On the Solvability of Euler Graphene Beam Subject to Axial Compressive Load

In this paper we formulate the equilibrium equation for a beam made of graphene subjected to some boundary conditions and acted upon by axial compression and nonlinear lateral constrains as a fourth-order nonlinear boundary value problem. We first study the nonlinear eigenvalue problem for buckling analysis of the beam. We show the solvability of the eigenvalue problem as an asymptotic expansion in a ratio of the elastoplastic parameters. We verify that the spectrum is a closed set bounded away from zero and contains a discrete infinite sequence of eigenvalues. In particular, we prove the existence of a minimal eigenvalue λ ∗ for the graphene beam corresponding to a Lipschitz continuous eigenfunction, providing a lower bound for the critical buckling load of the graphene beam column. We also proved that the eigenfunction corresponding to the minimal eigenvalue is positive and symmetric. For a certain range of lateral forces, we demonstrate the solvability of the general equation by using energy methods and a suitable iteration scheme.