Elgindi, Mohamed B. M.Wei, DongmingElgindi, T.M.2016-11-232016-11-232014Mohamed B. Elgindi, Dongming Wei and Tarek M. Elgindi; 2014; On the solvability of euler graphene beam subject to axial compressive load; arXiv.org; http://nur.nu.edu.kz/handle/123456789/1932http://nur.nu.edu.kz/handle/123456789/1932In 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.enAttribution-NonCommercial-ShareAlike 3.0 United StatesMaterials ScienceAnalysis of PDEsArticle