Advances in materials Research

  • 제6권1호
  • /
  • Pages.45-63
  • /
  • 2017
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  • 2234-0912(pISSN)
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  • 2234-179X(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Bending and stability analysis of size-dependent compositionally graded Timoshenko nanobeams with porosities

  • Bensaid, Ismail (IS2M Laboratory, Faculty of Technology, Mechanical engineering Department, University of Tlemcen) ;
  • Guenanou, Ahmed (Faculty of Technology, Mechanical engineering Department, University of Tlemcen)
  • 투고 : 2017.02.03
  • 심사 : 2017.04.26
  • 발행 : 2017.03.25
⟨ 이전 논문 다음 논문 ⟩

초록

In this article, static deflection and buckling of functionally graded (FG) nanoscale beams made of porous material are carried out based on the nonlocal Timoshenko beam model which captures the small scale influences. The exact position of neutral axis is fixed, to eliminate the stretching and bending coupling due to the unsymmetrical material change along the FG nanobeams thickness. The material properties of FG beam are graded through the thickness on the basis of the power-law form, which is modified to approximate the material properties with two models of porosity phases. By employing Hamilton's principle, the nonlocal governing equations of FG nanobeams are obtained and solved analytically for simply-supported boundary conditions via the Navier-type procedure. Numerical results for deflection and buckling of FG nanoscale beams are presented and validated with those existing in the literature. The influences of small scale parameter, power law index, porosity distribution and slenderness ratio on the static and stability responses of the FG nanobeams are all explored.

