## 2 Mechanical Properties  
Although unstable in free space, graphene is often considered as the strongest material ever found. This assertion needs to be quantified by parameters such as breaking strength, fracture toughness, covalent bond energy, and other mechanical data.  
The theoretical tensile strength of graphite in directions parallel to the basal plane is around 100 GPa. The strength of graphene can be estimated by multiplying the bulk value of graphite by the Van der Waals thickness of a single sp\({}^{2}\) carbon plane, to obtain 34 N/m. DFT calculations set up the ultimate tensile strength of graphene along the zigzag and armchair directions at 36 N/m and 40 N/m, respectively [10]. By comparison, the ultimate uniaxial strength of graphene extracted from nano-indentation amounts to 40 N/m [11; 12]. This latter value is the result of fitting experimental data with non-linear [PERSON]'s law \(\gamma=C\varepsilon+C_{2}\varepsilon^{2}\), here applied to an isotropic medium subject to uniaxial strain. In that expression, \(\gamma\) is the applied tension (N/m), \(\varepsilon\) is the strain, \(C\) is the first-order 2D Young's modulus introduced in Section 1, and \(C_{2}\) is a second-order coefficient [11]. With a negative \(C_{2}\), the stress-deformation curve has a maximum--by definition the ultimate strength, equal to \(-C^{2}/4C_{2}\). This represents the largest possible tensile strength graphene can support before breaking. As reviewed in Ref. [13], the introduction of defects (boundaries between large or small grains, vacancies, slits) decreases the strength [12] from 40 down to 1 N/m [4]. By comparison, the tensile strength of a PET (polyethylene terephthalate) film of 9 \(\upmu\)m thickness (equivalent to a stack of about 26,500 graphene planes) is 1800 N/m.  
Indications exist that graphene undergoes brittle failure [10; 14]. Consequently, [PERSON]'s model of crack propagation [15] is more relevant than the theoretical maximum of the \(\gamma\)-\(\varepsilon\) curve to address the breaking strength of a 2D material [4; 16]. [PERSON]'s criterion states that a fracture of initial length \(l_{0}\) will propagate in the medium when the tension applied perpendicular to the crack length exceeds the critical value \(\gamma_{c}=\sqrt{2\lambda C/\pi l_{0}}\). In that formula, \(\lambda\) is the edge excess energy in the direction of the crack (analog to the surface energy for a 2D material) and \(C\) is the 2D Hooke constant. The fracture toughness is by definition \(K_{c}=\sqrt{2\lambda C}\). Its value measured on free-standing bilayer graphene is \(2.7\times 10^{-3}\) Nm\({}^{-1/2}\)[14]. In few-layer graphene, \(K_{c}\) should increase linearly with the number of atomic planes, which means that the toughness of monolayer graphene can be estimated at about half the reported value--\(1.3\times 10^{-3}\) Nm\({}^{-1/2}\) (\(1.2\times 10^{-3}\) Nm\({}^{-1/2}\) according to Ref. [4]).  
From the energetics point of view, one may estimate the value of the bond breaking energy \(E_{b}\) from the cohesive energy of graphene (7.91 eV/atom [17]). The simple formula \(E_{c}=\frac{Z}{2}E_{b}\) with \(Z=3\) the number of nearest neighbors yields \(E_{b}=5.3\) eV, which corresponds to a breaking energy \(G=3.4\) nJ/m to cut the bonds perpendicular to a zigzag direction. We note the excellent agreement with the excess energy reported for the zigzag edge of graphene (half the breaking energy \(G\)), \(\lambda=1.7\) nJ/m [18]. Using this value of \(\lambda\) together with \(C=350\) N/m [5] yields \(K_{c}=1.1\times 10^{-3}\) Nm\({}^{-1/2}\), in fair agreement with the experimental value reported above for monolayer graphene.  
All the above reasoning was based on the implicit hypothesis of flat graphene. Out-of-plane distortions (ripples, wrinkles, etc.) are common features of mechanically transferred graphene [19]. How shape corrugation impacts the mechanical properties and the elastic constants of a 2D material is a new and growing field of research [20; 21; 22].