Berdichevsky Variational Principles of Continuum Mechanics

I. Fundamentals
1. Auflage 2009
ISBN: 978-3-540-88467-5
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 586 Seiten

Reihe: Engineering (R0)

Thereareabout500booksonvariationalprinciples. Theyareconcernedmostlywith the mathematical aspects of the topic. The major goal of this book is to discuss the physical origin of the variational principles and the intrinsic interrelations between them. For example, the Gibbs principles appear not as the rst principles of the theory of thermodynamic equilibrium but as a consequence of the Einstein formula for thermodynamic uctuations. The mathematical issues are considered as long as they shed light on the physical outcomes and/or provide a useful technique for direct study of variational problems. Thebookisacompletelyrewrittenversionoftheauthor’smonographVariational Principles of Continuum Mechanics which appeared in Russian in 1983. I have been postponing the English translation because I wished to include the variational pr- ciples of irreversible processes in the new edition. Reaching an understanding of this subject took longer than I expected. In its nal form, this book covers all aspects of the story. The part concerned with irreversible processes is tiny, but it determines the accents put on all the results presented. The other new issues included in the book are: entropy of microstructure, variational principles of vortex line dynamics, va- ational principles and integration in functional spaces, some stochastic variational problems, variational principle for probability densities of local elds in composites with random structure, variational theory of turbulence; these topics have not been covered previously in monographic literature.

Autoren/Hrsg.

Berdichevsky, Victor

Inhaltsverzeichnis

1;Preface;6
2;Acknowledgements;7
3;Contents - I. Fundamentals;8
4;Contents - II. Applications;12
5;Part I Fundamentals;16
5.1;1 Variational Principles ;17
5.1.1; Prehistory;17
5.1.2; Mopertuis Variational Principle;25
5.1.3; Euler's Calculus of Variations;29
5.1.4; Lagrange Variational Principle;34
5.1.5; Jacobi Variational Principle;40
5.1.6; Hamilton Variational Principle;40
5.1.7; Hamiltonian Equations;46
5.1.8; Physical Meaning of the Least Action Principle;50
5.2;2 Thermodynamics ;59
5.2.1; Thermodynamic Description;59
5.2.2; Temperature;61
5.2.3; Entropy;65
5.2.4; Entropy and Probability;73
5.2.5; Gibbs Principles;73
5.2.6; Nonequilibrium Processes;74
5.2.7; Secondary Thermodynamics and Higher Order Thermodynamics;78
5.3;3 Continuum Mechanics ;80
5.3.1; Continuum Kinematics;80
5.3.2; Basic Laws of Continuum Mechanics;106
5.3.3; Classical Continuum Models;111
5.3.4; Thermodynamic Formalism;125
5.4;4 Principle of Least Action in Continuum Mechanics ;129
5.4.1; Variation of Integral Functionals;129
5.4.2; Variations of Kinematic Parameters;133
5.4.3; Principle of Least Action;137
5.4.4; Variational Equations;140
5.4.5; Models with High Derivatives;146
5.4.6; Tensor Variations;148
5.5;5 Direct Methods of Calculus of Variations ;160
5.5.1; Introductory Remarks;161
5.5.2; Quadratic Functionals;174
5.5.3; Existence of the Minimizing Element;178
5.5.4; Uniqueness of the Minimizing Element;179
5.5.5; Upper and Lower Estimates;183
5.5.6; Dual Variational Principles;189
5.5.7; Legendre and Young-Fenchel Transformations;192
5.5.8; Examples of Dual Variational Principles;212
5.5.9; Hashin-Strikman Variational Principle;227
5.5.10; Variational Problems with Constraints;235
5.5.11; Variational-Asymptotic Method;254
5.5.12; Variational Problems and Functional Integrals;281
5.5.13; Miscellaneous;289
6;Part II Variational Features of Classical Continuum Models;294
6.1;6 Statics of a Geometrically Linear Elastic Body ;295
6.1.1; Gibbs Principle;295
6.1.2; Boundedness from Below;299
6.1.3; Complementary Energy;303
6.1.4; Reissner Variational Principle;304
6.1.5; Physically Linear Elastic Body;304
6.1.6; Castigliano Variational Principle;308
6.1.7; Hashin-Strikman Variational Principle;316
6.1.8; Internal Stresses;328
6.1.9; Thermoelasticity;331
6.1.10; Dislocations;333
6.1.11; Continuously Distributed Dislocations;338
6.2;7 Statics of a Geometrically Nonlinear Elastic Body ;350
6.2.1; Energy Functional;350
6.2.2; Gibbs Principle;359
6.2.3; Dual Variational Principle;364
6.2.4; Phase Equilibrium of Elastic Bodies;378
6.3;8 Dynamics of Elastic Bodies ;384
6.3.1; Least Action vs Stationary Action;384
6.3.2; Nonlinear Eigenvibrations;386
6.3.3; Linear Vibrations: The Rayleigh Principle;388
6.3.4; The Principle of Least Action in Eulerian Coordinates;390
6.4;9 Ideal Incompressible Fluid;397
6.4.1; Least Action Principle;397
6.4.2; General Features of Solutions of Momentum Equations;400
6.4.3; Variational Principles in Eulerian Coordinates;404
6.4.4; Potential Flows;413
6.4.5; Variational Features of Kinetic Energy in Vortex Flows;416
6.4.6; Dynamics of Vortex Lines;422
6.4.7; Quasi-Two-Dimensional and Two-Dimensional Vortex Flows;435
6.4.8; Dynamics of Vortex Filaments in Unbounded Space;441
6.4.9; Vortex Sheets;452
6.4.10; Symmetry of the Action Functional and the Integrals of Motion;454
6.4.11; Variational Principles for Open Flows;461
6.5;10 Ideal Compressible Fluid ;463
6.5.1; Variational Principles in Lagrangian Coordinates;463
6.5.2; General Features of Dynamics of Compressible Fluid;465
6.5.3; Variational Principles in Eulerian Coordinates;469
6.5.4; Potential Flows;476
6.5.5; Incompressible Fluid as a Limit Case of Compressible Fluid;478
6.6;11 Steady Motion of Ideal Fluid and Elastic Body;481
6.6.1; The Kinematics of Steady Flow;481
6.6.2; Steady Motion with Impenetrable Boundaries;483
6.6.3; Open Steady Flows of Ideal Fluid;487
6.6.4; Two-Dimensional Flows;491
6.6.5; Variational Principles on the Set of Equivortical Flows;492
6.6.6; Potential Flows;498
6.6.7; Regularization of Functionals in Unbounded Domains;501
6.7;12 Principle of Least Dissipation ;503
6.7.1; Heat Conduction;503
6.7.2; Creeping Motion of Viscous Fluid;506
6.7.3; Ideal Plasticity;510
6.7.4; Fluctuations and Variations in Steady Non-Equilibrium Processes;513
6.8;13 Motion of Rigid Bodies in Fluids;517
6.8.1; Motion of a Rigid Body in Creeping Flow of Viscous Fluid;517
6.8.2; Motion of a Body in Ideal Incompressible Fluid;522
6.8.3; Motion of a Body in a Viscous Fluid;529
6.9;Appendices;538
6.9.1; On Variational Formulation of Arbitrary Systems of Equations;545
6.9.2; A Variational Principle for Probability Density;550
6.9.3; Lagrange Variational Principle;556
6.9.4; Microdynamics Yielding Classical Thermodynamics;560
6.10;Bibliographic Comments;563
6.11;Bibliography;568
6.12;Index;581
6.13;Notation;587