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Galerkin’s approach for buckling analysis of functionally graded anisotropic nanoplates/different boundary conditions

  • Behrouz Karami
  • Maziar Janghorban
  • Abdelouahed Tounsi
Original Article
  • 18 Downloads

Abstract

For the first time, buckling behavior of functionally graded (FG) nanoplates made of anisotropic material (beryllium crystal as a hexagonal material) is investigated. Also, it is the first time that the size-dependent behavior of nanostructured systems is studied for buckling response of the graded anisotropic material. The properties of graded material are assumed vary exponentially through the z-direction. Nonlocal strain gradient theory is utilized to predicate the size-dependent buckling behavior of the nanoplate. The nanoplate is modeled by a higher order shear deformation refined plate theory in which any shear correction factor not used. Governing equations and boundary conditions are obtained using a virtual work of variational approach. To solve the buckling problem for different boundary conditions, Galerkin’s approach is utilized. Finally, the influences of different boundary conditions, small-scale parameters, geometry parameters and exponential factor are studied and discussed in detail. It is hoped that the present numerical results can help the engineers and designers to understand and predict the buckling response of FG anisotropic materials.

Keywords

Functionally graded anisotropic material Nanoplate Nonlocal strain gradient theory Buckling Boundary conditions 

Notes

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Kar VR, Panda SK (2015) Nonlinear flexural vibration of shear deformable functionally graded spherical shell panel. Steel Compos Struct 18(3):693–709Google Scholar
  2. 2.
    Ebrahimi F, Dabbagh A (2017) On flexural wave propagation responses of smart FG magneto-electro-elastic nanoplates via nonlocal strain gradient theory. Compos Struct 162:281–293Google Scholar
  3. 3.
    Shahsavari D, Shahsavari M, Li L, Karami B (2018) A novel quasi-3D hyperbolic theory for free vibration of FG plates with porosities resting on Winkler/Pasternak/Kerr foundation. Aerosp Sci Technol 72:134–149Google Scholar
  4. 4.
    Yang H, Wu H, Yao Z, Shi B, Xu Z, Cheng X, Pan F, Liu G, Jiang Z, Cao X (2018) Functionally graded membranes from nanoporous covalent organic frameworks for highly selective water permeation. J Mater Chem A 6(2):583–591Google Scholar
  5. 5.
    Kar VR, Panda SK (2015) Free vibration responses of temperature dependent functionally graded curved panels under thermal environment. Latin Am J Solids Struct 12(11):2006–2024Google Scholar
  6. 6.
    Kar VR, Panda SK (2015) Thermoelastic analysis of functionally graded doubly curved shell panels using nonlinear finite element method. Compos Struct 129:202–212Google Scholar
  7. 7.
    Karami B, Shahsavari D, Karami M, Li L (2018) Hygrothermal wave characteristic of nanobeam-type inhomogeneous materials with porosity under magnetic field. Proc Inst Mech Eng Part C J Mech Eng Sci.  https://doi.org/10.1177/0954406218781680 Google Scholar
  8. 8.
    Mindlin RD (1951) Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J Appl Mech 18:31–38zbMATHGoogle Scholar
  9. 9.
    Karami B, Janghorban M, Li L (2018) On guided wave propagation in fully clamped porous functionally graded nanoplates. Acta Astronaut 143:380–390Google Scholar
  10. 10.
    Kahya V, Turan M (2017) Finite element model for vibration and buckling of functionally graded beams based on the first-order shear deformation theory. Compos Part B Eng 109:108–115Google Scholar
  11. 11.
    Sofiyev A, Hui D, Haciyev V, Erdem H, Yuan G, Schnack E, Guldal V (2017) The nonlinear vibration of orthotropic functionally graded cylindrical shells surrounded by an elastic foundation within first order shear deformation theory. Compos Part B Eng 116:170–185Google Scholar
  12. 12.
