356 Index. NLSGM nanorods, 124

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1 Index A Ab initio pseudopotentials, 35 Acoustic phonon dispersion, 330, 336 Acoustic pressure, 337 Anodic alumina, 206 Armchair CNT chirality, 93 Asymptotic stiffness, 83 B Band gap, 174, 190, 201, 211, 243, 252, 290, 303 Bessel function, 338 Bi-Helmholtz equation, 151 Born-Karman model, 61, 85, 143, 151 Boussinesq problem, 51 Breathing modes, 336 Brillouin zone, 85, 151, 152 Buckling load, 74, 76 Bulk waves, 305 C Carbon nanotube, 3 armchair, 8, 71, 80, 86 chiral, 8, 71, 80, 101 chiral angle, 8 chirality, 40 in plane stiffness, 72 material property extimation, 71 multi-wall, 3, 8 nanoresonator, 38 properties, 10 roll-up vector, 8 rotating, 177 single-wall, 4, 7, 38 structure, 8 zigzag, 8, 71, 80, 94 Centrifugal stiffening, 179 Circumferential displacement, 326, 337 Circumferential wavenumber, 327, 331, 339 Classical elasticity theory, 123, 143, 205, 314 constitutive relation, 143 Classical plate theory, 271 COMPASS, 39 Contractional wave mode, 348 Coupling stiffness, 239, 262 Critical wavenumber, 134, 136 Crystal lattice, 66, 280 Crystallographic direction, 206, 310 Current density, 195 Cut-off frequency, 23, 28, 294 DNBS, 252 DNRS, 243, 246, 247 DWCNT, 226 fluid carrying SWCNT, 193 fluid filled nanoshell, 340 graphene, 275, 277, 278 graphene on silicon substrate, 289 higher order nanoshell, 346 magnetic filed effects, 200 nanoplate with surface effects, 309, 314 nanoplate with temperature effects, 301, 304 nanoshell, 329, 331, 332 SWCNT Timoshenko beam, 223 Timoshenko DNBS, 259, 261 Timoshenko nanobeam, 174 Cylindrical nanoshell, 324 S. Gopalakrishnan and S. Narendar, Wave Propagation in Nanostructures, NanoScience and Technology, DOI: / Ó Springer International Publishing Switzerland

2 356 Index D Density functional theory, 34 Dispersion relation DNBS, 251 DWCNT, 225 fluid filled nanoshell, 337 fourth order PDE, 26 graphene, 273 graphene on silicon substrate, 287 higher order nanoshell, 345 lateral inertia effects, 142 MWCNT, 219 nanoplate with surface effects, 309 nanoplate with temperature effects, 300 nanoshell, 327 NLSGM nanorod, 124 NLStGM nanorods, 135 second order PDE, 24 SWCNT, 222 TWCNT, 231 DNBS Euler-Bernoulli beam theory, 250 governing equations, 257 Timoshenko beam theory, 254 Double nanobeam system, 239, 248 governing equations, 250 Double nanorod system, 239 governing equations, 240 DWCNT, 220, 221 governing equation, 225 Dynamic stiffness, 153 E Effective density, 139 Eigenvalue problem, 92, 98, 113 Electric field, 195 Equivalent continuum structure, 200 Escape frequency DNRS, 243, 247 DWCNT, 227, 228 fluid carrying SWCNT, 193 graphene, 275, 277 higher order nanoshell, 347, 349 nanoplate with surface effects, 310, 312, 317 nanoshaft, 149, 151, 152 nanoshell, 329, 332 NLGSM rod, 126 NLSGM nanorods, 124 SWCNT Timoshenko beam, 224 Timoshenko beam theory, 174 Timoshenko DNBS, 259, 261 Timoshenko nanobeam, 174, 175, 234 Euler-Bernoulli beam theory, 72, 81, 165, 185, 203, 211, 239, 248, 249, 266 dispersion relation, 167 F First order shear deformation theory, 341 Flügge s shell theory, 324, 336, 352 Flexural wave graphene on silicon substrate, 290 nanoplate with surface effects, 312 Timoshenko DNBS, 260 Fluid carrying SWCNT, 187, 189 dispersion relation, 188 effect of fluid density, 192 governing equations, 187 Timoshenko beam theory, 189 Fluid filled nanoshell governing equations, 336 Fourier transform, 20, 168, 273 DFT, 21, 24, 153, 154, 172, 181, 199, 206, 274, 301, 345 FFT, 152, 274, 301 G Graphene, 60, 121, 269, 270, 289, 317 elastic shell model, 280 governing equations, 271, 273 Graphene on silicon substrate, 280 equilibrium & force constants, 281 force constant, 282, 287 governing equations, 284, 286 Graphite, 8 Green s function, 65 Group speeds, 22 DNBS, 252, 254 DWCNT, 227 graphene, 274, 277 graphene on silicon substrate, 290, 292 higher order nanoshell, 347 magnetic field effects, 200 nanoplate with surface effects, 309, 312 nanorods, 125 nanoshaft, 150, 152 NLGSM nanobeams, 169 NLStGM nanorods, 134 rotating nanotubes, 185 second order NLStGM nanorods, 133 Timoshenko beam theory, 174

