Antman S.S. — Nonlinear Problems of Elasticity :: Электронная библиотека попечительского совета мехмата МГУ

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Antman S.S. — Nonlinear Problems of Elasticity

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Название: Nonlinear Problems of Elasticity

Автор: Antman S.S.

Аннотация:

Within the past few decades, there has been an accelerating development of methods for studying nonlinear equations. Nonlinear analysis offers exciting prospects for certain specific areas of nonlinear problems with continuum mechanics. The objective of this book is to carry out such studies for problems in nonlinear elasticity. This book is directed toward scientists, engineers, and mathematicians who wish to see careful treatments of uncompromised problems. The author's aim is to retain the orientation toward fascinating problems that characterizes the best engineering texts on structural stability while retaining the precision of modern continuum mechanics and employing powerful methods of nonlinear analysis. The author's approach is to lay down a general theory for each kind of elastic body, carefully formulate specific problems, introduce the pertinent mathematical methods, and then conduct rigorous analyses of the problems.

Язык:

Рубрика: Физика/Классическая физика/Упругие среды/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1995

Количество страниц: 750

Добавлена в каталог: 17.06.2005

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Предметный указатель

Divergence theorem      382—383
Domain in Euclidean space      7
Domain of definition      3
Dot product      4 371
Dual space      237
Dyad (Dyadic product)      374
Dynamical problems for longitudinal motions of rods      629—649
Dynamical problems for rods in space      334—337
Dynamical problems for strings      33—39 77—83
Dynamical problems for three-dimensional bodies      521—527 649—664
Edge of a shell      565
Eigencurve      151—153
Eigenspace      141
Eigenvalue      141 373
Eigenvalue, algebraic multiplicity of      145 154 156 691
Eigenvalue, geometric multiplicity of      141
Eigenvalue, simple      154
Eigenvector      141 373
Elastic modulus      592
Elastic stability theory      170—173
Elastica, x      92 97 125—131 160—166 168—170 319 323
Elasticity, constitutive equations of linear      527 529
Elasticity, constitutive equations of, for axisymmetric shells      346—347
Elasticity, constitutive equations of, for induced theories of rods      545 551
Elasticity, constitutive equations of, for induced theories of shells      572—578
Elasticity, constitutive equations of, for intrinsic theories of shells      586
Elasticity, constitutive equations of, for rods in space      264 280—283 286—301
Elasticity, constitutive equations of, for rods in the plane      90—93 204—207
Elasticity, constitutive equations of, for strings      16—20
Elasticity, constitutive equations of, for three-dimensional bodies      418 448—449 457—469
Elastoplasticity      see “Plasticity”
Element of a set      7
Ellipticity      460—467
Empty set      7
End of a rod      535
Energy, balance of      443
Energy, criterion of stability      171
Energy, equation      see “Conservation of energy”
Energy, internal      see “Stored energy”
Energy, strain      see “Stored energy”
Engineering stress      407
entropy      444
Entropy conditions for shocks      634 639—640
Entropy conditions for shocks, Lax's      634
Entropy inequality      444—445 610—611
Equations of motion for extensible strings      11—48
Equations of motion for induced theories of rods      538—539 542—545
Equations of motion for induced theories of shells      567—572 574
Equations of motion for intrinsic theories of shells      583—586
Equations of motion for one-dimensional plasticity      615
Equations of motion for rods in space      263 265—272
Equations of motion for rods in the plane      90 95
Equations of motion for the mass center      402
Equations of motion for three-dimensional bodies      402—403 405—406
Equations of motion, Newton — Euler      402
Equilibrium problems for rods in space      325—334 339—342
Equilibrium problems for rods in the plane      96—123 116—123 125—131 160—166 168—170 197—215 218—230
