## Abstract

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 vacancies, 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 nanomaterials. This overview presents a useful reference for the defect manipulation and design of 2D materials toward engineering applications.

## 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 hexagonal boron nitride (h-BN) [5,6], phosphorene [7,8], molybdenum disulfide (MoS2) [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]. Experimentally, 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) [1417]. 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 [1821], which can cause either undesirable 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 [2224]. 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 [2529]. 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 prevention 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 functionalization), 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 fundamental 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 σcr can be expressed as [33] follows:
$σcr=2Eγπa$
(1)
where E is Young's modulus of the material without crack, a is the crack length, and γ is the surface energy. Applicability of the Griffith criterion to 2D materials is still a controversial issue. In some cases, this criterion has been validated both experimentally and theoretically at different length scales [3436]; while in many other cases, this criterion was shown to be invalid. A central issue is that at the nanoscale, these defects take various forms, shapes, and sizes, and as a result, it is difficult to define a crack. In addition, at the nanoscale, a variety of factors, such as atomic lattice, edge structure, defect concentration and interactions, intrinsic residual strain, out-of-plane wrinkling, rippling, defect concentration, and interaction, can affect the failure behavior of 2D materials [3743]. For example, the fracture mechanism in graphene is influenced by its lattice structure, which involves a fracture path that alternates the bond rotation and rupture [37]. Edge configurations and vacancy concentration in 2D materials can alter the fracture mode from brittle to ductile under tensile loading [38]. Prestretched bonds in defects, such as dislocations and S-W structure, may also influence the failure behavior [39,40]. Clearly, it is these features that are beyond the consideration of the Griffith theory, which result in its invalidation.

In order to understand and predict mechanical failures of 2D materials, the failure criteria based on nanoscale structural mechanics have also been proposed [4143]. Generally speaking, the structural mechanics is a discrete treatment that predicts the deformation and failure of bonds and bond angles in 2D materials by analyzing the deformation and failure of these basic structural elements. A clear advantage of the structural mechanics is that it can deal with failure phenomena in 2D materials even in the absence of a well-defined crack. Clearly, there is a need to discuss under what conditions the fracture mechanics criteria can be used and what conditions the structural mechanics criteria can be used.

Currently, there is still an ongoing debate whether nanoscale materials exhibit defect-tolerance or insensitivity. Previous studies have shown that in some cases, the presence of defects in 2D materials does not affect the mechanical strength and failure behaviors [4446], while in other cases, they may fail at much lower failure stress than their defect-free counterparts, that is, the fracture stress and strain are defect sensitive [30,47,48]. Hence, the questions are as follows: at the nanoscale, are mechanical failures of 2D materials defect sensitive? What are the factors that control the fracture behavior? Answers to these questions not only require the understanding of atomic structures of those defects but also their influences on the mechanical and failure behaviors. In this review, we would like to discuss these issues.

With these issues in mind, we primarily focus on four representative 2D materials, that is, graphene, h-BN, MoS2, and phosphorene, with emphasis on the aforementioned seven typical types of defects. Our aim is to provide a comprehensive overview of the recent development in the understanding of the effects of nanoscale defects on the failure behaviors of 2D materials from the viewpoints of fracture mechanics and structural mechanics and address the issues raised in Sec. 1. Our arrangement is as follows: Sec. 2 first introduces the atomistic configurations of seven types of defects based on the understandings from previous modeling, experiment, and simulation studies. The size, shape, and stability of these defects in the four representative 2D materials are summarized. Subsequently, the failure mechanisms due to the presence of defects are discussed in Sec. 3. In Sec. 4, brief accounts including the origins of mechanical failures, failure strength sensitivity, and failure criteria are also presented to understand the failure behaviors of 2D materials. In Sec. 5, we provide a summary and outlook for further studies related to defect reduction, defect engineering, and mechanical design principles for 2D materials.

## 2 Defect Types

### 2.1 Vacancies.

Vacancies (see Figs. 2(a) and 2(b)) are the most studied point defects in 2D materials [8,4951]. They can be either unintentionally formed during fabrication processes or deliberately introduced by processing. In atomistic simulations, these defects are generated by removing atoms from their two-dimensional lattices. A single vacancy involves an elimination of a single atom, while a double vacancy requires a deletion of a pair of adjacent atoms. Note that double vacancies with an even number of missing atoms allow the complete saturation of dangling bonds, and therefore, they are thermodynamically favored over single vacancy or triple vacancies with an odd number of missing atoms [52]. Subsequently, the atomic structure around the created vacancy is relaxed through geometry optimization.