    Sharma N, Mahapatra TR, Panda SK, Katariya P (2018) Thermo-acoustic analysis of higher order shear deformable laminated composite sandwich flat panel. J Sandwich Struct Mater.  https://doi.org/10.1177/1099636218784846 Google Scholar
  13. 13.
    Katariya PV, Das A, Panda SK (2018) Buckling analysis of SMA bonded sandwich structure-using FEM. In: IOP conference series: materials science and engineering, vol 1. IOP Publishing, p 012035Google Scholar
  14. 14.
    Katariya PV, Hirwani CK, Panda SK (2018) Geometrically nonlinear deflection and stress analysis of skew sandwich shell panel using higher order theory. Eng Comput.  https://doi.org/10.1007/s00366-018-0609-3 Google Scholar
  15. 15.
    Katariya PV, Panda SK, Hirwani CK, Mehar K, Thakare O (2017) Enhancement of thermal buckling strength of laminated sandwich composite panel structure embedded with shape memory alloy fibre. SMART STRUCTURES AND SYSTEMS 20(5):595–605Google Scholar
  16. 16.
    Katariya P, Panda SK (2017) Simulation study of transient responses of laminated composite sandwich plate. In: ASME 2017 gas turbine india conference, 2017. American Society of Mechanical Engineers, pp V002T005A031–V002T005A031Google Scholar
  17. 17.
    Hirwani CK, Panda SK (2018) Numerical and experimental validation of nonlinear deflection and stress responses of pre-damaged glass-fibre reinforced composite structure. Ocean Eng 159:237–252Google Scholar
  18. 18.
    Singh VK, Hirwani CK, Panda SK, Mahapatra TR, Mehar K (2018) Numerical and experimental nonlinear dynamic response reduction of smart composite curved structure using collocation and non-collocation configuration. Proc Inst Mech Eng Part C J Mech Eng Sci.  https://doi.org/10.1177/0954406218774362 Google Scholar
  19. 19.
    Mehar K, Panda SK, Mahapatra TR (2018) Large deformation bending responses of nanotube-reinforced polymer composite panel structure: Numerical and experimental analyses. Proc Inst Mech Eng Part G J Aerosp Eng.  https://doi.org/10.1177/0954410018761192 Google Scholar
  20. 20.
    Karami B, Shahsavari D, Janghorban M, Li L (2018) Wave dispersion of mounted graphene with initial stress. Thin Walled Struct 122:102–111Google Scholar
  21. 21.
    Katariya PV, Panda SK (2018) Numerical evaluation of transient deflection and frequency responses of sandwich shell structure using higher order theory and different mechanical loadings. Eng Comput.  https://doi.org/10.1007/s00366-018-0646-y Google Scholar
  22. 22.
    Kar V, Panda S (2016) Nonlinear thermomechanical behavior of functionally graded material cylindrical/hyperbolic/elliptical shell panel with temperature-dependent and temperature-independent properties. J Pressure Vessel Technol 138(6):061202Google Scholar
  23. 23.
    Kar VR, Panda SK (2013) Free vibration responses of functionally graded spherical shell panels using finite element method. In: ASME 2013 gas turbine India conference, 2013. American Society of Mechanical Engineers, pp V001T005A014–V001T005A014Google Scholar
  24. 24.
    Karami B, Janghorban M (2016) Effect of magnetic field on the wave propagation in nanoplates based on strain gradient theory with one parameter and two-variable refined plate theory. Mod Phys Lett B 30(36):1650421MathSciNetGoogle Scholar
  25. 25.
    Shahsavari D, Karami B, Mansouri S (2018) Shear buckling of single layer graphene sheets in hygrothermal environment resting on elastic foundation based on different nonlocal strain gradient theories. Eur J Mech A/Solids 67:200–214MathSciNetzbMATHGoogle Scholar
  26. 26.
    Sehoul M, Benguediab M, Bakora A, Tounsi A (2017) Free vibrations of laminated composite plates using a novel four variable refined plate theory. Steel Compos Struct 24(5):603–613Google Scholar
  27. 27.
    Shahsavari D, Janghorban M (2017) Bending and shearing responses for dynamic analysis of single-layer graphene sheets under moving load. J Braz Soc Mech Sci Eng 39(10):3849–3861Google Scholar
  28. 28.
    Karami B, Janghorban M, Shahsavari D, Tounsi A (2018) A size-dependent quasi-3D model for wave dispersion analysis of FG nanoplates. Steel Compos Struct 28(1):99–110Google Scholar
  29. 29.