3 Index 357 H Hamilton s principle, 130, 286 Helmholtz decomposition, 151 Hexagonal lattice, 8 Higher order nanoshell governing equations, 342 Hybrid lattice, 282 bond force, 282 Hyperbolic partial differential equation, 50 I In-plane stiffness, 71, 74, 77, 83 Integro-partial differential equations, 62 Inter-atomic repulsion, 39 Interatomic potentials, 31 AIREBO, 40 Mie, 281 Morse, 85 REBO, 39 Interpolation functions, 45 Inviscid fluid, 187 K Kirchhoff plate theory, 308 L Lagrange s equation, 140 Lamb waves, 306, 342 Laplace Young equation, 307 Laplacian operator, 122 Lattice dynamics, 62, 65, 85, 145, 323 dispersion relation, 145 Lattice parameter, 67 Length scales, 47, 60, 121 Lorentz force, 196 Love s equation, 140 M Magnetic field, 195 Magnetic field effects, 199 Euler-Bernoulli beam theory, 197 governing equation, 196 SWCNT, 194 Maxwell s relations, 195 MEMS, 121, 249 Metropolis algorithms, 42 Microcontinuum, 59 Micropolar theory, 59 Modeling ab initio methods, 32, 290, 330 continuum modeling, 43, 60 3-D elastodynamic continuum model, 330 finite element method, 44, 47, 152, 157, 310 integral type non local elasticity, 52 kinetic Monte Carlo simulations, 42 mixed spatial-temporal derivatives, 51 molecular dynamics, 36, 280 Monte Carlo methods, 42 multiscale modeling, 46 non local continuum mechanics, 49 Molecular dynamics simulations, 82, 270, 310 Molecular mechanics model, 84 Multi-layered graphene sheets, 270 MWCNT, 215, 239 N Nanoplate, 284 surface effect, 305 Nanoplate with surface effects governing equations, 307 Nanoplate with temperature effects governing equations, 297 Nanorods dynamic stiffness, 156 Nanoshaft governing equation, 147 torsional wavenumber, 148 Nanoshell governing equations, 326 Nanostructures, 41, 81, 121, 201, 249 fullerenes, 3, 121 hybridization, 4 hybridization index, 5 linear hybridization, 7 modeling nanofiber, 121 nanotube, 7, 40, 121 nanowire, 7, 121, 203 one-dimensional, 121 tertrahedral hybridization, 6 tridiagonal hybridization, 6 two-dimensional, 121, 280 wave propagation, 13 zero-dimensional, 121 Nanotechnology, 2 NEMS, 13, 165, 239, 249, 306 NLSGM, 122 NLSGM MWCNT governing equations, 218 NLSGM nanobeam constitutive relation, 170 dispersion relation, 172

