On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable model

  • Belkorissat, Ismahene ;
  • Houari, Mohammed Sid Ahmed ;
  • Tounsi, Abdelouahed ;
  • Bedia, E.A. Adda ;
  • Mahmoud, S.R.
  • Received : 2014.07.17
  • Accepted : 2014.11.06
  • Published : 2015.04.25


In this paper, a new nonlocal hyperbolic refined plate model is presented for free vibration properties of functionally graded (FG) plates. This nonlocal nano-plate model incorporates the length scale parameter which can capture the small scale effect. The displacement field of the present theory is chosen based on a hyperbolic variation in the in-plane displacements through the thickness of the nano-plate. By dividing the transverse displacement into the bending and shear parts, the number of unknowns and equations of motion of the present theory is reduced, significantly facilitating structural analysis. The material properties are assumed to vary only in the thickness direction and the effective properties for the FG nano-plate are computed using Mori-Tanaka homogenization scheme. The governing equations of motion are derived based on the nonlocal differential constitutive relations of Eringen in conjunction with the refined four variable plate theory via Hamilton's principle. Analytical solution for the simply supported FG nano-plates is obtained to verify the theory by comparing its results with other available solutions in the open literature. The effects of nonlocal parameter, the plate thickness, the plate aspect ratio, and various material compositions on the dynamic response of the FG nano-plate are discussed.


nonlocal elasticity theory;nano-plates;free vibration;refined plate theory;functionally graded materials


  1. Aghababaei, R. and Reddy, J.N. (2009), "Non-local third-order shear deformation plate theory with application to bending and vibration of plates" J. Sound Vib., 326(1-2), 227-289.
  2. Aifantis, E.C. (1999), "Strain gradient interpretation of size effects", Int. J. Fract., 95(1-4), 1-4.
  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.
  4. 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., Int. J., 53(6), 1143-1165.
  5. Bachir Bouiadjra, M., Houari, M.S.A. and Tounsi, A. (2012), "Thermal buckling of functionally graded plates according to a four-variable refined plate theory", J. Therm. Stress., 35(8), 677-694.
  6. Bachir Bouiadjra, R., Adda Bedia, E.A. and Tounsi, A. (2013), "Nonlinear thermal buckling behavior of functionally graded plates using an efficient sinusoidal shear deformation theory", Struct. Eng. Mech., Int. J., 48(4), 547-567.
  7. 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.
  8. Benachour, A., Daouadji Tahar, H., Ait Atmane, H., Tounsi, A. and Meftah, S.A. (2011), "A four variable refined plate theory for free vibrations of functionally graded plates with arbitrary gradient", Compos.: Part B, 42(6), 1386-1394.
  9. Benguediab, S., Tounsi, A., Zidour, M. and Semmah, A. (2014), "Chirality and scale rffects on mechanical buckling properties of zigzag double-walled carbon nanotubes", Compos.: Part B, 57, 21-24.
  10. 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.
  11. 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., Int. J., 48(3), 351-365.
  12. 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.
  13. 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., Int. J., 14(1), 85-104.
  14. 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.
  15. 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.
  16. Bouremana, M., Houari, M.S.A., Tounsi, A., Kaci, A. and Adda Bedia, E.A. (2013), "A new first shear deformation beam theory based on neutral surface position for functionally graded beams", Steel Compos. Struct., Int. J., 15(5), 467-479.
  17. Cheng, Z.Q. and Batra, R. (2000), "Three-dimensional thermoelastic deformations of a functionally graded elliptic plate", Compos. Part B: Eng., 31(2), 97-106.
  18. Daneshmehr, A., Rajabpoor, A. and Pourdavood, M. (2014), "Stability of size dependent functionally graded nanoplate based on nonlocal elasticity and higher order plate theories and different boundary conditions", Int. J. Eng. Sci., 82, 84-100.
  19. Draiche, K., Tounsi, A. and Khalfi, Y. (2014), "A trigonometric four variable plate theory for free vibration of rectangular composite plates with patch mass", Steel Compos. Struct., Int. J., 17(1), 69-81.
  20. 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", Int. J. Mech. Sci., 53(4), 237-247.
  21. Eringen, A.C. (1967), "Theory of micropolar plates", Z. Angew. Math. Phys., 18(1), 12-30.
  22. Eringen, A.C. (1972), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10(1), 1-16.
  23. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54, 4703-4710.
  24. 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.
