Steel and Composite Structures

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Assessment of various nonlocal higher order theories for the bending and buckling behavior of functionally graded nanobeams

  • Rahmani, O. (Smart Structures and New Advanced Materials Laboratory, Department of Mechanical Engineering, University of Zanjan) ;
  • Refaeinejad, V. (Smart Structures and New Advanced Materials Laboratory, Department of Mechanical Engineering, University of Zanjan) ;
  • Hosseini, S.A.H. (Smart Structures and New Advanced Materials Laboratory, Department of Mechanical Engineering, University of Zanjan)
  • Received : 2016.08.27
  • Accepted : 2017.01.11
  • Published : 2017.02.28
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Abstract

In this paper, various nonlocal higher-order shear deformation beam theories that consider the size dependent effects in Functionally Graded Material (FGM) beam are examined. The presented theories fulfill the zero traction boundary conditions on the top and bottom surface of the beam and a shear correction factor is not required. Hamilton's principle is used to derive equation of motion as well as related boundary condition. The Navier solution is applied to solve the simply supported boundary conditions and exact formulas are proposed for the bending and static buckling. A parametric study is also included to investigate the effect of gradient index, length scale parameter and length-to-thickness ratio (aspect ratio) on the bending and the static buckling characteristics of FG nanobeams.

Keywords

References

  1. Ahouel, M., Houari, M.S.A., Bedia, E. and Tounsi, A. (2016), "Size-dependent mechanical behavior of functionally graded trigonometric shear deformable nanobeams including neutral surface position concept", Steel Compos. Struct., Int. J., 20(5), 963-981. https://doi.org/10.12989/scs.2016.20.5.963
  2. Akgoz, B. and Civalek, O. (2013), "A size-dependent shear deformation beam model based on the strain gradient elasticity theory", Int. J. Eng. Sci., 70, 1-14. https://doi.org/10.1016/j.ijengsci.2013.04.004
  3. Akgoz, B. and Civalek, O. (2014a), "A new trigonometric beam model for buckling of strain gradient microbeams", Int. J. Mech. Sci., 81, 88-94. https://doi.org/10.1016/j.ijmecsci.2014.02.013
  4. Akgoz, B. and Civalek, O. (2014b), "Shear deformation beam models for functionally graded microbeams with new shear correction factors", Compos. Struct., 112, 214-225. https://doi.org/10.1016/j.compstruct.2014.02.022
  5. Aydogdu, M. (2009), "A new shear deformation theory for laminated composite plates", Compos. Struct., 89(10), 94. https://doi.org/10.1016/j.compstruct.2008.07.008
  6. Aydogdu, M. and Taskin, V. (2007), "Free vibration analysis of functionally graded beams with simply supported edges", Mater. Des., 28(5), 1651-1656. https://doi.org/10.1016/j.matdes.2006.02.007
  7. Benatta, M., Mechab, I., Tounsi, A. and Adda Bedia, E. (2008), "Static analysis of functionally graded short beams including warping and shear deformation effects", Comput. Mater. Sci., 44(2), 765-773. https://doi.org/10.1016/j.commatsci.2008.05.020
  8. Chakraborty, A., Gopalakrishnan, S. and Reddy, J. (2003), "A new beam finite element for the analysis of functionally graded materials", Int. J. Mech. Sci., 45(3), 519-539. https://doi.org/10.1016/S0020-7403(03)00058-4
  9. Duan, W.H., Challamel, N., Wang, C. and Ding, Z. (2013), "Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams", J. Appl. Phys., 114(10), 104312. https://doi.org/10.1063/1.4820565
  10. Ekinci, K. and Roukes, M. (2005), "Nanoelectromechanical systems", Review of Scientific Instruments, 76(6), 061101. https://doi.org/10.1063/1.1927327
  11. Eltaher, M.A., Emam, S.A. and Mahmoud, F.F. (2012), "Free vibration analysis of functionally graded size-dependent nanobeams", Appl. Math. Comput., 218(14), 7406-7420. https://doi.org/10.1016/j.amc.2011.12.090