키워드

참고문헌

  1. Adali, S. (2008), "Variational principles for multi-walled carbon nanotubes undergoing buckling based on nonlocal elasticity theory", Phys. Lett. A, 372(35), 5701-5705. https://doi.org/10.1016/j.physleta.2008.07.003
  2. 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", Braz. J. Phys., 45(2), 225-233. https://doi.org/10.1007/s13538-015-0306-2
  3. Aghababaei, R. and Reddy, J.N. (2009), "Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates", J. Sound Vibr., 326(1-2), 277-289. https://doi.org/10.1016/j.jsv.2009.04.044
  4. 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", Struct. Eng. Mech., 20(5), 963-981.
  5. Aissani, K., Bachir Bouiadjra, M., Ahouel, M. and Tounsi, A. (2015), "A new nonlocal hyperbolic shear deformation theory for nanobeams embedded in an elastic medium", Struct. Eng. Mech., 55(4), 743-762. https://doi.org/10.12989/sem.2015.55.4.743
  6. 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. Sand. Struct. Mater., 16(3), 293-318. https://doi.org/10.1177/1099636214526852
  7. 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
  8. Amara, K., Tounsi, A., Mechab, I. 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
  9. Ansari, R., Sahmani, S., and Rouhi, H. (2011), "Axial buckling analysis of single-walled carbon nanotubes in thermal environments via the Rayleigh-Ritz technique", Comput. Mater. Sci., 50(10), 3050-3055. https://doi.org/10.1016/j.commatsci.2011.05.027
  10. 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
  11. Aydogdu, M. (2009), "A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration", Phys. E, 41(9), 1651-1655. https://doi.org/10.1016/j.physe.2009.05.014
  12. Azimi, M., Seyed, S.M., Shafiei, N. and Hamouda, A.M.S. (2017a), "Thermo-mechanical vibration of rotating axially functionally graded nonlocal Timoshenko beam", Appl. Phys. A, 123(1), 104.
  13. Azimi, M., Seyed, S.M., Shafiei, N., Hamouda, A.M.S. and Davari, E. (2017b), "Vibration of rotating functionally graded (FG) Timoshenko nanobeam with nonlinear thermal distribution", Mech. Adv. Mater. Struct.
  14. 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
  15. 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
  16. 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
  17. Bensaid, I., Cheikh, A., Mangouchi, A. and Kerboua, B. (2017), "Static deflection and dynamic behavior of higher-order hyperbolic shear deformable compositionally graded beams", Adv. Mater. Res., 6(1), 13-26. https://doi.org/10.12989/amr.2017.6.1.013
  18. 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
  19. Benzair, A., Tounsi, A., Besseghier, A., Heireche, H., Moulay, N. and Boumia, L. (2008), "The thermaleffect 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
  20. 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
  21. Bessaim, A., Houari, M.S.A., Bernard, F. and Tounsi, A. (2015), "A nonlocal quasi-3D trigonometric plate model for free vibration behaviour of micro/nanoscale plates", Struct. Eng. Mech., 56(2), 223-240. https://doi.org/10.12989/sem.2015.56.2.223
  22. Besseghier, A., Heireche, H., Bousahla, A.A., Tounsi, A. and Benzair, A. (2015), "Nonlinear vibration properties of a zigzag single-walled carbon nanotube embedded in a polymer matrix", Adv. Nano Res., 3(1), 29-37. https://doi.org/10.12989/anr.2015.3.1.029
  23. 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
  24. Bouchafa, A., Bachir Bouiadjra, M., Houari, M.S.A. and Tounsi, A. (2015), "Thermal stresses and deflections of functionally graded sandwich plates using a new refined hyperbolic shear deformation theory", Steel Compos. Struct., 18(6), 1493-1515. https://doi.org/10.12989/scs.2015.18.6.1493
  25. 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
  26. 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), 227-279. https://doi.org/10.12989/scs.2016.20.2.227
  27. 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
  28. Bourada, M., Tounsi, A., Houari, M.S.A. and Adda Bedia, E.A. (2012), "A new four-variable refined plate theory for thermal buckling analysis of functionally graded sandwich plates", J. Sandw. Struct. Mater., 14(1), 5-33. https://doi.org/10.1177/1099636211426386
