Some Comments on the Mathematical Analysis of the Mechanical Properties of Graphene

ORAL

Abstract

Graphene sheets have mechanical properties which may be modeled using the classical theory of plates [1]. To obtain the associated mechanical parameters, data is fitted to the solutions of these mathematical models. While the fundamental equations are PDE's, an averaging over the space variables result in an ODE for which time is the independent variable. This ODE is the Duffing equation which describes1-dim, nonlinear oscillations. While the coefficients of this equation are constant, their explicit values depend on the functional form assumed for the fundamental mode of the grapheme sheet and other issues such as the particular functions used for the damping/dissipative terms. This work focuses on the coefficients of the derived Duffy equation and examines how their values are modified by selection of different functional forms for the fundamental mode.\\[4pt] [1] A. Eichler et. Al., ``Nonlinear damping in mechanical resonators made from carbon nanotubes and grapheme,'' NATURE Nanotechnology, Vol. 6 (June 2011), 339-342.

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Authors

  • Ronald E. Mickens

    Clark Atlanta University, Clark Atlanta University, Atlanta, GA 30314

  • J.K. Hwang

    Western Kentucky University, University of Pardubice, Francis Marion University, Clemson University Professor, Francis Marion University Professor, Undergraduate Administrator, Oak Ridge National Laboratory, Oak Ridge, Tennessee, Austin Peay State University, University Strenwarte-Muenchen, Seoul National University, Gatton Academy for Science and Mathematics, Alabama A\&M University, Cygnus, Center for Nanophase Materials Science at Oak Ridge National Laboratory, Vanderbilt University, Fisk Univ, 2Cornell High Energy Synchrotron Source, Cornell University, Ithaca, NY, NOVA Center, Western Kentucky University, Department of Physics, Florida A\&M University, Tallahassee, FL-32307, Correlated Electron Materials Group, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6061 USA, Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN 37235, USA, Department of Electrical and Computer Engineering, University of Rochester, Rochester, NY 14627, USA, The Institute of Optics, University of Rochester, Rochester, NY 14627, USA, Universidade Estadual Paulista (UNESP), Clark Atlanta University, Deapartment of Physics \& Astronomy, Georgia State University, USA, Department of Electrical and Computer Engineering, McGill University, Montreal, QC H3A 2A7, Canada, Oak Ridge National Laboratory, University of South Alabama, Samford University, University of Rochester, University of North Carolina, Chapel Hill, Sandia National Laboratories, New Mexico State University, University of Tennessee Space Institute, Shanghai Jiao Tong University, Shanghai, China, University of Leeds, Leeds, UK, Georgia State University, Atlanta GA, University of Alabama at Birmingham, National High Magnetic Field Laboratory, Prairie View A\&M University, Brookhaven National Laboratory, University of Southern Indiana, Center for Nanophase Materials Sciences at Oak Ridge National Laboratory, JINR(Dubna), Tsinghua Univ., LBNL, Vanderbilt Univ., Vanderbilt Univ./Univ. of Tennessee, Knoxville, Vanderbilt Univ./Univ. of Kentucky, GANIL, Vanderbilt Univ./Union Univ., JINR, ORAU, Tsinghua University, LNBL

  • J.K. Hwang

    Western Kentucky University, University of Pardubice, Francis Marion University, Clemson University Professor, Francis Marion University Professor, Undergraduate Administrator, Oak Ridge National Laboratory, Oak Ridge, Tennessee, Austin Peay State University, University Strenwarte-Muenchen, Seoul National University, Gatton Academy for Science and Mathematics, Alabama A\&M University, Cygnus, Center for Nanophase Materials Science at Oak Ridge National Laboratory, Vanderbilt University, Fisk Univ, 2Cornell High Energy Synchrotron Source, Cornell University, Ithaca, NY, NOVA Center, Western Kentucky University, Department of Physics, Florida A\&M University, Tallahassee, FL-32307, Correlated Electron Materials Group, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6061 USA, Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN 37235, USA, Department of Electrical and Computer Engineering, University of Rochester, Rochester, NY 14627, USA, The Institute of Optics, University of Rochester, Rochester, NY 14627, USA, Universidade Estadual Paulista (UNESP), Clark Atlanta University, Deapartment of Physics \& Astronomy, Georgia State University, USA, Department of Electrical and Computer Engineering, McGill University, Montreal, QC H3A 2A7, Canada, Oak Ridge National Laboratory, University of South Alabama, Samford University, University of Rochester, University of North Carolina, Chapel Hill, Sandia National Laboratories, New Mexico State University, University of Tennessee Space Institute, Shanghai Jiao Tong University, Shanghai, China, University of Leeds, Leeds, UK, Georgia State University, Atlanta GA, University of Alabama at Birmingham, National High Magnetic Field Laboratory, Prairie View A\&M University, Brookhaven National Laboratory, University of Southern Indiana, Center for Nanophase Materials Sciences at Oak Ridge National Laboratory, JINR(Dubna), Tsinghua Univ., LBNL, Vanderbilt Univ., Vanderbilt Univ./Univ. of Tennessee, Knoxville, Vanderbilt Univ./Univ. of Kentucky, GANIL, Vanderbilt Univ./Union Univ., JINR, ORAU, Tsinghua University, LNBL