**2.1. Mechanical properties**

Pristine graphene structures are found in 2D plane sheets. It has a hexagonal crystal lattice which resulted in covalent bonds between carbon atoms. In the environment, graphene are discovered in tow forms: "monolayer" and "free- standing." With the first form, we find graphene parts as a cover over a substrate material such as SiC. However, we are able to find graphene individually and independent from other materials in the environment which corresponds with the second form, "free-standing graphene" [33].

Mechanical properties for any crystal material are affected by pristine Lattice and defects are comprised of dislocations and grain boundaries [34, 35]. For example, we can mention elastic properties of materials that are affected by atoms interactions and lattice geometry, whereas strength and plastic flow stress as another properties of materials are affected by characteristics of defects. Indeed caused defects in the material severely decrease strength of it in comparison of ideal material. Anyway, we are always not able to impede the existence of defect and its effect in the materials. However, there is one exception; nano-materials can be discovered defect-free initially, and this is the main reason of superiority of strength for these materials [36]. Graphene as a nano-material is not excepted in this issue.

Lee and his co-workers performed the pioneer empirical analysis of elastic properties and strength of pristine graphene [37]. A deposited graphene membrane onto a substrate material that possesses some cavities on the surface is loaded by the tip of the atomic force microscope (Fig. 1) [37], and it was discovered that graphene brings out both nonlinear elastic behavior and brittle fracture. Thus, for nonlinear elastic behavior, we can write: *σ* =*Eε* + *Dε* <sup>2</sup> , where *σ* is the applied stress (the symmetric second Piola Kirchhoff stress), *E* is the Young modulus, *ε* is the elastic strain (uniaxial Lagrangian strain), and *D* is the third-order elastic stiffness. This experiment is convoyed by this result as follows: Young modulus of *E* = 1.0 TPa, and the thirdorder elastic stiffness of *D* = -2.0 TPa. The Young modulus they found is very close to the Young modulus of nanotubes. They also found that brittle fracture happens at an intrinsic stress as much as *σint* =130*GPA*, which is very huge and magnitude.

Simulating by computer [38] shows *E* = 1.05 TPa and *σint* =110 *GPa* for Young modulus and brittle fracture of graphene, which is compatible with explorations of Lee and his co-workers. All of these explorations prove that graphene can be very useful for structural applications and for the cases that we are dependent on high strength. Furthermore, graphene is flexible and can be bent easily, which make it more desirable and attractive.

The propagation of crack in monolayer graphene has been studied empirically and also analytically (molecular dynamics) by considering Crystallographic characteristics [39]. It has been evident that the sources of cracks in monolayer graphene membranes are unavoidable, mechanically applied stresses that are exerted during their processing. Cracks or tears propagate along the sides of the hexagonal crystal lattice and defray an occasional direction change as much as 30o in vertices of hexagonal.

**•** Graphene can be grown on metal surfaces by surface segregation of carbon or by

Pristine graphene structures are found in 2D plane sheets. It has a hexagonal crystal lattice which resulted in covalent bonds between carbon atoms. In the environment, graphene are discovered in tow forms: "monolayer" and "free- standing." With the first form, we find graphene parts as a cover over a substrate material such as SiC. However, we are able to find graphene individually and independent from other materials in the environment which

Mechanical properties for any crystal material are affected by pristine Lattice and defects are comprised of dislocations and grain boundaries [34, 35]. For example, we can mention elastic properties of materials that are affected by atoms interactions and lattice geometry, whereas strength and plastic flow stress as another properties of materials are affected by characteristics of defects. Indeed caused defects in the material severely decrease strength of it in comparison of ideal material. Anyway, we are always not able to impede the existence of defect and its effect in the materials. However, there is one exception; nano-materials can be discovered defect-free initially, and this is the main reason of superiority of strength for these materials

Lee and his co-workers performed the pioneer empirical analysis of elastic properties and strength of pristine graphene [37]. A deposited graphene membrane onto a substrate material that possesses some cavities on the surface is loaded by the tip of the atomic force microscope (Fig. 1) [37], and it was discovered that graphene brings out both nonlinear elastic behavior

is the applied stress (the symmetric second Piola Kirchhoff stress), *E* is the Young modulus, *ε* is the elastic strain (uniaxial Lagrangian strain), and *D* is the third-order elastic stiffness. This experiment is convoyed by this result as follows: Young modulus of *E* = 1.0 TPa, and the thirdorder elastic stiffness of *D* = -2.0 TPa. The Young modulus they found is very close to the Young modulus of nanotubes. They also found that brittle fracture happens at an intrinsic stress as

