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How graphene flexes and stretches under concomitant bending couples and tractions

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Abstract

We propose a geometrically and materially nonlinear discrete mechanical model of graphene that assigns an energetic cost to changes in bond lengths, bond angles, and dihedral angles. We formulate a variational equilibrium problem for a rectangular graphene sheet with assigned balanced forces and couples uniformly distributed over opposite side pairs. We show that the resulting combination of stretching and bending makes achiral graphene easier to bend and harder (easier) to stretch for small (large) traction loads. Our general developments hold for a wide class of REBO potentials; we illustrate them in detail by numerical calculations performed in the case of a widely used 2nd-generation Brenner potential.

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Notes

  1. In this connection, we note that, reversing the force distribution shown in Fig. 1 does not necessarily induce hardening, because the problem nonlinearity demands for a recalculation of the solution with a priori unpredictable effetcs.

  2. In fact, we repeat, our procedure is general enough to accommodate a variety of diehedral-angle sensitive REBO potentials; consequently, it can be adopted to find out whether an intermolecular potential in the class specified by (19–20) does predict the peculiar behavior of graphene predicted in [38].

  3. Couples and forces are uniformly distributed in a discrete sense.

  4. The value of this parameter depends slightly on the intermolecular potential of one’s choice; for the 2nd-generation Brenner potential we use later on in our computations, \(r_0=0.14204\) nm.

  5. Here we have taken relations (11)\(_{4,5}\) into account; later on, when we deal with zigzag bending, we shall use another specialization of (19) and (20).

  6. For an example of such assumptions, which are fulfilled by the stored-energy functional we will use to obtain the representative results reported in Sect. 6, see [10].

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Acknowledgments

N.M.P. is supported by the European Research Council (ERC StG Ideas 2011 BIHSNAM No. 279985, ERC PoC 2015 SILKENE No. 693670) and by the European Commission under the Graphene Flagship (WP14 Polymer Composites, No. 696656).

Funding

N.M.P. is supported by the European Research Council (ERC StG Ideas 2011 BIHSNAM No. 279985, ERC PoC 2015 SILKENE No. 693670) and by the European Commission under the Graphene Flagship (WP14 Polymer Composites, No. 696656).

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Authors and Affiliations

  1. Department of Structural and Geotechnical Engineering, Sapienza University of Rome, Rome, Italy

    Antonino Favata

  2. Dipartimento di Ingegneria Civile e Ingegneria Informatica, University of Rome Tor Vergata, Rome, Italy

    Andrea Micheletti

  3. Accademia Nazionale dei Lincei, Rome, Italy

    Paolo Podio-Guidugli

  4. Department of Mathematics, University of Rome Tor Vergata, Rome, Italy

    Paolo Podio-Guidugli

  5. Laboratory of Bio-Inspired and Graphene Nanomechanics, Department of Civil, Environmental and Mechanical Engineering, University of Trento, Trento, Italy

    Nicola M. Pugno

  6. Center for Materials and Microsystems, Fondazione Bruno Kessler, Trento, Italy

    Nicola M. Pugno

  7. School of Engineering and Materials Science, Queen Mary University of London, London, UK

    Nicola M. Pugno

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  1. Antonino Favata
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  2. Andrea Micheletti
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  4. Nicola M. Pugno
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Correspondence to Antonino Favata.

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Favata, A., Micheletti, A., Podio-Guidugli, P. et al. How graphene flexes and stretches under concomitant bending couples and tractions. Meccanica 52, 1601–1624 (2017). https://doi.org/10.1007/s11012-016-0503-2

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