Index

acoustic mode, 284 Bloch wave mode, 281 acoustic cloaking, 228, 244 Bloch wave vector, 269, 272 inertial, 256 Bloch-Floquet theorem, 205, 269, 285, acoustic intensity level, 23 288 acoustic metafluid, 220, 255, 256 body force, 13, 196 acoustic metamaterial, 191 bound electron, 130 acoustic power, 23 Brewster’s angle, 66, 76 acoustic tensor, 52, 347 Brillouin zone, 284 effective, 274 bubbly liquid, 275 acoustics, 16, 63, 64, 107, 267 bulk modulus, 17, 267, 275 adiabatic, 17 complex, 181 adiabatic index, 17 bulk viscosity, 18 adjoint, 199 effective, 278 Airy equation, 312 Airy function, 312 capacitance, 141 Ament’s relation, 153 Cartesian coordinates Ampere’s law, 26 acoustics, 19 angular momentum, 4, 177 elastodynamics, 7 anisotropic mass, 167 electrodynamics, 31 anisotropic mass density, 40, 167, 326 Cauchy principal value, 171 anisotropy, 324, 325 Cauchy stress, 4, 5, 7, 8, 11, 14, 17, anomalous cloaking, 243 18, 56, 62, 69, 114, 182, 186, antiplane shear, 13, 37, 38, 54, 190 189, 196, 203, 211, 252, 271, artificial magnetism, 137 272, 277, 301, 325, 335, 336, attenuation length, 282 347 auxetic material, 214 unsymmetric, 213 axial vector, 175 Cauchy’s integral formula, 169 Cauchy-Goursat theorem, 171 Backus upscaling, 334 Cauchy-Riemann equations, 169 band gap, 190, 285 causality, 5, 17, 20, 34, 41, 140, 169, Bessel function, 97, 122, 136, 141, 312 184, 241 spherical, 107 chiral medium, 203 Bessel’s equation, 97, 117 Christoffel symbols, 9 bi-anisotropic medium, 202 cloaking, 227 Biot-Savart law, 26 acoustic, 244 Bloch condition, 269, 291 elastodynamics, 251 Bloch wave, 208, 269, 272, 278 electromagnetism, 235

357 358 An Introduction to Metamaterials and Waves in Composites cloaking transformation, 238 easy strain, 217 coated sphere, 275 effective bulk modulus, 276 coherent potential approximation, 119, effective density, 333 192 effective dynamic mass, 159 conductivity effective field, 119 electrical, 231 effective mass, 163 conservation effective permeability, 190, 280 angular momentum, 4 effective permittivity, 280, 339 linear momentum, 4, 18 effective phase velocity, 333 mass, 16 effective property, 334 constitutive relation, 5, 196 eigen decomposition, 215, 328 convolution, 10, 197 eigenvalue, 166, 206, 215, 320 convolution theorem, 169 eigenvalue problem, 267, 272, 279, 281, Coulomb’s law, 25 319 CPA method, 119 eigenvector, 166, 320 curl, 31 Eikonal equation, 315 curvilinear coordinates, 32, 238 elastic wave equation, 40 acoustics, 19 elastodynamic power, 13 elastodynamics, 8 elastodynamic wave, 270 electrodynamics, 31 elastodynamics, 51, 112, 290 cylindrical coordinates, 96, 122, 124, 134, governing equations, 3 141 periodic, 270 quasistatic limit, 270 Willis equations, 194 damping constant, 162 electric current, 26 damping factor, 182 electric dipole, 105 dashpot, 160 electric displacement, 27 Debye potential, 112, 116 electric field, 25 deformation gradient, 230 electric permittivity, 27 deviatoric stress, 18 electric polarization, 27 dielectric constant, 27, 131 electric susceptibility, 27 diffraction, 145 electrical conductivity, 28, 41 dipole moment, 27, 130, 131 electrodynamic metamaterial, 129 dipole source, 99 electrodynamics, 24, 71, 72, 87, 104, Dirac delta function, 96, 98 115 Dirichlet to Neumann map, 233 periodic, 278 dispersion, 51, 110, 290 quasistatic limit, 278 dispersion relation, 51, 133, 206, 272, electromagnetic impedance 274, 281, 284, 289 characteristic, 79 displacement, 53, 197 electromotive force, 27 displacement gradient, 6 electrostatics, 25 dissipation, 13, 23, 41, 42 ensemble averaging, 194 divergence theorem, 14, 42, 248 ergodicity, 194 Drude model, 133 Euler-Lagrange equations, 163–165, 205 dynamic mass, 161 Eulerian description, 16 Index 359 evanescent wave, 89, 99, 133, 147, 329 characteristic, 67 extremal material, 214 effective, 331 elastic, 58, 165 