  25. Fu, Y., Du, H. and Zhang, S. (2003), "Functionally graded TiN/TiNi shape memory alloy films", Mater. Lett., 57(20), 2995-2999.
  26. 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.
  27. Hasanyan, D.J., Batra, R.C. and Harutyunyan, S. (2008), "Pull-in instabilities in functionally graded micro-thermo-electro-mechanical systems", J. Therm. Stress., 31(10), 1006-1021.
  28. 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", ASCE J. Eng. Mech., 140(2), 374-383.
  29. Heireche, H., Tounsi, A., Benzair, A., Maachou, M. and Adda Bedia, E.A. (2008a) "Sound wave propagation in single-walled carbon nanotubes using nonlocal elasticity", Physica E., 40(8), 2791-2799.
  30. Heireche, H., Tounsi, A. and Benzair, A. (2008b), "Scale effect on wave propagation of double-walled carbon nanotubes with initial axial loading", Nanotechnology, 19(18), 185703.
  31. Heireche, H., Tounsi, A., Benzair, A. and Mechab, I. (2008c), "Sound wave propagation in single - Carbon nanotubes with initial axial stress", J Appl. Phys., 104, 014301.
  32. Houari, M.S.A., Tounsi, A. and Anwar Beg, O. (2013), "Thermoelastic bending analysis of functionally graded sandwich plates using a new higher order shear and normal deformation theory", Int. J. Mech. Sci., 76, 102-111.
  33. Janghorban, M. and Zare, A. (2011), "Free vibration analysis of functionally graded carbon nanotubes with variable thickness by differential quadrature method", Physica E, 43(9), 1602-1604.
  34. Kettaf, F.Z., Houari, M.S.A., Benguediab, M. and Tounsi, A. (2013), "Thermal buckling of functionally graded sandwich plates using a new hyperbolic shear displacement model", Steel Compos. Struct. Int. J., 15(4), 399-423.
  35. Klouche Djedid, I., Benachour, A., Houari, M.S.A., Tounsi, A. and Ameur, M. (2014), "A n-order four variable refined theory for bending and free vibration of functionally graded plates", Steel Compos. Struct., Int. J., 17(1), 21-46.
  36. 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., Int .J., 18(2), 425-442.
  37. Lu, C.F., Lim, C.W. and Chen, W.Q. (2009), "Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory", Int. J. Solid. Struct., 46(5), 1176-1185.
  38. Mohammadi-Alasti, B., Rezazadeh, G., Borgheei, A.M., Minaei, S. and Habibifar, R. (2011), "On the mechanical behavior of a functionally graded micro-beamsubjected to a thermal moment and nonlinear electrostatic pressure", Compos. Struct., 93(6), 1516-1525.
  39. Natarajan, S., Baiz, P., Ganapathi, M., Kerfriden, P. and Bordas, S. (2011), "Linear free flexural vibration of cracked functionally graded plates in thermal environment", Comput. Struct., 89(15-16), 1535-1546.
  40. Natarajan, S., Chakraborty, S., Thangavel, M., Bordas, S. and Rabczuk, T. (2012), "Size-dependent free flexural vibration behavior of functionally graded nanoplates", Computat. Mater. Sci., 65, 74-80.
  41. Nedri, K., El Meiche, N. and Tounsi, A. (2014), "Free vibration analysis of laminated composite plates resting on elastic foundations by using a refined hyperbolic shear deformation theory", Mech. Compos. Mater., 49(6), 641-650.
  42. 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.
  43. Peddieson, J., Buchanan, G.R. and McNitt, R.P. (2003), "Application of nonlocal continuum models to nanotechnology", Int. J. Eng. Sci., 41(3-5), 305-312.
  44. Qian, L., Batra, R. and Chen, L. (2004), "Static and dynamic deformations of thick functionally graded elastic plates by using higher-order shear and normal deformable plate theory and meshless local Petrov-Galerkin method", Compos. Part B: Eng., 35(6-8), 685-697.
  45. Rahaeifard, M., Kahrobaiyan, M.H. and Ahmadian, M.T. (2009), "Sensitivity analysis of atomic force microscope cantilever made of functionally graded materials", DETC2009-86254, Proceedings of the 3rd International Conference on Micro- and Nanosystems (MNS3) 2009, San Diego, CA, USA, August-September.
  46. Reddy, J.N. (2002), Energy Principles and Variational Methods in Applied Mechanics, John Wiley & Sons Inc.
  47. Reddy, J.N. (2007), "Nonlocal theories for bending, buckling and vibration of beams", Int. J Eng. Sci., 45(2-8), 288-307.
