Fracture Size Effects in Nanoscale Materials: The Case of Graphene

Nanoscale materials display enhanced strength and toughness but also larger fluctuations and more pronounced size effects with respect to their macroscopic counterparts. Here we study the system size dependence of the failure strength distribution of a monolayer graphene sheet with a small concentration of vacancies by molecular dynamics simulations. We simulate sheets of varying size encompassing more than three decades and systematically study their deformation as a function of disorder, temperature, and loading rate. We generalize the weakest-link theory of fracture size effects to rate- and temperature-dependent failure and find quantitative agreement with the simulations. Our numerical and theoretical results explain the crossover of the fracture strength distribution between a thermal and rate-dependent regime and a disorder-dominated regime described by the extreme-value theory.

Figure 1

Failure of graphene sheets. The graphene sheet is composed of $N=50\times {10}^3\mathrm{atoms}$, with a vacancy concentration (porosity) $P=0.1\%$. The color bar indicates the ${\sigma }_{xx}$ component of the stress tensor per atom. (a) A graphical view of the failure process (from left to right). The crack nucleates from one of the defects already present in the material (not necessarily the most stressed) and rapidly grows until the graphene sheet complete failure is achieved. (b) The stress-strain curve displays temperature-dependent fracture strength. The prestressed initial condition ($ϵ=0$) is due to the constraint applied to the atoms belonging to the four outermost layers of the sheet, which are subject to the stretching along $X$.

Figure 2

Graphene fracture size effects. (a) The average failure stress for defected graphene depends on the system size $N$ and on the vacancy concentration $P$. Simulations are carried out with $T=300\mathrm{K}$ and $\dot{ϵ}=0.128\times {10}^8{\mathrm{s}}^{-1}$. The lines are the theoretical prediction as discussed in Appendix pp2. They do not arise as a direct fit of the numerical curves but result from the analytical evaluation of the integral expression of $⟨\sigma {⟩}_n$. (b) The failure stress survival distribution at $T=300\mathrm{K}$ and $\dot{ϵ}=0.128\times {10}^8{\mathrm{s}}^{-1}$ for different system sizes with a vacancy concentration equal to $P=0.1\%$ (blue), $P=0.2\%$ (green), and $P=0.5\%$ (red). When the survival probability distributions are rescaled by $N$ according to the predictions of the extreme-value theory, the data collapse into a single curve that depends only on the vacancy concentration $P$.

Figure 3

Temperature and rate effects of the graphene tensile strength distribution. The survival distribution of defected graphene sheets depends on (a) the temperature (with $P=0.2$, $\dot{ϵ}=1.28\times {10}^8{\mathrm{s}}^{-1}$, and $N={10}^4$) and (b) strain rate (with $P=0.2$, $T=300\mathrm{K}$, and $N={10}^4$). The dashed lines represent the best least-square fit according to the theory of breaking kinetics discussed in the text.

Figure 4

Survival distribution of defected graphene at a low temperature. We report the survival distribution obtained from simulations at $T=1\mathrm{K}$, $\dot{ϵ}=0.128\times {10}^8{\mathrm{s}}^{-1}$, and $N={10}^4$ for different values of the vacancy concentration $P$. The numerical data are fitted with the exponential function $A{e}^{\sigma /E{ϵ}_0}$ (solid lines), leading to a Gumbel distribution for the failure strains $\rho ({ϵ}_c)=A{e}^{({ϵ}_c/{ϵ}_0)-A{e}^{{ϵ}_c/{ϵ}_0}}$ [see Eq. (b1)]. For $P=0.1\%$, we obtain $A=7.92\pm 0.05\times {10}^{-38}$ and ${ϵ}_0=0.00125\pm 0.00004$. For $P=0.2\%$, $A=1.767\pm 0.005\times {10}^{-35}$ and ${ϵ}_0=0.001338\pm 0.000007$. For $P=0.5\%$, $A=1.804\pm 0.007\times {10}^{-28}$ and ${ϵ}_0=0.00167\pm 0.00004$.

Figure 5

Fitting parameters for graphene survival distributions. The best fitted values of (a) the representative volume $V_0$ and (b) the activation frequency as a function of temperature ($N={10}^4$ and $\dot{ϵ}=1.28\times {10}^8{\mathrm{s}}^{-1}$). These values are obtained by the least-square fit of the numerical survival probability distribution with the expression (b2) [dashed lines in Fig. 3]. The same values [(c),(d)] as a function of the strain rate ($N={10}^4$ and $T=300\mathrm{K}$), obtained from the best fits shown in Fig. 3 ($P=0.2\%$) and Fig. 8 ($P=0.5\%$) (dashed lines).

Figure 6

Tuning the AIREBO potential. The stress-strain curve is obtained as a function of the cutoff $r_c$. The simulated graphene sheet is composed by $N={10}^4$ carbon atoms at $T=300\mathrm{K}$, without in-built defects ($P=0$) and pulled at constant strain rate $\dot{ϵ}=0.128\times {10}^8{\mathrm{s}}^{-1}$. For $r_c<0.195\mathrm{nm}$ the stress displays a spurious increase, while for $r_c=2\mathrm{nm}$ the pair potential shows an unphysical singularity (not shown). The chosen value of $r_c$ is set to 0.195 nm.

Figure 7

Survival distribution of defected graphene at $P=0.5\%$. We report the survival distribution obtained from simulations at $T=300\mathrm{K}$, $N={10}^4$, and three strain rates $\dot{ϵ}$. The vacancy concentration is set to $P=0.5\%$. The fitting function is provided by Eq. (b2). The values of the fitting parameters $V_0$ and ${\omega }_0$ are reported in Fig. 5.

Figure 8

Survival distribution of defected graphene at different $P$. We report the survival distribution obtained from simulations at $T=300\mathrm{K}$, $\dot{ϵ}=0.128\times {10}^8{\mathrm{s}}^{-1}$, and $N={10}^4$ for three vacancy concentrations $P$. The numerical data are fitted with the expression (b2) (dashed lines) using the proper values of $A$ and ${ϵ}_0$ reported in Fig. 4. The fitted values of $V_0$ and ${\omega }_0$ are shown in Fig. 5 for $P=0.2\%$ and $P=0.5\%$. For $P=0.1\%$, the least-square fit gives $V_0=0.3806\pm 0.0003{\mathrm{nm}}^3$ and ${\omega }_0=4.1416\pm 0.0006\times {10}^7{\mathrm{s}}^{-1}$. The set of parameters $A$, ${ϵ}_0$, $V_0$, and ${\omega }_0$ which characterize uniquely the theoretical expression (b2) (dashed lines) are inserted into Eq. (b4) to calculate the mean average rupture stress $⟨\sigma {⟩}_n$ as a function of $N$, shown in Fig. 2.