Equilibrium problems for strings      29—32 39—42 49—77
Equilibrium problems for three-dimensional bodies      470—509 511—521
Ericksen's constraint      423 428 455
Euclidean space      4
Euler angles      300—301
Euler — Lagrange equations      45 244—245 590
Euler, L.      vii
Eulerian formulation      see “Spatial formulation”
Eversion of a three-dimensional elastic spherical shell      501—503
Eversion of an elastic tube      477
Existence theory for three-dimensional elasticity, discussion of      468—469
Extension of a fiber in a body      388
Extension of a string      16
Extension of an elastic block      480—488
Extension of an elastic cylinder      477 478
Extra stress      see “Active stress”
Face of a shell      565
Fading memory, materials with      449
First law of thermodynamics      443
Fixed point      675
Flexibility, perfect      16
Flexural strain for rods      88 261 272 275
Flexural strain for shells      344 573
Follower loads      217—218
Force on a body      401 (also see “Body force” “Contact
Fourier heat conduction inequality      448
Frame-indifference      see “Invariance under rigid motions”
Frechet derivatives and differentials      see under “Derivatives and differentials”
Fredholm alternative theorem      see “Alternative theorems”
Fredholm operator      672—673
Free energy      444 446 447 610
Frenet — Serret formulas      275
Function spaces      9
Functional      43
Functional, bounded (continuous) linear      237
Functions, conventions for      2—4
Fundamental Lemma of Calculus of Variations      21 244
Gateaux derivatives and differentials      see under “Derivatives and differentials”
Gauss's theorem      382—383
Geometric multiplicity of eigenvalues      141
Global Bifurcation Theorems      156—160 697—698
Global Continuation Theorem, Multiparameter      485—486
Gradient      380—381
Gram matrix      47
Green deformation tensor      388
Green's theorem      382—383
Green-elastic material      458
Greenberg, J.M.      33
Gronwall inequality      644
Growth and coercivity conditions for constitutive equations for axisymmetric shells      346—347
Growth and coercivity conditions for constitutive equations for calculus of variations      235
Growth and coercivity conditions for constitutive equations for rods in space      288—289 291
Growth and coercivity conditions for constitutive equations for rods in the plane      91—92
Growth and coercivity conditions for constitutive equations for strings      19—20 83
Growth and coercivity conditions for constitutive equations for three-dimensional bodies      467—468
Hamilton's Principle for elastic strings      45—46
Hamilton's Principle for hyperelastic rods      312—313
Hamilton's Principle for three-dimensional hyperelastic bodies      458—459
Heat      442
Heat flux vector      442
Helmholtz free energy      444 446 447 610
Hemitropic functions      284—285 436—440
Hilbert space      667
Historical notes      vii 11—12 83—84 190 436 449 521 599—601 603—604
History of a function up to time t      18 417
Holder continuity      10
Holder inequality      9 666
Homeomorphism      385
Homogenization      529
Homotopy      688 696
Hooke universal joint      305
Hydrostatic pressure      67 101—116 218—222 227—230 245—251 254—256 364—367 369
Hyperbolicity      463 465—467
Hyperelasticity for induced theories of rods      547
Hyperelasticity for three-dimensional bodies      428 447 458—459 466—467 528
Hyperelasticity of rods in space      282—283 290 312—313 315 317 330
Hyperelasticity of rods in the plane      93 228
Hyperelasticity of shells      586
Hypoelasticity      603—604
Identity tensor      5 373 383
Image of a function      3
Impenetrability of Matter, Principle of      386
Imperfection parameters      202
Imperfection sensitivity      131 172 200—204
Implicit Function Theorem (Local) of Hildebrandt & Graves      678
Implicit function theorem global      20 690
Impulse-Momentum Laws      see “Linear Impulse-Momentum Laws and Angular Impulse-Momentum Laws”
Incompressibility      423 428 452—453 455 457 459 467 471—477
Index, associated with degree      61 155 691 696
Indicial notation      382—383
Induced theories for rods and shells      531
Inelastic material      18 604
Inextensibility for three-dimensional bodies      423 428 433

