Abstract A multiscale graph theory-based approach is introduced here to predict the microscale crack path in polycrystalline materials. The crack path is represented as the boundary of the partition of a geometric graph. The partitioning is carried out by optimizing an Ising-type hamiltonian. The hamiltonian parameters are chosen such that each partition cost is the same as the energy of the corresponding crack. The interplay of the loading conditions on the specimen and the microstructure of the material near the crack tip determines the crack growth angle in polycrystalline materials. Two different length scales of macro and micro is incorporated for the crack path by defining the crack total energy as the summation of macroscopic energy release and microscopic surface energy. The former term represents the macroscopically favorable crack growth angle by directing the crack to propagate from the crack tip along the direction of the maximum energy release. The latter term guides the microscopic crack path along the macroscopically preferred direction. At this scale, the crack path naturally accommodates both the intergranular and transgranular fractures. In the case of intergranular fracture, the crack propagates along the grain boundaries, while in the case of transgranular fracture, the crack propagates along the crystallographic cleavage planes. The mixed-mode fracture in a thin foil specimen is studied, and the effect of the dihedral angle of the 2D crack is included in defining the effective surface energy. The model is validated using the analytical results for mixed-mode fracture in an isotropic medium and mode-I fracture in a medium with a preferred crack direction. The proposed method can be used to design materials microstructure with optimal fracture resistance.

中文翻译：

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多晶材料裂纹扩展多尺度建模和预测的图论方法

摘要 这里介绍了一种基于多尺度图论的方法来预测多晶材料中的微尺度裂纹路径。裂纹路径表示为几何图形分区的边界。划分是通过优化一个 Ising 型哈密顿量来进行的。选择哈密顿参数，使得每个分区成本与相应裂纹的能量相同。试样上的加载条件和裂纹尖端附近材料的微观结构的相互作用决定了多晶材料的裂纹扩展角。通过将裂纹总能量定义为宏观能量释放和微观表面能的总和，裂纹路径包含宏观和微观两种不同的长度尺度。前一项表示宏观上有利的裂纹扩展角，通过引导裂纹沿最大能量释放方向从裂纹尖端扩展。后一项沿着宏观优选方向引导微观裂纹路径。在这种规模下，裂纹路径自然地容纳了沿晶和穿晶断裂。在晶间断裂的情况下，裂纹沿晶界扩展，而在穿晶断裂的情况下，裂纹沿晶面解理。研究了薄箔试样中的混合模式断裂，并将二维裂纹的二面角的影响包括在定义有效表面能中。该模型使用各向同性介质中的混合模式断裂和具有优选裂纹方向的介质中的模式 I 断裂的分析结果进行验证。所提出的方法可用于设计具有最佳抗断裂性的材料微观结构。