Structural Engineering and Mechanics

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A unified formulation for static behavior of nonlocal curved beams

  • Tufekci, Ekrem (Istanbul Technical University, Faculty of Mechanical Engineering) ;
  • Aya, Serhan A. (Istanbul Technical University, Faculty of Mechanical Engineering) ;
  • Oldac, Olcay (Istanbul Technical University, Faculty of Mechanical Engineering)
  • Received : 2015.07.31
  • Accepted : 2016.04.19
  • Published : 2016.08.10
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Abstract

Nanobeams are widely used as a structural element for nanodevices and nanomachines. The development of nano-sized machines depends on proper understanding of mechanical behavior of these nano-sized beam elements. Small length scales such as lattice spacing between atoms, surface properties, grain size etc. are need to be considered when applying any classical continuum model. In this study, Eringen's nonlocal elasticity theory is incorporated into classical beam model considering the effects of axial extension and the shear deformation to capture unique static behavior of the nanobeams under continuum mechanics theory. The governing differential equations are obtained for curved beams and solved exactly by using the initial value method. Circular uniform beam with concentrated loads are considered. The displacements, slopes and the stress resultants are obtained analytically. A detailed parametric study is conducted to examine the effect of the nonlocal parameter, mechanical loadings, opening angle, boundary conditions, and slenderness ratio on the static behavior of the nanobeam.

Keywords

Acknowledgement

Supported by : Technological Council of Turkey (TUBITAK), Istanbul Technical University

