Variational Principles of Continuum Mechanics (eBook)

Preface 6
Acknowledgements 7
Contents - I. Fundamentals 8
Contents - II. Applications 12
Part I Fundamentals 16
1 Variational Principles 17
Prehistory 17
Mopertuis Variational Principle 25
Euler's Calculus of Variations 29
Lagrange Variational Principle 34
Jacobi Variational Principle 40
Hamilton Variational Principle 40
Hamiltonian Equations 46
Physical Meaning of the Least Action Principle 50
2 Thermodynamics 59
Thermodynamic Description 59
Temperature 61
Entropy 65
Entropy and Probability 73
Gibbs Principles 73
Nonequilibrium Processes 74
Secondary Thermodynamics and Higher Order Thermodynamics 78
3 Continuum Mechanics 80
Continuum Kinematics 80
Basic Laws of Continuum Mechanics 106
Classical Continuum Models 111
Thermodynamic Formalism 125
4 Principle of Least Action in Continuum Mechanics 129
Variation of Integral Functionals 129
Variations of Kinematic Parameters 133
Principle of Least Action 137
Variational Equations 140
Models with High Derivatives 146
Tensor Variations 148
5 Direct Methods of Calculus of Variations 160
Introductory Remarks 161
Quadratic Functionals 174
Existence of the Minimizing Element 178
Uniqueness of the Minimizing Element 179
Upper and Lower Estimates 183
Dual Variational Principles 189
Legendre and Young-Fenchel Transformations 192
Examples of Dual Variational Principles 212
Hashin-Strikman Variational Principle 227
Variational Problems with Constraints 235
Variational-Asymptotic Method 254
Variational Problems and Functional Integrals 281
Miscellaneous 289
Part II Variational Features of Classical Continuum Models 294
6 Statics of a Geometrically Linear Elastic Body 295
Gibbs Principle 295
Boundedness from Below 299
Complementary Energy 303
Reissner Variational Principle 304
Physically Linear Elastic Body 304
Castigliano Variational Principle 308
Hashin-Strikman Variational Principle 316
Internal Stresses 328
Thermoelasticity 331
Dislocations 333
Continuously Distributed Dislocations 338
7 Statics of a Geometrically Nonlinear Elastic Body 350
Energy Functional 350
Gibbs Principle 359
Dual Variational Principle 364
Phase Equilibrium of Elastic Bodies 378
8 Dynamics of Elastic Bodies 384
Least Action vs Stationary Action 384
Nonlinear Eigenvibrations 386
Linear Vibrations: The Rayleigh Principle 388
The Principle of Least Action in Eulerian Coordinates 390
9 Ideal Incompressible Fluid 397
Least Action Principle 397
General Features of Solutions of Momentum Equations 400
Variational Principles in Eulerian Coordinates 404
Potential Flows 413
Variational Features of Kinetic Energy in Vortex Flows 416
Dynamics of Vortex Lines 422
Quasi-Two-Dimensional and Two-Dimensional Vortex Flows 435
Dynamics of Vortex Filaments in Unbounded Space 441
Vortex Sheets 452
Symmetry of the Action Functional and the Integrals of Motion 454
Variational Principles for Open Flows 461
10 Ideal Compressible Fluid 463
Variational Principles in Lagrangian Coordinates 463
General Features of Dynamics of Compressible Fluid 465
Variational Principles in Eulerian Coordinates 469
Potential Flows 476
Incompressible Fluid as a Limit Case of Compressible Fluid 478
11 Steady Motion of Ideal Fluid and Elastic Body 481
The Kinematics of Steady Flow 481
Steady Motion with Impenetrable Boundaries 483
Open Steady Flows of Ideal Fluid 487
Two-Dimensional Flows 491
Variational Principles on the Set of Equivortical Flows 492
Potential Flows 498
Regularization of Functionals in Unbounded Domains 501
12 Principle of Least Dissipation 503
Heat Conduction 503
Creeping Motion of Viscous Fluid 506
Ideal Plasticity 510
Fluctuations and Variations in Steady Non-Equilibrium Processes 513
13 Motion of Rigid Bodies in Fluids 517
Motion of a Rigid Body in Creeping Flow of Viscous Fluid 517
Motion of a Body in Ideal Incompressible Fluid 522
Motion of a Body in a Viscous Fluid 529
Appendices 538
On Variational Formulation of Arbitrary Systems of Equations 545
A Variational Principle for Probability Density 550
Lagrange Variational Principle 556
Microdynamics Yielding Classical Thermodynamics 560
Bibliographic Comments 563
Bibliography 568
Index 581
Notation 587