A comprehensive lattice-stability limit surface for graphene

The limits of reversible deformation in graphene under various loadings are examined using lattice-dynamical stability analysis. This information is then used to construct a comprehensive lattice-stability limit surface for graphene, which provides an analytical description of incipient lattice instabilities of \textit{all kinds}, for arbitrary deformations, parametrized in terms of symmetry-invariants of strain/stress. Symmetry-invariants allow obtaining an accurate parametrization with a minimal number of coefficients. Based on this limit surface, we deduce a general continuum criterion for the onset of all kinds of lattice-stabilities in graphene: an instability appears when the magnitude of the deviatoric strain $\gamma$ reaches a critical value $\gamma^c$ which depends upon the mean hydrostatic strain $\bar {\mathcal E}$ and the directionality $\theta$ of the deviatoric stretch. We also distinguish between the distinct regions of the limit surface that correspond to fundamentally-different mechanisms of lattice instabilities in graphene, such as structural vs material instabilities, and long-wave (elastic) vs short-wave instabilities. Utility of this limit surface is demonstrated in assessment of incipient failures in defect-free graphene via its implementation in a continuum Finite Elements Analysis (FEA). The resulting scheme enables on-the-fly assessments of not only the macroscopic conditions (e.g., load; deflection) but also the microscopic conditions (e.g., local stress/strain; spatial location, temporal proximity, and nature of incipient lattice instability) at which an instability occurs in a defect-free graphene sheet subjected to an arbitrary loading condition.