Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

A nonlocal zeroth-order shear deformation theory for nonlinear postbuckling of nanobeams

  • Bellifa, Hichem (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Benrahou, Kouider Halim (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Bousahla, Abdelmoumen Anis (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Tounsi, Abdelouahed (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Mahmoud, S.R. (Department of Mathematics, Faculty of Science, King Abdulaziz University)
  • Received : 2015.02.24
  • Accepted : 2017.05.04
  • Published : 2017.06.25
⟨ Previous Next ⟩

Abstract

In this work, a nonlocal zeroth-order shear deformation theory is developed for the nonlinear postbuckling behavior of nanoscale beams. The beauty of this formulation is that, in addition to including the nonlocal effect according to the nonlocal elasticity theory of Eringen, the shear deformation effect is considered in the axial displacement within the use of shear forces instead of rotational displacement like in existing shear deformation theories. The principle of virtual work together of the nonlocal differential constitutive relations of Eringen, are considered to obtain the equations of equilibrium. Closed-form solutions for the critical buckling load and the amplitude of the static nonlinear response in the postbuckling state for simply supported and clamped clamped nanoscale beams are determined.

Keywords

References

  1. Adda Bedia, W., Benzair, A., Semmah, A., Tounsi, A. and Mahmoud, S.R. (2015), "On the thermal buckling characteristics of armchair single-walled carbon nanotube embedded in an elastic medium based on nonlocal continuum elasticity", Brazil. J. Phys., 45(2), 225-233. https://doi.org/10.1007/s13538-015-0306-2
  2. Ahouel, M., Houari, M.S.A., Adda Bedia, E.A. and Tounsi, A. (2016), "Size-dependent mechanical behavior of functionally graded trigonometric shear deformable nanobeams including neutral surface position concept", Steel Compos. Struct., 20(5), 963-981. https://doi.org/10.12989/scs.2016.20.5.963
  3. Ait Amar Meziane, M., Abdelaziz, H.H. and Tounsi, A. (2014), "An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions", J. Sandw. Struct. Mater., 16(3), 293-318. https://doi.org/10.1177/1099636214526852
  4. Ait Atmane, H., Tounsi, A., Bernard, F. and Mahmoud, S.R. (2015), "A computational shear displacement model for vibrational analysis of functionally graded beams with porosities", Steel Compos. Struct., 19(2), 369-384. https://doi.org/10.12989/scs.2015.19.2.369
  5. Ait Yahia, S., Ait Atmane, H., Houari, M.S.A. and Tounsi, A. (2015), "Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories", Struct. Eng. Mech., 53(6), 1143-1165. https://doi.org/10.12989/sem.2015.53.6.1143
  6. Amara, K., Tounsi, A. and Adda Bedia, E.A. (2010), "Nonlocal elasticity effect on column buckling of multiwalled carbon nanotubes under temperature field", Appl. Math. Model., 34(12), 3933-3942. https://doi.org/10.1016/j.apm.2010.03.029
  7. Attia, A., Tounsi, A., Adda Bedia, E.A. and Mahmoud, S.R. (2015), "Free vibration analysis of functionally graded plates with temperature-dependent properties using various four variable refined plate theories", Steel Compos. Struct., 18(1), 187-212. https://doi.org/10.12989/scs.2015.18.1.187
  8. Ball, P. (2001), "Roll up for the revolution", Nature, 414(6860), 142-144. https://doi.org/10.1038/35102721
  9. Baughman, R.H., Zakhidov, A.A. and de Heer, W.A. (2002), "Carbon nanotubes-the route toward applications", Science, 297(5582), 787-792. https://doi.org/10.1126/science.1060928
  10. Belabed, Z., Houari, M.S.A., Tounsi, A., Mahmoud, S.R. and Anwar Beg, O. (2014), "An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates", Compos.: Part B, 60, 274-283. https://doi.org/10.1016/j.compositesb.2013.12.057
  11. Beldjelili, Y., Tounsi, A. and Mahmoud, S.R. (2016), "Hygro-thermo-mechanical bending of S-FGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory", Smart Struct. Syst., 18(4), 755-786. https://doi.org/10.12989/sss.2016.18.4.755
  12. Bellifa, H., Benrahou, K.H., Hadji, L., Houari, M.S.A. and Tounsi, A. (2016), "Bending and free vibration analysis of functionally graded plates using a simple shear deformation theory and the concept the neutral surface position", J. Braz. Soc. Mech. Sci. Eng., 38(1), 265-275. https://doi.org/10.1007/s40430-015-0354-0
