Variational Models and Methods in Solid and Fluid Mechanics

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
PREFACE......Page 6
Table of Contents......Page 9
1.1 Metrodoron and his followers......Page 10
1.2 Why Variational Principles and Calculus of Variation?......Page 11
1.3 The problem of including dissipation......Page 13
2.1 Principle of Least Action......Page 14
2.3 La Cin`ematique d’Abord !......Page 15
2.5 Two possible choices for the set of admissible and accessible motions......Page 16
3 In other words: How to find “Real Motions”?......Page 17
4 Lagrangian Action Functionals: technical details......Page 18
4.1 Variation of the Action Functional......Page 19
4.2 The Space-Time Case (m = 4)......Page 20
5 Principle of Virtual Power and Principle of Least Action......Page 21
7 Conclusions......Page 22
Bibliography......Page 23
Beyond Euler-Cauchy Continua: The structure of contact actions in N-th gradient generalized continua: a generalization of the Cauchy tetrahedron argument.......Page 25
1 Introduction......Page 26
2 Second and Higher Gradient Continuum Theories......Page 28
2.1 A first method for extending Cauchy model for continuous bodies......Page 29
2.2 A second method for extending Cauchy model and its relationship with the first......Page 30
The mathematical difficulties presented by this second method......Page 31
3 Some commentaries about a recent paper on second gradient continua......Page 32
3.1 Concluding remarks......Page 34
4 Resum´e of some results in Differential Geometry of Riemannian manifolds embedded in the Euclidean Space.......Page 35
5.1 Piecewise regular surfaces embedded in E3.......Page 36
5.2 Local parametrizations for regular curves and surfaces. Local curvilinear coordinate systems in E3 adapted to surfaces and surface edges.......Page 38
6 Gauss Divergence Theorem for embedded Riemannian manifolds......Page 42
7 Power expended by internal or external actions......Page 43
7.1 Representations of distributions which are N th order derivatives of absolutely continuous measures concentrated on submanifolds of RH with border......Page 44
8 Principle of Virtual Powers applied to N   th gradient continua.......Page 48
9 Contact actions in N   th order strain-gradient multipolar continua: N   th order forces and stress multipoles.......Page 49
9.1 The terms of surface (1-)forces and surface L + 1  forces......Page 53
9.2 The terms of line (1-)forces and line J + 1  forces.......Page 55
9.3 The terms of (1-)forces and J + 1  forces concentrated on wedges......Page 56
9.4 The obtained expression for Internal Power in N  th Gradient Continua......Page 57
9.5 Some final considerations about obtained results:......Page 59
10 Bodies, Contact Surfaces and Shapes ??......Page 60
10.1 Contact Actions......Page 64
11 A Seeming Impossibility For Edge Forces......Page 70
11.1 Condition 142 implies that Forces on Dihedral Edges must vanish.......Page 73
11.2 Condition 142 implies that Forces on General Edges must vanish.......Page 75
12 Generalized Noll Theorem......Page 77
13 The structure imposed on stress states by the regularity assumptions 10.1 and quasi-balance of contact power......Page 79
14.1 On contact actions including forces of order greater than one.......Page 81
14.2 A Representation Theorem for surface 1 forces generalizing the Cauchy tetrahedron Theorem.......Page 87
14.3 Representation theorems for contact 1 forces. Cauchy stress tensor......Page 91
14.4 Surface N-th order Forces......Page 94
14.5 A Representation Theorem for surface (N   1) forces generalizing the Cauchy tetrahedron Theorem.......Page 96
14.6 Representation theorem for contact k forces with k < N.......Page 104
15 Conclusions......Page 106
15.1 REFERENCES......Page 107
1 From Griffith to the variational......Page 115
1.1 Griffith’s theory......Page 116
1.2 A variational equivalence......Page 120
1.3 Functional framework– A weak variational evolution......Page 123
2.1 1d traction......Page 127
2.2 A tearing experiment......Page 129
3.1 Discrete evolution......Page 132
3.2 Global minimality in the limit......Page 136
3.3 Energy balance in the limit......Page 140
3.4 The time-continuous evolution......Page 142
4 Numerics......Page 144
4.1 Numerical approximation of the energy......Page 146
4.2 Minimization algorithm......Page 155
4.3 The Tearing experiment......Page 158
Bibliography......Page 166
1 Introduction......Page 170
2 Variations......Page 171
3 Hamilton’s principle......Page 174
4.1 Incompressible fluids......Page 175
4.2 Compressible fluids......Page 178
4.3 Isotropic elastic bodies......Page 179
5 Multiphase flow modeling: general definitions......Page 185
6 Equilibrium one-velocity model......Page 186
6.1 What is the sound speed in multiphase flow models?......Page 188
7 Nonequilibrium one-velocity model......Page 190
8 Bubbly fluids......Page 193
8.1 Dispersive shallow water flows as an analogue of bubbly flows......Page 198
9 Two-velocity one-pressure model......Page 199
9.1 Analysis of the two-velocity one-pressure model......Page 200
10 Two-velocity two-pressure model......Page 202
11 Equilibrium diffuse interface model of solid-fluid interactions......Page 203
12 Non-equilibrium diffuse interface model of solid-fluid interactions......Page 208
12.1 Hyperbolicity of the non-equilibrium model......Page 210
12.2 Applications......Page 214
Bibliography......Page 215
0.1 Reminder from complex analysis......Page 218
0.2 Some facts about integrals......Page 231
0.3 Reminder from probability theory......Page 235
0.4 The central limit theorem and the law of large numbers......Page 239
0.5 Poisson distribution......Page 247
0.6 Stochastic variational problems......Page 250
1 A New Point of View about Dissipation: Introduction......Page 255
2.1 Static systems: the effect of the hidden variables......Page 259
2.2 Dynamic systems......Page 261
2.3 Pseudo-dissipative effects: motion about ω0......Page 264
2.4 Remainder term: return time and energy transfer rate......Page 274
2.5 A variational theorem for the minimum remainder term......Page 277
2.6 Examples of application: set of parallel resonators......Page 283
3 UNISAT: An Engineering Application......Page 300
Bibliography......Page 316
1 Introduction......Page 320
2 Properties of the Gradients of C2 Vector Fields......Page 328
3 Properties of the Gradients of Piecewise C1 Vector Fields......Page 331
4 Bulk Kinematical Identities and Hadamard Conditions at Moving Boundaries......Page 334
5 Balance Equations and Corresponding Jump Conditions in the Space-Time......Page 338
6.1 Kinematics......Page 339
6.2 Balance of Masses......Page 341
7.1 Action and Rayleigh Functionals......Page 343
7.2 Equations of Motion......Page 344
7.3 Galilean Invariance......Page 346
8 The Case of a Deformable Porous Medium Surrounded by a Pure Fluid......Page 348
9 Conclusions......Page 351
A Appendix: Some Preliminary Variations......Page 352
B Appendix: Variation of the Action Functional......Page 354
C Appendix: Computation of the Rayleigh-Hamilton Dissipation......Page 356
D Bibliography......Page 357