在于单个晶粒中，有时为单个滑移面（G1），有时为两个或三个滑移面（G2、G3），但始终位于{111}面上（通过利用 EBSD 晶粒取向数据的滑移迹线分析得到证实）。我们还观察到交叉滑移（G4）和晶粒内的更弥散应变（G5），后者可能为离散的，但如果是这样的话，则超出了应变数据的分辨率。重要的是，我们观察到跨越多个晶粒的弥散应变带，这些应变带与拉伸轴成∼ ±45 °，且不能归因于单个晶粒的行为。具有复杂滑移几何的晶粒（G4）通常表现出相应的晶格取向分布复杂，而具有弥散滑移的晶粒（G5）可能导致意外的晶格取向分布，例如与塑性滑移带不对应的取向带（G5）。

To investigate the impact of calculating misorientation (ϕ) in different ways, we have taken the EBSD data from the region of interested (ROI) in Fig. 2 and plotted nine different lattice misorientation measures, each calculated from the same EBSD data set, which we display in Fig. 3. The name of each calculation is given above the figure and these names relate to the descriptors detailed in Table 1. There are significant differences in the magnitude and spatial distribution of the lattice misorientation for the different methods of calculation. In order to quantify the differences in magnitude, we have plotted the frequency distribution of the EBSD misorientation values for the whole field of view in Fig. 2 and for each of the nine calculation types in Table 1. We plot these frequency distributions in Fig. 4 part a) together with the distribution of ɛ_eff (dashed black line) from the HRDIC map in Fig. 2. The distribution of plastic strain is significantly different from the distribution of misorientation at this field-of-view length scale. Some of the measures of misorientation provide good fits to the plastic strain data for low values of strain, and other measures of misorientation capture more information in the higher strain range, but in all cases some information is lost when comparing strain to misorientation. To quantify the similarity of the map distributions we can use the Kullback–Leibler deviation [47], which is a quantitative descriptor of the information lost when correlating two distributions. The deviation is calculated as in Eq. (3).
为了研究计算取向差（ϕ）的不同方法的影响，我们从图 2 中感兴趣区域（ROI）获取了 EBSD 数据，并绘制了九种不同的晶格取向差度量，这些度量均基于相同的 EBSD 数据集，如图 3 所示。每种计算方法的名称标在图上方，这些名称与表 1 中详细描述的描述符相关。不同计算方法的晶格取向差的数值和空间分布存在显著差异。为了量化数值上的差异，我们在图 2 的全视场和表 1 中的九种计算类型中绘制了 EBSD 取向值的频率分布，如图 4a)所示，同时绘制了来自图 2 HRDIC 图的ɛ _eff （虚线黑线）的分布。在这个视场长度尺度上，塑性应变的分布与取向差的分布存在显著差异。某些取向错配的度量在应变值较低时与塑性应变数据吻合良好，而其他取向错配的度量在高应变范围内捕获更多信息，但在所有情况下，当比较应变与取向错配时，总会丢失一些信息。为了量化地图分布的相似性，我们可以使用 Kullback–Leibler 偏差[47]，它是在关联两个分布时丢失信息的定量描述符。该偏差按公式(3)计算。(3)

D_{K L} (P ∥ Q) = \sum_{x \in χ} P (x) l o g \frac{P (x)}{Q (x)}

where P and Q are functions of the variable x over the probability space χ, in this case a measure of deformation as the plastic strain measured by HRDIC or the lattice misorientation calculated from EBSD orientation data, i.e. P(x) is the distribution of ɛ_eff and Q(x) the distribution of ϕ. While P(ɛ_eff) and Q(ϕ) scale at approximately 1:10 as shown in Fig. 4 part b), they do not populate the same probability space χ. Therefore, the D_KL statistic calculated here should be taken as a relative measure of the agreement between the distributions of strain and misorientation, as opposed to some absolute measure of the deviation. We calculate D_KL and plot the values in Fig. 4 part b). Lower D_KL are a measure of less information loss and so the 3 × 3 KAM provides the best representation of the strain values when considering the map as a whole, followed by GROD Mean. Conversely, the GROD Mean Max, Centroid and the original orientation give the poorest representation of the distribution of plastic strain values, which is primarily due to their lack of range; the GROD Mean Max and Centroid functions do not cover the low strain range and the PQ Max does not cover the high strain values
其中 P 和 Q 是变量 x 在概率空间χ上的函数，在此情况下，χ是变形的度量，如 HRDIC 测量的塑性应变或从 EBSD 取向数据计算出的晶格位向，即 P(x)是ɛ _eff 的分布，Q(x)是ϕ的分布。虽然 P(ɛ _eff )和 Q(ϕ)在图 4 部分 b)中大约按 1:10 的比例缩放，但它们并不占据相同的概率空间χ。因此，此处计算的 D _KL 统计量应被视为应变和位向分布之间一致性的相对度量，而不是偏差的绝对度量。我们计算 D _KL 并在图 4 部分 b)中绘制这些值。较低的 D _KL 是信息损失较少的度量，因此当将地图视为整体时，3 × 3 KAM 提供了应变值的最佳表示，其次是 GROD Mean。相反，GROD 均值最大值、质心和原始取向对塑性应变值分布的表征最差，这主要归因于它们缺乏范围；GROD 均值最大值和质心函数未覆盖低应变范围，而 PQ 最大值未覆盖高应变值

In Fig. 3 we show that how the misorientation is calculated affects the magnitude and localization of lattice misorientation within grains. We know that the map average misorientation is positively correlated to the macroscopic strain. To test whether this relationship holds at the grain level, we have taken the mean of the ɛ_eff values within a grain (

\bar{ε_{e f f}}

) and plotted it against the mean of the misorientation values within that grain (

\bar{ϕ}

) for each of the nine calculations for misorientation. In Fig. 5 we plot these grain mean correlations for ∼600 grains for each of the nine methods. Most correlations are positive, but some have a higher gradient than others and all of the data have significant scatter. In most of the correlations between

\bar{ε_{e f f}}

and

\bar{ϕ}

the larger grains show a higher

\bar{ϕ}

for a given

\bar{ε_{e f f}}

. However, in the correlation between strain and grain average 3 × 3 KAM in Fig. 3i) the largest grains have the smallest mean misorientations. Of all of the ϕ measures, the KAM is the most sensitive to step size and grain size because the kernel size must be defined with respect to these length scales. Grain distribution measurements that do not require a kernel, such as GROD, are much less sensitive to step size [8,28,30].
在图 3 中，我们展示了晶格取向的计算方式如何影响晶粒内晶格取向的幅度和分布。我们知道，地图平均取向与宏观应变呈正相关。为了测试这种关系是否在晶粒尺度上成立，我们取了每个晶粒内ɛ _eff 值的平均值（

\bar{ε_{e f f}}

），并将其与该晶粒内取向值的平均值（

\bar{ϕ}

）在九种取向计算方法中分别绘制出来。在图 5 中，我们绘制了九种方法中每种方法约 600 个晶粒的晶粒平均相关性。大多数相关性为正，但有些的梯度高于其他，所有数据都存在显著散布。在

\bar{ε_{e f f}}

和

\bar{ϕ}

之间的多数相关性中，较大的晶粒在给定的

\bar{ε_{e f f}}

下显示出更高的

\bar{ϕ}

。然而，在图 3i)中应变与晶粒平均 3 × 3 KAM 的相关性中，最大的晶粒具有最小的平均取向。在所有ϕ度量中，KAM 对步长和晶粒尺寸最为敏感，因为核尺寸必须相对于这些长度尺度来定义。无需核的晶粒分布测量方法（如 GROD）对步长变化非常不敏感[8,28,30]。

We can obtain values that reflect the sensitivity and reliability of each of the

\bar{ε_{e f f}}

\bar{ϕ}

correlations by performing a simple linear regression analysis. We define the sensitivity of

\bar{ϕ}

to changes in

\bar{ε_{e f f}}

by calculating the gradient of the

\bar{ε_{e f f}}

\bar{ϕ}

correlation (m) and we define the reliability of

\bar{ϕ}

as an indicator of

\bar{ε_{e f f}}

by calculating the correlation coefficient (R) and the standard error of the moving mean (SE). In this way, the

\bar{ε_{e f f}}

\bar{ϕ}

correlations with a high m, high R and low SE are the most sensitive and reliable correlations. While a significance test can be performed for linear regression, the resulting probability p values are very sensitive to the sample size and, in this case, results in unrealistically low values. Instead, we use the concept that a value of R > 0.3 indicates a low but detectable correlation [46]. In Fig. 6 part a) we show these three correlation descriptors for each of the nine correlations and include a dotted green line to show the R = 0.3 limit for detectable correlation. Only the

\bar{ε_{e f f}}

\bar{ϕ}

correlations using the GROD KAM Min, GROD PQ Max and the 3 × 3 KAM result in non-detectable correlations, which might be expected due to the sensitivity of the KAM to the grain/step/kernel size and the randomness with which the pixel with the highest pattern quality might be selected. The other

