Wave Propagation Analysis of Smart Nanostructures

263 Pages·2019·32.071 MB·\263

by  Farzad Ebrahimi (Author), Ali Dabbagh (Author)

Wave 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