Vibrations of Elastic Plates: Linear and Nonlinear Dynamical Modeling of Sandwiches, Laminated Composites, and Piezoelectric Layers

This book deals with the dynamical modeling of thin elastic structures, such as beams, plates and shells - particularly, the linear and nonlinear vibrations in these structures. The approach makes systematic use of variational equations of motion.

Table of Contents:
1 Nonlinear Elasticity Theory.- 1.1 Strains.- 1.2 Stresses.- 1.3 Strain Energy Function and Principle of Virtual Work.- 1.4 Hamilton’s Principle and Variational Equations of Motion.- 1.5 Pseudo-Variational Equations of Motion.- 1.6 Generalized Hamilton’s Principle and Variational Equation of Motion.- 1.7 Stress-Strain Relations in Nonlinear Elasticity.- References.- 2 Linear Vibrations of Plates Based on Elasticity Theory.- 2.1 Equations of Linear Elasticity Theory.- 2.2 Rayleigh-Lamb Solution for Plane-Strain Modes of Vibration in an Infinite Plate.- 2.3 Simple Thickness Modes in an Infinite Plate.- 2.4 Horizontal Shear Modes in an Infinite Plate.- 2.5 Modes in an Infinite Plate Involving Phase Reversals in Both x-and y-Directions.- 2.6 Plane-Strain Modes in an Infinite Sandwich Plate.- 2.7 Simple Thickness Modes in an Infinite Sandwich Plate.- References.- 3 Linear Modeling of Homogeneous Plates.- 3.1 Classical Equations for Flexure of a Homogeneous Plate.- 3.2 Refined Equations for Flexure of an Isotropic Plate: Mindlin Plate Equations and Timoshenko Beam Equations.- 3.3 Classical Equations for Extension of an Isotropic Plate.- 3.4 Refined Equations for Extension of an Isotropic Plate.- 3.5 Vibrations of an Infinite Plate: Useful Ranges of Plate Equations.- 3.6 General Equations of an Anisotropic Plate.- References.- 4 Linear Modeling of Sandwich Plates.- 4.1 Refined Equations for Flexure of a Sandwich Plate Including Transverse Shear Effects in All Layers.- 4.2 Simplified Refined Equations for Flexure of a Sandwich Plate with Membrane Facings.- 4.3 Classical Equations for Flexure of a Sandwich Plate.- 4.4 Flexural Vibration of an Infinite Sandwich Plate: Useful Ranges of Sandwich Plate Equations.- 4.5 Extensional Vibration of an Infinite Sandwich Plate Based onClassical Equations.- References.- 5 Linear Modeling of Laminated Composite Plates.- 5.1 Classical Equations of a Laminated Composite Plate.- 5.2 Refined Equations of a Laminated Composite Plate.- 5.3 Flexural Vibration of a Symmetric Laminate: Useful Ranges of Equations.- 5.4 Extensional Vibration of a Symmetric Laminate: Useful Ranges of Equations.- References.- 6 Linear Vibrations Based on Plate Equations.- 6.1 Free Flexural Vibration of Plates with Simply Supported Edges.- 6.2 Free Flexural Vibration of Plates with Clamped Edges.- 6.3 Forced Flexural Vibration of Homogeneous and Sandwich Plates in Plane Strain.- References.- 7 Nonlinear Modeling for Large Deflections of Beams, Plates, and Shallow Shells.- 7.1 Equations for Large Deflections of a Buckled Timoshenko Beam.- 7.2 Von Kármán Equations for Large Deflections of a Plate: Incorporation of Transverse Shear Effect.- 7.3 Marguerre Equations for Large Deflections of a Shallow Shell: Incorporation of Transverse Shear Effect.- 7.4 Remarks on the Variational Equations of Motion.- References.- 8 Nonlinear Modeling and Vibrations of Sandwiches and Laminated Composites.- 8.1 Equations for Large Deflections of a Sandwich Plate.- 8.2 Nonlinear Vibration of a Sandwich Plate.- 8.3 Equations for Large Deflections of a Laminated Composite Plate.- 8.4 Nonlinear Vibration of an Orthotropic Symmetric Laminate.- 8.5 Equations for Large Deflections of a Sandwich Beam with Laminated Composite Facings and an Orthotropic Core.- References.- 9 Chaotic Vibrations of Beams.- 9.1 A Numerical Study of Chaos According to Duffing’s Equation: Effect of Damping.- 9.2 More Poincaré Maps According to Duffing’s Equation for Small Damping.- 9.3 Spectral Analysis of Chaos.- 9.4 Acoustic Radiation from Chaotic Vibrations of a Beam.-References.- 10 Nonlinear Modeling of Piezoelectric Plates.- 10.1 From Elasticity to Peizoelectricity.- 10.2 Generalized Hamilton’s Principle and Variational Equation of Motion Including Piezoelectric Effect.- 10.3 Classical Equations for Large Deflections of a Piezoelectric Plate.- 10.4 Refined Equations for Large Deflections of a Piezoelectric Plate.- 10.5 Final Remarks on the Variational Equations of Motion.- References.