  48. Reddy, J.N. and Pang, S.D. (2008), "Nonlocal continuum theories of beams for the analysis of carbon nanotubes", J. Appl. Phys., 103, 023511.
  49. Sadoune, M., Tounsi, A., Houari, M.S.A. and Adda Bedia, E.A. (2014), "A novel first-order shear deformation theory for laminated composite plates", Steel Compos. Struct., Int. J., 17(3), 321-338.
  50. Saidi, H., Houari, M.S.A., Tounsi, A. and Adda Bedia, E.A. (2013), "Thermo-mechanical bending response with stretching effect of functionally graded sandwich plates using a novel shear deformation theory", Steel Compos. Struct., Int. J., 15(2), 221-245.
  51. Thai, C.H., Nguyen-Xuan, H., Nguyen-Thanh, N., Le, T.H., Nguyen-Thoi, T. and Rabczuk, T. (2012), "Static, free vibration and buckling analysis of laminated composite Reissner-Mindlin plates using NURBS-based isogeometric approach", Int. J. Numer. Method. Eng., 91(6), 571-603.
  52. 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, 104301.
  53. 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., Int. J., 1(1), 1-11.
  54. Tounsi, A., Semmah, A. and Bousahla, A.A. (2013b), "Thermal buckling behavior of nanobeam usin an efficient higher-order nonlocal beam theory", J. Nanomech. Micromech. (ASCE), 3(3), 37-42.
  55. Tounsi, A., Benguediab, S., Houari, M.S.A. and Semmah, A. (2013c), "A new nonlocal beam theory with thickness stretching effect for nanobeams", Int. J. Nanosci., 12(4), 1350025.
  56. Tounsi, A., Houari, M.S.A., Benyoucef, S. and Adda Bedia, E.A. (2013d), "A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates", Aerosp. Sci. Technol., 24(1), 209-220.
  57. Valizadeh, N., Natarajan, S., Gonzalez-Estrada, O.A., Rabczuk, T., Bui, T.Q. and Bordas, S.P.A. (2013), "NURBS-based finite element analysis of functionally graded plates: Static bending, vibration, buckling and flutter", Compos. Struct., 99, 309-326.
  58. Witvrouw, A. and Mehta, A. (2005), "The use of functionally graded Ploy-SiGe layers for MEMS applications", Mater. Sci. Forum, 492-493, 255-260.
  59. Xu, M. (2006), "Free transverse vibrations of nano-to-micronscale beams", Proceedings of the Royal Society, 462(2074), 2977-2995.
  60. Yaghoobi, H. and Torabi, M. (2013), "Post-buckling and nonlinear free vibration analysis of geometrically imperfect functionally graded beams resting on nonlinear elastic foundation", Appl. Math. Model, 37(18-19), 8324-8340.
  61. 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.
  62. 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.

Cited by

  1. Temperature distribution effects on buckling behavior of smart heterogeneous nanosize plates based on nonlocal four-variable refined plate theory vol.7, pp.3, 2016,
  2. Magneto-hygro-thermal vibration behavior of elastically coupled nanoplate systems incorporating nonlocal and strain gradient effects vol.39, pp.11, 2017,
  3. Free vibration of anisotropic single-walled carbon nanotube based on couple stress theory for different chirality vol.36, pp.3, 2017,
  4. Buckling analysis of isotropic and orthotropic plates using a novel four variable refined plate theory vol.21, pp.6, 2016,
  5. Porosity-dependent vibration and dynamic stability of compositionally gradient nanofilms using nonlocal strain gradient theory 2017,
  6. Forced vibration of porous functionally graded nanoplates under uniform dynamic load using general nonlocal stress–strain gradient theory 2017,
  7. Stability analysis of an embedded single-walled carbon nanotube with small initial curvature based on nonlocal theory vol.24, pp.11, 2017,
  8. Vibration analysis of functionally graded piezoelectric nanoscale plates by nonlocal elasticity theory: An analytical solution vol.100, 2016,
  9. Thermal buckling of embedded sandwich piezoelectric nanoplates with functionally graded core by a nonlocal second-order shear deformation theory 2018,
  10. Size-dependent electro-magneto-elastic bending analyses of the shear-deformable axisymmetric functionally graded circular nanoplates vol.132, pp.10, 2017,
  11. Critical buckling load of chiral double-walled carbon nanotube using non-local theory elasticity vol.3, pp.4, 2015,
  12. A novel four variable refined plate theory for bending, buckling, and vibration of functionally graded plates vol.22, pp.3, 2016,