fading memory, 185, 188 electrical, 231 far-field pattern, 123 electromagnetic, 79, 241 Faraday’s law, 27 normal, 79 folding transformation, 235 impedance matching, 68, 80, 148, 241 Foldy-Lax method, 119 impedance tube, 84, 85, 91, 163 Fourier transform, 10, 11, 98, 101 incident wave, 64, 72 free-body diagram, 158 independent scattering, 118 frequency, 64 inductance, 141 Fresnel equations, 75, 89 inertia function frequency dependent, 173 completely monotonic, 188 inertial cloak, 244 infinite mass, 249 gauge transformation, 30 infinitesimal strain, 6 Gauss’s law, 25 inhomogeneous wave, 103 Graf’s addition theorem, 124 integral matrix, 321 Green’s function, 96, 97, 199 intensity, 23 Greenleaf-Leonhardt-Pendry, 227 irrotational flow, 18 group velocity, 51, 147, 281, 284, 285 isotropic material, 6 gyrocontinua, 173 Jacobian, 229, 230 Hamiltonian, 163 Jacobian matrix, 103 Hankel function, 97, 99 spherical, 108 Kelvin moduli, 215 harmonic plane wave, 50 Korringa-Kohn-Rostoker method, 119 Hashin’s relations, 193, 276 Kramers-Kronig relation, 169 elastostatics, 275, 296 electrostatics, 296, 349 Lagrangian, 162, 165, 204 Helmholtz equation, 49 Lagrangian description, 16 Helmholtz potential Lame´ moduli, 6 body force, 6 Lame’s´ theorem, 6 Helmholtz resonator, 179, 192 Laplace-Beltrami operator, 9, 263 Hertz vector potentials, 30, 116 lattice spacing, 271 hierarchical laminates, 214 layered media, 299 homogeneous material, 6 periodic, 326 homogenization, 136, 139, 143, 169, 190, Legendre polynomial, 107, 111 266, 274, 299, 308, 334 associated, 117 Hooke’s law, 158 Legendre’s equation, 107 Leibniz rule, 42 ideal gas, 17 linear elastic, 196 image resolution, 145 micromorphic, 173 impedance, 141, 167 linear momentum, 4, 156, 176 acoustic, 67, 83, 164, 330 linear operator, 199, 267, 272 360 An Introduction to Metamaterials and Waves in Composites linear superposition, 25 multiple scales analysis, 273 linear viscoelasticity, 184 multiple scattering, 119, 193 locally resonant sonic material, see LRSM multipole, 120 Lorenz condition, 30 Love wave, 300 Nanson’s formula, 248 LRSM, 86, 154, 290 Navier-Stokes equations, 16 lumped-mass model, see spring-mass sys- negative density, 62 tem negative index, 145 negative mass, 40, 159 magnetic field, 28 negative modulus, 155, 179 magnetic flux density, 25 negative refractive index, 62, 67, 70, 80, magnetic induction, 25 129 magnetic moment density, 28 mirror, 80 magnetic monopole, 25 negative shear modulus, 62 magnetic permeability, 26, 28 Newton’s second law, 158 effective, 139 Newtonian fluid, 18 magnetization vector, 28 nonlocal, 5 magnetostatics, 25 normal strain, 220 magnification, 243 null space, 268 mass frequency dependent, 154, 159, 161, Ohm’s law, 28 169 Onsager reciprocity relations, 41 mass density, 197, 267, 271 optical mode, 285 anisotropic, 155 dynamic, 154 p-polarization, 38 effective, 153 P-wave, 6, 9, 55, 56, 60 negative, 154 pentamode material, 214, 217, 254–256 static, 153 perfect lens, 143, 241 transversely isotropic, 334 periodic media matricant, 326 one-dimensional, 326 Maxwell model, 160 periodic medium, 271 generalized, 184, 189 permittivity, 25, 130, 131, 161 Maxwell’s equations, 24, 28, 31–34, 37, negative, 137 39, 71, 72, 104, 116, 134, 227, perturbation, 272 235, 238, 254, 270, 278, 280, phase, 51 324 phase speed, 72 Mie solution, 115 phase velocity, 36, 51, 73, 147, 281 Milton’s relation, 346 phononic crystals, 265 molecule, 153 photonic crystals, 265 momentum, 196, 210 plane wave, 50 balance of, 196 acoustics, 63 linear, 209 elastodynamics, 51 monochromatic plane wave, 50 electrodynamics, 71 monodromy matrix, 326, 328 plane wave expansion, 95 multilayered cloak, 247 plasma frequency, 133 Index 361