### 2.2 Dislocations.

Dislocations are a different type of point defects in 2D materials (see Figs. 2(c)2(g)), which may be formed in the course of CVD growth, electron beam sputtering or as a combination of other point defects [53]. Dislocations can form grain boundaries and cause out-of-plane enfolding of monolayers [46]. They often have a significant impact on the mechanical, electronic, magnetic, and optical properties of 2D materials.

Dislocations in graphene are represented by pentagon-heptagon (5|7) pairs (see Fig. 2(c)). A (5|7) pair is the most energetically favorable edge dislocation with the smallest Burger vector [54]. In graphene, dislocations generate a long-range stress field with high strain energy, which is partially released by graphene out-of-plane buckling around them [55,56] (see Fig. 2(d)).

Dislocations in h-BN are represented by (5|7) and (4|8) pairs (see Fig. 2(e)). Two different atomic species in boronitrene result in a new type of (4|8) dislocations. The (4|8) dislocations maintain the alternating order of B- and N-atoms and thus have a lower electrostatic energy than (5|7) ones, but a higher strain energy. Analogous to graphene, the strain energy is partially reduced via out-of-plane buckling around dislocations.

In phosphorene, both (5|7) and (4|8) dislocations (see Figs. 2(f) and 2(g), respectively) are present due to the anisotropic puckered structure. Counter to the resilient sp2-hybridized bonds of graphene (or boronitrene), the much weaker sp3-hybridized P-P bonds of phosphorene create a flexible lattice, which is able to accommodate dislocations without noticeable out-of-plane buckling.

### 2.3 Stone-Wales Defects.

An extraordinary property of 2D materials with the hexagonal lattice structure is their capability to form nonhexagonal rings via local bond rotations. This new class of point defects in 2D materials are formed without addition or elimination of atoms. These defects are called topological defects, and the most extensively studied one is the S-W defects in graphene [57,58] (see Fig. 3(a)). An S-W defect is formed by a 90 deg rotation of a pair of adjacent carbon atoms around the midpoint of the connecting bond (see Fig. 3(a)). The rotation converts four adjacent neighboring hexagons into two pentagons and two heptagons (5|7|7|5 defect). The bonds between pentagon and adjacent hexagons, as well as the central rotated bond, are compressed, while the bonds between heptagons and pentagons are stretched. The strain energy of an S-W defect is further lowered by the out-of-plane buckling around it [59,60].

The S-W defects were also found in other 2D materials such as boronitrene, transition metal dichalcogenides, and phosphorene [8,9,6163]. In boronitrene, like in graphene, an S-W defect is made by the 90 deg rotation of two adjacent B- and N-atoms around the midpoint of the bond connecting them [61] (see Fig. 3(b)), although the bond rotation results in the creation of two homoelemental bonds, which are energetically unfavorable. Thus, the defect formation energy of an S-W defect in boronitrene is higher than that in graphene [64], while the lowest formation energy is in phosphorene [19] due to its peculiar puckered structure.

### 2.4 Chemical Functionalization.

Chemical functionalization is one of the main tools applied to examine the mechanisms of interaction of 2D materials with their environment [65]. Different chemical groups (hydrogen, chlorine, fluorine, and oxygen [52]) can be attached to 2D materials (see Fig. 3(c)) to modify their structures and properties [66]. Chemical functionalization makes possible solvent-assisted processing of 2D materials and prevents the agglomeration of monolayers. Functionalization of 2D materials is crucial for a variety of technological applications, such as hydrogen storage, spintronics, transistors, sensors, supercapacitors, drug delivery, flexible electrodes, and polymer nanocomposites [67,68].

There are two principal types of functionalization, the covalent one (with covalent bond formation) and noncovalent one (with physical adsorption of molecules on 2D materials). The covalent functionalization results in much stronger modification of the structures and properties of 2D materials. The structural alteration can occur at the edges and/or in the interior. In graphene and boronitrene, chemical functionalization is associated with re-hybridization of the sp2 configurations into the sp3 ones. The sp3 defects are very ubiquitous in functionalized 2D materials. The covalent modification of 2D materials can be achieved in different ways (nucleophilic substitution, electrophilic addition, and condensation) with different modifying agents [69].

### 2.5 Grain Boundary.

GB, an interface between two adjacent grains, is a line defect in 2D materials (see Figs. 3(d)3(g)). Large-scale graphene produced by CVD is constituted by single-crystalline grains with different lattice orientations. This variety in grain orientations inevitably leads to the presence of grain boundaries. A GB in 2D materials consists of an array of dislocations arranged in a linear way. Generally, GBs are classified by the value of the tilt angle separating two neighboring grains: small- and large-angle grain boundaries. In the small-angle GBs, dislocations are separated by a sufficiently large distance, while in the large-angle GBs, dislocations are closely spaced (and sometimes overlapped).