Inextensibility of rods      92
Inextensibility of strings      49 55
Infinitesimal stability      171—172
Inflation of a three-dimensional elastic shell      501
Inflation of an elastic ring      110—116 245—251
Inflation of an elastic shell      369
Inflation of an elastic tube      477
Initial conditions for strings      14 23—24
Initial conditions for three-dimensional bodies      410—411
Inner product for Hilbert space      237 667
Inner product of tensors      374
Inner product of vectors      see “Dot product”
Integral theorems      381—382
Integral type, materials of      419
Internal constraints      see “Material constraints”
Internal energy      see “Stored energy”
Internal-variable type, materials of      419 604—612
Intersection of sets      7
Invariance under rigid motions (Frame-indifference, Objectivity) for constitutive functions for rods      282—284 559
Invariance under rigid motions (Frame-indifference, Objectivity) for constitutive functions for shells      586
Invariance under rigid motions (Frame-indifference, Objectivity) for constitutive functions for strings      17
Invariance under rigid motions (Frame-indifference, Objectivity) for constitutive functions for three-dimensional bodies      420—422 460
Invariance under rigid motions (Frame-indifference, Objectivity) of strains and strain rates for rods      89 273
Invariance under rigid motions (Frame-indifference, Objectivity), lack of, for constitutive functions of linear elasticity      529
Invariants of tensors      377 438—441
Inverse of a linear operator      671—672
Isoenergetic deformations      449
Isolas      148
Isoperimetric problem      228 234
Isotropy for axisymmetric shells      349
Isotropy for rods in space      292—298
Isotropy for three-dimensional bodies      436—441 459
Isotropy, representation theorems for      438—441
Jump conditions      28—29
Kelvin (Stokes) Theorem      382
Kelvin temperature      443
Kernel of an operator      see “Null space”
Kinematics      see “Deformation”
Kinetic energy for longitudinal motion of rods      643
Kinetic energy for rods in space      313 559
Kinetic energy for strings      21 43 82
Kinetic energy for three-dimensional bodies      442
Kirchgassner, K.      146
Kirchhoff constraints      424 429—430 434 572 588
Kirchhoff shells      588—589
Kirchhoff's Kinetic Analogy      299 319 323
Kirchhoff's problem for helical equilibrium states of rods      327—330
Kirchhoff's Uniqueness Theorem      460
Kolodner's problem for whirling strings      176—185
Kronecker delta      37 259
Lagrange multipliers      231 427 540 671
Lagrange's Criterion for a surface to be a material surface      451
Lagrangian formulation      see “Material formulation”
Lagrangian functional for elastic rods      313
Lagrangian functional for elastic strings      44
Lame coefficients      528 592
Landau order symbols      6 210
Lateral instability      133 339—342
Lateral surface of a rod      535
Lax entropy conditions      634
Lebesgue measure      8
Lebesgue spaces      9 667
Left Cauchy — Green deformation tensor      452
Legendre transform      54 93 290
Legendre — Hadamard condition      461
Lin      372
Linear analysis, topic in      665—673
Linear elasticity      527—529
Linear Impulse-Momentum Laws for one-dimensional plasticity      615
Linear Impulse-Momentum Laws for rods      262 266—267 308—311
Linear Impulse-Momentum Laws for strings      22—28
Linear Impulse-Momentum Laws for three-dimensional bodies      411—417
Linear manifold      667
Linear momentum for rods      261—262 266—268
Linear momentum for shells      583
Linear momentum for strings      15 23
Linear momentum for three-dimensional bodies      402
Linear operator      668
Linear operator, bounded (continuous)      668
Linearization      142
Lintearia problem      49 71—74
Lipschitz continuity      10 53
Local bifurcation theorem      154—155
Lyapunov — Schmidt method      679
Lyapunov, A.M.      vii
Mass      400—401 453
Mass center      402
Material constraints for rods      314—317
Material constraints for three-dimensional bodies      423—436 455 464—467
Material constraints, generating rod theories      269—271 535—537 541
Material constraints, generating shell theories      424 429 434—436 566 568
Material formulation      387
Material points of a rod      260
Material points of a string      12
Material points of a three-dimensional body      385
Material strain tensor      390 649
Matrix of a tensor      375
Mean-value theorem      6
Measure of a set      8
Membranes      369—370 589—591
Memory, materials with      417 604
Metric tensor      383
Minimal surface equation      591
Minimization Theorem      239
Moments of inertia of cross-sections      270 559
Moments, bending for rods      263
Moments, twisting for rods      263
Monotonicity conditions for constitutive equations for axisymmetric deformations of shells      346 576
Monotonicity conditions for constitutive equations for induced theories of rods      547
Monotonicity conditions for constitutive equations for rods in space      286—287
Monotonicity conditions for constitutive equations for rods in the plane      91 630
Monotonicity conditions for constitutive equations for strings      18
Monotonicity conditions for constitutive equations for three-dimensional bodies      448 460
Mooney — Rivlin material      469 509
Motion of a (special Cosserat) rod in space      260
Motion of a shell      582
Motion of a three-dimensional body      386
Multiparameter bifurcation problems      151
Multiparameter Global Bifurcation Theorem      159
Multiparameter Global Continuation Theorem      485—486
Multiplicity of eigenvalues, algebraic      145 154—156 691
Multiplicity of eigenvalues, geometric      141
Multiplier rule      230—234 433 671
Multiplier stress for rods      537
Multiplier stress for three-dimensional bodies      427
Natural configuration for strings      13 19
Necking      514—517 554—558
Neo-Hookean material      469
Newton polygon      681
Newton — Euler Laws of Motion      402
Newton's Law of Action and Reaction      405
Nodal structure      163—165
Node      163
Nonconvex energies      257
Nonlinear analysis, global      683—698
Nonlinear analysis, local      675—680
Nonsingular tensor      373
Nontrivial solution branches      129
Notation      1 7 380—381
Null space      141 668
Objectivity      see under “Invariance under rigid motions”
Order, preservation of, in constitutive equations of elasticity      460—467 (also see “Monotonicity conditions for constitutive equations” “Strong
Orientation, preservation of, for a three-dimensional body      386—387 452—453
Orientation, preservation of, for axisymmetric shells      344—345
Orientation, preservation of, for induced theories of rods      261 276—280 536
Orientation, preservation of, for induced theories of shells      566
Orientation, preservation of, for rods in space      261 276—280
Orientation, preservation of, for rods in the plane      88
Orthogonal tensor      373
Orthonormal basis      372

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