First Course in Rational Continuum Mechanics V1 (eBook)

Front Cover 1
A First Course in Rational Continuum Mechanics 4
Copyright Page 5
Contents 10
Preface to the Second Edition 14
Preface to the First Edition 16
PART 1: GENERAL CONCEPTS 20
Chapter I. Bodies, Forces, and Motions 22
1. Rational Mechanics 24
2. Universes of Bodies 26
3. Example of Universes 34
4. Mass 35
5. Force 38
6. The Event World. Rigid Frames 48
7. Motions 53
8. Linear Momentum. Rotational Momentum. Kinetic Energy. Working. Torque 56
9. Changes of Frame 63
10. Rigid Motion 70
11. Frame-Indifference 75
12. Axioms of Mechanics 79
13. The Axioms of Inertia. Euler’s Laws of Motion 83
14. Power. Kinetic Energy. Potential Energy. 95
15. Internal Energy 98
General References 102
Chapter II. Kinematics 104
1. Placements. Universes of Shapes 105
2. Mass-Density 111
3. Reference Placement. Transplacement 113
4. Descriptions of Motion: Substantial, Referential, and Spatial 114
5. Transplacement Gradient 118
6. Substantial Time Rates and Gradients in the Spatial Description. Substantial Surfaces. Kinematic Boundaries 122
7. Change of Reference Placement 127
8. Present Placement as Reference 128
9. Stretch and Rotation 130
10. Histories 135
11. Stretching and Spin 136
12. Homogeneous Transplacement 148
13. Rates of Change of Integrals over Substantial Lines, Surfaces and Regions. Substantial Vector Lines. The Vorticity 152
14. Changes of Frame. Frame-Indifference 162
General References 166
Chapter III. The Stress Tensor 168
1. Forces and Torques. The Laws of Dynamics. Body Forces and Contact Forces 170
2. Reactions upon Containers and Submerged Obstacles 184
3. The Traction Field. The Cauchy Postulate and the Hamel–Noll Theorem 189
4. Cauchy’s Fundamental Theorem: Existence of the Stress Tensor 193
5. The General Balance 200
6. Cauchy’s Laws of Motion 201
7. Mean Values and Lower Bounds for the Stress Field 206
8. Load. Boundary Condition of Traction 211
9. Motion of a Free Body 212
General References 215
Chapter IV. Constitutive Relations 216
1. Dynamic Processes 217
2. Constitutive Relations. Noll’s Axioms 218
3. Simple Materials 223
4. Some Classical Instances. Specimens of the Effect of the Principle of Material Frame-Indifference 225
5. Material Frame–Indifference. Reduced Constitutive Relations 233
6. Internal Constraints 237
7. Principle of Determinism for Constrained Simple Materials 238
8. Simple Bodies. Equations of Motion. Homogeneous Universal Transplacements, Motions, and Flows 242
9. Universal Homogeneous Transplacements of Unconstrained Bodies 248
10. Universal Homogeneous Transplacements of Incompressible Bodies 252
11. Material Isomorphisms 255
12. The Peer Group 257
13. Comparison of Peer Groups with Respect to Different Reference Placements 260
14. Isotropic Materials 262
15. Universal Transplacements of Isotropic Incompressible Bodies 264
16. Solids 283
17. Fluids 289
18. Universal Flows of Homogeneous Incompressible Fluids 292
19. Steady Rotation of a Homogeneous Body of Incompressible Fluid Loaded by Surface Tension 295
20. Fluid Crystals 297
21. Monotonous Motions 298
22. Reduction of the Constitutive Relation for a Simple Material in a Monotonous Motion 311
General References 314
Appendix I. General Scheme of Notation 316
Appendix II. Some Definitions and Theorems of Algebra Geometry, and Calculus 320
A. Algebra 320
B. Geometry 342
C. Calculus 345
Appendix III. Solutions of the Exercises 362
Gazetter 396
Index 400