Interaction and multiscale mechanics

  • 제2권3호
  • /
  • Pages.223-233
  • /
  • 2009
  • /
  • 1976-0426(pISSN)
  • /
  • 2092-6200(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

The effects of stiffness strengthening nonlocal stress and axial tension on free vibration of cantilever nanobeams

  • Lim, C.W. (Department of Building and Construction, City University of Hong Kong) ;
  • Li, C. (Department of Building and Construction, City University of Hong Kong, Department of Modern Mechanics, University of Science and Technology of China) ;
  • Yu, J.L. (Department of Modern Mechanics, University of Science and Technology of China)
  • 투고 : 2009.05.25
  • 심사 : 2009.07.07
  • 발행 : 2009.09.25
⟨ 이전 논문 다음 논문 ⟩

초록

This paper presents a new nonlocal stress variational principle approach for the transverse free vibration of an Euler-Bernoulli cantilever nanobeam with an initial axial tension at its free end. The effects of a nanoscale at molecular level unavailable in classical mechanics are investigated and discussed. A sixth-order partial differential governing equation for transverse free vibration is derived via variational principle with nonlocal elastic stress field theory. Analytical solutions for natural frequencies and transverse vibration modes are determined by applying a numerical analysis. Examples conclude that nonlocal stress effect tends to significantly increase stiffness and natural frequencies of a nanobeam. The relationship between natural frequency and nanoscale is also presented and its significance on stiffness enhancement with respect to the classical elasticity theory is discussed in detail. The effect of an initial axial tension, which also tends to enhance the nanobeam stiffness, is also concluded. The model and approach show potential extension to studies in carbon nanotube and the new result is useful for future comparison.

