Buckling of magneto-electro-hygro-thermal piezoelectric nanoplates system embedded in a visco-Pasternak medium based on nonlocal theory

  • Mahsa Karimiasl
  • Kimiya Kargarfard
  • Farzad Ebrahimi
Technical Paper


In this present work the critical loading of magneto-electro-viscoelastic-hygro-thermal (MEVHT) piezoelectric nanoplates embedded in a viscoelastic foundation are investigated. Via two variable shear deformation plate theory displacement are obtained. The governing equations of nonlocal viscoelastic nanoplate are driven by using Hamilton’s principle and solved by an analytical solution. A parametric study is presented to examine the effect of the nonlocal parameter, hygro-thermo-magneto-electro-mechanical loadings and aspect ratio on the critical loading of MEVHT nanoplates. It is found that the critical loading is quite important to the mechanical loading, electric loading and magnetic loading, while it is insensitive to the hygro-thermal loading.



  1. Akbarzadeh AH, Chen ZT (2013) Hygrothermal stresses in one-dimensional functionally graded piezoelectric media in constant magnetic field. Compos Struct 97:317–331CrossRefGoogle Scholar
  2. Aksencer T, Aydogdu M (2011) Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory. Phys E 43(4):954–959CrossRefGoogle Scholar
  3. Alzahrani EO, Zenkour AM, Sobhy M (2013) Small scale effect on hygro-thermo-mechanical bending of nanoplates embedded in an elastic medium. Compos Struct 105:163–172CrossRefGoogle Scholar
  4. Ansari R, Gholami R (2016) Nonlocal free vibration in the pre-and post-buckled states of magneto-electro-thermo elastic rectangular nanoplates with various edge conditions. Smart Mater Struct 25(9):095033CrossRefGoogle Scholar
  5. Ansari R, Gholami R (2017) Size-dependent buckling and postbuckling analyses of first-order shear deformable magneto-electro-thermo elastic nanoplates based on the nonlocal elasticity theory. Int J Struct Stab Dyn. MathSciNetCrossRefGoogle Scholar
  6. Arani AG, Haghparast E, Zarei HB (2016) Nonlocal vibration of axially moving graphene sheet resting on orthotropic visco-Pasternak foundation under longitudinal magnetic field. Phys B 495:35–49CrossRefGoogle Scholar
  7. Arefi M, Soltan Arani A (2018) Higher-order shear deformation bending results of a magneto-electro-thermo-elastic functionally graded nano-beam in thermal, mechanical, electrical and magnetic environments. Mech Based Design Struct Mach. CrossRefGoogle Scholar
  8. Arefi M, Zenkour AM (2016) A simplified shear and normal deformations nonlocal theory for bending of functionally graded piezomagnetic sandwich nanobeams in magneto-thermo-electric environment. J Sandwich Struct Mater 18(5):624–651CrossRefGoogle Scholar
  9. Arefi M, Zenkour AM (2017) Nonlocal electro-thermo-mechanical analysis of a sandwich nanoplate containing a Kelvin–Voigt viscoelastic nanoplate and two piezoelectric layers. Acta Mech 228(2):475–493MathSciNetCrossRefGoogle Scholar
  10. Asemi HR, Asemi SR, Farajpour A, Mohammadi M (2015) Nanoscale mass detection based on vibrating piezoelectric ultrathin films under thermo-electro-mechanical loads. Phys E 68:112–122CrossRefGoogle Scholar
  11. Bellifa H, Benrahou KH, Hadji L, Houari MSA, 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):265–275CrossRefGoogle Scholar
  12. Bessaim A, Houari MSA, Bernard F, Tounsi A (2015) A nonlocal quasi-3D trigonometric plate model for free vibration behaviour of micro/nanoscale plates. Struct Eng Mech 56(2):223–240CrossRefGoogle Scholar
  13. Chaht FL, Kaci A, Houari MSA, Tounsi OA, Mahmoud SR (2015) Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect. Steel Compos Struct 18(2):425–442CrossRefGoogle Scholar
  14. Ebrahimi F, Barati MR (2016a) A nonlocal higher-order shear deformation beam theory for vibration analysis of size-dependent functionally graded nanobeams. Arab J Sci Eng 41(5):1679–1690MathSciNetCrossRefGoogle Scholar
  15. Ebrahimi F, Barati MR (2016b) Dynamic modeling of a thermo-piezo-electrically actuated nanosize beam subjected to a magnetic field. Appl Phys A 122(4):1–18CrossRefGoogle Scholar
