Non-local Structural Mechanics

Danilo Karlicic is a Lecturer at the Mechanical Engineering Faculty at the University of Niš, Serbia. Tony Murmu is a Lecturer of Mechanical Engineering at the University of the West of Scotland, United Kingdom. Sondipon Adhikari is the Chair of Aerospace Engineering at the College of Engineering at Swansea University, United Kingdom. Michael McCarthy is Professor of Aeronautical Engineering at the University of Limerick, Ireland.

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-w20) 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