Table Of ContentWave Propagation Analysis of
Smart Nanostructures
Wave Propagation Analysis of
Smart Nanostructures
Farzad Ebrahimi and Ali Dabbagh
CRC Press
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To my parents
Farzad Ebrahimi
To my parents and my sisters
Ali Dabbagh
Contents
Preface xiii
Acknowledgments xv
Authors xvii
1 An Introduction to Wave Theory and Propagation Analysis 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Practical Applications of Waves . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Wave Propagation Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.1 Beam-Type Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.1.1 Classical Beams’ Solution . . . . . . . . . . . . . . . . . . . 4
1.3.1.2 Shear Deformable Beams’ Solution . . . . . . . . . . . . . . 4
1.3.2 Plate-Type Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.2.1 Classical Plates’ Solution . . . . . . . . . . . . . . . . . . . 5
1.3.2.2 Shear Deformable Plates’ Solution . . . . . . . . . . . . . . 6
2 An Introduction to Nonlocal Elasticity Theories and
Scale-Dependent Analysis in Nanostructures 7
2.1 Size Dependency: Fundamentals and Literature Review . . . . . . . . . . . 7
2.2 Mathematical Formulation of the Nonlocal Elasticity . . . . . . . . . . . . 9
2.2.1 Constitutive Equation for Linear Elastic Solids . . . . . . . . . . . . 9
2.2.2 Constitutive Equations of Piezoelectric Materials . . . . . . . . . . . 9
2.2.3 Constitutive Equations of Magnetoelectroelastic (MEE)
Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Mathematical Formulation of the Nonlocal Strain Gradient Elasticity . . . 11
2.3.1 Constitutive Equation for Linear Elastic Solids . . . . . . . . . . . . 11
2.3.2 Constitutive Equations of Piezoelectric Materials . . . . . . . . . . . 12
2.3.3 Constitutive Equations of MEE Materials . . . . . . . . . . . . . . . 12
3 Size-Dependent Effects on Wave Propagation in Nanostructures 13
3.1 Importance of Wave Dispersion in Nanostructures . . . . . . . . . . . . . . 13
3.2 Wave Dispersion in Smart Nanodevices . . . . . . . . . . . . . . . . . . . . 14
3.3 Crucial Parameters in Accurate Approximation of the Wave
Propagation Responses in Nanostructures . . . . . . . . . . . . . . . . . . . 15
4 Wave Propagation Characteristics of Inhomogeneous Nanostructures 17
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.1.1 Functionally Graded Materials (FGMs) . . . . . . . . . . . . . . . . 17
4.1.2 FG Nanostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 Homogenization of FGMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2.1 Power-Law Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
vii
viii Contents
4.2.2 Mori–Tanaka Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3 Analysis of FG Nanobeams . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.3.1 Kinematic Relations of Beams . . . . . . . . . . . . . . . . . . . . . 20
4.3.1.1 Euler–Bernoulli Beam Theory . . . . . . . . . . . . . . . . 20
4.3.1.2 Refined Sinusoidal Beam Theory . . . . . . . . . . . . . . . 20
4.3.2 Derivation of the Equations of Motion for Beams . . . . . . . . . . . 21
4.3.2.1 Equations of Motion of Euler–Bernoulli Beams . . . . . . . 22
4.3.2.2 Equations of Motion of Refined Sinusoidal Beams . . . . . 22
4.3.3 Constitutive Equations of FG Nanobeams . . . . . . . . . . . . . . . 23
4.3.4 The Nonlocal Governing Equations of FG Nanobeams . . . . . . . . 24
4.3.5 Wave Solution for FG Nanobeams . . . . . . . . . . . . . . . . . . . 25
4.3.5.1 Solution of Euler–Bernoulli FG Nanobeams . . . . . . . . . 25
4.3.5.2 Solution of Refined Sinusoidal FG Nanobeams . . . . . . . 25
4.3.6 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . 26
4.4 Analysis of FG Nanoplates . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.4.1 Kinematic Relations of Plates . . . . . . . . . . . . . . . . . . . . . . 31
4.4.1.1 Classical Plate Theory. . . . . . . . . . . . . . . . . . . . . 31
4.4.1.2 Refined Sinusoidal Plate Theory . . . . . . . . . . . . . . . 32
4.4.2 Derivation of the Equations of Motion for Plates . . . . . . . . . . . 33
4.4.2.1 Equations of Motion for Classical Plates. . . . . . . . . . . 33
4.4.2.2 Equations of Motion for Refined Sinusoidal Plates . . . . . 34
4.4.3 Constitutive Equations of FG Nanoplates . . . . . . . . . . . . . . . 35
4.4.4 The Nonlocal Governing Equations of FG Nanoplates . . . . . . . . 37
4.4.5 Wave Solution for FG Nanoplates . . . . . . . . . . . . . . . . . . . 38
4.4.5.1 Solution of Classical FG Nanoplates . . . . . . . . . . . . . 39
4.4.5.2 Solution of Refined Sinusoidal FG Nanoplates . . . . . . . 39
4.4.6 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . 40
