The following article shows that the linear elasticity theory does
not predict well inextensional modes of vibration of a single-wall
zig-zag tube.

 Gupta, S.S., Bosco, F.  and 
Batra, R. C. (2009) Breakdown of structural models for vibrations of
single-wall zigzag carbon nanotubes, J.
Applied Physics, 106,
063527.

Myfeeling is that what we're trying to find are elastic constants of a continuum structure whose response in several (ideally all) deformations is the same as that of the carbon nanotube subjected to the same boundary conditions as the continuum structure.  We (A. Sears and R. C. Batra, Macroscopic Properties of Carbon Nanotubes from Molecular-Mechanics Simulations, Physical Reviews B, 69, 235406, 2004) have simulated simple tension and torsional deformations of a SWNT and its equivalent continuum structure defined as the one whose strain energy density is the same as that of the SWNT.  For an isotropic structure, the thickness of the equivalent structure was found to be~0.21 and it depends upon the MM potential used.  This has been validated by performing bending, buckling and combined loading tests on the SWNT and the equivalent continuum structure.