**2. Fraction anisotropy in graphene**

It has been known that graphene, graphene oxide and their composites exhibit mechanical anisotropy due to their characteristic of 2D extension [17, 18]. The investigations on the superlubricity of 2D nanomaterials have been also extensively conducted. A classic example is that M. Dienwiebel et al. studied the energy dissipation of a graphite at selective sliding directions on a Tribolever setup equipped with a tungsten tip and found the ultralow friction with the incommensurability nature [12]. Another example is M. Poot and H. S. J. Van der Zant adopted atomic force microscope (AFM) to measure force-distance relations on few-layer graphene and graphite flakes and discovered that a principle direction represents a stiffer direction than the others [19]. In contrast to those studies, a molecular dynamics simulation (MD) study is particularly selected to exhibit the anisotropic mechanical behaviors of graphene monolayers under uniaxial tensile condition along the zigzag and armchair directions [11]. 4.15 4.15 nm<sup>2</sup> square-shaped graphene monolayers with a thickness of 0.335 nm were fixed at one end and the tensile tests along the zigzag and armchair directions are present in **Figure 2a** and **b**. The relations between applied force and one unit cell are also present. The specific parameters of the non-equilibrium MD simulations can be found in the literature.

Regardless of the fracture patterns, the MD experiments first calculated the fracture stresses along the zigzag and armchair directions, which are 0.18 TPa at a strain of 32.48% and 0.21 TPa at 43.85%, respectively. In is worth noting that the predicted critical stresses and strains are anticipated to be higher than the empirical ones due to the idealism in the conditions of MD simulations. Prior to the crack formation, two test modes share similarity, that is, in the elastic region, the graphene monolayers regardless of testing directions could sustain large elastic deformation and upon crack formations, the crack propagated rapidly and led to the final fracture within 0.01% strain, suggestive of a brittle cleavage fracture. For the zigzag direction, within the strain from 32.484% to 32.489%, the crack propagated from one edge to the other edge, forming a zigzag-like fracture topography and the

#### **Figure 2.**

metal coatings or thin films have higher hardness and strength, abiding by the wellknown Hall–Petch relationship. However, the abundant columnar GBs with directionality are often the sites where the voids reside. The sluggish adatom kinetics and the shadowing effect from the surface e roughness lead to void formation residing at the columnar GBs and void-free GBs have lower cohesive energy when compared

It has been known that graphene, graphene oxide and their composites exhibit mechanical anisotropy due to their characteristic of 2D extension [17, 18]. The investigations on the superlubricity of 2D nanomaterials have been also extensively conducted. A classic example is that M. Dienwiebel et al. studied the energy dissipation of a graphite at selective sliding directions on a Tribolever setup equipped with a tungsten tip and found the ultralow friction with the incommensurability nature [12]. Another example is M. Poot and H. S. J. Van der Zant adopted atomic force microscope (AFM) to measure force-distance relations on few-layer graphene and graphite flakes and discovered that a principle direction represents a stiffer direction than the others [19]. In contrast to those studies, a molecular dynamics simulation (MD) study is particularly selected to exhibit the anisotropic mechanical behaviors of graphene monolayers under uniaxial tensile condition along the zigzag and armchair directions [11]. 4.15 4.15 nm<sup>2</sup> square-shaped graphene monolayers with a thickness of 0.335 nm were fixed at one end and the tensile tests along the zigzag and armchair directions are present in **Figure 2a** and **b**. The relations between applied force and one unit cell are also present. The specific parameters of

the non-equilibrium MD simulations can be found in the literature.

Regardless of the fracture patterns, the MD experiments first calculated the fracture stresses along the zigzag and armchair directions, which are 0.18 TPa at a strain of 32.48% and 0.21 TPa at 43.85%, respectively. In is worth noting that the predicted critical stresses and strains are anticipated to be higher than the empirical ones due to the idealism in the conditions of MD simulations. Prior to the crack formation, two test modes share similarity, that is, in the elastic region, the graphene monolayers regardless of testing directions could sustain large elastic deformation and upon crack formations, the crack propagated rapidly and led to the final fracture within 0.01% strain, suggestive of a brittle cleavage fracture. For the zigzag direction, within the strain from 32.484% to 32.489%, the crack propagated from one edge to the other edge, forming a zigzag-like fracture topography and the

to the grain interiors. Z. S. You et al. found that nanotwinned (NT) Cu with columnar grains packed with horizontal coherent twin boundaries (TBs) experienced inhomogeneous deformation and columnar GBs were subjected to much larger plastic strain, compared to the grain interiors [16]. This caused one ambiguous puzzle, that is, the constant in the Tabor equation expressed as *H=Cσ* which translates the indentation hardness to the tensile strength often remarkably fails to fall in the proper proportionality range [9]. The proportionality constant, *C*, is dependent on the deformation mode under indentation and it had been empirically determined that H/*σ*≈2*:*7 for materials with high strain hardening coefficient and yield strength (elastic–plastic transition mode). This indicates the 2D metallic thin films with columnar GBs possess substantial structural anisotropy, despite the crystal anisotropy governed by either Schmid factor or Taylor factor [2, 8]. Metallic coatings or thin films have been used as protective, reflective, conductive components on apparatuses and devices. Comprehension toward the anisotropy of 2D metallic materials would substantially help improve their reliability and realize

property optimization.

