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A finite deformation membrane based on inter-atomic potentials for the transverse mechanics of

Summary: A finite deformation membrane based on
inter-atomic potentials for the transverse mechanics of
Marino Arroyo, Ted Belytschko *
Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA
Received 31 May 2001; received in revised form 27 March 2002
A finite deformation hyper-elastic membrane theory based on inter-atomic potentials for crystalline films composed
of a single atomic layer is developed. For this purpose, an extension of the standard Born rule that exploits the dif-
ferential geometry concept of the exponential map is proposed to deal with the curvature of surfaces. The exponential
map is approximated locally and strain measures based on the stretch and the curvature of the membrane arise. The
methodology is first particularized to atomic chains in two dimensions, and then to graphene sheets. A reduced model
for the transverse mechanics of carbon nanotubes is developed in detail. This model is a hyper-elastic constrained
membrane which fully exploits the symmetry of the transverse deformation. Additionally, a continuum version of the
non-bonded interactions is provided. The continuum model is discretized using finite elements and very good agreement
with molecular mechanics simulations is obtained. Finally, several simulations illustrate the strong effect of the van der
Waals interactions in the transverse deformation of carbon nanotubes.
Ó 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Crystal elasticity; Quasicontinuum; Born rule; Exponential map; Membrane; Carbon nanotubes
1. Introduction


Source: Arroyo, Marino - Departament de Matemątica Aplicada III, Universitat Politčcnica de Catalunya


Collections: Engineering; Materials Science