Nonlinear Theory of Elastic Plates provides the theoretical materials necessary for the three plate models—Cosserat plates, Reissner-Mindlin plates and Kirchhoff-Love plates— in the context of finite elastic deformations. One separate chapter is devoted to the linearized theory of Kirchhoff-Love plates, which allows for the study of vibrations of a pre-stressed plate and the static buckling of a plate. All mathematical results in the tensor theory in curvilinear coordinates necessary to investigate the plate theory in finite deformations are provided, making this a self-contained resource.
1. Cover - p. ii 2. Title page - p. ii 3. Table of Contents - p. ii 4. Dedication - p. ii 5. Copyright - p. iv 6. Preface - p. ix 7. 1: Fundamentals of Tensor Theory - p. 1 8. 2: Initial Position of a Plate - p. 23 9. 3: Cosserat Plate Theory - p. 37 10. Abstract - p. 37 11. 3.1 Current position of the plate mid-surface - p. 38 12. 3.2 Current position of the plate - Displacement field - p. 39 13. 3.3 Displacement gradient - p. 46 14. 3.4 Strain tensor - p. 46 15. 3.5 Velocity field - p. 49 16. 3.6 Principle of Virtual Power (PVP) - p. 49 17. 3.7 Virtual velocity field - p. 50 18. 3.8 Virtual velocity gradient - p. 50 19. 3.9 Virtual power of inertia forces - p. 50 20. 3.10 Virtual power of internal forces - p. 52 21. 3.11 Virtual power of external forces - p. 54 22. 3.12 Equations of motion and boundary conditions - p. 59 23. 3.13 Static problems - p. 61 24. 3.14 Another method to obtain the equations - p. 61 25. 3.15 Overview of the equations and unknowns - p. 65 26. 4: Reissner-Mindlin Plate Theory - p. 67 27. 5: Kirchhoff-Love Plate Theory - p. 83 28. Abstract - p. 83 29. 5.1 Current position of the plate mid-surface - p. 83 30. 5.2 Current position of the plate - Displacement field - p. 89 31. 5.3 Strain tensor - p. 95 32. 5.4 Velocity field - p. 96 33. 5.5 Virtual velocity field - p. 96 34. 5.6 Virtual powers of inertia forces - p. 97 35. 5.7 Virtual power of internal forces - p. 97 36. 5.8 Virtual power of external forces - p. 105 37. 5.9 Equations of motion and boundary conditions - p. 111 38. 5.10 Static problems - p. 114 39. 5.11 Overview of equations and unknowns - p. 114 40. 5.12 Example: Kirchhoff-Love plate in cylindrical bending - p. 115 41. 6: Constitutive Law of Plates - p. 129 42. Abstract - p. 129 43. 6.1 Hyperelastic 3D constitutive law - p. 129 44. 6.2 Strains in terms of the Z-coordinate - p. 133 45. 6.3 Stress resultants for Cosserat plates - p. 134 46. 6.4 Zero normal stress hypothesis σ33 = 0 - p. 137 47. 6.5 Plane stress state - p. 137 48. 6.6 Reduced constitutive law - p. 140 49. 6.7 Stress resultants for Reissner-Mindlin plates - p. 146 50. 6.8 Stress resultants for Kirchhoff-Love plates - p. 147 51. 6.9 Review of the hypotheses used - p. 152 52. 7: Linearized Kirchhoff-Love Plate Theory - p. 157 53. Abstract - p. 157 54. 7.1 Statement of the problem - p. 157 55. 7.2 Linearization principle - p. 160 56. 7.3 Linearization of the vectors of the current natural basis - p. 164 57. 7.4 Linearized current curvatures - p. 166 58. 7.5 Linearized current Christoffel symbols - p. 167 59. 7.6 Linearized strain tensor - p. 167 60. 7.7 Linearized integrated constitutive laws - p. 168 61. 7.8 Linearized governing equations and boundary conditions - Vibrations of a pre-stressed plate - p. 170 62. 7.9 Overview of the equations and unknowns - p. 176 63. 7.10 Displacement equations - p. 176 64. 7.11 Equilibrium of a pre-stressed plate - p. 180 65. 7.12 Plate buckling problem - p. 183 66. 7.13 Example: Buckling of a simply-supported rectangular plate - p. 184 67. 7.14 Example: Buckling of a circular plate - p. 188 68. Appendix: Some Mechanical Relations in 3D Curvilinear Coordinates - p. 195 69. Bibliography - p. 199 70. Index - p. 201
Nonlinear Theory of Elastic Plates provides the theoretical materials necessary for the three plate models—Cosserat plates, Reissner-Mindlin plates and Kirchhoff-Love plates— in the context of finite elastic deformations. One separate chapter is devoted to the linearized theory of Kirchhoff-Love plates, which allows for the study of vibrations of a pre-stressed plate and the static buckling of a plate. All mathematical results in the tensor theory in curvilinear coordinates necessary to investigate the plate theory in finite deformations are provided, making this a self-contained resource.
