A micromechanical analysis of intergranular stress corrosion cracking of an irradiated austenitic stainless steel

doi:10.1016/j.actamat.2020.116482

Acta Materialia

Volume 204, 1 February 2021, 116482
卷 204，2021 年 2 月 1 日，116482

https://doi.org/10.1016/j.actamat.2020.116482

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Abstract 摘要

Irradiation Assisted Stress Corrosion Cracking (IASCC) is a material degradation phenomenon affecting austenitic stainless steels used in nuclear Pressurized Water Reactors (PWR), leading to the initiation and propagation of intergranular cracks. Such phenomenon belongs to the broader class of InterGranular Stress Corrosion Cracking (IGSCC). A micromechanical analysis of IGSCC of an irradiated austenitic stainless steel is performed in this study to assess local cracking conditions. A 304L proton irradiated sample tested in PWR environment and showing intergranular cracking is investigated. Serial sectioning, Electron BackScatter Diffraction (EBSD) and a two-step misalignment procedure are performed to reconstruct the 3D microstructure over an extended volume, to assess statistically cracking criteria. A methodology is also developed to compute Grain Boundary (GB) normal orientations based on the EBSD measurements. The statistical analysis shows that cracking occurs preferentially for GB normals aligned with the mechanical loading axis, but also for low values of the Luster-Morris slip transmission parameter. Micromechanical simulations based on the reconstructed 3D microstructure, FFT-based solver and crystal plasticity constitutive equations modified to account for slip transmission at grain boundaries are finally performed. These simulations rationalize the correlation obtained experimentally into a single stress-based criterion. The actual strengths and weaknesses of such micromechanical approach are discussed.
辐照辅助应力腐蚀开裂（IASCC）是一种影响核压水堆（PWR）中使用的奥氏体不锈钢的材料退化现象，导致沿晶裂纹的萌生和扩展。这种现象属于更广泛的沿晶应力腐蚀开裂（IGSCC）类别。本研究对辐照奥氏体不锈钢的 IGSCC 进行微观力学分析，以评估局部开裂条件。研究调查了在 PWR 环境中测试并显示沿晶开裂的 304L 质子辐照样品。通过连续切片、电子背散射衍射（EBSD）和两步错位程序，重建了扩展体积上的三维微观结构，以评估统计开裂标准。还开发了一种基于 EBSD 测量计算晶界（GB）法线方向的方法。统计分析表明，开裂优先发生在晶界法线与机械加载轴对齐的情况下，但也发生在 Luster-Morris 滑移传输参数值较低的情况下。基于重建的 3D 微观结构、基于 FFT 的求解器和考虑晶界滑移传递的晶体塑性本构方程修改的微观力学模拟最终被执行。这些模拟将实验获得的关联合理化为单一基于应力的判据。讨论了这种微观力学方法的实际优缺点。

Graphical abstract 图示摘要

Keywords 关键词

Stress corrosion cracking

Intergranular

Austenitic stainless steel

EBSD

FFT

应力腐蚀开裂晶间奥氏体不锈钢 EBSDFFT

1. Introduction 1. 引言

Failures of austenitic stainless steels Baffle-to-Former Bolts (BFB) in the core of nuclear Pressurized Water Reactors (PWR) have been reported since the 80's as a result of an InterGranular Stress Corrosion Cracking (IGSCC) degradation phenomenon [1]. As these materials are mainly immune to SCC in aqueous environment, the phenomenon is referred to as Irradiation Assisted Stress Corrosion Cracking (IASCC) as core materials in PWR are heavily irradiated with neutrons [2]. Neutron irradiation leads to the formation of irradiation defects as a result of the recombination of point defects generated by ballistic interactions between neutrons and the atoms of the material [3]. The main irradiation defect observed in austenitic steels in PWR conditions (300 °C) is Frank dislocation loop, but other irradiation defects are also observed such as Helium bubbles and precipitates [4]. In addition, irradiation induces depletion of Cr and enrichment of Ni at grain boundaries. Understanding and predicting IGSCC of irradiated austenitic stainless steels, but also of unirradiated alloys such as Nickel-based materials, still remains a challenge [5], [6]. The use of ion-irradiated materials (e.g., protons [7]) in the last decades, relieving the burden of dealing with neutron irradiated radioactive samples, allows a deeper understanding of the physical mechanisms involved in IGSCC of irradiated austenitic stainless steels [8], [9]. Both radiation-induced hardening and radiation-induced segregation at Grain Boundaries (GB) are observed and act synergistically to favor IGSCC [10], [11]. Moreover, irradiation induces changes in intragranular deformation mechanisms and dislocation channelling is often reported for irradiated materials. For low applied strain, which is relevant for IGSCC cracking initiation, deformation occurs only in narrow bands of about tens of nanometers width as a result of the presence of irradiation defects [12], [13]. The impingement of these dislocation channels on GB has been proposed to lead to high local stresses responsible for GB cracking [14]. More precisely, cracking has been shown to correlate with the discontinuity of traces of dislocation channels across GB on specimen surfaces (see, e.g., [15]). The presence of high local stresses at the impingement of an interrupted dislocation channel on a GB has been experimentally demonstrated through residual stress measurements [16], and in more details through the use of Molecular Dynamics (MD) [14], [16] or Finite Element (FE) simulations [17]. Experiments using heavy-ion irradiated samples, leading to less localization than proton irradiation, have however questioned this mechanism as cracking initiation is also observed [18]. In addition, GB well oriented with respect to the mechanical loading direction have been found to be more prone to cracking [19], emphasizing the role of intergranular normal stresses. Similar results have been obtained on irradiated austenitic steels in Super Critical Water (400 °C) [20] as well as a correlation between cracking and high Schmid factor mismatches. These observations have been rationalized through a Schmid-Modified Grain Boundary Stress (SMGBS) model. The experimental determinations of local cracking conditions have been mostly made using 2D measurements (except for example in [21]) such as traces of GB and dislocation channels on specimen surfaces. However 3D information about the microstructure is required to accurately compute GB normals, SMGBS model or slip transmission criteria. This may be done for selected cracks through ionic milling, as performed in [22] to assess SMGBS model on the IGSCC of a thermally sensitized austenitic steel. Experimental techniques are available to obtain full 3D microstructures, either destructive such as serial sectioning coupled with 2D Electron BackScatter Diffraction (EBSD) [23] or non destructive such as Diffraction Contrast Tomography (DCT) [24]. Both techniques require dedicated procedures to avoid reconstruction artefacts, but provide valuable information regarding grain shapes / crack patterns. Up to now, to the authors’ knowledge, no such analysis has been performed in the context of IGSCC of irradiated austenitic stainless steels to assess in more details local cracking conditions.
自 20 世纪 80 年代以来，报道了在核压水堆（PWR）堆芯中奥氏体不锈钢 Baffle-to-Former Bolts（BFB）因间晶应力腐蚀开裂（IGSCC）退化现象导致的失效[1]。由于这些材料主要对水环境中的应力腐蚀开裂（SCC）具有免疫力，因此该现象被称为辐照辅助应力腐蚀开裂（IASCC），因为 PWR 堆芯材料受到中子的大量辐照[2]。中子辐照导致由于中子与材料原子之间的弹道相互作用产生的点缺陷复合而形成辐照缺陷[3]。在 PWR 条件下（300 °C）观察到的奥氏体钢的主要辐照缺陷是 Frank 位错环，但还观察到其他辐照缺陷，如氦气泡和析出物[4]。此外，辐照导致晶界处的 Cr 减少和 Ni 富集。理解和预测辐照奥氏体不锈钢的 IGSCC，以及对未辐照合金如镍基材料，仍然是一个挑战[5],[6]。近几十年来，使用离子辐照材料（例如质子[7]）减轻了处理中子辐照放射性样品的负担，从而使得对辐照奥氏体不锈钢的 IGSCC（晶间应力腐蚀开裂）中涉及的物理机制有了更深入的理解[8][9]。观察到辐照引起的硬化以及晶界（GB）的辐照引起偏析，两者协同作用有利于 IGSCC[10][11]。此外，辐照会导致晶内变形机制的变化，并且辐照材料中常报道位错通道现象。对于与 IGSCC 开裂起始相关的低施加应变，由于辐照缺陷的存在，变形仅发生在宽度约为几十纳米的窄带中[12][13]。这些位错通道对晶界的冲击被认为会导致高局部应力，从而引起晶界开裂[14]。更具体地说，研究表明开裂与样品表面晶界处位错通道痕迹的不连续性相关（例如[15]）。在断裂位错通道与晶界的碰撞处存在高局部应力，这一现象已通过残余应力测量实验得到证实[16]，并通过分子动力学（MD）[14]、[16]或有限元（FE）模拟进行了更详细的阐述[17]。然而，使用重离子辐照样品的实验（与质子辐照相比，辐照后局部化程度较低）却质疑了这一机制，因为也观察到了裂纹的萌生[18]。此外，研究发现，与机械加载方向良好取向的晶界更易发生裂纹[19]，这突出了晶界间法向应力的作用。在超临界水（400 °C）中辐照的奥氏体钢中也获得了类似的结果[20]，以及裂纹与高 Schmid 因子失配之间的相关性。这些观察结果已通过 Schmid-修正晶界应力（SMGBS）模型得到解释。局部裂纹条件的实验测定主要采用二维测量（例如[21]除外），如样品表面上的晶界和位错通道痕迹。然而，为了准确计算晶界法线、SMGBS 模型或滑移传递标准，需要 3D 的微观结构信息。这可以通过离子研磨对选定裂纹进行处理来完成，正如文献[22]中评估热敏化奥氏体钢 IGSCC 的 SMGBS 模型所做的那样。目前有实验技术可以获取完整的 3D 微观结构，包括破坏性技术（如串联切片结合二维电子背散射衍射（EBSD）[23]）和非破坏性技术（如衍射衬度断层扫描（DCT）[24]）。这两种技术都需要专门的程序来避免重建伪影，但能提供有关晶粒形状/裂纹模式的宝贵信息。到目前为止，据作者所知，在辐照奥氏体不锈钢的 IGSCC 背景下，还没有进行过此类分析以更详细地评估局部开裂条件。

Theoretical and numerical modelling of IGSCC have been proposed in the literature. Using synthetic or realistic polycrystalline aggregates along with crystal plasticity constitutive equations, distributions of intergranular stresses have been determined for unirradiated and irradiated austenitic stainless steels [25], [26]. This approach requires estimations of GB strength to predict IGSCC which can now be assessed through the development of microtestings [27]. However, experimental data remains scarce, especially for irradiated materials. From a numerical perspective, cohesive zones modelling can be used to predict both initiation and propagation of intergranular cracks [28], [29]. The results of these simulations are heavily dependent on the accuracy of the crystal plasticity constitutive equations, which is still an ongoing work for irradiated stainless steels. In particular, reproduction of dislocation channelling phenomenon is still a challenge within the crystal plasticity framework, although significant advances have been made recently regarding modelling of strain localization at the intragranular scale [30], [31]. All the aforementioned numerical studies account only for the anisotropic mechanical behavior of grains through the use of phenomenological crystal plasticity constitutive equations [32]. Intergranular stresses arise mainly as a result of deformation mismatches, as no additional constraint is imposed at GB. As an example, plastic slip can take arbitrary value close to a GB, which is not in agreement with experimental observations such as GB pile-up. A model has been proposed recently [33] to handle that issue by combining standard crystal plasticity equations with slip transmission criteria. This model is simple to implement, and has been shown to be able to reproduce phenomenon associated with dislocation pile-up at GB such as Hall-Petch effect. Despite the current limitations of polycrystalline aggregates simulations to predict IGSCC, such simulations are nevertheless required to identify gaps where modelling efforts should be put. To the authors’ knowledge, no crystal plasticity simulations have been reported on realistic microstructure for the IGSCC of irradiated austenitic steels with direct comparisons to cracking initiation experimental data.
在文献中已提出了关于 IGSCC 的理论和数值模型。通过使用合成或真实的多晶集合体以及晶体塑性本构方程，已确定了未辐照和辐照奥氏体不锈钢的晶间应力分布[25], [26]。这种方法需要估算晶界强度来预测 IGSCC，而晶界强度现在可以通过微测试的发展来评估[27]。然而，实验数据仍然稀缺，尤其是对于辐照材料。从数值角度来看，内聚区模型可用于预测晶间裂纹的萌生和扩展[28], [29]。这些模拟结果严重依赖于晶体塑性本构方程的准确性，而辐照不锈钢的晶体塑性本构方程的准确性仍是一项正在进行的工作。特别是，在晶体塑性框架内再现位错通道现象仍然是一个挑战，尽管最近在模拟晶粒尺度上的应变局部化方面取得了显著进展[30], [31]。所有上述数值研究仅通过使用现象学晶体塑性本构方程[32]考虑了晶粒的各向异性力学行为。晶界间应力主要是由变形不匹配引起的，因为在晶界处没有施加额外的约束。例如，塑性滑移可以在靠近晶界处取任意值，这与晶界堆垛等实验观察结果不符。最近提出了一个模型[33]，通过结合标准晶体塑性方程与滑移传递准则来处理该问题。该模型易于实现，并已被证明能够再现与晶界位错堆垛相关的现象，如霍尔-佩奇效应。尽管多晶聚合物的模拟在预测 IGSCC 方面目前存在局限性，但这种模拟仍然需要用来识别建模工作应投入的空白。据作者所知，尚未有关于辐照奥氏体不锈钢 IGSCC 的晶体塑性模拟，且没有与裂纹萌生实验数据进行直接比较的报道。

Based on this literature review, the objectives of this study are twofold. The first one is to assess local cracking conditions based on a 3D microstructure obtained on an irradiated austenitic stainless steel tested in PWR environment and exhibiting intergranular cracks. The second objective is to perform crystal plasticity simulations based on the 3D microstructure to assess the strengths and weaknesses of such micromechanical approach, as well as directions for future research. The paper is organized as follows: in Section 2, the experimental characterization is detailed, including the description of the material, the reconstruction of the 3D microstructure, as well as the analysis of local cracking conditions. Numerical simulations are detailed in Section 3. The experimental and numerical results are finally discussed in Section 4.
基于这篇文献综述，本研究的目标有两个。第一个目标是基于在 PWR 环境下测试并表现出沿晶裂纹的辐照奥氏体不锈钢获得的 3D 微观结构，评估局部开裂条件。第二个目标是基于 3D 微观结构进行晶体塑性模拟，以评估这种微观力学方法的优缺点，以及未来研究的方向。本文的结构如下：在第二节中，详细介绍了实验表征，包括材料描述、3D 微观结构的重建以及局部开裂条件的分析。数值模拟在第三节中详细介绍。最后，在第四节中讨论了实验和数值结果。

2. Experimental characterization
2. 实验表征

2.1. Material 2.1. 材料

The material is a Solution Annealed (1050 °C/30 min followed by water quench) 304L austenitic stainless steel (18.75% Cr, 8.55% Ni, 0.02% Mo, 0.45% Si, 1.65% Mn, 0.012% C, wt), with a mean grain size of

27 μ

m. Solution annealing allows to bring back (intergranular) carbides into solid solution - hence reducing the susceptibility to IGSCC - and is the heat treatment performed on the materials used for PWRs baffles and formers. Flat tensile specimens (2mm x 2mm x 18mm gauge length) have been sampled using electrical discharge machining, and mirror polished on one side. The last polishing step is a vibratory polishing with colloidal silica solution (

0.05 μ

m, pH=7) to remove any surface hardening from previous polishing steps.
该材料是一种固溶退火（1050 °C/30 分钟随后水淬）的 304L 奥氏体不锈钢（18.75% Cr，8.55% Ni，0.02% Mo，0.45% Si，1.65% Mn，0.012% C，重量百分数），平均晶粒尺寸为

27 μ

m。固溶退火可以使（晶间）碳化物重新溶解到固溶体中——从而降低对 IGSCC 的敏感性——并且是用于 PWR 挡板和成型件的热处理工艺。通过电火花加工取样制备了平板拉伸试样（2mm x 2mm x 18mm 标距），并在一面进行抛光。最后一步抛光是使用胶体二氧化硅溶液（

0.05 μ

m，pH=7）进行振动抛光，以去除先前抛光步骤留下的任何表面硬化。

The samples have been irradiated with 2 MeV protons at the Michigan Ion Beam Laboratory (MIBL) at a temperature of 350

\pm 10^{\circ}

C. The mean flux was

1.39 10^{14} H^{+} . {cm}^{- 2} . s^{- 1},

and the irradiation time 95h. The irradiation dose profile - quantified with displacements per atoms (dpa) [34] to compare with irradiations with different particles - is shown in Fig. 1a: only the first 20μm of the samples are irradiated, with an approximately constant irradiation dose of about 2 dpa over the first 10μm. A tensile sample was subjected to a Slow Strain Rate Test (SSRT) at a strain rate of

5 . 10^{- 8} s^{- 1}

up to 4% plastic strain in PWR environment (340 °C, deoxygenated water with 1000 ppm B, 2 ppm Li, 25-35 cc

H_{2}

). After the SSRT, Scanning Electron Microscope (SEM) observations have confirmed the presence of intergranular cracking, as shown on Fig. 1b. Details about the material, the irradiation conditions and the SCC test can be found in [18]. This tensile sample was used to assess local conditions for intergranular cracking of irradiated austenitic steels, which first requires the reconstruction of the 3D microstructure.
样品在密歇根离子束实验室（MIBL）用 2 MeV 质子辐照，温度为 350°C。平均通量为

1.39 10^{14} H^{+} . {cm}^{- 2} . s^{- 1},

，辐照时间 95 小时。辐照剂量曲线——用原子位移（dpa）[34]量化，以与不同粒子辐照进行比较——如图 1a 所示：仅样品的前 20μm 被辐照，在前 10μm 内辐照剂量约为 2 dpa 且大致恒定。一个拉伸样品在 PWR 环境下（340°C，去氧水，含 1000 ppm B、2 ppm Li、25-35 cc

H_{2}

）进行慢应变速率试验（SSRT），应变速率为

5 . 10^{- 8} s^{- 1}

，直至产生 4%的塑性应变。SSRT 后，扫描电子显微镜（SEM）观察证实存在沿晶裂纹，如图 1b 所示。关于材料、辐照条件和应力腐蚀开裂（SCC）试验的详细信息可参见[18]。该拉伸样品用于评估辐照奥氏体钢沿晶裂纹的局部条件，这首先需要重建 3D 微观结构。

2.2. Microstructure 2.2. 微结构

The typical surface cracks density of the sample has been determined in [18] to be about

300 {mm}^{- 2}

. In order to get statistically relevant results for the local cracking conditions, the 3D microstructure has to be reconstructed on an area containing few tens of cracks at least, e.g., a typical surface of about 1 mm². Different techniques have been proposed in the literature for this purpose. Serial-sectioning FIB tomography is rather limited to small volumes - typically

{(10 μ m)}^{3}

- due to milling rate limitations [35]. DCT has the advantages to be non destructive and to allow large scan volumes, but the detection of intergranular cracks might be difficult [24]. Therefore, the technique used in the following combined serial-sectioning polishing and 2D EBSD measurements, as for example done in [23]. Note that an alternative technique has been proposed recently where serial-sectioning polishing is replaced by Broad-Ion-Beam (BIB) milling [36], that also allows to assess large volumes.
样品的典型表面裂纹密度在[18]中已被确定为约

300 {mm}^{- 2}

。为了获得局部裂纹条件的统计相关结果，必须在至少包含几十个裂纹的区域重建 3D 微观结构，例如一个典型的表面，面积约为 1 mm ² 。文献中已提出了多种技术用于此目的。由于铣削速率的限制，串联切片 FIB 断层扫描技术主要限于小体积——通常为

{(10 μ m)}^{3}

[35]。DCT 具有非破坏性和允许大扫描体积的优点，但检测晶间裂纹可能比较困难[24]。因此，以下将采用串联切片抛光和 2D EBSD 测量的技术组合，例如在[23]中所示。请注意，最近提出了一种替代技术，其中串联切片抛光被宽离子束（BIB）铣削所取代[36]，该技术同样允许评估大体积。

2.2.1. 3D EBSD

A portion of the sample has been mounted into a conductive resin. Vickers indents identify an area of 1 mm x 1 mm, as shown Fig. 2a. These indents are also used to measure the thickness removal after each polishing step and help to align the SEM observations, as detailed below.
样品的一部分已安装在导电树脂中。维氏压痕确定了 1 毫米×1 毫米的区域，如图 2a 所示。这些压痕还用于测量每次抛光步骤后的去除厚度，并有助于对 SEM 观察进行对准，如下所述。

The experimental methodology is as follows. Polishing is performed on an automatic polishing machine using colloidal silica solution (

0.03 μ

m) to remove about

1 μ

m after each step. The sample is then cleaned up to remove the polishing solution, and the sizes of all indents are measured to estimate the thickness removed locally¹ The use of an automatic polishing machine on a sample placed into a mounting resin allowed to keep the parallelism of the consecutive surfaces, with a typical angle lower than 0.1° (Appendix A 3D EBSD reconstruction, Fig. 21a). SEM observations of the area identified by the Vickers indents are then performed under Secondary Electron (SE) mode to locate intergranular cracks: 20 low magnification images are taken, as shown on Fig. 2b, to ensure a sufficient resolution for crack detection, and image stitching is finally used to obtain a high-resolution image over the complete surface. EBSD maps have been performed on a JEOL IT300 SEM with a tungsten filament equipped with an OXFORD EBSD detector. All EBSD maps are acquired under the same conditions at 20 kV with high current. The sample is tilted at 70° with respect to the beam axis. Diffraction patterns are indexed with the AZTEC software, leading to the crystallographic phases (FCC or BCC) and orientations (through the three Euler angles

ϕ_{1}

, Φ, 

ϕ_{2}

) using a measurement step of

2 μ

m. These steps are finally repeated down to a thickness of

20 μ

m, which corresponds to the typical grain size of the material and to the thickness of the irradiated layer. The surface preparation leads to rather good indexation ratio, still the software can not find the crystallographic phases and orientations for about 2% of the measurement points. Moreover, as shown on Fig. 1a, ferrite is present in the material (about 3% of the EBSD measurements correspond to Body Centered Cubic (BCC) phase). As numerical simulations presented in Section 3 and based on the experimental results require only FCC phase with full crystallographic orientations fields, the software MTEX [37] is used to remove the BCC phase and fill the unindexed points with FCC crystallographic orientations of the neighbouring points. These corrections are not expected to influence the results presented hereafter as the amount of ferrite is low (

\sim 3 %

). Moreover, no crack was observed at BCC-FCC interfaces, consistent with observations on IGSCC of unirradiated austenitic stainless steels [38].
实验方法如下。使用自动抛光机，以胶体二氧化硅溶液（

0.03 μ

m）进行抛光，每一步去除约

1 μ

m。然后清洗样品以去除抛光溶液，测量所有压痕的尺寸，以估计局部去除的厚度 ¹ 将样品放入粘合树脂中，使用自动抛光机保持连续表面的平行性，典型角度低于 0.1°（附录 A 3D EBSD 重建，图 21a）。然后在次级电子（SE）模式下对维氏压痕确定的区域进行 SEM 观察，以定位沿晶裂纹：拍摄 20 张低倍图像，如图 2b 所示，以确保裂纹检测的足够分辨率，并最终使用图像拼接获得完整表面的高分辨率图像。在配备 OXFORD EBSD 探测器的 JEOL IT300 SEM 上进行了 EBSD 图谱，使用钨丝。所有 EBSD 图谱均在 20 kV 高电流条件下以相同条件获取。样品相对于束轴倾斜 70°。衍射图谱使用 AZTEC 软件进行标定，通过测量步长为 5 m，确定了晶体学相（FCC 或 BCC）和取向（通过三个欧拉角α, Φ, β）。这些步骤最终重复进行，直到厚度为 6 m，这对应于材料的典型晶粒尺寸和辐照层的厚度。表面制备导致标定比例相当好，但软件仍无法为大约 2%的测量点找到晶体学相和取向。此外，如图 1a 所示，材料中存在铁素体（大约 3%的 EBSD 测量对应于体心立方（BCC）相）。由于第 3 节中所示的数值模拟以及基于实验结果仅需 FCC 相和完整的晶体学取向场，因此使用软件 MTEX [37]去除 BCC 相，并用邻近点的 FCC 晶体学取向填充未标定点。这些修正预计不会影响此处后续展示的结果，因为铁素体的含量较低（7）。此外，在 BCC-FCC 界面处未观察到裂纹，这与未辐照奥氏体不锈钢的 IGSCC 观察结果一致[38]。

As detailed in previous studies (see [39] and references therein), the main challenge of 3D EBSD through serial sectioning lies in the reconstruction procedure. First, each 2D EBSD map is potentially affected by at least two effects that may lead to geometrical distortions. Depending on the detector used, EBSD measurements over large scale areas can be quite long (typically few hours for the parameters used in this study), which can result in a beam drift over time due to thermal / mechanical changes of the SEM configuration. This effect has been assessed to be negligible in this study by comparing systematically SEM images taken before and after the EBSD map. However, the configuration used for EBSD measurements - which corresponds to a 70° tilted specimen with respect to the beam axis - leads to image distortion as can be seen on Fig. 3b where the Vickers indents are no longer forming a square. Such distortion appears for large scale EBSD maps performed using low magnification, and should be corrected to obtain correct grain shapes.
如前所述研究（参见[39]及其参考文献），3D EBSD 通过串行切片的主要挑战在于重建过程。首先，每个 2D EBSD 图可能至少受到两种可能导致几何畸变的影响。根据所用探测器，EBSD 测量大面积区域可能相当耗时（本研究中使用的参数通常需要几小时），这可能导致由于 SEM 配置的温热/机械变化而导致的束漂移。通过系统比较 EBSD 图前后拍摄的 SEM 图像，本研究评估了这种影响可以忽略不计。然而，用于 EBSD 测量的配置——即相对于束轴倾斜 70°的样品——会导致图像畸变，如图 3b 所示，维氏压痕不再形成方形。这种畸变出现在使用低倍率进行的大规模 EBSD 图中，并且需要校正以获得正确的晶粒形状。

Several algorithms have been proposed in the literature to deal with 3D EBSD reconstruction [40], [41], [42]. An in-house procedure is used in this study. In order to correct the geometrical distortion, the displacements of the four corners Vickers indents required to match a square of 1 mm size are determined (Fig. 3c), and the image is then corrected using a bilinear interpolation correction scheme based on these displacements. A typical result is shown on Fig. 3d where the actual EBSD zone scanned has in fact a trapezoidal shape. All 2D EBSD maps have been corrected according to this procedure. However, this does not ensure that the superposition of the 2D EBSD maps will lead to a 3D EBSD microstructure free of geometrical distortions [39]. The main reason is that slight differences in alignment and / or SEM acquisition parameters between each step are unavoidable since the sample is removed from the SEM for polishing. Therefore, an additional correction procedure is applied to EBSD maps, following the first correction procedure described above. The displacements of the Vickers indents are defined as the ones allowing maximizing the cross-correlation of Euler angles between the current EBSD map and the previous one. In practice, only one Euler angle is used for the cross-correlation, and a Nelder-Mead algorithm is used to perform the maximization. This method is valid as long as two successive EBSD maps are not too different, i.e., when the distance between the two planes is small compared to the grain size, and for materials with no morphological texture (as for example the method is expected to introduce bias for materials with tilted columnar grains). In addition, Euler angles should be mostly constant in each grain, i.e., without jumps from one voxel to another that can happen due to the FCC symmetry, which has been checked (Appendix A). In that case, an alternative strategy would be to minimize the local misorientation between consecutives maps [39]. The details about the EBSD corrections are given in Appendix A. Finally, the three Euler angles values are projected on a regular grid - required for the numerical simulations - using nearest value interpolation, allowing subsampling if necessary.
文献中已提出多种算法用于处理三维 EBSD 重建[40]、[41]、[42]。本研究采用内部程序。为校正几何畸变，需确定使四个角维氏压痕匹配 1 毫米边长的位移量（图 3c），然后基于这些位移量采用双线性插值校正方案对图像进行校正。典型结果如图 3d 所示，实际扫描的 EBSD 区域实际上呈梯形。所有二维 EBSD 图均按此程序进行了校正。然而，这并不能保证二维 EBSD 图的叠加将产生无几何畸变的 EBSD 三维显微结构[39]。主要原因在于，由于样品需从扫描电镜中取出进行抛光，每一步之间的对准和/或 SEM 采集参数存在微小差异是不可避免的。因此，在上述首次校正程序之后，对 EBSD 图应用了额外的校正程序。维氏压痕的位移被定义为能够最大化当前 EBSD 图谱与先前图谱之间欧拉角交叉相关性的位移。实际上，交叉相关性仅使用一个欧拉角进行计算，并采用 Nelder-Mead 算法进行最大化。只要两个连续的 EBSD 图谱差异不大，即两个平面之间的距离相对于晶粒尺寸较小，并且对于没有形态织构的材料（例如，该方法预计会对具有倾斜柱状晶粒的材料引入偏差），此方法都是有效的。此外，欧拉角在每个晶粒中应基本保持不变，即不会从一个体素跳到另一个体素，这种情况可能由于面心立方对称性引起，这一点已经得到验证（附录 A）。在这种情况下，可以采用替代策略，即最小化连续图谱之间的局部取向差[39]。关于 EBSD 校正的详细信息见附录 A。最后，三个欧拉角值使用最近值插值投影到规则网格上——这是数值模拟所需的——如果必要，可以进行子采样。

The 3D EBSD reconstruction is shown on Fig. 3e, and corresponds to a volume of about

800 μ m x 800 μ m x 20 μ m

. The correction procedure is assessed on Fig. 3f by looking at cross sections of the 3D microstructure, showing a strong reduction of geometrical artefacts for grain shapes compared to the case without corrections (Appendix A). Another artefact reported in the literature [42] is associated with the differences of crystallographic orientations in a given grain on two consecutive 2D EBSD maps due to slight misalignment of the sample inside the SEM. This has been checked in the 3D reconstructed microstructure by looking at in-plane and through thickness Euler angles profiles in several grains. The variations along the thickness - thus between consecutive 2D EBSD maps - are found to be of the same order (few degrees) as the in-plane variations (Appendix A), indicating that the problem reported in [42] is not significant in our study.
3D EBSD 重建结果如图 3e 所示，对应于约

800 μ m x 800 μ m x 20 μ m

的体积。校正过程通过观察 3D 微观结构的横截面在图 3f 中进行评估，与未进行校正的情况相比，晶粒形状的几何伪影显著减少（附录 A）。文献[42]中报道的另一种伪影与在连续的两个 2D EBSD 图中给定晶粒的晶体学取向差异有关，这是由于样品在 SEM 内部轻微错位造成的。通过观察几个晶粒的平面和厚度方向欧拉角分布，在 3D 重建的微观结构中对此进行了验证。沿厚度方向的变化（即连续的两个 2D EBSD 图之间）与平面方向的变化（少数度）具有相同的数量级（附录 A），表明文献[42]中报道的问题在我们的研究中并不显著。

2.2.2. Intergranular cracks
2.2.2. 晶间裂纹

The SEM cartography performed at the surface of the sample (Fig. 2a) allows detection, after automatic binarization and manual corrections, of intergranular cracks in the frame defined by the Vickers indents. The next step consists of locating these cracks at the surface of the 3D microstructure (Fig. 3e). First, the GB of the EBSD map are detected using the MTEX software (Fig. 4) using a misorientation threshold of 10°. As both the 3D microstructure and the intergranular cracks have been obtained in the same frame, it is possible to project intergranular cracks on the GB, which is shown on Fig. 4. The spatial resolution of EBSD maps is

Δ = 2 μ

m, and the maps have been corrected for distortions. Spatial resolution of SEM observations is much higher and distortion free because the sample is perpendicular to the beam axis. Hence, the projection of cracks on GB is not straightforward. For each position corresponding to a crack in the SEM images, the neighbouring pixels are also considered to correspond to a crack up to a distance of

P Δ,

to be consistent with the resolution of the EBSD maps. As detailed in Appendix B,

P = 1

is chosen so as to minimize the number of false cracked GB. The projection is then performed, leading to the results presented in Fig. 4.
在样品表面进行的 SEM 图像绘制（图 2a）经过自动二值化和手动校正后，可以在维氏压痕定义的框架内检测到沿晶裂纹。下一步是在 3D 微观结构的表面定位这些裂纹（图 3e）。首先，使用 MTEX 软件以 10°的取向差阈值为依据检测 EBSD 图谱中的晶界（图 4）。由于 3D 微观结构和沿晶裂纹都是在同一框架内获得的，因此可以将沿晶裂纹投影到晶界上，如图 4 所示。EBSD 图谱的空间分辨率为

Δ = 2 μ

m，并且已对变形进行了校正。SEM 观察的空间分辨率要高得多，且无变形，因为样品垂直于束轴。因此，裂纹在晶界上的投影并不直接。对于 SEM 图像中对应裂纹的位置，其邻近像素也被考虑为裂纹，距离可达

P Δ,

，以与 EBSD 图谱的分辨率保持一致。如附录 B 所述，

P = 1

的选择旨在最小化假裂纹晶界的数量。投影随后进行，得到图 4 所示的结果。

2.2.3. Grain boundaries 3D characterization
2.2.3. 晶界三维表征

The MTEX software [37] used in the previous section allows only detection of GBs from a 2D EBSD map, whereas the characterization of the 3D GB is required to compute GB normals. For the analysis of the cracking initiation conditions, at least the computation of GB normals for the free surface GB is needed, which is done with an in-house procedure. The 3D reconstruction of the free-surface GB is briefly described (more details can be found in Appendix B).
上一节中使用的 MTEX 软件[37]仅能从 2D EBSD 图中检测晶界，而计算晶界法线需要表征 3D 晶界。为了分析裂纹起始条件，至少需要计算自由表面晶界的法线，这通过内部程序完成。自由表面晶界的 3D 重建简要描述如下（更多细节可参见附录 B）。

The employed method reconstructs GB from two 2D images stacked atop each other to form a section of a 3D aggregate model of a sample. Each colored voxel forming a regular grid of a 2D image represents a local crystallographic orientation (using the correspondence between the Euler angles and RGB color scheme). On the top horizontal EBSD plane, the method identifies first GB locations by analyzing local image contrasts, see Fig. 5a,b, and GB slopes in a selected GB voxel is computed by accounting for nearest-neighbor voxels region of size

(2 N_{n} + 1) \times (2 N_{n} + 1)

(thin white square in Fig. 5b). For each GB voxel identified on a top plane four vertical cross-section planes are then formed and same-colored voxels identified within each plane in order to calculate the (average) out-of-plane GB slope, see Fig. 5c,d,e. Once the in-plane and out-of-plane GB slopes are known a 3D GB normal

n

is finally calculated and assigned to the corresponding free-surface GB voxel. It has been tested, using 2D images from Voronoi aggregate model with exactly known GB normals, that the accuracy of the above method improves with the increasing distance between the two considered 2D image planes. A sensitivity analysis detailed in Appendix B leads to consider

N_{n} = 2

and a distance between the top and bottom planes of

d = 4 μ

m in all computations. It should also be noted that triple points GB (and their close vicinities) are not considered (Fig. 5) as GB normal computation is not possible. Therefore, intergranular cracks (Fig. 4) are projected onto the GB locations where GB normals are available.
所采用的方法通过将两个堆叠在一起的 2D 图像重构为样品 3D 聚集模型的一个截面来重建晶界。每个形成 2D 图像规则网格的彩色体素代表一个局部晶体学取向（使用欧拉角与 RGB 颜色方案之间的对应关系）。在顶部的水平 EBSD 平面上，该方法通过分析局部图像对比度来识别晶界的初始位置，见图 5a,b，并通过考虑最近邻体素区域（图 5b 中的白色薄正方形）来计算选定晶界体素中的晶界斜率。然后，对于在顶平面上识别的每个晶界体素，形成四个垂直的横截面平面，并在每个平面中识别相同颜色的体素，以计算（平均）面外晶界斜率，见图 5c,d,e。一旦面内和面外晶界斜率已知，最终会计算并分配 3D 晶界法线

n

到相应的自由表面晶界体素。通过使用具有已知晶界法线的 Voronoi 聚集模型的 2D 图像进行测试表明，随着所考虑的两个 2D 图像平面之间距离的增加，上述方法的精度得到提高。附录 B 中详细进行的敏感性分析导致在所有计算中考虑

N_{n} = 2

和上下平面之间的距离为

d = 4 μ

米。还应注意，三重点 GB（及其附近区域）未被考虑（图 5），因为 GB 法向计算不可行。因此，沿晶裂纹（图 4）被投影到 GB 法向可用的位置。

The statistical analysis presented hereafter is based on the computation of probability density functions (noted pdf) and cumulative distribution functions (cdf). For the latter, 95% confidence bounds are also computed². pdf and cdf of the GB

n

normal components are plotted in Fig. 6a,b. For materials with no morphological texture (grain shapes), the theoretical pdf of the absolute values of the normal components

n_{i}

should be equal to 1, and the corresponding cdf linear with a slope equal to 1. In-plane components

n_{x}

and

n_{y}

distributions are found to be close to these values, showing that the material has no morphological texture. Out-of-plane component

n_{z}

exhibits an overrepresentation of values close to 0 and 0.5, which is probably due to the absence of smoothing for the out-of-plane component as compared to the in-plane components.
此处提供的统计分析基于概率密度函数（记为 pdf）和累积分布函数（cdf）的计算。对于后者，还计算了 95%置信区间 ² 。图 6a,b 中绘制了 GB

n

法向分量 pdf 和 cdf。对于没有形态织构（晶粒形状）的材料，法向分量绝对值的理论 pdf

n_{i}

应等于 1，相应的 cdf 斜率等于 1。面内分量

n_{x}

和

n_{y}

的分布接近这些值，表明该材料没有形态织构。面外分量

n_{z}

显示出接近 0 和 0.5 的值过度代表，这可能是由于与面内分量相比，面外分量没有进行平滑处理。

The experimental characterization has allowed reconstruction of the 3D microstructure - crystallographic orientations and grain shapes - over an extended area, to detect (un-)cracked GB as well as to evaluate GB slopes on the free-surface. The data are then examined to assess local cracking conditions in the next section, keeping in mind some unavoidable imperfections of the experimental characterization for the interpretation of the results. First, the characterization has been performed after the SSRT, on a deformed material, which could have affected both grain shapes and crystallographic orientations. However, the low level of applied strain (

\sim 4 %

) and the mechanical loading condition (uniaxial stress) is expected to lead to minor effects. For crystallographic orientations, the misorientation angle distribution computed by selecting random positions on the free-surface of the reconstructed microstructure does not show deviations from the theoretical McKenzie distribution for untextured materials (Fig. 7a). An outcome of having performed the EBSD measurements on the deformed material is that local crystallographic orientations are affected by residual stresses. MTEX software has been used to compute Geometrically Necessary Dislocations (GND) densities associated with orientations gradients at the free surface (Inset Fig. 7b). As GNDs can be seen as an indicator of local deformation incompatibilities, average values close to GB³ have been computed, and cumulative distribution functions for uncracked and cracked GB are shown on Fig. 7b. No significant difference is observed, thus cracking can not be associated with GNDs in this study. However, as GNDs computations depend strongly on the accuracy of crystallographic orientations and GND may arise at a distance of GB smaller than the spatial resolution of the EBSD maps, additional measurements are required.
实验表征使能够在较大区域内重建 3D 微观结构——晶体取向和晶粒形状——以检测（未）开裂晶界，并评估自由表面上的晶界斜率。随后将检查数据，以在下一节中评估局部开裂条件，同时考虑到实验表征在结果解释中不可避免的一些缺陷。首先，表征是在 SSRT 之后在变形材料上进行的，这可能影响了晶粒形状和晶体取向。然而，施加的应变水平（

\sim 4 %

）和机械加载条件（单轴应力）预计只会产生轻微影响。对于晶体取向，通过在重建的微观结构的自由表面上选择随机位置计算的角度分布，并未显示出与无织构材料的理论麦肯齐分布的偏差（图 7a）。在变形材料上进行 EBSD 测量的一个结果是，局部晶体取向受到残余应力的影响。 MTEX 软件已被用于计算自由表面处（插图图 7b）与取向梯度相关的几何必需位错（GND）密度。由于 GND 可以被视为局部变形不匹配的指标，因此计算了接近 GB ³ 的平均值，并在图 7b 中显示了未开裂和开裂 GB 的累积分布函数。未观察到显著差异，因此在本研究中裂纹不能与 GND 相关联。然而，由于 GND 计算高度依赖于晶体取向的准确性，并且 GND 可能出现在距离 GB 小于 EBSD 图空间分辨率的位置，因此需要额外的测量。

2.3. Analysis of experimental data
2.3. 分析实验数据

2.3.1. GB normals 2.3.1. 晶界法线

The cdf of GB normals are first assessed in this section. Fig. 8 shows the cdf of the normal component

n_{x},

the

x -

axis corresponding to the loading axis during the SSRT. As already shown in Fig. 6b, the cdf of uncracked GB (which correspond to the majority of the GB) almost follows a linear relationship of slope 1 with the absolute value of the normal component, as expected for materials with no morphological texture. Cracked GB exhibit a significant deviation from uncracked GB, where cracked GB have statistically higher values of

n_{x}

. Accordingly, the cdf of the two other normal components -

n_{y}

(Fig. 8b) and

n_{z}

- of cracked GB are significantly shifted towards the lower values as compared to uncracked GB.
本节首先评估了 GB 法向的 CDF。图 8 显示了在 SSRT 加载轴方向上对应于

x -

轴的法向分量

n_{x},

的 CDF。如图 6b 所示，未开裂 GB（对应大多数 GB）的 CDF 几乎与法向分量的绝对值呈斜率为 1 的线性关系，这与无形态织构材料的预期结果一致。开裂 GB 与未开裂 GB 存在显著差异，其中开裂 GB 的

n_{x}

值在统计上更高。因此，与未开裂 GB 相比，开裂 GB 的另外两个法向分量

n_{y}

（图 8b）和

n_{z}

的 CDF 显著向低值偏移。

These results, consistent with previous studies based only on 2D observations (see, e.g., [19]), support the dependence of IGSCC of irradiated stainless steels to the local stress conditions, as the normal stress acting on a GB is

σ_{n n} = σ n_{x}^{2}

(assuming a uniaxial stress state of magnitude

σ

). Of course, the local stress conditions may deviate from uniaxial stress conditions due to grain-grain interactions, but higher intergranular normal stresses are in majority related to well-oriented boundaries, thus with high values of

| n_{x} |

for uniaxial loading conditions [43]. Interestingly, Fig. 8 shows that a significant number of cracked GB are not well-oriented with respect to the mechanical loading axis, and thus that the GB orientation is not a sufficient condition for cracking.
这些结果与仅基于二维观察的研究结果一致（例如[19]），支持辐照不锈钢的晶间应力腐蚀开裂（IGSCC）与局部应力条件相关，因为在晶界上的正应力为

σ_{n n} = σ n_{x}^{2}

（假设为大小为

σ

的单轴应力状态）。当然，由于晶粒间的相互作用，局部应力条件可能偏离单轴应力条件，但更高的晶间正应力主要与取向良好的晶界相关，因此在单轴加载条件下

| n_{x} |

值较高[43]。有趣的是，图 8 显示，相当数量的开裂晶界相对于机械加载轴并非取向良好，因此晶界取向并不是开裂的充分条件。

2.3.2. Slip transmission criteria
2.3.2. 滑移传递准则

Experimental observations based on slip traces at the surface of irradiated austenitic stainless steels samples tested in PWR environment have been reported to show a correlation between slip discontinuity and cracking [14]. Several slip transmission criteria have been proposed in the literature to predict such discontinuities [44]. The use of a reliable slip transmission criterion is thus a potential way to predict GB that are most susceptible to cracking. However, assessment of these criteria with respect to experimental observations remains scarce [45], and, to the authors’ knowledge, no such assessment has been done yet for irradiated austenitic stainless steels. The experimental data obtained in this study do not allow to assess the reliability of slip transmission criteria, as the (dis-)continuity of slip traces at GB has not been examined systematically. However, as GB have been fully characterized with crystallographic orientations on each side and GB normal orientation, correlations between slip transmission criteria and cracking are assessed. Considering a GB of normal

n_{Γ}

between two grains

A

and

B,

each of them having a set of slip systems with corresponding slip direction

d_{i}^{A, B}

and slip plane normal

n_{i}^{A, B},

slip transmission parameters

N_{i j}^{A, B}

can be written as:
基于在 PWR 环境下测试的辐照奥氏体不锈钢样品表面滑移痕迹的实验观察，已有报道表明滑移不连续性与裂纹之间存在相关性[14]。文献中已提出了几种滑移传递准则来预测此类不连续性[44]。因此，使用可靠的滑移传递准则是预测最易开裂晶界的潜在方法。然而，这些准则与实验观察的评估仍然稀缺[45]，据作者所知，目前还没有针对辐照奥氏体不锈钢进行过此类评估。本研究获得的实验数据无法评估滑移传递准则的可靠性，因为晶界处滑移痕迹的（不）连续性尚未得到系统检查。然而，由于晶界两侧的晶体学取向和晶界法向取向已得到充分表征，因此评估了滑移传递准则与裂纹之间的相关性。考虑在两个晶粒

A

和

B,

之间的一个正常

n_{Γ}

晶界，每个晶粒都有一套滑移系统，具有相应的滑移方向

d_{i}^{A, B}

和滑移面法线

n_{i}^{A, B},

，滑移传递参数

N_{i j}^{A, B}

可以表示为：(1)

N_{i j}^{A, B} = F (n_{Γ}, d_{i, j}^{A, B}, n_{i, j}^{A, B})

The slip transmission criteria used in this study are summarized in Table 1. More complicated slip transmission criteria exist [44], but they depend on additional material parameters that need to be calibrated and are thus not used. Slip transmission is evaluated based on the 12 x 12 matrix

N_{i j}^{A, B}

coefficients defined for each GB. In the following, the maximal value

N_{\max}^{A, B} = m a x_{i, j} | N_{i j}^{A, B} |

is used as an indicator for slip transmission. In addition, Schmid factors are also computed as low values have been correlated with intergranular cracking [46].
本研究中使用的滑移传递标准总结在表 1 中。存在更复杂的滑移传递标准[44]，但它们依赖于需要校准的附加材料参数，因此未使用。滑移传递基于为每个晶界定义的 12 x 12 矩阵

N_{i j}^{A, B}

系数进行评估。在下文中，最大值

N_{\max}^{A, B} = m a x_{i, j} | N_{i j}^{A, B} |

被用作滑移传递的指标。此外，还计算了 Schmid 因子，因为低值与沿晶开裂相关[46]。

Table 1. Summary of the slip transmission parameters used in this study [44].
表 1. 本研究[44]中使用的滑移传递参数总结。

Empty Cell	Parameter $N_{i j}^{A, B}$ 参数 $N_{i j}^{A, B}$
Livingston-Chalmers 利文斯顿-查默斯	$(n_{i}^{A} \cdot n_{j}^{B}) (d_{i}^{A} \cdot d_{j}^{B}) + (n_{i}^{A} \cdot d_{j}^{B}) (n_{j}^{B} \cdot d_{i}^{A})$
Luster-Morris 拉斯特-莫里斯	$(n_{i}^{A} \cdot n_{j}^{B}) (d_{i}^{A} \cdot d_{j}^{B})$
Shen-Wagoner-Clark 沈-瓦格纳-克拉克	$(l_{i}^{A} \cdot l_{j}^{B}) (d_{i}^{A} \cdot d_{j}^{B})$ with $l_{i}^{A, B} = (n_{i}^{A, B} \times n_{Γ}) / \| n_{i}^{A, B} \times n_{Γ} \|$
Lee-Robertson-Birnbaum	$(l_{i}^{A} \cdot l_{j}^{B})$ with $l_{i}^{A, B} = (n_{i}^{A, B} \times n_{Γ}) / \| n_{i}^{A, B} \times n_{Γ} \|$

The cumulative distribution functions of

N_{\max}^{A, B}

are computed for both uncracked and cracked GB, and the results are shown on Fig. 9. For Livingston-Chalmers model (Fig. 9a), no significant difference between cracked and uncracked GB is observed, whereas a slight difference appears for Lee-Robertson-Birnbaum model (Fig. 9d). Significant differences are found for the Luster-Morris and Shen-Wagoner-Clark models (Fig. 9b,c), with lower values of

N_{\max}^{A, B}

for cracked GB. Luster-Morris and Shen-Wagoner-Clark models differ only by the fact that the former used the slip plane normals, while the latter the cross-product of the slip plane normal by the GB normal. The differences between cracked and uncracked GB for both models are rather similar, supporting the fact that GB normals may not be a key ingredient of the correlation with cracking. Thus, only Luster-Morris model is considered in the following.
计算了

N_{\max}^{A, B}

在未开裂和开裂晶界的累积分布函数，结果如图 9 所示。对于 Livingston-Chalmers 模型（图 9a），未观察到开裂和未开裂晶界之间存在显著差异，而 Lee-Robertson-Birnbaum 模型（图 9d）则出现轻微差异。Luster-Morris 和 Shen-Wagoner-Clark 模型（图 9b,c）存在显著差异，开裂晶界的

N_{\max}^{A, B}

值较低。Luster-Morris 和 Shen-Wagoner-Clark 模型的不同之处仅在于前者使用了滑移面法线，而后者使用了滑移面法线与晶界法线的叉积。两种模型的开裂和未开裂晶界之间的差异相当相似，支持晶界法线可能不是开裂相关性中的关键因素。因此，以下仅考虑 Luster-Morris 模型。

2.3.3. Intergranular cracking correlations
2.3.3. 晶间开裂相关性

The previous sections have shown correlations between intergranular cracking and both GB normal well-oriented with respect to the loading axis and low values of Luster-Morris parameters. These two criteria may be related to higher local intergranular normal stresses, which thus appear to play a major role for IGSCC of irradiated austenitic stainless steels in PWR environment. A correlation using both GB normal and Luster-Morris parameter is assessed in the following, defined as:
前几节已经展示了晶间开裂与晶界法线良好取向（相对于加载轴）以及 Luster-Morris 参数低值之间的相关性。这两个标准可能与更高的局部晶间法向应力有关，因此似乎在 PWR 环境下辐照奥氏体不锈钢的 IGSCC 中起着重要作用。以下评估了同时使用晶界法线和 Luster-Morris 参数的相关性，定义为：(2)

{\begin{matrix} | n_{x} | \geq α_{n} \\ N_{m a x}^{Luster - Morris} \leq α_{N} \end{matrix}

The percentage of GB that fulfil Eq. 2 as a function of the parameters

{α_{n}; α_{N}}

are plotted on Fig. 10a for uncracked GB for reference to be compared to the case of cracked GB in Fig. 10b. A significant difference is observed between the two distributions, which can be highlighted by looking at the difference between them, which is done on Fig. 10c. It is found that, for

{α_{n}; α_{N}} = {0.6; 0.95},

Eq. 2 is fulfilled by about 65% of the cracked GB, while only by 15% of the uncracked GB. Interestingly, the threshold value for the Luster-Morris parameter is similair to the one used in [33] and based on experimental observations of slip transmission. These results make clear that local mechanical state at GB is a factor influencing IGSCC of irradiated austenitic stainless in PWR environment, but is definitely not a sufficient condition for cracking as 35% of the cracked GBs do not fulfil Eq. 2. This has already been pointed out in early studies [20] and will be discussed in Section 4.
满足公式 2 的 GB 百分比随参数

{α_{n}; α_{N}}

的变化关系如图 10a 所示，用于参考未开裂 GB 的情况，以便与图 10b 中开裂 GB 的情况进行比较。观察两个分布之间存在显著差异，这一点可以通过查看它们之间的差异来突出显示，该差异在图 10c 中进行了展示。研究发现，对于

{α_{n}; α_{N}} = {0.6; 0.95},

，约 65%的开裂 GB 满足公式 2，而只有 15%的未开裂 GB 满足。有趣的是，Luster-Morris 参数的阈值与文献[33]中基于滑移传递的实验观察所使用的阈值相似。这些结果表明，界面的局部力学状态是影响 PWR 环境中辐照奥氏体不锈钢的 IGSCC 的一个因素，但绝不是开裂的充分条件，因为 35%的开裂 GB 不满足公式 2。这一点在早期研究[20]中已经指出，将在第 4 节中讨论。

In order to understand in more details the physical origin of the correlation observed between cracking and low values of the Luster-Morris parameter, additional correlations are assessed. As previous studies have found correlations between intergranular cracking and low values of Schmid factor on one side of the GB [20] based on the argument that such situation leads to higher stresses, the cdf of minimal Schmid factor (over the two grains forming the GB) has been computed and shown in Fig. 11a. The minimal Schmid factor for cracked GB appears to be also lower for these experimental data, although the difference is weak. More importantly, misorientation angle between grains defining GBs is computed and distributions are shown on Fig. 11b. For uncracked GB, about 60% have a misorientation angle close to 60°, which corresponds to

Σ_{3}

twin boundary. For cracked GB, this proportion is considerably lower, about 20%. This result is consistent with previous studies showing that special GB, and especially

Σ_{3},

are less susceptible to IGSCC in austenitic stainless steels [46], [47]. Based on this result, correlations between cracking and well-oriented GB (Figs. 8a) / low values of the Luster-Morris parameter (Fig. 9b) are reassessed by considering only non

Σ_{3}

GB, assuming that

Σ_{3}

GB are less susceptible to cracking. Distributions of GB normal component

n_{x}

(Fig. 12a) are similar to the ones obtained considering all GB (Fig. 8a). The distribution of Luster-Morris parameter for uncracked GB (Fig. 12b) is completely different to the one shown in Fig. 9b, while the distribution for cracked GB is weakly affected. This is due to the fact that Luster-Morris parameter

N_{\max}^{A, B} ≃ 1

for

Σ_{3}

GB. However, cracked GB still exhibit lower values.
为了更详细地了解裂纹与 Luster-Morris 参数低值之间观察到的相关性，评估了额外的相关性。正如先前研究基于这种情况会导致更高应力的论点，发现晶界[20]一侧的晶间裂纹与 Schmid 因子低值之间存在相关性。已计算并示于图 11a 的最小 Schmid 因子的累积分布函数。对于这些实验数据，裂纹晶界的最小 Schmid 因子似乎也较低，尽管差异很小。更重要的是，计算了定义晶界晶粒的取向角，并在图 11b 中显示了分布。对于未开裂的晶界，约 60%的取向角接近 60°，这对应于

Σ_{3}

孪晶界。对于开裂的晶界，这一比例要低得多，约为 20%。这一结果与先前研究一致，这些研究表明特殊晶界，尤其是

Σ_{3},

，在奥氏体不锈钢中不易发生晶间应力腐蚀开裂[46],[47]。基于这一结果，裂纹与良好取向晶界之间的相关性（如图所示） 8a) / 低 Luster-Morris 参数值（图 9b）通过仅考虑非

Σ_{3}

GB 并假设

Σ_{3}

GB 对裂纹的敏感性较低而重新评估。GB 法向分量

n_{x}

（图 12a）的分布与考虑所有 GB（图 8a）获得的分布相似。未开裂 GB 的 Luster-Morris 参数分布（图 12b）与图 9b 中所示完全不同，而开裂 GB 的分布受影响较弱。这是由于 Luster-Morris 参数

N_{\max}^{A, B} ≃ 1

对

Σ_{3}

GB。然而，开裂 GB 仍然表现出较低值。

The results shown in Fig. 12b indicate that the correlation between cracking and low values of the Luster-Morris parameter (Fig. 9b) has (at least) two different origins. The first one comes from the overrepresentation of

Σ_{3}

type GB on uncracked GB compared to cracked GB. Restricting to non

Σ_{3}

GB, significant statistical differences are still observed on Fig. 12b, that are attributed to the effect of slip discontinuity at GBs on cracking. As stated before, additional studies are still required to assess if Luster-Morris criterion allows effectively to predict slip (dis)-continuity at GBs in irradiated austenitic steels. Note that such assessment has been done for Aluminum [45], leading to good agreement with experimental observations associated with some additional criteria. Nevertheless, the experimental results obtained in this study confirm that IGSCC cracking of irradiated austenitic steels occurs preferentially on well oriented non

Σ_{3}

GB, as already shown in previous studies, but also on GB that have low values of Luster-Morris parameter (Fig. 12b), which is a new observation to the authors’ knowledge.
图 12b 所示结果表明，裂纹与 Luster-Morris 参数低值（图 9b）之间的相关性至少有两种不同的来源。第一种来源是未开裂晶界相对于开裂晶界中

Σ_{3}

型晶界的过代表现。仅限于非

Σ_{3}

型晶界时，图 12b 中仍观察到显著的统计差异，这归因于晶界处滑移不连续性的影响。如前所述，仍需进一步研究以评估 Luster-Morris 准则是否能够有效预测辐照奥氏体钢中晶界处的滑移（不）连续性。注意，此类评估已在铝中完成[45]，并与某些附加准则相关的实验观察结果吻合良好。然而，本研究获得的实验结果表明，辐照奥氏体钢的 IGSCC 裂纹优先发生在取向良好的非

Σ_{3}

型晶界上，这与先前研究的结果一致，但也发生在 Luster-Morris 参数值低的晶界上（图 12b），据作者所知，这是一个新的发现。

3. Numerical assessment 3. 数值评估

The micromechanical analysis performed in the previous section has shown that intergranular cracking of irradiated austenitic stainless in PWR environment is related to the local intergranular (normal) stresses, although they are neither sufficient nor necessary for cracking. In this section, numerical simulations based on the 3D microstructure obtained in the experimental part are performed to assess the strengths and weaknesses of such micromechanical modelling approach, as well as to provide directions for further studies.
前文进行的微观力学分析表明，在 PWR 环境中辐照奥氏体不锈钢的晶间开裂与局部晶间（法向）应力有关，尽管这些应力既不是开裂的充分条件也不是必要条件。在本节中，基于实验部分获得的 3D 微观结构，进行了数值模拟，以评估这种微观力学建模方法的优缺点，并为进一步研究提供方向。

3.1. Constitutive equations
3.1. 本构方程

Physically based Crystal Plasticity (CP) constitutive equations are widely used to describe plasticity of single crystals, accounting for the anisotropy induced by the slip systems and hardening mechanisms [32]. Simulations on polycrystalline aggregates using CP equations allow to estimate the homogenized behavior, or to evaluate intra- and inter-granular stresses, as for example done in [25], [26]. Dedicated physically-based crystal plasticity constitutive equations have been developed for irradiated materials, accounting for the presence of irradiation defects [48], [49], [50]. The key ingredients of such models are the influence of irradiation defects on hardening as well as the removal of irradiation defects by dislocations. Few constitutive equations have been proposed for irradiated austenitic stainless steels irradiated in PWR conditions [26], [51]. After calibration, these models reproduce the evolution of the polycrystalline behavior with irradiation. In the following, the model described in [26] is used. The three main sets of equations of the model are described below. For each slip system

α

corresponding to a slip direction

m^{α}

and a slip plane of normal

n^{α}

(defining a Schmid tensor

M^{α} = m^{α} \otimes n^{α}

), the evolution of shear strain

{\dot{γ}}^{α}

is given by:
基于物理的晶体塑性（CP）本构方程被广泛用于描述单晶的塑性，考虑了滑移系和硬化机制引起的各向异性[32]。使用 CP 方程对多晶聚集体进行模拟，可以估算均匀化行为，或评估晶内和晶间应力，例如文献[25]、[26]中所述。针对辐照材料，已开发出专门的基于物理的晶体塑性本构方程，考虑了辐照缺陷的存在[48]、[49]、[50]。此类模型的关键要素是辐照缺陷对硬化的影响以及位错对辐照缺陷的清除。针对在 PWR 条件下辐照的奥氏体不锈钢，提出的本构方程较少[26]、[51]。经过标定，这些模型能够再现多晶行为随辐照的演变。以下将使用文献[26]中描述的模型。该模型的主要三组方程如下所述。对于每个滑移系统

α

，对应滑移方向

m^{α}

和法向滑移面

n^{α}

（定义 Schmid 张量

M^{α} = m^{α} \otimes n^{α}

），剪切应变

{\dot{γ}}^{α}

的演化由下式给出：(3)

{\dot{γ}}^{α} = {〈 \frac{| τ^{α} | - τ_{c}^{α}}{K_{0}} 〉}^{n} sign (τ^{α}), with 〈 x 〉 = {\begin{matrix} x & ; x > 0 \\ 0 & ; x \leq 0 \end{matrix}

where

τ^{α} = σ : M^{α}

and

τ_{c}^{α}

are the resolved shear stress and the Critical Resolved Shear Stress (CRSS), respectively, and

σ

the Cauchy stress tensor. For FCC materials, 12 slip systems

< 111 > (110)

are considered. The viscoplastic regularization through the parameters

K_{0}

and

n

is used for numerical reasons, and chosen such as to have a nearly time-independent response. The evolution of the CRSS is given by:
其中

τ^{α} = σ : M^{α}

和

τ_{c}^{α}

分别为 resolved shear stress 和 Critical Resolved Shear Stress (CRSS)，

σ

为 Cauchy 应力张量。对于 FCC 材料，考虑 12 个滑移系统

< 111 > (110)

。通过参数

K_{0}

和

n

的粘塑性正则化用于数值原因，并选择使其具有近似时间不变的响应。CRSS 的演化由下式给出：(4)

τ_{c}^{α} = τ_{0} + τ_{a} \exp (- \frac{| γ^{α} |}{γ_{0}}) + μ \sqrt{\sum_{β = 1}^{12} a^{α β} r_{D}^{β}} + μ α_{L} \sqrt{\sum_{p = 1}^{4} r_{L}^{p}}

where

τ_{0}

is an effective lattice friction stress that may account for Hall-Petch effects when calibrated with respect to polycrystalline aggregates simulations. A so-called avalanche term introducing a softening after yielding is set by the parameters

τ_{a}

and

γ_{0}

r_{D}^{α}

is a normalized dislocation density in slip system

α

(normalization factor

b_{D}^{2},

with Burgers vector

b_{D} = 2.54 10^{- 10}

m),

μ

and

a^{α β}

are respectively the macroscopic shear modulus and

12 \times 12

matrix (with 6 independent parameters) of long-range interactions between dislocations.

r_{L}^{p}

is a normalized Frank loop density in slip plane

p

(normalization factor

b_{L}^{2} ϕ_{L},

with Burgers vector

b_{L} = 2.08 \times 10^{- 10}

m),

ϕ_{L}

the mean size of Frank loops that depends on irradiation level and

α_{L}

sets the relative contribution of Frank loops to hardening. The evolutions of dislocation density and Frank loops density are:
其中

τ_{0}

是一个有效晶格摩擦应力，当根据多晶聚合体模拟进行校准时，它可以解释霍尔-佩奇效应。一个引入屈服后软化的所谓雪崩项由参数

τ_{a}

和

γ_{0}

设定。

r_{D}^{α}

是滑移系统

α

中的归一化位错密度（归一化因子

b_{D}^{2},

，布氏矢量

b_{D} = 2.54 10^{- 10}

m），

μ

和

a^{α β}

分别是宏观剪切模量和位错之间长程相互作用的矩阵（具有 6 个独立参数）。

r_{L}^{p}

是滑移面

p

中的归一化弗兰克回线密度（归一化因子

b_{L}^{2} ϕ_{L},

，布氏矢量

b_{L} = 2.08 \times 10^{- 10}

m），

ϕ_{L}

是弗兰克回线的平均尺寸，它取决于辐照水平，

α_{L}

设定了弗兰克回线对强化的相对贡献。位错密度和弗兰克回线密度的演化如下：(5)

\begin{matrix} {\dot{r}}_{D}^{α} & = (\frac{1}{κ} \sqrt{\sum_{β = 1}^{12} b^{α β} r_{D}^{β}} + \frac{1}{κ} \sqrt{K_{d l} \sum_{p = 1}^{4} r_{L}^{p}} - G_{c} r_{D}^{α}) | {\dot{γ}}^{α} | \\ {\dot{r}}_{L}^{p} & = - A_{L} (r_{L}^{p} - r_{L}^{s a t}) (\sum_{α \in plane p}^{3} r_{D}^{α}) (\sum_{α \in plane p}^{3} | {\dot{γ}}^{α} |) \end{matrix}

where

b^{α β}

is a matrix of interactions between dislocations, being of the same shape as

a^{α β}

. Parameters

κ

and

G_{c}

set the multiplication and annihilation mechanisms, respectively. The irradiation effects are modelled by adding a term to the multiplication part, with

K_{d l}

being a coefficient of effective interaction between dislocations and Frank loops. The evolution of Frank loops depends on the parameter

A_{L}

which is the annihilation dimensionless area (rescaling factor

b_{L}^{3} / ϕ_{L}

) of Frank loops and

r_{L}^{s a t}

is a stabilized value of normalized defect density which depends on the irradiation dose. Since removal of Frank loops by mobile dislocations occurs only within the plane of the loop, only slipping in this plane can contribute to the evolution of defect density (

α \in plane p

). Anisotropic elasticity is finally considered, with non-zero parameters of the elastic fourth order tensor

C_{11} = C_{22} = C_{33},

C_{12} = C_{13} = C_{23}

and

C_{44} = C_{55} = C_{66}

in Voigt notations.
其中

b^{α β}

是位错间相互作用的矩阵，形状与

a^{α β}

相同。参数

κ

和

G_{c}

分别设定了乘积机制和湮灭机制。辐照效应通过在乘积部分添加一项来建模，其中

K_{d l}

是位错与弗兰克环之间有效相互作用的系数。弗兰克环的演化取决于参数

A_{L}

，它是弗兰克环的湮灭无量纲面积（缩放因子

b_{L}^{3} / ϕ_{L}

），而

r_{L}^{s a t}

是归一化缺陷密度的稳定值，该值取决于辐照剂量。由于移动位错移除弗兰克环仅发生在环的平面内，因此只有该平面内的滑移才能对缺陷密度的演化（

α \in plane p

）做出贡献。最后考虑各向异性弹性，Voigt 记号中弹性四阶张量的非零参数

C_{11} = C_{22} = C_{33},

C_{12} = C_{13} = C_{23}

和

C_{44} = C_{55} = C_{66}

。

Slip transmission at GB has been shown to be a key ingredient in IGSCC of irradiated austenitic stainless steels. The data obtained in this study supports this statement through the correlation observed between low values of the Luster-Morris parameter - associated with less slip transmission - and cracking (Fig. 12b). Using the constitutive equations detailed above in polycrystalline aggregates simulations without any additional modelling corresponds to the hypothesis of fully-transparent GB. Intergranular stresses will result only from grain-grain interactions, which is clearly a limitation. The model proposed recently in [33] considers a simple modification of the evolution law for dislocation density as:
在晶界处的滑移传递已被证明是辐照奥氏体不锈钢发生晶间应力腐蚀开裂的关键因素。本研究获得的数据通过观察 Luster-Morris 参数（与滑移传递较少相关）的低值与开裂之间的相关性，支持这一观点（图 12b）。使用上述详细给出的本构方程在多晶聚集体模拟中，不进行任何额外的建模，对应于完全透明晶界的假设。晶间应力将仅由晶粒间的相互作用产生，这显然是一个限制。最近在[33]中提出的模型考虑了对位错密度演化定律的简单修改，如下所示：(6)

{\dot{r}}_{D}^{α} = (\max [\frac{1}{κ} \sqrt{\sum_{β = 1}^{12} b^{α β} r_{D}^{β}}, \frac{K_{s}^{α}}{d_{b}^{α}}] + \frac{1}{κ} \sqrt{K_{d l} \sum_{p = 1}^{4} r_{L}^{p}} - G_{c} r_{D}^{α}) | {\dot{γ}}^{α} |

where

K_{s}^{α}

is an additional parameter and

d_{b}^{α}

is the normalized (with

b_{D}

) minimal distance to a GB along slip system

α

. The parameter

K_{s}^{α}

depends on the value of the Luster-Morris parameter such as:
其中

K_{s}^{α}

是一个附加参数，

d_{b}^{α}

是沿滑移系统

α

到晶界的归一化（用

b_{D}

）最小距离。参数

K_{s}^{α}

取决于 Luster-Morris 参数的值，例如：(7)

K_{s}^{α} = {\begin{matrix} K_{s}^{0} & if \forall β N_{α, β} < α_{N} \\ 0 & otherwise \end{matrix}

where

β

corresponds to the slip systems of the closest grain (at a distance

d_{b}^{α}

). Far from the GB, Eq. 6 and 7 tend to Eq. 5, i.e. no effect of GB on the local mechanical behavior. Close to the GB, a strong increase of the density of dislocations is obtained in slip systems where slip transmission is unlikely, leading to a strong hardening through Eq. 4. The parameters of the constitutive equations are taken from [26] for a dose of 2 dpa, consistent with the average dose of the irradiated layer of the sample (Fig. 1a). The parameter

K_{s}^{0}

was set to

K_{s}^{0} = 5

in [33].

K_{s}^{0} = 0

is also used to recover Eq. 5. From Eq. 6, it can be noticed that the modelling of GB has an effect for:

β

对应于最近晶粒的滑移系统（距离

d_{b}^{α}

）。远离晶界（GB）时，公式 6 和 7 趋向于公式 5，即晶界对局部力学行为无影响。靠近晶界时，在滑移传递不太可能的滑移系统中，位错密度显著增加，通过公式 4 导致强烈的硬化。本构方程的参数取自文献 [26]，对应 2 dpa 的剂量，与样品辐照层（图 1a）的平均剂量一致。参数

K_{s}^{0}

在文献 [33] 中被设置为

K_{s}^{0} = 5

。

K_{s}^{0} = 0

也用于恢复公式 5。从公式 6 可以看出，晶界建模对以下方面有影响：(8)

d_{b}^{α} \leq K_{s}^{0} κ {(\sqrt{\sum_{β = 1}^{12} b^{α β} r_{D}^{β}})}^{- 1}

thus the discretization used in simulations should be finer to have any effects. All the parameters are summarized in Table 2.
因此，模拟中使用的离散化应该更精细才能产生任何效果。所有参数汇总于表 2。

Table 2. Parameters of the crystal plasticity constitutive equations.
表 2. 晶体塑性本构方程的参数。

$C_{11}$ (GPa)	$C_{12}$ (GPa)	$C_{44}$ (GPa)	$n$	$K_{0}$ (MPa)	$τ_{0}$ (MPa)	$μ$ (GPa)	$κ$	$G_{c}$	$r_{D}^{0}$
199	136	105	15	10	88	65.6	42.8	10.4	$3.66 10^{- 11}$
$α_{L}$	$K_{d l}$	$A_{L}$	$r_{L}^{s a t}$	$r_{L}^{0}$	$τ_{a}$	$γ_{0}$	$K_{s}^{0}$
0.44	$0.25 10^{- 6}$	$4.48 10^{8}$	$3.78 10^{- 6}$	$4.72 10^{- 6}$	50	$5 10^{- 3}$	[0,5]

The constitutive equations have been implemented in the MFront code generator [52] under finite strain settings using a fully implicit integration scheme solved by a Newton-Raphson algorithm. Details about the numerical implementation can be found in [26].
本构方程已在 MFront 代码生成器[52]中采用完全隐式积分方案，通过牛顿-拉夫逊算法在有限应变设置下实现。关于数值实现的详细信息，请参见[26]。

3.2. FFT simulations 3.2. 快速傅里叶变换模拟

The numerical simulations have been performed using the 3D reconstructed microstructure obtained experimentally (Fig. 13a), using the Fast Fourier Transform (FFT) solver AMITEX_FFTP [53]. The FFT method relies on regular grids to compute the mechanical equilibrium based on Lippmann-Schwinger equations [54].
数值模拟是使用实验获得的 3D 重建微观结构（图 13a）进行的，采用了快速傅里叶变换（FFT）求解器 AMITEX_FFTP [53]。FFT 方法基于规则的网格，根据 Lippmann-Schwinger 方程[54]计算力学平衡。

The mechanical behavior of all voxels corresponds to the constitutive equations described in the previous section. The parameters of these equations are summarized in Table 2, and the crystallographic orientation of each voxel is given by the experimental values. Only the irradiated layer is considered, with constant materials parameters, thus neglecting the effect of the dose profile along the thickness. The distance

d_{b}^{α}

in Eq. 6 is shown in Fig. 13b, using the simplification that the distance to the nearest GB is computed in each plane separately. For each voxel and slip system

α,

the Luster-Morris parameters

N_{α β}

are computed, where

β

corresponds to the slip system of the adjacent grain, and Eq. 7 is used to define

K_{S}^{α}

. FFT methods only allow to apply macroscopic (volume-average) strain / stress (or mixed) loading conditions on periodic structures. Uniaxial stress loading conditions are applied as:
所有体素（voxels）的力学行为对应于上一节描述的本构方程。这些方程的参数汇总于表 2，每个体素的晶体学取向由实验值给出。仅考虑辐照层，其材料参数为常数，因此忽略了沿厚度方向的剂量分布效应。方程 6 中的距离

d_{b}^{α}

在图 13b 中显示，采用简化方法，即在每个平面上分别计算到最近晶界（GB）的距离。对于每个体素和滑移系统

α,

，计算 Luster-Morris 参数

N_{α β}

，其中

β

对应于相邻晶粒的滑移系统，并使用方程 7 定义

K_{S}^{α}

。FFT 方法仅允许对周期性结构施加宏观（体积平均）应变/应力（或混合）加载条件。单轴应力加载条件施加如下：(9)

\frac{1}{V} \int_{V} σ d V = Σ_{x x} e_{x} \otimes e_{x}

where the x-axis corresponds to the loading axis during the SSRT. Due to the fact that the microstructure is not periodic along the in-plane directions, such condition might create some artefacts close to the boundary. In addition, to simulate the free-surface effect, a stress-free layer of one voxel is added at the surface. Macroscopic deformation gradient

F_{x x}

is applied along the

x

axis up to 4% strain, as in the SSRT. All simulations have been performed using the full 3D microstructure

(800 μ m \times 800 μ m \times 20 μ m)

with an in-plane discretization of 2 μm and through thickness discretization of 0.5 μm, corresponding to 6M voxels. Additional simulations presented in Appendix C have shown that coarser discretizations or considering a larger thickness does not affect the conclusions presented hereafter. Simulations have been performed using

K_{s}^{0} \in [0; 5]

to evaluate the effect of slip transmission modelling on GB local stresses. The first case corresponds to fully transparent GB, the last to the value used in [33].
其中 x 轴对应 SSRT 期间的加载轴。由于微观结构在平面方向上不是周期性的，这种条件可能在边界附近产生一些伪影。此外，为了模拟自由表面效应，在表面添加了一层无应力的 1 个体素层。宏观变形梯度

F_{x x}

沿

x

轴施加，应变高达 4%，与 SSRT 相同。所有模拟均使用完整的 3D 微观结构

(800 μ m \times 800 μ m \times 20 μ m)

进行，平面方向离散化间距为 2μm，厚度方向离散化间距为 0.5μm，对应 6M 个体素。附录 C 中给出的额外模拟表明，更粗糙的离散化或考虑更大的厚度不会影响此处得出的结论。使用

K_{s}^{0} \in [0; 5]

进行模拟，以评估滑移传递建模对晶界局部应力的影响。第一个案例对应完全透明的晶界，最后一个案例对应[33]中使用的值。

3.3. Numerical results 3.3. 数值结果

Simulations have been performed on the Cobalt supercomputer (CCRT/CEA) providing a typical runtime of about 48h on 1624 processors for each simulation. Typical results at the free surface of the aggregate are shown on Figs. 14 and 15 for stress and strain fields, respectively. For

K_{s}^{0} = 0,

i.e., fully transparent GB, grain to grain variations of local stress (Fig. 14a) can be observed due to the different crystallographic orientations. Grain boundaries do not appear to have higher stresses than the interior of adjacent grains, consistent with the absence of any GB modelling. Interesting features appear on strain field as shown on Fig. 15a. At the scale of the aggregate, localization bands spanning multiple grains emerge, while at the grain scale, additional localization bands are present. The root cause of these localizations is crystal-scale (through the removal of irradiation defects) and aggregate-scale (as shown on Fig. 16a and discussed hereafter) softening mechanisms. Intragranular localization bands are expected to be mesh dependent, as no regularization is used in the model. Results are drastically different using GB modelling, as shown on Figs. 14b and 15 b for

K_{s}^{0} = 5

. Stresses are higher at GB than in the interior of the grains due to the absence of slip transmission and thus multiplication of dislocations close to these GB, as modelled by Eq. 6. Concurrently, strains are higher in grains’ interior than close to the boundary, as shown on Fig. 15b. Still some intragranular localization bands can be observed, but no localization at the aggregate scale.
在 Cobalt 超级计算机（CCRT/CEA）上进行了模拟，每个模拟在 1624 个处理器上提供约 48 小时的典型运行时间。在聚集体自由表面上，应力和应变场分别的典型结果如图 14 和图 15 所示。对于

K_{s}^{0} = 0,

，即完全透明的晶界，由于不同的晶体学取向，可以观察到局部应力（图 14a）的晶粒间变化。晶界似乎没有比相邻晶粒内部更高的应力，这与没有晶界建模一致。在应变场中出现了有趣的特征，如图 15a 所示。在聚集体尺度上，出现了跨越多个晶粒的局部化带，而在晶粒尺度上，还存在额外的局部化带。这些局部化的根本原因是晶体尺度（通过移除辐照缺陷）和聚集体尺度（如图 16a 所示，并将在下文中讨论）的软化机制。晶粒内局部化带预计与网格相关，因为模型中没有使用正则化。使用晶界建模时，结果差异很大，如图 14b 和图 15b 所示，对于

K_{s}^{0} = 5

。在晶界处的应力高于晶粒内部的应力，这是由于滑移传递的缺失，从而导致这些晶界附近的位错倍增，如式 6 所示。同时，晶粒内部的应变高于靠近边界处，如图 15b 所示。尽管仍可观察到一些晶粒内局部化带，但在聚集尺度上没有局部化现象。

The aggregate stress-strain curves are shown on Fig. 16a. For

K_{s}^{0} = 0,

an almost perfectly plastic behavior is obtained, which is consistent with a 2 dpa 304L mechanical behavior [26]. A slight softening is observed just after yielding, followed by a hardening behavior with a small hardening modulus. The overall stress magnitude is lower than the one reported in [26] using the same constitutive equations on a 3D aggregate, due to the small thickness of the aggregate used in this study where plane stress conditions prevail. As discussed before, it has been verified that decreasing the thickness does indeed change the macroscopic stress-strain curves, but not the values of the stresses at the free surface (Appendix C), which is the main interest for cracking initiation. For

K_{s}^{0} > 0,

a strong hardening behavior is observed due to the accumulation of dislocations at GB where slip transmission is unlikely [33]. As expected, the stress-strain curves for

K_{s}^{0} = 5

are not consistent with a 2 dpa austenitic stainless steels, as the other hardening parameters of the constitutive equations have been calibrated assuming

K_{s}^{0} = 0

. In order to get the desired almost perfectly plastic macroscopic behavior, a stronger softening due to the removal of irradiation defects should be considered, that will lead to more heterogeneous intragranular deformation behavior. Such calibration will be considered in future studies.
总应力-应变曲线如图 16a 所示。对于

K_{s}^{0} = 0,

，获得了几乎完美的塑性行为，这与 2 dpa 304L 的力学行为[26]一致。在屈服后观察到轻微的软化，随后呈现具有较小硬化模量的硬化行为。由于本研究中使用的集料厚度较小，平面应力条件占主导地位，因此整体应力幅度低于[26]中在 3D 集料上使用相同本构方程所报告的应力幅度。如前所述，已经验证了减小厚度确实会改变宏观应力-应变曲线，但不会改变自由表面上的应力值（附录 C），这是裂纹萌生的主要关注点。对于

K_{s}^{0} > 0,

，由于位错在晶界处累积，滑移传递不太可能，因此观察到强烈的硬化行为[33]。正如预期的那样，

K_{s}^{0} = 5

的应力-应变曲线与 2 dpa 奥氏体不锈钢不一致，因为本构方程的其他硬化参数是在假设

K_{s}^{0} = 0

的情况下校准的。为了获得理想的近似完全塑性的宏观行为，应考虑因辐照缺陷去除而导致的更强软化效应，这将导致更异质的晶内变形行为。这种校准将在未来的研究中进行。

Local stresses at the free surface along with GB locations and normals are used to compute intergranular normal stresses for a macroscopic applied strain value of

4 %

that corresponds to the maximal value at the end of the SSRT⁴ in PWR environment:
自由表面处的局部应力以及晶界位置和法线被用于计算宏观应变为

4 %

时的晶间正应力，该应变值对应于 PWR 环境下 SSRT ⁴ 结束时的最大值：(10)

σ_{n n} = n \cdot σ \cdot n

A typical result of the computation of intergranular normal stresses is shown on Fig. 16b, where the magnitude of

σ_{n n}

can be seen to strongly depend on GB positions. These results are used to compute cumulative distribution functions of

σ_{n n}

for both uncracked and cracked GB as well as to evaluate the effect of the GB modelling through the parameter

K_{s}^{0}

. As

Σ_{3}

GB have been shown in the literature and in this study to be resistant to cracking, distributions are computed only on non

Σ_{3}

GB.
晶间正应力的计算结果典型地显示在图 16b 中，其中

σ_{n n}

的幅度可以看出强烈依赖于晶界位置。这些结果被用于计算未开裂和开裂晶界的

σ_{n n}

累积分布函数，以及通过参数

K_{s}^{0}

评估晶界建模的影响。由于文献和本研究都表明

Σ_{3}

晶界具有抗开裂性，因此仅在非

Σ_{3}

晶界上计算分布。

For

K_{s}^{0} = 0,

i.e., fully transparent GB, normalized intergranular normal stress at cracked GB are found to be higher than for uncracked GB (Fig. 17a). This observation is consistent with the analysis of the experimental data shown in Fig. 8 where larger values of the normal component along the loading axis was found, which has been related to higher intergranular normal stresses. The difference is rather significant: about 60% of cracked GB have for example intergranular normal stresses

σ_{n n} / Σ_{x x} \geq 0.8,

compared to about 20% of uncracked GB. For

K_{s}^{0} = 5

(Fig. 17b), a shift towards the higher

σ_{n n}

values is observed for both uncracked and cracked GB, showing the effectiveness of such GB modelling. Differences are still observed between uncracked and cracked GB. However, for the current model, it is observed to affect similarly uncracked and cracked GB, showing that the difference pointed out in Fig. 12b is not sufficient to increase local stresses at cracked GB. The results shown on Fig. 17 indicate that cracked GB have statistically higher intergranular normal stresses, but the differences observed are not sufficient to define a cracking criterion. Accounting for GB modelling through Luster-Morris parameter, although found to correlate experimentally with cracking, does not allow to exacerbate the differences.
对于

K_{s}^{0} = 0,

，即完全透明的晶界，发现裂纹晶界处的归一化晶间法向应力高于未裂纹晶界（图 17a）。这一观察结果与图 8 所示实验数据分析一致，在该分析中，沿加载轴的法向分量值更大，这与更高的晶间法向应力相关。差异相当显著：例如，约 60%的裂纹晶界具有晶间法向应力

σ_{n n} / Σ_{x x} \geq 0.8,

，而约 20%的未裂纹晶界具有该应力。对于

K_{s}^{0} = 5

（图 17b），未裂纹和裂纹晶界均观察到向更高

σ_{n n}

值的偏移，显示了此类晶界建模的有效性。未裂纹和裂纹晶界之间仍存在差异。然而，对于当前模型，观察到其对未裂纹和裂纹晶界的影响相似，表明图 12b 中指出的差异不足以增加裂纹晶界处的局部应力。图 17 所示结果表明，裂纹晶界具有统计上更高的晶间法向应力，但观察到的差异不足以定义裂纹判据。通过 Luster-Morris 参数进行晶界建模，尽管实验上发现与裂纹相关，但并不能加剧差异。

This was somehow expected as the underlying assumption of this analysis is that all (non

Σ_{3}

) GB have the same strength, which is clearly a strong assumption. It is therefore interesting to assess the effect of considering different GB strengths on cracking predictions, based on these numerical results. Fig. 18 shows the percentage of cracking for GB defined as cracked and uncracked based on experimental observations, considering a differential GB strength

Δ {\bar{σ}}_{c}

between uncracked GB and cracked GB. A fully predictive model corresponds to a cracking fraction of 1 for cracked GB and 0 for uncracked GB. Considering a similar strength for all GB (

Δ {\bar{σ}}_{c} = 0

and

σ_{c}

set arbitrarily to the macroscopic stress), the difference between cracking fraction is small. Increasing

Δ {\bar{σ}}_{c}

leads to drastic evolutions of the cracking fractions, showing that such micromechanical approach coupled with some models that are able to predict accurately GB strength may be a promising approach.
这多少是意料之中的，因为这项分析的基本假设是所有（非

Σ_{3}

）晶界具有相同的强度，这显然是一个很强的假设。因此，基于这些数值结果，评估考虑不同晶界强度对裂纹预测的影响是有趣的。图 18 显示了根据实验观察将晶界定义为裂纹和未裂纹的百分比，考虑未裂纹晶界和裂纹晶界之间存在差异的晶界强度

Δ {\bar{σ}}_{c}

。完全预测模型对应于裂纹晶界的裂纹分数为 1，未裂纹晶界的裂纹分数为 0。考虑所有晶界具有相似强度（将

Δ {\bar{σ}}_{c} = 0

和

σ_{c}

任意设置为宏观应力），裂纹分数之间的差异很小。增加

Δ {\bar{σ}}_{c}

会导致裂纹分数发生剧烈变化，表明这种微观力学方法结合能够准确预测晶界强度的某些模型可能是一种有前景的方法。

4. Discussion 4. 讨论

The key experimental result of this study is the correlation observed between cracking initiation and the Luster-Morris slip transmission criterion. This result allows to go beyond correlations based on slip discontinuity observed after the tests, and opens the way for a micromechanical approach for IASCC that may not need to account explicitly for dislocation channelling phenomenon. The strengths and weaknesses of the current micromechanical approach are now detailed.
这项研究的关键实验结果是观察到裂纹萌生与 Luster-Morris 滑移传输准则之间的相关性。这一结果使得能够超越基于试验后观察到的滑移不连续性的相关性，并为可能无需明确考虑位错通道现象的 IGSCC 的微观力学方法开辟了道路。当前微观力学方法的优缺点现在将详细说明。

4.1. Towards a micromechanical approach for IGSCC
4.1. 迈向 IGSCC 的微观力学方法

Crystal plasticity simulations on realistic polycrystalline aggregates allow evaluating local stresses / strains fields accounting for the geometry - due to grain shapes - and anisotropy - due to the crystallographic orientations in a way which is both physical and efficient, using the ability of massively parallel FFT-based solvers to deal with large-scale simulations. This allows to account naturally for both the effects of GB normals and grain-grain interactions on intergranular stresses, which is clearly the main strength of such approach as demonstrated by the results of this study. The main weakness lies in the crystal scale constitutive equations and GB modelling. First, even if the equations used to describe hardening and evolutions of dislocations and defects are physically based, the calibration and validation of the parameters remain incomplete due to the lack of experimental results on single crystals. This is particularly true for irradiated austenitic stainless steels where only few data are available in the literature [55], [56], [57], most of them showing size effects due to the small-scale samples used. While some parameters of the models described in [26] are taken from lower-scale simulations (e.g., Discrete Dislocation Dynamics for the coefficients of the interaction matrix), others (especially those related to hardening) are calibrated against polycrystalline aggregates results, which does not ensure that the constitutive equations would lead to satisfactory results for single crystals. Secondly, although the modelling of GB effect proposed in [33] and used in this study is both physically-based and numerically efficient, it remains to be supported by dedicated experimental observations of slip transmission in irradiated stainless steels, as well as by experimental local stresses measurements. Although requiring a lot of experimental observations, these two weaknesses could be overcome in the near future. The modelling of intragranular strain localization in polycrystalline aggregates is a bigger challenge. It has been shown how the impingement of a well-defined slip band on a given GB could lead to fracture [17], but applying the same criterion requires to simulate all slip bands appearing at the crystal scale. Gradient enhanced crystal plasticity constitutive equations are needed to regularize strain localization and to ensure mesh independence in numerical simulations. Moreover, as the softening behavior is a necessary condition for strain localization, improving the calibration of constitutive equations is also necessary. Besides, although numerical tools are available in the literature to perform large-scale simulations exhibiting regularized intragranular strain localization [30], [31], as observed experimentally for irradiated austenitic stainless steels, it remains unknown how to control the distance between slip bands (although theoretical models are developed to predict such distance [58]). In the meantime, accounting for slip transmission through the model proposed in [33], once properly calibrated to affect only GB prone to cracking, is a promising mesoscopic approach to detect GB where slip continuity is unlikely and thus more prone to cracking, without having to explicitly account for dislocation channelling.
基于 FFT 的并行求解器能够处理大规模模拟，使得晶体塑性模拟在真实多晶集合体上评估局部应力/应变场成为可能，同时考虑了晶粒形状导致的几何形状和晶体学取向导致的各向异性，这种方式既物理又高效。这允许自然地考虑晶界法向和晶粒间相互作用对晶间应力的影响，这显然是这种方法的优点，正如本研究的结果所证明的那样。主要弱点在于晶体尺度的本构方程和晶界建模。首先，尽管描述位错和缺陷硬化及演化的方程基于物理原理，但由于缺乏单晶的实验结果，参数的校准和验证仍不完整。这一点对于辐照奥氏体不锈钢尤为突出，文献中可用的数据很少[55]，[56]，[57]，其中大多数数据显示出由于使用小尺寸样品而导致的尺寸效应。在[26]中描述的模型中，部分参数取自低尺度模拟（例如，用于相互作用矩阵系数的离散位错动力学），而其他参数（尤其是与硬化相关的参数）则通过与多晶聚集体结果进行校准，这并不能保证本构方程会为单晶材料带来令人满意的结果。其次，尽管[33]中提出的界面效应建模方法在本研究中被采用，该方法既基于物理原理又具有数值效率，但仍需通过专门的实验观察来支持，包括辐照不锈钢中的滑移传递实验观察以及局部应力测量的实验数据。尽管需要大量实验观察，但这两种弱点有望在不久的将来得到克服。多晶聚集体中晶粒内的应变局部化建模是一个更大的挑战。已有研究表明，一条定义明确的滑移带与特定界面的相互作用可能导致断裂[17]，但应用相同标准需要模拟晶体尺度上出现的所有滑移带。需要梯度增强晶体塑性本构方程来规范应变局部化和确保数值模拟中的网格无关性。此外，由于软化行为是应变局部化的必要条件，因此改进本构方程的标定也是必要的。除此之外，尽管文献中已有数值工具可用于执行表现出规范晶粒内应变局部化的大规模模拟[30][31]，但正如实验观察到的辐照奥氏体不锈钢的情况，如何控制滑移带之间的距离仍然未知（尽管已开发出理论模型来预测这种距离[58]）。与此同时，通过[33]中提出的模型考虑滑移传递，一旦正确标定仅影响易开裂的晶界，这是一种有前景的介观方法，用于检测滑移连续性不太可能且因此更易开裂的晶界，而无需显式考虑位错通道。

All the aforementioned strengths and weaknesses of the micromechanical approach deal with accurate predictions of local stresses. Intergranular cracking will of course depends also on GB strength that may depend on irradiation, oxidation time, GB type. The analysis of the experimental and numerical results presented in the previous sections make it clear that a high local intergranular stress is not a sufficient condition for cracking, and that differential GB strength may have an important effect on cracking prediction (Fig. 18). GB strength and intergranular oxidation are discussed in the next section.
上述所有关于微观力学方法的优点和缺点都涉及局部应力的准确预测。当然，沿晶断裂也取决于晶界强度，而晶界强度可能受辐照、氧化时间、晶界类型等因素影响。前几节中展示的实验和数值分析结果清楚地表明，高局部沿晶应力并不是断裂的充分条件，并且晶界强度的差异可能对断裂预测有重要影响（图 18）。晶界强度和沿晶氧化将在下一节讨论。

4.2. Grain boundary oxidation
4.2. 晶界氧化

The strength of a GB, defined in the previous section as the critical normal stress above which a GB will fracture, is not an intrinsic property. It should be considered as an effective property that depends on GB type - as shown for example by the fact that

Σ_{3}

GB are considered to be resistant to cracking [47] - and on the environmental embrittlement. As such, GB oxidation is expected to play a key role in IGSCC. For some Nickel-based alloys, dedicated experiments allowed to quantify intergranular oxidation in high temperature aqueous environment [59] as well as the dramatic degradation of GB strength with oxidation [60]. However, even for unirradiated austenitic stainless steels, experimental quantification of intergranular oxidation in PWR environment is scarce. In [61], TEM observations performed on several GB showed that oxidation strongly depends on the GB considered, which could explain the fact that for two GB having the same estimated local stresses, only one of them (or none) will fail due to different oxidation states. For irradiated austenitic stainless steels, quantification of intergranular oxidation in PWR environment is an identified research gap. Indirect observations are available based on the effect of GB type on irradiation-induced segregation. In particular,

Σ_{3}

GB have been shown to be resistant to segregation [62]. Dedicated experimental programs are definitely required to quantify, as a function of oxidation time, irradiation level and GB type, intergranular oxidation and correlations between cracking and microchemistry. Although experimental techniques - such as TEM on FIB samples - are available, it will require a huge amount of time, but appears unavoidable to improve the micromechanical approach of IGSCC. Regarding the effect of GB type on intergranular oxidation in austenitic stainless steel, a recently proposed model based on the definition of an Atomic Packing Density (APD) showed a promising correlation with intergranular corrosion in highly corroding environment [63]. For the experimental data obtained in this study, APD was computed for each grains, and the minimal value of the two grains adjacent to a GB was used as the GB APD. However, no difference is observed between uncracked and cracked GB, indicating that GB APD, as described in [63], may not be able to predict GB oxidation / strength of austenitic stainless steels in PWR environment.
晶界强度，在上一节中被定义为超过其发生断裂的临界正应力，并非固有属性。应将其视为一种有效属性，该属性取决于晶界类型——例如，

Σ_{3}

晶界被认为具有抗裂性能[47]——以及环境脆化作用。因此，晶界氧化预计在沿晶应力腐蚀开裂（IGSCC）中发挥关键作用。对于某些镍基合金，专门实验已能量化高温水环境中的沿晶氧化[59]以及氧化导致的晶界强度急剧下降[60]。然而，即使对于未辐照的奥氏体不锈钢，在压水堆（PWR）环境中的沿晶氧化实验量化数据仍然稀缺。在[61]中，对多个晶界进行的透射电子显微镜（TEM）观察表明，氧化程度强烈依赖于所考虑的晶界类型，这可以解释为什么对于两个具有相同估计局部应力的晶界，只有一个（或没有）会因不同的氧化状态而失效。对于辐照奥氏体不锈钢，在 PWR 环境中的沿晶氧化量化是一个已识别的研究空白。基于 GB 类型对辐照诱导偏析的影响，已有间接观察结果。特别是，

Σ_{3}

GB 已被证明具有抗偏析性[62]。确实需要专门的实验计划来量化，作为氧化时间、辐照水平和 GB 类型函数的晶间氧化以及裂纹与微观化学成分之间的相关性。尽管实验技术（如 FIB 样品的 TEM）是可用的，但这将需要大量时间，但似乎为了改进 IGSCC 的微观力学方法不可避免。关于 GB 类型对奥氏体不锈钢晶间氧化的影响，最近基于原子堆积密度（APD）定义提出的模型在高腐蚀环境中与晶间腐蚀显示出有希望的关联[63]。对于本研究获得的实验数据，计算了每个晶粒的 APD，并使用 GB 相邻的两个晶粒中的最小值作为 GB APD。然而，在未开裂和开裂的晶界之间未观察到差异，表明正如[63]中所述的晶界原子偏析可能无法预测压水堆环境中奥氏体不锈钢的晶界氧化/强度。

5. Conclusion and perspectives
5. 结论与展望

A micromechanical analysis of IGSCC of an irradiated austenitic stainless steel has been proposed, based on the reconstruction (through sequential polishing and 2D EBSD) of the 3D microstructure of a 304L proton-irradiated sample tested in PWR environment. This analysis allows to go beyond the analysis often reported in the literature based only on 2D observations, in particular with respect to the effect of GB normal and slip transmission criteria. Moreover, the large volume considered (and thus large number of cracks) has made possible a statistical analysis of the data. A correlation is observed between intergranular cracking and GB well-oriented with respect to the mechanical loading applied during the IGSCC test. In addition, evaluation of several slip transmission criteria has shown a correlation between the Luster-Morris parameter - involving only the crystallographic orientations on both sides of the GB - and cracking. Interestingly, this slip transmission criterion has been put forward for other FCC materials in recent studies showing its ability to predict slip (dis-)continuity. Micromechanical simulations based on the reconstructed 3D microstructure and crystal plasticity constitutive equations modified to account for slip transmission at GB have been performed. The two main outcomes of these simulations are first higher local stresses for cracked GB, and secondly evidence that considering differential GB strength has a strong effect on cracking predictions. In addition, GB modelling described in [33] appears numerically efficient and effective to increase local stresses where slip transmission is unlikely, and a promising mesoscopic way to account for the observed dependence of cracking on slip discontinuity without having to describe explicitly dislocation channels. These simulations allow to point out the main research gap that lies in finding the dependence of GB strength on GB type and oxidation, setting the path for future studies.
基于对在 PWR 环境下测试的 304L 质子辐照样品的三维微观结构（通过逐级抛光和二维 EBSD 重建）的分析，提出了一种辐照奥氏体不锈钢的 IGSCC 微观力学分析。这种分析超越了文献中通常仅基于二维观察的分析，特别是在晶界法向和滑移传递标准方面。此外，所考虑的大体积（因此裂纹数量多）使得对数据进行统计分析成为可能。观察到沿晶裂纹与晶界相对于 IGSCC 测试期间施加的机械载荷的优取向之间存在相关性。此外，对几种滑移传递标准的评估显示，Luster-Morris 参数（仅涉及晶界两侧的晶体学取向）与裂纹之间存在相关性。有趣的是，最近的研究中，这种滑移传递标准已被提出用于其他 FCC 材料，并显示出预测滑移（不）连续性的能力。基于重建的 3D 微观结构和考虑晶界滑移传递的晶体塑性本构方程修改的微观力学模拟已被执行。这些模拟的两个主要结果是：首先，裂纹晶界的局部应力更高；其次，证明考虑晶界强度差异对裂纹预测有显著影响。此外，文献[33]中描述的晶界建模在数值上高效且有效，能够增加滑移传递不太可能发生的局部应力，并是一种有前景的中尺度方法，用于解释裂纹与滑移不连续性依赖关系，而无需明确描述位错通道。这些模拟指出了主要的研究差距，即在于寻找晶界强度与晶界类型和氧化的依赖关系，为未来的研究指明了方向。

CRediT authorship contribution statement
CRediT 作者贡献声明

D. Liang: Investigation, Software, Formal analysis, Writing - review & editing. J. Hure: Investigation, Software, Formal analysis, Writing - original draft, Supervision, Conceptualization. A. Courcelle: Investigation, Writing - review & editing. S. El Shawish: Investigation, Software, Writing - review & editing. B. Tanguy: Conceptualization, Writing - review & editing.
D. Liang：研究、软件、形式分析、写作 - 审稿与编辑。J. Hure：研究、软件、形式分析、写作 - 初稿、指导、概念化。A. Courcelle：研究、写作 - 审稿与编辑。S. El Shawish：研究、软件、写作 - 审稿与编辑。B. Tanguy：概念化、写作 - 审稿与编辑。

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。

Acknowledgements 致谢

The authors gratefully acknowledge financial support provided by Slovenian Research Agency (grant P2-0026) and French Atomic Energy Commission.
作者们由衷感谢斯洛文尼亚研究机构（项目 P2-0026）和法国原子能委员会提供的资金支持。

Appendix A. 3D EBSD reconstruction
附录 A. 3D EBSD 重建

This section provides the technical details regarding the EBSD data corrections and 3D EBSD reconstruction. For each EBSD map, the crystallographic phases and orientations are obtained in the frame

{e_{1}, e_{2}}

shown in Fig. 19a. The Euler angles

Ψ_{i}

are defined at locations:
本节提供了有关 EBSD 数据校正和 3D EBSD 重建的技术细节。对于每个 EBSD 图像，晶体学相和取向是在图 19a 中显示的框架

{e_{1}, e_{2}}

中获得的。欧拉角

Ψ_{i}

定义在以下位置：(11)

Ψ_{i} (x, y) = Ψ_{i} (X)

Corrections of the coordinates are necessary to remove image distortions such as the positions of the indents:
需要对坐标进行校正以消除图像畸变，例如压痕的位置：(12)

X^{A} = (\begin{matrix} 0 \\ 0 \end{matrix}) X^{B} = (\begin{matrix} α_{1} \\ 0 \end{matrix}) X^{C} = (\begin{matrix} α_{2} \\ α_{3} \end{matrix}) X^{D} = (\begin{matrix} α_{4} \\ α_{5} \end{matrix})

go back to their initial positions and forming a square of size 1 (Fig. 19b):
使其回到初始位置并形成边长为 1 的正方形（图 19b）：(13)

X_{0}^{A} = (\begin{matrix} 0 \\ 0 \end{matrix}) X_{0}^{B} = (\begin{matrix} 1 \\ 0 \end{matrix}) X_{0}^{C} = (\begin{matrix} 0 \\ 1 \end{matrix}) X_{0}^{D} = (\begin{matrix} 1 \\ 1 \end{matrix})

which allows to define piecewise linear transformations in the triangles (ACD) and (ABD) and to determine the real position of the Euler angles measurements:
这允许在三角形（ACD）和（ABD）中定义分段线性变换，并确定欧拉角测量的实际位置：(14)

X_{0} = M \cdot X Ψ_{i} (X_{0}) = Ψ_{i} (M \cdot X)

The coefficients

α_{i}

corresponding to the positions of the Vickers indents are measured on SEM images. However, as shown on Fig. 20a, applying separately corrections to each 2D EBSD map does not prevent for 3D reconstruction artefacts. Corrections of the positions of each Vickers indents

Δ α_{i}

are thus allowed, and determined through maximizing the cross-correlation of the first Euler angles

ϕ_{1}

between the current EBSD map and the previous one:
在 SEM 图像上测量与维氏压痕位置对应的系数

α_{i}

。然而，如图 20a 所示，对每个 2D EBSD 图分别应用校正并不能防止 3D 重建伪影。因此允许对每个维氏压痕的位置

Δ α_{i}

进行校正，并通过最大化当前 EBSD 图与前一图之间的第一欧拉角

ϕ_{1}

的互相关来确定：(15)

Δ α_{i}^{n} = argmax (ϕ_{1}^{n} ★ ϕ_{1}^{n - 1})

where

★

denotes the convolution product. Derivative-free Nelder-Mead algorithm is used to compute Eq. 15, allowing to successfully align successive EBSD maps, as shown in Fig. 20b. All computations have been performed using Matlab software.
其中

★

表示卷积积。使用无导数的 Nelder-Mead 算法计算公式 15，成功地对连续的 EBSD 图进行对齐，如图 20b 所示。所有计算均使用 Matlab 软件完成。

The reconstruction algorithm allows to correct in-plane misalignments of successive EBSD scans, but no correction is done regarding potential out-of-plane misalignments. Fig. 21a shows the evolutions of the thickness removed measured at the four Vickers indents located at the corners of a

l = 1

mm square. The maximum absolute difference is about

Δ h = 2 μ

m, which corresponds to an angle of

θ \approx Δ h / l \approx 0 . 1^{\circ}

with respect to the initial plane. Thus, out-of-plane misalignment induced by the polishing procedure is negligible.
重建算法允许校正连续 EBSD 扫描的平面内错位，但未对潜在的平面外错位进行校正。图 21a 显示了在

l = 1

mm 正方形的四个维氏压痕处测量的去除厚度随抛光步骤的变化。最大绝对差约为

Δ h = 2 μ

m，这对应于相对于初始平面的

θ \approx Δ h / l \approx 0 . 1^{\circ}

角度。因此，抛光程序引起的平面外错位可以忽略不计。

Still, misalignment coming from the incorrect mounting of the sample in the SEM sample holder needs to be quantified. Points are selected at the surface of the reconstructed microstructure (Fig. 20b) in the middle of large grains, and the evolutions of the Euler angles along the three axis are shown on Fig. 21b. The variations along the thickness (z-axis) are of the same order as the in-plane variations, indicating that the out-of-plane misalignment is negligible.
然而，SEM 样品台上样品安装不正确导致的错位需要量化。在重构的微观结构表面（图 20b）的大晶粒中心选择点，沿三个轴的欧拉角变化如图 21b 所示。沿厚度（z 轴）的变化与平面内的变化属于同一数量级，表明面外错位可以忽略不计。

Appendix B. Detection of grain boundaries
附录 B. 晶界检测

This section provides a more detailed technical description of the approach employed in this study to reconstruct free-surface GB of a polycrystalline sample. A focus is set to the calculation of in-plane and out-of-plane GB slopes given the two EBSD images obtained on two parallel surfaces initially stacked atop each other in a specimen. The first step consists of identifying all GB voxels on a top plane grid. A voxel is defined to be a GB voxel if there is exactly one additional color recognized (within some prescribed tolerance) in its immediate neighborhood composed of four voxels (up, down, left, right neighbors). In this way, each identified GB voxel is characterized by the corresponding color pair. However, if two or more additional colors are found in the neighborhood, a voxel is defined to be a triple point voxel.
本节提供了本研究中用于重建多晶样品自由表面晶界的更详细技术描述。重点在于根据在两个最初堆叠在样品中的平行表面上获得的两个 EBSD 图像，计算平面内和面外晶界斜率。第一步包括在顶平面网格上识别所有晶界体素。如果体素在其由上、下、左、右四个体素组成的直接邻域中（在规定的容差范围内）恰好识别出一种额外的颜色，则该体素被定义为晶界体素。通过这种方式，每个识别出的晶界体素都由相应的颜色对表征。然而，如果在邻域中发现两种或更多种额外颜色，则该体素被定义为三重点体素。

In the second step, each GB voxel is assigned an in-plane GB slope by accounting for its neighborhood of same-like GB voxels (having the same color pair). The centers of these neighboring same-like voxels compose a 2D discrete object which is further used to calculate the (in-plane) axis with the smallest moment of inertia. Such an axis is taken to be the principal direction of the (complex) object surrounding the corresponding GB voxel and is therefore used here to represent also the in-plane GB slope (denoted by

k_{0}

). The above step is demonstrated in Fig. 22 for three different ranges of the neighborhood. Here, the range is characterized by parameter

N_{n} = 1, 2, 3

which counts the number of voxel layers surrounding a GB voxel.
在第二步中，每个晶界体素被分配一个平面晶界斜率，这是通过考虑其周围具有相同颜色对的同类晶界体素邻域来实现的。这些相邻同类体素的中心组成一个二维离散对象，该对象进一步用于计算（平面）惯性矩最小的轴。这样的轴被视为围绕相应晶界体素的（复杂）对象的主方向，因此在此处也用于表示平面晶界斜率（用

k_{0}

表示）。上述步骤在图 22 中针对三个不同的邻域范围进行了演示。在这里，范围由参数

N_{n} = 1, 2, 3

表征，该参数计算围绕一个晶界体素周围的体素层数。

In Fig. 23 the in-plane GB slopes are shown for all GB voxels using again three different ranges of the considered neighborhoods (

N_{n} = 1, 2, 3

). Depending on the grain size, the performance of the method varies locally for a given

N_{n}

. Generally, smoother slopes are produced for larger

N_{n},

while better recognition of smaller grains is found for smaller

N_{n}

. Here,

N_{n} = 2

or 3 seems to be the optimum choice.
图 23 显示了所有晶界体素平面晶界斜率，再次使用了三种不同的考虑邻域范围（

N_{n} = 1, 2, 3

）。根据晶粒尺寸，该方法在给定

N_{n}

时局部性能有所不同。通常，较大的

N_{n},

会产生更平滑的斜率，而较小的

N_{n}

能更好地识别小晶粒。在此，

N_{n} = 2

或 3 似乎是最佳选择。

In the third step of the method, four vertical cross sections are produced for each GB voxel on the top plane, see Fig. 24. By identifying same GB types (color pairs) on the bottom plane, the out-of-plane GB slopes

k_{i}

(

i = 1, 2, 3, 4

) can be calculated in each cross section. It is clear, moreover, that the accuracy of

k_{i}

improves with increasing distance between the two planes.
在方法的第三步中，对顶平面上的每个晶界体素生成四个垂直横截面，见图 24。通过识别底平面上的相同晶界类型（颜色对），可以在每个横截面上计算面外晶界斜率

k_{i}

(

i = 1, 2, 3, 4

)。此外，很明显，

k_{i}

的精度随着两个平面之间距离的增加而提高。

In the last step, GB normals are finally calculated given the in-plane and out-of-plane slopes of all the GB voxels. A GB normal

n,

assigned to one particular GB voxel, is calculated from the in-plane slope

k_{0}

and out-of-plane slope

k_{i},

following the definitions in Fig. 25, as (omitting the normalization)
在最后一步，根据所有晶界体素的平面外和平面内斜率，最终计算晶界法线。分配给某个特定晶界体素的晶界法线

n,

，根据图 25 中的定义，由平面内斜率

k_{0}

和平面外斜率

k_{i},

计算得出（省略归一化）(16)

\begin{matrix} n \sim (- k_{0}, 1, \frac{k_{0} - 1}{\sqrt{2} k_{1}}) \sim (- k_{0}, 1, \frac{k_{0}}{k_{2}}) \sim (- k_{0}, 1, \frac{k_{0} + 1}{\sqrt{2} k_{3}}) \\ \sim (- k_{0}, 1, \frac{1}{k_{4}}) . \end{matrix}

In principle, one

k_{i}

and

k_{0}

are enough to determine a GB normal

n

unambiguously. However, if more than one

k_{i}

is available, an average out-of-plane slope

〈 k_{4} 〉

is calculated first to reduce the error employed in the estimation of

k_{i}

. In this respect, all available

k_{i}

are first transformed to one common cross section (labeled D) to obtain
原则上，一个

k_{i}

和一个

k_{0}

就足够明确地确定晶界法线

n

。然而，如果有多个

k_{i}

可用，则首先计算平均平面外斜率

〈 k_{4} 〉

，以减少在

k_{i}

估计中使用的误差。在这方面，所有可用的

k_{i}

首先被转换到一个共同的截面（标记为 D）以获得(17)

k_{4, 1} = \frac{\sqrt{2} k_{1}}{k_{0} - 1}, k_{4, 2} = \frac{k_{2}}{k_{0}}, k_{4, 3} = \frac{\sqrt{2} k_{3}}{k_{0} + 1}, k_{4, 4} = k_{4},

using Eq. (16). The average slope

〈 k_{4} 〉

is then calculated as
使用公式(16)计算平均斜率

〈 k_{4} 〉

。(18)

〈 k_{4} 〉 = \frac{\sum_{j} \sin (\arctan (k_{4, j}))}{\sum_{j} \cos (\arctan (k_{4, j}))},

using the rule for calculating the mean of angles. Finally, a GB normal is calculated as
使用计算角度平均值的规则。最后，计算出一个 GB 法线。(19)

n \sim (- k_{0}, 1, \frac{1}{〈 k_{4} 〉})

followed by a proper normalization.
随后进行适当的归一化处理。

The effects of varying the distance between the top and bottom planes

h

and the smoothing parameter

N_{n}

on the GB normal components are shown on Figs. 26, 27. The parameter

h

affects mostly the

n_{z}

component (Fig. 26c), while the effect on the other components is weaker (Fig. 26a,b). Due to the discrete nature of the EBSD data, the accuracy of the determination of GB out-of-plane component increases with higher values of

h,

while lower values are needed to assess GB normals close to the surface. In the following,

h = 4 μ

m is chosen as a compromise, keeping in mind that the GB normal computed corresponds to an average value over a thickness

h

.
图 26、27 展示了顶底平面

h

之间距离和平滑参数

N_{n}

变化对晶界法向分量的影响。参数

h

主要影响

n_{z}

分量（图 26c），而对其他分量的影响较弱（图 26a,b）。由于 EBSD 数据的离散性，晶界平面外分量的确定精度随

h,

值增大而提高，而评估表面附近晶界法向时需要较低值。在下文中，选择

h = 4 μ

m 作为折衷值，同时考虑到计算出的晶界法向对应于厚度

h

的平均值。

The effect of the smoothing parameter

N_{n}

is more limited, as shown on Fig. 27. As a compromise between the detection of small grains and smoothing of the 2D GB,

N_{n} = 2

is chosen for the distributions reported in the paper.
平滑参数

N_{n}

的影响更为有限，如图 27 所示。为了在小晶粒检测和平滑二维 GB 之间取得平衡，论文中报告的分布选择了

N_{n} = 2

作为平滑参数。

Finally, the effect of the projection of cracks on GB is assessed in Fig. 28. As discussed in Section 2.2.2, a voxel at a position

{x, y}

of the EBSD map is considered to correspond to a crack if

\sqrt{{(x - x_{c})}^{2} + {(y - y_{c})}^{2}} \leq P Δ,

where

{x_{c}, y_{c}}

is the location of a crack determined from SEM observations,

Δ

the step used for ESBD analysis and

P

is a parameter controlling the projection process. The effect of

P

on the cumulative distribution functions is shown on Fig. 28. Due to the relatively small number of cracks, the distributions for uncracked GB are not affected by the choice of the parameter

P

. For cracked GB, higher values of

P

lead to distributions closer to the ones for uncracked GB, which is consistent with the fact that higher number of false cracked GB will be considered. The fraction of cracked GB depends on the choice of the parameter

P

and is equal to

1.6 %, 3.2 %, 4.9 %

for

P = 1, 2, 3,

respectively. In order to minimize the number of false cracked GB, the smallest value of the parameter

P = 1

is considered for all cdf presented in the main part of the manuscript. This leads to rather small fraction of cracked GB and thus to uncertainties that are evaluated by computed cdf 95% confidence bounds.
最后，图 28 评估了裂纹投影对晶界的影响。如 2.2.2 节所述，如果 EBSD 图中位置

{x, y}

的体素满足条件

\sqrt{{(x - x_{c})}^{2} + {(y - y_{c})}^{2}} \leq P Δ,

，则认为其对应裂纹，其中

{x_{c}, y_{c}}

为 SEM 观察确定的裂纹位置，

Δ

为 EBSD 分析所用的步长，

P

为控制投影过程的参数。图 28 展示了

P

对累积分布函数的影响。由于裂纹数量相对较少，未开裂晶界的分布不受参数

P

选择的影响。对于开裂晶界，

P

的更高值导致分布更接近未开裂晶界的分布，这与考虑更多假开裂晶界的事实一致。开裂晶界的比例取决于参数

P

的选择，对于

P = 1, 2, 3,

分别等于

1.6 %, 3.2 %, 4.9 %

。为了最小化假开裂晶界的数量，在论文主体部分呈现的所有 CDF 中，均采用参数

P = 1

的最小值。这导致开裂晶界的比例较小，因此通过计算 CDF 的 95%置信区间评估不确定性。

Appendix C. Convergence analysis of numerical simulations
附录 C. 数值模拟的收敛性分析

The convergence of the numerical results presented in Section 3 are assessed with respect to the aggregate size and voxel size. Figures 29a,b show the macroscopic stress-strain curves and distributions of intergranular normal stresses computed at the free surface, for the various parameters considered. Convergence of the numerical results is already achieved for an in-plane voxel size of

4 μ

m for

K_{s}^{0} = 0,

larger than the value of

2 μ

m used for all simulation results reported in Section 3. A slight effect is however noticed for

K_{s}^{0} = 5

. A more significant effect is observed on the stress-strain curves (Fig. 29a) between aggregates of thickness

20 μ

m and

10 μ

m, coming from the influence of the free surface. The key point is that the influence of the aggregate’s thickness (and in-plane voxel size) on the intergranular normal stresses at the surface is weak, as shown on Fig. 29b, justifying the convergence of the results reported in Section 3.
第 3 节中数值结果的收敛性被评估了，评估依据是总体尺寸和体素尺寸。图 29a,b 展示了在自由表面上计算的宏观应力-应变曲线和晶间法向应力分布，针对所考虑的各种参数。对于平面体素尺寸为

4 μ

m，比第 3 节中所有模拟结果所使用的

2 μ

m 的值大

K_{s}^{0} = 0,

，数值结果的收敛性已经实现。然而，对于

K_{s}^{0} = 5

有轻微的影响。在厚度为

20 μ

m 和

10 μ

m 的聚集体之间的应力-应变曲线（图 29a）上观察到更显著的影响，这是由自由表面的影响引起的。关键点在于，聚集体厚度（和平面体素尺寸）对表面晶间法向应力的影响很弱，如图 29b 所示，这证明了第 3 节中报告的结果的收敛性。

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Insights into stress corrosion cracking in scratched area of alloy 690TT steam generator tubes
合金 690TT 蒸汽发生器管材刮擦区域应力腐蚀开裂的见解
2023, Acta Materialia
To investigate the stress corrosion cracking behaviors and microscopic mechanism in a scratched area of alloy 690TT, the microstructures of typical cracks are systematically characterized. There is a severe structural mismatch at the interface between fine grain zone and severe deformation zone, and hence cracks tend to propagate continually along this interface. In addition, the scratching leads to fracture and displacement of carbides, which changes their morphologies and distribution. Carbides with small sizes are engulfed by the crack oxidation zone, while large carbides can change the propagation path, and thus the crack grows into the deformation layer.
Substructure-sensitive crystal plasticity with material-invariant parameters
具有材料不变参数的亚结构敏感晶体塑性
2022, International Journal of Plasticity
Even though crystal plasticity models have been available for decades, the quantification of material parameters is still a matter of debate. Polycrystalline experimental results can normally be reproduced by multiple sets of parameters, raising concerns about the best parameterization to predict the grain-level response. This work presents a novel physics-based crystal plasticity model based on mesoscale dislocation substructures, which are used to characterize material parameters independently. We employ a unique set of parameters with known uncertainty to reproduce the mechanical response of FCC single- and poly-crystals. We demonstrate that mesoscale parameters are material-invariant and can be used to model FCC metals with similar dislocation substructures such as for Cu, Ni and Al. Furthermore, the model is validated by comparing to experimental single- and poly-crystalline stress–strain curves and mesoscale dislocation substructure images. This novel modeling approach is intrinsically designed to predict the response of materials with similar dislocation substructures without the need of single crystal experimental data for calibration.
An experimental investigation on the effect of gas tungsten arc welding current modes upon the microstructure, mechanical, and fractography properties of welded joints of two grades of AISI 316L and AISI310S alloy metal sheets
关于气体保护钨极电弧焊电流模式对两种 AISI 316L 和 AISI310S 合金金属板材焊接接头的微观结构、力学性能和断口形貌特性的实验研究
2022, Materials Science and Engineering A
In this investigation, dissimilar welded joints of AISI 316 L and AISI 310S stainless steels were produced using continuous and pulsed modes current of the gas tungsten arc welding process. A filler metal type ER309L was used to strengthen the welded joints. The fracture mode of the tensile and Charpy impact test samples was studied using a field emission scanning electron microscope (FE-SEM). Results showed that the welded joints were broken in the 316 L steel side during the tensile test due to the presence of lower alloying elements in this steel compared with the AISI 310S stainless steel. As well, microhardness and Charpy impact tests results showed that changing the welding current from continuous to the pulsed one increased the values of these two mentioned attributes. Fractography analysis, performed on the fracture surfaces of both joints, showed a completely ductile fracture under both tensile and Charpy impact tests. Moreover, microstructural observations showed that the weld metal (WM) structure was austenitic-ferritic (AF) and contained columnar and equiaxed dendrites. Changing the welding current from the continuous to the pulsed one led to the transformation of the columnar dendrites to the very fine equiaxed dendrites. This welding current variation reduced the dendrite size of the WM and decreased the area of the unmixed zone (UMZ). Finally, XRD results indicated that austenite was the predominant phase in the welded joints.
On the effect of slip transfer at grain boundaries on the strength of FCC polycrystals
关于晶界滑移转移对 FCC 多晶材料强度的影响
2022, European Journal of Mechanics A Solids
The effect of slip transfer on the flow strength of various FCC polycrystals was analyzed by means of computational homogenization of a representative volume element of the microstructure. The crystal behavior was governed by a physically-based crystal plasticity model in the framework of finite strains where slip transfer at grain boundaries was allowed between slip systems suitably oriented according to geometrical criteria. Conversely, slip transfer was blocked if the conditions for slip transfer were not fulfilled, leading to the formation of dislocation pile-ups. All the model parameters for each material were identified from either dislocation dynamics simulations or experimental data from the literature. Slip transfer led to a reduction in the flow stress of the polycrystals (as compared with the simulations with opaque grain boundaries) which was dependent on the fraction of translucent and transparent grain boundaries in the microstructure. Moreover, dislocation densities and Von Mises stresses were much higher around opaque grain boundaries, which become suitable places for damage nucleation. Finally, predictions of the Hall–Petch effect in Al, Ni, Cu and Ag polycrystals including slip transfer were in better agreement with the literature results, as compared with predictions assuming that all grain boundaries are opaque, particularly for small grain sizes ( $<$ $20 μ m$ ).
The effect of pulse current changes in PCGTAW on microstructural evolution, drastic improvement in mechanical properties, and fracture mode of dissimilar welded joint of AISI 316L-AISI 310S stainless steels
脉冲电流变化对 PCGTAW 对 AISI 316L-AISI 310S 不锈钢异种焊接接头微观结构演变、机械性能显著改善及断裂模式的影响
2021, Materials Science and Engineering A
2021, 材料科学与工程 A
In this paper, the effect of pulse current changes in the Pulsed Current Gas Tungsten Arc Welding (PCGTAW) on the various properties of dissimilar welding of AISI 316L-AISI 310S stainless steels was investigated. 10 mm thickness steel sheets were joined by the PCGTAW process with the background current (I_b) of 55, 70 and 85A, as well as the peak current (I_p) of 110, 130 and 150A. Then, optical microscopy (OM) and Field Emission Scanning Electron Microscopy (FE-SEM) techniques were used to study the microstructural evolution in different areas of the welded joints. Also, tensile, Charpy impact and Vickers microhardness tests were used to evaluate the effect of the pulsed current changes on mechanical properties. Finally, the fracture surfaces of Charpy impact and tensile tests samples were studied by FE-SEM. The weld metal (WM) microstructure consisted of austenite dendrites together with a low amount of delta ferrite in the grain-boundaries. Results also showed that by increasing I_b and decreasing I_p, the microstructure of the WM was changed from columnar dendritic to coaxial, and a very fine, dendritic one. This condition led to the reduction of the size of the dendrites and the amount of delta ferrite ingrain boundaries of the WM, as well as a reduction in the width of Unmixed Zone (UMZ) area. Moreover, all the welded joints were fractured from the AISI 316L stainless steels side. However, the results of the Charpy impact and microhardness tests showed that with the above variation in the welding parameters, hardness value and fracture energy of the WM increased significantly. Fractography of the surfaces showed a completely ductile fracture for both tensile and Charpy impact tests samples.
Insights into Plastic Localization by Crystallographic Slip from Emerging Experimental and Numerical Approaches
基于晶体学滑移的塑性局部化研究：新兴实验与数值方法的见解
2023, Annual Review of Materials Research
2023, 《材料研究年度评论》

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¹: As the depth of the initial Vickers indents is smaller than the total thickness removed, new indents are regularly performed at the same locations than the previous ones.
由于初始维氏压痕的深度小于去除的总厚度，因此会在与先前相同的位置定期进行新的压痕。

²: as implemented in the Matlab built-in function ecdf
如 Matlab 内置函数 ecdf 中实现的那样

³: The averaging is performed using the first neighbors, thus on a disk of 2 $μ m$ radius.
平均处理使用的是最近邻，因此是在半径为 2 的圆盘上进行。

⁴: Analysis of the numerical data at other timesteps does not change the results reported hereafter.
对其他时间步的数值数据的分析不会改变此处报告的结果。

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Outline 概述

Cited by (31) 被引用 (31)

Figures (30) 图 (30)

Tables (2) 表格（2）

Acta Materialia

Abstract 摘要

Graphical abstract 图示摘要

Keywords 关键词

1. Introduction 1. 引言

2. Experimental characterization
2. 实验表征

2.1. Material 2.1. 材料

2.2. Microstructure 2.2. 微结构

2.2.1. 3D EBSD

2.2.2. Intergranular cracks
2.2.2. 晶间裂纹

2.2.3. Grain boundaries 3D characterization
2.2.3. 晶界三维表征

2.3. Analysis of experimental data
2.3. 分析实验数据

2.3.1. GB normals 2.3.1. 晶界法线

2.3.2. Slip transmission criteria
2.3.2. 滑移传递准则

2.3.3. Intergranular cracking correlations
2.3.3. 晶间开裂相关性

3. Numerical assessment 3. 数值评估

3.1. Constitutive equations
3.1. 本构方程

3.2. FFT simulations 3.2. 快速傅里叶变换模拟

3.3. Numerical results 3.3. 数值结果

4. Discussion 4. 讨论

4.1. Towards a micromechanical approach for IGSCC
4.1. 迈向 IGSCC 的微观力学方法

4.2. Grain boundary oxidation
4.2. 晶界氧化

5. Conclusion and perspectives
5. 结论与展望

CRediT authorship contribution statement
CRediT 作者贡献声明

Declaration of Competing Interest
利益冲突声明

Acknowledgements 致谢

Appendix A. 3D EBSD reconstruction
附录 A. 3D EBSD 重建

Appendix B. Detection of grain boundaries
附录 B. 晶界检测

Appendix C. Convergence analysis of numerical simulations
附录 C. 数值模拟的收敛性分析

References

Cited by (31)

Insights into stress corrosion cracking in scratched area of alloy 690TT steam generator tubes
合金 690TT 蒸汽发生器管材刮擦区域应力腐蚀开裂的见解

Substructure-sensitive crystal plasticity with material-invariant parameters
具有材料不变参数的亚结构敏感晶体塑性

On the effect of slip transfer at grain boundaries on the strength of FCC polycrystals
关于晶界滑移转移对 FCC 多晶材料强度的影响

Insights into Plastic Localization by Crystallographic Slip from Emerging Experimental and Numerical Approaches
基于晶体学滑移的塑性局部化研究：新兴实验与数值方法的见解

A micromechanical analysis of intergranular stress corrosion cracking of an irradiated austenitic stainless steel一种辐照奥氏体不锈钢的晶间应力腐蚀开裂的微观力学分析

Abstract 摘要

Graphical abstract 图示摘要

Keywords 关键词

1. Introduction 1. 引言

2. Experimental characterization2. 实验表征

2.1. Material 2.1. 材料

2.2. Microstructure 2.2. 微结构

2.2.1. 3D EBSD

2.2.2. Intergranular cracks2.2.2. 晶间裂纹

2.2.3. Grain boundaries 3D characterization2.2.3. 晶界三维表征

2.3. Analysis of experimental data2.3. 分析实验数据

2.3.1. GB normals 2.3.1. 晶界法线

2.3.2. Slip transmission criteria2.3.2. 滑移传递准则

2.3.3. Intergranular cracking correlations2.3.3. 晶间开裂相关性

3. Numerical assessment 3. 数值评估

3.1. Constitutive equations3.1. 本构方程

3.2. FFT simulations 3.2. 快速傅里叶变换模拟

3.3. Numerical results 3.3. 数值结果

4. Discussion 4. 讨论

4.1. Towards a micromechanical approach for IGSCC4.1. 迈向 IGSCC 的微观力学方法

4.2. Grain boundary oxidation4.2. 晶界氧化

5. Conclusion and perspectives5. 结论与展望

CRediT authorship contribution statementCRediT 作者贡献声明

Declaration of Competing Interest利益冲突声明

Acknowledgements 致谢

Appendix A. 3D EBSD reconstruction附录 A. 3D EBSD 重建

Appendix B. Detection of grain boundaries附录 B. 晶界检测

Appendix C. Convergence analysis of numerical simulations附录 C. 数值模拟的收敛性分析

References

A micromechanical analysis of intergranular stress corrosion cracking of an irradiated austenitic stainless steel
一种辐照奥氏体不锈钢的晶间应力腐蚀开裂的微观力学分析

2. Experimental characterization
2. 实验表征

2.2.2. Intergranular cracks
2.2.2. 晶间裂纹

2.2.3. Grain boundaries 3D characterization
2.2.3. 晶界三维表征

2.3. Analysis of experimental data
2.3. 分析实验数据

2.3.2. Slip transmission criteria
2.3.2. 滑移传递准则

2.3.3. Intergranular cracking correlations
2.3.3. 晶间开裂相关性

3.1. Constitutive equations
3.1. 本构方程

4.1. Towards a micromechanical approach for IGSCC
4.1. 迈向 IGSCC 的微观力学方法

4.2. Grain boundary oxidation
4.2. 晶界氧化

5. Conclusion and perspectives
5. 结论与展望

CRediT authorship contribution statement
CRediT 作者贡献声明

Declaration of Competing Interest
利益冲突声明

Appendix A. 3D EBSD reconstruction
附录 A. 3D EBSD 重建

Appendix B. Detection of grain boundaries
附录 B. 晶界检测

Appendix C. Convergence analysis of numerical simulations
附录 C. 数值模拟的收敛性分析