tailieunhanh - Wave Propagation 2011 Part 9

Tham khảo tài liệu 'wave propagation 2011 part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Wave Propagation in Carbon Nanotubes 227 local elastic model of Timoshenko beam. Section 5 turns to the dispersion relation of flexural waves in a multi-walled carbon nanotube from a non-local elastic model of multiTimoshenko beams which also takes the second order gradient of strain into account. Similarly Section 6 gives the dispersion relation of longitudinal waves in a multi-walled carbon nanotube on the basis of a non-local elastic model of multi-cylindrical shells. Finally the chapter ends with some concluding remarks made in Section 7. 2. Molecular dynamics model for carbon nanotubes This section presents the molecular dynamics models for wave propagation in a carbon nanotube respectively for a wide range of wave numbers. Molecular dynamics simulation consists of the numerical solution of the classical equations of motion which for a simple atomic system may be written mr F dV . dr 1 For this purpose the force F acting on the atoms are derived from a potential energy V rij where rij is the distance from atom i to atom j. In the molecular dynamics models of this chapter the interatomic interactions are described by the Tersoff-Brenner potential Brenner 1990 which has been proved applicable to the description of mechanical properties of carbon nanotubes. The structure of the Tersoff-Brenner potential is as follows V r vv Vrr -BV r j 2 i j i VR r and VA rij are the repulsive and attractive terms given by .Di VR r j - f j r j -7exp -J2sỊjfl j r - r0 3a S-1 VA r j - f r ệD-exp -ự2 s Pj r -r . 3b Sij 1 Here Sij D j P j 15nm-1 r0 fj Dij Sij P j are scalars fij rij is a switch function used to confine the pair potential in a neighborhood with radius of r2 as following f r - f1 1l n r- r1 Ì 1 cos I I 2 L I r2 - r1 0 r r1 r1 - r j - r2 r r2 3c o o where r1 A r2 . In Equation 2 B reads 228 Wave Propagation in Materials for Modern Applications - 1 . B - 1 b b i 2 i r 3d b - 1 2 G 9 1 fikbik k i i b - 1 2 G 0tji fik rjk k i i 3e _ k c2 G