Andrew J. Kurdila - Vibrations of Linear Piezostructures

1 Cover

2 Title Page Vibrations of Linear Piezostructures Andrew J. Kurdila and Pablo A. Tarazaga Virginia Polytechnic Institute and State University

3 Copyright

4 Foreword

5 Preface

6 Acknowledgments

7 List of Symbols

8 1 Introduction1.1 The Piezoelectric Effect 1.2 Applications 1.3 Outline of the Book

9 2 Mathematical Background 2.1 Vectors, Bases, and Frames 2.2 Tensors 2.3 Symmetry, Crystals, and Tensor Invariance 2.4 Problems

10 3 Review of Continuum Mechanics 3.1 Stress 3.2 Displacement and Strain 3.3 Strain Energy 3.4 Constitutive Laws for Linear Elastic Materials 3.5 The Initial‐Boundary Value Problem of Linear Elasticity 3.6 Problems

11 4 Review of Continuum Electrodynamics 4.1 Charge and Current 4.2 The Electric and Magnetic Fields 4.3 Maxwell's Equations 4.4 Problems

12 5 Linear Piezoelectricity 5.1 Constitutive Laws of Linear Piezoelectricity 5.2 The Initial‐Value Boundary Problem of Linear Piezoelectricity 5.3 Thermodynamics of Constitutive Laws 5.4 Symmetry of Constitutive Laws for Linear Piezoelectricity 5.5 Problems

13 6 Newton's Method for Piezoelectric Systems 6.1 An Axial Actuator Model 6.2 An Axial, Linear Potential, Actuator Model 6.3 A Linear Potential, Beam Actuator 6.4 Composite Plate Bending 6.5 Problems

14 7 Variational Methods 7.1 A Review of Variational Calculus 7.2 Hamilton's Principle 7.3 Hamilton's Principle for Piezoelectricity 7.4 Bernoulli–Euler Beam with a Shunt Circuit 7.5 Relationship to other Variational Principles 7.6 Lagrangian Densities 7.7 Problems

15 8 Approximations 8.1 Classical, Strong, and Weak Formulations 8.2 Modeling Damping and Dissipation 8.3 Galerkin Approximations 8.4 Problems

16 Supplementary MaterialS.1 A Review of Vibrations S.2 Tensor Analysis S.3 Distributional and Weak Derivatives

17 Bibliography

18 Index

19 End User License Agreement

1 Chapter 2Table 2.1 The crystal systems and their point groups. See [13] Table A.2 or [...

2 Chapter 6Table 6.1 Boundary conditions for piezoelectric composite beam driven by two vol...

3 Chapter 8Table 8.1 Axial composite piezoelectric actuator, system properties.Table 8.2 A piezoelectric composite beam material and system properties.Table 8.3 Material and system properties for the piezoelectric composite beam wi...

1 Chapter 1 Figure 1.1 Barium titanate and lead zirconate titanate. (Left) Barium titana... Figure 1.2 Polarization versus applied electrical field for ferroelectric ab... Figure 1.3 and domains in , [31]. Figure 1.4 Polarization versus electrical field hysteresis below the Curie t... Figure 1.5 The direct piezoelectric effect. Figure 1.6 The converse piezoelectric effect. Figure 1.7 Piezoelectrically based microphones from PCB®, Figure 1.8 Piezoelectric stack actuators available from PI ceramic®, Source:... Figure 1.9 Piezoelectric bender actuators available from PI ceramic®, Source...

2 Chapter 2 Figure 2.1 Frames generated by basis vectors and , , and their cyclic pe...Figure 2.2 Unit cell and lattice parameters.Figure 2.3 The unit cell of the triclinic unit cell.Figure 2.4 Crystallographic coordinates, directions, and planes.Figure 2.5 The fourteen Bravais lattices, seven crystal systems, and associa...Figure 2.6 Rotoinversion of the point . The rotation of is followed by ...Figure 2.7 Examples of objects with two, three, and fourfold symmetry about ...Figure 2.8 Unit cell of monoclinic crystal system.Figure 2.9 Invariance of the monoclinic lattice with respect to reflection a...Figure 2.10 Invariance of the monoclinic lattice with respect to rotation th...

3 Chapter 3Figure 3.1 (Left) Continuum body having surface and the stress vector ...Figure 3.2 Tetrahedron with surface normal used to derive Cauchy's formula...Figure 3.3 (Left) Differential cube with surface stresses, (Right) All stres...Figure 3.4 Undeformed configuration , deformed configuration , and the def...Figure 3.5 Axial rod geometry, coordinate alignment, and displacementFigure 3.6 Beam geometry, coordinate alignment, and displacementFigure 3.7 Geometry of the thin, rectangular, Kirchoff plateFigure 3.8 Linearly elastic body , applied external stress field , boundar...

4 Chapter 4Figure 4.1 Point charges located at points , respectively, position vecto...Figure 4.2 Wire loops carrying the currents and , vectors and , and di...Figure 4.3 A typical crystal lattice, an asymmetric unit cell, the centers o...Figure 4.4 A dielectric parallel plate capacitor.Figure 4.5 A volume that straddles the top electrodes surface of the volum...Figure 4.6 A planar loop of wire carrying current , having area , and norm...Figure 4.7 A lattice with point charges at each corner.Figure 4.8 A finite wire with uniform linear charge density.Figure 4.9 An infinite duct.Figure 4.10 A nonplanar current carrying wire.