Slides of my talk at IISc Bangalore on nanomechanics and finite element analysis for statics and dynamics of nanoscale structures such as carbon nanotube, graphene, ZnO nanotube and BN nano sheet.

Published on: **Mar 3, 2016**

- 1. Atomistic Mechanics ofNanoscale Structures: Static& Dynamic AnalysesSondipon Adhikari
- 2. Where is Swansea? London – UK Capital Less than three hours by car or train (192 miles) Wales Swansea Cardiff - Welsh Capital Less than an hour away by car or train
- 3. Title ofpresentationClick to edit subtitle style
- 4. Swansea University • 29th UK university to be established • King George V laid the foundation stone of the University in July 1920 • Now over 12,500 students - 1,800 international
- 5. OverviewØ IntroductionØ Atomistic finite element methodØ Carbon nanotubes: static and dynamic analysis, bucklingØ Fullerenes: vibration spectraØ Graphene: static and dynamic analysis, compositesØ Nanobio sensors: vibrating nanotube and graphene based mass sensorØ DNA mechanicsØ Conclusions
- 6. Research Areasu Atomistic finite element methodu Nonlocal continuum mechanics for nanoscale objectsu Nanoscale bio sensorsu Uncertainty quantification in modelling and simulationu Dynamic analysis of complex structuresu Vibration energy harvesting
- 7. Collaborators u Prof F Scarpa (University of Bristol) u Dr R Chowdhury, Dr C Wang, Dr A Gil, Prof P Rees (Swansea University). u Dr T Murmu, Prof M McCarthy (University of Limerick)Acknowledgements
- 8. Carbon NanotubesTitle ofpresentationClick to edit subtitle style
- 9. Can we use continuum mechanics at thenanoslace? u What about the “holes”? u Can we have an “equivalent” continuum model with “correct” properties? u How defects can be taken into account ?
- 10. Which Youngs modulus? Effective Longitudinal Surface F F FY11 = 2 Y11 = s Y11 = πR ε11 2πRd ε11 2πRε11
- 11. Which Youngs modulus?
- 12. Yakobson’s paradox (Wang CY, Zhang LC, 2008. Nanotechnology 19, 075705) (Huang Y, Wu J, Hwang K C, 2006. Phys. Rev. B 74, 245413)
- 13. Atomistic finite element methodu Atomic bonds are represented by beam elementsu Beam properties are obtained by energy balance Utotal = Ur +Uθ +Uτ 1 EA U axial = K axial (ΔL) 2 = (ΔL) 2 2 2L 1 GJ U torsion = K torsion (Δβ ) 2 = (Δβ ) 1 2 1 2 1 2 2 2LUr = kr ( Δr ) Uθ = kθ ( Δθ ) Uτ = kτ ( Δφ ) 2 2 2 1 EI 4 + Φ U bending = K bending (2α ) 2 = (2α ) 2 2 2L 1+ ΦScarpa, F. and Adhikari, S., "A mechanical equivalence for the Poissons ratio and thickness of C-C bonds in single wallcarbon nanotubes", Journal of Physics D: Applied Physics, 41 (2008) 085306
- 14. Atomistic finite element method u All parameters of the beam can be obtained in closed-form: 4kr L 32kτ L E= , G= πd2 πd4 3kr d 4 ( 6 +12ν + 6ν 2 ) Φ= 32kτ L2 ( 7 +12ν + 4ν 2 ) kr d 2 4Α + Β kτ kθ = d<2 6 16 Α + Β kr Α = 112L2 kτ +192L2 kτ ν + 64L2 kτ ν 2 Β = 9kr d 4 +18kr d 4ν + 9kr d 4ν 2Scarpa, F. and Adhikari, S., "A mechanical equivalence for the Poissons ratio and thickness of C-C bonds in single wallcarbon nanotubes", Journal of Physics D: Applied Physics, 41 (2008) 085306