GBs in graphene are composed of linear arrays of aligned (5|7) edge dislocations (see Fig. 3(d)). This symmetric atomistic configuration is energetically favorable: deviation from the bisect increases the interface energy. Introduction of GBs in free-standing graphene often results in out-of-plane warping near the low-angle GBs, which relieves the in-plane strain at GBs [55,70,71].

GBs in boronitrene are formed by linear arrays of (4|8) dislocations (see Fig. 3(e), left). The strain and the type of bonding determine the formation energy of GBs in boronitrene. Energetically costly homoelemental B–B and N–N bondings in the (5|7) dislocations (see Fig. 3(e), right) make the (4|8) dislocations with heteroelemental B–N bonding energetically preferable [8] despite an increase in the lattice strain energy [72,73].

GBs in MoS2 are formed by arrays of (5|7) dislocations (see Fig. 3(f)). In contrast to boronitrene, GBs in MoS2 contain only by (5|7) dislocations since the high flexural rigidity of triple-layered MoS2 precludes the possibility of the out-of-plane buckling along GBs [54].

GBs in phosphorene are formed by (4|8) dislocations along the ZZ direction (see Fig. 3(g), left panel) and (5|7) dislocations along the AC direction (see Fig. 3(g), right panel). Phosphorene with GBs preserves its planar shape avoiding out-of-plane wrapping due to its unique puckered structure [74,75].

### 2.6 Holes.

Holes in 2D materials are a type of structural defects that can be formed either by coalescence of a number of vacancies or by electron beam irradiation via knocking atomic clusters from the lattice (see Figs. 3(h) and 3(i)). Once a hole is formed, it continuously grows under the electron beam. Thus, the size of hole defects can be monitored by the electron beam irradiation time at a given acceleration voltage.

Holes in graphene (see Fig. 3(h)) have mixed AC and ZZ atomic structures, as well as reconstructed 5–7 zigzag configurations at the edges [44]. Graphene is highly reactive at the edges of the holes that have dangling bonds. The hole edges in graphene are filled with nearby C adatoms within a few hours even under ultra-high vacuum conditions [76]. Therefore, stabilization of hole edges via passivation is a key issue for the application of graphene as membranes with nanosized pores.

In boronitrene, hole defects often have a triangular shape with N-terminated edges (see Fig. 3(i)), which is a unique feature as boronitrene contains two types of atoms. The triangular shape is preserved as it grows under the prolonged electron beam irradiation. The edges have an exclusively zigzag configuration, which is another unique feature of boronitrene compared with the edges of other 2D materials [76].

### 2.7 Crack.

A crack defect may appear in 2D materials after a pore was formed either spontaneously or with electron beam irradiation. They may acquire different sizes and orientations, and their role in 2D material failure is often critical: this type of defects degrade the mechanical properties of 2D materials often in a dramatic manner, which results in sudden, premature failure.

## 3 Failure Behavior Due to Defects

### 3.1 Mechanical Failure Due to Vacancy.

Vacancies play an essential role in mechanical failure of 2D materials: They are often the sources for heterogeneous crack nucleation. As opposed to pristine graphene, the crack nucleation in defective graphene is often through the rupture of bonds in the vicinity of a vacancy. Consequently, a nano-sized cavity nucleates around the broken bonds. At a low vacancy concentration (nv ≲ 0.08), graphene undergoes a brittle failure as a crack initiated around a vacancy propagates under the applied tensile load [77]. This process ultimately leads to the failure of graphene. At a high vacancy concentration (nv > 0.08), graphene becomes more ductile [78].

Similar to graphene, atomic vacancies in phosphorene lead to a significant degradation of its mechanical properties. Under tension, stress distribution in phosphorene is almost uniform, except at the small regions around vacancies where the stress is concentrated [79]. The stress concentration causes much earlier bond breaking (compared with defect-free phosphorene) at the defect region during deformation process. As shown in Fig. 4(a), the bond-breaking results in crack nucleation around a vacancy, which spontaneously propagates perpendicular to the loading direction, leading to the ultimate failure of defective phosphorene [79]. The reduction in the fracture strength depends on the type of vacancies and loading direction. Atomistic simulations show that divacancies cause a larger reduction in the fracture strength of phosphorene than monovacancies along the armchair direction [79].