Simulating by computer [38] shows *E* = 1.05 TPa and *σint* =110 *GPa* for Young modulus and brittle fracture of graphene, which is compatible with explorations of Lee and his co-workers. All of these explorations prove that graphene can be very useful for structural applications and for the cases that we are dependent on high strength. Furthermore, graphene is flexible

The propagation of crack in monolayer graphene has been studied empirically and also analytically (molecular dynamics) by considering Crystallographic characteristics [39]. It has been evident that the sources of cracks in monolayer graphene membranes are unavoidable, mechanically applied stresses that are exerted during their processing. Cracks or tears

, where *σ*

and brittle fracture. Thus, for nonlinear elastic behavior, we can write: *σ* =*Eε* + *Dε* <sup>2</sup>

decomposition of hydrocarbons.

corresponds with the second form, "free-standing graphene" [33].

[36]. Graphene as a nano-material is not excepted in this issue.

much as *σint* =130*GPA*, which is very huge and magnitude.

and can be bent easily, which make it more desirable and attractive.

**•** etc.

**2.1. Mechanical properties**

4 Graphene - New Trends and Developments

Reprinted with permission from C. Lee, X. Wei, J.W. Kysar, J. Hone, Science, Volume 321, 385 388, 2008. Copyright (2008) by the American Association for the Advancement of Science.

**Figure 1.** Mechanical testing of graphene. Schematic of nanoindentation on suspended graphene membrane (left fig‐ ure). Atomic force microscope image of a fractured graphene membrane (right figure).

It is sometimes seen that propagation can go under the TEM electron beam [40]. Kim et al. [39] used this assumption that the propagation of crack is motivated by incorporating the effects of stress concentrations at the tip of the crack and the ionization influence of electron beam. Under the simultaneous effect of these problems, atomic bonds break in the vicinities of a crack tip, and propagation takes place.

Novoselov et al. [40] performed a computer simulation on the exfoliation of graphene sheets from adhesive substrate to examine crystallographic features of cracks which happens. The idea of this simulation developed from this fact that graphene ribbons became tapered as they were produced by exfoliation process (Fig. 2). They found that tear angle is affected by adhesive strength. Their simulation showed that, with the low strength adhesive, tearing takes place in the conqueror armchair direction of the hexagonal crystal lattice of graphene, and meanwhile, occasional change in direction is observed rarely. They also concluded that any increase in adhesive strength results in more tear angel; hence, in pretty high adhesive strength a change of 60o in the direction of tearing will present.

As a material, graphene is not excluded from defects. The defects that graphene may obsess are: vacancies [41], Stone Wales defects [41], dislocations [42], and grand boundaries of GBs. Among these defects, dislocations and GBs are very common and play a prominent role in mechanical properties of graphene. For instance, dislocations can cause plastic flow in graphene, whereas GBs decrease its strength characteristics [44, 45]. Dislocations can also violate translational symmetry of graphene [46].

There is no getting around GBs in graphene specimens when it is produced in a large area; thus, their effects on mechanical properties have been always noticeable from both funda‐ mental and applied sides. Empirical data [44] have proved that free-standing polycrystalline graphene under concentrated load has severe low failure strength when found in polycrys‐ talline state than when produced as a single crystal. In this experiment, they used a tip of atomic force microscopy (AFM) to exert a concentrated load on a polycrystalline graphene membrane, and it was observed that the force needed to cause a tear in the graphene along GBs is an amount of 100 nN [44], this is while the force for tearing a single-crystal exfoliated graphene is not more than 1.7 mN [37].

Vargas et al. [45] did similar test using atomic force microscopy and molecular dynamics simulations to study the mechanical characteristics of a graphene with polycrystalline structure that is obtained by chemical vapor deposition. They used nanoindentation meas‐ urements and found that out-of-plane ripples effectively decrease in-plane stiffness in the mentioned graphene. They also found that GBs effectively decrease the breaking strength of the graphene. Molecular dynamics showed them voids can significantly weaken the graphene membranes. In fact, GB is a place where amorphous carbon and iron oxide nanoparticles are absorbed [45].

Reprinted with permission from D. Sen, K.S. Novoselov, P. Reis, M.J. Buehler, Small, Volume 6, 1108,2010.

**Figure 2.** Schematic diagram of the setup for the tearing studies of graphene: side and top views. The inset shows the sheet orientation. An initial flap of 8 nm in width is cut in the sheet, folded back, and moved at a constant speed.