- 15. Atomistic Structural Mechanics For space frames: [K ] {u} = {f } [K] à stiffness matrix {u} à nodal displacement vector {f} à nodal force vector (Weaver Jr., W. and Gere, J.M., 1990. Matrix Analysis ofFramed Structures. (third ed.),, Van Nostrand Reinhold, New York)
- 16. Atomistic FE – bending deformation ofSWCNTs (F Scarpa and S Adhikari, 2008. J. Phys. D: App. Phys., 41, 085306)
- 17. Atomistic FE – bending deformation of SWCNTs bundles Similarity between hexagonal SWCNT packing bundle and structural idealization for wing boxesPolar moment of inertia for each CNT: 4 4 π ⎡⎛d ⎞ ⎛ d ⎞ ⎤ I c = ⎢⎜ R + ⎟ − ⎜ R − ⎟ ⎥ (1) 4 ⎢⎝ ⎣ 2 ⎠ ⎝ 2 ⎠ ⎥ ⎦ Polar moment of inertia for hexagonal packing: 2 ⎛ ⎛ l 02 ⎞ ⎞I hex = 4⎜ I c + ⎜ 3 ⎟ Ac ⎟ + 3I c ⎜ 2 ⎟ (2) ⎜ ⎝ ⎠ ⎟ ⎝ ⎠ ⎡ 20 3πR 3 d 7 3πRd 3 64 3πR 2 tVdW d 16 3πRtVdW d ⎤ 2 Y f = Y ⎢ 4 + 4 + 4 + 4 ⎥ Flexural modulus of the nanbundle: ⎢ 3(2 R + tVdW ) 33(2 R + tVdW ) 11(2R + tVdW ) 11(2R + tVdW ) ⎥ ⎣ ⎦ I hex Yf = Y (3) Ih (F Scarpa and S Adhikari, 2008. J. Phys. D: App. Phys., 41, 085306)
- 18. Atomistic FE – bending deformation ofSWCNTs bundles (F Scarpa and S Adhikari, 2008. J. Phys. D: App. Phys., 41, 085306)
- 19. Buckling of Carbon nanotubes (a) Molecular dynamics (b) Hyperplastic atomistic FE (Ogden strain energy density function )Comparison of bucklingmechanisms in a (5,5) SWCNTwith 5.0 nm length. Flores, E. I. S., Adhikari, S., Friswell, M. I. and Scarpa, F., "Hyperelastic axial buckling of single wall carbon nanotubes", Physica E: Low-dimensional Systems and Nanostructures, 44[2] (2011), pp. 525-529.
- 20. Carbon nanotubes with defectsu We are interested in the changes in the mechanical properties Scarpa, F., Adhikari, S. and Wang, C. Y., "Mechanical properties of non reconstructed defective single wall carbon nanotubes", Journal of Physics D: Applied Physics, 42 (2009) 142002 !
- 21. Carbon nanotubes with defects(a) Ratio between mean of axial Young’s modulus and pristine stiffness and (b)between standard deviation of the Young’s modulus against pristine Young’smodulus for armchair (n,n). Pristine Young’s modulus Y0: 2.9, 1.36, 0.91, 0.67TPa for a thickness d = 0.084 nm. ● = 2 % NRV; ■ = 1.5 % NRV; ▲= 1 % NRV;♦= 0.5 % NRV
- 22. FullereneTitle ofpresentationClick to edit subtitle style
- 23. Vibration spectra of fullerene family The variation of the first natural frequency across the the complete range of fullerenes. The 8 spherical type of fullerenes include C60 , C80 , C180 , C60 , C240 , C260 , C320 , C500 , C720 and the 8 ellipsoidal (or non-spherical) type of fullerenes include C20 , C30 , C40 , C50 , C70 , C90 , C100 and C540 (fullerenes are not drawn in scale). Adhikari, S. and Chowdhury, R., "Vibration spectra of fullerene family", Physics Letters A, 375[22] (2011), pp. 1276-1280.