### 3.2 Mechanical Failure due to Dislocation.

The fracture strength and failure mode of 2D materials are strongly affected by the presence of dislocations. Among the various defects, dislocations are particularly important due to their long-range stress field. If dislocations are located in the vicinity of a crack, they modify the stress distribution at the crack tip and consequently the critical stress intensity factor required for crack propagation [80]. Combining atomistic simulations and continuum modeling, Meng et al. [80] studied dislocation shielding of a crack in a single-layer graphene. They examined the shielding effects on the threshold stress intensity for crack propagation along the AC and ZZ directions and found that the shielding effect of dislocations effectively reduces the crack-tip driving force. The shielding effect is a direct demonstration of the interaction between the stress fields of dislocations and crack. The shielding effect increases as the distance between the crack tip and dislocation decreases. When the distance between the crack tip and dislocation changes (with respect to the crack size), the shielding effect displays two dissimilar dependencies on the crack-tip dislocation distance: the near-tip one and the far-field dependencies, which arise from an explicit manifestation of the long-range stress field of dislocation in graphene. In addition, they found that dislocation shielding is insensitive to the shape of the crack. In the presence of dislocations, fracture of graphene is brittle, and the shielding effect can be accurately described by continuum linear-elastic fracture mechanics [80].

### 3.3 Mechanical Failure Due to Stone-Wales Defects.

Stone-Wales topological defects (nonhexagonal rings generated by reconstruction of hexagonal lattice) may play a significant role in the mechanical failure of 2D materials. Yet their role in the mechanical failure of 2D materials has long been intensely debated [8183]. Sun et al. [81] investigated the in-plane deformation of graphene with S-W defects under tensile loading using MD simulations. Surprisingly, they found that nearly all the embedded S-W defects can be eliminated via the inverse C–C bond rotations in the stretched graphene. As a consequence, they found that the strength of defective graphene with S-W defects is nearly indistinguishable from the pristine one [81]. The total annihilation of S-W defects is particularly evident at low strain rate in which the deformation rate of the graphene lattice is by several orders of magnitude smaller than the velocity of C–C bond rotation. In contrast, Ansari et al. [84] found that the presence of S-W defects, similar to single and double vacancies in graphene, leads to the local stress concentration, which in turn results in crack nucleation at a lower critical strain (in comparison with defect-free graphene). Fracture starts when a nanocrack appears at an S-W defect, where one of the C–C bonds shared by the heptagon and nearby hexagon breaks at a critical strain (see Fig. 4(b)). As the neighboring bonds break, the growing crack propagates perpendicular to the loading direction, as shown in Fig. 4(b). The ultimate failure of graphene occurs when the last connecting carbon chain fails [50,84,85]. By using MD simulations, Xu et al. [50] also demonstrated that the effect of S-W defects (at low defect concentration) is practically indistinguishable from the effect of vacancies as shown in Fig. 4(b). At higher concentration (≥7%) of S-W defects, defective graphene turns out to be more ductile. This conversion from a brittle-to-ductile fracture is due to the relocation of atoms around the S-W defects, which trap and blunt the advancing crack tip [50,82,83]. Similarly, Wang et al. [83] found that at a high defect concentration, the interaction between S-W defects becomes sufficiently strong so that they may aggregate and block crack-tip propagation, which leads to a more complicated pattern of fracture of defective graphene.

Similar to graphene, fracture of defective boroniterene containing S-W defects is brittle [48]. Brittle fracture begins near the S-W defects when boronitrene is strained along the AC direction. However, if it is stretched along the ZZ direction, the fracture begins far away from the S-W defects. The reason for the difference is that the tensile strain applied in parallel to the orientation of the most compressed central bond of the S-W defect reduces significantly the likelihood of fracture onset at the defect site [48].

### 3.4 Mechanical Failure Due to Chemical Functionalization.

Influence of chemical functionalization on the mechanical properties and failure of graphene was explored by Pei et al. [86]. They systematically investigated the mechanical failure of hydrogen functionalized graphene using MD simulations. It was found that the strength of the functionalized graphene significantly deteriorates due to the sp2-to-sp3 conversion of C–C bonds with the increasing H-coverage. It is well known that a sp3 bond is weaker than a sp2 bond, and it ruptures at a much lower tensile strain [86]. Besides, the newly created sp3 bonds can easily rotate and reorient themselves in the stretched graphene. The latter mechanism is unique to the 2D geometry of graphene and does not have a counterpart in 3D systems [87]. As a result of the sp3 bond breaking, a crack nucleates around the hydrogen passivated atoms. Then, the bonds outside of the hydrogenated regions start to break, leading to the formation and propagation of a crack and ultimately to the tearing of graphene (see Fig. 4(c)).

Functionalization of graphene with methyl group CH3 leads to similar results: In the limit of large strains, a fracture is primarily controlled by breaking of the weakest bonds [88]. Even a small amount of CH3 coverage can cause local bonds to change their hybridization from sp2 to sp3, which in turn triggers the sp3 bond breaking at a much earlier stage during deformation and therefore significantly reduces the strength of graphene.