\bar{ε_{e f f}}

\bar{ϕ}

correlations, and especially where

\bar{ϕ}

is taken from the GROD Mean, KAM and CMA, give sensitive and reliable correlations. While the

\bar{ϕ}

taken from the GROD Mean Max gives the most sensitive correlation, it has a large standard error and is not as reliable. To answer the question of which is the best grain mean correlation in Fig. 5, we have combined these correlation descriptors as Rm and Rm/SE, the latter giving more weight to the standard error, which is also included in the calculation of R. In Fig. 6 part b) show how Rm and Rm/SE vary for each of the nine

\bar{ε_{e f f}}

\bar{ϕ}

correlations. The greater the value of this combined statistic, the better

\bar{ϕ}

acts as a predictor of

\bar{ε_{e f f}}

. If considering Rm/SE, the best predictors of grain mean strain are the GROD Mean and the CMA; note that the GROD Mean Min and GROD Centroid have similar values to the GROD Mean due to their similarity in spatial distribution in the grain. However, if considering Rm then the best measure of

\bar{ϕ}

is the GROD Mean Max, largely due to its steep correlation gradient. The GROD Mean Max is calculated using a reference orientation that is the most severely rotated away from the mean orientation, which therefore highlights misorientation heterogeneity; the GROD Mean Max may indeed be the most suitable indicator of strain field because the ɛ_eff distribution is highly heterogeneous.
我们可以通过执行简单的线性回归分析来获得反映每个

\bar{ε_{e f f}}

\bar{ϕ}

相关性的敏感性和可靠性数值。我们通过计算

\bar{ε_{e f f}}

\bar{ϕ}

3.2. Correlations between plastic strain and lattice misorientation at the sub-grain scale
3.2. 亚晶尺度上的塑性应变与晶格倾斜的相关性

To approach the problem of what measures of misorientation can tell us about damage mechanisms, we must study the spatial relationship between strain and misorientation within the grain. We do this here through a pixel-by-pixel comparison within each grain between ɛ_eff and ϕ. In correlating the mean grain values

\bar{ε_{e f f}}

and

\bar{ϕ}

, the GROD Mean Max and GROD Mean give the most sensitive and reliable correlations and so we will use these measures for the sub-grain correlations. We will also use the KAM because it is itself spatially resolved due to its kernel averaging and it is so frequently used in the literature [8]. Within grains the plastic strain (ɛ_eff) is highly localised in slip bands but the misorientation varies smoothly. We have performed a smoothing and coarsening function to the ɛ_eff data, resulting in a continuum of strain (

ε_{e f f}^{C}

) in an attempt to improve the pixel-to-pixel comparisons. This was achieved by applying a 2D Gaussian filter to the ɛ_eff over a square kernel of 11 × 11 pixels. We have also calculated the gradient of this coarsened strain (

\nabla ε_{e f f}^{C}

) as the root of the sum of the squares of the gradient in the horizontal and vertical directions.
为了探究晶粒内部取向差异的度量如何反映损伤机制，我们需要研究应变与取向差异在空间上的关系。我们通过在每个晶粒内部对ɛ和ϕ进行逐像素比较来实现这一点。在关联平均晶粒值

\bar{ε_{e f f}}

和

\bar{ϕ}

时，GROD Mean Max 和 GROD Mean 给出了最敏感和可靠的关联，因此我们将使用这些度量进行亚晶粒关联。我们还将使用 KAM，因为它本身由于核平均而具有空间分辨率，并且在文献中频繁使用[8]。在晶粒内部，塑性应变（ɛ _eff ）高度集中在滑移带中，但取向差异则平滑变化。我们对ɛ _eff 数据进行了平滑和粗化处理，以产生连续应变（

ε_{e f f}^{C}

），试图改进逐像素比较。这是通过将 2D 高斯滤波器应用于 11×11 像素的方核上的ɛ _eff 实现的。我们还计算了这种粗化应变的梯度（

\nabla ε_{e f f}^{C}

），作为水平和垂直方向梯度平方和的平方根。

Although some authors have reported no correlation between the plastic strain and the lattice misorientation at the sub-grain scale [8], we see strong spatial correlations in some cases, depending on how the data is treated and the slip characteristics in a particular grain. Some grains deform relatively homogeneously and an example of such a grain is shown in Fig. 7; in this grain the slip bands are evenly spaced and of similar intensity but they fade or change in character as they approach the grain boundary. The strain component inserts in Fig. 7 show that the strain is reasonably homogeneous when smoothed and the gradient of that strain field is highest at the grain boundaries. For this analysis we use the Spearman's rank (ρ) as a measure of correlation strength, which, unlike Pearson coefficient, is non-parametric and does not assume normal distribution nor linearity, only monotonic differences. The pixel-to-pixel correlations for this grain show that the spatial correlation is strongest and positive between the strain gradient and the KAM (ρ = 0.64), closely followed by the GROD Mean (ρ = 0.62). Therefore, it is the interaction of the slip with the microstructural grain boundaries that govern the relationship between plastic strain and lattice misorientation in this grain. In some grains the strongest correlation is a negative correlation, especially between the coarsened strain and the GROD Mean or KAM. This negative correlation occurs largely because the misorientation is positively correlated to the strain gradient. Some grains deform more heterogeneously and an example of such a grain is shown in Fig. 8; in this grain the slip bands bifurcate in the grain centroid in the vicinity of a penetrating twin. The bifurcation is associated with greater values of strain in the grain centroid, which is highlighted in the coarsened strain map. The pixel-to-pixel correlations for this grain show that the spatial correlation is strongest and positive between the coarsened strain and the GROD Mean Max (ρ = 0.75). For this grain, the measure of misorientation that creates the greatest amount of grain-interior misorientation heterogeneity is the best indicator of local plastic strain. In other grains there can be large amount of strain in one region that fades towards a grain boundary, associated with a high GROD Mean at the boundary, and no strain in another region of the grain; in this case the GROD Mean Max will be strongly but inversely correlated to the strain.
尽管一些作者报告在亚晶尺度上塑性应变与晶格畸变之间没有相关性[8]，但在某些情况下，我们观察到强烈的空间相关性，这取决于数据的处理方式以及特定晶粒的滑移特性。有些晶粒变形相对均匀，图 7 展示了一个这样的晶粒；在该晶粒中，滑移带均匀分布且强度相似，但在接近晶界时它们会逐渐减弱或改变特性。图 7 中的应变分量显示，当进行平滑处理后，应变是相当均匀的，该应变场的梯度在晶界处最高。对于这项分析，我们使用 Spearman 等级相关系数(ρ)作为相关性强度的度量，与 Pearson 系数不同，它是一种非参数方法，不假设正态分布或线性关系，仅假设单调差异。该晶粒的像素间相关性表明，应变梯度与 KAM 之间的空间相关性最强且为正相关（ρ = 0.64），其次为 GROD 均值（ρ = 0.62）。因此，滑移与微观结构晶界之间的相互作用决定了该晶粒中塑性应变与晶格取向之间的关系。在某些晶粒中，最强的相关性为负相关，尤其是在粗化应变与 GROD 均值或 KAM 之间。这种负相关主要因为晶格取向与应变梯度呈正相关。一些晶粒变形更加异质，图 8 展示了一个这样的晶粒示例；在该晶粒中，滑移带在晶粒中心附近穿透孪晶处分叉。这种分叉与晶粒中心应变值更大有关，这在粗化应变图中得到突出显示。该晶粒的像素间相关性表明，粗化应变与 GROD 均值最大值之间的空间相关性最强且为正（ρ = 0.75）。对于该晶粒，造成晶粒内部取向异质性的最大取向偏差指标是局部塑性应变的最佳指示器。在其他晶粒中，一个区域可能存在大量应变，随着靠近晶界而逐渐减弱，同时晶界处 GROD 均值较高，而晶粒的另一区域则无应变；在这种情况下，GROD 均值最大值将强烈但呈负相关。

We have performed the sub-grain pixel-to-pixel analysis for 613 grains and show the distribution of the Spearman's rank for each of the nine correlations (ɛ_eff,

ε_{e f f}^{C}

and

\nabla ε_{e f f}^{C}

correlated each to GROD Mean, KAM and GROD Mean Max) in Fig. 9. Correlations are stronger as the Spearman's rank approaches ±1 and the positions of ρ = ±0.3 are shown by vertical lines to indicate where a correlation becomes detectable. It is immediately clear that the correlations between the ɛ_eff and misorientation are the poorest with a high fraction of grains showing ρ → 0. The widest distribution and therefore the best sub-grain spatial correlation is between

