Non-local Structural Mechanics

Serving as a review on non-local mechanics, this book provides an introduction to non-local elasticity theory for static, dynamic and stability analysis in a wide range of nanostructures. The authors draw on their own research experience to present fundamental and complex theories that are relevant across a wide range of nanomechanical systems, from the fundamentals of non-local mechanics to the latest research applications.

Auteur

Danilo Karlicic is a Lecturer at the Mechanical Engineering Faculty at the University of Ni, 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.

Contenu
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 EulerBernoulli 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:&#10...