STATIC AND DYNAMIC PROBLEMS OF NANOBEAMS AND NANOPLATES

by Snehashish Chakraverty, Laxmi Behera

Overview

This book will deal with different sections associated with bending, buckling and vibration of nanobeams and nanoplates along with systematic description of handling the complexities when nanoscales are considered. The introduction includes basic ideas concerned with nanostructures, the algorithms and iterations followed in numerical methods and introduction to beam and plate theories in conjunction with nonlocal elasticity theory applied in nanostructures.Next, the investigation of nanobeams and nanoplates subjected to different sets of boundary conditions based on various nonlocal theories will be included. The varieties of physical and geometrical parameters that influence the bending, buckling and vibration mechanisms will be summarized.Finally, effect of environments such as thermal environment, Winkler-Pasternak elastic foundations and non-uniformity etc. on the buckling and vibration mechanisms will be illustrated.

Product Details

ISBN-13:	9789813143937
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	10/14/2016
Sold by:	Barnes & Noble
Format:	eBook
Pages:	212
File size:	20 MB
Note:	This product may take a few minutes to download.

Table of Contents

Preface v

About the Authors ix

1 Introduction 1

1.1 Preliminaries of Nanomaterials and Structural Members 1

1.1.1 Review of nonlocal elasticity theory 4

1.2 Overview of Beam Theories 5

1.2.1 Beam theories 6

1.3 Overview of Plate Theories 12

1.3.1 Plate theory 13

2 Analytical Methods 17

2.1 Euler-Bernoulli Beam Theory (EBT) 18

2.2 Timoshenko Beam Theory (TBT) 19

2.3 Solution using Navier's Approach 20

3 Numerical Methods 21

3.1 Rayleigh-Ritz Method 21

3.1.1 Bending problems 21

3.1.2 Vibration problems 25

3.2 Plate Theory 27

3.2.1 Classical plate theory (CPT) 28

3.3 Differential Quadrature Method (DQM) 32

3.3.1 Buckling problems 33

3.3.2 Euler-Bernoulli beam theory (EBT) 33

3.3.3 Timoshenko beam theory (TBT) 33

3.3.4 Reddy-Bickford beam theory (RBT) 33

3.3.5 Levinson beam theory (LBT) 34

3.3.6 Vibration problems 34

3.3.7 Euler-Bernoulli beam theory (EBT) 35

3.3.8 Timoshenko beam theory (TBT) 35

3.3.9 Reddy-Bickford beam theory (RBT) 35

3.3.10 Levinson beam theory (LBT) 36

4 Bending of Nanobeams 43

4.1 Numerical Results and Discussions 44

4.1.1 Effect of aspect ratio 45

4.1.2 Effect of scale coefficient 46

4.1.3 Effect of boundary conditions 47

4.1.4 Deflection and rotation shapes 47

4.2 Conclusions 51

5 Buckling of Nanobeams 53

5.1 Numerical Results and Discussions 55

5.1.1 Convergence 55

5.1.2 Validation 55

5.1.3 Effect of small scale 57

5.1.4 Effect of non-uniform parameter 60

5.1.5 Effect of aspect ratio 61

5.1.6 Effect of various beam theories 64

5.1.7 Effect of boundary condition 64

5.2 Numerical Results and Discussions 72

5.2.1 Convergence 72

5.2.2 Validation 73

5.2.3 Effect of Winkler modulus parameter 74

5.2.4 Effect of Pasternak shear modulus parameter 77

5.2.5 Effect of temperature 79

5.2.6 Effect of aspect ratio 82

5.3 Conclusions 83

6 Vibration of Nanobeams 85

6.1 Vibration of Nanobeams using Rayleigh-Ritz Method 85

6.2 Numerical Results and Discussions 87

6.2.1 Vibration of nanobeams using DQM 94

6.3 Numerical Results and Discussions 96

6.3.1 Convergence 96

6.3.2 Validation 96

6.3.3 Effect of nonlocal parameter 98

6.3.4 Effect of various beam theories 105

6.3.5 Effect of boundary conditions 107

6.3.6 Effect of aspect ratio (L/h) 108

6.4 Conclusions 109

7 Vibration of Nanobeams with Complicating Effects 111

7.1 Vibration of Nanobeams with Non-uniform Material Properties 111

7.2 Numerical Results and Discussions 113

7.2.1 Convergence of the method 113

7.2.2 Validation 114

7.2.3 Effect of non-uniform parameter 115

7.2.4 Effect of small scale parameter 123

7.2.5 Effect of boundary condition 126

7.2.6 Effect of aspect ratio 128

7.2.7 Mode shapes 129

7.2.8 Vibration analysis of nanobeams embedded in elastic foundations 129

7.3 Euler-Bemoulli Beam Theory (EBT) 131

7.4 Timoshenko Beam Theory (TBT) 132

7.5 Reddy-Bickford Beam Theory (RBT) 135

7.6 Numerical Results and Discussions 137

7.6.1 Convergence 137

7.6.2 Validation 138

7.6.3 Effect of Winkler modulus parameter 140

7.6.4 Effect of Pasternak shear modulus parameter 142

7.6.5 Effect of temperature 145

7.6.6 Effect of aspect ratio 147

7.7 Conclusions 148

8 Bending and Buckling of Nanoplates 151

8.1 Bending of Nanoplates 151

8.2 Numerical Results and Discussions 152

8.2.1 Convergence 152

8.2.2 Validation 153

8.2.3 Effect of aspect ratio 154

8.2.4 Effect of length 154

8.2.5 Effect of nonlocal parameter 155

8.3 Conclusions 156

9 Vibration of Nanoplates 157

9.1 Numerical Results and Discussions 158

9.1.1 Convergence 158

9.1.2 Validation 159

9.1.3 Frequency parameters for different boundary conditions 160

9.1.4 Effect of aspect ratio 160

9.1.5 Effect of nonlocal parameter 163

9.1.6 Effect of length 164

9.1.7 Mode shapes 166

9.2 Conclusions 167

10 Vibration of Nanoplates with Complicating Effects 169

10.1 Numerical Results and Discussions 171

10.1.1 Convergence 171

10.1.2 Validation 172

10.1.3 Effect of non-uniform parameter 174

10.1.4 Effect of length 176

10.1.5 Effect of aspect ratio 180

10.1.6 Effect of nonlocal parameter 181

10.1.7 Effect of elastic foundation 182

10.1.8 Mode shapes 184

10.2 Conclusion 184

Bibliography 185

Index 193