Nonlinear elasticity of monolayer graphene (1006.0594v1)

Abstract: By combining continuum elasticity theory and tight-binding atomistic simulations, we work out the constitutive nonlinear stress-strain relation for graphene stretching elasticity and we calculate all the corresponding nonlinear elastic moduli. Present results represent a robust picture on elastic behavior of one-atom thick carbon sheets and provide the proper interpretation of recent experiments. In particular, we discuss the physical meaning of the effective nonlinear elastic modulus there introduced and we predict its value in good agreement with available data. Finally, a hyperelastic softening behavior is observed and discussed, so determining the failure properties of graphene.

• The paper establishes a constitutive nonlinear stress-strain relation for monolayer graphene using continuum elasticity theory and TB-AS simulations.
• It computes critical elastic constants, including a linear Young's modulus of 312 Nm⁻¹ and a nonlinear modulus of approximately -583 Nm⁻¹, which align with experimental observations.
• The simulation results confirm a hyperelastic softening behavior that informs graphene’s potential in advanced composite materials and nano-engineered systems.

Nonlinear Elasticity of Monolayer Graphene: A Quantitative Analysis

The paper titled "Nonlinear elasticity of monolayer graphene" by Emiliano Cadelano and colleagues stands as a substantial advancement in the comprehension of graphene's mechanical properties. The authors employ a combination of continuum elasticity theory and tight-binding atomistic simulations (TB-AS) to derive the nonlinear stress-strain relationship specific to graphene's stretching behavior. This research unequivocally identifies the nonlinear elastic moduli of a singular layer of carbon atoms, shedding light on experimental observations from recent studies.

Main Contributions

This paper meticulously investigates the constitutive nonlinear stress-strain relationship of monolayer graphene. Utilizing an advanced theoretical framework, the authors transition from the effective parameter $D$ acknowledged in phenomenological models to a more rigorous definition involving three independent third-order elastic constants $C_{ijk}$ as predicted by continuum elasticity theory.

Key elements include:

1. Constitutive Nonlinear Stress-Strain Relation: The paper establishes a relation encompassing small strain and Lagrangian formulations. By deriving expressions for the nonlinear elastic coefficients $\Lambda_i$, the authors articulate the relation between third-order elastic constants and the observed nonlinear elasticity.
2. Simulation Methodology: The use of TB-AS provides an accurate computational approach to model graphene's response under different homogeneous deformation modes. The reliable representation by Xu et al. enables precise simulation of graphene's energy-strain curves without replicating complex experimental nanoindentation setups.
3. Numerical Validation: The tight correspondence between the computed value of $\langle D_{\vec{n}} \rangle$ and the experimental values from Lee et al. enhances the credibility of the simulation results, affirming the small strain nonlinearity regime relevant in the tested conditions.

Numerical Results

The numerical results demonstrate that the nonlinear elastic modulus ($D$) reflects a hyperelastic softening behavior of graphene, with a predicted failure stress $\sigma_f$ closely matching experimental observations. The TB-AS calculated values for the linear Young's modulus $E = 312 \, \text{Nm}^{-1}$ and $\nu = 0.31$ fit within the spectrum of existing literature. More notably, the nonlinear moduli $$\langle D_{\vec{n}} \rangle = -582.9 \, \text{Nm}^{-1}$$ are consistent with experimental results, reflecting effective softening in graphene's elasticity.

Implications and Future Directions

The implications of these findings are broad. Practically, the accurate characterization of graphene's mechanical properties informs its integration as a reinforcement component in composite materials and nano-engineered systems. The ability of graphene to endure significant strain before failure predicates its utility in advanced material applications withstanding extreme mechanical loads.

Theoretically, this paper opens avenues for deeper exploration into the nonlinear mechanical behavior of two-dimensional materials. Further work could involve extending these elasticity models to capture temperature-dependent behavior or exploring the effects of external perturbations such as defect introduction or chemical functionalization.

In summary, this research comprehensively enhances our understanding of graphene's nonlinear elastic properties. As the paper meticulously verifies through theoretical and computational validation, the defined stress-strain relationships are critical for pioneering future applications and theoretical investigations in nano-material mechanics.