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In a recent paper [1] Adati and Miyazawa studied conformally recurrent spaces, that is, Riemannian spaces defined by where is the conformal curvature tensor: λi is a non-zero vector and comma denotes covariant differentiation with respect to the metric tensor gij.

Molecular dynamics (MD) simulations are performed using adaptive intermolecular reactive bond order potential to analyze single-walled and double-walled carbon nanotubes. These carbon nanotubes were analyzed for buckling under compression and under torsion. The MD simulations create a comprehensive database for the critical buckling loads/strains and critical buckling torques/twist angles for armchair SWCNTs and DWCNTs of varying diameters and lengths. Using MD results as a computational benchmark, an equivalent thick shell model of CNT is proposed, which is amenable for analysis using a commercially available software ABAQUS. Based on our MD results, an empirical equation that describes the size-dependent Young's modulus for a single-walled carbon nanotube is established. Buckling analysis of CNT under compression and under torsion are performed with the equivalent shell model using size-dependent Young's modulus, Poisson's ratio = 0.19 and shell thickness h = 0.066 nm. We show that the equivalent shell model gives good estimate of critical buckling load/strain and critical buckling torque with respect to the MD results. Variation of critical twist angle with length of CNT, predicted by the shell model is in good qualitative agreement with MD simulation. However, the equivalent shell model underestimates the critical twist angle by 30% because the continuum shell model overestimates torsional stiffness of CNT compared to an atomistic model of CNT. The equivalent shell model is less computational intensive to implement as compared with MD. Its accuracy for predicting the buckling states for long carbon nanotubes allows it to be used for moderately long CNTs under compression/torsion, in-lieu of MD simulations.

Carbon nanotubes (CNTs) have potential applications in various fields of science and engineering due to their extremely high elasticity, strength, and thermal and electrical conductivity. Owing to their hollow and slender nature, these tubes are susceptible to buckling under a compressive axial load. As CNTs can undergo large, reversible post-buckling deformation, one may utilize this postbuckling response of CNT to manufacture mechanical energy storage devices at the nano-scale, or use it as a nano-knife or nano-pump. It is therefore important to understand the buckling behavior of CNTs under a compressive axial load. Experimental investigations on CNT buckling are very expensive and difficult to perform. As such, researchers often rely on molecular dynamics (MD) simulations, or continuum mechanics modeling to study their mechanical behaviors. In order to develop a good continuum mechanics model for buckling analysis of CNTs, one needs to possess adequate experimental or MD simulation data for its calibration. For ''short'' CNTs with small aspect ratios (B10), researchers have reported different critical buckling loads/strains for the same CNTs based on MD simulations. Moreover, existing MD simulation data are not sufficiently comprehensive to allow rigorous benchmarking of continuum-based models. This chapter presents extensive sets of MD critical buckling loads/strains for armchair single-walled CNT (SWCNTs) and double-walled CNTs (DWCNTs), with various aspect ratios less than 10. These results serve to address the discrepancies found in the existing MD simulations, as well as to offer a comprehensive database for the critical buckling loads/strains for various armchair SWCNTs and DWCNTs. The Adaptive Intermolecular Reactive Bond Order (AIREBO) potential was adopted for MD simulations. Based on the MD results, the Young's modulus, Poisson's ratio and thickness for an equivalent continuum cylindrical shell model of CNTs are calibrated. The equivalent continuum shell model may be used to calculate the buckling loads of CNTs, in-lieu of MD simulations.

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