  13. Belkorissat, I., Houari, M.S.A., Tounsi, A., Adda Bedia, E.A. and Mahmoud, S.R. (2015), "On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable model", Steel Compos. Struct., 18(4), 1063-1081. https://doi.org/10.12989/scs.2015.18.4.1063
  14. Benguediab, S., Tounsi, A., Zidour, M. and Semmah, A. (2014), "Chirality and scale effects on mechanical buckling properties of zigzag double-walled carbon nanotubes", Compos. Part B., 57, 21-24. https://doi.org/10.1016/j.compositesb.2013.08.020
  15. Bennoun, M., Houari, M.S.A. and Tounsi, A. (2016), "A novel five variable refined plate theory for vibration analysis of functionally graded sandwich plates", Mech. Adv. Mater. Struct., 23(4), 423-431. https://doi.org/10.1080/15376494.2014.984088
  16. Bessaim, A., Houari, M.S.A., Tounsi, A., Mahmoud, S.R. and Adda Bedia, E.A. (2013), "A new higher-order shear and normal deformation theory for the static and free vibration analysis of sandwich plates with functionally graded isotropic face sheets", J. Sandw. Struct. Mater., 15(6), 671-703. https://doi.org/10.1177/1099636213498888
  17. Benzair, A., Tounsi, A., Besseghier, A., Heireche, H., Moulay, N. and Boumia, L. (2008), "The thermal effect on vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory", J. Phys. D: Appl. Phys., 41(22), 225404. https://doi.org/10.1088/0022-3727/41/22/225404
  18. Berrabah, H.M., Tounsi, A., Semmah, A. and Adda Bedia, E.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
  19. Besseghier, A., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2017), "Free vibration analysis of embedded nanosize FG plates using a new nonlocal trigonometric shear deformation theory", Smart Struct. Syst., 19(6), 601-614. https://doi.org/10.12989/SSS.2017.19.6.601
  20. Bodily, B.H. and Sun, C.T. (2003), "Structural and equivalent continuum properties of single-walled carbon nanotubes", Int. J. Mater. Product Technol., 18(4-6), 381-397. https://doi.org/10.1504/IJMPT.2003.002498
  21. Bouafia, K., Kaci, A., Houari, M.S.A., Benzair, A. and Tounsi, A. (2017), "A nonlocal quasi-3D theory for bending and free flexural vibration behaviors of functionally graded nanobeams", Smart Struct. Syst., 19(2), 115-126. https://doi.org/10.12989/sss.2017.19.2.115
  22. Bouderba, B., Houari, M.S.A. and Tounsi, A. (2013), "Thermomechanical bending response of FGM thick plates resting on Winkler-Pasternak elastic foundations", Steel Compos. Struct., 14(1), 85-104. https://doi.org/10.12989/scs.2013.14.1.085
  23. Bouderba, B., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2016), "Thermal stability of functionally graded sandwich plates using a simple shear deformation theory", Struct. Eng. Mech., 58(3), 397-422. https://doi.org/10.12989/sem.2016.58.3.397
  24. Boukhari, A., Ait Atmane, H., Tounsi, A., Adda Bedia, E.A. and Mahmoud, S.R. (2016), "An efficient shear deformation theory for wave propagation of functionally graded material plates", Struct. Eng. Mech., 57(5), 837-859. https://doi.org/10.12989/sem.2016.57.5.837
  25. Bounouara, F., Benrahou, K.H., Belkorissat, I. and Tounsi, A. (2016), "A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundation", Steel Compos. Struct., 20(2), 227-249. https://doi.org/10.12989/scs.2016.20.2.227
  26. Bourada, M., Kaci, A., Houari, M.S.A. and Tounsi, A. (2015), "A new simple shear and normal deformations theory for functionally graded beams", Steel Compos. Struct., 18(2), 409-423. https://doi.org/10.12989/scs.2015.18.2.409
  27. Bousahla, A.A., Houari, M.S.A., Tounsi, A. and Adda Bedia, E.A. (2014), "A novel higher order shear and normal deformation theory based on neutral surface position for bending analysis of advanced composite plates", Int. J. Comput. Meth., 11(6), 1350082. https://doi.org/10.1142/S0219876213500825
  28. Bousahla, A.A., Benyoucef, S., Tounsi, A. and Mahmoud, S.R. (2016), "On thermal stability of plates with functionally graded coefficient of thermal expansion", Struct. Eng. Mech., 60(2), 313-335. https://doi.org/10.12989/sem.2016.60.2.313
  29. Chikh, A., Tounsi, A., Hebali, H. and Mahmoud, S.R. (2017), "Thermal buckling analysis of cross-ply laminated plates using a simplified HSDT", Smart Struct. Syst., 19(3), 289-297. https://doi.org/10.12989/sss.2017.19.3.289
