Stability and nonlinear solid mechanics:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English French |
| Published: |
Chichester [u.a.]
Wiley
2000
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVII, 398 S. Ill., graph. Darst. |
| ISBN: | 0471492884 |
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MARC
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| 245 | 1 | 0 | |a Stability and nonlinear solid mechanics |c Quoc Son Nguyen |
| 264 | 1 | |a Chichester [u.a.] |b Wiley |c 2000 | |
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| adam_text | Titel: Stability and nonlinear solid mechanics
Autor: Nguyen, Quoc-Son
Jahr: 2000
Preface xv
1 Basic elements of statics and dynamics 1
1.1 Discrete systems 1
1.1.1 Discrete mechanical systems 1
1.1.2 Fundamental law of dynamics 2
1.1.3 Necessity of supplementary relations 3
1.1.4 Lagrange s formalism 4
1.2 Continuous systems 5
1.2.1 Deformation of a continuum 5
1.2.2 Fundamental law of dynamics 8
1.2.3 Equations of motion in Eulerian description 11
1.2.4 Equations of motion in Lagrangian description 13
1.2.5 Common deformations 15
2 Constitutive relations and thermodynamics 17
2.1 Overview of thermodynamics 17
2.1.1 Thermodynamic equilibrium 17
2.1.2 Reversible and irreversible transformations 18
v
2.1.3 First principle 18
2.1.4 Second principle 18
2.2 State variables and common thermodynamic functions 21
2.2.1 State variables 22
2.2.2 Heat capacity 23
2.2.3 Free energy and energy equation 23
2.2.4 State equation 24
2.3 Irreversible processes 24
2.3.1 Thermodynamic force and flux 24
2.3.2 Onsager s reciprocity 25
2.3.3 Dissipation potential 26
2.3.4 Examples 26
2.4 Continuum thermodynamics 29
2.4.1 Local state postulate 29
2.4.2 First principle 30
2.4.3 Second principle 30
2.4.4 Discontinuity relations 31
2.4.5 Irreversible processes 32
2.5 Common constitutive laws 33
2.5.1 Common models of fluids 33
2.5.2 Common models of solids 34
2.5.3 Generalized standard models 37
2.6 Thermo-mechanical evolution 37
2.6.1 Thermo-mechanical coupling 37
2.6.2 Equations of evolution 38
3 Elements of mathematics 41
3.1 Directional derivative and derivative 41
3.2 Common functional spaces 42
3.3 Calculus of variations 44
3.3.1 Stationary condition 44
3.3.2 Examples 45
3.3.3 Local minimum and second variation 46
3.4 Convex analysis 47
3.4.1 Convex sets and convex functions 47
3.4.2 Sub-gradient and Legendre-Fenchel s transform 48
3.4.3 Minimization of convex functions 48
3.5 Variational inequality and linear complementarity 50
3.5.1 Quadratic variational inequality 50
3.5.2 Linear complementarity problem 50
3.6 Eigenvalue problem 52
3.7 Implicit function theorem 53
4 Elastic problems in small deformation 55
4.1 Elastic laws 55
4.1.1 Linear elasticity 55
4.1.2 Nonlinear elasticity 56
4.2 Static and dynamic elasticity 57
4.2.1 Dynamic equations 58
4.2.2 Static equations 59
4.3 Minimum principles and static equilibrium 59
4.3.1 Preliminary definitions 59
4.3.2 Minimum of total potential energy 60
4.3.3 Minimum of total complementary energy 60
4.3.4 The mixed principle 61
4.3.5 Illustrative example 61
4.4 Existence and uniqueness 63
4.4.1 Uniqueness 63
4.4.2 Existence 63
4.5 Elastic solids in perfect unilateral contact 64
5 Elastic-plastic problems in small deformation 67
5.1 Elastic-plastic constitutive laws 67
5.1.1 Perfect plasticity 68
5.1.2 Hardening plasticity 72
5.2 Static and dynamic responses of an elastic-plastic solid 77
5.2.1 Dynamic or quasi-static evolution 77
5.2.2 Example: bending of an elastic-plastic beam 78
5.3 Some general results on the elastic-plastic evolution 80
5.3.1 Uniqueness 81
5.3.2 Shake-down 83
5.4 Quasi-static rate problems 87
5.4.1 Complements on incremental laws 88
5.4.2 Rate problems 89
6 Stability of an equilibrium
