Stability in viscoelasticity:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Amsterdam u.a.
Elsevier
1994
|
| Schriftenreihe: | North-Holland series in applied mathematics and mechanics
38 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXI, 600 S. graph. Darst. |
| ISBN: | 0444819517 |
Internformat
MARC
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| 100 | 1 | |a Drozdov, Aleksey D. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Stability in viscoelasticity |c Aleksey D. Drozdov ; Vladimir B. Kolmanovskii |
| 264 | 1 | |a Amsterdam u.a. |b Elsevier |c 1994 | |
| 300 | |a XXI, 600 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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| 650 | 7 | |a Stabiliteit |2 gtt | |
| 650 | 7 | |a Stabilité |2 ram | |
| 650 | 7 | |a Visco-elasticiteit |2 gtt | |
| 650 | 7 | |a Viscoélasticité |2 ram | |
| 650 | 7 | |a Élasticité |2 ram | |
| 650 | 4 | |a Stability | |
| 650 | 4 | |a Viscoelasticity | |
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Datensatz im Suchindex
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| adam_text | xiii
CONTENTS
Preface vii
Chapter 1. Constitutive models of viscoelastic materials 1
1. Kinematics of motion 2
1.1. Description of motion 2
1.2. Deformation gradient 3
1.3. Deformation measures and strain tensors 6
1.4. Stretch tensors 8
1.5. Rigid motion 9
1.6. Volume deformation 11
1.7. Piola s identity 12
1.8. Deformation of the surface element 12
2. Dynamics of continuum 14
2.1. Forces 14
2.2. Mass conservation law 16
2.3. Principle of linear momentum 16
2.4. Principle of angular momentum 18
2.5. The Piola stress tensor 19
3. Small perturbations of the actual configuration 20
3.1. Small perturbations of the basic vectors 21
3.2. Small perturbations of the Cauchy measure 22
3.3. Small perturbations of the principal invariants
of the Cauchy measure 23
3.4. Small perturbations of the Finger measure 24
3.5. Small perturbations of the surface element 25
3.6. Small perturbations of the motion equations 26
3.7. Small perturbations of boundary conditions 28
4. Constitutive theory 29
4.1. Axioms of the constitutive theory 29
4.2. Materials with constrains 35
5. Constitutive equations for viscoelastic materials with
infinitesimal strains 36
5.1. Linear viscoelastic solid 36
5.1.1. Uniaxial stresses 36
5.1.2. Three dimensional stresses 46
5.2. Nonlinear viscoelastic solid 49
5.3. Physical theories of viscoelasticity 54
xiv
6. Creep and relaxation kernels 58
6.1. Some examples of kernels 59
6.2. Properties of relaxation measures 64
6.2.1. Relaxation measures for non ageing
materials 64
6.2.2. Relaxation measures for ageing materials 68
6.2.3. Relaxation kernels for ageing materials 70
7. Thermodynamic potentials and variational principles
in linear viscoelasticity 71
7.1. Thermodynamic potentials for ageing
viscoelastic media 72
7.2. Variational principles in viscoelasticity 74
7.3. Gibbs principle and the second law of
thermodynamics 77
7.4. Thermodynamic inequalities in linear
viscoelasticity 79
8. Hyperelasticity theory 80
8.1. Specific potential energy 81
8.2. Examples of strain energy densities 82
8.3. The Lagrange variational principle 85
8.4. Constitutive equations for hyperelastic
materials 86
8.5. Constitutive restrictions 90
8.6. Small perturbations of constitutive laws 91
9. Constitutive equations for viscoelastic materials
with finite strains 93
9.1. Brief survey of constitutive models 94
9.2. New constitutive model 104
9.2.1. Spring model for a viscoelastic
material 104
9.2.2. The Lagrange variational principle 105
9.2.3. Constitutive equations for
viscoelastic media 109
9.2.4. Tension of a viscoelastic bar 111
References 117
Chapter 2. Linear stability problems 133
1. Stability of viscoelastic bars 134
1.1. Governing equations 134
1.2. Stability of viscoelastic bars under
quasi static loading 135
1.2.1. Integral estimates 136
1.2.2. The Euler problem of stability 137
XV
1.2.3. Remarks 139
1.3. Stability of viscoelastic bars under
dynamic loading 142
1.3.1. Lyapunov s functionals 143
1.3.2. Remarks 146
1.4. Stability of an integro differential equation
