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Index

C-irreducible representation, 340, 349 Ce, 197 Coo, 286, 288 Cooh, 286 Coov, 285, 286, 288, 293, 296, 340 Coov X Coov, 314, 320, 321, 324, 326,

331, 340, 344 C;, 286 Cn, 197, 206, 286, 288 Cnh, 286, 288 Cnv, 286, 288, 296 Din;;,, 317 D3, 194, 245, 247 Ds, 269 De, 98, 246 D2h, 30, 296, 297, 359 Doo, 286, 288 Doooo, 353 Dooh, 285-288, 293, 298, 299 Dn, 189, 198, 234, 251, 262 Dn (Schoenflies notation), 286, 288,

294 Dnd, 286, 288, 289, 299, 305, 306 Dnh, 286, 288, 299, 303, 304 EC~~kll 319 ECnnkll 317, 318, 320, 321, 324, 329,

355

G (group) equivariant system, 164 invariance, 160, 162, 167, 168 symmetric, 164, 165

OB~~, 319, 352 OBn;;,, 317, 321, 324 0(2) X 0(2), 28, 314, 325, 326, 328,

344 0(2) X S0(2), 344 0(2) x z2, 28, 30, 313 R-irreducible representation, 340, 349 Sn, 286, 288 Z2 x z2, 326

absolutely irreducible, 160, 186 representation, 173, 174

aloof path, 218 asymptote, 213 asymptotic

analysis, 42, 44, 64 behavior, 72 distribution, 110 generalized law, 129, 133, 134 law, 139 potential, 209 stability, 40

atomic crystal lattice, 383

406 Index

autocorrelation function, 116 average, 104 axial strain, 144 axisymmetry, 286

Benard convection, 311 basis, 46, 158

change, 165, 172 vectors, 24 7

beam on a nonlinear foundation, 115, 387, 388

bifurcation, 38 direct, 321, 324, 369, 370 equation, 41, 45, 47, 69, 126, 343,

350, 353 explosive, 389 mode, 64, 388

harmonic, 387 parameter, 1, 2, 37 problem, 1 secondary, 32, 321, 361, 369, 370

bifurcation behavior imperfect, 43, 64, 212, 382 local, 42, 61, 204

bifurcation diagram, 17, 224 experimentally observed, 18 imperfect, 61, 132 mathematical, 18, 125, 224

bifurcation point asymmetric, 43, 52 cascade, 391 clustered, 379 double, 12, 99, 179, 195, 197, 236,

244, 259, 269 unstable, 254

hilltop, 383-385, 391 n-fold, 390 pitchfork, 4, 57, 62, 68, 71, 88, 103,

121, 189, 342 stable, 59, 75 unstable, 10, 59, 71, 76, 84, 93,

96, 117, 126, 13~ 214, 383 quadruple, 344 search, 144, 294 simple, 74, 224, 350 symmetric, 52, 63 transcritical, 43, 56, 68, 70, 75, 88,

103, 120

block-diagonal form, 160, 162, 172, 173, 252

block-diagonalization, 162, 173-175 branch, 131, 190, 204, 391

Cam clay model, 324 canonical coordinate, 17 cantilever, 2, 8 central limit theorem, 107 classification

simple critical point, 52 commutativity condition, 163 complementary invariant subspace,

160 complementary subspace, 168 complex coordinates, 198 concrete

cylindrical specimen, 262, 263, 380 conjugate, 176 correlation coefficient, 271 Couette-Taylor problem, 311 critical

eigenvector, 7, 40, 43, 46, 64, 180, 197, 246

left, 38, 84 right, 38, 84

initial imperfection, 10, 11, 14, 82, 83, 86-88, 93, 96, 98, 234, 235, 246

load, 7, 383, 388 normalized, 104 random variation, 117, 259, 272 variation, 251

load increment normalized, 254, 257

point, 1, 3, 38-40, 53, 388 double, quadruple (see

bifurcation point) simple, 46, 105, 236

criticality condition, 47, 69, 170, 199 cumulative distribution function, 105,

109, 110, 258 cylindrical domain, 26 cylindrical specimen

concrete, 262, 263, 380 sand, 16, 23, 26, 143, 145, 262, 272,

274, 287, 289, 308, 309, 381, 382, 385

soil, 26, 27, 308, 309, 314, 322-324

degenerate, 54 deviatoric stress, 144, 274 deviatoric stress versus axial strain

curve, 145, 294, 297, 322, 382 deviatoric stress versus volumetric

strain curve, 145 diagonal block, 173 diagonal shear band, 369 diamond pattern, 28, 315, 347 diffuse necking, 364 direct product, 285, 313, 326, 348 direct sum, 179

