Аннотация:
In this paper we formulate the equilibrium equation for a beam made of graphene sub-
jected 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 eigen-
value 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 in-
finite sequence of eigenvalues. In particular, we prove the existence of a minimal eigenvalue
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.