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

TL;DR

This paper analyzes the mathematical solvability of a graphene beam under axial compression, formulating nonlinear boundary value and eigenvalue problems, and verifying solutions using asymptotic and energy methods.

Contribution

It introduces a novel formulation of the buckling problem for graphene beams and proves solvability under specific boundary conditions and parameter ranges.

Findings

Spectrum is bounded away from zero.

Existence of an infinite discrete eigenvalue sequence.

Solvability confirmed via energy methods and iteration schemes.

Abstract

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.