A new approach of multi-scale simulation via coupling of boundary element and finite element is proposed to predict fracture toughness and crack propagation of a single layer graphene sheet. In this simulation a molecular dynamics-based nonlinear beam element is developed for the atomistic model near the crack tip, whereas a special two-dimensional boundary element is employed for the continuum model in the remaining field of the cracked specimen. The material and section properties required in the nonlinear beam element are estimated through the equivalence between the potential energy of molecular dynamics and the elastic strain energy of continuum mechanics. With the estimated properties of beam element, the material properties of boundary element are further estimated by applying loads on the specimen of graphene, which is formed by a hexagonal lattice of carbon atoms. Coupling the nonlinear beam elements by boundary elements with the local-global concept of multi-scale modeling, a vast of computational time can be saved. The accuracy of near tip stresses obtained in our simulation remedies the inaccuracy of linear elastic fracture mechanics, and can be used for the prediction of atomic bond-breaking, which leads to crack propagation. The associated critical load can then be applied in the continuum model for the cracked specimen to predict mode I and mode II fracture toughness of graphene. The obtained values are then verified by the published results measured or predicted by the other methods. By varying the size of cracks and orientation of applied loads, several interesting phenomena have been observed and discussed in this paper.

中文翻译：

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通过边界元与非线性梁元耦合预测石墨烯的断裂韧性和裂纹扩展

提出了一种通过边界元和有限元耦合的多尺度模拟新方法来预测单层石墨烯片的断裂韧性和裂纹扩展。在该模拟中，为裂纹尖端附近的原子模型开发了基于分子动力学的非线性梁单元，而在裂纹试样的剩余区域中，为连续介质模型采用了特殊的二维边界单元。非线性梁单元所需的材料和截面特性是通过分子动力学的势能和连续介质力学的弹性应变能之间的等价性来估计的。利用梁单元的估计特性，通过对石墨烯试样施加载荷，进一步估计边界单元的材料特性，它由碳原子的六边形晶格形成。将非线性梁单元通过边界单元与多尺度建模的局部-全局概念耦合，可以节省大量的计算时间。在我们的模拟中获得的近尖端应力的准确性弥补了线弹性断裂力学的不准确性，并可用于预测原子键断裂，从而导致裂纹扩展。然后可以将相关的临界载荷应用于裂纹试样的连续模型中，以预测石墨烯的 I 型和 II 型断裂韧性。然后通过其他方法测量或预测的公布结果验证获得的值。通过改变裂纹的大小和施加载荷的方向，本文观察到并讨论了几个有趣的现象。