An introduction to the mathematical theory of vibrations of elastic plates

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.

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

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