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Browsing Mathematics by Subject "Analysis of PDEs"

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    On the solvability of euler graphene beam subject to axial compressive load
    (arXiv.org, 2014) Elgindi, Mohamed B. M.; Wei, Dongming; Elgindi, T.M.
    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.