ε_{e f f}^{C}

and GROD Mean Max but there are also some strong positive correlations between

\nabla ε_{e f f}^{C}

and GROD Mean and the KAM. Each of the 613 grains has a maximum positive correlation ρ → 1 and a minimum inverse correlation ρ → −1. The distribution of these maximum positive and inverse correlations is shown in Fig. 10a), together with guide lines for the ρ = ±0.3 condition for a detectable spatial correlation. From this histogram we find that the positive correlation is stronger than the inverse correlation in 60.2% of grains and that a total of 87.9% of grains have a detectable correlation (ρ > 0.3, ρ < −0.3) between one of the measure pairs: 52.2% positive correlations and 35.7% inverse correlations. We show which correlation pair gave the strongest positive and inverse correlation in Fig. 10b) as a bar chart for each of the nine pairs. From this bar chart it is clear that the strongest sub-grain pixel-to-pixel correlations are most common between the coarsened strain and the GROD Mean Max. While Fig. 8 is an example of this positive correlation, the bar chart in Fig. 10b) shows that the correlation can be both positive and inverse with almost equal probability. Therefore, while GROD Mean Max might give a good indication of the magnitude of strain and its heterogeneity within a grain, it cannot be relied upon for the spatial location of strain; for a positive spatial correlation the one should use the GROD Mean or the KAM to infer the spatial location of the gradient of plastic deformation.
我们对 613 个晶粒进行了亚晶粒像素到像素的分析，并展示了九种相关性（ɛ _eff 、

ε_{e f f}^{C}

和

\nabla ε_{e f f}^{C}

分别与 GROD Mean、KAM 和 GROD Mean Max 相关）的斯皮尔曼等级分布，如图 9 所示。随着斯皮尔曼等级接近±1，相关性增强，垂直线标出了ρ = ±0.3 的位置，表示相关性变得可检测。很明显，ɛ _eff 与取向差的相关性最差，有大量晶粒显示ρ → 0。分布最宽，因此亚晶粒空间相关性最好的是

ε_{e f f}^{C}

与 GROD Mean Max，但

\nabla ε_{e f f}^{C}

与 GROD Mean 和 KAM 之间也存在一些强正相关。每个 613 个晶粒的最大正相关ρ → 1 和最小负相关ρ → −1 的分布如图 10a)所示，同时显示了ρ = ±0.3 的条件下的空间相关性可检测性引导线。从这个直方图我们可以发现，在 60.2%的晶粒中，正相关强于负相关，并且总共有 87.9%的晶粒在测量对之一（ρ > 0.3, ρ < −0.3）之间存在可检测的相关性：52.2%为正相关，35.7%为负相关。我们在图 10b)中展示了哪一对相关性给出了最强的正相关和负相关，以条形图形式呈现九对中的每一对。从这个条形图可以看出，最常见的是粗化应变与 GROD Mean Max 之间的亚晶粒像素间相关性最强。虽然图 8 是这种正相关的例子，但图 10b)中的条形图表明，这种相关性可能是正相关也可能是负相关，概率几乎相等。因此，虽然 GROD Mean Max 可能可以很好地指示晶粒内应变的幅度及其异质性，但它不能被依赖来指示应变的空间位置；对于正的空间相关性，应该使用 GROD Mean 或 KAM 来推断塑性变形梯度的空间位置。

We have shown that while the GROD Mean and KAM measures of misorientation are the most reliable indicators of local strain gradients, the GROD Mean Max gives the strongest local positive and inverse correlations due to its inherent measure of heterogeneity. The suitability of one particular EBSD-based measure of misorientation in representing the distribution plastic deformation depends upon the number and relative strength of slip modes active in that grain and their interaction with microstructural features such as grain boundaries. Relative slip activity is affected by crystal orientations and constraint and will be different at the surface and in the bulk.
我们证明了，虽然 GROD 均值和 KAM 取向偏差度量是局部应变梯度的最可靠指标，但由于其固有的异质性度量，GROD 均值最大值给出了最强的局部正相关和负相关。基于 EBSD 的某一特定取向偏差度量在表征塑性变形分布的适用性取决于该晶粒中活跃的滑移模式的数量及其相对强度，以及它们与晶界等微观结构特征的相互作用。相对滑移活性受晶体取向和约束的影响，在表面和体相中会不同。

3.3. The influence of grain orientation on grain deformation
3.3. 晶粒取向对晶粒变形的影响

Since the relative slip activity is affected by orientation, it is plausible that grain orientation correlates with measures of local strain and misorientation. To assess the influence of grain orientation on the magnitude of plastic strain, we have plotted the grain mean strain on a colour scale within an IPF (loading direction) based on the scatter orientations in Fig. 1 part a). The texture in this material is random and we have smoothed the scatter plots by local linear interpolations to produce the grain mean

\bar{ε_{e f f}}

IPF in Fig. 11. The IPF plot suggests that the grain mean

\bar{ε_{e f f}}

is independent of the orientation of the grain. Since grain mean

\bar{ϕ}

is positively correlated to the

\bar{ε_{e f f}}

, Fig. 5, we might also expect the misorientation IPFs to be random. To confirm this we have plotted the

\bar{ϕ}

IPFs (loading direction) in Fig. 12 for each of the nine measures of

\bar{ϕ}

, and, indeed, there is no obvious correlation between the orientation of a grain and the magnitude of

\bar{ϕ}

. This is an important result because orientation-based predictors of deformation based on the global boundary condition, such as the Taylor factor or the Schmid factor, are often used for the explanation of observed localized deformation [48], [49], [50], [51], [52] but we find that on the surface of this alloy and after a strain of 0.02 there is no relationship between the orientation of a grain and its magnitude of deformation. To demonstrate this explicitly, we have calculated the Taylor factor and Schmid factor for each grain in its original, non-deformed orientation and plotted this on the IPFs (loading direction) in Fig. 13 parts a) and b), respectively.
由于相对滑移活性受取向影响，因此晶粒取向可能与局部应变和取向差度量相关。为评估晶粒取向对塑性应变幅值的影响，我们基于图 1a)中的散乱取向，在 IPF（加载方向）内用彩色标尺绘制了晶粒平均应变。该材料的织构是随机的，我们通过局部线性插值平滑了散点图，生成了图 11 中的晶粒平均 IPF。IPF 图表明晶粒平均

\bar{ε_{e f f}}

与晶粒取向无关。由于晶粒平均

\bar{ε_{e f f}}

与

\bar{ϕ}

呈正相关（图 5），我们也可以预期取向差 IPF 是随机的。为确认这一点，我们为九种

\bar{ϕ}

度量中的每一种绘制了

\bar{ϕ}

IPF（加载方向）（图 12），确实，晶粒取向与

\bar{ϕ}

幅值之间没有明显的相关性。这是一个重要结果，因为基于全局边界条件的变形取向预测器，如泰勒因子或施密特因子，通常用于解释观察到的局部变形[48]、[49]、[50]、[51]、[52]，但我们发现，在该合金表面和 0.02 的应变后，晶粒的取向与其变形程度之间没有关系。为了明确地证明这一点，我们计算了每个晶粒在原始、未变形取向下的泰勒因子和施密特因子，并分别绘制在 IPFs（加载方向）的图 13 a)和 b)中。

The Taylor factor of a grain is a prediction of the amount of work required to deform that grain; all grains are assumed to undergo the same strain and so a higher Taylor factor indicates that a greater amount of work should be necessary to deform that grain [53,54]. Therefore, grains with a higher Taylor factor are often considered harder than grains with a lower Taylor factor, and would be expected to have lower values of strain. Similarly, a higher Schmid factor corresponds to a higher resolved shear stress for slip and therefore a grain with a higher maximum Schmid factor is often considered softer than a grain with a lower maximum Schmid factor. As Fig. 13 shows, however, the distribution of Taylor factor and Schmid factor in Fig. 13 shows no resemblance to the strain distribution in Fig. 11 nor any of the nine misorientation distributions in Fig. 12. To solidify this point, in Fig. 14 we plot the grain mean HRDIC plastic effective strain (data point positions) against the Taylor factor in part a) and the Schmid factor in part b) as calculated by the grain's original, non-deformed orientation, where the colour and size of the data points relate to the GROD Mean misorientation. By plotting these values for 590 grains we are able to show definitively that, at a macroscopic axial strain of ε_xx ∼0.02, there is no relationship between the Taylor factor as calculated using macroscopic or local boundary conditions (see Supplementary material for the latter) and the amount of plastic strain or lattice misorientation. There is also only a weak relationship between Schmid factor and effective strain: grains with low Schmid factor tend to deform less and while grains with a higher strain tend to have a higher Schmid factor, grains with a high Schmid factor do not necessarily undergo large strain.
晶粒的泰勒因子是预测使该晶粒变形所需工作量的一种指标；所有晶粒都被假定为承受相同的应变，因此较高的泰勒因子表明变形该晶粒需要更多的工作量[53,54]。因此，具有较高泰勒因子的晶粒通常被认为比具有较低泰勒因子的晶粒更硬，并且预计其应变值较低。类似地，较高的施密特因子对应于较高的滑移 Resolved 剪切应力，因此具有较高最大施密特因子的晶粒通常被认为比具有较低最大施密特因子的晶粒更软。然而，如图 13 所示，图 13 中泰勒因子和施密特因子的分布与图 11 中的应变分布以及图 12 中的九种取向分布均无相似之处。为了巩固这一点，在图 14 中，我们将晶粒平均 HRDIC 塑性有效应变（数据点位置）与晶粒原始、未变形取向计算的泰勒因子（a 部分）和施密特因子（b 部分）进行绘制，其中数据点的颜色和大小与 GROD 平均取向相关。通过绘制 590 个晶粒的这些值，我们能够明确地表明，在宏观轴向应变ε _xx ∼0.02 时，使用宏观或局部边界条件（后者参见补充材料）计算的泰勒因子与塑性应变或晶格畸变量之间没有关系。此外，Schmid 因子与有效应变之间也仅存在微弱的关系：Schmid 因子较低的晶粒倾向于变形较小，而应变较高的晶粒往往具有较高的 Schmid 因子，但 Schmid 因子较高的晶粒并不一定会发生较大应变。