키워드

참고문헌

  1. Cagin, T., Che, J.W., Gardos, M.N., Fijany, A. and Goddard, W.A. (1999), "Simulation and experiments on friction and wear of diamond: a material for MEMS and NEMS application", Nanotechnology, 10(3), 278-284. https://doi.org/10.1088/0957-4484/10/3/310
  2. Chen, C.S., Wang, C.K. and Chang, S.W. (2008), "Atomistic simulation and investigation of nanoindentation, contact pressure and nanohardness", Interact. Multi. Mech., 1(4), 411-422. https://doi.org/10.12989/imm.2008.1.4.411
  3. Eringen, A.C. (1972), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10(1), 1-16. https://doi.org/10.1016/0020-7225(72)90070-5
  4. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54(9), 4703-4710. https://doi.org/10.1063/1.332803
  5. Eringen, A.C. and Edelen, D.G.B. (1972), "On nonlocal elasticity", Int. J. Eng. Sci., 10(3), 233-248. https://doi.org/10.1016/0020-7225(72)90039-0
  6. Gao, J.G. and Zhao. Y.P. (2006), "Dynamic stability of electrostatic torsional actuators with van der Waals effect", Int. J. Solids Struct., 43(3-4), 675-685. https://doi.org/10.1016/j.ijsolstr.2005.03.073
  7. He, L.H., Lim, C.W. and Wu, B.S. (2004), "A continuum model for size-dependent deformation of elastic films of nano-scale thickness", Int. J. Solids Struct., 41(3-4), 847-857. https://doi.org/10.1016/j.ijsolstr.2003.10.001
  8. Hu, Y.G., Liew, K.M., Wang, Q., He, X.Q. and Yakobson, B.I. (2008), "Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes", Int. J. Solids Struct., 56(12), 3475-3485.
  9. Jonsson, L.M., Santandrea, F., Gorelik, Y.K., Shekhter, R.I. and Jonson, M. (2008), "Self-organization of irregular nanoelectromechanical vibrations in multimode shuttle structures", Phys. Rev. Lett., 100(18), 186802. https://doi.org/10.1103/PhysRevLett.100.186802
  10. Lim, C.W. (2008), "A discussion on the nonlocal elastic stress field theory for nanobeams", The 11th East Asia-Pacific Conference on Structural Engineering & Construction (EASEC-11), Taipei, November.
  11. Lim, C.W. and Wang, C.M. (2007), "Exact variational nonlocal stress modeling with asymptotic higher-order strain gradients for nanobeams", J. Appl. Phys., 101(5), 054312. https://doi.org/10.1063/1.2435878
  12. Liu, Y.Z., Chen, W.L. and Chen, L.Q. (1998), Vibration Mechanics. Higher Education Press, Beijing.
  13. Lu, P., Lee, H.P., Lu, C. and Zhang, P.Q. (2006), "Dynamic properties of flexural beams using a nonlocal elasticity model", J. Appl. Phys., 99(7), 073510. https://doi.org/10.1063/1.2189213
  14. Mikkelsen, L.P. and Tvergaard, V. (1999), "A nonlocal two-dimensional analysis of instabilities in tubes under internal pressure", J. Mech. Phys. Solids, 47(4), 953-969. https://doi.org/10.1016/S0022-5096(98)00062-3
  15. Na, S., Librescu, L. and Shim, J.K. (2003), "Modeling and bending vibration control of nonuniform thin-walled rotating beams incorporating adaptive capabilities", Int. J. Mech. Sci., 45(8), 1347-1367. https://doi.org/10.1016/j.ijmecsci.2003.09.015
  16. Oz, H.R., Pakdemirli, M. and Boyaci, H. (2001), "Non-linear vibrations and stability of an axially moving beam with time-dependent velocity", Int. J. Nonlin. Mech., 36, 107-115. https://doi.org/10.1016/S0020-7462(99)00090-6
  17. Parker, R.G. and Orloske, K. (2006), "Flexural-torsional buckling of misaligned axially moving beams: vibration and stability analysis", Int. J. Solids Struct., 43(14-15), 4323-4341. https://doi.org/10.1016/j.ijsolstr.2005.08.015
  18. Reddy, J.N. and Wang, C.M. (1998), "Deflection relationships between classical and third-order plate theories", Acta Mech. Sinica, 130 (3-4), 199-208.
  19. Sato, M. and Shima, H. (2008), "Buckling characteristics of multiwalled carbon nanotubes under external pressure", Interact. Multi. Mech., 2(2), 209-222.
  20. Shibutani, Y., Vitek, V. and Bassani, J.L. (1998), "Nonlocal properties of inhomogeneous structures by linking approach of generalized continuum to atomistic model", Int. J. Mech. Sci., 40(2-3), 129-137. https://doi.org/10.1016/S0020-7403(97)00042-8
  21. Sudak, L.J. (2003), "Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics", J. Appl. Phys., 94(11), 7281-7287. https://doi.org/10.1063/1.1625437
  22. Tounsi, A., Heireche, H., Berrabah, H.M., Benzair, A. and Boumia, L. (2008), "Effect of small size on wave propagation in double-walled carbon nanotubes under temperature field", J. Appl. Phys., 104(10), 104310. https://doi.org/10.1063/1.3021158
  23. Unnikrishnan, V.U., Reddy, J.N., Banerjee, D. and Rostam-Abadi, F. (2008), "Thermal characteristics of defective carbon nanotube-polymer nanocomposites", Interact. Multi. Mech., 1(4), 397-409. https://doi.org/10.12989/imm.2008.1.4.397
  24. Wang, Y.F., Huang, L.H. and Liu, X.T. (2005), "Eigenvalue and stability analysis for transverse vibrations of axially moving strings based on Hamiltonian dynamics", Acta Mech. Sinica, 21, 485-494. https://doi.org/10.1007/s10409-005-0066-2
  25. Yakobson, B.I., Brabec, C.I. and Bernholc, J. (1996), "Nanomechanics of carbon tubes: Instabilities beyond linear response", Phys. Rev. Lett., 76(14), 2511-2514. https://doi.org/10.1103/PhysRevLett.76.2511
  26. Yu, J.L. (1985), "Progress and applications of solid mechanics considering microstructure", Adv. Mech., 15(1), 82-89.
  27. Zhang, Y.Q., Liu, G.R. and Xie, X.Y. (2005), "Free transverse vibration of double-walled carbon nanotubes using a theory of nonlocal elasticity", Phys. Rev. B., 71(19), 195404. https://doi.org/10.1103/PhysRevB.71.195404
  28. Zhao, H. and Aluru, N.R. (2008), "Molecular dynamics simulation of bulk silicon under strain", Interact. Multi. Mech., 1(2), 303-315. https://doi.org/10.12989/imm.2008.1.2.303