  16. Ebrahimi F, Barati MR (2016c) Small-scale effects on hygro-thermo-mechanical vibration of temperature-dependent nonhomogeneous nanoscale beams. Mech Adv Mater Struct. CrossRefGoogle Scholar
  17. Ebrahimi F, Barati MR (2016d) Buckling analysis of smart size-dependent higher order magneto-electro-thermo-elastic functionally graded nanosize beams. J Mech 2016:1–11Google Scholar
  18. Ebrahimi F, Barati MR (2016e) A nonlocal higher-order refined magneto-electro-viscoelastic beam model for dynamic analysis of smart nanostructures. Eng Sci 107:183–196CrossRefGoogle Scholar
  19. Ebrahimi F, Barati MR (2016f) Wave propagation analysis of quasi-3D FG nanobeams in thermal environment based on nonlocal strain gradient theory. Appl Phys A 122(9):843CrossRefGoogle Scholar
  20. Ebrahimi F, Barati MR (2017) A nonlocal strain gradient refined beam model for buckling analysis of size-dependent shear-deformable curved FG nanobeams. Compos Struct 159:174–182CrossRefGoogle Scholar
  21. Ebrahimi F, Dabbagh A (2017) Wave dispersion characteristics of rotating heterogeneous magneto-electro-elastic nanobeams based on nonlocal strain gradient elasticity theory. Electromagn Waves Appl 32(2):138–169CrossRefGoogle Scholar
  22. Ebrahimi F, Barati M, Dabbagh A (2016) A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates. Int J Eng Sci 107:169–182CrossRefGoogle Scholar
  23. El Meiche N, Tounsi A, Ziane N, Mechab I (2011) A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate. Int J Mech Sci 53(4):237–247CrossRefGoogle Scholar
  24. Farajpour A, Yazdi MH, Rastgoo A, Loghmani M, Mohammadi M (2016) Nonlocal nonlinear plate model for large amplitude vibration of magneto-electro-elastic nanoplates. Compos Struct 140:323–336CrossRefGoogle Scholar
  25. Hashemi SH, Mehrabani H, Ahmadi-Savadkoohi A (2015) Exact solution for free vibration of coupled double viscoelastic graphene sheets by viscoPasternak medium. Compos B Eng 78:377–383CrossRefGoogle Scholar
  26. Hosseini M, Jamalpoor A (2015) Analytical solution for thermomechanical vibration of double-viscoelastic nanoplate-systems made of functionally graded materials. J Therm Stresses 38(12):1428–1456CrossRefGoogle Scholar
  27. Jamalpoor A, Ahmadi-Savadkoohi A, Hossein M, Hosseini-Hashemi S (2016) Free vibration and biaxial buckling analysis of double magneto-electro-elastic nanoplate-systems coupled by a visco-Pasternak medium via nonlocal elasticity theory. Eur J Mech A Solid 49:183–196zbMATHGoogle Scholar
  28. Ke LL, Wang YS, Yang J, Kitipornchai S (2014) Free vibration of size-dependent magneto-electro-elastic nanoplates based on the nonlocal theory. Acta Mech Sin 30(4):516–525MathSciNetCrossRefGoogle Scholar
  29. Ke LL, Liu C, Wang YS (2015) Free vibration of nonlocal piezoelectric nanoplates under various boundary conditions. Phys E 66:93–106CrossRefGoogle Scholar
  30. Lee CY, Kim JH (2013) Hygrothermal postbuckling behavior of functionally graded plates. Compos Struct 95:278–282CrossRefGoogle Scholar
  31. Lei Y, Adhikari S, Friswell MI (2013) Vibration of nonlocal Kelvin–Voigt viscoelastic damped Timoshenko beams. Int J Eng Sci 66:1–13MathSciNetCrossRefGoogle Scholar
  32. Li YS, Cai ZY, Shi SY (2014) Buckling and free vibration of magnetoelectroelastic nanoplate based on nonlocal theory. Compos Struct 111:522–529CrossRefGoogle Scholar
  33. Liang X, Hu S, Shengping S (2015) Size-dependent buckling and vibration behaviors of piezoelectric nanostructures due to flexoelectricity. Smart Mater Struct 24:105012CrossRefGoogle Scholar
  34. Liu C, Ke LL, Wang YS, Yang J, Kitipornchai S (2014) Buckling and post-buckling of size-dependent piezoelectric Timoshenko nanobeams subject to thermo-electro-mechanical loadings. Int J Struct Stab Dyn 14(03):1350067MathSciNetCrossRefGoogle Scholar
  35. Martin LW, Crane SP, Chu YH et al (2008) Multiferroics and magnetoelectrics: thin films and nanostructures. J Phys Condens Matter 20:434220CrossRefGoogle Scholar
  36. Mechab I, Atmane HA, Tounsi A, Belhadj HA (2010) A two variable refined plate theory for the bending analysis of functionally graded plates. Acta Mech Sin 26(6):941–949MathSciNetCrossRefGoogle Scholar