decomposition, 161, 167 irreducible representation, 161 isotypic components, 173 representation spaces, 161

discriminant, 72 displacement

experimentally observed, 19, 133, 224

nonsymmetric, 139 symmetric, 132, 138

distribution asymptotic, 110 conditional, 103 critical load, 102, 108, 253 cumulative, 105, 109, 110, 254, 258 double exponential, 111 exponential, 253, 254 Gumbel, 111 limit, 110, 111 minimum value, 108, 110, 258 multivariate normal, 16, 102, 112,

260, 262 normal, J.01~103, 116, 119, 250,

252, 271 potential, 5 standard normal, 104, 105, 252, 253 uniform, 107, 119, 265 Weibull, 102, 252 x2 , 254

double Fourier series, 30, 327, 333 dynamical system, 39

echelon mode, 28, 314, 317, 320, 323 elimination of passive coordinates, 45 equilibrium equation, 6, 41, 387 equilibrium path, 7 equivalence, 4 7

Index 407

equivalent, 160 representation, 179

equivariance, 157, 164, 354 bifurcation equation, 169, 343, 345 Jacobian matrix, 172

expected value, 105, 254 explosive bifurcation, 389 exponential distribution, 254 extreme order statistic, 109 extreme value, 109

five-bar truss tent, 244, 249 fold bifurcation, 52 Fourier series, 116 Fnkhet derivative, 40 fracture mechanics, 380 fundamental path, 3, 57, 213, 387

Galerkin-type solution, 116 gamma function, 105 Gaussian, 116 geometrical symmetry, 18, 279, 285,

326, 359 geometrical transformation, 20 governing equation, 1, 37, 40, 45, 87 gradient system, 39 granite, 381 group, 20, 156, 158

compact, 163, 242 cyclic, 163, 183, 187, 193 dihedral, 21, 159, 179, 183, 186, 288 equivariance, 163, 168, 193 finite, 158, 163 inversion, 286 representation theory, 172 symmetry, 164, 166, 235, 252, 341,

359 group-theoretic, 166, 202

bifurcation theory, 20, 26, 163 critical point, 166, 252 double bifurcation poin , 225, 251,

342, 350, 353 Gumbel distribution, 111

half-rotation, 315, 327, 350 symmetry, 294, 317, -~, 324, 336

hierarchy of subgrou, 20, 24, 30, 177, 192, 287 C\

histogram, 271

408 Index

homogeneous, 161 hyperbolic umbilic catastrophe, 166

identity element, 158 identity transformation, 157 image simulation, 331, 340

kaolin, 30, 331, 334, 336, 338 steel, 331, 337, 339

imperfect system, 1 imperfection, 4, 269, 289

category, 91 data bank, 250 generalized sensitivity law, 126 influence factor, 89 magnitude, 7, 39 parameter vector, 1, 37, 42, 84,

102, 178, 243, 259 pattern vector, 11, 38, 78, 201, 235 random, 16, 101, 102, 108, 113,

245, 251, 258, 273 sensitivity, 381

piecewise linear, 384 sensitivity law, 10, 68, 72, 75, 252,

385 sensitivity matrix, 11, 51, 86, 87,

112, 180 implicit function theorem, 46, 168,

175 index, 185, 219, 236, 252 inequivalent, 160, 162

irreducible representation, 161, 173, 186, 299, 341

infinite-periodic-cylinder approximation, 314

infinite-periodic-domain approximation, 326

inheritance group symmetry, 167 reciprocity, 167, 171

initial imperfection (see imperfection) invariance, 157 invariant subspace, 160, 162 inverse, 158 inversion, 359 irreducible, 160, 163

component, 161 matrix representation, 162, 344 representation, 161, 186, 187, 340,

