Poly-, Quasi- and Rank-One Convexity in Applied Mechanics (eBook)

Jörg Schröder, Patrizio Neff (Herausgeber)

eBook Download: PDF

2010 | 2010
VII, 361 Seiten
Springer Wien (Verlag)
978-3-7091-0174-2 (ISBN)

Infos

Generalized convexity conditions play a major role in many modern mechanical applications. They serve as the basis for existence proofs and allow for the design of advanced algorithms. Moreover, understanding these convexity conditions helps in deriving reliable mechanical models.
The book summarizes the well established as well as the newest results in the field of poly-, quasi and rank-one convexity. Special emphasis is put on the construction of anisotropic polyconvex energy functions with applications to biomechanics and thin shells. In addition, phase transitions with interfacial energy and the relaxation of nematic elastomers are discussed.

Inhalt

PREFACE 6
Table of Contents 7
Progress and puzzles in nonlinear elasticity 8
1 Introduction 8
1.1 Function Spaces 9
1.2 Properties of W 10
1.3 Roles of quasiconvexity in the calculus of variations 12
1.4 Open problems in elastostatics 15
1.5 Dynamics 19
Bibliography 20
Quasiconvex envelopes in nonlinear elasticity 23
1 The Saint Venant-Kinchhoff stored energy function 23
1.1 Non quasiconvexity of the Saint Venant-Kirchhoff stored energy function 23
1.2 The quasiconvex envelope of the Saint Venant-kinchhoff stored energy function 26
2 Quasiconvexity in the derivation of slender structure models 32
2.1 The three-dimensional and rescaled problems 33
2.2 Computation of the G-limit of the rescaled energies 34
2.3 Convergence of the recaled deformations and the nonlinear memnrane model 40
3 Quasiconvexity in the derivation of lattice homogenized models 43
3.1 The lattice energy 43
3.2 Convergence results 46
Bibliography 55
Anisotropic polyconvex energies 58
1 Introduction 58
2 Mechanical Foundations 62
2.1 Fundamental Kinematical Relations 62
2.2 Hyperelasticity and Invariance Requiements 64
3 Invariant Theory - Some Remarks 67
3.1 Representation of Tensor Functions 67
3.2 Principal/Main and MIxed Invariants 69
4 Generalized Convexity Conditions 73
5 Polyconvex Functions 79
5.1 Isotropic Polyconvex Functions 80
5.2 Transversely Isotropic Polyconvex Functions 82
5.3 Orthotropic Polyconvex Functions 92
5.4 Generalization to Lower Symmetries 93
6 Examples 95
6.1 Accompanying Localization Analysis 97
6.2 Homogeneous Biaxial Tension Test 101
6.3 Numerical Example: 3D-Analysis of a Tapered Cantilever 104
7 Conclusion 106
Bibliography 106
Construction of polyconvex energies for non-trivial anisotropy classes 111
1 Introduction 111
2 Non-Trivial Anisotropy Groups 113
3 Construction of Anisotropic Polyconvex Energies 117
3.1 Specific Anisotropic Polyconvex Energies 118
4 Fitting to Referential Data 121
5 Conclusion 128
Bibliography 128
Applications of anisotropic polyconvex energies: thin shells and biomechanics of arterial walls 135
1 Introduction 135
2 Continuum-Mechanical Preliminaries 136
3 Polyconvexity and Fiber-Reinforced Matrerials 139
3.1 Isotropic Polyconvex Strain Energy Functions 140
3.2 Fudamental Transversely Isotropic Polyconvex Functions 142
3.3 Polyconvex Framework for Anisotropic Functions Satisfying a priori the Natural State Condition 144
3.4 Transversely Isotropic Polyconvex Strain Energy Functions 147
4 Polyconvex Energues Applied to Biomechanics 149
4.1 Experimental Data for a Human Carotid Artery 149
4.2 Method for Parameter Adjustment 150
4.3 Comparative Analysis of Polyconvex Models 153
5 Application of Polyconvex Energies to Thin Shells 159
5.1 Thin Shell Kinematics 159
5.2 Variational form of the Shell Formulation 160
5.3 Interface to General 3D-Constitutive Laws 162
5.4 Examples: Influence of Anisotropy 163
6 Conclusion 168
Bibliography 169
Phase transitions with interfacial energy: convexity conditions and the existence of minimizers 180
1 Introduction 180
2 Constitutive theory 182