4. Discussion 4. 讨论

The global average misorientation has been shown to correlate linearly with macroscopic plastic strain [6,7,9,55] up to strains of ε_xx < 0.125 [28], after which there is deviation from linearity. However, at a given strain level, there is wide variation in the minimum and maximum average grain misorientations [7]. This is both because the strain is different in different grains but also because strain and misorientation at the grain scale only correlate weakly [18]. Our results are consistent with these findings and add that the scatter in the values of the misorientation depends largely on the misorientation measure used. The scatter in the grain mean misorientations is on the scale of 1°, which is in agreement with scatter reported for single strain steps of a similar magnitude in austenitic stainless steel [6,7] and in Ni alloy A600 [6]. However, the sub-grain misorientation can exceed 5°, depending on the measure of misorientation used.
全球平均晶格倾斜已被证明与宏观塑性应变线性相关[6,7,9,55]，直到应变为ε _xx < 0.125[28]之后，这种线性关系出现偏差。然而，在给定的应变水平下，最小和最大平均晶粒倾斜存在很大差异[7]。这是因为不同晶粒的应变不同，而且晶粒尺度的应变和倾斜度之间仅存在微弱相关性[18]。我们的结果与这些发现一致，并补充指出，倾斜度值的离散程度很大程度上取决于所使用的倾斜度测量方法。晶粒平均倾斜度的离散程度在 1°的量级，这与奥氏体不锈钢中类似幅度的单步应变所报告的离散程度[6,7]以及 Ni 合金 A600 中的离散程度[6]一致。然而，亚晶粒倾斜度可能超过 5°，具体取决于所使用的倾斜度测量方法。

The HRDIC measurements reveal that the magnitude and localization of the lattice misorientation within a grain depends on not only the magnitude of the plastic strain, but also the number and relative strength of the slip modes active in that grain and the interaction of those slip modes with microstructural features such as grain boundaries. We therefore argue that the local deformation characteristics of plastic strain and lattice rotation emerge from the complex interaction between grains during deformation and cannot be determined by simple orientation-based predictors of plasticity relative to the global deformation tensor, such as the Taylor factor or Schmid factor. This non-crystallographic strain localization is especially evident when we consider the strain bands at the mesoscale that span many grains in Fig. 2 at ± 45° to the loading direction, which must arise from effects that relate to deformation compatibility and the interactions between many grains. These bands of high strain, which are also predicted by crystal plasticity finite element models [56], cross many grains with different orientations and are consistent with a lack of correlation between local strain and crystal orientation at the grain level. Further, the suitability of the lattice misorientation to describe the plastic strain depends upon the way in which the misorientation is calculated.
HRDIC 测量表明，晶粒内晶格转位的幅度和位置不仅取决于塑性应变的幅度，还取决于该晶粒中活跃的滑移模式的数量和相对强度，以及这些滑移模式与晶界等微观结构特征的相互作用。因此，我们认为塑性应变和晶格旋转的局部变形特征源于变形过程中晶粒之间的复杂相互作用，而无法通过基于取向的全局变形张量的塑性简单预测因子（如泰勒因子或施密特因子）来确定。这种非晶体学应变局部化在考虑图 2 中±45°方向于加载方向跨越多个晶粒的中尺度应变带时尤为明显，这些应变带必然源于与变形相容性以及许多晶粒之间相互作用相关的效应。这些高应变带，正如晶体塑性有限元模型[56]所预测的，穿过许多不同取向的晶粒，并且与晶粒尺度上局部应变和晶体取向之间缺乏相关性一致。此外，晶格取向适合描述塑性应变，取决于取向的计算方式。

KAM is an indicator of local geometrically necessary dislocation (GND) density and is often converted to a GND density using a Nye tensor model [57]. Strictly speaking, the GND density depends on gradients of plastic spin on active slip systems. In polycrystalline deformation the gradient in plastic strain will usually imply gradients in plastic spin, and hence gradients in plastic strain are expected to correlate with KAM. Our measurements show this sub-grain correlation (e.g. between

\nabla ε_{e f f}^{C}

and the KAM) but it is not always the strongest correlation, depending on the strain heterogeneity within the grain and the interaction of slip with microstructural features.
KAM 是局部几何必需位错（GND）密度的指标，通常使用 Nye 张量模型[57]将其转换为 GND 密度。严格来说，GND 密度取决于活动滑移系统上塑性自旋的梯度。在多晶变形中，塑性应变的梯度通常意味着塑性自旋的梯度，因此预计塑性应变的梯度将与 KAM 相关。我们的测量显示了这种亚晶粒相关性（例如

\nabla ε_{e f f}^{C}

与 KAM 之间），但它并不总是最强的相关性，这取决于晶粒内的应变异质性和滑移与微观结构特征的相互作用。

4.1. Orientation-based predictors of deformation
4.1. 基于取向的变形预测

The discrete slip bands observed within grains are thought to arise in this alloy due to the low stacking fault energy of Ni-based superalloys and the resulting dissociation of dislocations into partials, which cannot cross-slip easily [58]. Moreover, the local shearing of the γ’ strengthening precipitates results in glide plane softening [59] and so locally the shear bands contain high strain (ɛ_eff ∼0.4). The high spatial resolution employed in the present work allows us to measure discrete plastic events and so we are able to validate or contest traditional assumptions employed in continuum theories of plasticity.
在晶粒内部观察到的离散滑移带被认为是由镍基高温合金的低层错能引起的，这导致位错分解成部分位错，而部分位错难以跨滑移[58]。此外，γ'强化析出物的局部剪切导致滑移面软化[59]，因此局部剪切带包含高应变（ɛ _eff ∼0.4）。本研究采用的高空间分辨率使我们能够测量离散的塑性事件，因此我们能够验证或挑战塑性连续理论中使用的传统假设。

Taylor theory [53], after von Mises [60], assumes that all grains undergo the same shape change by activating a minimum of five slip systems. While it has been noted that different parts of a grain may exhibit single slip Sachs-like behaviour, it is generally accepted that a grain will meet this five slip system criteria overall [61]. However, we observe many grains in which only one slip plane is active, incurring likely a single slip system but a maximum of three. The limited number of slip systems observed is in part due to the plane stress state experienced at the surface as is predicted by full field crystal plasticity models [62] but also because, unlike the core assumption of the full constraints of Taylor theory, the grains deform differently from one another and from the macroscopic deformation. This is not new observation, and other crystal plasticity models, be it self-consistent or full-field, account for this. Nevertheless, orientation-based predictors of deformation, such as the Taylor factor and Schmid factor, have been used to, for example, explain fatigue pre-crack initiation at Σ3 annealing twin boundaries [48], to characterize the plastic zone ahead of crack tips [49] and as related to local crack propagation rates [50] and to the degree of local lattice misorientation [51] or local plastic strain directly [52]. In all these, it is assumed that, at small strains, the local strain correlates positively with increasing Schmid factor or decreasing Taylor factor. Our data shows that this simple predictor based on global boundary conditions is not suitable for this system at low levels of deformation. While it is has been shown that local conditions for the Schmid factor can better predict predicting strain accumulation at this length scale [63], our results show that even local constraint would be unable to account for the magnitude of plastic strain due to the large mesoscale banding that we observe, which is not crystallographic. To explore this point further we have performed the Taylor factor calculation using the local constraint of the mean grain strain and have still observed a poor correlation to the magnitude of plastic strain (see Supplementary material).
泰勒理论[53]在冯·米塞斯[60]之后假设所有晶粒通过激活至少五个滑移系统经历相同的形状变化。虽然已经注意到晶粒的不同部分可能表现出类似 Sachs 的单滑移行为，但通常认为晶粒总体上会满足这一五个滑移系统标准[61]。然而，我们观察到许多晶粒中只有一个滑移面是活跃的，这可能导致一个单滑移系统，最多三个。观察到的滑移系统数量有限，部分原因是表面承受的平面应力状态，正如全场晶体塑性模型[62]所预测的那样，但也因为与泰勒理论的核心假设——所有晶粒都受到完全约束——不同，晶粒彼此之间以及与宏观变形的变形方式各不相同。这并非新的观察结果，其他晶体塑性模型，无论是自洽的还是全场的，都考虑到了这一点。然而，基于取向的变形预测因子，例如泰勒因子和施密特因子，已被用于解释例如Σ3 退火孪晶界处的疲劳预制裂纹萌生[48]，表征裂纹尖端的塑性区[49]，以及与局部裂纹扩展速率[50]和局部晶格取向度[51]或局部塑性应变直接相关[52]。在所有这些情况下，都假设在小应变下，局部应变与施密特因子增加或泰勒因子减少呈正相关。我们的数据显示，这种基于全局边界条件的简单预测方法不适用于该系统在低变形水平的情况。虽然已有研究表明，施密特因子的局部条件可以更好地预测该尺度上的应变累积[63]，但我们的结果表明，即使局部约束也无法解释由于我们观察到的较大中尺度带状结构（非晶体学特征）导致的塑性应变幅度。为了进一步探讨这一点，我们使用平均晶粒应变局部约束进行了泰勒因子计算，仍然观察到与塑性应变幅值的相关性较差（参见补充材料）。