348, 353

four-dimensional, 344 one-dimensional, 179, 186, 224,

299, 340, 349 two-dimensional, 179, 186, 198,

225, 299, 333, 342, 349 isola center, 56 isomorphic, 288, 375 isotypic component, 161, 173 isotypic decomposition, 161, 173

Jacobian matrix, 37, 228, 388

kaolin, 26, 29, 30, 331, 334 cross-checker pattern, 26 echelon mode, 335 image simulation, 30, 331, 334, 336,

338 kernel, 38, 166, 198

space, 38, 167, 180 symmetry, 341, 342, 351

Koiter law, 126

leading term, 42, 68 Lennard-Jones constitutive law, 389 Liapunov-Schmidt reduction, 45, 85,

166, 167, 176 Liapunov-Schmidt-Koiter reduction,

45 limit distribution, 111 limit point, 4, 52, 54, 69, 76, 93, 118 linear

map, 160 operator, 38 representation, 158 stability, 50 transformation, 158

linearly stable, 39, 40 linearly unstable, 39, 40 load versus elongation curve, 372 loading parameter, 3, 10

Maschke theorem, 160, 161 material softening, 145 matrix representation, 158, 159 maximum

load, 364 point, 59, 79, 218 stress, 274

mean, 103

minimum critical load, 108, 258 point, 59, 79, 218

mode interference, 334 mode switching, 23, 289, 297, 380,

385, 387 Monte Carlo method, 117 multiplicity, 38, 161, 272, 274, 353 multivariate normal distribution, 260,

262

Newton polygon, 60, 73 nondegeneracy condition, 54 nondegenerate, 204 nonreciprocal system, 39, 157 nonshallow truss arch, 76, 83, 93, 96 normal distribution, 103, 252

oblique stripe pattern, 28, 314, 317, 347

odd function, 59, 157 one-half power law, 70, 76, 246 orbit, 164, 176, 197 order, 185 ordinary point, 38, 40, 54 orthogonal complement, 160, 162 orthogonal matrix, 173, 17 4 orthonormal basis, 168, 171, 173 orthonormal eigenvectors, 47

parametric, 166, 378 path, 3

aloof, 218 bifurcated, 3, 57, 96, 112, 189, 193,

299, 329, 379 fundamental, 3, 57, 96, 112, 193,

247, 324, 372, 387 post-bifurcation, 3 primary, 3 secondary, 3 stable, 5, 40 trivial, 3 unstable, 5, 40

pattern, 26, 322 perfect system, 1, 39 periodic boundary, 26, 314, 325, 328,

333 periodic symmetry, 279, 361 permutation, 17 4

representation, 159 pitchfork, 52

Index 409

Poincare's exchange of stability, 57 point group, 288, 303 point of accumulation, 379 polar coordinates, 203, 213 potential, 39, 157 power series, 51, 200 principle

minimum potential energy, 4 stationary potential energy, 3, 39

probability density function, 16, 103, 108, 117, 254

semiempirical evaluation, 108 theoretical evaluation, 108

projection, 167, 168 propped cantilever, 41, 62, 75, 112,

125

range space, 46 rank deficiency, 38, 166 reciprocal system, 39, 47, 167, 192,

193, 202, 208, 287, 352 reciprocity, 39 recovering perfect system, 134 rectangular domain, 30, 325 rectangular parallelepiped domain,

359, 375 rectangular parallelepiped steel

specimen, 361 rectangular plate, 141 rectangular steel specimen, 384 recursive bifurcation, 28, 30, 192, 268,

289, 296, 298, 320, 324, 334, 361, 369, 372

reflection, 179, 243, 285, 313, 326, 359 symmetry, 206, 269

regular-hexagonal truss dome, 97, 137, 139, 140, 246

regular-hexagonal truss tent, 12 regular-pentagonal truss dome, 269 regular-polygonal truss dome, 259 regular-triangular truss dome, 20, 193 reliability function, 105, 254 representation, 158, 160

four-dimensional, 344 matrix, 157, 160, 340 one-dimensional, 159 space, 158, 160, 173

