E.T. Thostenson et al. Composites Science and Technology 61(2001)1899-1912 potential and molecular dynamics simulations. Their molecular-dynamics simulations show that carbon nano- tubes, when subjected to large deformations, reversibly switch into different morphological patterns. Each shape change corresponds to an abrupt release of energy and singularity in the stress/strain curve. These transforma- tions are explained well by a continuum shell model. With properly chosen parameters, their model provided a very accurate'roadmap'of nanotube behavior beyond the lin ear elastic regime. They also made molecular dynamics simulations to single- and double-walled nanotubes of different chirality and at different temperatures [45]. Their emulations show that nanotubes have an extremely large breaking strain (in the range 30-40%)and the breaking train decreases with temperature. Yakobson [11] also applied dislocation theory to carbon nanotubes for Fig. 12. TEM micrograph and computer simulation of nanotube describing their main routes of mechanical relaxation buckling [47]. under tension. It was concluded that the yield strength of a nanotube depends on its symmetry and it was believed Vaccarini et al. [49] investigated the influence of that there exists an intra-molecular plastic flow. Under anotube structure and chirality on the elastic proper- high stress, this plastic flow corresponds to a motion of ties in tension, bending, and torsion. They found that dislocations along helical paths within the nanotube wall the chirality played a small influence on the nanotube and causes a stepwise necking, a well-defined new sym tensile modulus. However, the chiral tubes exhibit metry, as the domains of different chiral symmetry are asymmetric torsional behavior with respect to left and formed. As a result, both the mechanical and electronic right twist, whereas the armchair and ziz-zag tubes do properties of carbon nanotubes are changed not exhibit this asymmetric torsional behavior. The single walled nanotubes produced by laser abla- A relatively comprehensive study of the elastic prop- tion and arc-discharge techniques have a greater ten- erties of single-walled nanotubes was reported by Lu dency to form ropes'or aligned bundles [15, 23]. Thus, [44]. In this study, Lu adopted an empirical lattice- theoretical studies have been made to investigate the dynamics model [50], which has been successfully adopted mechanical properties of these nanotube bundles. Ru in calculating the phonon spectrum and elastic properties [52] presented a modified elastic-honeycomb model to of graphite. In this lattice-dynamics model, atomic inter tudy elastic buckling of nanotube ropes under high actions in a single carbon layer are approximated by a pressure. Ru gave a simple formula for the critical pres sum of pair-wise harmonic potentials between atoms. sure as a function of nanotube Youngs modulus and The local structure of a nanotube layer is constructed wall thickness-to-radius ratio. It was concluded that sin from conformal mapping of a graphite sheet on to a gle-walled ropes are susceptible to elastic buckling under cylindrical surface. Lu's work attempted to answer such high pressure and elastic buckling is responsible for the nanotubes depend on the structural details, such as size electrical resistivity of single walled nanotubes odes and basic questions as: (a)how do elastic properties of pressure-induced abnormalities of vibration mo and chirality? and(b) how do elastic properties of Popov et al. [53] studied the elastic properties of tri- nanotubes compare with those of graphite and diamond? angular crystal lattices formed by single-walled nano- Lu concluded that the elastic properties of nanotubes are tubes by using analytical expressions based on a force- insensitive to size and chirality. The predicted Youngs constant lattice dynamics model [54]. They calculated modulus (I TPa), shear modulus(0.45 TPa), and various elastic constants of nanotube crystals for nano- bulk modulus(0.74 TPa)are comparable to those of tube types, such as armchair and zigzag. It was shown diamond. Hernandez and co-workers [51] performed cal- that the elastic modulus, Poissons ratio and bulk mod culations similar to those of Lu and found slightly higher ulus clearly exhibit strong dependence on the tube values( 1. 