Our observation that the Taylor factor does not correlate to the amount of local plastic strain is in agreement with HRDIC and EBSD measurements in dual phase ferritic-martensitic steels [64]. However, authors working with FCC commercially pure Al [65] and a BCC ferritic-martensitic steel [66] have grouped grains into orientation families to show that grain families with a high Taylor factor tend to experience greater lattice misorientation after large monotonic plastic strains up to ∼ 0.3. The authors assumed that all grains experience the same strain to argue that higher Taylor factor grains require more slip to accommodate the same strain and so they incur greater lattice misorientation. In other studies, grain family Taylor factor has been correlated to the amount of grain breakup but not the misorientation [67]. We therefore conclude that whilst Taylor theory may be useful in determining overall hardening behaviour of a macroscopic specimen, it is more appropriate for the high strain regime and it does not scale down to the grain-to-grain level, especially at relatively low macroscopic plastic strains.
我们的观察表明泰勒因子与局部塑性应变量不相关，这与双相铁素体-马氏体钢中的 HRDIC 和 EBSD 测量结果一致[64]。然而，在面心立方纯铝[65]和 BCC 铁素体-马氏体钢[66]的研究中，作者将晶粒分组为取向族，表明具有高泰勒因子的晶粒族在经历高达~0.3 的大单调塑性应变后，倾向于产生更大的晶格畸变。作者假设所有晶粒都经历相同的应变，以此论证具有更高泰勒因子的晶粒需要更多的滑移来适应相同的应变，因此它们会产生更大的晶格畸变。在其他研究中，晶粒族泰勒因子与晶粒破碎程度相关，但与畸变无关[67]。因此，我们得出结论，尽管泰勒理论可能有助于确定宏观试样的整体硬化行为，但它更适用于高应变范围，并且不能缩小到晶粒与晶粒的级别，尤其是在相对较低宏观塑性应变的情况下。

The maximum Schmid factor is well-used orientation-based predictor of deformation and, likewise, we see poor correlation between this and the local plastic strain or the lattice misorientation. There is a strong correlation between the slip system activated and the maximum Schmid factor, in agreement with previous work [17], but no correlation with the magnitude of strain. This is in agreement with studies that combine EBSD with lower resolution DIC than that shown here [64,[68], [69], [70]] and correlative EBSD and atomic force microscopy for the measurement of slip step height [71]. The use of Schmid factor may be appropriate for more anisotropic crystal systems and has been shown as useful in hexagonal Mg alloys to describe local crack growth rates [50] and in hexagonal Ti alloys [72] to describe the relative strain between grains that deform on different slip systems. We therefore conclude that the concept of “hard” and “soft grains” based on their orientation [73] is not relevant to FCC γ/γ’ Ni-based superalloys at small global plastic deformation. The effect of grain size has been suggested as more significant than that of orientation [64] and as important for fatigue crack initiation sites [74]. In Fig. 5 the colour and size of the grain mean strain-misorientation correlation data points relate to the grain size. We observe a positive correlation between the lattice misorientation and the grain size, but this relationship is less clear for the plastic strain. Interestingly, the misorientation as measured by the grain mean KAM has an inverse relationship with grain size. This is because, in this material, many grains contain relatively large regions of single slip bands, resulting in almost no local lattice misorientation within the grain interior. Larger grains, therefore, have larger areas that correspond to these regions, which results in lower mean KAM values. Importantly, because the KAM is a very local measure of deformation it is step-size dependent and as the number of steps in a grain is proportional to the grain size, the KAM is also proportional to the grain size. The GROD variants and the CMA measure do not suffer from this issue [28,30].
最大 Schmid 因子是变形的常用基于取向的预测因子，同样地，我们看到它与局部塑性应变或晶格畸变之间没有很好的相关性。激活的滑移系与最大 Schmid 因子之间存在强相关性，这与先前的工作[17]一致，但没有与应变幅值的相关性。这与结合了 EBSD 与比这里显示的更低分辨率的 DIC 的研究[64,[68], [69], [70]]以及用于测量滑移步高的相关 EBSD 和原子力显微镜的研究[71]一致。Schmid 因子的使用可能适用于更各向异性的晶体系统，并且已被证明在六方 Mg 合金中描述局部裂纹扩展速率[50]以及在六方 Ti 合金[72]中描述在不同滑移系上变形的晶粒之间的相对应变是有用的。因此，我们得出结论，基于取向的“硬”和“软”晶粒的概念[73]在 FCC γ/γ’镍基高温合金的小范围整体塑性变形下并不相关。晶粒尺寸的影响被认为比取向的影响更为显著[64]，并且对疲劳裂纹萌生位点同样重要[74]。在图 5 中，晶粒平均应变-晶格位向相关性数据点的颜色和大小与晶粒尺寸相关。我们观察到晶格位向与晶粒尺寸之间存在正相关关系，但这种关系在塑性应变上不太明显。有趣的是，通过晶粒平均 KAM 测量的晶格位向与晶粒尺寸呈负相关关系。这是因为在该材料中，许多晶粒包含相对较大的单一滑移带区域，导致晶粒内部几乎没有局部晶格位向。因此，较大的晶粒具有更大的对应这些区域的面积，这导致平均 KAM 值较低。重要的是，由于 KAM 是对变形的非常局部的测量，因此它依赖于步长，并且由于晶粒中的步数与晶粒尺寸成正比，KAM 也与晶粒尺寸成正比。GROD 变体和 CMA 测量不受此问题的影响[28,30]。

4.2. Sub-grain deformation localisation
4.2. 亚晶粒变形局部化

Through specific examples of grain plasticity and lattice misorientation in Fig. 2, Fig. 7 and Fig. 8 we have shown that the lattice misorientation within a grain does not only depend on the magnitude of the strain but also on the slip characteristics specific to that grain. This implies that a grain may experience low but heterogeneous plastic strain and that this would result in a large lattice misorientation. Likewise, there may be significant plastic strain but on only one slip system, resulting in grain lattice rotation but low lattice misorientation within the grain. There can be significant misorientation where there is very little plastic strain, especially in the vicinity of grain boundaries. This is because the slip bands fade before the boundary, resulting in a sharp change in the plastic strain gradient that requires an associated lattice reorientation in order to maintain deformation compatibility with both the grain centroid and the neighbouring grain. In terms of plastic strain, previous lower-resolution DIC studies on the proximity of plastic strain to grain boundaries have demonstrated a wider range in strain values closer to grain boundaries [68,75], but that the mean plastic strain is not significantly different at the grain boundary relative to its centroid. Analysis that details the change in strain along individual slip traces towards the grain boundary is necessary to explore this further [76].
通过图 2、图 7 和图 8 中晶粒塑性变形和晶格转位的具体实例，我们表明晶粒内的晶格转位不仅取决于应变的大小，还取决于该晶粒特有的滑移特性。这意味着晶粒可能经历低但异质的塑性应变，这将导致较大的晶格转位。同样，可能存在显著的塑性应变，但仅在单个滑移系上，导致晶粒晶格旋转，但晶粒内晶格转位较低。在晶界附近，即使塑性应变很小，也可能存在显著的晶格转位。这是因为滑移带在晶界前消失，导致塑性应变梯度发生急剧变化，需要相应的晶格重新取向，以保持与晶粒中心和相邻晶粒的变形兼容性。在塑性应变方面，先前关于塑性应变与晶界邻近关系的低分辨率 DIC 研究已经表明，靠近晶界的应变值范围更广[68,75]，但相对于其质心，晶界处的平均塑性应变没有显著差异。为了进一步探索这一点，需要分析沿着单个滑移迹线朝向晶界时应变的变化[76]。

