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
  • Fig. 1 Simulation models of SWNTs with embedded defects.

    (A) Definition of the simulation protocol for a defective SWNT (length, 170 Å) under axial tension: a constant strain rate Embedded Image is applied at the CNT free end, with resulting far-field stress σff and maximum stress σmax. (B) Simulation models of armchair SWNTs [(10, 10) chirality] with Stone-Wales (a) and vacancy (b to i) defects. The number of missing atoms in vacancies is indicated. (C) Simulation models of zigzag SWNTs [(17, 0) chirality] with Stone-Wales (a) and vacancy (b to i) defects.

  • Fig. 2 Simulated stress and strength of SWNTs under tension.

    (A) Stress-strain curves of defect-free zigzag and armchair SWNTs. The images show the SWNTs’ deformation at 20% strain. (B) Maximum stress concentration Kt in armchair and zigzag SWNTs containing defects of various shapes and sizes (Fig. 1, B and C), at a strain of ~10%. The number of missing atoms in vacancies is indicated at the top of the figure. (C) Ultimate tensile strength σf in armchair and zigzag SWNTs containing defects of various shapes and sizes. Kt and σf data are provided in note S4.

  • Fig. 3 High stresses around a defect lead to crack propagation.

    (A) Nonuniform stress distribution in an armchair SWNT containing a two-atom vacancy defect [Fig. 1B(c)], under tensile loading at 10% strain, expressed in terms of atom forces in the loading direction. The highest stressed atoms are colored red. (B) Spontaneous crack propagation at a fixed strain of 11%. The loading direction is vertical, and the crack is viewed from the side. (C) Atom forces around a crack (front view).

  • Fig. 4 Comparison of simulation with solid mechanics theories.

    (A) Simulated SWNT maximum stress concentration factor Kt = σmaxff at ~10% elongation versus the relative defect semilength a/r0, compared with linear elasticity predictions for an infinite-width thin plate having an elliptical hole with fixed tip radius (ρ = r0 and ρ = 2r0). r0 ≅ 1.42 Å is the C═C bond length. The SWNTs have one-atom, two-atom, and three-atom vacancy defects (Fig. 1, B and C); the filled and unfilled symbols are for the simulated armchair (AC) and zigzag (ZZ) tubes, respectively, and the letter labels designate the defect types per Fig. 1. Refer to Eq. 1 in the text. The defect dimensions and the loading direction (indicated by the arrow) are defined in the inset drawing. (B) Simulated SWNT relative fracture strength σf/σ* versus the relative defect semilength a/r0, compared with LEFM prediction for a sharp crack in an infinite-width thin plate (KIc = 2.69 MPa m0.5) and with QFM predictions for sharp (ρ = 0, q = r0) and blunt (Embedded Image, Embedded Image) cracks. Refer to Eqs. 2 and 4 in the text. The far-field stress σff and the fracture strength σf in these plots are the nominal values at the defect’s cross section, adjusted for the infinite-width plate assumption (Materials and Methods). The data are provided in note S4.

  • Fig. 5 Defect orientation, shape, and scale.

    (A) Effect of chirality on the defect orientation and length 2a. The direction of the load is indicated by the solid arrow. (B) Vacancy defects of equivalent SWNT strength, sharing the same length 2a and tip curvature r0, but with different width 2b, shape, and orientation. The dashed arrows indicate possible crack propagation paths. (C) Comparison of defects at the microscale and nanoscale. Microscale defects are surrounded by a continuum and, if sharp, can cause infinite stress concentrations. Nanoscale defects (that is, SWNT vacancies) are surrounded by a truss-like lattice (they are not sharp), cause finite stress concentrations, and cannot be smaller than a minimal size 2amin (the length of a monovacancy).

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/2/2/e1500969/DC1

    Note S1. Molecular dynamics.

    Note S2. Bond failure criterion.

    Note S3. Defect-free SWNT under tension.

    Note S4. SWNT with defects under tension.

    Note S5. Effect of prestresses in topological defects.

    Note S6. Comparison with solid mechanics theories.

    Note S7. Alternative estimation of the SWNT fracture toughness.

    Table S1. Defect dimensions and simulation results.

    Fig. S1. Simulated stresses around a Stone-Wales defect.

    Movie S1. Crack propagation illustration (separate file).

    References (5461)

  • Supplementary Materials

    This PDF file includes:

    • Note S1. Molecular dynamics.
    • Note S2. Bond failure criterion.
    • Note S3. Defect-free SWNT under tension.
    • Note S4. SWNT with defects under tension.
    • Note S5. Effect of prestresses in topological defects.
    • Note S6. Comparison with solid mechanics theories.
    • Note S7. Alternative estimation of the SWNT fracture toughness.
    • Table S1. Defect dimensions and simulation results.
    • Fig. S1. Simulated stresses around a Stone-Wales defect.
    • References (54–61)

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    Other Supplementary Material for this manuscript includes the following:

    • Movie S1. Crack propagation illustration (separate file).

    Files in this Data Supplement:

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