410 Index

theory, 163 two-dimensional, 163 unit, 159, 189, 340

resonance of symmetry, 242, 246, 247 restriction, 160, 161 resultant, 72 ribbon, 311 rock, 381

echelon mode, 308, 310 rotation, 178, 183, 285, 313

saddle node, 52 sample

mean, 108, 262 size, 113 standard deviation, 113 variance, 108, 262

sand cylindrical specimen, 16, 23, 26,

143, 145, 262, 272, 274, 287, 289, 382, 385

diamond pattern, 308, 309 mode switching, 297 oblique stripe pattern, 308, 309 recursive bifurcation, 289 size effect, 381

hollow cylindrical specimen, 26, 27 cross-checker pattern, 26, 27

Schoenflies notation, 184, 362 Schur complement, 170 Schur's lemma, 162 Schwedler dome, 197 shallow truss arch, 76, 93 shape effect, 359, 362, 364 shear band, 28, 293, 364 singular point, 38 size effect, 378, 380, 381, 383, 385,

388, 389 negative, 380 positive, 381

soil cylindrical specimen, 26, 27, 314,

322 echelon mode, 26, 27, 308, 309,

323 oblique stripe pattern, 324 pattern, 322, 323 shear-band formation, 324

spatial symmetry, 24, 286, 317

spiral, 311 stability, 4, 39, 57, 210 stable, 39, 40

bifurcation point, 210 branch, 212 linearly, 40 path, 5, 40 point, 5, 40

standard normal distribution, 104, 105, 252

state variable vector, 1, 10 stationary point, 54 statistical property, 116, 255 steel, 30, 331, 337

cross-checker pattern, 308, 310 image simulation, 331, 337, 339 recursive bifurcation, 369 shape effect, 364 shear band, 364 specimen, 30, 31, 362

stress ratio versus axial strain curve, 381

stress versus strain curve, 16, 19, 24, 379

subgroup, 20, 185, 286, 342 abelian, 348

subrepresentation, 160, 167, 180 symmetric displacement, 226 symmetric system, 20 symmetry, 20, 156, 165, 192, 242

bilateral, 59, 156, 317, 322 breaking, 22, 314 condition, 157 kernel, 341, 342, 351 solution, 175 upside-down, 142, 262, 288, 317,

326 symmetry group (see group)

tangent stiffness matrix, 37 tensor product of representations,

340, 341, 349 three-bar truss tent, 178, 243, 245 torus, 314 total potential energy, 3, 6, 39, 325,

387 transcritical, 52 transformation, 165, 17 4 transformation matrix, 17 4

translation, 30, 314 translational symmetry, 26, 28 triaxial compression test, 23, 143,

287, 289, 314, 322, 385 trivial solution, 78, 157, 203 truss

arch, 99 nonshallow, 66, 76, 81, 83, 93,

96, 149 shallow, 76, 93

dome regular-hexagonal, 97, 100, 137,

139, 140, 231, 246 regular-triangular, 20, 181, 193,

249 rotation-symmetric, 193

member, 41, 99 tent

five-bar, 244, 249 n-bar, 231, 243 regular-hexagonal, 12 three-bar, 178, 243, 245

turning point, 52 two rigid bars, 5, 12 two-thirds power law, 10, 71, 80, 84,

86, 126, 245, 383

unfolding parameter, 72 uniaxial compression test, 331 uniform material, 321 unitary, 159, 160

matrix representation, 164, 235, 251

representation, 160, 17 4, 179, 186, 251

universal unfolding, 61, 71 unlinked cellular structural form, 389 unstable, 39, 40

bifurcation point, 210 branch, 212 linearly, 40 path, 5, 40 point, 5, 40

variance, 103, 105, 254 variance-covariance matrix, 103, 117,

252

Weibull distribution, 255

Index 411

weight matrix, 11, 85, 235, 243 width-thickness ratio, 362

x2 distribution, 254

Applied Mathematical Sciences (continued from page ii)

60. GhiVChildress: Topics in Geophysical Dynamics:

Atmospheric Dynamics, Dynamo Theory and

Climate Dynamics.

