Axisymmetric and beamlike vibrations of multiwall carbon nanotubes
Bibliographic record
Abstract
Several vibration problems of multiwall carbon nanotubes (MWNTs) are studied in detail based on a multiple-elastic shell model. According to recent data available in the literature, an updated value of bending stiffness for single-wall carbon nanotubes (SWNTs) is suggested, which is in a much better agreement with atomistic model for phonon-dispersion relation of SWNTs. For axisymmetric vibrations (with circumferential wave number $n=0$), it is found that longitudinal $(L)$ modes of individual tubes of a MWNT have almost identical frequencies and are usually coupled with each other through Poisson-ratio effect-induced radial $(R)$ vibrations and interlayer van der Waals interaction. Especially in the transition zone of $R$- and $L$ modes, the significant Poisson-ratio effect leads to mixed $R--L$ modes with comparable longitudinal and radial displacements. On the other hand, for beamlike vibrations (with $n=1$), the present multiple-shell model is found to be in good agreement with the multiple-beam model for almost coaxial bending $(B)$ modes of large- and small-radius MWNTs and noncoaxial $B$ modes of small-radius MWNTs (e.g., of the outermost radius less than $2\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$), with relative errors less than 10%. However, for high-order noncoaxial modes of large-radius MWNTs, the relative errors between the two models increase up to 50% in extreme cases due to larger non-beamlike deformation of the cross section while both models give similar overall vibration modes through the entire length of MWNTs. In particular, for lower circumferential wave numbers $(n=0--10)$, the lowest frequency always corresponds to the minimum half-axial wave number $m=1$ for simply supported end conditions. When the wave vector decreases, the lowest frequency decreases and the associated mode shifts from an $R$ mode with larger $n$ to a coaxial $B$ mode with $n=1$.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".