91
6.1 Evolution and stability 91
6.1.1 Equilibrium and evolution 91
6.1.2 Stability of an equilibrium 92
6.1.3 Liapunov s functional 93
6.2 Linearization method 93
6.2.1 Liapunov s theorem 93
6.2.2 Linearized equations in mechanics 96
6.2.3 Equilibrium depending on a load parameter 96
6.3 Conservative mechanical systems 97
6.3.1 Lejeune-Dirichlet s theorem 98
6.3.2 Second variation criterion 99
6.4 Examples 101
6.4.1 Two-bar system under a follower force 101
6.4.2 Two-bar system under a conservative load 102
6.4.3 Elastica: stability of a flexible ruler tr 103
6.4.4 Exercise: stability of a system of two balloons 105
7 Static and dynamic bifurcation 107
7.1 Equilibrium curve of an evolving system 107
7.2 Bifurcation point and limit point 108
7.2.1 Bifurcation point 108
7.2.2 Rate problem 108
7.2.3 Limit point 109
7.3 Equilibrium curve(s) going through a given point 110
7.3.1 Regular point 111
7.3.2 Singular point 112
7.3.3 Stability exchange 114
7.3.4 Stability analysis at a singular equilibrium 114
7.4 Static bifurcation in mechanics 115
7.4.1 Static and dynamic evolutions 115
7.4.2 Illustration 116
7.5 Dynamic bifurcation 119
7.5.1 Hopf bifurcation 119
7.5.2 Post-critical analysis in Hopf s bifurcation 1 119
7.5.3 Stability exchange 124
7.6 Examples of Hopf s bifurcation 127
7.6.1 Elementary example 127
7.6.2 Van der Pol s oscillator 127
7.6.3 Follower load
129
8 Bifurcation analysis of conservative systems 133
8.1 Bifurcation analysis of an equilibrium curve 133
8.1.1 Detection of bifurcation points 133
8.1.2 Liapunov-Schmidt s method 134
8.1.3 Energy levels 138
8.1.4 Stability exchange 138
8.2 Examples 139
8.2.1 Buckling of a two-bar system 139
8.2.2 Buckling of a ruler 140
8.3 Multiple modes 141
8.3.1 Bifurcation analysis under multiple modes 141
8.3.2 Stability analysis under the assumption of multiple modes 143
8.4 Examples 145
8.4.1 Buckling of a triangular frame 145
8.4.2 Exercise: bifurcation of a cable-stayed column 147
9 Buckling of elastic structures 149
9.1 Elastic structures 149
9.1.1 The small strain-finite rotation assumption 149
9.1.2 Bifurcation analysis 150
9.2 Elastic beams under axial compression 151
9.3 Plates and shells 154
9.3.1 Shallow shells 154
9.3.2 Buckling of a rectangular plate under compression 155
9.4 Three-dimensional solids 157
9.4.1 Elastic laws in finite deformation 157
9.4.2 Static problems 159
9.5 Buckling under pressure 161
9.5.1 Contribution of the pressure to tangent stiffness 161
9.5.2 Example 162
9.6 Stability of a thermo-elastic equilibrium 163
9.6.1 Thermo-elastic evolution 163
9.6.2 Second variation criterion 164
10 Complements of conservative systems
10.1 Influence of imperfections
167
167
10.1.1 Geometric imperfection 167
10.1.2 Equilibrium curve with imperfection 169
10.1.3 Illustration 172
10.1.4 Multiple modes 173
10.1.5 Load imperfection 173
10.1.6 Elements of catastrophe theory 174
10.1.7 Exercise on cavitation pressure 177
10.2 System under perfect constraints 180
10.2.1 Perfect constraints 180
10.2.2 Equilibrium of a conservative system under perfect
constraints 181
10.2.3 Stability under perfect constraints 182
11 Plastic buckling of beams 185
11.1 Shanley s model 185
11.1.1 Discrete model 185
11.1.2 Continuous model 188
11.2 Elastic-plastic buckling of beams 191
11.2.1 Governing equations 192
11.2.2 Bifurcation at the tangent critical load 194
11.2.3 Bifurcation at the reduced critical load 198
11.2.4 Bifurcation at an intermediate critical load 199
12 Rate problems and Hill s criteria 201
12.1 Plasticity at finite strains 201
12.2 Quasi-static evolution of solids 204
12.2.1 Governing equations 205
12.2.2 Rate problem 205
12.2.3 Rate uniqueness 206
12.3 Bifurcation analysis 206
12.3.1 IMon-bifurcation criterion 206
