An introduction to the mathematical theory of vibrations of elastic plates

R. D. Mindlin, Jiashi Yang9812703810, 9789812703811

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.

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Table of contents :
Contents……Page 18
Foreword……Page 8
Preface……Page 12
1.01 Notation……Page 21
1.02 Principle of Conservation of Energy……Page 26
1.03 Hooke’s Law……Page 27
1.04 Constants of Elasticity……Page 30
1.05 Uniqueness of Solutions……Page 35
1.06 Variational Equation of Motion……Page 39
1.07 Displacement-Equations of Motion……Page 40
2.02 Simple Thickness-Modes in an Infinite Plate……Page 43
2.03 Simple Thickness-Modes in an Infinite, Isotropic Plate……Page 45
2.04 Simple Thickness-Modes in an Infinite, Monoclinic Plate……Page 49
2.05 Simple Thickness-Modes in an Infinite, Triclinic Plate……Page 53
2.06 Plane Strain in an Isotropic Body……Page 54
2.07 Equivoluminal Modes……Page 55
2.08 Wave-Nature of Equivoluminal Modes……Page 58
2.09 Infinite, Isotropic Plate Held between Smooth, Rigid Surfaces (Plane Strain)……Page 62
2.10 Infinite, Isotropic Plate Held between Smooth, Elastic Surfaces (Plane Strain)……Page 68
2.11 Coupled Dilatational and Equivoluminal Modes in an Infinite, Isotropic Plate with Free Faces (Plane Strain)……Page 73
2.12 Three-Dimensional Coupled Dilatational and Equivoluminal Modes in an Infinite Isotropic Plate with Free Faces……Page 93
2.13 Solutions in Cylindrical Coordinates……Page 95
2.14 Additional Boundaries……Page 97
3.01 Introductory……Page 99
3.02 Stress-Equations of Motion……Page 101
3.03 Strain……Page 106
3.04 Stress-Strain Relations……Page 110
3.05 Strain-Energy and Kinetic Energy……Page 111
3.06 Uniqueness of Solutions……Page 114
3.07 Plane Tensors……Page 118
4.01 Separation of Zero-Order Terms from Series……Page 121
4.02 Uniqueness of Solutions……Page 125
4.03 Stress-Strain Relations……Page 128
4.04 Displacement-Equations of Motion……Page 130
4.05 Useful Range of Zero-Order Approximation……Page 132
5.01 Separation of Zero- and First-Order Terms from Series……Page 135
5.02 Adjustment of Upper Modes……Page 141
5.03 Uniqueness of Solutions……Page 147
5.04 Stress-Strain Relations……Page 149
5.05 Stress-Displacement Relations……Page 153
5.06 Displacement-Equations of Motion……Page 157
5.07 Useful Range of First-Order Approximation……Page 165
6.01 Introductory……Page 173
6.02 Thickness-Shear, Thickness-Flexure and Face-Extension……Page 174
6.03 Thickness-Shear and Thickness-Flexure……Page 181
6.04 Classical Theory of Low-Frequency Vibrations of Thin Plates……Page 184
6.05 Moderately-High-Frequency Vibrations of Thin Plates……Page 191
References……Page 195
Appendix Applications of the First-Order Approximation……Page 199
Biographical Sketch of R. D. Mindlin……Page 201
Students of R. D. Mindlin……Page 204
Presidential Medal for Merit……Page 206
National Medal of Science……Page 207
Handwritten Equations from the 1955 Monograph……Page 208
Index……Page 209