  30. Draiche, K., Tounsi, A. and Mahmoud, S.R. (2016), "A refined theory with stretching effect for the flexure analysis of laminated composite plates", Geomech. Eng., 11(5), 671-690. https://doi.org/10.12989/gae.2016.11.5.671
  31. Ebrahimi, F. and Salari, E. (2015), "Effect of various thermal loadings on buckling and vibrational characteristics of nonlocal temperature dependent FG nanobeams", Mech. Adv. Mater. Struct., 23(12), 1379-1397.
  32. Ebrahimi, F., Salari, E. and Hosseini, S.A.H. (2015), "Thermomechanical vibration behavior of FG nanobeams subjected to linear and nonlinear temperature distributions", J. Therm. Stress., 38(12), 1362-1388.
  33. Ebrahimi, F. and Barati, M.R. (2016a), "Nonlocal strain gradient theory for damping vibration analysis of viscoelastic inhomogeneous nano-scale beams embedded in visco-Pasternak foundation", J. Vib. Control, 1077546316678511.
  34. Ebrahimi, F. and Barati, M.R. (2016b), "A unified formulation for dynamic analysis of nonlocal heterogeneous nanobeams in hygro-thermal environment", Appl. Phys. A, 122(9), 792.
  35. 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
  36. Emam, S.A. (2013), "A general nonlocal nonlinear model for buckling of nanobeams", Appl. Math. Model., 37(10), 6929-6939. https://doi.org/10.1016/j.apm.2013.01.043
  37. 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
  38. Eringen, A.C. (2002), Nonlocal continuum field theories, Springer, New York.
  39. Fahsi, A., Tounsi, A., Hebali, H., Chikh, A., Adda Bedia, E.A. and Mahmoud, S.R. (2017), "A four variable refined nth-order shear deformation theory for mechanical and thermal buckling analysis of functionally graded plates", Geomech. Eng., (In press).
  40. Fekrar, A., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2014), "A new five-unknown refined theory based on neutral surface position for bending analysis of exponential graded plates", Meccanica, 49(4), 795-810. https://doi.org/10.1007/s11012-013-9827-3
  41. Hadji, L., Khelifa, Z. and Adda Bedia, E.A. (2015), "A new higher order shear deformation model for functionally graded beams", KSCE J. Civ. Eng., 20(5), 1835-1841.
  42. Hamidi, A., Houari, M.S.A., Mahmoud, S.R. and Tounsi, A. (2015), "A sinusoidal plate theory with 5-unknowns and stretching effect for thermomechanical bending of functionally graded sandwich plates", Steel Compos. Struct., 18(1), 235-253. https://doi.org/10.12989/scs.2015.18.1.235
  43. Hebali, H., Tounsi, A., Houari, M.S.A., Bessaim, A. and Adda Bedia, E.A. (2014), "A new quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates", J. Eng. Mech., ASCE, 140(2), 374-383. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000665
  44. Heireche, H., Tounsi, A., Benzair, A., Maachou, M. and Adda Bedia, E.A. (2008), "Sound wave propagation in single- walled carbon nanotubes using nonlocal elasticity", Physica E, 40(8), 2791-2799. https://doi.org/10.1016/j.physe.2007.12.021
  45. Houari, M.S.A., Tounsi, A., Bessaim, A. and Mahmoud, S.R. (2016), "A new simple three -unknown sinusoidal shear deformation theory for functionally graded plates", Steel Compos. Struct., 22(2), 257-276. https://doi.org/10.12989/scs.2016.22.2.257
  46. Larbi Chaht, F., Kaci, A., Houari, M.S.A., Tounsi, A., Anwar Beg, O. and Mahmoud, S.R. (2015), "Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect", Steel Compos. Struct., 18(2), 425-442. https://doi.org/10.12989/scs.2015.18.2.425
  47. Li, C. and Chou, T.W. (2003a), "A structural mechanics approach for the analysis of carbon nanotubes", Int. J. Solid. Struct., 40(10), 2487-2499. https://doi.org/10.1016/S0020-7683(03)00056-8
  48. Li, C. and Chou, T.W. (2003b), "Single-walled nanotubes as ultra high frequency nanomechanical oscillators", Phys. Rev. B, 68, 073405. https://doi.org/10.1103/PhysRevB.68.073405
  49. Lu, P., Lee, H.P., Lu, C. and Zhang, P.Q. (2007), "Application of nonlocal beam models for carbon nanotubes", Int. J. Solid. Struct., 44(16), 5289-5300. https://doi.org/10.1016/j.ijsolstr.2006.12.034
  50. Mahi, A., Adda Bedia, E.A. and Tounsi, A. (2015), "A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates", Appl. Math. Model., 39(9), 2489-2508. https://doi.org/10.1016/j.apm.2014.10.045
  51. Meksi, R., Benyoucef, S., Mahmoudi, A., Tounsi, A., Adda Bedia, E.A. and Mahmoud, S.R. (2017), "An analytical solution for bending, buckling and vibration responses of FGM sandwich plates", J. Sandw. Struct. Mater., 1099636217698443.
