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# Derivation of the bending stiffness from REBO potential -- Contribution from the dihedral term

Thanks to Marino, I have found the reason for the difference in our bending stiffness calculation. The original discussion is here:

node/4029

The reason why we have a higher bending stiffness is due to the dihedral term. This dihedral term does have a significant contribution to the bending stiffness. However, in Ref. [26], apparently, this dihedral term was ignored.

I have written a short document showing the contribution of the dihedral term to the bending stiffness. Please take a look at the attachment.

I received great help from Dr. Huang and Marino. Thank you very much.

Attachment | Size |
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Derivation of the bending stiffness from REBO potential.pdf | 116.62 KB |

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## Comments

## bending modulus of graphene

We have written a short paper on bending modulus of graphene, following the prior discussions. The manuscript is now available at

arXiv:0901.4370v1, which is also under review for journal publication.

Update on September 7, 2009: This work has been published as: Q. Lu, M. Arroyo, R. Huang, Elastic bending modulus of monolayer graphene. J. Phys. D: Appl. Phys. 42, 102002 (2009).

Update on May 10, 2010: This article has been selected to be part of the Journal of Physics D Highlights of 2009 collection. As part of this collection, this article will be free to read until the end of 2010.

RH

## bending modulus of double layers and multiple layers of graphene

Hi Qiang and Rui,

wondering if you have extended work on the bending modulus for multiple layers of graphene. So far I found one paper (doi:10.1016/j.tsf.2005.08.317) on that yet the results for multi-layered graphene is quite doubtable (the authors argue that the bending modulus is the same as that of bi-layered graphene). Another related paper (doi:10.1063/1.3223783) from Prof. Buehler's group indicated the bending modulus is proportional to the cube of thickness of multi-layered graphene. Maybe you can figure out what's the relationship between these parameters.

Xinghua

## bending modulus of few-layered graphene

An analytical form of the bending modulus for N-layered graphene has been obtained by Koskinen and Kit . The same formula has also been developed by an analytical approach (unpublished work). For N > 10, the bending modulus is in close agreement with the classical formula (proportional to N^3). For N < 10, however, the error increases from about 1% (N = 9) to about 30% (N = 2, bilayer). For N = 1, the formula recovers the bending modulus of monolayer graphene, which is about two orders of magnitude smaller than that of a bilayer.

RH

## bending modulus of stretched graphene

Thanks Rui! It really helps me a lot!

I have another question that puzzled me recently. In your J. Phys. D: Appl. Phys paper you studied the bending stiffness of graphene in a free state. If the graphene is under uniaxial stretching, how about the bending stiffness? I think it should change, yet how to prove it? According to the procedure you described in the paper, we can calculate the strain energy of a series of CNTs that under axial stretching. Since in this case, the total strain energy consists of bending energy, strain energy in axial direction and in circumferential direction. It seems diffucult to decouple the three strain energies since the CNT is no longer in a free state. Do you have any suggestions?

## Re: bending modulus of stretched graphene

Hi Xinghua,

In another paper (Int. J. Appl. Mechanics 1, 443-467, 2009), we described a general approach to define the tangent modulus (stiffness) under combined stretching and bending (Section 2.3). In addition to the in-plane and bending stiffness, there is also a coupling modulus in general. For CNTs under axial stretching, you may follow the procedure in Section 2.5 but add an axial stretch in the X2 direction. As you expected, you may not be able to decouple the three energy terms because of the coupling between bending and stretch.

Hope it helps.

RH