Huasong Qin 1 Institute of High Performance Computing, A*STAR, Singapore 138632; State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, China e-mail: qinhuasong1992@stu.xjtu.edu.cn Viachesla Sorkin 1 Institute of High Performance Computing, A*STAR, Singapore 138632 e-mail: sorkinv@ihpc.a-star.edu.sg Qing-Xiang Pei 2 Institute of High Performance Computing, A*STAR, Singapore 138632 e-mail: peiqx@ihpc.a-star.edu.sg Yilun Liu State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, China e-mail: yilunliu@mail.xjtu.edu.cn Yong-Wei Zhang 2 Institute of High Performance Computing, A*STAR, Singapore 138632 e-mail: zhangyw@ihpc.a-star.edu.sg Failure in Two-Dimensional Materials: Defect Sensitivity and Failure Criteria Two-dimensional (2D) materials have attracted a great deal of attention recently owing to their fascinating structural, mechanical, and electronic properties. The failure phenomena in 2D materials can be diverse and manifested in different forms due to the presence of defects. Here, we review the structural features of seven types of defects, including vacan- cies, dislocations, Stone-Wales (S-W) defects, chemical functionalization, grain boundary, holes, and cracks in 2D materials, as well as their diverse mechanical failure mechanisms. It is shown that in general, the failure behaviors of 2D materials are highly sensitive to the presence of defects, and their size, shape, and orientation also matter. It is also shown that the failure behaviors originated from these defects can be captured by the maximum bond- stretching criterion, where structural mechanics is suitable to describe the deformation and failure of 2D materials. While for a well-established crack, fracture mechanics-based failure criteria are still valid. It is expected that these findings may also hold for other nano- materials. This overview presents a useful reference for the defect manipulation and design of 2D materials toward engineering applications. [DOI: 10.1115/1.4045005] Keywords: computational mechanics, failure criteria, mechanical properties of materials, structures 1 Introduction Over the last two decades, two-dimensional (2D) materials have emerged as one of the most exciting classes of materials [1,2]. Members in this class include graphene [3,4], boronitrene or hexag- onal boron nitride (h-BN) [5,6], phosphorene [7,8], molybdenum disulfide (MoS 2 )[9,10], and silicene [11], etc., which have been shown to exhibit excellent properties and potential applications in electronics, energy conversion, and composites [12,13]. Experimen- tally, 2D materials can be fabricated by two main approaches: “top-down” (e.g., mechanical cleavage, liquid exfoliation, and ion intercalation) and “bottom-up” (e.g., chemical vapor deposition (CVD) and wet chemical synthesis) [14–17]. These 2D materials render a unique combination of mechanical properties, with high in-plane stiffness and strength but extremely low flexural rigidity. In thermodynamic equilibrium, the second law of thermodynamics predicts the inevitable existence of defects. In a nonequilibrium state, however, defects can be introduced either unintentionally or intentionally into 2D materials [18–21], which can cause either unde- sirable or desirable effects on their physical properties. For example, the actual performance of the graphene-based devices was found to be well below its intrinsic behavior due to inherent defects [22–24]. Although much effort has been devoted to the elimination of defects in 2D materials, it is still a significant challenge to accurately control the type, location, and density of those defects [25–29]. Importantly, due to the intrinsically brittle nature of 2D materials, understanding the consequence of those defects on failure behavior and the development of corresponding failure pre- vention strategies become crucial for the design, fabrication, and operation of 2D material-based nanodevices and nanocomposites. In 2D materials, the types of defects are diverse, including point defects (vacancy, dislocation, Stone-Wales (S-W) defect), line defect (grain boundary (GB)), pattern defect (chemical functionali- zation), areal defect (hole), and crack [21,30], as shown in Fig. 1. Significant efforts have been directed toward investigating the failure mechanisms in 2D materials under the influence of these defects. Due to the differences in lattice structures and bonding energies in 2D materials, atomistic configurations of these defects may take different forms. As a result, the defect energetics notably vary with the geometry, orientation, and system size, which in turn may significantly affect the failure behavior of 2D materials. It has been shown that the fracture modes in 2D materials can be very diverse and manifested in different forms, including tensile, shear, tear, chemical, and irradiation failures [30]. In this review, we would like to discuss these issues. Various mechanical theories have been developed to predict the failure of engineering materials, for example, the maximum stress criterion [31], the maximum strain criterion [32], and the Griffith theory [33]. With regard to the failure in 2D materials, a few funda- mental questions have yet to be addressed from both continuum and atomistic points of view. The Griffith theory, which was developed based on the linear-elastic fracture mechanics for brittle materials, is frequently used to predict the onset fracture in 2D materials. The essence of this theory is that when the increase of surface energy (or edge energy) for an infinitesimal extension of a crack is smaller than the decrease of strain energy, failure should occur. Based on this theory, the fracture strength σ cr can be expressed as [33] follows: σ cr =  2Eγ πa  (1) 1 These two authors contributed equally to this work. 2 Corresponding authors. Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 30, 2019; final manuscript received September 24, 2019; published online September 30, 2019. Assoc. Editor: Yonggang Huang. Journal of Applied Mechanics MARCH 2020, Vol. 87 / 030802-1 Copyright © 2020 by ASME Downloaded from https://asmedigitalcollection.asme.org/appliedmechanics/article-pdf/87/3/030802/6482882/jam_87_3_030802.pdf by guest on 05 July 2020