61. Sa/linger/Weaver: Lie Groups and Algebras with

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62. LaSalle: The Stability and Control of Discrete

Processes. 63. Grasman: Asymptotic Methods of Relaxation

Oscillations and Applications. 64. Hsu: Cell-to-Cell Mapping: A Method of Global

Analysis for Nonlinear Systems. 65. Rand/ Armbruster: Perturbation Methods,

Bifurcation Theory and Computer Algebra.

66. Hlawicek/Haslinger/NecasVLovfsek: Solution of

Variational Inequalities in Mechanics.

67. Cercignani: The Boltzmann Equation and Its

Applications. 68. Temam: Infinite-Dimensional Dynamical Systems

in Mechanics and Physics, 2nd ed.

69. Golubitsky/Stewart/Schaejfer: Singularities and

Groups in Bifurcation Theory, Vol. II. 70. Constantin/Foia;/Nicolaenko/femam: Integral

Manifolds and Inertial Manifolds for Dissipative

Partial Differential Equations. 71. Catlin: Estimation, Control, and the Discrete

Kalman Filter. 72. Lochak/Meunier: Multiphase Averaging for

Classical Systems.

73. Wiggins: Global Bifurcations and Chaos.

74. Mawhin!Willem: Critical Point Theory and

Hamiltonian Systems. 75. Abraham/Marsden!Ratiu: Manifolds, Tensor

Analysis, and Applications, 2nd ed. 76. Lagerstrom: Matched Asymptotic Expansions:

Ideas and Techniques. 77. Aldous: Probability Approximations via the

Poisson Clumping Heuristic. 78. Dacorogna: Direct Methods in the Calculus of

Variations. 79. Hernandez-Lerma: Adaptive Markov Processes.

80. Lawden: Elliptic Functions and Applications.

81. Bluman/Kumei: Symmetries and Differential

Equations. 82. Kress: Linear Integral Equations, 2nd ed.

83. Beberne>IEberly: Mathematical Problems from

Combustion Theory. 84. Joseph: Fluid Dynamics of Viscoelastic Fluids.

85. Yang: Wave Packets and Their Bifurcations in Geophysical Fluid Dynamics.

86. Dendrinos/Sonis: Chaos and Socio-Spatial Dynamics.

87. Weder: Spectral and Scattering Theory for Wave Propagation in Perturbed Stratified Media.

88. Bogaevski/Povzner: Algebraic Methods in Nonlinear Perturbation Theory.

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90. Meyer/Hall: Introduction to Hamiltonian Dynamical Systems and theN-body Problem

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Nonlinear Convection. 92. Naber: The Geometry of Minkowski Spacetime. 93. Colton/Kress: Inverse Acoustic and

Electromagnetic Scattering Theory, 2nd ed.

94. Hoppensteadt: Analysis and Simulation of Chaotic

Systems, 2nd ed. 95. Hackbusch: Iterative Solution of Large Sparse

Systems of Equations. 96. Marchioro/Pulvirenti: Mathematical Theory of

Incompressible Nonviscous Fluids.

97. Lasota/Mackey: Chaos, Fractals, and Noise:

Stochastic Aspects of Dynamics, 2nd ed.

98. de Boor/Hollig/Riemenschneider: Box Splines.

99. Hale/Lunel: Introduction to Functional Differential

Equations. 100. Sirovich (ed): Trends and Perspectives in

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2nd ed. 102. Chossat/Iooss: The Couette-Taylor Problem

I 03. Chorin: Vorticity and Turbulence. 104. Farkas: Periodic Motions.

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108. Zeidler: Applied Functional Analysis: Applications to Mathematical Physics.

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Delay Equations: Functional-, Complex-, and

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126. Hoppensteadtllzhikevich: Weakly Connected Neural Networks.

127. lsakov: Inverse Problems for Partial Differential Equations.

128. LVWiggins: Invariant Manifolds and Fibrations for Perturbed Nonlinear Schrildinger Equations.

129. Miiller: Analysis of Spherical Symmetries in Euclidean !ipaces.

130. Feintuch: Robust Control Theory in Hilbert Space.

131. Ericksen: Introduction to the Thermodynamics of Solids, Revised ed.

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Flnids. 136. Cioranescu/Saint Jean Paulin: Homogenization of

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147. Aubert!Komprobst: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations.

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154. Bluman/Anco: Symmetry and Integration Methods for Differential Equations.