Phonon, Cauchy-Born and homogenized stability criteria for a free-standing monolayer graphene at the continuum level

Abstract

Abstract The purpose of the present work is the study of three stability criteria for a free standing monolayer graphene modelled as a monoatomic hexagonal 2-lattice at the continuum level. The criteria treated are: the phonon stability, the Cauchy-Born stability and the homogenized stability criteria. Phonon stability requires plane progressive waves to propagate along the material with real velocities. The outcome consists of necessary and sufficient conditions for the wave speed to be real in terms of graphene's acoustic tensors. Requiring energy's second variation to be positive render the Cauchy-Born stability criterion. The outcome consist of a Hessian matrix whose components are derivatives of the energy with respect to its arguments. The Cauchy-Born stability criterion stipulate this Hessian matrix to be positive semi-definite. Solving the equations ruling the shift vector, enable to rule the shift vector out of the energy. The Cauchy-Born stability criterion for this homogenized energy renders the homogenized stability criterion. These three stability criteria are lied down for graphene for both the geometrically and materially linear and the nonlinear case. For the phonon stability study the equation ruling the shift vector has two alternatives: an equilibrium equation and a rate equation according to a gradient flow law.