12.3.2 Energy considerations 207
12.3.3 Energy comparison after bifurcation 209
12.4 Stability analysis 210
12.4.1 Stability criterion 210
12.4.2 Illustration 211
12.4.3 Justification of the stability criterion 212
13 Plastic bifurcation 215
13.1 Bifurcation analysis of a curve 215
13.1.1 Singular and regular points 215
13.1.2 Bifurcation under monotone loading 216
13.2 Bifurcation at the tangent critical load 217
13.2.1 Assumptions 217
13.2.2 Hutchinson s post-bifurcation analysis 219
13.2.3 Energy considerations 224
13.3 Tangent bifurcation 226
13.3.1 Triantafyllidis example 227
13.3.2 Remarks 231
14 Materials and structures in finite deformation 233
14.1 Constitutive equations at finite strains 234
14.1.1 Eulerian and updated Lagrangian descriptions 234
14.1.2 Elastic-plastic constitutive equations 235
14.1.3 The relaxed configuration model 236
14.1.4 Rate problem and major symmetry 238
14.2 Strain localization problem 239
14.2.1 Strain localization 239
14.2.2 Bifurcation by localization 239
14.2.3 Localization in plate under biaxial tension 241
14.2.4 Instability by localization 246
14.3 Buckling of elastic-plastic structures 247
14.3.1 Common tangent modulus 247
14.3.2 Buckling under compression 249
15 Standard dissipative systems 253
15.1 Standard dissipative systems 254
15.1.1 Biot s equation and generalized standard models 254
15.1.2 Invariance aspects 255
15.1.3 Standard dissipative systems 255
15.2 Time-independent behaviour 258
15.3 Rate problems 259
15.3.1 Rate equations 259
15.3.2 General results 260
15.4 Stability and non-bifurcation analysis 261
15.4.1 Stability criterion 261
15.4.2 Non-bifurcation criterion 2b3
15.5 Illustrations ^63
15.5.1 Example of Shanley s model 263
15.5.2 Elastic-plastic solids with multiple plastic potentials 265
15.6 Some mathematical results 267
15.6.1 Visco-plastic regularization method 268
15.6.2 Continuity and uniqueness 271
16 Stability of a quasi-static evolution 273
16.1 Stability of an evolution 274
16.1.1 Stability of the solution of a non-autonomous differential
equation 274
16.1.2 Linearization method 274
16.2 Stability of a visco-elastic evolution 276
16.2.1 Visco-elastic Shanley s model 276
16.2.2 Visco-elastic solids 278
16.2.3 Asymptotic stability of a quasi-static evolution 280
16.3 Stability of a visco-plastic or elastic-plastic evolution 281
16.3.1 Visco-plastic solids 281
16.3.2 Stability of a visco-plastic evolution 282
16.3.3 Stability of an elastic-plastic evolution 284
16.4 Criterion of second variation of free energy 284
16.4.1 Limit modulus 285
16.4.2 Tangent modulus versus limit modulus 286
17 Crack propagation and stability 287
17.1 Propagation of a system of interacting linear cracks 288
17.1.1 Crack propagation, stability and bifurcation 288
17.1.2 Example 290
17.2 Determination of the energy release rate 293
17.2.1 Method of geometry change 293
17.2.2 Integrals J and Ge 294
17.3 Variations of G and second derivatives of energy 295
17.3.1 Variations of G with respect to domain 295
17.3.2 Second derivatives of energy 297
17.4 Numerical aspects 300
17.4.1 Numerical determination of G in plane elasticity 300
17.4.2 Computation of the second derivatives of energy 301
17.5 Thermo-mechanical aspects 304
17.6 The curved crack problem 308
17.6.1 Some analytical results 308
17.6.2 Mode-I criterion 311
17.6.3 Criterion of maximum release rate 312
18 Plane cracks 315
18.1 Plane cracks of arbitrary shape 315
18.2 Brittle law of propagation 318
18.3 Debonding of a membrane 319
18.3.1 Membrane bonded on a rigid substrate 319
18.3.2 Instability of a circular crack 321
18.3.3 Configurational instability of a tunnel crack 322
18.4 Viscous law of propagation 325
18.5 Delamination cracks 326
18.5.1 A problem of bonded plates 326