  13. On guided wave propagation in fully clamped porous functionally graded nanoplates vol.143, 2018,
  14. Nonlinear analysis of size-dependent and material-dependent nonlocal CNTs vol.153, 2016,
  15. Frequency analysis of porous nano-mechanical mass sensors made of multi-phase nanocrystalline silicon materials vol.4, pp.7, 2017,
  16. Nonlinear bending of a two-dimensionally functionally graded beam vol.184, 2018,
  17. Vibration analysis of multi-phase nanocrystalline silicon nanoplates considering the size and surface energies of nanograins/nanovoids vol.119, 2017,
  18. Wave propagation analysis of size-dependent rotating inhomogeneous nanobeams based on nonlocal elasticity theory 2017,
  19. An efficient shear deformation theory for wave propagation of functionally graded material plates vol.57, pp.5, 2016,
  20. Thermomechanical vibration analysis of FGM viscoelastic multi-nanoplate system incorporating the surface effects via nonlocal elasticity theory vol.23, pp.8, 2017,
  21. A new nonlocal hyperbolic shear deformation theory for nanobeams embedded in an elastic medium vol.55, pp.4, 2015,
  22. Thermo-mechanical vibration of rotating axially functionally graded nonlocal Timoshenko beam vol.123, pp.1, 2017,
  23. A review of continuum mechanics models for size-dependent analysis of beams and plates vol.177, 2017,
  24. Buckling analysis of tapered nanobeams using nonlocal strain gradient theory and a generalized differential quadrature method vol.4, pp.6, 2017,
  25. Thermal post-buckling behavior of imperfect temperature-dependent sandwich FGM plates resting on Pasternak elastic foundation vol.22, pp.1, 2016,
  26. A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundation vol.20, pp.2, 2016,
  27. A simple hyperbolic shear deformation theory for vibration analysis of thick functionally graded rectangular plates resting on elastic foundations vol.11, pp.2, 2016,
  28. A refined theory with stretching effect for the flexure analysis of laminated composite plates vol.11, pp.5, 2016,
  29. Free vibration analysis of chiral double-walled carbon nanotube using non-local elasticity theory vol.4, pp.1, 2016,
  30. Nonlinear electroelastic vibration analysis of NEMS consisting of double-viscoelastic nanoplates vol.122, pp.10, 2016,
  31. Wave propagation in embedded inhomogeneous nanoscale plates incorporating thermal effects 2017,
  32. Wave propagation analysis of quasi-3D FG nanobeams in thermal environment based on nonlocal strain gradient theory vol.122, pp.9, 2016,
  33. Dynamic modeling and vibration analysis of double-layered multi-phase porous nanocrystalline silicon nanoplate systems vol.66, 2017,
  34. Vibration analysis of porous functionally graded nanoplates vol.120, 2017,
  35. Bending analysis of different material distributions of functionally graded beam vol.123, pp.4, 2017,
  36. An efficient and simple shear deformation theory for free vibration of functionally graded rectangular plates on Winkler-Pasternak elastic foundations vol.22, pp.3, 2016,
  37. Dynamic modeling of porous heterogeneous micro/nanobeams vol.132, pp.12, 2017,
  38. Investigating dynamic characteristics of porous double-layered FG nanoplates in elastic medium via generalized nonlocal strain gradient elasticity vol.132, pp.9, 2017,
  39. Nonlocal Thermal Buckling Analysis of Embedded Magneto-Electro-Thermo-Elastic Nonhomogeneous Nanoplates vol.40, pp.4, 2016,
  40. A general bi-Helmholtz nonlocal strain-gradient elasticity for wave propagation in nanoporous graded double-nanobeam systems on elastic substrate vol.168, 2017,
  41. Bending, buckling and vibration analyses of MSGT microcomposite circular-annular sandwich plate under hydro-thermo-magneto-mechanical loadings using DQM 2017,
  42. Hygro-thermal vibration analysis of graded double-refined-nanoplate systems using hybrid nonlocal stress-strain gradient theory vol.176, 2017,
  43. Investigating post-buckling of geometrically imperfect metal foam nanobeams with symmetric and asymmetric porosity distributions vol.182, 2017,
  44. Coupled twist–bending static and dynamic behavior of a curved single-walled carbon nanotube based on nonlocal theory vol.23, pp.7, 2017,
  45. Thermal stability of functionally graded sandwich plates using a simple shear deformation theory vol.58, pp.3, 2016,