- 24. Thin shell theoryThe natural frequencies of spherical fullerenes can be given by
- 25. Atomistic Simulation vs Shell theory 30 30 b2 C80 C b2 60 25 a0 25 a0 Frequency (THz)Frequency (THz) 20 a1 20 C260 C180 C20 a2 C240 15 C320 15 C30 C500 C50 10 C 10 740 C90 C40 5 5 C C70 540 C100 0 0 0.01 0.02 0.03 0.04 0 0.02 0.04 0.06 0.08 1/2 1/2 1/M (amu 1/2) 1/2 1/M (amu ) Spherical type fullerenes Ellipsoidal type fullerenes
- 26. GrapheneTitle ofpresentationClick to edit subtitle style
- 27. Atomistic FE – in-plane SLGS (F Scarpa, S Adhikari, A S Phani, 2009. Nanotechnology 20, 065709)
- 28. Atomistic FE vs Continuum – SLGSCircular SLGS (R = 9: 5 nm) Deformation of rectangular SLGSunder central loading. Distribution (15.1 x 13.03 nm2) under centralof equivalent membrane stresses. loading. Scarpa, F., Adhikari, S., Gil, A. J. and Remillat, C., "The bending of single layer graphene sheets: Lattice versus continuum approach", Nanotechnology, 21[12] (2010), pp. 125702:1-9.
- 29. Axtomistic FE vs Continuum – SLGS 35 Lattice R = 2.5 nm 35 Lattice a = 3.88 nm 30 Continuum R = 2.5 nm Continuum a = 3.88 nm Lattice R = 5.0 nm 30 Lattice a = 5.0 nm Continuum R = 5.0 nm Continuum a = 5.0 nm 25 25 Lattice a = 15.1 nm Lattice R = 9.5 nm Continuum a = 15.1 nm 20 Continuum R = 9.5 nm Eq. (18) F a b/Y/d3FR2/Y/d3 20 Eq. (17) 15 15 10 10 5 5 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 w/d w/d Comparison of the nondimensional force vs. nondimensional out-of-plane displacement for circular and rectangular lattice and continuum SLGS.
- 30. Analytical approach for SLGS –honeycomb structure C-C bonds deform under stretching and hinging 8kτ Kh = Hinging constant related to thickness d d2 Applying averaging of stretching and hinging deformation over unit cell: 4 3k r K h 4 3k r K h E1 = E2 = 3d (k r + 3K h ) 3d (k r + 3K h ) 1 − K h kr 3K h k r ν 21 = ν 12 = G12 = 1 + 3 K h kr 3d (k r + K h ) • Isotropic for “infinite” graphene sheet • Orthotropic for finite size graphene and considering edge effects F Scarpa, S Adhikari, A S Phani, 2009. Nanotechnology 20, 065709
- 31. Analytical approach for SLGS –honeycomb structure Unit cell made by rods withstanding axial and bending deformation 4 Lk r Equivalent Young’s modulus for Ya −f = axial members πd s2 16kθ Equivalent Young’s modulus for Yg − n = πLd b2 axial members A Rigidity matrix is obtained using a lattice continuum modelling of space frames à equivalence with plane stress formulation for a plane sheet: 1 4 3 1 E= ( k r L2 + 12 kθ ) ν = d max 9 L2 3 F Scarpa, S Adhikari, A S Phani, 2009. Nanotechnology 20, 065709 (L Kollár and I Hegedús. Analysis and design of space frames by the Continuum Method. Developments in Civil Engineering, 10. Elsevier, Amsterdam, 1985)
- 32. Atomistic FE – Bilayer Graphene • Equivalent to structural “sandwich” beams • C-C bonds in graphene layers represented with classical equivalent beam models • “Core” represented by Lennard-Jones potential interactions: rmin = 0.383 nm ε = 2.39 meV Ef =0.5 TPa (I.W. Frank, D.M. Tanenbaum, A.M. van der Zande, P.L. McEuen, J. Vac. Sci. Technol. B 25 (2007) 2558) Scarpa, F., Adhikari, S. and Chowdhury, R., "The transverse elasticity of bilayer graphene", Physics Letters A, 374[19-20] (2010), pp. 2053-2057.
