Research ArticleNANOMATERIALS

Toughness of carbon nanotubes conforms to classic fracture mechanics

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Science Advances  05 Feb 2016:
Vol. 2, no. 2, e1500969
DOI: 10.1126/sciadv.1500969

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  • RE: Best fit of QFM
    • Israel Greenfeld, Academic Consultant, Weizmann Institute of Science
    • Other Contributors:
      • H. Daniel Wagner, Professor, Weizmann Institute of Science

    Thank you. As noted in your 2004 paper, at the atomic scale the fracture quantum length is expected to be close to the atomic spacing. At the same time, you noted in your previous eLetter that LEFM represents the limiting case of QFM for the fracture quantum tending to zero. We checked the applicability of these assertions to our study by running the best fit of QFM to our CNT simulation results, leaving the fracture quantum and the crack tip radius as free parameters. The fit yields a fracture quantum very close to zero (several orders of magnitude smaller than the atomic spacing), and as a result the QFM fitting function coincides with the LEFM model. In other words, for the CNT vacancy defects investigated in our study, the fracture quantum tends to zero and hence QFM reduces to LEFM. Therefore, it seems to us that LEFM provides a complete and sufficient description for vacancy defects in CNTs.

    Competing Interests: None declared.
  • RE:

    Thanks. Obviously one has to make the "best fit" for deriving the proper value of the fracture quantum q, not just a "good fit" with q=0 (LEFM) or a bad fit using an a priori value of q (this is not QFM but a wrong application of QFM, that btw is not "quantum" but "quantized" fracture mechanics) since it depends on details, e.g. the used potentials.

    Competing Interests: None declared.
  • RE: clarification on seeming inconsistency
    • Israel Greenfeld, Academic Consultant, Weizmann Institute of Science
    • Other Contributors:
      • H. Daniel Wagner, Professor, Weizmann Institute of Science

    As the authors of this paper, we wish to thank Prof. Pugno for the comment in his eLetter from 9-May 2016. Indeed, as proposed in his past papers, quantum fracture mechanics (QFM) reduces to linear elastic fracture mechanics (LEFM) when the fracture quantum length tends to zero. This is in fact clearly pointed out in our paper, relieving any potential concern among readers about a seeming inconsistency between QFM and LEFM. That said, when using a fracture quantum length of the same scale as the bond length or cell size, as indicated in Pugno and Ruoff’s 2004 paper, we see that QFM does not fit well the results of our simulation for small defect sizes, and this is the inconsistency reported in our paper. We could obviously have used QFM with the fracture quantum length zeroed, but then we would not have gained any useful insight on CNTs fracture toughness in the presence of small defects. By contrast, we do see that LEFM provides a good fit for both short and long defects, and does not diverge (that is, tends to infinity) even for the smallest possible vacancy defect.

    Competing Interests: None declared.
  • RE: evidence of self-inconsistent sections in this paper

    Dear Colleagues,
    only now I have noted this paper,
    and in particular the two sections:
    Toughness at the nanoscale conforms to classic fracture mechanics &
    Quantum fracture mechanics is inconsistent with the simulation of small
    defects,
    that are not self-consistent by definition: a part from details (e.g. we
    already reported in 2004 the fracture toughness of cnt and graphene,
    only recently measured), the self-inconsistency is evident noting that classical (I would
    better say, linear elastic) fracture mechanics (i.e. LEFM) represents
    the limiting case of quantized fracture mechanics (QFM) for fracture
    quantum tending to zero, thus QFM includes LEFM.
    As an author (with Rod Ruoff) of the Quantized fracture mechanics
    theory/paper I am writing this eLetter in order
    to avoid the propagation of some discrepancies reported in
    those sections of this paper.
    Sincerely, Nicola Pugno
    http://www.ing.unitn.it/~pugno/

    Competing Interests: None declared.

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