  46. Critical Buckling Load of Chiral Double-Walled Carbon Nanotubes Embedded in an Elastic Medium vol.53, pp.6, 2018,
  47. Size-dependent thermal stability analysis of graded piezomagnetic nanoplates on elastic medium subjected to various thermal environments vol.122, pp.10, 2016,
  48. Size-dependent nonlinear vibration of beam-type porous materials with an initial geometrical curvature vol.184, 2018,
  49. Influence of various temperature distributions on critical speed and vibrational characteristics of rotating cylindrical microshells with modified lengthscale parameter vol.132, pp.6, 2017,
  50. Buckling and free vibration of shallow curved micro/nano-beam based on strain gradient theory under thermal loading with temperature-dependent properties vol.123, pp.1, 2017,
  51. Electro-magnetic effects on nonlocal dynamic behavior of embedded piezoelectric nanoscale beams vol.28, pp.15, 2017,
  52. An analytical method for free vibration analysis of functionally graded sandwich beams vol.23, pp.1, 2016,
  53. Nonlinear vibration of nonlocal four-variable graded plates with porosities implementing homotopy perturbation and Hamiltonian methods vol.229, pp.1, 2018,
  54. 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,
  55. Surface effects on nonlinear vibration of embedded functionally graded nanoplates via higher order shear deformation plate theory 2018,
  56. A nonlocal strain gradient refined beam model for buckling analysis of size-dependent shear-deformable curved FG nanobeams vol.159, 2017,
  57. Free vibration investigation of nano mass sensor using differential transformation method vol.123, pp.3, 2017,
  58. A new simple three-unknown sinusoidal shear deformation theory for functionally graded plates vol.22, pp.2, 2016,
  59. A new refined nonlocal beam theory accounting for effect of thickness stretching in nanoscale beams vol.4, pp.4, 2016,
  60. A novel four variable refined plate theory for laminated composite plates vol.22, pp.4, 2016,
  61. Thermal stability analysis of solar functionally graded plates on elastic foundation using an efficient hyperbolic shear deformation theory vol.10, pp.3, 2016,
  62. Small scale effect on the vibration of non-uniform nanoplates vol.55, pp.3, 2015,
  63. A nonlocal quasi-3D trigonometric plate model for free vibration behaviour of micro/nanoscale plates vol.56, pp.2, 2015,
  64. Dynamic behavior of FGM beam using a new first shear deformation theory vol.10, pp.2, 2016,
  65. Investigating physical field effects on the size-dependent dynamic behavior of inhomogeneous nanoscale plates vol.132, pp.2, 2017,
  66. Frequency analysis of nanoporous mass sensors based on a vibrating heterogeneous nanoplate and nonlocal strain gradient theory 2017,
  67. Effect of Longitudinal Magnetic Field on Vibration Characteristics of Single-Walled Carbon Nanotubes in a Viscoelastic Medium vol.47, pp.6, 2017,
  68. Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects vol.120, 2017,
  69. On the bending and stability of nanowire using various HSDTs vol.3, pp.4, 2015,
  70. Nonlocal strain gradient theory calibration using molecular dynamics simulation based on small scale vibration of nanotubes vol.514, 2017,
  71. Nonlocal Timoshenko Beam for Vibrations of Magnetically Affected Inclined Single-Walled Carbon Nanotubes as Nanofluidic Conveyors vol.131, pp.6, 2017,
  72. A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams vol.40, pp.5-6, 2016,
  73. A general nonlocal stress-strain gradient theory for forced vibration analysis of heterogeneous porous nanoplates vol.67, 2018,
  74. A new higher-order shear and normal deformation theory for functionally graded sandwich beams vol.19, pp.3, 2015,
  75. Wave propagation analysis of a size-dependent magneto-electro-elastic heterogeneous nanoplate vol.131, pp.12, 2016,
  76. Thermomechanical effects on the bending of antisymmetric cross-ply composite plates using a four variable sinusoidal theory vol.19, pp.1, 2015,
  77. Application of nonlocal strain gradient theory and various shear deformation theories to nonlinear vibration analysis of sandwich nano-beam with FG-CNTRCs face-sheets in electro-thermal environment vol.123, pp.5, 2017,
  78. Thermo-mechanical postbuckling of symmetric S-FGM plates resting on Pasternak elastic foundations using hyperbolic shear deformation theory vol.57, pp.4, 2016,