### 3.5 Mechanical Failure Due to Grain Boundary.

The role of GBs in mechanical failure is of particular importance for polycrystalline 2D materials. A number of studies revealed many intriguing aspects related to the role GBs play in the failure process [89]. In general, GBs reduce the bonding strength across the interface between adjacent grains. However, the strength reduction in graphene with high-angle GBs (that have a high density of edge dislocations per unit length) is unexpectedly lower than that due to low-angle GBs [46,90]. This was at first observed in atomistic simulations and then confirmed experimentally. The graphene with high-angle tilt boundaries is almost as strong as the pristine one and is much stronger than that with low-angle boundaries [46,90], which is contrary to conventional reasoning. This unusual effect is explained by the fact that edge dislocations, which form the GBs, are evenly spaced and located sufficiently close to each other in high-angle GBs, and as a result, their strain fields overlap and partially cancel each other. Consequently, the critical bonds (the bonds which rupture first at a critical strain) in the high-angle GBs rupture at a significantly higher critical strain than those in the low-angle tilt GBs (see Fig. 4(d)). The large-angle boundaries are stronger since they can better accommodate these strained rings than the low-angle GBs [46,90].

Like graphene, a high defect concentration at GBs in phosphorene does not lead to a larger decline of its mechanical properties [8,75]. In addition to the six-atom ridged rings, phosphorene may also contain four-, five-, seven-, and eight-atom rings as defects. These defects can form arrays of linearly arranged (5|7) and (4|8) edge dislocations, which constitute GBs separating phosphorene grains with different orientations. By using tight-binding simulations, Sorkin and Zhang [75] showed that the first bond breaking at the beginning of failure in phosphorene is the most stretched bond in heptagons of (5|7) and octagons of (4|8) and oriented along the tensile direction (see Figs. 5(a) and 5(b), correspondingly). Like in graphene, phosphorene with high-angle GBs formed by densely packed edge dislocations is much stronger than those with low-angle GBs. The high tilt angle boundaries are better to accommodate strain. The shorter the distance between the edge dislocations is, the stronger the mutual cancellation of their overlapping stress fields is. For this reason, the tensile prestrain of the critical bond is lowered in the high-angle GBs. Once a crack is formed at the GB, it will propagate along the GB under applied tensile strain perpendicular to the GB as demonstrated Guo et al. [91] using MD simulations. They also found that both fracture strength and strain decrease with increasing temperature, making fracture more probable at relatively high temperatures. The effect of strain rate on both fracture strength and strain is insignificant, demonstrating that phosphorene is a typical brittle 2D material [91].

The high-angle symmetrical GBs of boronitrene are also stronger than the low-angle GBs [72]. Atomistic simulations revealed that failure of boronitrene is similar to that of graphene and phosphorene. At a critical strain, boronitrene with GBs goes through a brittle fracture: the rupture occurs sharply without any indications [72].

The failure behavior of polycrystalline MoS2 monolayers with GBs under uniaxial tensile was studied by Dang and Spearot [92] using MD simulations. It was found that the failure initiates at GBs [92], and the nucleated crack predominantly propagates along the ZZ direction.

The above discussion on the effect of GBs on the mechanical failure of 2D materials only concerns GBs themselves. However, in polycrystalline samples, the terminations of GBs (GB junctions where one-dimensional regions where three or more adjacent GBs meet) are crucial for failure: they generally serve as accumulators of prestrain (stress concentrators) and sites for crack nucleation, thus lowering the fracture strength of 2D materials. Due to relatively small grain size in 2D materials, the dislocations generally cannot be generated within the grain interior, and therefore, these materials fail often at or along GBs.

MD simulations predict that the bond breaking may start at GB junctions [45,46,89,93] in graphene. According to the study by Sha et al. [45], GB junctions serve as crack nucleation sites. The crack nucleation occurs at various types of GB junctions (triple, quadruple, or higher junctions) in polycrystalline graphene provided that one of the connecting GBs of the junction is oriented perpendicular to the loading direction (see Fig. 5(c)). Since GB junctions in polycrystalline graphene are generally more defected and contain a combination of five, seven, and eight-membered rings, they are more compliant than defect-free grain interior and thus expected to have a lower load-carrying capacity [45]. As a result, the connecting GB, which is perpendicular (or nearly perpendicular) to the direction of applied strain, carries extra load and thus is most prone to cracking (see Fig. 5(c)). A crack preferentially nucleates at this specific type of GB junction via a direct rupture of the sp2 C–C bonds, rather than the motion of edge dislocations. It propagates initially along the GB (where it appeared) and then branches out along other connected GBs or sometimes across the grain interior. The predominant intergranular propagation is due to the fact that GBs are generally weaker than the crystalline grain [45]. Ultimately the crack propagation leads to the failure of polycrystalline graphene. This failure mechanism is explained by the “weakest-link” model [94], which uses a statistical model to describe and understand the distribution of strength and toughness of polycrystalline graphene. This model shows that the statistical variation in strength and toughness of polycrystalline graphene can be understood with “weakest-link” statistics. It takes into account the system size, grain size, and strain rate dependence for statistical fluctuation of strength and toughness, and thus, it is regarded as a failure criterion for graphene and other 2D brittle materials.