5 Porosity Effects on Wave Propagation Characteristics of
Inhomogeneous Nanostructures 47
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.1.1 Porous FGM Structures . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.1.2 Porous FGM Nanostructures . . . . . . . . . . . . . . . . . . . . . . 48
5.2 Homogenization of Porous FGMs . . . . . . . . . . . . . . . . . . . . . . . 48
5.2.1 Modified Power-Law Porous Model . . . . . . . . . . . . . . . . . . . 48
5.2.2 Coupled Elastic–Kinetic Porous Model . . . . . . . . . . . . . . . . . 49
5.3 Wave Propagation in Porous FG Nanostructures . . . . . . . . . . . . . . . 52
5.3.1 Analysis of Porous FG Nanobeams . . . . . . . . . . . . . . . . . . . 52
5.3.2 Analysis of Porous FG Nanoplates . . . . . . . . . . . . . . . . . . . 53
6 Wave Propagation Analysis of Smart Heterogeneous
Piezoelectric Nanostructures 59
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.2 Analysis of Piezoelectric FG Nanobeams . . . . . . . . . . . . . . . . . . . 61
6.2.1 Euler–Bernoulli Piezoelectric Nanobeams . . . . . . . . . . . . . . . 61
6.2.1.1 Motion Equations of Piezoelectric Euler–Bernoulli
Beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.2.1.2 NonlocalStrainGradientPiezoelectricityfor
Euler–Bernoulli Nanobeams . . . . . . . . . . . . . . . . . 63
6.2.1.3 GoverningEquationsofPiezoelectricEuler–Bernoulli
Nanobeams . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Contents ix
6.2.1.4 WaveSolutionoftheEuler–BernoulliPiezoelectric
Nanobeams . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.2.2 Refined Sinusoidal Piezoelectric Nanobeams . . . . . . . . . . . . . . 65
6.2.2.1 MotionEquationsofPiezoelectricRefinedShear
Deformable Beams . . . . . . . . . . . . . . . . . . . . . . . 65
6.2.2.2 Nonlocal Strain Gradient Piezoelectricity for Refined
Sinusoidal Nanobeams . . . . . . . . . . . . . . . . . . . . . 66
6.2.2.3 Governing Equations of Piezoelectric Refined Sinusoidal
Nanobeams . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.2.2.4 Wave Solution of the Refined Piezoelectric Nanobeams . . 67
6.2.3 Numerical Results for Piezoelectric Nanobeams . . . . . . . . . . . . 68
6.3 Analysis of FG Piezoelectric Nanoplates . . . . . . . . . . . . . . . . . . . . 72
6.3.1 Classical Piezoelectric Nanoplates . . . . . . . . . . . . . . . . . . . 73
6.3.1.1 MotionEquationsofPiezoelectricKirchhoff–Love
Nanoplates . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.3.1.2 NonlocalStrainGradientPiezoelectricityforKirchhoff–Love
Nanoplates . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.3.1.3 GoverningEquationsofPiezoelectricKirchhoff–Love
Nanoplates . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.3.1.4 WaveSolutionfortheKirchhoff–LovePiezoelectric
Nanoplates . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.3.2 Refined Sinusoidal Piezoelectric Nanoplates . . . . . . . . . . . . . . 77
6.3.2.1 Equations of Motion of Piezoelectric Refined Sinusoidal
Nanoplates . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.3.2.2 Nonlocal Strain Gradient Piezoelectricity for Refined
Sinusoidal Nanoplates . . . . . . . . . . . . . . . . . . . . . 78
6.3.2.3 Governing Equations of Piezoelectric Refined Sinusoidal
Nanoplates . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.3.2.4 Wave Solution for the Refined Sinusoidal Piezoelectric
Nanoplates . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.3.3 Numerical Results for Piezoelectric Nanoplates . . . . . . . . . . . . 82
7 Wave Dispersion Characteristics of Magnetostrictive Nanostructures 89
7.1 Magnetostriction and Magnetostrictive Materials . . . . . . . . . . . . . . . 89
7.2 Velocity Feedback Control System . . . . . . . . . . . . . . . . . . . . . . . 90
7.3 Constitutive Equations of Magnetostrictive Nanostructures . . . . . . . . . 91
7.4 Derivation of the Governing Equations . . . . . . . . . . . . . . . . . . . . 95
7.4.1 Governing Equations of Magnetostrictive Nanobeams . . . . . . . . 95
7.4.2 Governing Equations of Magnetostrictive Nanoplates . . . . . . . . . 96
7.5 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.5.1 Arrays of Damping Matrix of Nanobeams . . . . . . . . . . . . . . . 100
7.5.2 Arrays of Damping Matrix of Nanoplates . . . . . . . . . . . . . . . 100
7.6 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 101
8 Wave Propagation Analysis of Magnetoelectroelastic Heterogeneous
Nanostructures 105
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.2 Analysis of MEE-FG Nanobeams . . . . . . . . . . . . . . . . . . . . . . . 107
8.2.1 Euler–Bernoulli MEE Nanobeams . . . . . . . . . . . . . . . . . . . 107
8.2.1.1 Motion Equations of MEE Euler–Bernoulli Beams . . . . . 107
