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 xiChapter 1. Introduction to Non-Local Elasticity 11.1. Why the non-local elasticity method for nanostructures? 11.2. General modeling of nanostructures  31.3. Overview of popular nanostructures  41.4. Popular approaches for understanding nanostructures  81.5. Experimental methods  91.6. Molecular dynamics simulations  91.7. Continuum mechanics approach  91.8. Failure of classical continuum mechanics 101.9. Size effects in properties of small-scale structures  111.10. Evolution of size-dependent continuum theories  121.11. Concept of non-local elasticity 141.12. Mathematical formulation of non-local elasticity  151.12.1. Integral form 151.12.2. Non-local modulus 171.12.3. Differential form equation of non-local elasticity 171.13. Non-local parameter 181.14. Non-local elasticity theory versus molecular dynamics  19Chapter 2. Non-local Elastic Rod Theory  212.1.Background 212.2. Governing equation of motion of the nanorod  242.3.Results and discussions  29Chapter 3. Non-local Elastic Beam Theories 333.1. Background 333.2. Non-local nanobeam model 363.2.1. Non-local Euler–Bernoulli beam theory 363.2.2. Non-local Timoshenko beam theory  433.2.3. Non-local Reddy beam theory 513.3. Torsional vibration of nanobeam  603.4. Comparison of the non-local beam theories  64Chapter 4. Non-local Elastic Plate Theories  694.1. Non-local plate for graphene sheets  694.2. Non-local plate constitutive relations 694.3. Free vibration of single-layer graphene sheets 724.3.1. Transverse-free vibration  734.3.2. Graphene sheets embedded in an elastic medium  754.4. Axially stressed nanoplate non-local theory  784.5. In-plane vibration 794.6. Buckling of graphene sheets 804.6.1. Uniaxial buckling 814.6.2. Graphene sheets embedded in an elastic medium  824.7. Summary  84Chapter 5. One-Dimensional Double-Nanostructure-Systems 875.1. Background 875.2. Revisiting non-local rod theory 905.2.1. Equations of motion of double-nanorod-system 915.2.2. Solution methodology  945.2.3. Clamped-clamped boundary condition  955.2.4. Clamped-free (cantilever) boundary condition  965.2.5. Longitudinal vibration of auxiliary (secondary) nanorod 985.3. Axial vibration of double-rod system 995.3.1. Effect of the non-local parameter in the clamped-type DNRS 1005.3.2. Coupling spring stiffness in DNRS  1025.3.3. Higher modes of vibration in DNRS 1025.3.4. Effect of non-local parameter, spring stiffness and higher modes in cantilever-type-DNRS  1035.4. Summary  1045.5. Transverse vibration of double-nanobeam-systems 1045.5.1. Background  1055.5.2. Non-local double-nanobeam-system 1075.6. Vibration of non-local double-nanobeam-system  1105.7. Boundary conditions in non-local double-nanobeam-system  1115.8. Exact solutions of the frequency equations  1135.9. Discussions 1165.9.1. Effect of small scale on vibrating NDNBS  1175.9.2. Effect of the stiffness of the coupling springs on NDNBS 1205.9.3. Analysis of higher modes of NDNBS 1205.10. Summary 1215.11. Axial instability of double-nanobeam-systems 1225.11.1. Background 1235.11.2. Buckling equations of non-local doublenanobeam-systems  1245.12. Non-local boundary conditions of NDNBS 1265.13. Buckling states of double-nanobeam-system  1285.13.1. Out-of-phase buckling load: (w1-w20) 1285.13.2. In-phase buckling state: (w1– w2=0)  1295.13.3. One nanobeam is fixed: 1305.14. Coupled carbon nanotube systems  1305.15. Results and discussions on the scale-dependent buckling phenomenon 1315.15. Summary 136Chapter 6. Double-Nanoplate-Systems 1376.1. Double-nanoplate-system  1376.2. Vibration of double-nanoplate-system 1396.3. Equations of