Similar to graphene, fracture in boronitrene and MoS2 is also brittle: Once a bond ruptures near a triple junction or a grain boundary, the released stress from the broken bond raises the stress on the nearby bonds, which consequently rupture [95,96]. This process proliferates, causing a cascade of bond breaking and resulting in brittle fracture of polycrystalline boronitrene and MoS2 [95,96].

### 3.6 Mechanical Failure Due to Hole.

The presence of holes in 2D materials modifies their strength and affects the failure mechanisms. One of the key questions is the hole size effect on the fracture mechanism. Using atomistic simulations, Zhang et al. [44] revealed that there is a critical size of a hole (embedded in graphene nanoribbon) below which the material is insensitive to the presence of the hole defect (flaw insensitivity). This means that when the hole size is below the critical one, the mechanical failure is not initiated at the hole defect and the strength of graphene nanoribbon is independent of the hole size [44].

Sha et al. [89] further explored the flaw insensitivity in large-area polycrystalline graphene containing a pre-existing circular hole. They confirmed that there is a critical hole size separating the flaw-sensitive and flaw-insensitive failure modes [89]. They found that the critical hole size is related to the average grain size of polycrystalline graphene. In the case of the flaw sensitive failure, when the hole size is above the critical one, the fracture in polycrystalline graphene is initiated at a hole defect (see Fig. 5(d), top) in line with the classical theory of stress concentration [89]. If a hole and its nearby GB are located sufficiently close to each other, then there is an overlap in their stress fields. In addition, if the GB is oriented perpendicular (or almost perpendicular) to the loading direction, then a small crack appears at this GB (see Fig. 5(d)). When the small crack arises at the hole, the stress concentration in the adjacent region is released, while the highest stress localized at the crack tip is driving the crack to grow further away from the hole (see Fig. 5(d)). When the hole size is below the critical one and the hole defect is located within the grain interior, the polycrystalline graphene becomes flaw insensitive [89]. In this case, the fracture does not initiate near the hole, but rather at some distance away from it (see Fig. 5(d), bottom). Although there is a stress concentration zone at the hole, the stress is not large enough to break the strong sp2 C–C bonds in the grain interior. Therefore, the failure initiates at the weakest GB elsewhere, as the GBs are weaker than the pristine crystalline interior of the grains [89].

### 3.7 Mechanical Failure Due to Crack.

Similar to the holes, the pre-existing cracks in 2D materials are the sources of stress concentration and the regions where the fracture can be initiated. The fracture process in this case is the growth of the pre-existing cracks due to an applied load that eventually results in the failure of 2D materials.

For graphene containing a pre-existing crack, the fracture mechanism involves alternating bond breaking and bond rotation at the tip of a crack [37]. The stress intensity factor near the crack tip (caused by a remote load) is proportional to Young's modulus and edge energy. Since Young's modulus is isotropic for graphene, the crack propagates along the directions with the minimal edge energy that is along AC and ZZ edges. The ZZ direction is more preferable since its edge energy is slightly lower than that of the AC edge [80]. Experimental observations [30] and MD simulations [80] indicate that the crack path in graphene is mostly represented by piecewise-straight AC or ZZ edge with sharp turns. Thus, anisotropic properties lead to different crack propagation paths in graphene under tensile loading. In addition, temperature and other defects may also affect the crack propagation [50,97].

Two representative stress–strain curves for a crack propagating along the AC and ZZ directions [80] are shown in Figs. 5(e) and 5(f). Under the applied tensile load, the stress increases monotonically until the onset of crack propagation corresponding to an abrupt stress drop in the stress–strain curve. The linear stress–strain response before the onset of fracture and the subsequent sharp decrease indicate that the fracture is brittle. The insets of Figs. 5(e) and 5(f) illustrate the atomic configurations near the crack tip before, at, and after crack initiation [80]. The bond at the crack tip is subjected to the highest tensile stress immediately before the crack initiation. This bond inevitably ruptures upon additional loading, followed by crack propagation. The crack running along the ZZ direction is characterized by a clean unidirectional cleavage (see Fig. 5(e)), whereas the jagged crack path is typical for the crack propagating along the AC direction (see Fig. 5(f)).

