Strain-dependent elastic constants of solid membranes

Computer simulations were conducted to explore the equilibrium states of a planar crystalline membrane as a function of external strain and temperature. The employed models include the continuous elastic Nelson-Peliti Hamiltonian, which describes the anharmonic coupling between in-plane and out-of-plane membrane fluctuations, and the atomistic long-range carbon bond-order potential (LCBOPII), which is designed to describe carbon interactions. The primary focus of the simulations is on studying the elastic constants of graphene: the compressibility modulus $B$ and the Lam\'e coefficient $\ensuremath{\mu}$. By analyzing the stability conditions in terms of the elastic constants, we can determine the region of thermodynamic stability for the planar morphology of the membrane as a function of external strain and temperature. The instability is associated with the vanishing of one of the elastic coefficients. Depending on the elastic response of the system, thermodynamic instability may be caused either by the condition $B=0$ or $\ensuremath{\mu}=0.$ The symmetry of the intrinsic anharmonicity of the model determines its elastic response to decreasing tensile stress. The two studied models provide different predictions for the membrane properties near the critical point. The continuous Nelson-Peliti model predicts auxetic behavior, with a Poisson's ratio of $\ensuremath{\nu}=\ensuremath{-}1$ at the critical point. In contrast, the atomistic LCBOPII model exhibits a positive Poisson's ratio, which becomes $\ensuremath{\nu}=1$ at the critical point. Our findings challenge the generally accepted view that a Poisson's ratio of $\ensuremath{\nu}=\ensuremath{-}1/3$ is a universal property of unstrained crystalline membranes. Instead, our results highlight that anharmonic effects may cause significant differences in the mechanical responses, especially near the critical point of instability.