motion for non-local doublenanoplate-system  1396.4. Boundary conditions in non-local doublenanoplate-system  1426.5. Exact solutions of the frequency equations  1446.5.1. Both nanoplates of NDNPS are vibrating out-of-phase:   1446.5.2. Both nanoplates of NDNPS are vibrating in-phase: 1466.5.3. One nanoplate of NDNPS is stationary:   1476.5.4. Discussions  1486.5.5. Non-local double-nanobeam-system versus non-local double-nanoplate-system  1566.5.6. Summary  1576.6. Buckling behavior of double-nanoplate-systems 1586.6.1. Background  1596.6.2. Uniaxially compressed double-nanoplate-system  1606.6.3. Buckling states of double-nanoplate-system 1636.7. Results and discussion  1676.7.1. Coupled double-graphene-sheet-system 1676.7.2. Effect of small scale on NDNPS undergoing compression  1686.7.3. Effect of stiffness of coupling springs in NDNPS 1706.7.4. Effect of aspect ratio on NDNPS  1736.8. Summary  177Chapter 7. Multiple Nanostructure Systems  1797.1. Longitudinal vibration of a multi-nanorod system  1807.1.1. The governing equations of motion  1827.1.2. Exact solution 1857.1.3. Asymptotic analysis 1917.1.4. Numerical examples and discussions 1927.2. Transversal vibration and stability of a multiplenanobeam system  1977.2.1. The governing equations of motion  1997.2.2. Exact solution 2027.2.3. Asymptotic analysis 2097.2.4. Numerical examples and discussions 2107.3. Transversal vibration and buckling of the multinanoplate system  2157.3.1. The governing equations of motion  2177.3.2. Exact solutions  2217.3.3 Asymptotic analysis  2277.3.4. Numerical results and discussions 2277.4. Summary  232Chapter 8. Finite Element Method for Dynamics of Nonlocal Systems  2358.1. Introduction 2368.2. Finite element modeling of non-local dynamic systems  2398.2.1. Axial vibration of nanorods 2398.2.2. Bending vibration of nanobeams  2418.2.3. Transverse vibration of nanoplates 2438.3. Modal analysis of non-local dynamical systems 2478.3.1. Conditions for classical normal modes  2488.3.2. Non-local normal modes  2508.3.3. Approximate non-local normal modes  2518.4. Dynamics of damped non-local systems  2548.5. Numerical examples  2568.5.1. Axial vibration of a single-walled carbon nanotube 2568.5.2. Bending vibration of a double-walled carbon nanotube 2618.5.3. Transverse vibration of a single-layer graphene sheet  2658.6. Summary  269Chapter 9. Dynamic Finite Element Analysis of Nonlocal Rods: Axial Vibration 2719.1. Introduction 2729.2. Axial vibration of damped non-local rods 2759.2.1. Equation of motion  2759.2.2. Analysis of damped natural frequencies 2779.2.3. Asymptotic analysis of natural frequencies  2799.3. Dynamic finite element matrix 2819.3.1. Classical finite element of non-local rods 2819.3.2. Dynamic finite element for damped non-local rod 2829.4. Numerical results and discussions 2859.5. Summary  291Chapter 10. Non-local Nanosensor Based on Vibrating Graphene Sheets 29310.1. Introduction  29410.2. Free vibration of graphene sheets 29510.2.1. Vibration of SLGS without attached mass 29710.3. Natural vibration of SLGS with biofragment  29910.3.1. Attached masses are at the cantilever tip  30110.3.2. Attached masses arranged in a line along the width  30110.3.3. Attached masses arranged in a line along the length  30210.3.4. Attached masses arranged with arbitrary angle 30210.4. Sensor equations and sensitivity analysis  30310.5. Analysis of numerical results 30510.6. Summary 311Chapter 11. Introduction to Molecular Dynamics for Small-scale Structures  31311.1. Background  31311.2. Overview of the molecular dynamics simulation method 31411.3. Acknowledgement  325Bibliography  327Index 353