Failure Processes in Embedded Monolayer Graphene under Axial Compression

Exfoliated monolayer graphene flakes were embedded in a polymer matrix and loaded under axial compression. By monitoring the shifts of the 2D Raman phonons of rectangular flakes of various sizes under load, the critical strain to failure was determined. Prior to loading care was taken for the examined area of the flake to be free of residual stresses. The critical strain values for first failure were found to be independent of flake size at a mean value of –0.60% corresponding to a yield stress up to -6 GPa. By combining Euler mechanics with a Winkler approach, we show that unlike buckling in air, the presence of the polymer constraint results in graphene buckling at a fixed value of strain with an estimated wrinkle wavelength of the order of 1–2 nm. These results were compared with DFT computations performed on analogue coronene/PMMA oligomers and a reasonable agreement was obtained.


Mathematical Analysis
According to the Euler stability criterion, the critical state for compression failure is reached when the work of the external forces equals the change in strain energy of the body 1,2 : (S1) Furthermore, the work of the compressive forces acting on the middle plane of the plate is given by 2 : where u is the out-of-plane displacement, A is the area of the plate (graphene) and N x is the compressive force along the x-direction. The bending energy is given by 2 : where D is the bending stiffness and v the Poisson ratio of graphene. The surrounding polymer matrix is assumed to contribute to the system through the deformation energy: where K w is the Winkler's modulus. From the physical point of view the last term describes the interaction between the plate and the elastic foundation which is loaded as the plate bends at the critical point.
By equating the work of the external forces with the change in the strain energy we obtain: For the out-of-plane displacement it is common to assume that it has a sinusoidal form [2][3][4][5][6] which models adequately the form of buckling that appears at the critical strain. For our purposes we make the following assumption for u: where l, w is the length and the width of the flake, respectively. Under the above assumption we obtain: Using the expression on eq 5 we obtain the critical force N x : (S10) Following the reasoning similar to reference 2, the critical force N x , being a sum of positive quantities, is minimized when only one term α mn is different than zero. In such a case we have: (S11) If we further make the physically plausible assumption 2 that there are several half waves in the direction of compression but only one half wave in the perpendicular direction (n=1) and use the formula N x =εC, where C is the tension rigidity, we finally arrive at the following expression for the critical strain for buckling: where (S13) The determination of the half waves, m, stem from equating the force expression 11, for m and m+1: (S14) This way we obtain the following equation for m: The evaluation of the half wavelength and the amplitude relies on the inextensibility assumption. We assume that for small values of applied strain, the length of the specimen remains the same after buckling initiates. The wavelength that corresponds to the half-wave number is evaluated according to the formula: The projection of the buckled length divided by the number of half waves, m, corresponds to the wave length. The length of the projection can be calculated by: , figure S1 where The incompressibility constraint is a plausible assumption since we are at the regime of low strains (approximately ~-0.5 %). The line integral: (S17) corresponds to the length of the flake. In the above relation u is the out-of-plane displacement that now takes the form: (S18) A its amplitude, while the term (1-ε cr ) describes the contractions due to buckling as the figures shows. Figure S1. The initial length of the specimen and the length after buckling occurs.

Stress transfer from the PMMA to the graphene flake
We apply a compressive loading to the system graphene-PMMA. This kind of loading results to a shear stress at the interface between the graphene flake and the surrounding medium which is responsible for transferring the stress to the inclusion (graphene) 8 . The shear stresses require a specific length to reach the maximum value of the stress that is possible to be transferred from the polymer to the graphene. This is the transfer length L t . So, if the length of the graphene is smaller than the critical transfer length, L c (L c =2L t ) then graphene is not stressed to the maximum value which is the externally applied stress through the flexure of PMMA beams. Indeed, only a fraction of the load is transferred to graphene. Figure S2. Building of stress transfer in graphene.
In the case of flakes with lengths not large enough compared to the required critical length (L c ), the strain/ stress that is transferred to the flake will never reach the maximum value (applied strain). Thus, the actual strain developed in the graphene must be corrected as it is not the same with the external strain. Our experimental results indicate that the flakes with length of up to about ~4 μm are not able to obtain the maximum values of the stresses applied to the system. This is observed by the slope of the curve Pos2D vs Strain (here the strain is that applied to the beam). In all these cases the slope is much smaller than the critical slope, ~60 cm -1 /% (estimate based on the present work as well as values from the literature, see Refs [9][10][11]). Thus, a simple correction can be implemented by shifting the Pos2D vs Strain slope near the origin (zero strain) and recalibrating the actual strain through the formula: (S20)

Fitting
In figure 2

Two-parameter PES scan
We have additionally performed a two-parameter PES scan on a coronene with two MMA molecules, one on each of the coronene surfaces. The configuration corresponds to that used in PES1, and the scanning parameters are the distances R 1 , R 2 between each monomer and the coronene plane. The resulting scan is shown in figure S5. The presence of the second MMA 11 molecule reduces the interaction very slightly, and the overall interaction energy is 1.2 % less compared to twice the interaction energy of the corresponding single monomer case. For U sh : ε = 0.25209(2) kJmol -1 Å -2 , σ = 2.96414(2) Å, and C = 0.67282 (5) For U sf : ε = 0.3014(1) kJmol -1 Å -2 , σ = 2.7495(1) Å, and C = 0.7093 (3) For U ih : ε = 0.3401(1) kJmol -1 Å -2 , σ = 2.75851(9) Å, and C = 0.7236 (2) These primitive and composite potentials can be used to create mesoscopic models for the interaction of graphene and PMMA with regions of arbitrary tacticity.