The correlation that we observe between the strain gradient and the misorientation, especially at grain boundaries, is supported by reports of misorientation localisation in the literature [69,75,[77], [78], [79], [80]]. This relationship is in agreement with Ashby's explanation of GND pile up at grain boundaries that results in local lattice misorientation [13] and back-stresses that contribute to local slip resistance [81]. However, there are ways in which lattice misorientation can occur at grain boundaries other than that according to Ashby, and these deformation modes have been well described in HRDIC-EBSD sub-grain comparisons in an austenitic stainless steel [17]. For instance, slip band impingement on a grain boundary can cause lattice misorientation in the neighbouring grain close to the boundary [14,15], which is especially prevalent if slip transmission is not favourable [14]. Further, special types of boundaries, such as the Σ3 annealing twin boundaries, and their interaction with high angle grain boundaries are known to be important in strain localisation that develop towards fatigue crack initiation sites [74]. Whilst many of the grains studied here exhibit a strong spatial correlation between the strain gradient and the KAM or GROD Mean, most do not. Therefore, it is likely that the phenomena of slip band fading, slip band impingement on grain boundaries and grain break up are all active throughout this material. Grain neighbourhoods, how they deform and how they interact with longer-range localisations like shear bands all determine the strain localisations that are often used to explain localised failure.
我们观察到的应变梯度与晶格转位的关联，尤其是在晶界处，得到了文献中关于晶格转位局部化的报告的支持[69,75,[77], [78], [79], [80]]。这种关系与 Ashby 对晶界处 GND 堆积导致局部晶格转位的解释[13]以及导致局部滑移抗力的背应力[81]相一致。然而，晶界处晶格转位的发生方式除了 Ashby 所述之外还有其他方式，这些变形模式已在奥氏体不锈钢的 HRDIC-EBSD 亚晶粒比较中得到充分描述[17]。例如，滑移带与晶界的碰撞可以在靠近晶界的邻近晶粒中引起晶格转位[14,15]，如果滑移传递不顺利，这种现象尤为普遍[14]。此外，特殊类型的晶界，如Σ3 退火孪晶界，及其与高角度晶界相互作用，在疲劳裂纹萌生部位发展过程中对应变局部化具有重要意义[74]。尽管这里研究的许多晶粒在应变梯度与 KAM 或 GROD 均值之间表现出强烈的空间相关性，但大多数并不如此。因此，滑移带消退、滑移带与晶界碰撞以及晶粒破碎等现象很可能在整个材料中都活跃着。晶粒邻域如何变形以及它们如何与剪切带等长程局部化相互作用，都决定了通常用于解释局部失效的应变局部化。

4.3. The suitability of different lattice misorientation calculations as a measure of local plasticity
4.3. 不同晶格取向计算方法作为局部塑性的适用性

Lattice misorientation is often used to calculate the density of geometrically necessary dislocations as a measure of residual plastic deformation [82] and also as a validation for full field models of crystal plasticity [63,70,[83], [84], [85], [86], [87]]. However, the different measures of lattice misorientation shown in Fig. 3 demonstrate the wide range of magnitude and spatial variation that is possible from the same EBSD data set. A deviation in the spatial variation of lattice misorientation has been shown for a number of GROD variants [29] to highlight that the GROD reference orientation as the original orientation results in a lattice misorientation that contains both the rigid body rotations and the rotations that arise from plastic strain, whereas the other GROD variants only describe the latter. In Fig. 5 and Fig. 6 part a) we show the significance of this difference; the GROD Original measure of misorientation is poorly correlated to the in-plane measure of deformation and this is due to the three dimensional nature of the rigid body and plastic rotations. In Fig. 6 part a) we show that the GROD Mean and the CMA give the strongest correlation to the magnitude of grain mean plastic strain and that the GROD Mean Max gives the most sensitive correlation and is more indicative of heterogeneity.
晶格倾斜常用于计算几何必要位错密度，作为残余塑性变形的度量[82]，同时也用于验证晶体塑性全场模型[63,70,[83], [84], [85], [86], [87]]。然而，图 3 中所示的不同晶格倾斜度量表明，从相同的 EBSD 数据集中可以得到广泛的幅度和空间变化范围。研究表明，对于多种 GROD 变体[29]，晶格倾斜的空间变化存在偏差，以强调 GROD 参考方向作为原始方向会导致晶格倾斜包含刚体旋转和塑性应变产生的旋转，而其他 GROD 变体仅描述后者。在图 5 和图 6a)中，我们展示了这一差异的重要性；GROD 原始度量与平面内的变形度量相关性较差，这是由于刚体和塑性旋转的三维性质所致。在图 6 a) 我们表明 GROD 均值和 CMA 与晶粒平均塑性应变的幅度具有最强的相关性，而 GROD 均值最大值给出了最敏感的相关性，并且更能指示异质性。

It has been shown that calculating the misorientation from the GROD Mean is much less sensitive to EBSD step size than using the mean KAM value [8,28,30]. The CMA as a measure of grain misorientation cannot be mapped the sub-grain scale, but it is even less sensitive to step size than the GROD Mean and more sensitive to changes in global plastic deformation [30]. This improved sensitivity is reflected in the statistical correlation analysis at the grain scale in Fig. 6. In Fig. 4 we show the distributions of plastic strain and misorientation for the whole field-of-view, ∼1 × 0.5 mm. Whilst the GROD Mean and the KAM distributions have the lowest Kullback–Leibler deviations and therefore best represent the distribution of the plastic strain, the similarity in distributions is predominantly in the low strain and low misorientation regime. The high strain regime, which is thought of as important for crack initiation and localised degradation mechanisms [74], is best represented by the GROD Mean Max. We must therefore conclude that strain localisation is difficult to quantify using only a single EBSD lattice misorientation measure and so any interpretation of these data sets for such a purpose should be done with caution. This is especially the case when local lattice misorientations or GND densities are used exclusively to explain local failure mechanisms. As our results show, slip bands produce almost no lattice misorientation within grains, and yet corrosion and cracking occur preferentially along slip bands in stainless steel [25,88].
研究表明，通过 GROD 均值计算晶格取向差比使用 KAM 均值更不敏感于 EBSD 步长[8,28,30]。作为晶粒取向差的度量，CMA 无法映射到亚晶粒尺度，但比 GROD 均值更不敏感于步长，且对整体塑性变形的变化更敏感[30]。这种改进的敏感性体现在图 6 中晶粒尺度的统计相关性分析中。在图 4 中，我们展示了整个视场（约 1×0.5 毫米）的塑性应变和取向差分布。虽然 GROD 均值和 KAM 分布具有最低的 Kullback-Leibler 偏差，因此最能代表塑性应变分布，但分布的相似性主要体现在低应变和低取向差状态。高应变状态被认为是裂纹萌生和局部退化机制的重要影响因素[74]，而 GROD Mean Max 最能代表这一状态。我们必须因此得出结论，仅使用单一的 EBSD 晶格取向差测量指标来量化应变局部化是困难的，因此，为这种目的对这些数据集进行任何解释都应谨慎进行。当仅使用局部晶格取向差或 GND 密度来解释局部失效机制时，这种情况尤其如此。正如我们的结果所示，滑移带在晶粒内部几乎不产生晶格取向差，然而在不锈钢中，腐蚀和裂纹却优先沿滑移带发生[25,88]。