with operator coefficients 147
2. Quasi static stability of viscoelastic bars under
non conservative loading 150
2.1. Formulation of the problem 152
2.2. Analysis of stability 153
2.3. Stability of a viscoelastic bar under
dead loading 159
2.4. Stability of a viscoelastic bar under
follower loading 164
3. Stability of viscoelastic bars under the action
of follower forces 164
3.1. Formulation of the problem 165
3.2. Stability conditions 167
3.3. Some properties of coefficient A 171
3.4. Discussion 174
4. Stability of an integro differential equation with
non commuting operator coefficients 175
4.1. Formulation of the problem and basic assumptions 175
4.2. Proof of Theorem 4.1 177
4.2.1. Remark 182
4.3. Discussion 182
4.4. Supersonic flutter of a viscoelastic panel 186
5. Stability of bars made of elastic materials with voids 197
5.1. Formulation of the problem 197
5.1.1. Nonperturbed deformation 198
5.1.2. Perturbed deformation 199
5.1.3. Definition of the bar stability 202
5.2. Development of stability conditions 203
5.2.1. Estimation of Ln 203
5.2.2. Stability conditions 205
5.3. Discussion 206
6. Concluding remarks 208
References 211
Chapter 3. Stability of viscoelastic structural members
under periodic and random loads 217
1. Stability of a viscoelastic shell under time varying loads 218
xvi
1.1. Formulation of the problem and governing equations 218
1.2. Stability conditions 220
1.3. Example 221
2. Stability of a linear integro differential equation with
periodic coefficients 223
2.1. Formulation of the problem and basic assumptions 224
2.2. Stability conditions 226
2.3. Stability of the integro differential equation and
the corresponding ordinary differential equation 232
2.4. Stability of a viscoelastic bar under periodic
compressive load 234
3. Stability of a viscoelastic shell driven by random loads 237
3.1. Stability conditions 238
3.2. Example 239
4. Stability of a viscoelastic bar driven by random
compressive loads 241
4.1. Formulation of the problem and basic assumptions 241
4.2. Transformation of the governing equations 242
4.3. Stability conditions 244
4.3.1. Remarks 244
4.3.2. Proof 245
4.4. Instability of an elastic bar under random
compressive load 248
5. Stability of a class of stochastic integro differential
equations 250
5.1. Formulation of the problem 250
5.2. Stability conditions 252
5.3. Examples 257
6. Concluding remarks 261
References 263
Chapter 4. Nonlinear problems of stability for viscoelastic
structural members 268
1. Stability of a non homogeneous, ageing, viscoelastic bar
under conditions of nonlinear creep 269
1.1. Formulation of the problem and governing equations 269
1.2. Development of stability conditions 271
1.3. Some particular cases 274
2. Stability of a nonlinear operator integro differential
equation 277
2.1. Formulation of the problem and auxiliary results 278
2.2. A priori estimates for solutions of linear
integro differential equations 283
xvii
2.3. Stability conditions 286
2.4. Stability of a viscoelastic bar on a nonlinear
elastic foundation 288
3. Stability of a system of nonlinear integro differential
equations with operator coefficients 291
3.1. Formulation of the problem and basic notation 291
3.2. Linear non homogeneous system 295
3.3. Linear homogeneous system 297
3.4. Proof of Theorem 3.1 298
3.5. Example 299
4. Stability of a class of nonlinear integro differential
equations 305
4.1. Formulation of the problem and main results 305
4.2. Proof of Theorem 4.1 307
5. Concluding remarks 316
References 318
Chapter 5. Applied problems of stability 322
1. Stability of growing viscoelastic bars in a finite time
interval 323
1.1. Formulation of the stability problem for a
non homogeneous viscoelastic bar 323
1.2. Stability of a non homogeneous, ageing,
cantilevered bar 324
1.3. Stability in an infinite time interval 328
1.4. Stability of a growing bar 329
2. Stability of viscoelastic bars with finite shear rigidity 338
2.1. Formulation of the problem 338
2.2. Stability of a cantilevered bar 341
2.3. Stability of a simply supported bar 347
2.4. Stability of a bar with clamped ends 348
2.5. Stability of a reinforced bar 350
