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

In this paper we formulate the equilibrium equation of a beam made of graphene material subjected to some boundary conditions and acted upon by axial compression and nonlinear lateral constrains as a fourth-order nonlinear boundary value problem. We also formulate the nonlinear eigenvalue for buckling analysis of the beam. We verify the solvability of the buckling problem as an asymptotic expansion in a ratio of the elastoplastic parameters, that the spectrum is bounded away from zero and contains a discrete infinite sequence of eigenvalues.We also verify, for certain ranges of the lateral forces, the solvability of the general equations using energy methods and a suitable iteration scheme.