References

  1. Abramowitz, M. and Stegun, I.A. (1972), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th Printing, Dover, New York, NY, USA.
  2. Akgoz, B. and Civalek, O. (2013), "Buckling analysis of linearly tapered micro-columns based on strain gradient elasticity", Struct. Eng. Mech., 48(2), 195-205. https://doi.org/10.12989/sem.2013.48.2.195
  3. Alotta, G., Failla, G. and Zingales, M. (2014), "Finite element method for a nonlocal Timoshenko beam model", Finite Elem. Anal. Des., 89, 77-92. https://doi.org/10.1016/j.finel.2014.05.011
  4. Arash, B., Wang, Q. and Duan, W.H. (2011), "Detection of gas atoms via vibration of graphenes", Phys. Lett. A, 375(24), 2411-2415. https://doi.org/10.1016/j.physleta.2011.05.009
  5. Behera, L. and Chakraverty, S. (2014), "Free vibration of nonhomogeneous Timoshenko nanobeams", Meccanica, 49(1), 51-67. https://doi.org/10.1007/s11012-013-9771-2
  6. Berrabah, H.M., Tounsi, A., Semmah, A. and Bedia, E.A.A. (2013), "Comparison of various refined nonlocal beam theories for bending, vibration and buckling analysis of nanobeams", Struct. Eng. Mech., 48(3), 351-365. https://doi.org/10.12989/sem.2013.48.3.351
  7. Bradshaw, R.D., Fisher, F.T. and Brinson, L.C. (2003), "Fiber waviness in nanotube-reinforced polymer composites-II: modeling via numerical approximation of the dilute strain concentration tensor", Compos. Sci. Technol., 63(11), 1705-1722. https://doi.org/10.1016/S0266-3538(03)00070-8
  8. Craighead, H.G. (2000), "Nanoelectromechanical systems", Science, 290(5496), 1532-153. https://doi.org/10.1126/science.290.5496.1532
  9. Ekinci, K.L. (2005), "Electromechanical transducers at the nanoscale: Actuation and sensing of motion in nanoelectromechanical systems (NEMS)", Small, 1(8-9), 786-797. https://doi.org/10.1002/smll.200500077
  10. Eringen, A.C. (1983), "Linear theory of nonlocal elasticity and dispersion of plane waves", J. Appl. Phys., 54, 4703-4710. https://doi.org/10.1063/1.332803
  11. Fisher, F.T., Bradshaw, R.D. and Brinson, L.C. (2003), "Fiber waviness in nanotube-reinforced polymer composites-I: Modulus predictions using effective nanotube properties", Compos. Sci. Technol., 63(11), 1689-1703. https://doi.org/10.1016/S0266-3538(03)00069-1
  12. Guo, R., Barisci, J.N., Innis, P.C., Too, C.O., Wallace, G.G. and Zhou, D. (2000), "Electrohydrodynamic polymerization of 2-methoxyaniline-5-sulfonic acid", Synthetic Met., 114(3), 267-272. https://doi.org/10.1016/S0379-6779(00)00242-3
  13. Hadjesfandiari, A.R. and Dargush, G.F. (2011), "Couple stress theory for solids", Int. J. Solid. Struct., 48(18), 2496-2510. https://doi.org/10.1016/j.ijsolstr.2011.05.002
  14. Hu, Y.G., Liew, K.M. and Wang, Q. (2009), "Nonlocal elastic beam models for flexural wave propagation in double-walled carbon nanotubes", J. Appl. Phys., 106(4):044301. https://doi.org/10.1063/1.3197857
  15. Huang, C., Ye, C., Wang, S., Stakenborg, T. and Lagae, L. (2012), "Gold nanoring as a sensitive plasmonic biosensor for on-chip DNA detection", Appl. Phys. Lett., 100, 173114. https://doi.org/10.1063/1.4707382
  16. Joshi, A.Y., Sharma, S.C. and Harsha, S.P. (2010) "Dynamic analysis of a clamped wavy single walled carbon nanotube based nanomechanical sensors", J. Nanotechnol. Eng. Med., 1, 031007-7. https://doi.org/10.1115/1.4002072
  17. Kong, J., Franklin, N.R., Zhou, C.W., Chapline, M.G., Peng, S., Cho, K. and Dai, H.J. (2000), "Nanotube molecular wires as chemical sensors", Science, 287, 622-625. https://doi.org/10.1126/science.287.5453.622
  18. Kong, X.Y., Ding, Y., Yang, R. and Wang, Z.L. (2004), "Single-crystal nanorings formed by epitaxial self-coiling of polar nanobelts", Science, 303, 1348-1351. https://doi.org/10.1126/science.1092356
  19. Li, C. (2013), "Size-dependent thermal behaviors of axially traveling nanobeams based on a strain gradient theory", Struct. Eng. Mech., 48(3), 415-434. https://doi.org/10.12989/sem.2013.48.3.415
  20. Li, C. (2014), "A nonlocal analytical approach for torsion of cylindrical nanostructures and the existence of higher-order stress and geometric boundaries", Compos. Struct., 118, 607-621. https://doi.org/10.1016/j.compstruct.2014.08.008
  21. Li, C. and Chou, T.W. (2003), "Single-walled carbon nanotubes as ultra-high frequency nanomechanical resonators", Phys. Rev. B, 68(7), 073405. https://doi.org/10.1103/PhysRevB.68.073405
