On the Buckling of Euler Graphene Beams Subject to Axial Compressive Load

Abstract

In this paper, we consider the buckling of an Euler-Bernoulli graphene beam due to an axial compressive load. We formulate the problem as a non-linear (eigenvalue) two-point boundary value problem, prove some properties of the eigenpairs and introduce a suitable numerical shooting method scheme for approximating them. We present the perturbation and the numerical approximations of the first and second buckling loads and the corresponding shapes.

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Elgindi, M. , Wei, D. , Soukiassian, Y. and Liu, Y. (2014) On the Buckling of Euler Graphene Beams Subject to Axial Compressive Load. World Journal of Engineering and Technology, 2, 149-158. doi: 10.4236/wjet.2014.22016.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Zhu, Y., Murali, S., Cai, W., Li, X., Suk, J.W., Potts, J.R. and Ruoff, R.S. (2010) Graphene and Graphene Oxide: Synthesis, Properties, and Applications. Advanced Materials, 22, 3906-3924.
http://dx.doi.org/10.1002/adma.201001068
[2] Galiotis, C. (2011) Mechanical Properties of Graphene. Graphene 2020/Brussels.
[3] Kim, M.T. (1996) Influence of Substrates on the Elastic Reaction of Films for the Microindentation Tests. Thin Solid Films, 283, 12-16.
http://dx.doi.org/10.1016/0040-6090(95)08498-3
[4] Lee, C., Wei, X., Kysar, J.W. and Hone, J. (2008) Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene. Science, 321, 385-388.
http://dx.doi.org/10.1126/science.1157996
[5] Lee, C., Wei, X., Li, Q., Carpick, R., Kysar, J.W. and Hone, J. (2009) Elastic and Frictional Properties of Graphene. Physica Status Solidi (B), 246, 2562-2567.
http://dx.doi.org/10.1002/pssb.200982329
[6] Sansoz, F. and Gang, T. (2010) A Force-Matching Method for Quantitative Hardness Measurements by Atomic Force Microscopy with Diamond-Tipped Sapphire Canti-levers. Ultramicroscopy, 111, 11-19.
http://dx.doi.org/10.1016/j.ultramic.2010.09.012
[7] Malina, E.W. (2011) Mechanical Behavior of Atomically Thin Graphene Sheets Using Atomic Force Microscopy Nanoindentation. Master’s Thesis, University of Vermont, Vermont.
[8] Elgindi, M.B.M. and Wei, D.M. (2012) On the Global Solvability of a Class of Fourth-Order Nonlinear Boundary Value Problems. Applied Mathematical Sciences, 6, 5981-5992.
[9] Li, P., You, Z. and Cui, T.H. (2012) Graphene Catilever Beams for Nano Switches. Applied Physics Letters, 101, 093111.
http://dx.doi.org/10.1063/1.4738891
[10] Wei, D., Sarria, A. and Elgindi, M.B.M. (2012) Critical Buckling Loads of the Perfect Hollomon’s Power-law Columns, Mechanics Research Communications. 1-12.
[11] Barkat, A.B. and Matti, V. (2012) On Generalized Trigonometric Functions with Two Parameters.
[12] Drbek, P. and Mansevich, R. (1999) On the Closed Solution to Some pLaplacian Nonho-Mogeneous Eigenvalue Problems. Differential Integral Equations, 12, 773-788.
[13] Elgindi, M.B.M., Wei, D. and Elgindi, T.M. (in Press) On the Solvability of an Euler Graphene Beam Subject to Axial Compressive Load.
[14] Malina, E.W. (2011) Mechanical Behavior of Atomically Thin Graphene Sheets Using Atomic Force Microscopy Nanoin-dentation. Master Thesis, University of Vermont, Vermont.

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