  12. Eltaher, M.A., Khairy, A., Sadoun, A.M. and Omar, F.-A. (2014), "Static and buckling analysis of functionally graded Timoshenko nanobeams", Appl. Math. Comput., 229, 283-295.
  13. 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
  14. 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
  15. Fallah, A. and Aghdam, M. (2011), "Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation", Eur. J. Mech.-A/Solids, 30(4), 571-583. https://doi.org/10.1016/j.euromechsol.2011.01.005
  16. Fu, Y., Du, H. and Zhang, S. (2003), "Functionally graded TiN/TiNi shape memory alloy films", Mater Lett., 57(20), 2995-2999. https://doi.org/10.1016/S0167-577X(02)01419-2
  17. Hasanyan, D., Batra, R. and Harutyunyan, S. (2008), "Pull-in instabilities in functionally graded microthermoelectromechanical systems", J. Therm. Stress., 31(10), 1006-1021. https://doi.org/10.1080/01495730802250714
  18. Hosseini, S. and Rahmani, O. (2016a), "Exact solution for axial and transverse dynamic response of functionally graded nanobeam under moving constant load based on nonlocal elasticity theory", Meccanica, 1-17.
  19. Hosseini, S. and Rahmani, O. (2016b), "Free vibration of shallow and deep curved FG nanobeam via nonlocal Timoshenko curved beam model", Appl. Physics A, 122(3), 1-11.
  20. Hosseini, S. and Rahmani, O. (2016c), "Thermomechanical vibration of curved functionally graded nanobeam based on nonlocal elasticity", J. Therm. Stress., 39(10), 1252-1267. https://doi.org/10.1080/01495739.2016.1215731
  21. Hosseini, A.H., Rahmani, O., Nikmehr, M. and Golpayegani, I.F. (2016), "Axial Vibration of Cracked Nanorods Embedded in Elastic Foundation Based on a Nonlocal Elasticity Model", Sensor Letters, 14(10), 1019-1025. https://doi.org/10.1166/sl.2016.3575
  22. Janghorban, M. (2012), "Static analysis of tapered nanowires based on nonlocal Euler-Bernoulli beam theory via differential quadrature method", Latin Am. J. Solids Struct., 9(2), 1-10.
  23. Jandaghian, A.A. and Rahmani, O. (2016), "Vibration analysis of functionally graded piezoelectric nanoscale plates by nonlocal elasticity theory: An analytical solution", Superlatt. Microstruct., 100, 57-75. https://doi.org/10.1016/j.spmi.2016.08.046
  24. Janghorban, M. and Zare, A. (2011), "Free vibration analysis of functionally graded carbon nanotubes with variable thickness by differential quadrature method", Physica E: Low-Dimens. Syst. Nanostruct., 43(9), 1602-1604. https://doi.org/10.1016/j.physe.2011.05.002
  25. Jha, D., Kant, T. and Singh, R. (2013), "Free vibration of functionally graded plates with a higher-order shear and normal deformation theory", Int. J. Struct. Stab. Dyn., 13(1), 1350004. https://doi.org/10.1142/S0219455413500041
  26. Karama, M., Afaq, K. and Mistou, S. (2003), "Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity", Int. J. Solids Struct., 40(6), 1525-1546. https://doi.org/10.1016/S0020-7683(02)00647-9
  27. Ke, L.-L., Yang, J., Kitipornchai, S. and Xiang, Y. (2009), "Flexural vibration and elastic buckling of a cracked Timoshenko beam made of functionally graded materials", Mech. Adv. Mater. Struct., 16(6), 488-502. https://doi.org/10.1080/15376490902781175
  28. Khalili, S., Jafari, A. and Eftekhari, S. (2010), "A mixed Ritz-DQ method for forced vibration of functionally graded beams carrying moving loads", Compos. Struct., 92(10), 2497-2511. https://doi.org/10.1016/j.compstruct.2010.02.012
  29. Kocaturk, T., Simsek, M. and Akbas, S.D. (2011), "Large displacement static analysis of a cantilever Timoshenko beam composed of functionally graded material", Sci. Eng. Compos. Mater., 18(1-2), 21-34. https://doi.org/10.1515/secm.2011.005
  30. Kong, S., Zhou, S., Nie, Z. and Wang, K. (2008), "The size-dependent natural frequency of Bernoulli-Euler micro-beams", Int. J. Eng. Sci., 46(5), 427-437. https://doi.org/10.1016/j.ijengsci.2007.10.002