*Material Flow Analysis*

**58**

**2. Fraction anisotropy in graphene**

*Molecular dynamics simulation of tensile tests on 4.15 4.15 nm<sup>2</sup> square-shaped graphene monolayer along (a) zigzag and (b) armchair directions and the relations between applied force and one unit cell are present. Reprinted with permission from reference [20].*

topological defects, whereas a rather smooth fracture feature was monitored as the strain varied from 43.859% to 43.866% under the test along armchair direction and the process left limited topological defects. It should be noted the five significant digits might be trivial in the real experiments but it was non-trivial in the MD simulations to capture detailed fracture process. The fracture evolutions along two directions were captured using snapshots in **Figure 3**. Since the C-C bonds have a critical strength, i.e. *σ<sup>C</sup><sup>C</sup>*, it is anticipated that the direction of the applied force with respect to the hexagonal honeycomb lattice eventually governed the fracture mode and the analysis on the evolution of bond angles during the straining under two testing conditions is essential to decipher the different fracture mechanisms. Along the ziazag direction in **Figure 3a**, two 120° bond angles that evolved in a symmetrical pattern with the increase in the strain declined down to <90 ° and sustained substantial external strain, while the bonds in parallel to the tensile

#### **Figure 3.**

*Tensile strain-induced fracture process (a) along the zigzag direction and (b) along the armchair direction at various strain levels. Reprinted with permission from reference [20].*

direction were elastically deformed until their critical strength was reached and then they broke, forming the broken hanging chains. In the other hand, the deformation along the armchair direction in **Figure 3b** caused the four 120° bond angles with a � 30 ° relation with the applied force to decrease, transferring the hexagonal lattice into the quasi-rectangular shape, until any of the bonds except the two bonds normal to the testing direction broke. In the armchair mode, once triggered, the crack front would lead to a sequential instantaneous bond-breaking along the same direction and this left a 60° rupture along the armchair direction. Therefore, the two bonds parallel to the zigzag testing direction and the four bonds perpendicular to the armchair testing directions underwent the larger stress. As a result, the critical stress along the armchair is calculated to be ffiffiffi <sup>3</sup> <sup>p</sup> *<sup>σ</sup>C*�*C*, 1.73 times of the *<sup>σ</sup>C*�*<sup>C</sup>* of the zigzag direction. In reality, the 0.21 TPa of the armchair direction is approximately 1.2 times of the 0.18 TPa of the ziazag direction, which is attributed to the evolving geometrical changes of the hexagonal structures amid the elongation processes. In addition, the geometrical changes of the hexagonal units during the straining determined the critical fracture stains for two testing directions. In order to validate the observation, the simulated dimensions were investigated to verify if the mechanical anisotropy has a size effect and it was found that the size effect was negligible.

microscopic (HRTEM) image shows different mosaic lattice domains as a result of 15° and 30° relative rotations between MoS2 nanosheets [21]. Commonly, six-fold and two-fold symmetry of the friction behaviors have been captured on empirical and computational researches. Some two-fold symmetry of friction behaviors, namely 180° periodicity, have been attributed to the oriented linear wrinkles induced by the elastic deformation of the substrate and the testing conditions, one of which is the direction-dependent friction measured by an AFM tip with rotation. It was hypothesized that the tip rotation generated a variety of possible combination of the tip-specimen interfaces and the friction results might be able to reflect the

A study involving experimental and MD simulation results on the friction property of MoS2 was present [22]. The direction-dependent friction behaviors were measured by changing the scanning direction and a 5 nm travel distance was applied to preclude the influence from the nanowrinkles. The atomic configuration and the scanning direction with respect to the lattice are illustrated in **Figure 5a**. **Figure 5b** presents the two friction loops consisting of forward and backward lateral scans, measured by AFM along zigzag and armchair directions, and it shows that the energy dissipated in each scan cycle of the tests along the armchair direction was 11 times higher than that of the tests along the zigzag direction. **Figure 5c** shows comparable simulation results and the quantitative discrepancy between the empirical and simulation results originates from the difference in tip conditions and the magnitude of the scanning speed and force. The Prandtl-Tomlinson model alleged that the friction at the atomic level relies on the height of the surface energy barrier and longer scanning length along the armchair direction would result in accumulated energy dissipation in comparison with the zigzag direction. Hence, the direction-dependent friction behaviors were examined using potential energy surface (PES). **Figure 6a** reveals a six-fold symmetry of the friction force in nN in comparison with the two-fold symmetry. To further comprehend the friction symmetry, PEC at various angular positions was observed with a 10° interval. **Figure 6b** reflects the cross-section energy profiles for the scans at 0°, 10° and 50°. **Figure 6c**-**j** show that PES possessed a 60° periodicity, e.g. the energy surfaces of the 0° and 60° being identical. Therefore, a friction anisotropy was explored at an atomic level, proving that the testing direction and tip-specimen contact quality greatly play significant roles in changing the energy landscape and affecting the friction behaviors. X. Cao et al. have exhibited that the friction behaviors of MoS2 had a thickness effect [13]. In brief, the decrease in MoS2 thickness down to a few nanometers could progressively weaken the anisotropy phenomenon and be more governed by the

*(a) Atomic configuration of a MoS2 monolayer in the simulations, indicating the armchair (30°) and zigzag directions (60°). (b) the experimental friction loops consisting of the forward and backward scanning along the armchair and zigzag directions. (c) the friction traces, due to tip-specimen contacts, predicted by the MD simulations along the armchair and zigzag directions. Reprinted with permission from reference [22].*

genuine crystallographic pattern of the tested materials.

*Anisotropic Mechanical Properties of 2-D Materials DOI: http://dx.doi.org/10.5772/intechopen.96598*

puckering effect.

**Figure 5.**

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