24 TPa)for the Youngs moduli of tubes. But radius. The bulk modulus was found to have a max unlike Lu, they found that elastic moduli are sensitive to imum value of 38 GPa for crystals composed of single- both tube diameter and structure walled nanotubes with 0.6 nm radius Besides their unique elastic properties behavior of nanotubes has also received considerable 5.2. Multi-walled carbon nanotubes attention. Yakobson and co-workers [10, 46] examined the instability behavior of carbon nanotubes beyond linear Multi-walled nanotubes are composed of a number of response by using a realistic many-body Tersoff-Brenner concentric single walled nanotubes held together withVaccarini et al. [49] investigated the influence of nanotube structure and chirality on the elastic proper￾ties in tension, bending, and torsion. They found that the chirality played a small influence on the nanotube tensile modulus. However, the chiral tubes exhibit asymmetric torsional behavior with respect to left and right twist, whereas the armchair and ziz-zag tubes do not exhibit this asymmetric torsional behavior. A relatively comprehensive study of the elastic prop￾erties of single-walled nanotubes was reported by Lu [44]. In this study, Lu adopted an empirical lattice￾dynamics model [50], which has been successfully adopted in calculating the phonon spectrum and elastic properties of graphite. In this lattice-dynamics model, atomic inter￾actions in a single carbon layer are approximated by a sum of pair-wise harmonic potentials between atoms. The local structure of a nanotube layer is constructed from conformal mapping of a graphite sheet on to a cylindrical surface. Lu’s work attempted to answer such basic questions as: (a) how do elastic properties of nanotubes depend on the structural details, such as size and chirality? and (b) how do elastic properties of nanotubes compare with those of graphite and diamond? Lu concluded that the elastic properties of nanotubes are insensitive to size and chirality. The predicted Young’s modulus (1 TPa), shear modulus (0.45 TPa), and bulk modulus (0.74 TPa) are comparable to those of diamond. Hernandez and co-workers [51] performed cal￾culations similar to those of Lu and found slightly higher values (1.24 TPa) for the Young’s moduli of tubes. But unlike Lu, they found that elastic moduli are sensitive to both tube diameter and structure. Besides their unique elastic properties, the inelastic behavior of nanotubes has also received considerable attention. Yakobson and co-workers [10,46] examined the instability behavior of carbon nanotubes beyond linear response by using a realistic many-body Tersoff-Brenner potential and molecular dynamics simulations. Their molecular-dynamics simulations show that carbon nano￾tubes, when subjected to large deformations, reversibly switch into different morphological patterns. Each shape change corresponds to an abrupt release of energy and a singularity in the stress/strain curve. These transforma￾tions are explained well by a continuum shell model. With properly chosen parameters, their model provided a very accurate ‘roadmap’ of nanotube behavior beyond the lin￾ear elastic regime. They also made molecular dynamics simulations to single- and double-walled nanotubes of different chirality and at different temperatures [45]. Their simulations show that nanotubes have an extremely large breaking strain (in the range 30–40%) and the breaking strain decreases with temperature. Yakobson [11] also applied dislocation theory to carbon nanotubes for describing their main routes of mechanical relaxation under tension. It was concluded that the yield strength of a nanotube depends on its symmetry and it was believed that there exists an intra-molecular plastic flow. Under high stress, this plastic flow corresponds to a motion of dislocations along helical paths within the nanotube wall and causes a stepwise necking, a well-defined new sym￾metry, as the domains of different chiral symmetry are formed. As a result, both the mechanical and electronic properties of carbon nanotubes are changed. The single walled nanotubes produced by laser abla￾tion and arc-discharge techniques have a greater ten￾dency to form ‘ropes’ or aligned bundles [15,23]. Thus, theoretical studies have been made to investigate the mechanical properties of these nanotube bundles. Ru [52] presented a modified elastic-honeycomb model to study elastic buckling of nanotube ropes under high pressure. Ru gave a simple formula for the critical pres￾sure as a function of nanotube Young’s modulus and wall thickness-to-radius ratio. It was concluded that sin￾gle-walled ropes are susceptible to elastic buckling under high pressure and elastic buckling is responsible for the pressure-induced abnormalities of vibration modes and electrical resistivity of single walled nanotubes. Popov et al. [53] studied the elastic properties of tri￾angular crystal lattices formed by single-walled nano￾tubes by using analytical expressions based on a force￾constant lattice dynamics model [54]. They calculated various elastic constants of nanotube crystals for nano￾tube types, such as armchair and zigzag. It was shown that the elastic modulus, Poisson’s ratio and bulk mod￾ulus clearly exhibit strong dependence on the tube radius. The bulk modulus was found to have a max￾imum value of 38 GPa for crystals composed of single￾walled nanotubes with 0.6 nm radius. 5.2. Multi-walled carbon nanotubes Multi-walled nanotubes are composed of a number of concentric single walled nanotubes held together with Fig. 12. TEM micrograph and computer simulation of nanotube buckling [47]. 1906 E.T. Thostenson et al. / Composites Science and Technology 61 (2001) 1899–1912