  52. Meradjah, M., Kaci, A., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2015), "A new higher order shear and normal deformation theory for functionally graded beams", Steel Compos. Struct., 18(3), 793-809. https://doi.org/10.12989/scs.2015.18.3.793
  53. Murmu, T. and Adhikari, S. (2012), "Non local elasticity based vibration of initially pre-stressed coupled nanobeam systems", Eur. J. Mech. A-Solids, 34, 52-62. https://doi.org/10.1016/j.euromechsol.2011.11.010
  54. Mustapha, K.B. and Zhong, Z.W. (2010a), "The thermo-mechanical vibration of a single-walled carbon nanotube studied using the Bubnov-Galerkin method", Physica E., 43(1), 375-381. https://doi.org/10.1016/j.physe.2010.08.012
  55. Naceri, M., Zidour, M., Semmah, A., Houari, M.S.A., Benzair, A. and Tounsi, A. (2011), "Sound wave propagation in armchair single walled carbon nanotubes under thermal environment", J. Appl. Phys., 110(12), 124322. https://doi.org/10.1063/1.3671636
  56. Pradhan, S.C. and Phadikar, J.K. (2008), "Nonlinear analysis of carbon nanotubes", Proceedings of Fifth International Conference on Smart Materials, Structures and Systems, Indian Institute of Science, Bangalore, 24-26 July 2008, paper ID 19.
  57. Ray, M.C. (2003), "Zeroth-order shear deformation theory for laminated composite plates", J. Appl. Mech., 70(3), 374-380. https://doi.org/10.1115/1.1558077
  58. Reddy, J.N. (2007), "Nonlocal theories for buckling bending and vibration of beams", Int. Eng. Sci., 45(2), 288-307. https://doi.org/10.1016/j.ijengsci.2007.04.004
  59. Roque, C.M.C., Ferreira, A.J.M. and Reddy, J.N. (2011), "Analysis of Timoshenko nanobeams with a nonlocal formulation and meshless method", Int. J. Eng. Sci., 49(9), 976-984. https://doi.org/10.1016/j.ijengsci.2011.05.010
  60. Sayyad, A.S. and Ghugal, Y.M. (2014), "Flexure of cross-ply laminated plates using equivalent single layer trigonometric shear deformation theory", Struct. Eng. Mech., 51(5), 867-891. https://doi.org/10.12989/sem.2014.51.5.867
  61. Song, J., Shen, J. and Li, X.F. (2010), "Effects of initial axial stress on waves propagating in carbon nanotubes using a generalized nonlocal model", Comput. Mater. Sci., 49(3), 518-523. https://doi.org/10.1016/j.commatsci.2010.05.043
  62. Swaminathan, K. and Naveenkumar, D.T. (2014), "Higher order refined computational models for the stability analysis of FGM plates-Analytical solutions", Eur. J. Mech. A/Solid., 47, 349-361. https://doi.org/10.1016/j.euromechsol.2014.06.003
  63. Taibi, F.Z., Benyoucef, S., Tounsi, A., Bachir Bouiadjra, R., Adda Bedia, E.A. and Mahmoud, S.R. (2015), "A simple shear deformation theory for thermo-mechanical behaviour of functionally graded sandwich plates on elastic foundations", J. Sandw. Struct. Mater., 17(2), 99-129. https://doi.org/10.1177/1099636214554904
  64. 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), 104301. https://doi.org/10.1063/1.3018330
  65. Tounsi, A., Benguediab, S., Adda Bedia, E.A., Semmah, A. and Zidour, M. (2013a), "Nonlocal effects on thermal buckling properties of double-walled carbon nanotubes", Adv. Nano Res., 1(1), 1-11. https://doi.org/10.12989/anr.2013.1.1.001
  66. Tounsi, A., Houari, M.S.A., Benyoucef, S. and Adda Bedia, E.A. (2013b), "A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates", Aerosp. Sci. Technol., 24(1), 209-220. https://doi.org/10.1016/j.ast.2011.11.009
  67. Tounsi, A., Houari, M.S.A. and Bessaim, A. (2016), "A new 3-unknowns non-polynomial plate theory for buckling and vibration of functionally graded sandwich plate", Struct. Eng. Mech., 60(4), 547-565. https://doi.org/10.12989/sem.2016.60.4.547
  68. Wang, Q., Zhou, G.Y. and Lin, K.C. (2006a), "Scale effect on wave propagation of double walled carbon nanotubes", Int. J. Solid. Struct., 43(20), 6071-6084. https://doi.org/10.1016/j.ijsolstr.2005.11.005
  69. Wang, Q. and Varadan, V.K. (2006b), "Vibration of carbon nanotubes studied using nonlocal continuum mechanics", Smart Mater. Struct., 15(2), 659-666. https://doi.org/10.1088/0964-1726/15/2/050
  70. 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
  71. Zidi, M., Tounsi, A., Houari, M.S.A., Adda Bedia, E.A. and Anwar Beg, O. (2014), "Bending analysis of FGM plates under hygro-thermo-mechanical loading using a four variable refined plate theory", Aerosp. Sci. Technol., 34, 24-34. https://doi.org/10.1016/j.ast.2014.02.001

