E.T. Thostenson et al. Composites Science and Technology 61(2001)1899-1912 Fig. 11. Micrograph phs showing(a) the apparatus for tensile loading of MWCNTs and(b) the telescoping, ""sword and sheath"fracture behavior of the MWCNT [38] al. [40] also tested ropes of multi-walled nanotubes in with parameters determined from first principles. A tension. In their experiments, the obtained tensile comparison of the bending stiffnesses of single-walled strength and modulus were 3.6 and 450 GPa, respec- nanotubes and an iridium beam was presented. The tively. It was suggested that the lower values for bending stiffness of the iridium beam was deduced by strength and stiffness may be a consequence of defects using the continuum Bernoulli-Euler theory of beam in the CVD-grown nanotubes bending. Overney and co-workers concluded that the beam bending rigidity of a nanotube exceeds the highest values found in any other presently available materials 5. Mechanics of carbon nanotubes Besides their experimental observations, Iijima et al [47] examined response of nanotubes under compression As discussed in the previous section, nanotube defor- using molecular dynamics simulations. They simulated mation has been examined experimentally. Recent the deformation properties of single- and multi-walled investigations have shown that carbon nanotubes pos- nanotubes bent to large angles. Their experimental and sess remarkable mechanical properties, such as excep- theoretical results show that nanotubes are remarkably tionally high elastic modulus [34,35, large elastic strain flexible. The bending is completely reversible up to and fracture strain sustaining capability [41, 42]. Similar angles in excess of 110, despite the formation of com conclusions have also been reached through some theo- plex kink shapes. Fig. 12 shows their numerical and retical studies [43-46], although very few correlations experimental results, demonstrating the exceptional between theoretical predictions and experimental studies resilience of carbon nanotubes at large strain. have been made In this section we examine the mechan. Ru [48] noticed that actual bending stiffness of single- ics of both single walled and multi-walled nanotubes walled nanotubes is much lower than that given by the elastic-continuum shell model if the commonly defined 5. Single-walled carbon nanotubes representative thickness is used. Ru proposed the use of an effective nanotube bending stifness as a material Theoretical studies concerning the mechanical prop parameter not related to the representative thickness rties of single-walled nanotubes have been pursued With the aid of this concept, the elastic shell equations extensively. Overney et al. [43] studied the low-frequency can be readily modified and then applied to single-wal- vibrational modes and structural rigidity of long nano- led nanotubes. The computational results based on this tubes consisting of 100, 200 and 400 atoms. The calcula- concept show a good agreement with the results from tions were based on an empirical Keating Hamiltonian molecular dynamics simulationsal. [40] also tested ropes of multi-walled nanotubes in tension. In their experiments, the obtained tensile strength and modulus were 3.6 and 450 GPa, respec￾tively. It was suggested that the lower values for strength and stiffness may be a consequence of defects in the CVD-grown nanotubes. 5. Mechanics of carbon nanotubes As discussed in the previous section, nanotube defor￾mation has been examined experimentally. Recent investigations have shown that carbon nanotubes pos￾sess remarkable mechanical properties, such as excep￾tionally high elastic modulus [34,35], large elastic strain and fracture strain sustaining capability [41,42]. Similar conclusions have also been reached through some theo￾retical studies [43–46], although very few correlations between theoretical predictions and experimental studies have been made. In this section we examine the mechan￾ics of both single walled and multi-walled nanotubes. 5.1. Single-walled carbon nanotubes Theoretical studies concerning the mechanical prop￾erties of single-walled nanotubes have been pursued extensively. Overney et al. [43] studied the low-frequency vibrational modes and structural rigidity of long nano￾tubes consisting of 100, 200 and 400 atoms. The calcula￾tions were based on an empirical Keating Hamiltonian with parameters determined from first principles. A comparison of the bending stiffnesses of single-walled nanotubes and an iridium beam was presented. The bending stiffness of the iridium beam was deduced by using the continuum Bernoulli-Euler theory of beam bending. Overney and co-workers concluded that the beam bending rigidity of a nanotube exceeds the highest values found in any other presently available materials. Besides their experimental observations, Iijima et al. [47] examined response of nanotubes under compression using molecular dynamics simulations. They simulated the deformation properties of single- and multi-walled nanotubes bent to large angles. Their experimental and theoretical results show that nanotubes are remarkably flexible. The bending is completely reversible up to angles in excess of 110
, despite the formation of com￾plex kink shapes. Fig. 12 shows their numerical and experimental results, demonstrating the exceptional resilience of carbon nanotubes at large strain. Ru [48] noticed that actual bending stiffness of single￾walled nanotubes is much lower than that given by the elastic-continuum shell model if the commonly defined representative thickness is used. Ru proposed the use of an effective nanotube bending stiffness as a material parameter not related to the representative thickness. With the aid of this concept, the elastic shell equations can be readily modified and then applied to single-wal￾led nanotubes. The computational results based on this concept show a good agreement with the results from molecular dynamics simulations. Fig. 11. Micrographs showing (a) the apparatus for tensile loading of MWCNTs and (b) the telescoping, ‘‘sword and sheath’’ fracture behavior of the MWCNT [38]. E.T. Thostenson et al. / Composites Science and Technology 61 (2001) 1899–1912 1905