Preface xi
Chapter 1. Introduction to Non-Local Elasticity 1
1.1. Why the non-local elasticity method for nanostructures? 1
1.2. General modeling of nanostructures 3
1.3. Overview of popular nanostructures 4
1.4. Popular approaches for understanding nanostructures 8
1.5. Experimental methods 9
1.6. Molecular dynamics simulations 9
1.7. Continuum mechanics approach 9
1.8. Failure of classical continuum mechanics 10
1.9. Size effects in properties of small-scale structures 11
1.10. Evolution of size-dependent continuum theories 12
1.11. Concept of non-local elasticity 14
1.12. Mathematical formulation of non-local elasticity 15
1.12.1. Integral form 15
1.12.2. Non-local modulus 17
1.12.3. Differential form equation of non-local elasticity 17
1.13. Non-local parameter 18
1.14. Non-local elasticity theory versus molecular dynamics 19
Chapter 2. Non-local Elastic Rod Theory 21
2.1.Background 21
2.2. Governing equation of motion of the nanorod 24
2.3.Results and discussions 29
Chapter 3. Non-local Elastic Beam Theories 33
3.1. Background 33
3.2. Non-local nanobeam model 36
3.2.1. Non-local Euler–Bernoulli beam theory 36
3.2.2. Non-local Timoshenko beam theory 43
3.2.3. Non-local Reddy beam theory 51
3.3. Torsional vibration of nanobeam 60
3.4. Comparison of the non-local beam theories 64
Chapter 4. Non-local Elastic Plate Theories 69
4.1. Non-local plate for graphene sheets 69
4.2. Non-local plate constitutive relations 69
4.3. Free vibration of single-layer graphene sheets 72
4.3.1. Transverse-free vibration 73
4.3.2. Graphene sheets embedded in an elastic medium 75
4.4. Axially stressed nanoplate non-local theory 78
4.5. In-plane vibration 79
4.6. Buckling of graphene sheets 80
4.6.1. Uniaxial buckling 81
4.6.2. Graphene sheets embedded in an elastic medium 82
4.7. Summary 84
Chapter 5. One-Dimensional Double-Nanostructure-Systems 87
5.1. Background 87
5.2. Revisiting non-local rod theory 90
5.2.1. Equations of motion of double-nanorod-system 91
5.2.2. Solution methodology 94
5.2.3. Clamped-clamped boundary condition 95
5.2.4. Clamped-free (cantilever) boundary condition 96
5.2.5. Longitudinal vibration of auxiliary (secondary) nanorod 98
5.3. Axial vibration of double-rod system 99
5.3.1. Effect of the non-local parameter in the clamped-type DNRS 100
5.3.2. Coupling spring stiffness in DNRS 102
5.3.3. Higher modes of vibration in DNRS 102
5.3.4. Effect of non-local parameter, spring stiffness and higher modes in cantilever-type-DNRS 103
5.4. Summary 104
5.5. Transverse vibration of double-nanobeam-systems 104
5.5.1. Background 105
5.5.2. Non-local double-nanobeam-system 107
5.6. Vibration of non-local double-nanobeam-system 110
5.7. Boundary conditions in non-local double-nanobeam-system 111
5.8. Exact solutions of the frequency equations 113
5.9. Discussions 116
5.9.1. Effect of small scale on vibrating NDNBS 117
5.9.2. Effect of the stiffness of the coupling springs on NDNBS 120
5.9.3. Analysis of higher modes of NDNBS 120
5.10. Summary 121
5.11. Axial instability of double-nanobeam-systems 122
5.11.1. Background 123
5.11.2. Buckling equations of non-local doublenanobeam-systems 124
5.12. Non-local boundary conditions of NDNBS 126
5.13. Buckling states of double-nanobeam-system 128
5.13.1. Out-of-phase buckling load: (w1-w2�0) 128
5.13.2. In-phase buckling state: (w1– w2=0) 129
5.13.3. One nanobeam is fixed:??? ? ?? 130
5.14. Coupled carbon nanotube systems 130
5.15. Results and discussions on the scale-dependent buckling phenomenon 131
5.15. Summary 136
Chapter 6. Double-Nanoplate-Systems 137
6.1. Double-nanoplate-system 137
6.2. Vibration of double-nanoplate-system 139
6.3. Equations of motion for non-local doublenanoplate-system 139