  79. Bending analysis of FGM plates using a sinusoidal shear deformation theory vol.23, pp.6, 2016,
  80. A new 3-unknowns non-polynomial plate theory for buckling and vibration of functionally graded sandwich plate vol.60, pp.4, 2016,
  81. Buckling analysis of nonuniform nonlocal strain gradient beams using generalized differential quadrature method 2017,
  82. Effect of porosity on vibrational characteristics of non-homogeneous plates using hyperbolic shear deformation theory vol.22, pp.4, 2016,
  83. Parametric excitation analysis of a piezoelectric-nanotube conveying fluid under multi-physics field 2017,
  84. Vibration analysis of bonded double-FGM viscoelastic nanoplate systems based on a modified strain gradient theory incorporating surface effects vol.123, pp.3, 2017,
  85. Size-dependent mechanical behavior of functionally graded trigonometric shear deformable nanobeams including neutral surface position concept vol.20, pp.5, 2016,
  86. Size-dependent thermally affected wave propagation analysis in nonlocal strain gradient functionally graded nanoplates via a quasi-3D plate theory vol.232, pp.1, 2018,
  87. Analytical solution for bending analysis of functionally graded beam vol.19, pp.4, 2015,
  88. Thermo-mechanical analysis of FG nanobeam with attached tip mass: an exact solution vol.122, pp.12, 2016,
  89. Nonlocal microstructure-dependent dynamic stability of refined porous FG nanoplates in hygro-thermal environments vol.132, pp.10, 2017,
  90. Static and dynamic behavior of FGM plate using a new first shear deformation plate theory vol.57, pp.1, 2016,
  91. Nonlinear atomic vibrations and structural phase transitions in strained carbon chains vol.138, 2017,
  92. On thermal stability of plates with functionally graded coefficient of thermal expansion vol.60, pp.2, 2016,
  93. Dynamic response of a single-walled carbon nanotube under a moving harmonic load by considering modified nonlocal elasticity theory vol.133, pp.2, 2018,
  94. Size-dependent vibration analysis of viscoelastic nanocrystalline silicon nanobeams with porosities based on a higher order refined beam theory vol.166, 2017,
  95. Vibration of two-dimensional imperfect functionally graded (2D-FG) porous nano-/micro-beams vol.322, 2017,
  96. Nonlocal stress-strain gradient vibration analysis of heterogeneous double-layered plates under hygro-thermal and linearly varying in-plane loads 2017,
  97. Hygro-thermo-mechanical bending of S-FGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory vol.18, pp.4, 2016,
  98. Nonlinear thermal vibration analysis of refined shear deformable FG nanoplates: two semi-analytical solutions vol.40, pp.2, 2018,
  99. Non-Local Buckling Analysis of Functionally Graded Nanoporous Metal Foam Nanoplates vol.8, pp.11, 2018,
  100. Effect of van der Waals force on wave propagation in viscoelastic double-walled carbon nanotubes vol.32, pp.24, 2018,
  101. Vibration Analysis of Nano Beam Using Differential Transform Method Including Thermal Effect vol.54, pp.1661-9897, 2018,
  102. Stabilities and electronic properties of nanowires made of single atomic sulfur chains encapsulated in zigzag carbon nanotubes vol.29, pp.41, 2018,
  103. Forced vibration analysis of cracked nanobeams vol.40, pp.8, 2018,
  104. Vibration and buckling analysis of a rotary functionally graded piezomagnetic nanoshell embedded in viscoelastic media vol.29, pp.11, 2018,
  105. An analytical study on the size dependent longitudinal vibration analysis of thick nanorods vol.5, pp.7, 2018,
  106. Hygrothermal wave characteristic of nanobeam-type inhomogeneous materials with porosity under magnetic field pp.2041-2983, 2018,
  107. Temperature and porosity effects on wave propagation in nanobeams using bi-Helmholtz nonlocal strain-gradient elasticity vol.133, pp.5, 2018,
  108. Thermal and Small-Scale Effects on Vibration of Embedded Armchair Single-Walled Carbon Nanotubes vol.51, pp.1661-9897, 2018,
  109. Nonlinear vibration analysis of different types of functionally graded beams using nonlocal strain gradient theory and a two-step perturbation method vol.134, pp.1, 2019,
  110. 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,
  111. Modal participation of fixed–fixed single-walled carbon nanotube with vacancies pp.2008-6695, 2019,