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Research Papers

Mathematical Treatise to Model Dihedral Energy in the Multiscale Modeling of Two-Dimensional Nanomaterials

[+] Author and Article Information
Sandeep Singh

Department of Mechanical Engineering,
Birla Institute of Technology and Science Pilani,
K. K. Birla Goa Campus,
Goa 403726, India
e-mail: mechmehal@gmail.com

B. P. Patel

Department of Applied Mechanics,
Indian Institute of Technology Delhi,
New Delhi 110016, India
e-mail: badripatel@hotmail.com

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received December 7, 2017; final manuscript received February 21, 2018; published online March 23, 2018. Assoc. Editor: N. R. Aluru.

J. Appl. Mech 85(6), 061003 (Mar 23, 2018) (10 pages) Paper No: JAM-17-1668; doi: 10.1115/1.4039437 History: Received December 07, 2017; Revised February 21, 2018

An approximate mathematical treatise is proposed to improve the accuracy of multiscale models for nonlinear mechanics of two-dimensional (2D) nanomaterials by taking into account the contribution of dihedral energy term in the nonlinear constitutive model for the generalized deformation (three nonzero components of each strain and curvature tensors) of the corresponding continuum. Twelve dihedral angles per unit cell of graphene sheet are expressed as functions of strain and curvature tensor components. The proposed model is employed to study the bending modulus of graphene sheets under finite curvature. The atomic interactions are modeled using first- and second-generation reactive empirical bond order (REBO) potentials with the modifications in the former to include dihedral energy term for accurate prediction of bending stiffness coefficients. The constitutive law is obtained by coupling the atomistic and continuum deformations through Cauchy–Born rule. The present model will facilitate the investigations on the nonlinear mechanics of graphene sheets and carbon nanotubes (CNTs) with greater accuracy as compared to those reported in the literature without considering dihedral energy term in multiscale modeling.

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Figures

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Fig. 1

The schematic arrangement of carbon atoms in a unit cell, A and B represent atoms from two different sublattices

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Fig. 2

Schematic description for calculating the out of plane angles corresponding to r12 bond

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Fig. 3

Energy per atom of the zigzag graphene sheet versus curvature K11 curve for REBO-II potential with and without dihedral energy term

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Fig. 4

The variation of the bending modulus D11 of graphene sheet with curvature K11 for REBO-II potential

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Fig. 5

The softening of bending modulus D11 of graphene sheet with uniaxial tensile strain E11 for REBO-II potential

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Fig. 6

Energy per atom of the zigzag graphene sheet versus curvature K curves with and without dihedral energy term

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Fig. 7

The variation of bending modulus D11 of graphene sheet with curvature K11 with and without dihedral energy term for: (a) REBO-I and (b) REBO-II

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Fig. 8

The variation of bending modulus D11 and D22 of graphene sheet with curvatures with and without dihedral energy term under spherical bending

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Fig. 9

The variation of bending modulus D11 of graphene sheet with twisting curvatures K12 with and without dihedral term

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Fig. 10

The softening of bending modulus D11 of graphene sheet with strain E11 with and without dihedral energy term for: (a) REBO-I and (b) REBO-II

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Fig. 11

The softening of bending modulus D22 of graphene sheet with strain E11 with and without dihedral energy term for: (a) REBO-I and (b) REBO-II

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Fig. 12

The hardening of extensional modulus A11 of graphene sheet with curvature K11 with and without dihedral energy term for: (a) REBO-I and (b) REBO-II

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Fig. 13

Description of the graphene sheet considered in the present study (L = 1.968 nm, b = 4.111 nm). The dimensions are calculated using the bond length corresponding to REBO-II potential.

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Fig. 14

In-plane nominal strain versus central deflection curves of the CFCF graphene sheets obtained with and without considering dihedral energy term

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Fig. 15

Critical buckling load of the graphene sheet with CFCF boundary conditions (λWD: without dihedral energy term, λD: with dihedral energy term)

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