Introduction to Continuum Mechanics (eBook)

Front Cover 1
An Introduction to Continuum Mechanics 4
Copyright Page 5
Contents 6
Preface 10
Acknowledgments 12
Chapter I. Tensor Algebra 14
1. Points. Vectors. Tensors 14
2. Spectral Theorem. Cayley–Hamilton Theorem. Polar Decomposition Theorem 24
Selected References 30
Chapter II. Tensor Analysis 32
3. Differentiation 32
4. Gradient. Divergence. Curl 42
5. The Divergence Theorem. Stokes' Theorem 50
Selected References 53
Chapter III. Kinematics 54
6. Bodies. Deformations. Strain 54
7. Small Deformations 67
8. Motions 71
9. Types of Motions. Spin. Rate of Stretching 80
10. Transport Theorems. Volume. Isochoric Motions 90
11. Spin. Circulation. Vorticity 93
Selected References 98
Chapter IV. Mass. Momentum 100
12. Conservation of Mass 100
13. Linear and Angular Momentum. Center of Mass 105
Selected Reference 108
Chapter V. Force 110
14. Force. Stress. Balance of Momentum 110
15. Consequences of Momentum Balance 121
Selected References 126
Chapter VI. Constitutive Assumptions. Inviscid Fluids 128
16. Constitutive Assumptions 128
17. Ideal Fluids 132
18. Steady, Plane, Irrotational Flow of an Ideal Fluid 135
19. Elastic Fluids 143
Selected References 150
Chapter VII. Change in Observer. Invariance of Material Response 152
20. Change in Observer 152
21. Invariance under a Change in Observer 156
Selected References 158
Chapter VIII. Newtonian Fluids. The Navier–Stokes Equations 160
22. Newtonian Fluids 160
23. Some Simple Solutions for Plane Steady Flow 169
24. Uniqueness and Stability 172
Selected References 177
Chapter IX. Finite Elasticity 178
25. Elastic Bodies 178
26. Simple Shear of a Homogeneous and Isotropic Elastic Body 188
27. The Piola–Kirchhoff Stress 191
28. Hyperelastic Bodies 197
29. The Elasticity Tensor 207
Selected References 211
Chapter X. Linear Elasticity 212
30. Derivation of the Linear Theory 212
31. Some Simple Solutions 214
32. Linear Elastostatics 218
33. Bending and Torsion 227
34. Linear Elastodynamics 233
35. Progressive Waves 236
Selected References 239
Appendix 240
36. The Exponential Function 240
37. Isotropic Functions 242
38. General Scheme of Notation 252
References 256
Hints for Selected Exercises 260
Index 274