3. Stability of non homogeneous, ageing, viscoelastic
plates 350
3.1. Formulation of the problem 350
3.2. Governing equations 351
3.3. Development of stability conditions 354
3.4. An explicit stability condition 356
3.5. Remarks 361
3.6. Stability in a finite time interval 363
4. Stability of an elastic vertical casing in a viscoelastic
medium subjected to ageing 367
4.1. Formulation of the problem 367
xviii
4.2. Governing equations 368
4.3. Stresses in the viscoelastic medium 369
4.4. Stability conditions 374
4.5. Numerical analysis 377
5. Stability of an elastic stiffening for a horizontal mine
working in an ageing viscoelastic medium 378
5.1. Formulation of the problem 378
5.2. Governing equations for the fitting 381
5.3. Governing equations for the medium 383
5.4. Determination of the Airy function F 386
5.5. Stability conditions 388
5.6. Numerical analysis 392
6. Concluding remarks 393
References 396
Chapter 6. Stability of elastic and viscoelastic
three dimensional bodies 400
1. Quasi static stability of an ageing, linearly viscoelastic
body with infinitesimal strains 401
1.1. Formulation of the problem 401
1.2. Stability conditions 403
1.2.1. Governing equations 403
1.2.2. Development of stability conditions 404
1.2.3. Remarks 411
1.3. Some examples of stability conditions for
thin walled structural members 412
1.3.1. Stability of a viscoelastic bar 412
1.3.2. Stability of a viscoelastic plate
(the Kirchhoff Love hypotheses) 414
1.3.3. Stability of a viscoelastic plate
(the hypothesis of straight normals) 416
1.4. Stability of a viscoelastic body under a follower
pressure 417
1.5. Stability of an incompressible viscoelastic solid
under the action of dead loads 419
1.6. Stability of a self gravitating, incompressible,
viscoelastic sphere 419
1.6.1. Formulation of the problem 419
1.6.2. Stability conditions 422
2. Dynamic stability of a viscoelastic body with
infinitesimal strains 426
2.1. Formulation of the problem 426
2.2. Estimation of non perturbed stresses 428
xix
2.3. Stability conditions 435
2.4. Stability of a non homogeneously ageing,
viscoelastic plate 441
3. Stability of a viscoelastic body with finite strains 444
3.1. Formulation of the problem 444
3.2. Perturbations of constitutive equations 446
3.3. Stability conditions 449
3.4. Stability of a compressed viscoelastic bar 455
3.4.1. Determination of non perturbed stresses 456
3.4.2. Development of stability conditions 459
3.4.3. Some particular cases 461
3.5. Stability of a hydrostatically compressed
viscoelastic body 464
4. Stability of hyper viscoelastic solids 469
4.1. Concepts of stability 469
4.2. Formulation of the problem 471
4.3. Thermodynamic stability 473
4.4. Comparison of conditions for thermodynamic
and Lyapunov s stability 476
4.5. Stability of a homogeneously deformed
hyperelastic body 477
5. Stability of thin walled structural members 481
5.1. Variational asymptotic method 481
5.2. Stability of a compressed bar 482
5.3. Stability of a cylindrical shell 487
6. Concluding remarks 492
References 496
Chapter 7. Stability of functional differential equations 501
0. Introduction 502
1. Stability of linear stationary systems 505
1.1. Formulation of the problem and stability
conditions 505
1.2. Some generalizations 509
1.3. Stability of linear systems of the second order 510
1.4. Stability of linear equations of neutral type 514
2. Stability of linear equations with time varying
coefficients 516
2.1. Scalar equation with one delay 516
2.2. Scalar equation with several delays 519
2.3. System of two scalar equations 520
2.4. System with distributed delay 523
3. Stability of nonlinear equations 525
XX
3.1. Examples of stability functionals 525
3.2. Dissipative systems 527
3.2.1. Asymptotic stability 527
3.2.2. Exponential contractivity 530
3.3. Stability of quasilinear equations 532
4. Stability of a chemostat 534
5. Stability of a predator prey system 538
5.1. Formulation of the problem 538
5.2. Stability conditions 540
5.3. Stability of a system of n populations 544
References 547
Appendix 1. Theory of tensors 550
1. Definition of tensor 551
1.1. Coordinate frames and tangent vectors 551