6.4. Boundary conditions in non-local doublenanoplate-system 142
6.5. Exact solutions of the frequency equations 144
6.5.1. Both nanoplates of NDNPS are vibrating out-of-phase: ??? ? ?? ? ?? 144
6.5.2. Both nanoplates of NDNPS are vibrating in-phase: ??? ? ?? ? ?? 146
6.5.3. One nanoplate of NDNPS is stationary: ?????? ?? ?? ? ?? 147
6.5.4. Discussions 148
6.5.5. Non-local double-nanobeam-system versus non-local double-nanoplate-system 156
6.5.6. Summary 157
6.6. Buckling behavior of double-nanoplate-systems 158
6.6.1. Background 159
6.6.2. Uniaxially compressed double-nanoplate-system 160
6.6.3. Buckling states of double-nanoplate-system 163
6.7. Results and discussion 167
6.7.1. Coupled double-graphene-sheet-system 167
6.7.2. Effect of small scale on NDNPS undergoing compression 168
6.7.3. Effect of stiffness of coupling springs in NDNPS 170
6.7.4. Effect of aspect ratio on NDNPS 173
6.8. Summary 177
Chapter 7. Multiple Nanostructure Systems 179
7.1. Longitudinal vibration of a multi-nanorod system 180
7.1.1. The governing equations of motion 182
7.1.2. Exact solution 185
7.1.3. Asymptotic analysis 191
7.1.4. Numerical examples and discussions 192
7.2. Transversal vibration and stability of a multiplenanobeam system 197
7.2.1. The governing equations of motion 199
7.2.2. Exact solution 202
7.2.3. Asymptotic analysis 209
7.2.4. Numerical examples and discussions 210
7.3. Transversal vibration and buckling of the multinanoplate system 215
7.3.1. The governing equations of motion 217
7.3.2. Exact solutions 221
7.3.3 Asymptotic analysis 227
7.3.4. Numerical results and discussions 227
7.4. Summary 232
Chapter 8. Finite Element Method for Dynamics of Nonlocal Systems 235
8.1. Introduction 236
8.2. Finite element modeling of non-local dynamic systems 239
8.2.1. Axial vibration of nanorods 239
8.2.2. Bending vibration of nanobeams 241
8.2.3. Transverse vibration of nanoplates 243
8.3. Modal analysis of non-local dynamical systems 247
8.3.1. Conditions for classical normal modes 248
8.3.2. Non-local normal modes 250
8.3.3. Approximate non-local normal modes 251
8.4. Dynamics of damped non-local systems 254
8.5. Numerical examples 256
8.5.1. Axial vibration of a single-walled carbon nanotube 256
8.5.2. Bending vibration of a double-walled carbon nanotube 261
8.5.3. Transverse vibration of a single-layer graphene sheet 265
8.6. Summary 269
Chapter 9. Dynamic Finite Element Analysis of Nonlocal Rods: Axial Vibration 271
9.1. Introduction 272
9.2. Axial vibration of damped non-local rods 275
9.2.1. Equation of motion 275
9.2.2. Analysis of damped natural frequencies 277
9.2.3. Asymptotic analysis of natural frequencies 279
9.3. Dynamic finite element matrix 281
9.3.1. Classical finite element of non-local rods 281
9.3.2. Dynamic finite element for damped non-local rod 282
9.4. Numerical results and discussions 285
9.5. Summary 291
Chapter 10. Non-local Nanosensor Based on Vibrating Graphene Sheets 293
10.1. Introduction 294
10.2. Free vibration of graphene sheets 295
10.2.1. Vibration of SLGS without attached mass 297
10.3. Natural vibration of SLGS with biofragment 299
10.3.1. Attached masses are at the cantilever tip 301
10.3.2. Attached masses arranged in a line along the width 301
10.3.3. Attached masses arranged in a line along the length 302
10.3.4. Attached masses arranged with arbitrary angle 302
10.4. Sensor equations and sensitivity analysis 303
10.5. Analysis of numerical results 305
10.6. Summary 311
Chapter 11. Introduction to Molecular Dynamics for Small-scale Structures 313
11.1. Background 313
11.2. Overview of the molecular dynamics simulation method 314
11.3. Acknowledgement 325
Bibliography 327
Index 353