4.4. The spatial distribution of slip and lattice misorientation
4.4. 滑移和晶格取向差的空間分布

One of the observations here is the difference in the spatial distribution of slip and lattice misorientation when sampled at the same spatial resolution in the pixel-to-pixel sub-grain analyses. In this alloy, slip is highly localised in slip bands and therefore discrete in nature. Conversely, the misorientation shown here varies smoothly at the same scale and slip bands can only be seen in EBSD studies with very high spatial and angular resolution [15]. For the majority of grains, slip bands carry most of the strain and yet produce almost no local misorientation. This difference in the nature of spatial distribution implies that the dislocations producing the strain are different from those created to accommodate the deformation gradients. In Ashby's terminology, dislocations stored along slip bands are statistically stored dislocations (SSDs), whereas the dislocations producing misorientations are geometrically stored dislocations (GNDs). If slip were homogeneous, then GNDs could be assumed to have originated as SSDs that collect at regions of strain heterogeneity. But since slip occurs in bands, most of the GNDs produced are not involved in slip, although their development is affected by the slip activity. This is very different from how most continuum mechanics based crystal plasticity frameworks model polycrystalline deformation. In these models slip is assumed to be homogeneous and lattice rotation continuity is not enforced [89], [90], [91]. Given that the slip band spacing is of the order of 1 µm, this difference between how deformation is modelled and how it occurs in practice might limit the ability of these modelling approaches to predict damage nucleation or the detailed descriptions of the deformed state needed to model annealing. Certainly, there is currently no way to model different degrees of slip localisation, which as recent experiments on a proton irradiated zirconium alloy show [92], can produce very different patterns of lattice misorientation.
这里的一个观察是，在相同空间分辨率下进行像素到像素的亚晶粒分析时，滑移和晶格转位的空间分布存在差异。在这种合金中，滑移高度局部化于滑移带，因此具有离散性。相反，这里显示的转位在同一尺度上变化平滑，只有在具有非常高的空间和角度分辨率的 EBSD 研究中才能看到滑移带[15]。对于大多数晶粒而言，滑移带承载了大部分应变，但几乎不产生局部转位。这种空间分布的性质差异表明，产生应变的位错与适应应变梯度的位错是不同的。在阿什比（Ashby）的术语中，沿滑移带储存的位错是统计储存位错（SSD），而引起转位的位错是几何储存位错（GND）。如果滑移是均匀的，那么可以假设 GND 起源于 SSD，这些 SSD 在应变异质性区域聚集。但由于滑移以带状形式发生，大多数产生的位错环并未参与滑移，尽管它们的形成受到滑移活动的影响。这与基于连续介质力学的晶体塑性框架模拟多晶变形的方式有很大不同。在这些模型中，滑移被假定为均匀的，且不强制执行晶格旋转连续性[89][90][91]。考虑到滑移带的间距在 1 微米量级，这种建模方式与实际变形过程的差异可能会限制这些建模方法预测损伤萌生或模拟退火所需的变形状态详细描述的能力。当然，目前还没有方法能够模拟不同程度的滑移局部化，正如最近对质子辐照锆合金的实验所示[92]，这可以产生非常不同的晶格旋转模式。

5. Conclusions 5. 结论

At the sub-grain scale, the lattice misorientation is the result of the number, strength and spatial distribution of slip modes within the grain and their interaction with microstructural features such as grain boundaries. Further, the way in which the misorientation is calculated has a drastic effect on the magnitude and spatial distribution of the lattice curvature. We find that the strongest spatial correlation between plastic strain and misorientation within a grain is obtained when the plastic strain is coarsened to smooth out the strain in the slip bands and the misorientation is calculated using the GROD Mean Max to reflect heterogeneity. However, in practise, this correlation can be positive or inverse with equal probability and so it is recommended to also use the GROD Mean or KAM to identify the region of highest strain gradient.
在亚晶尺度上，晶格位错是晶粒内滑移模式的数量、强度和空间分布及其与晶界等微观结构特征相互作用的结果。此外，位错计算的方式对晶格曲率的幅度和空间分布有巨大影响。我们发现，当将塑性应变粗化以平滑滑移带中的应变，并使用 GROD Mean Max 计算位错以反映异质性时，晶粒内塑性应变与位错之间的空间相关性最强。然而，在实际中，这种相关性以相同的概率可以是正向或反向，因此建议同时使用 GROD Mean 或 KAM 来识别最高应变梯度区域。

The Taylor factor and the Schmid factor cannot be used as a predictor of the plastic strain of a grain or its lattice misorientation. This is because of grain interaction effects at the microscale, such as slip band fading towards grain boundaries, slip impingement on boundaries from neighbouring grains and the interaction of slip bands with annealing twins, all of which do not depend on grain orientation alone, and further because of mesoscale deformation banding, which is not crystallographic.
泰勒因子和施密特因子不能作为预测晶粒塑性应变或其晶格转位的指标。这是因为微观尺度上的晶粒相互作用效应，例如滑移带向晶界处的衰减、邻近晶粒对晶界的滑移阻碍以及滑移带与退火孪晶的相互作用，这些效应并不完全依赖于晶粒取向，此外还因为中尺度变形带，这种变形带并非晶体学特征。

The wide scatter in the positive correlation between the plastic strain in a grain and its lattice misorientation does not correlate with its orientation: local grain interactions and the development transgranular bands of high strain erase any relationship between local plastic strain and crystallographic orientation.
晶粒塑性应变与其晶格转位之间的正相关性存在广泛离散，这种离散与其取向无关：局部晶粒相互作用以及高应变穿晶带的发育会消除局部塑性应变与晶体学取向之间的任何关系。

The spatial distribution of slip and lattice misorientation are very different: slip is localised in sharp bands whereas misorientation varies smoothly over the same scale. This is very different from how deformation is modelled in continuum crystal plasticity models, which could have implications for their applicability to model damage and the deformed state at the microstructural scale.
滑移和晶格转位的空间分布差异很大：滑移集中在锐利的带状区域，而晶格转位在同一尺度上平滑变化。这与连续介质晶体塑性模型中如何模拟变形非常不同，这可能对其在微观结构尺度上模拟损伤和变形状态的应用性产生影响。

Declaration of Competing Interest
利益冲突声明

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
作者声明他们没有已知的利益冲突或个人关系可能影响本论文中报告的工作。

Acknowledgments 致谢

The authors would like to acknowledge funding for their time from EPSRC [EP/I005420/1, EP/M005607/1, EP/M000737/1]. We would also like to thank the following people for useful discussions: Philippa Reed, Fabrice Pierron and Rong Jiang from the University of Southampton, Liuguo Zhao from the University of Loughborough, Mark Hardy and Duncan Maclachlan from Rolls Royce Plc. and Gordon McColvin from GE Power.
作者感谢 EPSRC 资助其时间 [EP/I005420/1, EP/M005607/1, EP/M000737/1]。我们还要感谢以下人士进行有益的讨论：南安普顿大学的 Philippa Reed、Fabrice Pierron 和 Rong Jiang，拉夫堡大学的 Liuguo Zhao，罗尔斯·罗伊斯公司的 Mark Hardy 和 Duncan Maclachlan，以及通用电气动力公司的 Gordon McColvin。

Appendix. Supplementary materials
附录。补充材料

What’s this? 这是什么？

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Research data for this article
本文研究数据

High resolution deformation data from the surface of a Nickel-based superalloy: Coarse precipitates
镍基高温合金表面高分辨率变形数据：粗大析出物

Original Data 原始数据

High resolution digital image correlation (HRDIC) and electron backscattered diffraction (EBSD) data provided that quantifies the deformation on the surface of Nickel-based superalloy with coarse gamma prime precipitates (250 nm diameter) after 2% strain in tension.
高分辨率数字图像相关（HRDIC）和背散射电子衍射（EBSD）数据，量化了含有粗大γ'析出物（250 nm 直径）的镍基高温合金在 2%拉伸应变后的表面变形情况。

Get data from Zenodo 从 Zenodo 获取数据

Further information on research data
关于研究数据的更多信息

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Cited by (84)