Cited by

  1. Nonlocal three-dimensional theory of elasticity for buckling behavior of functionally graded porous nanoplates using volume integrals vol.5, pp.9, 2018, https://doi.org/10.1088/2053-1591/aad4c3
  2. A novel approach for nonlinear bending response of macro- and nanoplates with irregular variable thickness under nonuniform loading in thermal environment pp.1539-7742, 2019, https://doi.org/10.1080/15397734.2018.1557529
  3. Thermodynamic Model for Calculation the Position of Quitting Surface of Supersonic Beam Based on Speed Measurements vol.122, pp.46, 2017, https://doi.org/10.1021/acs.jpca.8b09149
  4. Influence of shear preload on wave propagation in small-scale plates with nanofibers vol.70, pp.4, 2017, https://doi.org/10.12989/sem.2019.70.4.407
  5. A simple quasi-3D HSDT for the dynamics analysis of FG thick plate on elastic foundation vol.31, pp.5, 2017, https://doi.org/10.12989/scs.2019.31.5.503
  6. A new higher-order shear and normal deformation theory for the buckling analysis of new type of FGM sandwich plates vol.72, pp.5, 2019, https://doi.org/10.12989/sem.2019.72.5.653
  7. On the modeling of dynamic behavior of composite plates using a simple nth-HSDT vol.29, pp.6, 2017, https://doi.org/10.12989/was.2019.29.6.371
  8. A numerical method for dynamic characteristics of nonlocal porous metal-ceramic plates under periodic dynamic loads vol.7, pp.1, 2020, https://doi.org/10.12989/smm.2020.7.1.027
  9. Buckling response of smart plates reinforced by nanoparticles utilizing analytical method vol.35, pp.1, 2020, https://doi.org/10.12989/scs.2020.35.1.001
  10. Axial dynamic analysis of a Bishop nanorod with arbitrary boundary conditions vol.100, pp.12, 2020, https://doi.org/10.1002/zamm.202000039
  11. On bending analysis of perforated microbeams including the microstructure effects vol.76, pp.6, 2020, https://doi.org/10.12989/sem.2020.76.6.765
  12. Nonlinear vibration and stability of FG nanotubes conveying fluid via nonlocal strain gradient theory vol.78, pp.1, 2021, https://doi.org/10.12989/sem.2021.78.1.103
  13. Mechanical analysis of bi-functionally graded sandwich nanobeams vol.11, pp.1, 2017, https://doi.org/10.12989/anr.2021.11.1.055
  14. On static buckling of multilayered carbon nanotubes reinforced composite nanobeams supported on non-linear elastic foundations vol.40, pp.3, 2021, https://doi.org/10.12989/scs.2021.40.3.389