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Summary
Summary
This book by the late R D Mindlin is destined to become a classic introduction to the mathematical aspects of two-dimensional theories of elastic plates. It systematically derives the two-dimensional theories of anisotropic elastic plates from the variational formulation of the three-dimensional theory of elasticity by power series expansions. The uniqueness of two-dimensional problems is also examined from the variational viewpoint. The accuracy of the two-dimensional equations is judged by comparing the dispersion relations of the waves that the two-dimensional theories can describe with prediction from the three-dimensional theory. Discussing mainly high-frequency dynamic problems, it is also useful in traditional applications in structural engineering as well as provides the theoretical foundation for acoustic wave devices.
Table of Contents
Foreword | p. vii |
Preface | p. xi |
Chapter 1 Elements of the Linear Theory of Elasticity | p. 1 |
1.01 Notation | p. 1 |
1.02 Principle of Conservation of Energy | p. 6 |
1.03 Hooke's Law | p. 7 |
1.04 Constants of Elasticity | p. 10 |
1.05 Uniqueness of Solutions | p. 15 |
1.06 Variational Equation of Motion | p. 19 |
1.07 Displacement-Equations of Motion | p. 20 |
Chapter 2 Solutions of the Three-Dimensional Equations | p. 23 |
2.01 Introductory | p. 23 |
2.02 Simple Thickness-Modes in an Infinite Plate | p. 23 |
2.03 Simple Thickness-Modes in an Infinite, Isotropic Plate | p. 25 |
2.04 Simple Thickness-Modes in an Infinite, Monoclinic Plate | p. 29 |
2.05 Simple Thickness-Modes in an Infinite, Triclinic Plate | p. 33 |
2.06 Plane Strain in an Isotropic Body | p. 34 |
2.07 Equivoluminal Modes | p. 35 |
2.08 Wave-Nature of Equivoluminal Modes | p. 38 |
2.09 Infinite, Isotropic Plate Held between Smooth, Rigid Surfaces (Plane Strain) | p. 42 |
2.10 Infinite, Isotropic Plate Held between Smooth, Elastic Surfaces (Plane Strain) | p. 48 |
2.11 Coupled Dilatational and Equivoluminal Modes in an Infinite, Isotropic Plate with Free Faces (Plane Strain) | p. 53 |
2.12 Three-Dimensional, Coupled Dilatational and Equivoluminal Modes in an Infinite, Isotropic Plate with Free Faces | p. 73 |
2.13 Solutions in Cylindrical Coordinates | p. 75 |
2.14 Additional Boundaries | p. 77 |
Chapter 3 Infinite Power Series of Two-Dimensional Equations | p. 79 |
3.01 Introductory | p. 79 |
3.02 Stress-Equations of Motion | p. 81 |
3.03 Strain | p. 86 |
3.04 Stress-Strain Relations | p. 90 |
3.05 Strain-Energy and Kinetic Energy | p. 91 |
3.06 Uniqueness of Solutions | p. 94 |
3.07 Plane Tensors | p. 98 |
Chapter 4 Zero-Order Approximation | p. 101 |
4.01 Separation of Zero-Order Terms from Series | p. 101 |
4.02 Uniqueness of Solutions | p. 105 |
4.03 Stress-Strain Relations | p. 108 |
4.04 Displacement-Equations of Motion | p. 110 |
4.05 Useful Range of Zero-Order Approximation | p. 112 |
Chapter 5 First-Order Approximation | p. 115 |
5.01 Separation of Zero-and First-Order Terms from Series | p. 115 |
5.02 Adjustment of Upper Modes | p. 121 |
5.03 Uniqueness of Solutions | p. 127 |
5.04 Stress-Strain Relations | p. 129 |
5.05 Stress-Displacement Relations | p. 133 |
5.06 Displacement-Equations of Motion | p. 137 |
5.07 Useful Range of First-Order Approximation | p. 145 |
Chapter 6 Intermediate Approximations | p. 153 |
6.01 Introductory | p. 153 |
6.02 Thickness-Shear, Thickness-Flexure and Face-Extension | p. 154 |
6.03 Thickness-Shear and Thickness-Flexure | p. 161 |
6.04 Classical Theory of Low-Frequency Vibrations of Thin Plates | p. 164 |
6.05 Moderately-High-Frequency Vibrations of Thin Plates | p. 171 |
References | p. 175 |
Appendix Applications of the First-Order Approximation | p. 179 |
Biographical Sketch of R. D. Mindlin | p. 181 |
Students of R. D. Mindlin | p. 184 |
Presidential Medal for Merit | p. 186 |
National Medal of Science | p. 187 |
Handwritten Equations from the 1955 Monograph | p. 188 |
Index | p. 189 |