- 33. Mechanical vibration of SLGSLumped mass matrix:Minimisation of the Hamiltonian for the ith mode:Comparison against Molecular Mechanicsmodel based on the eigenvalue analysis ofthe system Hessian matrix Scarpa, F., Chowdhury, R., Kam, K., Adhikari, S. and Ruzzene, M., "Wave propagation in graphene nanoribbons", Nanoscale Research Letters, 6 (2011), pp. 430:1-10. Chowdhury, R., Adhikari, S., Scarpa, F. and Friswell, M. I., "Transverse vibration of single layer graphene sheets", Journal of Physics D: Applied Physics, 44[20] (2011), pp. 205401:1-11.
- 34. Mechanical vibration of SLGS • (8,0) nanoribbons with different lengths • Errors between 2 and 3 % • Average thickness d = 0.077 nm
- 35. Graphene composites Polymer Matrix Graphene sheet van der Waals interaction 250 Armchair GRP2 Zigzag GRP4 200 150(GHz) 1 100 50 0 0 5 10 15 20 Length (nm) Chandra, Y., Chowdhury, R., Scarpa, F., Adhikari, S. and Seinz, J., "Multiscale modeling on dynamic behaviour of graphene based composites", Materials Science and Engineering B, in press.
- 36. Nanobio SensorsTitle ofpresentationClick to edit subtitle style
- 37. Vibration based mass sensor: CNT Point mass Distributed massChowdhury, R., Adhikari, S. and Mitchell, J., "Vibrating carbonnanotube based bio-sensors", Physica E: Low-dimensional Adhikari, S. and Chowdhury, R., "The calibration of carbonSystems and Nanostructures, 42[2] (2009), pp. 104-109. nanotube based bio-nano sensors", Journal of Applied Physics, 107[12] (2010), pp. 124322:1-8
- 38. Vibration based mass sensor: CNTThe equation of motion of free-vibration:The resonance frequencies:The Mode shapes:We use energy principles to obtain the frequency shift due to theadded mass.
- 39. Vibration based mass sensor: CNTNatural frequency with the added mass: Identification of the added mass
- 40. Vibration based mass sensor: CNT Mass of a nano object can be detected from the frequency shift ΔfAdhikari, S. and Chowdhury, R., "The calibration of carbon nanotube basedbio-nano sensors", Journal of Applied Physics, 107[12] (2010), pp.124322:1-8
- 41. Vibration based mass sensor: CNT Mass of a nano object can be detected from the frequency shift Δf 0.45 0.4 Molecular mechanics Exact solution 0.35 Calibration constant based approach AL 0.3 Normalized added mass: M / 0.25 0.2 0.15CNT with deoxythymidine 0.1 0.05Adhikari, S. and Chowdhury, R., "The calibration of carbon nanotube based 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4bio-nano sensors", Journal of Applied Physics, 107[12] (2010), pp. Relative frequency shift: f / f124322:1-8 n0
- 42. Vibration based mass sensor: Graphene Vibrating graphene sheets can be used as sensors with different mass arrangements
- 43. Vibration based mass sensor: Graphene Relative added mass:
- 44. Vibration based mass sensor: Graphene Vibrating graphene sheets can be used as sensors with different mass arrangements 1 edgeFixed Molecular mechanics 0.9 Proposed approach Normalized added mass: µ = M / ( ab) 0.8 0.7 0.6 0.5 0.4 SLGS with adenosine 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Relative frequency shift: f / f0
- 45. Boron Nitride Nanotube and NanosheetsTitle ofpresentationClick to edit subtitle style
- 46. Axial vibration of BNNT (a) (a) Axial vibration and itsassociated frequency of (b)zigzag and (c) armchairBNNTs given by the MMsimulations (discrete dots)and a column model withYoung’s modulus 1TPa !(solid lines). Chowdhury, R., Wang, C. W., Adhikari, S. and Scarpa, F., "Vibration and symmetry-breaking of boron nitride nanotubes", Nanotechnology, 21[36] (2010), pp. 365702:1-9.