  29. 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", J. Comput. Meth., 11(6), 1350082. https://doi.org/10.1142/S0219876213500825
  30. Chakraverty, S. and Pradhan, K.K. (2014), "Free vibration of exponential functionally graded rectangular plates in thermal environment with general boundary conditions", Aeros. Sci. Technol., 36, 132-156. https://doi.org/10.1016/j.ast.2014.04.005
  31. Chikh, A., Bakora, A., Heireche, H., Houari, M.S.A., Tounsi, A. and Adda Bedia, E.A. (2016), "Thermomechanical postbuckling of symmetric S-FGM plates resting on Pasternak elastic foundations using hyperbolic shear deformation theory", Struct. Eng. Mech., 57(4), 617-639. https://doi.org/10.12989/sem.2016.57.4.617
  32. Dresselhaus, M.S., Dresselhaus, G., Charlier, J.C. and Hernandez, E. (2004), "Electronic, thermal and mechanical properties of carbon nanotubes", Philos. Trans. Royal Soc. A, 362(1823), 2065-2098. https://doi.org/10.1098/rsta.2004.1430
  33. Ebrahimi, F. and Barati, M.R. (2016), "A unified formulation for dynamic analysis of nonlocal heterogeneous nanobeams in hygro-thermal environment", Appl. Phys. A, 122(9), 792. https://doi.org/10.1007/s00339-016-0322-2
  34. Ebrahimi, F. and Daman, M. (2017), "Analytical investigation of the surface effects on nonlocal vibration behavior of nanosize curved beams", Adv. Nano Res., 4(1), 35-47.
  35. Ebrahimi, F. and Salari, E. (2015), "Thermal buckling and free vibration analysis of size dependent Timoshenko FG nanobeams in thermal environments", Compos. Struct., 128, 363-380. https://doi.org/10.1016/j.compstruct.2015.03.023
  36. Ebrahimi, F., Ghasemi, F. and Salari, E. (2016), "Investigating thermal effects on vibration behavior of temperaturedependent compositionally graded euler beams with porosities", Meccan., 51(1), 223-249. https://doi.org/10.1007/s11012-015-0208-y
  37. Ebrahimi, F. and Jafari, A. (2017), "A four-variable refined shear-deformation beam theory for thermo mechanical vibration analysis of temperature-dependent FGM beams with porosities", Mech. Adv. Mater. Struct., 1-13.
  38. El Meiche, N., Tounsi, A., Ziane, N., Mechab, I. and Adda Bedia, E.A. (2011), "A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate", J. Mech. Sci., 53(4), 237-247. https://doi.org/10.1016/j.ijmecsci.2011.01.004
  39. Eltaher, M.A., Emam, S.A. and Mahmoud, F.F. (2012), "Free vibration analysis of functionally graded sizedependent nanobeams", Appl. Math. Comput., 218(14), 7406-7420. https://doi.org/10.1016/j.amc.2011.12.090
  40. Eringen, A.C. (1972), "Nonlocal polar elastic continua", J. Eng. Sci., 10(1), 1-16. https://doi.org/10.1016/0020-7225(72)90070-5
  41. 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
  42. Fekrar, A., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2014), "A new five-unknown refined theorybased on neutral surface position for bending analysis of exponential graded plates", Meccan., 49(4), 795-810. https://doi.org/10.1007/s11012-013-9827-3
  43. Hadj Elmerabet, A., Heireche, A., Tounsi, A. and Semmah, A. (2017), "Buckling temperature of a single-walled boron nitride nanotubes using a novel nonlocal beam model", Adv. Nano Res., 5(1), 1-12. https://doi.org/10.12989/anr.2017.5.1.001
  44. 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
  45. Hebali, H., Bakora, A., Tounsi, A. and Kaci, A. (2016), "A novel four variable refined plate theory for bending, buckling, and vibration of functionally graded plates", Steel Compos. Struct., 22(3), 473-495. https://doi.org/10.12989/scs.2016.22.3.473
  46. Hebali, H., Tounsi, A., Houari, M.S.A., Bessaim, A. and Adda Bedia, E.A. (2014), "New Quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates", ASCE J. Eng. Mech., 140(2), 374-383. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000665
  47. Heireche, H., Tounsi, A. and Benzair, A. (2008b), "Scale Effect on wave propagation of double-walled carbon nanotubes with initial axial loading", Nanotechnol., 19(18), 185703. https://doi.org/10.1088/0957-4484/19/18/185703
  48. Heireche, H., Tounsi, A., Benzair, A. and Adda Bedia, E.A. (2008a), "Sound wave propagation in singlewalled carbon nanotubes using nonlocal elasticity", Phys. E, 40(8), 2791-2799. https://doi.org/10.1016/j.physe.2007.12.021