1.2. Transformation of coordinate frames 554
1.3. Definition of tensor 555
2. Tensor algebra 556
2.1. Operations on tensors 556
2.1.1. Operation on one tensor 557
2.1.2. Operations on two tensors 557
2.2. Unit tensor 558
2.3. Inverse tensor 559
2.4. Principal invariants of a tensor 559
2.5. Eigenvectors and eigenvalues of a tensor 560
2.6. Positive definite tensors 561
2.7. Orthogonal tensors 562
2^8. Polar decomposition 562
3. Tensor analysis 563
3.1. Nabla operator 563
3.2. Operators connected with nabla operator 564
3.3. Properties of nabla operator 565
3.4. Christoffel s symbols 565
3.5. Derivatives of dual tangent vectors 567
3.6. Derivative of the elementary volume 567
3.7. Covariant derivative of a vector 568
3.8. Covariant derivative of a tensor 568
3.9. Ricci s theorem 569
3.10. Divergence of vectors and tensors 570
3.11. Second covariant derivative 570
3.12. The Stokes formula 571
4. Tensor functions 572
4.1. Scalar function of a tensor argument 572
xxi
4.2. Derivatives of the principal invariants 573
4.3. Finger s formula 575
4.4. Derivatives with respect to the inverse tensor 575
Appendix 2. Elements of functional analysis 577
1. Banach and Hilbert spaces 578
1.1. Metric spaces 578
1.2. Banach and Hilbert spaces 580
2. Linear functional and operators 582
2.1. Linear functional 582
2.2. Linear operators 584
3. Sobolev spaces 589
3.1. Generalized functions and Sobolev spaces 589
3.2. Embedding theorems 591
3.3. Korn s inequality 593
References 595
Index 597
|
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| indexdate | 2024-07-09T17:53:09Z |
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| series2 | North-Holland series in applied mathematics and mechanics |
| spelling | Drozdov, Aleksey D. Verfasser aut Stability in viscoelasticity Aleksey D. Drozdov ; Vladimir B. Kolmanovskii Amsterdam u.a. Elsevier 1994 XXI, 600 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier North-Holland series in applied mathematics and mechanics 38 Stabiliteit gtt Stabilité ram Visco-elasticiteit gtt Viscoélasticité ram Élasticité ram Stability Viscoelasticity Stabilität (DE-588)4056693-6 gnd rswk-swf Festkörper (DE-588)4016918-2 gnd rswk-swf Viskoelastizität (DE-588)4063621-5 gnd rswk-swf Viskoelastizität (DE-588)4063621-5 s Festkörper (DE-588)4016918-2 s Stabilität (DE-588)4056693-6 s DE-604 Kolmanovskij, Vladimir B. Verfasser aut North-Holland series in applied mathematics and mechanics 38 (DE-604)BV001900898 38 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006980632&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Drozdov, Aleksey D. Kolmanovskij, Vladimir B. Stability in viscoelasticity North-Holland series in applied mathematics and mechanics Stabiliteit gtt Stabilité ram Visco-elasticiteit gtt Viscoélasticité ram Élasticité ram Stability Viscoelasticity Stabilität (DE-588)4056693-6 gnd Festkörper (DE-588)4016918-2 gnd Viskoelastizität (DE-588)4063621-5 gnd |
| subject_GND | (DE-588)4056693-6 (DE-588)4016918-2 (DE-588)4063621-5 |
| title | Stability in viscoelasticity |
| title_auth | Stability in viscoelasticity |
| title_exact_search | Stability in viscoelasticity |
| title_full | Stability in viscoelasticity Aleksey D. Drozdov ; Vladimir B. Kolmanovskii |
| title_fullStr | Stability in viscoelasticity Aleksey D. Drozdov ; Vladimir B. Kolmanovskii |
| title_full_unstemmed | Stability in viscoelasticity Aleksey D. Drozdov ; Vladimir B. Kolmanovskii |
| title_short | Stability in viscoelasticity |
| title_sort | stability in viscoelasticity |
| topic | Stabiliteit gtt Stabilité ram Visco-elasticiteit gtt Viscoélasticité ram Élasticité ram Stability Viscoelasticity Stabilität (DE-588)4056693-6 gnd Festkörper (DE-588)4016918-2 gnd Viskoelastizität (DE-588)4063621-5 gnd |
| topic_facet | Stabiliteit Stabilité Visco-elasticiteit Viscoélasticité Élasticité Stability Viscoelasticity Stabilität Festkörper Viskoelastizität |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006980632&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV001900898 |
| work_keys_str_mv | AT drozdovalekseyd stabilityinviscoelasticity AT kolmanovskijvladimirb stabilityinviscoelasticity |