For many mechanicians and materials scientists one of the most confounding things (in the ever increasing literature on carbon nanotubes) is the reported theoretical value of the nanotube elastic modulus. Depending upon the specific paper at hand, the reported numerical values range from 1 -6 TPa! This scatter is known as “Yakobson’s Paradox” after Boris Yakobson (Rice University) who first discussed this. In a recent paper, Young Huang and co-workers address this issue and provide a resolution.

Atomistic calculations (whether empirical molecular dynamics or ab initio approach) provide only an estimate of the “tension rigidity” i.e. “Eh”, where E is the Young’s modulus and “h” is the thickness of the single walled nanotube or the graphene sheet. Most properly done atomistic calculations (from various sources) all seem to agree on the numerical value of Eh. There are many cases where there may be no need to know E. However, if a specific value is desired then an estimate of h is required to compute it. According to Huang et. al., therein lies the source for the wide scatter in reported values of E. If a thickness equivalent to that of graphite interlayer spacing is assumed, E turns out to be roughly 1 TPa. Afew works employ a shell model. In the latter case, one computes both the tension rigidity T=Eh/(1-v2), as well as the bending rigidity B=Eh3/12(1-v2). Their ratio leads to an expression for the thickness h=√12B/T. This exercise results in an estimate of 5-6 TPa for the elastic modulus. Interested reader can get the facts directly from the paper, however briefly the central physical insight is that due to the monolayer nature of graphene, torsion rigidity is zero and the ratio of the bending to tension rigidity is not a constant (depending in fact on the type of loading). This conclusion is in fact quite general for any material (and not just graphene) and will apply to all monolayers with hexagonal symmetry (who have thus ill-defined thickness). Some of these conclusions become quite transparent since Huang’s paper relates the relevant rigidities analytically to the interatomic potential.