In-situ tailoring microstructures to promote strength-ductility synergy in laser powder bed fusion of NiCoCr medium-entropy alloy
在激光粉末床熔融 NiCoCr 中熵合金中通过原位调控微观结构促进强度-延展性协同
2023, Additive Manufacturing
2023, 增材制造
Laser powder bed fusion (LPBF) has received widespread attention owing to its digital, flexible, and controllable fabrication process, which opens new possibilities for the direct and fast production of metal components with extremely complex geometries and good performance. Generally, post-treatments are conducted to further regulate the microstructure and performance of as-deposited geometrically-complex metal components. In this work, instead of applying post-treatments, we demonstrate a novel in-situ remelting strategy to effectively enhance the strength without sacrificing the ductility of LPBF-fabricated NiCoCr medium-entropy alloy by in-process tailoring microstructures. We find that in-situ remelting processing not only changes the melt pool geometry in favor of optimizing densification and developing unique crystallographic lamellar microstructures, but also promotes inherent thermal distortions and heat treatments during manufacturing resulting in increased dislocation density and grain refinement in the LPBF-remelted specimen. The present work paves a new way in additively manufacturing metal materials to tailor microstructures for enhanced mechanical performance without additional treatments.
Heterogeneous slip localization in an additively manufactured 316L stainless steel
增材制造 316L 不锈钢中的非均匀滑移局部化
2022, International Journal of Plasticity
2022, 国际塑性学杂志
Additively manufactured 316L stainless steels display significantly higher yield strength than their as-cast or wrought counterparts. This is associated with the micro-scale cellular structure and complex grain and sub-grain structure, resulting from high cooling rates occurring during the additive manufacturing process. The consequences of these peculiar microstructural features on plastic localization early in the plastic regime at the sub-grain scale are investigated. The plastic localization involved during monotonic deformation of conventional and additive manufactured 316L stainless steels is investigated using high-resolution digital image correlation. Significant heterogeneous slip localization is observed in the additively manufactured 316L stainless steels compared to the wrought 316L stainless steels. The cellular structure and low-angle grain boundaries are observed to control the incipient plasticity. In addition, slip localization characteristics indicate that the additional strengthening in the AM material is mainly related to the cellular structure acting as a dislocation forest-type obstacle.
Simulated effects of sample size and grain neighborhood on the modeling of extreme value fatigue response
样本尺寸和晶粒邻域对极端值疲劳响应建模的模拟效应
2022, Acta Materialia
Citation Excerpt : 引用摘录 :
They later extended this method to consider individual and secondary slip systems [36]. Harte et al. [39] used EBSD and high-resolution DIC to measure plastic strain in Ni-based superalloy RR1000 strained to 2%. They found a weak correlation between grain plasticity and grain orientation derived metrics such as the Schmid Factor or Taylor Factor due to deformation bands at the mesoscale that were not crystallographic and complex interactions between grains.
他们后来将该方法扩展到考虑单个和次级滑移系统[36]。Harte 等人[39]使用 EBSD 和高分辨率 DIC 测量了应变至 2%的镍基高温合金 RR1000 中的塑性应变。他们发现由于中尺度变形带并非晶体学特征以及晶粒间复杂的相互作用，晶粒塑性与晶粒取向衍生指标（如 Schmid 因子或 Taylor 因子）之间存在微弱相关性。
Assessing the size of representative volume elements (RVEs) for fatigue-related applications is challenging. A RVE relevant to random microstructure requires a volume of material that is sufficiently large to capture the grain/phase heterogeneity that captures all statistical moments of the distribution of the driving force for fatigue crack formation at “hot spot” grains. Consequently, the large size of a microstructure RVE required to study fatigue phenomena is largely computationally intractable and difficult to explore. A more realistic objective in this work is to systematically study, as a function of the size of a statistical sample of microstructure, trends towards convergence of the simulated distribution of driving force for fatigue crack formation. The present work accordingly leverages the recently developed open-source PRISMS-Fatigue framework [Yaghoobi et al., npj Comput. Mater., 7, 38 (2021)] to examine the trends in convergence of extreme value distributions (EVD) of Fatigue Indicator Parameters (FIPs) in progressively larger polycrystalline microstructure realizations of FCC Al alloy 7075-T6 using crystal plasticity finite element method simulations. The results are compared to the traditional method in which ensembles of statistical volume elements (SVEs) are simulated to build up statistics intended to approximate those associated with a larger volume of material. The convergence of EVDs with increase of size of a SVE of microstructure is closely related to the extent of grain nearest neighbor (NN) interactions. Accordingly, the sensitivity of the local micromechanical response at hot spot grains is quantitatively investigated by systematically varying the orientations of NN grains. Results indicate that SVEs with cubic crystallographic texture tend towards convergence of the EVD of FIPs with tens of thousands of grains while the random and rolled textures require larger volumes. Simple relationships based on microstructure parameters (e.g., Schmid Factor, grain size, NN misorientation) do not completely correlate to fatigue hot spot grains. Finally, the sensitivity of the extreme value fatigue response at hot spot grains extends to the 3^rd NN when a single neighborhood grain orientation is altered.
Origins of the mechanical property heterogeneity in a hybrid additive manufactured Hastelloy X
混合增材制造 Hastelloy X 的力学性能异质性的起源
2021, Materials Science and Engineering A
2021，材料科学与工程 A
Hybrid additive manufacturing (hybrid-AM) technique is a new processing technique in which the large part of a component is fabricated by conventional manufacturing techniques and the complex parts are added flexibly via AM. This technique provides an efficient and economical method for manufacturing metallic components with different degrees of complexity. However, the mechanical properties of the substrate and the additive manufactured zone usually failed to match well while the underlying mechanism is still not fully understood. To address this problem, here we revealed the mechanisms of mechanical property heterogeneity in a hybrid-AMed Hastelloy X by microstructural characterization, thermo-mechanical simulations, and strengthening mechanism analyzing. The results demonstrate that the tensile properties and micro-hardness along the building direction show distinctive heterogeneity in the hybrid-AMed Hastelloy X, which is directly related to the evident microstructure changes. We attributed the microstructural variation to the different thermal histories of different regions during AM. Strengthening mechanism analysis indicated that the inhomogeneity of mechanical properties in the hybrid-AM Hastelloy X was mainly determined by the heterogeneous distribution of grain size, dislocation density, and crystallographic texture. EBSD, ECCI, and residual stress simulation all provided strong evidence for this conclusion. These findings shed new insights not only for achieving more homogeneous mechanical properties during the hybrid-AM process but also for laser additive repair and other fields.
Effects of precipitate on the slip activity and plastic heterogeneity of Mg-11Y-5Gd-2Zn-0.5Zr (wt. %) during room temperature compression
析出物对 Mg-11Y-5Gd-2Zn-0.5Zr（wt. %）在室温压缩过程中的滑移活性和塑性非均匀性的影响
2021, Materials Science and Engineering A
2021, 材料科学与工程 A
Citation Excerpt : 引用摘录 :
The deviations were reasonably small, and they may be caused by lattice rotation during the deformation and misalignment of SEM/EBSD experiment. Lattice misorientation is developed by plastic strain gradients, thus it is an important parameter to evaluate plastic heterogeneity [48]. Therefore, the misorientation based parameters including intragranular misorientation angle (IGM) and kernel average misorientation (KAM) were used to evaluate the plastic heterogeneity at grain/subgrain scale in this work.
偏差合理地小，它们可能是由变形过程中的晶格旋转和 SEM/EBSD 实验的错位引起的。晶格旋转是由塑性应变梯度产生的，因此它是评估塑性非均匀性的重要参数[48]。因此，本研究使用了基于晶格旋转的参数，包括晶粒内旋转角（IGM）和核平均旋转角（KAM），来评估晶粒/亚晶尺度上的塑性非均匀性。
To understand the effects of precipitate on the deformation mechanisms and plastic heterogeneity of a cast Mg–11Y-5Gd-2Zn-0.5Zr (wt.%) alloy, quantitative comparison studies were performed on the solid solution (T4) and peak-aged (T6) samples during room temperature compression based on slip trace analysis and electron backscatter diffraction (EBSD) analysis. At 10% plastic strain: no twins were observed in both T4 and T6 samples; the dominant slip mode transitioned from 2^nd order pyramidal <c+a> slip (34.1%) to basal <a> slip (50.5%) after the T6 treatment, which indicated that the precipitate-hardening effect for pyramidal slip was stronger than that for basal slip. Substantial active slip systems (up to 40%) exhibited low normalized Schmid factors (<0.1), implying that local stress deviated from the globally applied stress. Compared to the T4 sample, a larger magnitude of geometrically necessary dislocation (GND) density was found in the T6 sample, which indicated that the precipitates enhanced the plastic heterogeneity in terms of GND density. Most of the high GND density areas were near grain boundaries (GBs), but they also appeared in grain interiors. Almost every near-GB area with larger GND density tended to follow the conditions: (1) large orientation discrepancy between neighborhood grains; (2) at least one grain exhibiting slip traces; (3) the GB with the X phase. The GBs where slip transfer occurs tended to exhibit a homogeneous GND density distribution.
Investigation on Slip Activity and Plastic Heterogeneity of Aged Mg–10Y Sheets During Compression
研究老化 Mg–10Y 板材在压缩过程中的滑移活性和塑性非均匀性
2022, Metallurgical and Materials Transactions A Physical Metallurgy and Materials Science
2022, 《金属与材料学报 A：物理冶金与材料科学》

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Full length article 全文A statistical study of the relationship between plastic strain and lattice misorientation on the surface of a deformed Ni-based superalloy一项关于变形镍基高温合金表面塑性应变与晶格取向关系的统计分析

Abstract 摘要

Graphical abstract 图示摘要

Keywords 关键词

1. Introduction 1. 引言

2. Material and experimental methodology2. 材料与实验方法

2.1. Material and sample 2.1. 材料和样品

2.2. Orientation mapping 2.2. 取向映射

2.3. High resolution digital image correlation2.3. 高分辨率数字图像相关

2.3.1. Gold remodelling 2.3.1. 金重塑

2.3.2. Image acquisition 2.3.2. 图像采集

2.3.3. Mechanical testing2.3.3. 力学测试

2.3.4. Image correlation 2.3.4. 图像相关

2.4. Data analysis routines2.4. 数据分析程序

3. Results 3. 结果

3.1. The spatial relationship between strain and misorientation at the mesoscale and mean grain scale3.1. 中尺度及平均晶粒尺度下应变与取向的时空关系

3.2. Correlations between plastic strain and lattice misorientation at the sub-grain scale3.2. 亚晶尺度上的塑性应变与晶格倾斜的相关性

3.3. The influence of grain orientation on grain deformation3.3. 晶粒取向对晶粒变形的影响

4. Discussion 4. 讨论

4.1. Orientation-based predictors of deformation4.1. 基于取向的变形预测

4.2. Sub-grain deformation localisation4.2. 亚晶粒变形局部化

4.3. The suitability of different lattice misorientation calculations as a measure of local plasticity4.3. 不同晶格取向计算方法作为局部塑性的适用性

4.4. The spatial distribution of slip and lattice misorientation4.4. 滑移和晶格取向差的空間分布