  37. Meziane MAA, Abdelaziz HH, Tounsi A (2014) An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions. J Sandwich Struct Mater 16(3):293–318CrossRefGoogle Scholar
  38. Mohammadi M, Farajpour A, Moradi A, Ghayour M (2014) Shear buckling of orthotropic rectangular graphene sheet embedded in an elastic medium in thermal environment. Compos B Eng 56:629–637CrossRefGoogle Scholar
  39. Mohammadsalehi M, Zargar O, Baghani M (2016) Study of non-uniform viscoelastic nanoplates vibration based on nonlocal first-order shear deformation theory. Meccanica 52:1063–1077MathSciNetCrossRefGoogle Scholar
  40. Murmu T, McCarthy MA, Adhikari S (2013) In-plane magnetic field affected transverse vibration of embedded single-layer graphene sheets using equivalent nonlocal elasticity approach. Compos Struct 96:57–63CrossRefGoogle Scholar
  41. Pouresmaeeli S, Ghavanloo E, Fazelzadeh SA (2013) Vibration analysis of viscoelastic orthotropic nanoplates resting on viscoelastic medium. Compos Struct 96:405–410CrossRefGoogle Scholar
  42. Prashanthi K, Shaibani PM, Sohrabi A et al (2012) Nanoscale magnetoelectric coupling in multiferroic BiFeO3 nanowires. Phys Status Solidi R 6:244–246CrossRefGoogle Scholar
  43. Schatz GC (2007) Using theory and computation to model nanoscale properties. PNAS 104(17):6885–6892CrossRefGoogle Scholar
  44. Shen ZB, Tang HL, Li DK, Tang GJ (2012) Vibration of single-layered graphene sheet-based nanomechanical sensor via nonlocal Kirchhoff plate theory. Comput Mater Sci 61:200–205CrossRefGoogle Scholar
  45. Shokrani MH, Karimi M, Tehrani MS, Mirdamadi HR (2016) Buckling analysis of double-orthotropic nanoplates embedded in elastic media based on non-local two-variable refined plate theory using the GDQ method. J Braz Soc Mech Sci Eng 38(8):2589–2606CrossRefGoogle Scholar
  46. Sobhy M (2015) Hygrothermal vibration of orthotropic double-layered graphene sheets embedded in an elastic medium using the two-variable plate theory. Appl Math Model 40:85–99MathSciNetCrossRefGoogle Scholar
  47. van den Boomgard J, Terrell DR, Born RAJ et al (1974) An in situ grown eutectic magnetoelectric composite material. J Mater Sci 9:1705–1709CrossRefGoogle Scholar
  48. Wang KF, Wang BL (2011) Vibration of nanoscale plates with surface energy via nonlocal elasticity. Phys E 44(2):448–453CrossRefGoogle Scholar
  49. Wang Y, Hu JM, Lin YH et al (2010) Multiferroic magnetoelectric composite nanostructures. NPG Asia Mater 2:61–68CrossRefGoogle Scholar
  50. Yahia SA, Atmane HA, Houari MSA, Tounsi A (2015) Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories. Struct Eng Mech 53(6):1143–1165CrossRefGoogle Scholar
  51. Zenkour AM (2016) Nonlocal transient thermal analysis of a single-layered graphene sheet embedded in viscoelastic medium. Phys E 79:87–97CrossRefGoogle Scholar
  52. Zenkour AM, Sobhy M (2015) A simplified shear and normal deformations nonlocal theory for bending of nanobeams in thermal environment. Phys E 70:121–128CrossRefGoogle Scholar
  53. Zhang Z, Jiang L (2014) Size effects on electromechanical coupling fields of a bending piezoelectric nanoplate due to surface effects and flexoelectricity. J Appl Phys 116(13):134308CrossRefGoogle Scholar
  54. Zhang DP, Lei YJ, Adhikari S (2018) Flexoelectric effect on vibration responses of piezoelectric nanobeams embedded in viscoelastic medium based on nonlocal elasticity theory. Acta Mech 229:2379–2392MathSciNetCrossRefGoogle Scholar
  55. Zheng H, Wang J, Lofland SE et al (2004) Multiferroic BaTiO3–CoFe2O4 nanostructures. Science 303:661–663CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018
corrected publication September 2018

Authors and Affiliations

  • Mahsa Karimiasl
    • 1
  • Kimiya Kargarfard
    • 1
    • 2
  • Farzad Ebrahimi
    • 1
  1. 1.Department of Mechanical Engineering, Faculty of EngineeringImam Khomeini International UniversityQazvinIran
  2. 2.Department of Civil Engineering, Faculty of EngineeringImam Khomeini International UniversityQazvinIran

Personalised recommendations