  31. Lavrik, N.V., Sepaniak, M.J. and Datskos, P.G. (2004), "Cantilever transducers as a platform for chemical and biological sensors", Rev. Sci. Instru., 75(7), 2229-2253. https://doi.org/10.1063/1.1763252
  32. Li, X.F. (2008), "A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler-Bernoulli beams", J. Sound Vib., 318(4-5), 1210-1229. https://doi.org/10.1016/j.jsv.2008.04.056
  33. Lu, C., Lim, C. and Chen, W. (2009), "Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory", Int. J. Solids Struct., 46(5), 1176-1185. https://doi.org/10.1016/j.ijsolstr.2008.10.012
  34. McFarland, A.W. and Colton, J.S. (2005), "Role of material microstructure in plate stiffness with relevance to microcantilever sensors", J. Micromech. Microeng., 15(5), 1060. https://doi.org/10.1088/0960-1317/15/5/024
  35. Mesut, S. (2011), "Forced vibration of an embedded single-walled carbon nanotube traversed by a moving load using nonlocal Timoshenko theory", Steel Compos. Struct., Int. J., 11(1), 59-76. https://doi.org/10.12989/scs.2011.11.1.059
  36. Mohammadi-Alasti, B., Rezazadeh, G., Borgheei, A.-M., Minaei, S. and Habibifar, R. (2011), "On the mechanical behavior of a functionally graded micro-beam subjected to a thermal moment and nonlinear electrostatic pressure", Compos. Struct., 93(6), 1516-1525. https://doi.org/10.1016/j.compstruct.2010.11.013
  37. Mohanty, S., Dash, R. and Rout, T. (2012), "Static and dynamic stability analysis of a functionally graded Timoshenko beam", Int. J. Struct. Stab. Dyn., 12(4), 1250025. https://doi.org/10.1142/S0219455412500253
  38. Nguyen, N.-T., Kim, N.-I. and Lee, J. (2014), "Analytical solutions for bending of transversely or axially FG nonlocal beams", Steel Compos. Struct., Int. J., 17(5), 641-665. https://doi.org/10.12989/scs.2014.17.5.641
  39. Rahaeifard, M., Kahrobaiyan, M. and Ahmadian, M. (2009), "Sensitivity analysis of atomic force microscope cantilever made of functionally graded materials", ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference.
  40. Rahmani, O., Hosseini, S., Noroozi Moghaddam, M. and Fakhari Golpayegani, I. (2015), "Torsional vibration of cracked nanobeam based on nonlocal stress theory with various boundary conditions: An analytical study", Int. J. Appl. Mech., 7(3), 1550036. https://doi.org/10.1142/S1758825115500362
  41. Rahmani, O., Hosseini, S. and Parhizkari, M. (2016a), "Buckling of double functionally-graded nanobeam system under axial load based on nonlocal theory: An analytical approach", Microsyst. Technol., 1-13.
  42. Rahmani, O., Hosseini, S. and Hayati, H. (2016b), "Frequency analysis of curved nano-sandwich structure based on a nonlocal model", Modern Phys. Lett. B, 30(10), 1650136.
  43. Reddy, J.N. (1984), "A simple higher-order theory for laminated composite plates", J. Appl. Mech., 51(4), 745-752. https://doi.org/10.1115/1.3167719
  44. Reddy, J. (2007), "Nonlocal theories for bending, buckling and vibration of beams", Int. J. Eng. Sci., 45(2), 288-307. https://doi.org/10.1016/j.ijengsci.2007.04.004
  45. Refaeinejad, V., Rahmani, O. and Hosseini, S.A.H. (2016a), "An analytical solution for bending, buckling, and free vibration of FG nanobeam lying on Winkler-Pasternak elastic foundation using different nonlocal higher order shear deformation beam theories", Int. J. Sci. Iranica.
  46. Refaeinejad, V., Rahmani, O. and Hosseini, S.A.H. (2016b), "Evaluation of nonlocal higher order shear deformation models for the vibrational analysis of functionally graded nanostructures", Mech. Adv. Mater. Struct.
  47. Saggam, N. (2012), "Scale effects on coupled wave propagation in single walled carbon nanotubes", Latin Am. J. Solids Struct., ABCM J., 9(4), 497. https://doi.org/10.1590/S1679-78252012000400005
  48. Sallai, B.O., Tounsi, A., Mechab, I., Bachir, B.M., Meradjah, M. and Adda Bedia, E.A. (2009), "A theoretical analysis of flexional bending of Al/Al2O3O S-FGM thick beams", Comput. Mater. Sci., 44(4), 1344-1350. https://doi.org/10.1016/j.commatsci.2008.09.001
