Index

K -matrix, 46diatomic chain, 48EAM potential, 297fcc lattice, 73hexagonal lattice, 49monoatomic chain, 47

N -scale material, 230

acoustic frequency, 55adiabatic approximation, 43adiabatic system, 81advanced materials, 225amount of substance, 83angular momentum, 12associate cell, 44associate substructure, see associate cellatomic force constant, 16, 46austenite phase, 246autocorrelation function, 14Avogadro number, 83

barometric formula, 107Berendsen equation, 116binomial coefficient, 87biomimicry, 244blood plasma, 276Boltzmann constant, 83Boltzmann distribution, 106Boltzmann equation, 87bond length, 21bond stiffness, 16bond-order functions, 25boundary condition operator, 69boundary value problem, 62, 63Brenner potential, see potential

bridging scale, 131coarse scale, 132comparison to other multiple scale

methods, 173fine scale, 132future research directions, 220governing equations, 172impedance force (1D), 142impedance force (3D), 148, 172Lagrangian, 133numerical examples, 176

1D wave propagation, 1761D wave propagation

(non-nearest neighborpotential), 212

2D dynamic fracture, 1872D wave propagation, 1843D dynamic fracture, 195

numerical implementationcoupling force, 151impedance force, 150

random force (1D), 142staggered time integration, 170

Brownian motion, 13

canonical distribution, 97, 101ideal gas, 103periodic lattice, 109

canonical ensemble, 101carbon nanotube (CNT), 29, 202, 287

bending and buckling, 200Cauchy–Born rule, 127

bridging scale, 158quasi-continuum (QC) method,

127

Nano Mechanics and Materials: Theory, Multiscale Methods and Applications W. K. Liu, E. G. Karpov and H. S. Park 2006 John Wiley & Sons, Ltd. ISBN: 0-470-01851-8

316 INDEX

Cauchy–Born rule (continued )comparison to virtual atom cluster

method, 178central field, 13characteristic stiffness, 108CNT, see carbon nanotubecollision diameter, 22composite material, 246compressibility, 82, 84, 90concentration, 107constraints, 8

holonomic, 8nonholonomic, 8

coordinate, 8generalized, 8

Coulomb potential, see potentialcrystal lattice, 23, 37crystal plane, 42crystallographic axis, 42cubic lattice, 39cutoff function, 24cutoff radius, 23

damping kernel, see time historykernel

Debye layer, 284decohesion energy, 238deterministic method, 259dielectrophoresis (DEP), 283

driving force, 283, 284positive and negative DEP, 283

diffusion constant, 96Dirac delta function, 51discrete Fourier transform, 53

convolution theorem, 54numerical inversion, 54

dislocation energy, 22dispersion branch, 55dispersion law, 55drift of particle, 96ductile fracture simulator, 240

Einstein model, 110electric charge, 20electric displacement, 282electric permittivity, 282electric potential, 282

electro-manipulation, 283dielectrophoresis, 283electroosmosis, 284electrophoresis, 283

driving force, 284electron cloud, 21electron-volt, 22electroosmotic flow, see electroosmosiselementary cell, 38elementary charge, e0, 20elimination of degrees of freedom, 69embedded atom potential, see potentialembedding energy function, 26energy

potential energy, 9energy fluctuation, 101ensemble average, 94enthalpy, 85, 89entropy, 86, 96

ideal gas, 100microcanonical ensemble, 98probabilistic interpretation, 87

equation of motiondiatomic chain, 48free 1D lattice, 120free lattice, 54general lattice, 46Hamilton, 92hexagonal lattice, 49Lagrange, 10Newtonian, 13

equation of stateideal gas, 83, 101in thermodynamics, 82

equilibrium bond length, see bond lengthergodic hypothesis, 95ergodic system, 95

Fahraeus–Lindqvist effect, 280fcc lattice, 39field strength, 282finite difference equation, 46finite elements, 123first-principles calculation, 236fluctuation, 80fluctuation-dissipation theorem, 113fluid–structure interaction, 29 265

INDEX 317

force, 7conservative, 8external, 7generalized, 10internal, 7nonconservative, 8, 11

dissipative, 9gyroscopic, 9

Fourier transform, 50fracture toughness simulation, 239frame invariance principle, 10, 12free energy, 89, 91, 96

mechanical interpretation, 91free lattice, 54functional transform, 49

gas-structure interaction, 29Gauss theorem, 282generalized coordinate, see coordinategeneralized Langevin equation, 14generalized momentum, 92generalized strain, 230generalized stress, 230generalized velocity, see velocityGibbs canonical distribution, see

canonical distributionGibbs potential, 89, 91granular material, 232Green’s function, 59

dynamic, 60monoatomic chain lattice, 65

quasi-static, 67Green’s function method, 59

Hamilton function, see HamiltonianHamiltonian, 91, 92

ideal gas, 99periodic lattice, 108

handshake region, 68harmonic approximation, 46heat bath, 101, 102

Berendsen thermostat, 112Hoover-Nose method, 118numerical techniques, 111phonon method, 119

heat capacity, 84, 90numerical calculation, 90

periodic lattice, 110heat conductivity, 96Heaviside step function, 99hierarchical modeling, 234, 237, 240