Other hexagonal graphene-like 2D materials, including boronitrene [98], phosphorene [91], and silicene [11,99] containing pre-existing cracks, behave in a similar way under a tensile load. These single-layer materials fail in a brittle way, and their failure is governed by the strength of the atomic bonds at the crack tip. Similar to graphene, the ZZ direction is the preferable direction of crack propagation in these materials, which indicates that the bonds in the AC direction have the stronger capability to resist crack propagation than the ZZ direction [21,78,98100].

## 4 Discussion

The failure behaviors of 2D materials can be very diverse and manifested in different forms. The defects are the sources of stress concentration and ultimately the regions where the fracture can be initiated. Previous studies have shown that fracture behavior can be shape dependent or shape independent, size dependent or size independent, and defect sensitive or defect insensitive. Our analyses have shown that the mechanical failure arises from the competition among different defects. The types of defect, together with their chirality, size, distribution, as well as loading direction, are crucial factors that affect the failure modes.

The fracture behavior of 2D materials may exhibit some unique features. For examples, the divacancies cause easier failure than monvacancies along the armchair direction in phosphorene [79]. The presence of S-W defect in high concentration makes the defective graphene more ductile due to trapping through atomic arrangements and blunting at a propagating crack tip [50]. Besides, the fracture of graphene ruga with the periodic arrangement of pentagons and heptagons was systematically studied [101], showing that the fracture toughness of graphene with distributed topological defects is twice that of pristine graphene. Such toughness enhancement arises from nanocrack shielding and atomic scale crack bridging. For functionalization on graphene surfaces, during the tensile loading, the newly created sp3 bonds can easily rotate and reorient themselves in the stretched graphene [86,88]. Besides, the patterned chemical functionalization arranged in lines perpendicular to the tensile direction leads to a larger strength deterioration than that parallel to the tensile direction. Furthermore, graphene and phosphorene containing large-angle tilt boundaries with an array of dislocations in a high density are nearly as strong as the pristine material but much stronger than those with low-angle boundaries. The fracture toughness of polycrystalline graphene and graphene with randomly distributed GBs can also be 50% and 100% that of pristine graphene, respectively [39,102].

For 2D materials containing both dislocations and cracks, the shielding effect on the threshold stress intensity for crack propagation along the AC and ZZ directions was observed. The shielding effect of dislocations was found to effectively reduce the crack-tip driving force. As the separation between the crack tip and dislocation varies (with respect to crack size), the shielding effect exhibits two different dependences on crack-tip dislocation distance: the near-tip one and the far-field dependency, which are an explicit manifestation of the long-ranges stress field of dislocation in graphene [80].

In addition, the dislocation shielding is insensitive to the shape of the crack. For 2D materials containing both GBs and holes (or notches), if the hole size is smaller than the grain size and the hole is located within the grain interior, the material will be insensitive to the presence of the hole defect [89]. Besides, recent experimental studies showed that in the graphene with high density of atomic defects, the crack propagation is also strongly retarded due to the shielding of defects [103]. It is noted that both dislocations and GBs have relatively long-range effects. However, the structural mechanics approach proposed in this work is based on the maximum bond-stretching criterion, where the critical bond is subjected to the highest tensile stress, and thus, this criterion works for different types of defects with either local or long-range effect.

In 2D materials, the mechanical strength and the mechanical failure are generally sensitive to flaws. If the Griffith criterion written in Eq. (1) can be applied to a crack in the nanoscale scale, then we can have the following relation:
$Δσcr=−2Eγπ1a3Δa$
(2)

Equation (2) is obtained by differentiating Eq. (1). Equation (2) shows that the variation rate of the fracture strength is a function of $1/a3$. When the crack length becomes smaller, the value $1/a3$ becomes increasingly larger. As a result, for the same Δa, the fluctuations of fracture strength also becomes increasingly larger, suggesting that at the nanoscale, the fracture strength is becoming increasingly more sensitive to the smaller flaw size [36,104]. It should be noted that since Eq. (2) is derived from the Griffith rule, which is valid for a well-defined crack, this means that Eq. (2) is only applicable for a well-defined crack in 2D materials.

Our analyses have shown that the introduction of atomic defects in small quantity can lead to a drastic reduction in failure strength and/or a change in failure patterns. In 2D materials, there exist many different types of defects, and these defects may interact either in a constructive or a destructive manner. When they interact in a constructive manner, the failure strength can be decreased. However, if they interact in a destructive manner, the failure strength can be increased or kept unchanged. Also the defects may play a competitive role in a failure process. The strengths of defective 2D materials are generally lower than those of perfect ones, which indicates that the mechanical strength is generally sensitive to flaws. If we focus our attention on specific defects, but neglect other defects that actually control the failure of a 2D material, we may come to the wrong conclusion.