피인용 문헌

  1. Flapwise bending vibration analysis of rotary tapered functionally graded nanobeam in thermal environment 2017, https://doi.org/10.1080/15376494.2017.1365982
  2. Chaos prediction in nano-resonators based on nonlocal elasticity theory vol.148, 2018, https://doi.org/10.1051/matecconf/201814807006
  3. Nonlinear bending vibration of a rotating nanobeam based on nonlocal Eringen’s theory using differential quadrature method vol.22, pp.12, 2016, https://doi.org/10.1007/s00542-015-2662-9
  4. Is a nanorod (or nanotube) with a lower Young’s modulus stiffer? Is not Young’s modulus a stiffness indicator? vol.53, pp.4, 2010, https://doi.org/10.1007/s11433-010-0170-6
  5. A Variational Principle Approach for Vibration of Non-Uniform Nanocantilever Using Nonlocal Elasticity Theory vol.10, 2015, https://doi.org/10.1016/j.mspro.2015.06.087
  6. Vibration analysis of rotating functionally graded Timoshenko microbeam based on modified couple stress theory under different temperature distributions vol.121, 2016, https://doi.org/10.1016/j.actaastro.2016.01.003
  7. Dynamic analysis of functionally graded multi-walled carbon nanotube-polystyrene nanocomposite beams subjected to multi-moving loads vol.49, 2013, https://doi.org/10.1016/j.matdes.2013.01.073
  8. Non-linear transverse vibrations of tensioned nanobeams using nonlocal beam theory vol.55, pp.2, 2015, https://doi.org/10.12989/sem.2015.55.2.281
  9. Influence of size effect on flapwise vibration behavior of rotary microbeam and its analysis through spectral meshless radial point interpolation vol.123, pp.5, 2017, https://doi.org/10.1007/s00339-017-0955-9
  10. Size-dependent thermal behaviors of axially traveling nanobeams based on a strain gradient theory vol.48, pp.3, 2013, https://doi.org/10.12989/sem.2013.48.3.415
  11. Thermo-mechanical vibration analysis of rotating nonlocal nanoplates applying generalized differential quadrature method vol.24, pp.15, 2017, https://doi.org/10.1080/15376494.2016.1227499
  12. Nonlocal finite element analysis and small scale effects of CNTs with Timoshenko beam theory vol.50, 2012, https://doi.org/10.1016/j.finel.2011.08.008
  13. Influence of thermal and surface effects on vibration behavior of nonlocal rotating Timoshenko nanobeam vol.122, pp.7, 2016, https://doi.org/10.1007/s00339-016-0196-3
  14. Comparison of modeling of the rotating tapered axially functionally graded Timoshenko and Euler–Bernoulli microbeams vol.83, 2016, https://doi.org/10.1016/j.physe.2016.04.011
  15. ANALYTICAL SOLUTIONS FOR VIBRATION OF SIMPLY SUPPORTED NONLOCAL NANOBEAMS WITH AN AXIAL FORCE vol.11, pp.02, 2011, https://doi.org/10.1142/S0219455411004087
  16. Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams vol.49, pp.11, 2011, https://doi.org/10.1016/j.ijengsci.2010.12.009
  17. Vibration analysis of a nano-turbine blade based on Eringen nonlocal elasticity applying the differential quadrature method vol.23, pp.19, 2017, https://doi.org/10.1177/1077546315627723
  18. Vibration of rotating functionally graded Timoshenko nano-beams with nonlinear thermal distribution 2017, https://doi.org/10.1080/15376494.2017.1285455
  19. Nonlocal and strain gradient based model for electrostatically actuated silicon nano-beams vol.21, pp.2, 2015, https://doi.org/10.1007/s00542-014-2110-2
  20. Nonlocal thermal-elasticity for nanobeam deformation: Exact solutions with stiffness enhancement effects vol.110, pp.1, 2011, https://doi.org/10.1063/1.3596568