18.5.2 Delamination in multi-layered plates 329
18.5.3 Exercise 331
18.6 Plane cracks in a three-dimensional solid 332
18.6.1 Expression of the energy release rate 332
18.6.2 Examples 333
19 Contact with friction 335
19.1 Coulomb s law of dry friction 335
19.2 Quasi-static problems 336
19.3 Rate problems 339
19.3.1 Rate relations 339
19.3.2 Variational formulation of the rate problem 340
19.3.3 Reduced quasi-static equations 341
19.3.4 Klarbring s example 341
19.4 Important particular cases 343
19.4.1 Small deformation assumption 343
19.4.2 Hencky-type friction 344
19.4.3 Effective contact 344
19.5 Dynamic problems 345
19.5.1 Bouncing contact in a discrete system 345
19.5.2 Stick-slip vibrations of a simple oscillator 348
19.5.3 On dynamic evolutions 348
19.6 Equilibrium of a solid in contact with a moving obstacle 352
19.6.1 Solid in contact with a moving half-space 352
19.6.2 Stability of an equilibrium 353
19.6.3 Instability of the steady sliding motion 354
20 Nonlinear numerical analysis 359
20.1 Static responses of elastic solids 360
20.1.1 Incremental methods 360
20.1.2 Asymptotic methods 361
20.1.3 Passing beyond a singular point 361
20.2 Quasi-static responses of elastic-plastic and visco-plastic solids 364
20.2.1 Implicit and explicit incremental methods 364
20.2.2 Elastic-plastic solids 365
20.2.3 Viscous solids 367
20.2.4 Large time-increment methods 369
20.2.5 Remarks on steady responses 371
20.2.6 Illustrations of cyclic plasticity 371
20.3 Eigenvalue problems 374
20.3.1 Symmetric matrices 374
20.3.2 Unsymmetric matrices 376
References 377
Index
395
|
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| id | DE-604.BV013745027 |
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| indexdate | 2024-07-09T18:51:15Z |
| institution | BVB |
| isbn | 0471492884 |
| language | English French |
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| physical | XVII, 398 S. Ill., graph. Darst. |
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| publisher | Wiley |
| record_format | marc |
| spelling | Nguyen, Quoc-Son Verfasser (DE-588)11494573X aut Stabilité et mécanique non linéaire Stability and nonlinear solid mechanics Quoc Son Nguyen Chichester [u.a.] Wiley 2000 XVII, 398 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Elasticity Nonlinear mechanics Stability Structural analysis (Engineering) Festkörpermechanik (DE-588)4129367-8 gnd rswk-swf Festkörpermechanik (DE-588)4129367-8 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009396037&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Nguyen, Quoc-Son Stability and nonlinear solid mechanics Elasticity Nonlinear mechanics Stability Structural analysis (Engineering) Festkörpermechanik (DE-588)4129367-8 gnd |
| subject_GND | (DE-588)4129367-8 |
| title | Stability and nonlinear solid mechanics |
| title_alt | Stabilité et mécanique non linéaire |
| title_auth | Stability and nonlinear solid mechanics |
| title_exact_search | Stability and nonlinear solid mechanics |
| title_full | Stability and nonlinear solid mechanics Quoc Son Nguyen |
| title_fullStr | Stability and nonlinear solid mechanics Quoc Son Nguyen |
| title_full_unstemmed | Stability and nonlinear solid mechanics Quoc Son Nguyen |
| title_short | Stability and nonlinear solid mechanics |
| title_sort | stability and nonlinear solid mechanics |
| topic | Elasticity Nonlinear mechanics Stability Structural analysis (Engineering) Festkörpermechanik (DE-588)4129367-8 gnd |
| topic_facet | Elasticity Nonlinear mechanics Stability Structural analysis (Engineering) Festkörpermechanik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009396037&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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