2.1 Informal description 182
2.2 Response functions 184
2.3 The exchange of the actual and reference configurations 186
2.4 Frame indifference 187
2.5 Change in the reference configuration and the symmetry group 188
2.6 Particular cases of interface reponse 191
3 Equilibrium states 197
3.1 State, minimizers of energy and equilibrium equations 199
3.2 Interface quasiconvexity, null lagrangian and polyconvexity 203
3.3 The existence of equilibrium states 207
4 Graphs, currents, and quasiconvexity of degree r 212
4.1 Rectifiable currents 213
4.2 Parametric intergrands and semiellipticity 216
4.3 Standard integrand adn degree r quasiconvexity 219
4.4 Graphs of maps on rectifiable currents 223
4.5 Degree r null lagrangians and degree r polyconvexity 228
4.6 Convergence of graphs 232
A Appendices 236
A.1 Differentiation on manifolds and on rectifiable sets 236
A.2 Multilinear algebra 238
Bibliography 241
Nematic elastomers: modelling, analysis, and numerical simulations 243
1 Introduction 243
2 A finite-dimensional warm-up 244
3 Nematic elastomers: a minimal model 246
4 Stripe-domain paterns: uniaxial stretching 253
5 Stripe-domain patterns: complex loading conditions 256
6 Dynamics 260
7 Conclusion and Outlook 262
Bibliography 264
Applications of polyconvexity and strong ellipticity to nonlinear elasticity and elastic plate theory 267
1 Introduction 267
2 Invariants 268
3 Derivative formulae 269
4 Elastic moduli 272
5 Convexity of the stretch in variants 273
6 Polyconvexity for isotropic materials 274
7 Extension to transverse isotropy 276
8 Thickness-wise expansion of the energy 283
9 Construction of the order - .3 plate enegy 288
10 Ill-prosedness of the model and a substitute model 294
11 Equilibrium equations 297
Bibliography 299
G-convergence for a geometrically exact Cosserat shell-model of defective elastic crystals 302
1 Introduction 302
1.1 Aspects of shell theory 302
1.2 Outline of this contribution 304
1.3 Notation 306
2 The underlying three-dimensional Cosserat model 308
2.1 Problem statement in variational form 308
2.2 Mathematical results for the Cosserat bullk problem 310
3 Dimensional reduction of the Cosserat bulk model 311
3.1 The three-dimensional Cosserat problem on a thin domain 311
3.2 Transformation on a fixed domain 312
3.3 The rescaled variational Cosserat bulk problem 314
3.4 On the choice of the scaling 315
4 Some facts form G-convergence 316
5 The "two-field" Cosserat G-limit 318
5.1 The spaces adn admissible sets 318
5.2 The G-limit vatiational "menbrane" problem 320
5.3 The defective elastic crystal limit case µc = 0 322
5.4 The formal limit µc = 8 324
6 Proof for positive Cosserat couple modulus µc > 0
6.1 Equiocoercivity of I. hj, compactness and dimensional reduction 326
6.2 Lower bound-the lim inf-condition 328
6.3 Upper bound-the recovery sequence 333
7 Proof for zero Cosserat couple modulus µc =0 339
7.1 The "membrane" lower bound for µc = 0 340
7.2 A lower bound for the "membrane" lower bound 341
8 Comparison with the formal finite-strain Cosserat thin plate model with size effects 347
Statement of the formal Cosserat plate model 347
8.1 Mathematical results for the formal Cosserat thin plate model 351
9 The form of the transverse shear enegy for non-vanishing thickness and the shear correction factor k 351
10 Consequences for the Cosserat couple modules µc 354
11 Conclusion 356
Bilbliography 356

Produkt-Details

• Erscheint lt. Verlag: 4.8.2010
• Reihe/Serie: CISM International Centre for Mechanical Sciences
• Reihe/Serie: CISM International Centre for Mechanical Sciences
• Zusatzinfo: VIII, 362 p. 39 illus.
• Verlagsort: Vienna
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik ► Maschinenbau
• Schlagworte: algorithms • Applied Mechanics • Biomechanics • Convexity • linear optimization • Mechanics • Shells • thin shell
• ISBN-10: 3-7091-0174-3 / 3709101743
• ISBN-13: 978-3-7091-0174-2 / 9783709101742