    Shahsavari D, Karami B, Li L (2018) A high-order gradient model for wave propagation analysis of porous FG nanoplates. Steel Compos Struct 29(1):53–66Google Scholar
  30. 30.
    Karami B, Shahsavari D, Janghorban M (2018) A Comprehensive Analytical Study on Functionally Graded Carbon Nanotube-Reinforced Composite Plates. Aerosp Sci Technol 82:499–512Google Scholar
  31. 31.
    Karami B, Shahsavari D, Li L, Karami M, Janghorban M (2018) Thermal buckling of embedded sandwich piezoelectric nanoplates with functionally graded core by a nonlocal second-order shear deformation theory. Proc Inst Mech Eng Part C J Mech Eng Sci.  https://doi.org/10.1177/0954406218756451 Google Scholar
  32. 32.
    She G-L, Yuan F-G, Ren Y-R, Xiao W-S (2017) On buckling and postbuckling behavior of nanotubes. Int J Eng Sci 121:130–142MathSciNetzbMATHGoogle Scholar
  33. 33.
    She G-L, Yuan F-G, Ren Y-R (2017) Thermal buckling and post-buckling analysis of functionally graded beams based on a general higher order shear deformation theory. Appl Math Model 47:340–357MathSciNetGoogle Scholar
  34. 34.
    She G-L, Ren Y-R, Xiao W-S, Liu H (2018) Study on thermal buckling and post-buckling behaviors of FGM tubes resting on elastic foundations. Struct Eng Mech 66(6):729–736Google Scholar
  35. 35.
    Kar VR, Panda SK (2016) Post-buckling behaviour of shear deformable functionally graded curved shell panel under edge compression. Int J Mech Sci 115:318–324Google Scholar
  36. 36.
    Kar VR, Panda SK, Mahapatra TR (2016) Thermal buckling behaviour of shear deformable functionally graded single/doubly curved shell panel with TD and TID properties. Adv Mater Res Int J 5(4):205–221Google Scholar
  37. 37.
    Kar VR, Mahapatra TR, Panda SK (2017) Effect of different temperature load on thermal postbuckling behaviour of functionally graded shallow curved shell panels. Compos Struct 160:1236–1247Google Scholar
  38. 38.
    Kar VR, Panda SK (2015) Effect of temperature on stability behaviour of functionally graded spherical panel. In: IOP conference series: materials science and engineering, vol 1. IOP Publishing, p 012014Google Scholar
  39. 39.
    Mohammadi M, Saidi AR, Jomehzadeh E (2010) Levy solution for buckling analysis of functionally graded rectangular plates. Appl Compos Mater 17(2):81–93Google Scholar
  40. 40.
    Saidi A, Rasouli A, Sahraee S (2009) Axisymmetric bending and buckling analysis of thick functionally graded circular plates using unconstrained third-order shear deformation plate theory. Compos Struct 89(1):110–119Google Scholar
  41. 41.
    Zhang L, Lei Z, Liew K (2015) Buckling analysis of FG-CNT reinforced composite thick skew plates using an element-free approach. Compos Part B Eng 75:36–46Google Scholar
  42. 42.
    Lei Z, Liew KM, Yu J (2013) Buckling analysis of functionally graded carbon nanotube-reinforced composite plates using the element-free kp-Ritz method. Compos Struct 98:160–168zbMATHGoogle Scholar
  43. 43.
    Zenkour A, Sobhy M (2010) Thermal buckling of various types of FGM sandwich plates. Compos Struct 93(1):93–102Google Scholar
  44. 44.
    Malekzadeh P (2011) Three-dimensional thermal buckling analysis of functionally graded arbitrary straight-sided quadrilateral plates using differential quadrature method. Compos Struct 93(4):1246–1254Google Scholar
  45. 45.
    Sobhy M, Zenkour AM (2018) Thermal buckling of double-layered graphene system in humid environment. Mater Res Express 5(1):015028Google Scholar
  46. 46.
    Moita JS, Araújo AL, Correia VF, Soares CMM, Herskovits J (2018) Buckling and nonlinear response of functionally graded plates under thermo-mechanical loading. Compos Struct 202:719–730Google Scholar
  47. 47.
    Thai S, Thai H-T, Vo TP, Lee S (2018) Postbuckling analysis of functionally graded nanoplates based on nonlocal theory and isogeometric analysis. Compos StructGoogle Scholar
  48. 48.
    Eringen AC, Edelen D (1972) On nonlocal elasticity. Int J Eng Sci 10(3):233–248MathSciNetzbMATHGoogle Scholar
  49. 49.
    Shahsavari D, Karami B, