Distribution of normal stress at grain boundaries in multicrystals: application to an intergranular damage modeling

doi:10.1016/S0927-0256(02)00251-3

Computational Materials Science
计算材料科学

Volume 25, Issues 1–2, September 2002, Pages 73-84
第 25 卷，第 1-2 期，2002 年 9 月，第 73-84 页

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Abstract 摘要

Under transient power conditions in pressurized water reactor, zircaloy-4 fuel claddings are possibly submitted to stress corrosion cracking by volatile fission products. The localization of stress and strain in the inner surface of the cladding and the local aspects of the damage phenomena incite to consider a modeling at the granular scale. At this scale, the behavior of multicrystals is described by a crystal plasticity model including the local orientation of each grain and the Zy-4 slip-system families. Representative microstructures are meshed (2D and 3D) in order to evaluate intergranular but also intragranular heterogenities of the stress and strain fields. Large strain heterogenities appear due to deformation incompatibilities between grains, which induce over-stresses at the grain boundaries. 3D computations of multicrystalline aggregates are performed in order to compute the distribution of the normal stresses at the grain boundaries with respect to the angle between the load direction and the normal to the grain boundary. Effects of neighborhood is evaluated. In addition, an intergranular damage model is proposed. The formulation of this model is based on a decomposition of the strength at grain boundaries into normal and shear components. Finally, results on 2D aggregates are presented and show examples of anisotropic damage patterns.
在压水堆的瞬态功率条件下，锆合金-4 燃料包壳可能因挥发性裂变产物而遭受应力腐蚀开裂。包壳内表面的应力和应变分布以及损伤现象的局部特征促使人们考虑在颗粒尺度上进行建模。在此尺度上，多晶体的行为由一个晶体塑性模型描述，该模型包括每个晶粒的局部取向和 Zy-4 滑移系家族。代表性微观结构被网格化（2D 和 3D），以评估晶间和晶内的应力与应变场的异质性。由于晶粒间变形不兼容，会产生大应变异质性，这会在晶界处引起过应力。对多晶体聚集体进行 3D 计算，以计算晶界处正应力的分布，该分布与载荷方向和晶界法线之间的夹角有关。评估了邻域效应。此外，还提出了一个晶间损伤模型。该模型的建立基于晶界强度分解为法向和剪切分量的方法。最后，展示了二维聚集体结果，并呈现了各向异性损伤模式的示例。

PACS

81.40.Np

62.20.Mk

02.60

07.05.T

Keywords 关键词

Multicrystalline aggregates

Crystal plasticity

Finite element

Stress corrosion cracking

Zirconium alloy

Modeling

Damage

多晶集合体晶体塑性有限元应力腐蚀开裂锆合金建模损伤

1. Introduction 1. 引言

Fuel pins of typical bundles of pressurized water reactor are composed of uranium dioxide pellets set in zircaloy-4 claddings. In case of power ramp, a rapid increase of the linear heat generation rate (i.e., thermal power released per unit length) induces large dilatations of fuel pellets. Fragments of uranium dioxide may indent the inner surface of the cladding which can lead to local stress concentrations. This phenomenon of pellet-cladding interaction occurs in a corrosive environment, composed of accumulated volatile fission products such as iodine, and may lead to stress corrosion cracking. According to observations of failure surfaces, failure can be divided into three stages, including initiation and intergranular growth, transgranular growth by fluting and quasi-cleavage, and finally ductile failure by plastic instability. According to several authors, the first stage corresponds to the most important part of time to failure.
压水堆典型组件的燃料棒由二氧化铀圆柱体组成，这些圆柱体被锆合金-4 包壳所包围。在功率阶跃情况下，线性发热率（即单位长度的释放热功率）的快速增加会导致燃料圆柱体发生大幅度膨胀。二氧化铀碎片可能会压入包壳内表面，从而引发局部应力集中。这种燃料-包壳相互作用现象发生在腐蚀环境中，该环境由碘等累积的挥发性裂变产物组成，可能导致应力腐蚀开裂。根据失效表面的观察，失效过程可分为三个阶段，包括起始和沿晶生长、通过波纹化和准解理的穿晶生长，以及最终由于塑性不稳定性导致的延性失效。据多位研究者指出，第一阶段对应于失效过程中最关键的持续时间。

The risk of cladding failure is still a limitation for the management of the nuclear network. For this reason, power ramps have been simulated to establish a correlation between technical parameters, such as maximum power levels and rate of power increases, and mechanical variables, which control the damage kinetics, such as local stress and strain fields in the inner surface of the cladding. Those simulations on structures provide good estimations of the mechanical loading but they give gradients of the stress and strains fields with a characteristic length which is of the same order than the grain size, which is not consistent with the theoretical concept of representative volume element. In order to avoid this limitation, it is quite relevant to develop an approach at the granular scale for damage modeling. This method allows to take the chemical aspects of the damage into account with a physical support (mechanical–chemical weak-coupled calculation method [1]), and to investigate the stress heterogenities in the grains and at the grain boundaries. At this scale, the behavior of multicrystalline aggregates is described by crystallographic plasticity and the intergranular damage model proposed in this paper has been developed in respect with this crystallographic framework.
包覆层失效的风险仍然是核网管理的限制因素。为此，已对功率斜坡进行了模拟，以建立技术参数（如最大功率水平和功率增长率）与控制损伤动力学的力学变量（如包覆层内表面的局部应力场和应变场）之间的相关性。这些结构模拟能提供良好的力学载荷估计，但它们给出的应力场和应变场梯度具有与晶粒尺寸同数量级的特征长度，这与代表性体积元素的理论概念不一致。为了避免这一限制，开发一种颗粒尺度上的损伤建模方法非常相关。该方法允许通过物理支撑（机械-化学弱耦合计算方法[1]）考虑损伤的化学方面，并研究晶粒和晶界处的应力异质性。在这个尺度上，多晶聚合物的行为由晶体塑性描述，而本文提出的晶界间损伤模型是在这个晶体框架下开发的。

2. Metallurgical properties of zircaloy-4
2. 锆合金-4 的冶金性能

Zircaloy-4 is a zirconium based alloy (hexagonal lattice), with tin, iron, chrome and oxygen as main alloying elements. The composition of the alloy studied in this paper is reported in Table 1.
锆合金-4 是一种基于锆的合金（六方晶格），主要合金元素包括锡、铁、铬和氧。本文研究的合金成分见表 1。

Table 1. Weight composition of Zy-4 [2]
表 1. Zy-4 的重量组成[2]

Alloying elements (wt.%) 合金元素（wt.%）		Impurities (ppm) 杂质（ppm）
Sn	1.26	C	125–145
Fe	0.21	N	30–35
Cr	0.11	Al	30–35
O	0.132	Hf	–
Zr	Balance 平衡	Si	50
		H	9–11

Zircaloy claddings are machined by rolling and present high morphological and textural anisotropies. Tubes undergo a thermal treatment to reduce this anisotropy and two metallurgical states have to be distinguished: the stress relieved state and the recrystallized state. Zircaloy of fuel pins is actually used in the cold worked stress relieved state but in this case, the microstructure presents morphological anisotropy and a high density of dislocations which do not allow to develop easily a crystallographic approach. For this reason, the recrystallized state have been studied. In this case, the treatment consists in 4–5 h at 700±30 °C and the material does not undergo the last step of rolling. The microstructure is composed of equiaxial grains of about 10 μm, as shown in (Fig. 1), which reproduces a (r,θ) plan (polished and chemically attacked). The crystallographic texture of those tubes presents nevertheless preferential orientations of the {0 0 0 2} axis at ±35° in the (r,θ) plane [3].
锆合金包壳通过轧制加工，表现出高度形态和织构各向异性。管材经过热处理以减少这种各向异性，必须区分两种冶金状态：应力消除状态和再结晶状态。燃料棒用锆合金实际上是在冷加工应力消除状态下使用，但在这种情况下，微观结构表现出形态各向异性，并且位错密度高，这不允许轻易发展晶体学方法。因此，研究了再结晶状态。在这种情况下，处理过程包括在 700±30 °C 下 4–5 小时，并且材料不经历最后的轧制步骤。微观结构由约 10 μm 的等轴晶粒组成，如图 1 所示，该图再现了(r,θ)平面（抛光和化学侵蚀）。这些管材的晶体学织构仍然在(r,θ)平面中表现出{0 0 0 2}轴在±35°的择优取向[3]。

3. Mechanical modeling of zircaloy-4
3. 锆合金-4 的力学建模

3.1. Deformation modes 3.1. 变形模式

Zirconium presents an hexagonal lattice under 800 °C (α phase) with a c/a ratio of 1.594 (Fig. 2), which is lower than the theoretical ratio (1.633). The active slip system families of recrystallized Zy-4 have been determined by TEM observations [2]. For unirradiated material, the prismatic slip systems are the most important, with some secondary slip system families, such as the basal one, the

π1 〈a〉

and the

π1 〈c+a〉

to allow deformation in the 〈c〉 direction. In addition, no twinning is observed above 20 °C.
锆在 800 °C 以下呈现六方晶格（α相），其 c/a 比值为 1.594（图 2），低于理论比值（1.633）。通过透射电子显微镜观察确定了再结晶锆合金-4 的活性滑移系统[2]。对于未辐照材料，棱柱滑移系统最为重要，同时存在一些次要滑移系统，如基面滑移系统、

π1 〈a〉

和

π1 〈c+a〉

滑移系统，以允许在〈c〉方向上的变形。此外，在 20 °C 以上未观察到孪晶现象。

3.2. Crystal plasticity model
3.2. 晶体塑性模型

At the granular scale, the viscoplastic behavior of multicrystalline aggregates can be described, grain by grain, by the amount of plastic slip for each slip system. In addition, a flow rule is chosen to compute the plastic flow rate with respect to the local stress field.
在颗粒尺度上，多晶聚合物的粘塑性行为可以通过每个滑移系统的塑性滑移量逐粒描述。此外，选择一个流动法则来计算相对于局部应力场的塑性流速。

It is assumed that Schmid's law is valid to have the expression of the resolved shear stress τ_s (Eq. (1) where

n_{s}

and

l_{s}

respectively are the normal to the slip plane and the slip direction). The associated slip system s is activated when (|τ_s|−τ^c_s)>0. In addition of this criterion (extensively used for single crystal modeling in plasticity and viscoplasticity), several formulations have been proposed for the flow rule and the hardening law. Cailletaud proposed a phenomenological formulation [4] as an extension of the classical crystallographic approach for single crystal, first developed by Taylor [5], Mandel [6] and Asaro [7]. The strain rate

γ ̇_{s}

is a power law of the viscoplastic stress and the evolution of the critical resolved shear stress τ_s^c is described by a nonlinear isotropic hardening law (Eq. (2)). Self hardening and latent hardening due to interactions between slip systems are introduced by the matrix h_rs, according to the Mandel's hardening rule. In the case of zircaloy, latent hardening is neglected [8].
假设 Schmid 定律适用，则解析剪切应力τ _s 的表达式为（式(1)，其中

n_{s}

和

l_{s}

分别代表滑移面和滑移方向的法向）。当(|τ _s |−τ ^c _s )>0 时，相关滑移系统 s 被激活。除了这一标准（广泛用于塑性及粘塑性单晶建模），还提出了几种关于流动法则和硬化定律的公式。Cailletaud 提出了一个唯象公式[4]，作为对 Taylor[5]、Mandel[6]和 Asaro[7]首次发展的经典晶体学方法的单晶扩展。应变率

γ ̇_{s}

是粘塑性应力的幂律函数，临界解析剪切应力τ _s ^c 的发展由非线性各向同性硬化定律（式(2)）描述。根据 Mandel 的硬化法则，矩阵 h _rs 引入了自硬化以及由于滑移系统相互作用而产生的潜硬化。对于锆合金，潜硬化被忽略[8]。(1)

m ∼_{s} = 12 (n_{s} ⊗ l_{s} + l_{s} ⊗ n_{s}) and τ_{s} = σ ∼ : m ∼_{s}

(2)

γ ̇_{s} = |τ_{s} |−τ_{s}^{c} K_{1}^{n_{I}} sign (τ_{s}) with τ_{s}^{c} =τ_{0I}^{c} +Q_{I} ∑_{r} h_{rs} (1− exp (−b_{I} v_{r}))

(3)

ε ̇ ∼^{p} =∑_{s} m_{s} ∼ γ ̇_{s}

v^s is the cumulated slip on the system s

(v^{s} (t)=∫^{t}_{0} | γ ̇_{s} (u)| d u)

. For each slip system family I, a set of material coefficients characterizes the initial critical resolved shear stress (τ_0I), the hardening (Q_I, b_I), and the viscosity (n_I, K_I). As illustrated in Table 2, four families can be activated in Zy-4 [2]. This model has been implemented in a FE code [9], [10], and it has been widely used for the FE simulation of copper specimens, and of Ni-base single crystal components.
v ^s 是系统 s

(v^{s} (t)=∫^{t}_{0} | γ ̇_{s} (u)| d u)

的累积滑移量。对于每个滑移系族 I，一组材料系数表征了初始临界 resolved shear stress (τ _0I )、硬化 (Q, b) 和粘度 (n, K)。如表 2 所示，Zy-4 中可以激活四个族 [2]。该模型已在有限元代码 [9]、[10] 中实现，并已广泛用于铜试样的有限元模拟和镍基单晶部件的有限元模拟。

Table 2. Material parameters for recrystallized Zy-4 at room temperature [2]
表 2. 室温下再结晶 Zy-4 的材料参数[2]

Empty Cell	Prismatic 棱柱体	Basal 基底层	$π1 〈a〉$	$π1 〈c+a〉$
τ₀ (MPa)	53	197	144	292
K (MPa s^1/n)	178	100	94	184
n	6.5	3.2	2.6	2.1
Q (MPa)	17.6	20	62.2	169.2
b	103	17	94	101

4. Description of the intergranular damage
4. 间隙损伤的描述

Extensive studies have been set up to understand damage phenomena associated to the cladding failure. In the particular case of intergranular damage, observations of samples surfaces show that damage preferentially nucleates at grain boundaries, perpendicular to the load direction [11] (Fig. 3).
为理解包覆层失效相关的损伤现象，已开展大量研究。在颗粒间损伤的特定情况下，样品表面的观察表明，损伤优先在垂直于载荷方向的晶界处形核[11]（图 3）。

At this scale, the intergranular stresses, and particularly, the normal stress, appear as the driving force for the damage kinetics. Those observations are consistent with an iodine adsorption enhanced damage phenomenon, which seems to be quite relevant [12]. Adsorption on free surface can locally reduce the interatomic cohesion force at the grain boundaries and can lead to chemically assisted damage. Furthermore, plastic strain is quite heterogeneous, with undeformed grains, and localized plastic slip in slip bands for other grains. Observations of samples surfaces, in more details, seem to correlate large differences of the plastic deformation, from one grain to an other, or intersection of slip bands and grain boundaries, and the appearance of intergranular or transgranular microcracks. The role of the plastic strain and the interaction between the slip bands and the grain boundaries are still unknown. Actually, on one hand, plasticity at the crack tip can also increase adsorption by creating free surfaces, and on the other hand, deformation incompatibilities may induce over-stresses at the grain boundaries.
在这个尺度上，晶界间的应力，尤其是正应力，表现为损伤动力学的驱动力。这些观察结果与碘吸附增强损伤现象一致，该现象似乎相当相关[12]。在自由表面上吸附可以局部降低晶界处的原子间结合力，并可能导致化学辅助损伤。此外，塑性应变非常不均匀，部分晶粒未变形，而其他晶粒则在滑移带中发生局部塑性滑移。更详细地观察样品表面似乎与塑性变形的显著差异相关，这些差异存在于不同晶粒之间，或滑移带与晶界的交点，以及晶界间或穿晶微裂纹的出现。塑性应变的作用以及滑移带与晶界之间的相互作用仍不清楚。实际上，一方面，裂纹尖端的塑性变形可以通过形成自由表面来增加吸附，另一方面，变形不匹配可能导致晶界处产生过应力。

The aim of this paper is to investigate the second point, i.e., the stress and strain fields at the granular scale to evaluate the contribution of the macroscopic stress (or mean stress) and the contribution due to local deformation incompatibilities on the local stress field.
本文旨在研究第二点，即颗粒尺度上的应力和应变场，以评估宏观应力（或平均应力）以及局部变形不协调对局部应力场的贡献。

5. Local stress and strain fields analysis for multicrystals
5. 多晶体的局部应力与应变场分析

5.1. Realistic microstructures generation
5.1. 真实微观结构的生成

In order to capture the effects of strain and stress heterogenities on damage kinetics at the granular scale some realistic microstructures are meshed including morphological information and local orientations of the lattice. 2D and 3D aggregates are generated with a random tessellation method. The numerical method used for this work has been developed by Decker and Jeulin [13] and is based on the Voronoı̈ polyhedra model. This representation reproduces correctly microstructures resulting from a typical solidification process with constant and isotropic growth rate from a random distribution of germs. Firstly, a specific procedure is used to build a partition of a discrete domain, made of 3D voxel, with some Voronoı̈ polyhedra. In a second time, this representation is linked with a 3D mesher to generate the corresponding meshes. Examples of 2D and 3D structures with about a 100 grains are reported in Fig. 4, Fig. 5(a).
为了捕捉应变和应力异质性对颗粒尺度损伤动力学的影响，一些包含形态信息和晶格局部取向的真实微观结构被网格化。通过随机镶嵌方法生成二维和三维聚集体。本工作中使用的数值方法由 Decker 和 Jeulin [13] 开发，基于 Voronoi 多面体模型。这种表示正确地再现了由随机分布的晶核以恒定且各向同性的生长速率凝固产生的微观结构。首先，使用特定方法构建一个由三维体素组成的离散域的划分，其中包含一些 Voronoi 多面体。其次，将这种表示与三维网格生成器链接起来以生成相应的网格。图 4、图 5(a)展示了约 100 个晶粒的二维和三维结构的示例。

Moreover, the lattice orientation of each grain is defined and the distribution of those crystallographic orientations are numerically generated in order to respect the macroscopic texture of the tubes (Fig. 4(b)). In the 2D case, the grain boundaries are meshed with some volumic elements (Fig. 5(b)) to introduce intergranular damage.
此外，每个晶粒的晶格取向被定义，并且这些晶体学取向的分布被数值生成，以尊重管材的宏观织构（图 4(b)）。在 2D 情况下，晶界使用一些体积单元进行网格划分（图 5(b)）以引入晶间损伤。

5.2. Deformation incompatibilities effects analysis
5.2. 变形不兼容效应分析

2D aggregates have been simulated, with generalized plane strain conditions. The normal displacements are imposed on two faces (top and bottom), in order to obtain a constant deformation rate

2×10^{−5} s^{−1}

) in one direction and two lateral surfaces are free (nevertheless, the lateral displacement is imposed identical on the face: multipoints constraint). The distribution of the total cumulated prismatic slip, reported in Fig. 6, shows large intergranular heterogenities but also transgranular heterogenities.
对 2D 聚集体进行了模拟，采用广义平面应变条件。在两个面上施加法向位移（顶部和底部），以获得一个方向上的恒定变形速率

2×10^{−5} s^{−1}

)，并且两个侧表面是自由的（尽管侧向位移在面上施加相同：多点约束）。图 6 报道的总累积棱柱滑移分布显示出大的晶间异质性，同时也显示出晶内异质性。

The intergranular scatter can easily be described by the distribution of the Schmid's factors for the prismatic slip systems but the gradient of plastic strain within each grain is due to geometrical compatibility conditions which cannot be satisfied with homogeneous strain fields in each grain. Because of those deformation incompatibilities, the local stress field can be very different from the macroscopic stress field. At the grain boundaries, the ratio of the local maximum principal stress over the macroscopic uniaxial stress (intergranular over-stress) can reach 2.6 for a mean total strain of 10%. This stress field allows the activation of secondary slip systems, such as the basal one or the

π1 〈a〉

. The activation of those systems promotes the accommodation of the deformation and reduces the tangential modulus between stress and total strain (Fig. 7).
晶界间的散射可以通过棱柱滑移系统的 Schmid 因子分布来轻易描述，但每个晶粒内的塑性应变梯度是由于几何兼容性条件造成的，这些条件无法通过每个晶粒中的均匀应变场来满足。由于这些变形不兼容性，局部应力场可能与宏观应力场非常不同。在晶界处，局部最大主应力与宏观单轴应力的比值（晶界过应力）在平均总应变为 10% 时可达到 2.6。这种应力场允许次级滑移系统的激活，例如基面或

π1 〈a〉

。这些系统的激活促进了变形的适应，并降低了应力与总应变之间的切向模量（图 7）。

Those results illustrate the importance of a good description of the stress and strain fields at the granular scale for an intergranular damage model. Other simulations of 3D aggregates have been performed to investigate the respective contribution of the mean stress and of deformation incompatibilities on the local stresses, for several amount of total strain. The 3D aggregate is composed of about 100 grains (Fig. 4(c)).
这些结果说明了在晶粒尺度上对应力和应变场进行良好描述对于晶间损伤模型的重要性。已经进行了其他 3D 聚集体模拟，以研究平均应力和变形不兼容性对局部应力的各自贡献，对于不同的总应变量。3D 聚集体由约 100 个晶粒组成（图 4(c)）。

Homogeneous strain boundary conditions, corresponding to a mean deformation tensor

E ∼

, are imposed. The displacement vector at each point of the outer surface is defined as:
施加均匀应变边界条件，对应于平均变形张量

E ∼

。每个外表面点的位移矢量定义为：(4)

u = E ∼ · x where x is the position with, in

addition E ∼ =〈 ε ∼ 〉

The brackets 〈·〉 denote the mean value in the aggregate. In this case,

E ∼

is diagonal, with E₁₁=E₃₃=−(1/2) and E₂₂=1, with a strain rate of 1×10⁻⁵ s⁻¹ Those homogeneous strain conditions on the surface allow to use the mean value of the stress components on the faces (which is the macroscopic stress

Σ ∼

) to compute the mean value of the stress tensor in the aggregate, since Hill Mandel's lemma reduces to

Σ ∼ =〈 σ ∼ 〉

.
括号〈·〉表示集合的平均值。在这种情况下，

E ∼

是对角矩阵，其中 E ₁₁ =E ₃₃ =−(1/2) 和 E ₂₂ =1，应变速率为 1×10 ⁻⁵ s ⁻¹ 。这些均匀应变条件在表面上允许使用面（即宏观应力

Σ ∼

）上应力分量的平均值来计算集合中应力张量的平均值，因为希尔曼德尔引理简化为

Σ ∼ =〈 σ ∼ 〉

。

Curves on Fig. 8 give distributions of the ratio––normal stress at the grain boundary over the mean stress in the loading direction––with respect to the angle between the load direction and the normal to the grain boundary. Each point represents the mean value of the normal stress on a grain boundary. The solid lines in the same diagram represent the range of the normal stress with respect to the angle in the case of a uniform stress field. In this case:
图 8 中的曲线给出了晶界法向应力与加载方向平均应力的比值随加载方向与晶界法线之间夹角变化的分布。每个点表示晶界上法向应力的平均值。同一图中的实线表示在均匀应力场情况下法向应力随角度变化的范围。在这种情况下：(5)

σ_{n} = Σ ∼ : n ⊗ n

where

Σ ∼

is the mean stress tensor.
其中

Σ ∼

是平均应力张量。

In the elastic range, intergranular over-stresses are quite low, even with anisotropic elasticity, so that about no scatter can be seen. On the other hand, plastic flow will produce important mismatches in the microstructure. The scatter obtained for the contribution of the mean stress is due to the mean plastic anisotropy of the aggregate and does not describe correctly the dispersion. Even for a small amount of plasticity (total strain below 0.5%), the mismatch between the local stress and the stress computed with an homogeneous hypothesis can reach 25%. For more than 3% of plastic strain, the mismatch reach 50% with a saturation effect above 3%.
在弹性范围内，晶界间的超应力相当低，即使存在各向异性弹性，也几乎看不到散射现象。另一方面，塑性变形会在微观结构中产生重要的不匹配。所获得的平均应力贡献的散射是由于集体的平均塑性各向异性，并不能正确描述分散情况。即使对于少量的塑性变形（总应变低于 0.5%），局部应力与基于均匀假设计算的应力之间的不匹配可能达到 25%。对于超过 3%的塑性应变，不匹配程度达到 50%，且在 3%以上存在饱和效应。

6. Intergranular damage model
6. 界面间损伤模型

Observations of small samples, submitted to stress corrosion cracking by iodine, show that initiation often occurs at grain boundaries perpendicular to the loading direction. Furthermore, the correlations established between the macroscopic strain and the local over-stresses at grain boundaries incite to consider the normal stress at grain boundaries as a critical parameter for the damage kinetics.
对经碘应力腐蚀开裂的小样品的观察表明，裂纹往往在垂直于加载方向的晶界处萌生。此外，宏观应变与晶界局部超应力之间建立的相关性，促使人们将晶界处的正应力视为损伤动力学的一个关键参数。

The natural anisotropy of the boundary leads to decompose the strength into a normal component N and a shear component T. This normal direction is considered as a principal damage direction. One way to distinguish this direction is to introduce damage as a tensor and to couple elasticity and damage with an effective stress [14]. With this formulation, the stiffness moduli have to reach down to very small values for an effective opening of 100% without stress. In addition, this method provides well known numerical difficulties when the damage variable approaches 1. Actually it can be efficient to distinguish the damage of the boundary and its effective opening but to couple them. One method consists in introducing an inelastic strain due to damage. This method allows to keep an isotropic damage variable and to control the anisotropy of the damage with this inelastic flow. Coupling between damage (due to shear and normal loading) and opening of the boundary (only under normal loading) depends on the dissipation potential.
边界天然的各向异性导致将强度分解为一个法向分量 N 和一个剪切分量 T。这个法向方向被视为一个主要损伤方向。区分这个方向的一种方法是将损伤引入为张量，并将弹性与损伤通过有效应力耦合起来[14]。在这种公式中，刚度模量必须达到非常小的值，才能在没有应力的情况下实现 100%的有效张开。此外，当损伤变量接近 1 时，这种方法会提供众所周知的数值困难。实际上，区分边界损伤及其有效张开但将它们耦合起来可能是有效的。一种方法包括引入由损伤引起的非弹性应变。这种方法允许保持各向同性的损伤变量，并通过这种非弹性流动来控制损伤的各向异性。损伤（由剪切和法向载荷引起）与边界张开（仅在法向载荷下）之间的耦合取决于耗散势。

The decomposition can also be justified by the damage mechanisms. It introduces a first elastic damage due to chemical weakening of the boundary (including removal of metal) and an opening of the grain by local plasticity at the tip of the crack.
这种分解也可以通过损伤机制来证明。它引入了由于边界化学弱化（包括去除金属）导致的初始弹性损伤，以及裂纹尖端局部塑性变形引起的晶粒开裂。

6.1. Description of the model
6.1. 模型描述

6.1.1. Choice of an equivalent stress for the damage kinetic
6.1.1. 损伤动力学的等效应力选择

In order to be consistent with the framework of the thermodynamics of irreversible processes, equations of the model are divided into state equations, which allows to define the driving force associated to damage and constitutive equations based on a dissipation potential.
为了与不可逆过程热力学框架保持一致，模型的方程被分为状态方程，这允许定义与损伤相关的驱动力和基于耗散势的构型方程。

The specific Helmholtz free energy Ψ_e is taken as state potential. One assumes that no coupling occurs between the elastic and hardening parts. Assuming that damage can be described by a scalar variable D (corresponding to the environment enhanced degradation of the grain boundary), the energy density ρΨ_e can be written as:
特定亥姆霍兹自由能Ψ _e 被取作状态势。假设弹性和硬化部分之间没有耦合发生。假设损伤可以用标量变量 D（对应于环境增强的晶界退化）来描述，能量密度ρΨ _e 可以表示为：(6)

ρΨ_{e} = 12 (1−D)(ε_{e} ∼ − k ∼ ΔT): C ≈ :(ε_{e} ∼ − k ∼ ΔT)

where

C ≈

is the elastic tensor,

ε_{e} ∼

the elastic strain tensor,

k ∼

the termal expansion tensor and T is the temperature in Kelvin. This allows to define the effective modulus

C ̃ ≈

and the effective stress

σ ̃ ∼

:
其中

C ≈

是弹性张量，

ε_{e} ∼

是弹性应变张量，

k ∼

是热膨胀张量，T 是开尔文温度。这允许定义有效模量

C ̃ ≈

和有效应力

σ ̃ ∼

：(7)

σ ∼ =ρ ∂ Ψ_{e} ∂ ε_{e} ∼ =(1−D) C ≈ :(ε_{e} ∼ − k ∼ ΔT)

(8)

σ ∼ = C ̃ ≈ :(ε_{e} ∼ − k ∼ ΔT) or σ ̃ ∼ = C ≈ :(ε_{e} ∼ − k ∼ ΔT)

Derivation of the free energy versus D classically defines an equivalent energy in terms of elastic damage for isotropic materials [15].
自由能相对于 D 的经典推导定义了各向同性材料的等效能量[15]。

In the following, the expression , will be preserved to represent the coupling between elasticity and damage, but other critical variable are used to represent damage evolution, in order to account for anisotropy.
在下文中，表达式将保持不变以表示弹性和损伤之间的耦合，但使用其他关键变量来表示损伤演化，以考虑各向异性。

The equivalent stress is defined by means of σ_n and τ, which respectively denote, the positive part of the normal stress to the grain boundary and the shear stress component in the plane of the grain boundary. They are given by:
等效应力通过σ _n 和 τ 定义，分别表示晶界法向应力的正值和晶界平面内的剪切应力分量。它们由下式给出：(9)

σ_{n} = Max (0, σ ∼ :(n ⊗ n))

(10)

τ^{2} = σ ∼ : A ≈_{T} : σ ∼

with 其中(11)

A ≈_{T} =({n ⊗ e_{2}}_{s} ⊗{n ⊗ e_{2}}_{s} +{n ⊗ e_{3}}_{s} ⊗{n ⊗ e_{3}}_{s})

where (

e_{2}, e_{3}

) is any base in the plane of the grain boundary. A quadratic expression is chosen to combine the two components, using a positive scalar β:
其中 (

e_{2}, e_{3}

) 是晶界平面内的任意基。选择二次表达式，使用正标量 β 将两个分量组合：(12)

σ_{eq} = σ^{2}_{n} +βτ^{2}^{1/2}

In the following, the fourth order tensor

A ≈_{N}

will also be used:
在下文中，还将使用第四阶张量

A ≈_{N}

：(13)

A ≈_{N} = n ⊗ n ⊗ n ⊗ n

6.1.2. Definition of the flow rate
6.1.2. 流动速率的定义

The flow rate is the most specific part of this model. The shear component of

σ ∼

does not have to modify explicitly the opening kinetics because opening occurs only under normal loading. In addition, the shear component of the plastic flow must be proportional to the shear component of

σ ∼

.
流动速率是此模型中最具体的部分。

σ ∼

的剪切分量无需显式地修改开动学，因为开动仅在正载荷下发生。此外，塑性流动的剪切分量必须与

σ ∼

的剪切分量成正比。

The same dissipation potential could be used to derive the damage kinetics and the inelastic flow rate. But this formulation would introduce too much coupling between the normal and shear components in the plastic flow.
可以使用相同的耗散势来推导损伤动力学和非弹性流动速率。但此公式会在塑性流动的正应力和剪应力分量之间引入过多的耦合。

The strain rate due to damage is evaluated from a potential using the values of normal and shear stresses computed with the effective stress tensor:
损伤引起的应变率通过有效应力张量计算的应力和剪应力值从势能中评估：(14)

Φ^{*} = K n+1 σ ̃_{n} −R K^{n+1} + τ ̃ −R K^{n+1}

so that 因此(15)

ε ̇ ∼^{D} = ∂ Φ^{*} ∂ σ ̃_{n} ∂ σ ̃_{n} ∂ σ ∼ + ∂ Φ^{*} ∂ τ ̃ ∂ τ ̃ ∂ σ ∼

(16)

ε ̇ ∼^{D} = 1 1−D σ ̃_{n} −R K^{n} A ≈_{N} : σ ∼ σ_{n} +β τ ̃ −R K^{n} A ≈_{T} : σ ∼ τ

Note that the directions of the flow are respectively

n ⊗ n

for the first term and

{n ⊗ m}_{s}

for the second term, m being the direction of the shear component τ.
请注意，流动的方向分别为第一项的

n ⊗ n

和第二项的

{n ⊗ m}_{s}

，m 是剪应力分量 τ 的方向。

In order to simplify the identification of the material parameters, the set of coefficients (K, n, R) is the same for both terms.
为了简化材料参数的识别，系数组 (K, n, R) 对两项都是相同的。

On the other hand, damage evolution is simply inspired from the kinetics initially introduced by Kachanov [16] and modified by Rabotnov [17], using the equivalent stress defined in Eq. (12), as proposed in [18]:
另一方面，损伤演化简单地借鉴了 Kachanov 最初引入[16]并由 Rabotnov 修改[17]的动力学，使用式(12)中定义的等效应力，如[18]中提出：(17)

D ̇ = σ_{eq} A^{r} (1−D)^{−k}

This formulation provides an opening of the grain boundary only under tensile normal loading, but a shear preloading promotes this opening through the effect of the effective stress. Such a preloading increases the damage variable which introduces a shear memory effect on the opening kinetics. For instance in Fig. 9 the solid line corresponds to a pure tension on a grain boundary, and the dashed line represents the subsequent tension after a shear preloading of 460 MPa. The macroscopic strain rate is 2×10⁻⁴ s⁻¹. One can observe the effect of the elasticity-damage coupling, which reduces the Young's modulus, and the effect of damage on inelastic tension, which induces a stress drop during opening.
这种公式仅在拉伸正应力加载下开启晶界，但剪切预载通过有效应力效应促进这种开启。这种预载增加了损伤变量，从而在开启动力学中引入剪切记忆效应。例如在图 9 中，实线对应晶界上的纯拉伸，虚线表示在 460 MPa 的剪切预载后的后续拉伸。宏观应变速率为 2×10 ⁻⁴ s ⁻¹ 。可以观察到弹塑性耦合效应，该效应降低了杨氏模量，以及损伤对非弹性拉伸的影响，该影响在开启过程中导致应力下降。

In addition, a softening term in the flow rate promotes numerical integration of the model and describes final opening of the boundary (Table 3). This macroscopic opening (over the critical strain) is generally not possible with classical damage mechanics formulations.
此外，流量中的软化项促进了模型的数值积分，并描述了边界的最终开度（表 3）。这种宏观开度（超过临界应变）通常无法通过经典损伤力学公式实现。

Table 3. Coefficients of the model used in this paper
表 3. 本文使用的模型系数

Behavior 行为	E (MPa)	ν	K (MPa s^1/n) K (MPa·s)	n	R₀ (MPa) R (MPa)
	80,000	3.2	100	30	1000−999(D/0.95)⁸ 1000−999(D/0.95)

Damage 损伤	β	A (MPa s^1/r)	r	k
	0.4	3500	3	4

6.2. First results 6.2. 初步结果

2D aggregates with 115 grains and a volumic meshing of the grain boundaries have been simulated with a simplified version of the model presented in this paper (no contribution of the shear component). Generalized plane strain conditions are imposed. The normal displacements are prescribed on two faces (top and bottom), in order to obtain a constant deformation rate of 1×10⁻⁴ s⁻¹ in one direction and two lateral surfaces are free (Fig. 10).
对具有 115 个晶粒的 2D 聚集体和晶界体积网格进行了模拟，使用了本文提出的模型的简化版本（剪切分量无贡献）。施加了广义平面应变条件。在两个面上（顶部和底部）规定了法向位移，以获得一个方向上的恒定变形速率 1×10 ⁻⁴ s ⁻¹ ，两个侧表面是自由的（图 10）。

Local orientations of the grain boundaries are used which produces a macroscopically anisotropic damage pattern with this microscopically isotropic damage model. Damage occurs mostly on planes perpendicular to the macroscopic stress. This indicates that the effect of macroscopic stress in the aggregate remains predominant. Nevertheless, intergranular over-stresses due to deformation incompatibilities between neighbors can also locally modify the stress field and one can observe damaged grain boundaries making an angle of 15–20° with the principal direction, while other grain boundaries with smaller angles seem still resistant.
晶界局部取向被采用，这使得微观上各向同性的损伤模型在宏观上呈现出各向异性的损伤模式。损伤主要发生在垂直于宏观应力的平面上。这表明宏观应力在集合体中的影响仍然占主导地位。然而，由于相邻晶粒变形不匹配导致的晶界超应力也能局部改变应力场，可以观察到受损晶界与主方向成 15-20°角，而其他角度较小的晶界似乎仍然具有抵抗力。

According to those results, the anisotropy of the damage patterns should be different for damage with homogeneous nucleation in the aggregate (case of this paper) and damage with propagation. In the first case, nucleation and most of growth and coalescence occur for small plastic strains. In the second case, nucleation occurs later, with more mean plastic strain, and more plastic strain at the tip of a crack. Propagation can also follow a crack path which is locally not perpendicular to the load direction.
根据这些结果，损伤模式的各向异性应该对于集合体中均匀形核的损伤（本文的情况）和扩展型损伤是不同的。在第一种情况下，形核以及大部分生长和合并发生在小塑性应变下。在第二种情况下，形核发生较晚，平均塑性应变更大，裂纹尖端处的塑性应变也更大。扩展也可能沿着一条局部不垂直于载荷方向的裂纹路径进行。

7. Conclusion 7. 结论

Representative microstructures have been computed and a special attention was paid for respecting the global texture and the shape of the grains, especially in the 3D case. Those computations provide a good description of the stress and strain fields at the granular scale and show large intergranular and intragranular heterogenities of the plastic slip. Plasticity of grains has been analyzed, to explain the evolution of the stress at the grain boundaries with respect to the activation of the slip systems. A 3D analysis of the distribution of the normal stress at grain boundaries has shown that the contribution of the neighborhood (due to the deformation incompatibilities and the intergranular heterogenities described above) is very important. An intergranular damage model is proposed at the granular scale in order to take those heterogenities into account, in a natural way. The damage variable is isotropic but this formulation allows to describe anisotropic damage patterns. Coefficients of the model have been identified with some computations of DCB samples in order to reproduce the experimental crack growth rate. The transposition of the numerical crack growth rate from DCB to aggregates is quite good, which simplify the identification. Those computations illustrate homogeneous damage nucleation, but propagation of intergranular cracks will be investigated in the future, especially in order to study the branching phenomenon. The crystal plasticity model should be improved in order to introduce a physical length and to take the intragranular strain gradient effects into account.
已经计算了具有代表性的微观结构，并特别注意保持整体织构和晶粒的形状，特别是在 3D 情况下。这些计算能够很好地描述颗粒尺度上的应力和应变场，并显示出塑性滑移在晶界间和晶粒内的显著异质性。通过分析晶粒的塑性，解释了滑移系统激活时晶界处应力的演变。对晶界处正应力分布的 3D 分析表明，邻域（由于上述变形不兼容性和晶界间异质性）的贡献非常重要。在颗粒尺度上提出了一个晶界间损伤模型，以自然的方式考虑这些异质性。该损伤变量是各向同性的，但该公式允许描述各向异性损伤模式。模型的系数通过一些 DCB 样品的计算来确定，以再现实验中的裂纹扩展速率。将数值裂纹扩展速率从 DCB 转移到颗粒聚集体相当好，这简化了识别。这些计算说明了均匀损伤成核，但未来将研究晶间裂纹的扩展，特别是为了研究分叉现象。晶体塑性模型应该得到改进，以便引入一个物理长度，并考虑晶内应变梯度效应。

Acknowledgements 致谢

The authors would like to thank the Research department of Electricité de France for a financial and technical support during this study.
作者感谢法国电力公司的研究部门在本研究中提供的资金和技术支持。

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Cited by (54)

A crystal plasticity-based approach for creep-fatigue life prediction and damage evaluation in a nickel-based superalloy
基于晶体塑性的方法，用于镍基高温合金的蠕变疲劳寿命预测和损伤评估
2021, International Journal of Fatigue
2021，国际疲劳杂志
Citation Excerpt : 引用摘录 :
There are two common numerical methods of Voronoi Tessellation (VT) technique and EBSD-based approach to generate representative volume element (RVE) models prior to CPFE simulation. Although the traditional VT technique has been widely used, many researchers realized that the method has some limitations on representing the actual lattice orientations within grains and misorientations between adjacent grains [47–49]. As compared to the VT technique, the EBSD-based approach can provide more accurate prediction results for fatigue crack initiation by combining microscopic observations [50].
有两种常见的 Voronoi 分形（VT）技术和基于 EBSD 的方法在 CPFE 模拟之前生成代表性体积元素（RVE）模型。尽管传统的 VT 技术已被广泛使用，但许多研究人员意识到该方法在表示晶粒内的实际晶格取向和相邻晶粒间的取向差方面存在一些局限性[47-49]。与 VT 技术相比，基于 EBSD 的方法通过结合微观观察，可以更准确地预测疲劳裂纹萌生的结果[50]。
A numerical process based on crystal plasticity finite element (CPFE) was implemented to predict creep-fatigue crack initiation life. CPFE-based model can describe the macroscopic cyclic deformation and reveal grain-level damage mechanism. A new life prediction approach was then constructed by introducing fatigue and creep indicator parameters. Furthermore, a series of strain-controlled creep-fatigue tests in GH4169 superalloy at 650℃ were used to validate predicted accuracy of this model, where most of the data points lied within ±1.5 error band. Finally, a potential methodology for conservative creep-fatigue life evaluation was provided by flexible creep-fatigue damage summation rule in engineering applications.
Strain gradient continuum plasticity theories: Theoretical, numerical and experimental investigations
应变梯度连续介质塑性理论：理论、数值和实验研究
2019, International Journal of Plasticity
2019，《国际塑性力学杂志》
Over the last three decades, a number of researchers have used the gradient-enhanced plasticity theory to model the microstructural material behavior, which is not able to be captured by the conventional plasticity theory, within the continuum based framework. This paper aims to review the strain gradient continuum plasticity in the context of the theoretical, experimental and numerical investigations. A review of the theoretical developments by the several research groups and their applications to the finite element method is presented in this paper. In addition, a review of the various experimental methodologies to study the size effects is presented. Lastly, some open issues in the gradient-enhanced plasticity such as the physical nature of the strain gradients and the physical interpretation of the material length scales are discussed.
Modeling the microstructural evolution of metallic polycrystalline materials under localization conditions
在局部化条件下模拟金属多晶材料的微观结构演化
2007, Journal of the Mechanics and Physics of Solids
2007，《固体力学与物理学杂志》
In general, the shear localization process involves initiation and growth. Initiation is expected to be a stochastic process in material space where anisotropy in the elastic–plastic behavior of single crystals and inter-crystalline interactions serve to form natural perturbations to the material's local stability. A hat-shaped sample geometry was used to study shear localization growth. It is an axi-symmetric sample with an upper “hat” portion and a lower “brim” portion with the shear zone located between the hat and brim. The shear zone length was 870–890 μm with deformation imposed through a Split-Hopkinson Pressure Bar system at maximum top-to-bottom velocity in the range of 8–25 m/s. The deformation behavior of tantalum tophat samples is modeled through direct polycrystal simulations. An embedded Voronoï-tessellated two-dimensional microstructure is used to represent the material within the shear zone of the sample. A thermo-mechanically coupled elasto-viscoplastic single crystal model is presented and used to represent the response of the grains within the aggregate shear zone. In the shoulder regions away from the shear zone where strain levels remain on the order of 0.05, the material is represented by an isotropic J₂ flow theory based upon the elasto-viscoplastic Mechanical Threshold Stress (MTS) model for flow strength. The top surface stress versus displacement results were compared to those of the experiments and over-all the simulated stress magnitude is over-predicted. It is believed that the reason for this is that the simulations are two-dimensional. A region within the numerical shear zone was isolated for statistical examination. The vonMises stress state within this isolated shear zone region suggests an approximate normal distribution with a factor of two difference between the minimum and maximum points in the distribution. The equivalent plastic strain distribution within this same region has values ranging between 0.4 and 1.5 and is not symmetric. Other material state distributions are also given. The crystallographic texture within this isolated shear zone is also compared to the experimental texture and found to match reasonably well considering the simulations are two-dimensional.
Coupling between experimental measurements and polycrystal finite element calculations for micromechanical study of metallic materials
实验测量与多晶有限元计算在金属材料微观力学研究中的耦合
2007, International Journal of Plasticity
2007，《国际塑性力学杂志》
This paper presents a methodology for multiscale coupling between the morphology and texture of a microstructure as has been characterised experimentally, and the results of mechanical strain field analysis. This methodology is based on a coupling between experimental characterisation of the microstructure, in situ and/or ex situ mechanical tests, local strain field measurements performed at the grain scale, and finite element simulations. First, with orientation imaging microscopy, a map of the microstructure is generated that can be meshed. Then, finite element calculations are carried out on this mesh, using a constitutive law which takes into account the crystallographic orientation of each grain, as has been determined by the orientation imaging itself. These numerical results are then compared to the experimental strain field as obtained by digital image correlation at the scale of the grains.
After a review of the different aspects of the coupling, the paper characterises and analyses possible sources of error of the measurements, as well as the differences in the simulation results with respect to mesh refinement and boundary conditions.
Then, a definition of a cost function is proposed in order to optimise the parameters of the crystallographic constitutive law. Finally, this method is applied to the studies of zirconium and titanium aluminide alloys in order to improve the understanding of their mechanical behaviour in relation with their microstructures, which is a key requirement for their use in the nuclear and aeronautic industries, respectively.
Micromorphic approach for gradient elasticity, viscoplasticity, and damage
梯度弹性、粘塑性及损伤的微形变方法
2009, Journal of Engineering Mechanics
2009, 工程力学杂志
Observations of intergranular stress corrosion cracking in a grain-mapped polycrystal
观察多晶粒映射中的晶间应力腐蚀开裂
2008, Science 2008, 科学

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Outline 概要

Cited by (54) 被引用次数 (54)

Figures (10) 图 (10)

Tables (3) 表(3)

Computational Materials Science
计算材料科学

Abstract 摘要

PACS

Keywords 关键词

1. Introduction 1. 引言

2. Metallurgical properties of zircaloy-4
2. 锆合金-4 的冶金性能

3. Mechanical modeling of zircaloy-4
3. 锆合金-4 的力学建模

3.1. Deformation modes 3.1. 变形模式

3.2. Crystal plasticity model
3.2. 晶体塑性模型

4. Description of the intergranular damage
4. 间隙损伤的描述

5. Local stress and strain fields analysis for multicrystals
5. 多晶体的局部应力与应变场分析

5.1. Realistic microstructures generation
5.1. 真实微观结构的生成

5.2. Deformation incompatibilities effects analysis
5.2. 变形不兼容效应分析

6. Intergranular damage model
6. 界面间损伤模型

6.1. Description of the model
6.1. 模型描述

6.1.1. Choice of an equivalent stress for the damage kinetic
6.1.1. 损伤动力学的等效应力选择

6.1.2. Definition of the flow rate
6.1.2. 流动速率的定义

6.2. First results 6.2. 初步结果

7. Conclusion 7. 结论

Acknowledgements 致谢

References

Cited by (54)

A crystal plasticity-based approach for creep-fatigue life prediction and damage evaluation in a nickel-based superalloy
基于晶体塑性的方法，用于镍基高温合金的蠕变疲劳寿命预测和损伤评估

Strain gradient continuum plasticity theories: Theoretical, numerical and experimental investigations
应变梯度连续介质塑性理论：理论、数值和实验研究

Modeling the microstructural evolution of metallic polycrystalline materials under localization conditions
在局部化条件下模拟金属多晶材料的微观结构演化

Coupling between experimental measurements and polycrystal finite element calculations for micromechanical study of metallic materials
实验测量与多晶有限元计算在金属材料微观力学研究中的耦合

Micromorphic approach for gradient elasticity, viscoplasticity, and damage
梯度弹性、粘塑性及损伤的微形变方法

Observations of intergranular stress corrosion cracking in a grain-mapped polycrystal
观察多晶粒映射中的晶间应力腐蚀开裂

Distribution of normal stress at grain boundaries in multicrystals: application to an intergranular damage modeling多晶中晶界处正应力的分布：应用于晶间损伤建模

Abstract 摘要

PACS

Keywords 关键词

1. Introduction 1. 引言

2. Metallurgical properties of zircaloy-42. 锆合金-4 的冶金性能

3. Mechanical modeling of zircaloy-43. 锆合金-4 的力学建模

3.1. Deformation modes 3.1. 变形模式

3.2. Crystal plasticity model3.2. 晶体塑性模型

4. Description of the intergranular damage4. 间隙损伤的描述

5. Local stress and strain fields analysis for multicrystals5. 多晶体的局部应力与应变场分析

5.1. Realistic microstructures generation5.1. 真实微观结构的生成

5.2. Deformation incompatibilities effects analysis5.2. 变形不兼容效应分析

6. Intergranular damage model6. 界面间损伤模型

6.1. Description of the model6.1. 模型描述

6.1.1. Choice of an equivalent stress for the damage kinetic6.1.1. 损伤动力学的等效应力选择

6.1.2. Definition of the flow rate6.1.2. 流动速率的定义

6.2. First results 6.2. 初步结果

7. Conclusion 7. 结论

Acknowledgements 致谢

References

Distribution of normal stress at grain boundaries in multicrystals: application to an intergranular damage modeling
多晶中晶界处正应力的分布：应用于晶间损伤建模

2. Metallurgical properties of zircaloy-4
2. 锆合金-4 的冶金性能

3. Mechanical modeling of zircaloy-4
3. 锆合金-4 的力学建模

3.2. Crystal plasticity model
3.2. 晶体塑性模型

4. Description of the intergranular damage
4. 间隙损伤的描述

5. Local stress and strain fields analysis for multicrystals
5. 多晶体的局部应力与应变场分析

5.1. Realistic microstructures generation
5.1. 真实微观结构的生成

5.2. Deformation incompatibilities effects analysis
5.2. 变形不兼容效应分析

6. Intergranular damage model
6. 界面间损伤模型

6.1. Description of the model
6.1. 模型描述

6.1.1. Choice of an equivalent stress for the damage kinetic
6.1.1. 损伤动力学的等效应力选择

6.1.2. Definition of the flow rate
6.1.2. 流动速率的定义