  21. Nonlinear vibration behavior of a rotating nanobeam under thermal stress using Eringen’s nonlocal elasticity and DQM vol.122, pp.8, 2016, https://doi.org/10.1007/s00339-016-0245-y
  22. Thermal buckling of nanorod based on non-local elasticity theory vol.47, pp.5, 2012, https://doi.org/10.1016/j.ijnonlinmec.2011.09.023
  23. Nonlinear Constitutive Model for Axisymmetric Bending of Annular Graphene-Like Nanoplate with Gradient Elasticity Enhancement Effects vol.139, pp.8, 2013, https://doi.org/10.1061/(ASCE)EM.1943-7889.0000625
  24. Application of strain gradient elasticity theory for buckling analysis of protein microtubules vol.11, pp.5, 2011, https://doi.org/10.1016/j.cap.2011.02.006
  25. Wave propagation in double-walled carbon nanotubes on a novel analytically nonlocal Timoshenko-beam model vol.330, pp.8, 2011, https://doi.org/10.1016/j.jsv.2010.10.028
  26. INVESTIGATION OF SIZE EFFECTS ON STATIC RESPONSE OF SINGLE-WALLED CARBON NANOTUBES BASED ON STRAIN GRADIENT ELASTICITY vol.09, pp.02, 2012, https://doi.org/10.1142/S0219876212400324
  27. Forced vibration of an embedded single-walled carbon nanotube traversed by a moving load using nonlocal Timoshenko beam theory vol.11, pp.1, 2011, https://doi.org/10.12989/scs.2011.11.1.059
  28. Transverse vibration of rotary tapered microbeam based on modified couple stress theory and generalized differential quadrature element method vol.24, pp.3, 2017, https://doi.org/10.1080/15376494.2015.1128025
  29. On size-dependent vibration of rotary axially functionally graded microbeam vol.101, 2016, https://doi.org/10.1016/j.ijengsci.2015.12.008
  30. Dynamic behaviour of axially moving nanobeams based on nonlocal elasticity approach vol.26, pp.5, 2010, https://doi.org/10.1007/s10409-010-0374-z
  31. Non-classical stiffness strengthening size effects for free vibration of a nonlocal nanostructure vol.54, pp.1, 2012, https://doi.org/10.1016/j.ijmecsci.2011.09.007
  32. Recent Researches on Nonlocal Elasticity Theory in the Vibration of Carbon Nanotubes Using Beam Models: A Review vol.24, pp.3, 2017, https://doi.org/10.1007/s11831-016-9179-y
  33. Application of Eringen's nonlocal elasticity theory for vibration analysis of rotating functionally graded nanobeams vol.17, pp.5, 2016, https://doi.org/10.12989/sss.2016.17.5.837
  34. Nano-resonator dynamic behavior based on nonlocal elasticity theory vol.229, pp.14, 2015, https://doi.org/10.1177/0954406214562058
  35. Vibration analysis of Nano-Rotor's Blade applying Eringen nonlocal elasticity and generalized differential quadrature method vol.43, 2017, https://doi.org/10.1016/j.apm.2016.10.061
  36. Design and simulation of resonance based DC current sensor vol.3, pp.3, 2009, https://doi.org/10.12989/imm.2010.3.3.257
  37. Effective mechanical properties of micro/nano-scale porous materials considering surface effects vol.4, pp.2, 2011, https://doi.org/10.12989/imm.2011.4.2.107
  38. Numerical investigation of mechanical properties of nanowires: a review vol.5, pp.2, 2009, https://doi.org/10.12989/imm.2012.5.2.115
  39. Nonlocal Finite Element Analysis of CNTs with Timoshenko Beam Theory and Thermal Environment vol.93, pp.4, 2009, https://doi.org/10.1007/s40032-012-0041-1
  40. Dynamic stiffness matrix method for axially moving micro-beam vol.5, pp.4, 2009, https://doi.org/10.12989/imm.2012.5.4.385
  41. Wave propagation analysis of smart strain gradient piezo-magneto-elastic nonlocal beams vol.66, pp.2, 2018, https://doi.org/10.12989/sem.2018.66.2.237