  22. Li, C., Li, S., Yao, L.Q. and Zhu, Z.K. (2015a),"Nonlocal theoretical approaches and atomistic simulations for longitudinal free vibration of nanorods/nanotubes and verification of different nonlocal models", Appl. Math. Model., 39, 4570-4585. https://doi.org/10.1016/j.apm.2015.01.013
  23. Li, C., Yao, L.Q., Chen, W.Q and Li, S. (2015b), "Comments on nonlocal effects in nano-cantilever beams", Int. J. Eng. Sci., 87, 47-57. https://doi.org/10.1016/j.ijengsci.2014.11.006
  24. Liu, Y.P. and Reddy, J.N. (2011), "A nonlocal curved beam model based on a modified couple stress theory", Int. J. Struct. Stab. Dyn., 11(3), 495-512.
  25. Mayoof, F.N. and Hawwa, M.A. (2009), "Chaotic behavior of a curved carbon nanotube under harmonic excitation", Chaos Solit. Fract., 42(3), 1860-1867. https://doi.org/10.1016/j.chaos.2009.03.104
  26. McFarland, A.W. and Colton, J.S. (2005), "Role of material microstructure in plate stiffness with relevance to microcantilever sensors", J. Micromech. Microeng., 15, 1060-1067. https://doi.org/10.1088/0960-1317/15/5/024
  27. Paola, M.D., Failla, G. and Zingales, M. (2013), "Non-local stiffness and damping models for shear-deformable beams", Eur. J. Mech. A-Solid., 40, 69-83. https://doi.org/10.1016/j.euromechsol.2012.12.009
  28. Peddieson, J., Buchanan, G.R. and McNitt, R.P. (2003), "Application of nonlocal continuum models to nanotechnology", Int. J. Solid. Struct., 41, 305-312.
  29. Polizzotto, C., Fuschi, P. and Pisano, A.A. (2006), "A nonhomogeneous nonlocal elasticity model", Eur. J. Mech. A-Solid., 25(2), 308-333. https://doi.org/10.1016/j.euromechsol.2005.09.007
  30. Povstenko, Y.Z. (1995), "Straight disclinations in nonlocal elasticity", Int. J. Eng. Sci., 33(4), 575-582. https://doi.org/10.1016/0020-7225(94)00070-0
  31. Pradhan, S.C. and Sarkar, A. (2009), "Analyses of tapered fgm beams with nonlocal theory", Struct. Eng. Mech., 32(6), 811-833. https://doi.org/10.12989/sem.2009.32.6.811
  32. Reddy, J.N. (2007), "Nonlocal theories for bending, buckling and vibration of beams", Int. J. Eng. Sci., 45, 288-307. https://doi.org/10.1016/j.ijengsci.2007.04.004
  33. Roukes, M. (2001), "Nanoelectromechanical systems face the future", Phys. World, 14, 25-31
  34. Sudak, L.J. (2003), "Column buckling of multi-walled carbon nanotubes using nonlocal elasticity", J. Appl. Phys., 94, 7281. https://doi.org/10.1063/1.1625437
  35. Taghizadeh, M., Ovesy, H.R. and Ghannadpour, S.A.M. (2015), "Nonlocal integral elasticity analysis of beam bending by using finite element method", Struct. Eng. Mech., 54(4) 755-769. https://doi.org/10.12989/sem.2015.54.4.755
  36. Treacy, M.M.J., Ebbesen, T.W. and Gibson, J.W. (1996), "Exceptionally high Young's modulus observed for individual carbon nanotubes", Nature, 381(6584), 678-680. https://doi.org/10.1038/381678a0
  37. Tufekci, E. (2001), "Exact solution of free in-plane vibration of shallow circular arches", Int. J. Struct. Stab. Dyn., 1, 409-428. https://doi.org/10.1142/S0219455401000226
  38. Tufekci, E. and Arpaci, A. (2006), "Analytical solutions of in-plane static problems for non-uniform curved beams including axial and shear deformations", Struct. Eng. Mech., 22(2), 131-150. https://doi.org/10.12989/sem.2006.22.2.131
  39. Wang, L.F. and Hu, H.Y. (2005), "Flexural wave propagation in single-walled carbon nanotubes", Phys. Rev. B, 71(19) 195412. https://doi.org/10.1103/PhysRevB.71.195412
  40. Wang, Q. (2005), "Wave propagation in carbon nanotubes via nonlocal continuum mechanics", J. Appl. Phys., 89, 124301.
  41. Wang, Q. and Shindo, Y. (2006), "Nonlocal continuum models for carbon nanotubes subjected to static loading", J. Mech. Mater. Struct., 1(4), 663-680. https://doi.org/10.2140/jomms.2006.1.663
  42. Zemri, A., Houari, M.S.A., Bousahla, A.A. and Tounsi, A. (2015), "A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory", Struct. Eng. Mech., 54(4), 693-710. https://doi.org/10.12989/sem.2015.54.4.693
  43. Zhang, Z., Wang C.M. and Challamel, N. (2015), "Eringen's length-scale coefficients for vibration and buckling of nonlocal rectangular plates with simply supported edges", J. Eng. Mech., 141(2), 04014117. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000838
  44. Zhao, Q., Gan, Z.H. and Zhuang, O.K. (2002), "Electrochemical sensors based on carbon nanotubes", Electroanal., 14(23), 1609-13. https://doi.org/10.1002/elan.200290000

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