  49. Simsek, M. (2010), "Non-linear vibration analysis of a functionally graded Timoshenko beam under action of a moving harmonic load", Compos. Struct., 92(10), 2532-2546. https://doi.org/10.1016/j.compstruct.2010.02.008
  50. Simsek, M. (2012), "Nonlocal effects in the free longitudinal vibration of axially functionally graded tapered nanorods", Comput. Mater. Sci., 61, 257-265. https://doi.org/10.1016/j.commatsci.2012.04.001
  51. Simsek, M. and Cansiz, S. (2012), "Dynamics of elastically connected double-functionally graded beam systems with different boundary conditions under action of a moving harmonic load", Compos. Struct., 94(9), 2861-2878. https://doi.org/10.1016/j.compstruct.2012.03.016
  52. Simsek, M. and Kocaturk, T. (2009), "Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load", Compos. Struct., 90(4), 465-473. https://doi.org/10.1016/j.compstruct.2009.04.024
  53. Simsek, M. and Reddy, J.N. (2013), "Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory", Int. J. Eng. Sci., 64, 37-53. https://doi.org/10.1016/j.ijengsci.2012.12.002
  54. Simsek, M. and Yurtcu, H. (2013), "Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory", Compos. Struct., 97, 378-386. https://doi.org/10.1016/j.compstruct.2012.10.038
  55. Sina, S., Navazi, H. and Haddadpour, H. (2009), "An analytical method for free vibration analysis of functionally graded beams", Mater. Des., 30(3), 741-747. https://doi.org/10.1016/j.matdes.2008.05.015
  56. Soldatos, K. (1992), "A transverse shear deformation theory for homogeneous monoclinic plates", Acta Mechanica, 94(3-4), 195-220. https://doi.org/10.1007/BF01176650
  57. Sudak, L. (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
  58. Thai, H.-T. (2012), "A nonlocal beam theory for bending, buckling, and vibration of nanobeams", Int. J. Eng. Sci., 52, 56-64. https://doi.org/10.1016/j.ijengsci.2011.11.011
  59. Thai, H.-T. and Vo, T.P. (2012a), "A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams", Int. J. Eng. Sci., 54, 58-66. https://doi.org/10.1016/j.ijengsci.2012.01.009
  60. Thai, H.-T. and Vo, T.P. (2012b), "Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories", Int. J. Mech. Sci., 62(1), 57-66. https://doi.org/10.1016/j.ijmecsci.2012.05.014
  61. Touratier, M. (1991), "An efficient standard plate theory", Int. J. Eng. Sci., 29(8), 901-916. https://doi.org/10.1016/0020-7225(91)90165-Y
  62. Wang, L.F. and Hu, H. (2005), "Flexural wave propagation in single-walled carbon nanotubes", Phys. Rev. B: Condens. Matter., 71(19), 195412. https://doi.org/10.1103/PhysRevB.71.195412
  63. Wang, C., Zhang, Y. and He, X. (2007), "Vibration of nonlocal Timoshenko beams", Nanotechnol., 18(10), 105401. https://doi.org/10.1088/0957-4484/18/10/105401
  64. Wang, C., Zhang, Y., Ramesh, S.S. and Kitipornchai, S. (2006), "Buckling analysis of micro-and nano-rods/tubes based on nonlocal Timoshenko beam theory", J. Phys. D: Appl. Phys., 39(17), 3904. https://doi.org/10.1088/0022-3727/39/17/029
  65. Wang, C., Kitipornchai, S., Lim, C. and Eisenberger, M. (2008), "Beam bending solutions based on nonlocal Timoshenko beam theory", J. Eng. Mech., 134(6), 475-481. https://doi.org/10.1061/(ASCE)0733-9399(2008)134:6(475)
  66. Witvrouw, A. and Mehta, A. (2005), "The use of functionally graded poly-SiGe layers for MEMS applications", Materials Science Forum, pp. 255-260.
  67. Zhang, J. and Fu, Y. (2012), "Pull-in analysis of electrically actuated viscoelastic microbeams based on a modified couple stress theory", Meccanica, 47(7), 1649-1658. https://doi.org/10.1007/s11012-012-9545-2
  68. Zhang, Z., Challamel, N. and Wang, C. (2013), "Eringen's small length scale coefficient for buckling of nonlocal Timoshenko beam based on microstructured beam model", J. Appl. Phys., 114(11), 114902. https://doi.org/10.1063/1.4821246

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