IB method, see immersed boundarymethod

ideal gas, 99, 103IFEM, see immersed finite element

methodimmersed boundary method, 264immersed finite element method (IFEM),

263, 265impedance boundary conditions, 146impedance force, 146indentation, 31initial-value problem, 62integral of motion, 12integral transform, 50interaction between scales, 231interatomic potential, see potentialinternal energy, 89, 96, 98

periodic lattice, 110inverse transform, 50isothermal system, 80, 91, 101

kinematic constraints, 268kinetic energy, 10

distribution, 105kinetic Monte Carlo method, 259Kronecker delta, 53

Lagrange equation, 10conservative systems, 10nonconservative systems, 10

Lagrange function, see LagrangianLagrangian, 10

in Cartesian coordinates, 11in generalized coordinates, 11periodic lattice, 45

Langevin equation, 13, 14stochastic force, 14

Langevin particle, 14Laplace transform, 51

convolution theorem, 51numerical inversion, 52time-derivative rule, 51

318 INDEX

lattice mechanics, 37, 137relationship to finite elements, 139

lattice origin vector, 38lattice potential energy, 43lattice site index, 43lattice site vector, 38lattice standing waves, 56lattice stiffness matrix, 175

1D Lennard-Jones, 1763D fcc lattice/EAM gold, 297

lattice symmetry, 37lattice thermal vibration, 110Lennard–Jones potential, 22linear transform, 50Liouville theorem, 94local environment potential, see

potentiallong-range interaction, 21

martensite phase, 246material design, 223, 233, 234

properties vs. microstructure, 223self-healing composites, 254

Maxwell equations, 281Maxwell stress tensor, 284Maxwell–Boltzmann distribution, 105mean free path, 96mean square force, 96mechanical system, see systemmechanisms of self-healing, 247microcanonical distribution, 97microcanonical ensemble, 97micromorphic material, see multiscale

materialmolar heat capacity, 84molecular dynamics, 7, 28, 123

evaluation of thermodynamicparameters, 96

simulation, 7visualization, 28

momentum, 12distribution, 104

monocrystal, 37Morse potential, 22multiple scale methods, xi, 124

bridging domain, 129bridging scale, 131

coarse-grained molecular dynamics(CGMD), 126

coupled atomistic discretedislocation (CADD), 128

macroscopic atomistic ab initiodynamics (MAAD), 124

quasi-continuum (QC) method, 126multiresolution constitutive law, 231multiresolution framework, 227multiscale boundary conditions, 69multiscale equations of motion, 143multiscale internal power, 230multiscale material, 227multiscale simulation, 68

nano-electromechanical system, 290nanodeposition, 33nanoindentation, 31, 71nanomanipulation, 283nanomechanics, xiNernst’s postulate, see third law of

thermodynamicsNewton equation, 13

dissipative system, 13nondeterministic method, 259normal amplitude, 17, 56normal coordinate, 16, 17normal frequencies, 56normal frequency, 17, 55, 108normal mode, 17, 56, 108

mean energy, 109

one-particle distribution, 104optical frequency, 55

pair-wise potential, see potentialpartition function, 92, 102

ideal gas, 103periodic lattice, 110

periodic cell model, 240permittivity constant, ε0, 21phase point, 93phase space, 93phase space integral, 98phase trajectory, 93phase vector, 93phonon, 109

INDEX 319

phonon gas, 109point symmetry, 37Poisson equation, 282polarization vector, 56, 282polycrystal, 37polymer chain, see polymerspolymers, 37polytropic process, 85porous materials, 227postprocessing, 92potential, 7, 9

Brenner, 26Coulomb, 21cutoff radius, 23embedded atom, 26interatomic, 18Lennard–Jones, 22local environment, 25Morse, 22multibody, 18skin, 25

potential barrier, 19potential well, 19pressure, 96, 98pressure coefficient, 83, 89, 90probabilistic FEM, 260pseudoelasticity, 248

quantity of heat, 84quasi-static approximation, 66

in thermodynamics, 81quasi-static process, 81

RBC, see red blood cellred blood cell (RBC), 5, 266,

275–279regular lattice, 37relaxation, 81representative volume element,

228

second law of thermodynamics,86

self-healing material, 244serial coupling, 226shape-memory alloy, 246

constitutive law, 250

shape-memory composite, 248spring-mass oscillator, 98, 108state equation, see equation of statestate function, 80, 89state vector

in Hamiltonian mechanics, 93in Lagrangian mechanics, 11in thermodynamics, 82

statistical ensemble, 93, 97, 101statistical mechanics, 79

systems in thermodynamicequilibrium, 92

statistical weight of a microstate,87

Stirling’s formula, 88Stokes’ friction, see viscous frictionsuperelasticity, 248symmetric field, 12system

closed, see isolatedconservative, 12constrained, 8isolated, 8nonconstrained, 8, 10nonisolated, 8

system nonconstrained, 10

temperature, 81, 96ideal gas, 100microcanonical ensemble, 98

thermal expansion coefficient, 84, 89thermal vibration, 54thermodynamic equilibrium, 80thermodynamic parameter, 79, 96, 97,

101internal or external, 82state parameter, 80, 90system parameter, 80, 90

thermodynamic potential, see statefunction

thermodynamic system, 81isolated, 81, 97nonisolated, 81

thermodynamics, 79fundamental equation, 86

thermostat, see heat baththird law of thermodynamics, 89

320 INDEX

time average, 95time history, 9time history kernel, 14, 64, 142

1D chain/Lennard-Jones, 1752D hexagonal

lattice/Lennard-Jones, 1543D FCC/Lennard-Jones, 156comparison to previous methods,

152extension to non-nearest neighbors,

203monoatomic chain lattice, 66programming flowchart, 157truncation for numerical simulation,

179time history damping kernel, see time

history kerneltoughening effect, 243trajectory, 7, 11transformation equation, 8

translation vector, 38translational symmetry, 37

uniform distribution, 109unit cell, 43

van der Waals equation, 83velocity

distribution, 105generalized, 8mean values, 105

virtual atom cluster (VAC) method, 159comparison to Cauchy–Born rule,

178viscosity, 96viscous friction, 13volume expansion coefficient, 82

wall function, 19

zeta potential, 284