Variational principles and natural boundary conditions for multilayered orthotropic graphene sheets undergoing vibrations and based on nonlocal elastic theory

Variational principles are derived for multilayered orthotropic graphene sheets undergoing transverse vibrations based on the nonlocal elastic theory of orthotropic plates which provide a continuum model for graphene sheets. The variational formulation allows the derivation of natural boundary conditions which are expressed in the form of a set of coupled equations for multilayered sheets as opposed to uncoupled boundary conditions applicable to simply supported and clamped boundaries and also in the case of a formulation based on the local (classical) elasticity theory. For the free vibrations case, the Rayleigh quotient is derived. The methods for the variational formulation use techniques of calculus of variations and the semi-inverse method for deriving variational integrals. Variational formulations provide the basis for a number of approximate and numerical methods of solutions and improve the understanding of the physical phenomena.