- 47. Torsional vibration of BNNT (a) (a) Torsional vibration and itsassociated frequency of (b) zigzagand (c) armchair BNNTs given bythe MM simulations (discrete dots)and a column model with shearmodulus 0.41TPa (solid lines). ! Chowdhury, R., Wang, C. W., Adhikari, S. and Scarpa, F., "Vibration and symmetry-breaking of boron nitride nanotubes", Nanotechnology, 21[36] (2010), pp. 365702:1-9.
- 48. Optimised shape of BNNT (a) (d) Optimized configuration (b) of armchair BNNTs: (a) (3, 3), (b) (4,4) and (c) (6,6) with the aspect ratio 15, and (d) short (c) (6, 6) with the aspect ratio 2.
- 49. Mechanical property of BN Sheets Example of armchair (4, 0) BN sheet. Boron atoms are in red, nitrogen atoms are in green. Boldrin, L., Scarpa, F., Chowdhury, R., Adhikari, S. and Ruzzene, M., "Effective mechanical properties of hexagonal boron nitride nanosheets", Nanotechnology, 22[50] (2011), pp. 505702:1-7.
- 50. DNA MechanicsTitle ofpresentationClick to edit subtitle style
- 51. Atomistic FE of DNA From protein data bank file to ANSYS input file – a new code for automatic translation
- 52. Atomistic FE of DNA Material properties of the beams are obtained depending on the nature of the bonds Mode 6 (MM:111.696; FE 112.71 GHz) Mode 3 (MM:33.679; FE 38.768 GHz)
- 53. ConclusionsØ Atomistic finite element method is developed for general nanosalce structures: § Carbon nanotube § Fullerenes § Graphene § Nanoscale bio sensorsØ Programs have been written to convert pdb files to Finite Element geometry file and material propertiesØ Encouraging results compared to MM simulation were obtainedØ Future: nonlinearity, large-scale problems such as proteins & nanocomposites, molecular dynamic simulations, experimental validation
- 54. References1. Murmu, T. and Adhikari, S., "Nonlocal frequency analysis of nanoscale biosensors", Sensors & Actuators: A. Physical, 173[1] (2012), pp. 41-48.2. Boldrin, L., Scarpa, F., Chowdhury, R., Adhikari, S. and Ruzzene, M., "Effective mechanical properties of hexagonal boron nitride nanosheets", Nanotechnology, 22[50] (2011), pp. 505702:1-7.3. Murmu, T., Seinz, J., Adhikari, S. and Arnold, C., "Nonlocal buckling behaviour of bonded double-nanoplate- system", Journal of Applied Physics, 110[8] (2011), pp. 084316:1-8.4. Flores, E. I. S., Adhikari, S., Friswell, M. I. and Scarpa, F., "Hyperelastic axial buckling of single wall carbon nanotubes", Physica E: Low-dimensional Systems and Nanostructures, 44[2] (2011), pp. 525-529.5. Murmu, T. and Adhikari, S., "Nonlocal vibration of bonded double-nanoplate-systems", Composites Part B: Engineering, 42[7] (2011), pp. 1901-1911.6. Chandra, Y., Chowdhury, R., Adhikari, S. and Scarpa, F., "Elastic instability of bilayer graphene using atomistic finite element", Physica E: Low-dimensional Systems and Nanostructures, 44[1] (2011), pp. 12-16. 7. Scarpa, F., Chowdhury, R., Kam, K., Adhikari, S. and Ruzzene, M., "Wave propagation in graphene nanoribbons", Nanoscale Research Letters, 6 (2011), pp. 430:1-10. 8. Chowdhury, R. and Adhikari, S., "Boron nitride nanotubes as zeptogram-scale bio-nano sensors: Theoretical investigations", IEEE Transactions on Nanotechnology, 10[4] (2011), pp. 659-667. 9. Chandra, Y., Chowdhury, R., Scarpa, F. and Adhikari, S., "Vibrational characteristics of bilayer graphene sheets", Thin Solid Films, 519[18] (2011), pp. 6026-6032. 10. Adhikari, S. and Chowdhury, R., "Vibration spectra of fullerene family", Physics Letters A, 375[22] (2011), pp. 1276-1280.