1. Cover - p. ii 2. Title page - p. ii 3. Table of Contents - p. ii 4. Dedication - p. ii 5. Copyright - p. iv 6. Preface - p. ix 7. 1: Fundamentals of Tensor Theory - p. 1 8. 2: Initial Position of a Plate - p. 23 9. 3: Cosserat Plate Theory - p. 37 10. Abstract - p. 37 11. 3.1 Current position of the plate mid-surface - p. 38 12. 3.2 Current position of the plate - Displacement field - p. 39 13. 3.3 Displacement gradient - p. 46 14. 3.4 Strain tensor - p. 46 15. 3.5 Velocity field - p. 49 16. 3.6 Principle of Virtual Power (PVP) - p. 49 17. 3.7 Virtual velocity field - p. 50 18. 3.8 Virtual velocity gradient - p. 50 19. 3.9 Virtual power of inertia forces - p. 50 20. 3.10 Virtual power of internal forces - p. 52 21. 3.11 Virtual power of external forces - p. 54 22. 3.12 Equations of motion and boundary conditions - p. 59 23. 3.13 Static problems - p. 61 24. 3.14 Another method to obtain the equations - p. 61 25. 3.15 Overview of the equations and unknowns - p. 65 26. 4: Reissner-Mindlin Plate Theory - p. 67 27. 5: Kirchhoff-Love Plate Theory - p. 83 28. Abstract - p. 83 29. 5.1 Current position of the plate mid-surface - p. 83 30. 5.2 Current position of the plate - Displacement field - p. 89 31. 5.3 Strain tensor - p. 95 32. 5.4 Velocity field - p. 96 33. 5.5 Virtual velocity field - p. 96 34. 5.6 Virtual powers of inertia forces - p. 97 35. 5.7 Virtual power of internal forces - p. 97 36. 5.8 Virtual power of external forces - p. 105 37. 5.9 Equations of motion and boundary conditions - p. 111 38. 5.10 Static problems - p. 114 39. 5.11 Overview of equations and unknowns - p. 114 40. 5.12 Example: Kirchhoff-Love plate in cylindrical bending - p. 115 41. 6: Constitutive Law of Plates - p. 129 42. Abstract - p. 129 43. 6.1 Hyperelastic 3D constitutive law - p. 129 44. 6.2 Strains in terms of the Z-coordinate - p. 133 45. 6.3 Stress resultants for Cosserat plates - p. 134 46. 6.4 Zero normal stress hypothesis σ33 = 0 - p. 137 47. 6.5 Plane stress state - p. 137 48. 6.6 Reduced constitutive law - p. 140 49. 6.7 Stress resultants for Reissner-Mindlin plates - p. 146 50. 6.8 Stress resultants for Kirchhoff-Love plates - p. 147 51. 6.9 Review of the hypotheses used - p. 152 52. 7: Linearized Kirchhoff-Love Plate Theory - p. 157 53. Abstract - p. 157 54. 7.1 Statement of the problem - p. 157 55. 7.2 Linearization principle - p. 160 56. 7.3 Linearization of the vectors of the current natural basis - p. 164 57. 7.4 Linearized current curvatures - p. 166 58. 7.5 Linearized current Christoffel symbols - p. 167 59. 7.6 Linearized strain tensor - p. 167 60. 7.7 Linearized integrated constitutive laws - p. 168 61. 7.8 Linearized governing equations and boundary conditions - Vibrations of a pre-stressed plate - p. 170 62. 7.9 Overview of the equations and unknowns - p. 176 63. 7.10 Displacement equations - p. 176 64. 7.11 Equilibrium of a pre-stressed plate - p. 180 65. 7.12 Plate buckling problem - p. 183 66. 7.13 Example: Buckling of a simply-supported rectangular plate - p. 184 67. 7.14 Example: Buckling of a circular plate - p. 188 68. Appendix: Some Mechanical Relations in 3D Curvilinear Coordinates - p. 195 69. Bibliography - p. 199 70. Index - p. 201
