Effective coarse-grained simulations of super-thick multi-walled carbon nanotubes under torsion

Under torsion and beyond the buckling point, multi-walled carbon nanotubes (MWCNTs) develop a periodic wave-like rippling morphology. Here, we show that torsional rippling deformations can be accurately described by a simple sinusoidal shape function. Combining this observation with the geometry optimization, we develop an effective coarse-grained model that reproduces the complex nonlinear mechanical responses of thick MWCNTs under torsion predicted by large-scale atomistic simulations. Furthermore, the model allows us to simulate super-thick tubes, inaccessible by other coarse-grained methods. With this effective coarse-grained model, we show from an energetic analysis that the rippling deformation is a result of in-plane strain energy relaxation, penalized by the increase in the interlayer van der Waals interaction energy. Our simulations reveal that the torsional response of MWCNTs with up to 100 layers approximately follows a simple bilinear law, and the ratio of the torsional rigidities in the pre- and post-buckling regimes is nearly a constant, independent of the tube radius. In contrast, the bifurcation torsional strain powerly scales with the tube radius. We also find that the wave number in the circumferential direction linearly increases with tube radius, while the wavelength monotonically increases with tube radius, and approaches a constant in the limit of bulk graphite. The bilinear constitutive relation, together with the scaling law of the bifurcation torsional strain, furnishes a simple nonlinear beam theory, which facilitates the analysis of MWCNT bundles and networks.