- 55. References11. Murmu, T., Adhikari, S. and Wang, C. W., "Torsional vibration of carbon nanotube-buckyball systems based on nonlocal elasticity theory", Physica E: Low-dimensional Systems and Nanostructures, 43[6] (2011), pp. 1276-1280. 12. Wang, C. W. and Adhikari, S., "ZnO-CNT composite nanowires as nanoresonators", Physics Letters A, 375[22] (2011), pp. 2171-2175. 13. Chowdhury, R., Adhikari, S., Scarpa, F. and Friswell, M. I., "Transverse vibration of single layer graphene sheets", Journal of Physics D: Applied Physics, 44[20] (2011), pp. 205401:1-11. 14. Scarpa, F., Chowdhury, R., and Adhikari, S., "Thickness and in-plane elasticity of Graphane", Physics Letters A, 375[20] (2011), pp. 2071-2074. 15. Wang, C. W., Murmu, T. and Adhikari, S., "Mechanisms of nonlocal effect on the vibration of nanoplates", Applied Physics Letters, 98[15] (2011), pp. 153101:1-3. 16. Murmu, T. and Adhikari, S., "Nonlocal vibration of carbon nanotubes with attached buckyballs at tip", Mechanics Research Communications, 38[1] (2011), pp. 62-67. 17. Chowdhury, R., Adhikari, S. and Scarpa, F., "Vibrational analysis of ZnO nanotubes: A molecular mechanics approach", Applied Physics A, 102[2] (2011), pp. 301-308. 18. Chowdhury, R., Adhikari, S., Rees, P., Scarpa, F., and Wilks, S.P., "Graphene based bio-sensor using transport properties", Physical Review B, 83[4] (2011), pp. 045401:1-8. 19. Murmu, T. and Adhikari, S., "Axial instability of double-nanobeam-systems", Physics Letters A, 375[3] (2011), pp. 601-608. 20. Flores, E. I. S., Adhikari, S., Friswell, M. I. and F. Scarpa, "Hyperelastic finite element model for single wall carbon nanotubes in tension", Computational Materials Science, 50[3] (2011), pp. 1083-1087.
- 56. References21. Murmu, T. and Adhikari, S., "Scale-dependent vibration analysis of prestressed carbon nanotubes undergoing rotation", Journal of Applied Physics, 108[12] (2010), pp. 123507:1-7. 22. Scarpa, F., Adhikari, S. and Phani, A. Srikanth, "Auxeticity in single layer graphene sheets", International Journal of Novel Materials, 1[2] (2010), pp. 39-43. 23. Murmu, T. and Adhikari, S., "Nonlocal effects in the longitudinal vibration of double-nanorod systems", Physica E: Low-dimensional Systems and Nanostructures, 43[1] (2010), pp. 415-422. 24. Murmu, T. and Adhikari, S., "Nonlocal transverse vibration of double-nanobeam-systems", Journal Applied Physics, 108[8] (2010), pp. 083514:1-9. 25. Scarpa, F., Peng, H. X., Boldri, L., Remillat, C. D. L., Adhikari, S., "Coupled thermo-mechanics of single-wall carbon nanotubes", Applied Physics Letters, 97[15] (2010), pp. 151903:1-3. 26. Chowdhury, R., Adhikari, S., Rees, P., "Optical properties of silicon doped ZnO", Physica B: Condensed Matter, 405[23] (2010), pp. 4763-4767. 27. Chowdhury, R., Wang, C. W., Adhikari, S. and Scarpa, F., "Vibration and symmetry-breaking of boron nitride nanotubes", Nanotechnology, 21[36] (2010), pp. 365702:1-9. 28. Wang, C. Y., Zhao, Y., Adhikari, S. and Feng, Y. T., "Vibration of axially strained triple-wall carbon nanotubes", Journal of Computational and Theoretical Nanoscience, 7[11] (2010), pp. 2176-2185. 29. Adhikari, S. and Chowdhury, R., "The calibration of carbon nanotube based bio-nano sensors", Journal of Applied Physics, 107[12] (2010), pp. 124322:1-8. 30. Chowdhury, R., Wang, C. Y., Adhikari, S., and Tong, F. M., "Sliding oscillations of multiwall carbon nanotubes", Physica E: Low-dimensional Systems and Nanostructures, 42[9] (2010), pp. 2295-2300.