4 358 Index NLSGM nanobeam (cont.) Euler Bernoulli beam theory, 165, 169 flexural wave, 170 Timoshenko beam theory, 170 NLSGM nanorod, 137, 143 effect of lateral innertia, 139 governing equation, 122, 124 NLSGM SWCNT governing equations, 218 NLStGM, 122, 129 NLStGM fourth order nanorod, 144 governing equation, 144 NLStGM nanorod, 137 constitutive relation, 129 fourth order, 143 fourth order nanorods, 134 governing equation, 129 second order nanorods, 133 uniqueness and stability, 131 NLStGM second order nanorod, 144 governing equation, 144 NOMS, 239, 249 Non local theory constitutive relation, 60, 73, 86, 91, 111, 147, 205, 218, 272 cylinrical shell problem, 68 introduction, 60 Kernel function, 69 mixed nonlocality, 63 need, 59 non local modulus, 61, 65 nonlocal scale parameter, 71 one dimensional modulus, 66 properties of kernels, 65 scale parameter estimation, 84 spatial nonlocality, 63 temporal nonlocality, 63 three dimensional modulus, 67 two dimensional modulus, 66 types, 63 Nonlocal continuum mechanics, 85, 101 Nyquist frequency, 168, 274, 299 P Pasternak foundation model, 196, 201 Phase speed, 22 DNBS, 252, 254 DNRS, 243, 246 fluid filled nanoshell, 338 graphene, 277 graphene on silicon substrate, 290 higher order nanoshell, 347 magnetic field effects, 200 nanoplate with surface effects, 309, 312 nanoplate with temperature effects, 301 nanorods, 125 nanoshaft, 150, 152 NLGSM nanobeams, 169 NLStGM nanorods, 134, 136 rotating nanotube, 183 second order NLStGM nanorods, 133 Phonon, 323 Poisson s ratio effect, 139, 342 Polynomial eigenvalue problem, 29, 174, 189, 190, 200, 221, 226, 232, 233, 252, 258, 259, 289, 290, 327, 330, 337, 345 Potential functions, 37 Principle of virtual work, 308 Q Quantum coherence, 1 confinement, 1 dots, 121 mechanical potentials, 37 surface effects, 2 R Radial displacement, 326, 337 Residual surface tension, 307 Rotating nanomotor, 177 Rotating nanotubes dispersion relation, 181 Euler-Bernoulli theory, 179 governing equations, 178 S Scale effect, 41 Schrödinger equation, 34, 36 Shape functions, 154, 155, 157 Shear wave mode, 349 Single-layered graphene sheets, 270 Singular value decomposition, 28 Sound speed, 337 Spectral analysis, 21 Spectral finite element, 20, 152, 156, 161, 249, 323

5 Index 359 Spectral stiffness matrix, 152 Stick-spiral system, 87, 94 Strain-gradient elasticity, 50 Structural dynamics, 19 Surface effects in nanobeam, 201 governing equations, 205 Surface residual stress, 316 SWCNT, 126, 135, 1433, 220, 244, 254 governing equations, 72, 222 bond angle, 86, 106 bond length, 86, 110 chiral angle, 101 chiral vector, 101 in plane stiffness estimation, 75 material property estimation, 79 nanobeam, 170 properties, 115 shell model, 330 solution of equations, 74 T Thermal expansion coefficient, 300 Thermal stress, 297 Three-body interaction, 281 Throw-off element, 153 Timoshenko beam theory, 165, 203, 211, 216, 218, 222, 236, 239, 250, 254, 259, 266 fluid carrying SWCNT, 187 governing equations, 254 Translation vector, 102 Transverse inertia, 140 TWCNT, 221 governing equations, 231 U Universal force field, 39 V Van der Waals force, 102, , 239, 241, 250, 257, 270 W Wave propagation coupled systems, D nanostructures, 269 DNBS, 248 DNRS, 242 double nanobeam system, 251 fluid filled nanoshell, 336 graphene, 271 graphene on silicon substrate, 284 higher order nanoshell, 341 introduction, 19, 20 molecular dynamics for CNT, 38 molecular dynamics for graphene, 41 nanobeam with surface effects, 206 nanobeams, 165 nanoplate with surface effects, 305 nanoplate with temperature effects, 295 nanorod with lateral inertia effects, 139, 142 nanorods, 121 nanoshaft, 147 NLSGM nanorods, 122 NLStGM nanorods, 128 terminologies, 21 TWCNT, 231 Waveguide, 21 Wavenumber, 21, 86 characteristic equation, 24 DNBS, 252, 253 DNRS, 242, 244 DWCNT, 226 evanescent mode, 27 fluid carrying SWCNT, 189 fluid filled nanoshell, 337 fourth order NLStGM nanorod, 134 graphene, 274, 275 graphene on silicon substrate, 287, 289 higher order nanoshell, 345, 346, 348 magnetic field effects, 199 nanobeam with surface effects, 206 nanoplate with surface effects, 309, 310 nanoplate with temperature effects, 301, 303 nanorod with lateral inertia effects, 143 nanoshell, 327, 330 NLGSM nanobeams, 168 NLSGM nanorods, 124 NLStGM nanorod, 133, 134 propagating mode, 28 rotating nanotubes, 185 SWCNT Timoshenko beam, 222 Timoshenko DNBS, 258 Timoshenko beam theory, 173 Weak form of the governing equation, 154 Winkler foundation model, 196, 250, 257 Winkler foundation parameter, 200, 244 Z Zigzag CNT chirality, 100

Contents. Part I Basic Thermodynamics and Kinetics of Phase Transformations 1

Contents. Part I Basic Thermodynamics and Kinetics of Phase Transformations 1 Contents Preface List of tables Notation page iii xiii xiv Part I Basic Thermodynamics and Kinetics of Phase Transformations 1 1 Introduction 3 1.1 What Is a Phase Transition? 3 1.2 Atoms and Materials