magnetic, 143 Runge-Kutta method, 310 plasmon, 133 RVE, 136, 137, 139, 156, 203, 205, 207, Poisson’s ratio 208, 210–212, 266, 270, 272, negative, 214 274, 276, 280, 282, 283, 286, polar decomposition theorem, 249 287, 291 polarization, 54, 73 polarization field, 198, 341 s-polarized wave, 37 porous media, 153 S-wave, 6 Poynting vector, 147 scalar potential, 41 principle of minimum dissipation, 42 acoustic, 19 projection operator, 250, 341, 346 elastodynamics, 6 propagating wave, 99 electrodynamics, 29 propagator matrix, 302, 318, 320, 326 scattering, 106, 265 pseudo-pressure, 220, 256 acoustic, 107, 110, 122 pull-back, 236, 252 cross-section, 110 elastodynamic, 112 quasistatic limit, 273 electrodynamic, 115 scattering cross-section, 115 radial map, 240 Schoenberg-Sen model, 192, 326 rank-1 laminate, 342 self-consistent method, 119 ray parameter, 58 SH-wave, 13, 54, 61, 62, 189, 190, 247, ray theory, 315 292 Rayleigh multipole method, 120 shear modulus, 6, 190, 275, 300 reciprocal lattice vector, 269 complex, 277 recursion relation, 306 negative, 191 reflected wave, 65, 72 shear strain, 220 reflection coefficient, 57, 65, 75, 84, 89, shear viscosity, 18, 160, 277 147, 306, 322, 331 single scattering, 106, 118, 241 apparent, 89 slab, 82 generalized, 308 slowness surface, 220 reflectivity, 87 slowness vector, 52, 53, 55, 56, 332, refractive index, 62, 70, 73 333 negative, 191 Snell’s law, 58, 66, 73 relaxation spectrum, 185, 188 spectral decomposition, 214 relaxation time, 185 spherical coordinates, 96, 100, 103, 108, representative volume element, see RVE 109, 113, 116, 118 residue, 101 spherical harmonics, 117 resistance, 141 split-ring resonator, 137, 138, 140, 142, resonance, 27, 87, 93, 127, 129, 131, 143, 189, 190 133, 148, 156, 159, 162, 169, spring constant 183, 192, 214 complex, 160 anomalous, 242 effective, 185 frequency, 159, 163 real, 160 Riccati equation, 310 spring-mass system, 158, 159, 162, 165, rotating ring, 173 167, 179, 189, 190, 192, 203, 362 An Introduction to Metamaterials and Waves in Composites

204, 265, 283, 289 transmission coefficient, 65, 75, 84, 89, SRR, see split-ring resonator 147, 323, 331 state equation, 299, 302, 319, 325, 326 generalized, 308 state vector, 302, 324, 325 transmission line model, 192 stiffness tensor, 5, 6, 196, 197, 270, 271, transmission loss, 164 291 transmissivity, 87 strain, see strain tensor transmitted wave, 65, 72 strain energy, 219, 254, 255, 257, 258 transverse electric field, see TE wave strain tensor, 5–7, 11–14, 182, 186, 189, transverse magnetic field, see TM wave 197, 198, 202, 216–218, 271– 273, 277, 301, 335, 336, 346 unit cell, 136, 137, 139, 156, 203, 205, stress, see Cauchy stress 207, 208, 210–212, 266, 270, Stroh formalism, 325 272, 274, 276, 280, 282, 283, Stroh matrix, 325 286, 287, 291 superpotentials, 30 supporting stress, 217 variation, 42 surface traction, 4, 13 vector potential SV-wave, 54, 56–59, 61, 68, 69, 112, elastodynamics, 6 127 electrodynamics, 29 Sylvester’s formula, 321 Hertz, 30 Voigt notation, 215 T-matrix, 123 volume average, 194, 335, 337, 339 Tartar formula, 349 volumetric strain, 6 Tartar-Murat-Lurie-Cherkaev formula, see volumetric stress, 18 TMLC formula wave equation TE wave, 37, 39, 77, 88, 96, 147, 189, acoustic, 19 191, 227, 244, 303–309, 319, elastodynamics, 12 324 pentamode, 220 TM wave, 37–39, 74, 96, 191, 304, 308 wave number, 50 TMLC formula, 343, 346 wave vector, 50, 64, 72, 208 traction vector, 4, 209 wavenumber vector, see wave vector transfer matrix, 318, 326 weak form, 291 transformation optics, 235 Wentzel-Kramers-Brillouin-Jeffreys method, transformation rule see WKBJ method acoustics, 244, 246 Weyl identity, 103 Chen-Chan, 246 Weyl integral, 99 conductivity, 232 Willis equations, 194, 197, 254, 326 Cummer-Schurig, 244 Willis materials, 194 current, 233 WKBJ method, 313, 316 elastodynamics, 254 Wronskian, 97 electromagnetism, 237 tensor, 229 Young’s modulus, 182 vector, 228 frequency dependent, 185 transition matrix, 318, 320 translation operator, 268