- 57. References31. Chowdhury, R., Adhikari, S., Wang, C. W. and Scarpa, F., "A molecular mechanics approach for the vibration of single walled carbon nanotubes", Computational Materials Science, 48[4] (2010), pp. 730-735. 32. Wang, C. Y., Li, C. F., and Adhikari, S., "Axisymmetric vibration of singlewall carbon nanotubes in water", Physics Letters A, 374[24] (2010), pp. 2467-2474. 33. Chowdhury, R., Adhikari, S. and Scarpa, F., "Elasticity and piezoelectricity of zinc oxide nanostructure", Physica E: Low-dimensional Systems and Nanostructures, 42[8] (2010), pp. 2036-2040. 34. Gil, A. J., Adhikari, S., Scarpa, F., and Bonet, J., "The formation of wrinkles in single layer graphene sheets under nanoindentation", Journal of Physics: Condensed Matter, 22[14] (2010), pp. 145302:1-6. 35. Scarpa, F., Adhikari, S. and Chowdhury, R., "The transverse elasticity of bilayer graphene", Physics Letters A, 374[19-20] (2010), pp. 2053-2057. 36. Scarpa, F., Adhikari, S., Gil, A. J. and Remillat, C., "The bending of single layer graphene sheets: Lattice versus continuum approach", Nanotechnology, 21[12] (2010), pp. 125702:1-9. 37. Chowdhury, R., Rees, P., Adhikari, S., Scarpa, F., and Wilks, S.P., "Electronic structures of silicon doped ZnO", Physica B: Condensed Matter, 405[8] (2010), pp. 1980-1985. 38. Scarpa, F., Adhikari, S. and Wang, C. Y., "Nanocomposites with auxetic nanotubes", International Journal of Smart and Nanomaterials, 1[2] (2010), pp. 83-94. 39. Chowdhury, R., Wang, C. Y. and Adhikari, S., "Low-frequency vibration of multiwall carbon nanotubes with heterogeneous boundaries", Journal of Physics D: Applied Physics, 43[11] (2010), pp. 085405:1-8. 40. Chowdhury, R., Adhikari, S. and Mitchell, J., "Vibrating carbon nanotube based bio-sensors", Physica E: Low- dimensional Systems and Nanostructures, 42[2] (2009), pp. 104-109.
- 58. References41. Scarpa, F., Adhikari, S. and Wang, C. Y., "Mechanical properties of non reconstructed defective single wall carbon nanotubes", Journal of Physics D: Applied Physics, 42 (2009) 142002 (6pp). 42. Wang, C. Y., Li, C. F., and Adhikari, S., "Dynamic behaviors of microtubules in cytosol", Journal of Biomechanics, 42[9] (2009), pp. 1270-1274. 43. Tong, F. M., Wang, C. Y., and Adhikari, S., "Axial buckling of multiwall carbon nanotubes with heterogeneous boundary conditions", Journal of Applied Physics, 105 (2009), pp. 094325:1-7. 44. Scarpa, F., Adhikari, S. and Phani, A. Srikanth, "Effective mechanical properties of single graphene sheets", Nanotechnology, 20[1-2] (2009), pp. 065709:1-11. 45. Scarpa, F. and Adhikari, S., "Uncertainty modelling of carbon nanotube terahertz oscillators", Journal of Non- Crystalline Solids, 354[35-39] (2008), pp. 4151-4156. 46. Scarpa, F. and Adhikari, S., "A mechanical equivalence for the Poissons ratio and thickness of C-C bonds in single wall carbon nanotubes", Journal of Physics D: Applied Physics, 41 (2008) 085306 (5pp).