  49. Hong, S. and Myung, S. (2007), "Nanotube electronics: A flexible approach to mobility", Nat. Nanotechnol., 2(4), 207-208. https://doi.org/10.1038/nnano.2007.89
  50. 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
  51. Khalfi, Y., Houari, M.S.A. and Tounsi, A. (2014), "A refined and simple shear deformation theory for thermal buckling of solar functionally graded plates on elastic foundation", J. Comput. Meth., 11(5), 135007.
  52. Kheroubi, B., Benzair, A., Tounsi, A. and Semmah, A. (2016), "A new refined nonlocal beam theory accounting for effect of thickness stretching in nanoscale beams", Adv. Nano Res., 4(4), 35-47.
  53. Koizumi, M. (1997), "FGM activities in Japan", Compos. Part B, 28(1-2), 251-264. https://doi.org/10.1016/S1359-835X(96)00117-0
  54. Krenich, F., Heireche, H., Houari, M.S.A. and Tounsi, A. (2017), "A novel nonlocal four variable plate theory for thermal stability of single-layered graphene sheets embedded in an elastic substrate medium", Curr. Nanomater., 1(3), 215-222. https://doi.org/10.2174/2405461501666161130121643
  55. 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
  56. Li, L., Li, X. and Hu, Y. (2016), "Free vibration analysis of nonlocal strain gradient beams made of functionally graded material", J. Eng. Sci., 102, 77-92. https://doi.org/10.1016/j.ijengsci.2016.02.010
  57. Liang, X., Wang, Z., Wang, L. and Liu, G. (2014), "Semi-analytical solution for three-dimensional transient response of functionally graded annular plate on a two parameter viscoelastic foundation", J. Sound Vibr., 333(12), 2649-2663. https://doi.org/10.1016/j.jsv.2014.01.021
  58. Mouaici, F., Benyoucef, S., Atmane, H.A. and Tounsi, A. (2016), "Effect of porosity on vibrational characteristics of non-homogeneous plates using hyperbolic shear deformation theory", Wind Struct., 22(4), 429-454. https://doi.org/10.12989/was.2016.22.4.429
  59. Murmu, T. and Pradhan, S.C. (2009a), "Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM", Phys. E, 41(7), 1232-1239. https://doi.org/10.1016/j.physe.2009.02.004
  60. Murmu, T. and Adhikari, S. (2011), "Axial instability of double-nanobeam-systems", Phys. Lett. A, 375(3), 601-608. https://doi.org/10.1016/j.physleta.2010.11.007
  61. Ould Larbi, L., Kaci, A., Houari, M.S.A. and Tounsi, A. (2013), "An efficient shear deformation beam theory based on neutral surface position for bending and free vibration of functionally graded beams", Mech. Based Des. Struct. Mach., 41(4), 421-433. https://doi.org/10.1080/15397734.2013.763713
  62. Ould Youcef, D., Kaci, A., Houari, M.S.A., Tounsi, A., Benzair, A. and Heireche, H. (2015), "On the bending and stability of nanowire using various HSDTs", Adv. Nano Res., 3(4), 177-191. https://doi.org/10.12989/anr.2015.3.4.177
  63. Peddieson, J., Buchanan, G.R. and McNitt, R.P. (2003), "Application of nonlocal continuum models to nanotechnology", J. Eng. Sci., 41(3-5), 305-312. https://doi.org/10.1016/S0020-7225(02)00210-0
  64. Pradhan, S.C. and Reddy, G.K. (2011), "Buckling analysis of single walled carbon nanotube on Winkler foundation using nonlocal elasticity theory and DTM", Comput. Mater. Sci., 50(3), 1052-1056. https://doi.org/10.1016/j.commatsci.2010.11.001
  65. Reddy, J.N. (2007), "Nonlocal theories for bending, buckling and vibration of beams", J. Eng. Sci., 45(2-8), 288-307. https://doi.org/10.1016/j.ijengsci.2007.04.004
  66. Reddy, J.N. and Pang, S.D. (2008), "Nonlocal continuum theories of beams for the analysis of carbon nanotubes", J. Appl. Phys., 103(2), 023511. https://doi.org/10.1063/1.2833431
  67. Shafiei, N., Mirjavadi, S.S., Afshari, B.M., Rabby, S. and Hamouda, A.M.S. (2017), "Nonlinear thermal buckling of axially functionally graded micro and nanobeams", Compos. Struct., 168, 428-439. https://doi.org/10.1016/j.compstruct.2017.02.048
  68. Simsek, M. (2010), "Vibration analysis of a single-walled carbon nanotube under action of a moving harmonic load based on nonlocal elasticity theory", Phys. E, 43(1), 182-191. https://doi.org/10.1016/j.physe.2010.07.003