Obviously, some of the fracture behaviors in 2D materials may not be described by Griffith-type fracture mechanics criteria since the Griffith theory does not take into account many factors important at the nanoscale, such as atomic lattice, edge structure, defect concentration and interactions, and intrinsic residual strain, out-of-plane wrinkling and rippling, which can affect the failure behavior of 2D materials. All these nanoscale factors that are beyond the consideration of the Griffith theory may result in its invalidation. Hence, the Griffith theory can be applied to predict the failure behavior only under the condition that there is a well-defined crack. On the other hand, the structural mechanics predicts the failure in 2D materials by analyzing the bond deformation and failure, which depends on the bond length, bond angle, and surrounding environment. An evident advantage of the structural mechanics is that it can be used to investigate the failure phenomena in 2D materials even in the absence of a well-defined crack. Since these kinds of mechanical failures are common in many nanomaterials, the structural mechanics approach based on the maximum bond-stretching criterion could be widely applicable to other types of nanomaterials.

## 5 Summary and Perspectives

We presented an overview of the current understandings of seven different types of defects, that is, vacancies, dislocations, S-W defects, chemical functionalization, grain boundary, holes, and crack. Focusing on graphene, h-BN, MoS2, phosphorene, and their defective structures, various failure behaviors were reviewed and discussed in details. At the nanoscale, defects can take various forms, shapes, and sizes, and thus, it is often difficult to define a crack. We showed that the failure strength and mode of 2D materials are generally sensitive to flaws since the strengths of real defective 2D materials are generally lower than those of perfect 2D materials. Besides, due to the presence of nanoscale factors, such atomic lattice, and bond prestretching, the Griffith theory fails to capture many of the fracture behaviors in 2D materials. While the structural mechanics analysis based on the maximum bond-stretching criterion, on the other hand, is able to explain many failure behaviors of 2D materials, where the critical bond length is a gauge for the bond failure, regardless of diverse types of defects. A few examples have been demonstrated and discussed in details, including high-angle tilt boundaries in single GB graphene and phosphorene, triple junction in polycrystalline graphene, and a hole in polycrystalline graphene, where the failure occurs through the rupture of the most prestretched chemical bonds in the atomic structure [45,46,75,89,90]. The present review not only reveals insights into the effects of defects on the mechanical failure behavior of 2D materials and serve as a guide for the development of new research directions in atomic level structural mechanics but also provides a useful reference for the design and fabrication of 2D material-based nanodevices.

The presence of various defects has a profound influence on the mechanical properties of 2D materials. However, up to now, most of the relevant research works are based on theoretical methods, such as MD simulations. Experimental studies are very limited due to the challenges in nanoscale experiments. Here, we suggest several issues that need to be addressed by experiments for better understanding of the mechanical behaviors of defective 2D materials. Since it is still difficult to control the size, type, shape, distribution, and concentration of defects in 2D materials during their growth and fabrication, a greater effort is needed to further reduce or heal these defects to achieve large-scale, high-quality production of 2D materials. On the other hand, if defects cannot be avoided, then defect engineering approach can be used to manipulate the defects to reduce their effect. For example, during the functionalization of 2D material or the introduction of certain intrinsic defects, we could utilize prestraining construction or destruction concepts to control the overall prestrain level. In addition, according to the maximum bond-stretching criterion, the tensile loading should be applied along the bond precompressing direction to achieve the optimized fracture strength and fracture strain.

With the rapid development of experimental capabilities, it is expected that the fracture behavior of 2D materials can be fully understood in the near future by manipulating the various defects, so that the mechanical properties of 2D materials can be better controlled for future applications of 2D materials in nanodevices. Defects in 2D materials could be engineered to realize new structural configurations or certain exciting properties, which may provide new opportunities for novel applications, such as in electronics and optoelectronics, quantum computing, thermoelectrics, thermal management, chemical sensing, biological sensing, energy storage, and conversion. Thus, experimental methods may be further developed to uncover the coupling phenomena and characterize the failure mechanisms of 2D materials. In such applications, these defects are subjected not only to mechanical loading but also to electrical field, magnetic field, irradiation, high temperature field, harsh chemical, and/or biological environments. An interesting and yet challenging direction is to study the failure behavior and failure criterion under these combined actions of various factors. For example, graphene with some particular chemical functionalizations or ultra-fine pores could be utilized for specialized filtering and selective membranes in chemical and biological applications [105,106]. Clearly, comprehensive studies are needed to further understand the failure behaviors in defective 2D materials under complex conditions and to explore their corresponding failure criteria.

## Acknowledgment

V. Sorkin, Q. X. Pei, and Y. W. Zhang acknowledge the support of a grant from the Science and Engineering Research Council, A*STAR, Singapore (152-70-00017). H. Qin acknowledges support from the China Scholarship Council.

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