  69. Simsek, M. (2011), "Nonlocal effects in the forced vibration of an elastically connected double-carbon nanotube system under a moving nanoparticle", Comput. Mater. Sci., 50(7), 2112-2123. https://doi.org/10.1016/j.commatsci.2011.02.017
  70. 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
  71. Simsek, M. (2016), "Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach", J. Eng. Sci., 105, 12-27. https://doi.org/10.1016/j.ijengsci.2016.04.013
  72. Simsek, M. and Yurtcu, H.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
  73. Thai, H.T. (2012a), "A nonlocal beam theory for bending, buckling, and vibration of nanobeams", J. Eng. Sci., 52, 56-64. https://doi.org/10.1016/j.ijengsci.2011.11.011
  74. Thai, H.T. and Thuc, P.V. (2012b), "A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams", J. Eng. Sci., 54, 58-66. https://doi.org/10.1016/j.ijengsci.2012.01.009
  75. Tounsi, A, Semmah, A. and Bousahla, A.A. (2013c), "Thermal buckling behavior of nanobeams using an efficient higher-order nonlocal beam theory", ASCE J. Nanomech. Micromech., 3(3), 37-42. https://doi.org/10.1061/(ASCE)NM.2153-5477.0000057
  76. Tounsi, A., Benguediab, S., Adda Bedia, E.A., Semmah, A. and Zidour, M. (2013b), "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
  77. Tounsi, A., Benguediab, S., Houari, M.S.A. and Semmah, A. (2013d), "A new nonlocal beam theory with thickness stretching effect for nanobeams", J. Nanosci., 12(4), 1-8.
  78. 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
  79. Tounsi, A., Houari, M.S.A., Benyoucef, S. and Adda Bedia, E.A. (2013e), "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
  80. Tounsi, A., Houari, M.S.A., Benyoucef, S., Adda Bedia, E.A. (2013a), "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
  81. Wang, C.Y., Zhang, Y.Y., Wang, C.M. and Tan, V.B.C. (2007), "Buckling of carbon nanotubes: A literature survey", J. Nanosci. Nanotechnol., 7(12), 4221-4247. https://doi.org/10.1166/jnn.2007.924
  82. Wang, Q. (2005), "Wave propagation in carbon nanotubes via nonlocal continuum mechanics", J. Appl. Phys., 98(12), 124301. https://doi.org/10.1063/1.2141648
  83. Wattanasakulpong, N. and Chaikittiratana, A. (2015), "Flexural vibration of imperfect functionally graded beams based on Timoshenko beam theory: Chebyshev collocation method", Meccan., 50(5), 1331-1342. https://doi.org/10.1007/s11012-014-0094-8
  84. Yaghoobi, H. and Torabi, M. (2013), "Exact solution for thermal buckling of functionally graded plates resting on elastic foundations with various boundary conditions", J. Therm. Stress., 36(9), 869-894. https://doi.org/10.1080/01495739.2013.770356
  85. 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
  86. Zhu, J., Lai, Z., Yin, Z., Jeon, J. and Lee, S. (2001), "Fabrication of ZrO2-NiCr functionally graded material by powder metallurgy", Mater. Chem. Phys., 68(1), 130-135. https://doi.org/10.1016/S0254-0584(00)00355-2
  87. Ziane, N., Meftah, S.A., Ruta, G., Tounsi, A. and Adda Bedia, E.A. (2015), "Investigation of the Instability of FGM box beams", Struct. Eng. Mech., 54(3), 579-595. https://doi.org/10.12989/sem.2015.54.3.579
  88. 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

피인용 문헌

  1. Quasi-3D Refined Theory for Functionally Graded Porous Plates: Displacements and Stresses vol.23, pp.1, 2017, https://doi.org/10.1134/s1029959920010051
  2. A GN-based modified model for size-dependent coupled thermoelasticity analysis in nano scale, considering nonlocality in heat conduction and elasticity: An analytical solution for a nano beam with ene vol.73, pp.3, 2017, https://doi.org/10.12989/sem.2020.73.3.287
  3. Size-dependent free vibration and buckling analysis of sigmoid and power law functionally graded sandwich nanobeams with microstructural defects vol.234, pp.18, 2017, https://doi.org/10.1177/0954406220916481
  4. A compressive study for porous FG curved nanobeam under various boundary conditions via a nonlocal strain gradient theory vol.136, pp.2, 2017, https://doi.org/10.1140/epjp/s13360-021-01238-w
  5. Closed-form expressions for bending and buckling of functionally graded nanobeams by the Laplace transform vol.10, pp.2, 2017, https://doi.org/10.1142/s2047684121500123
  6. 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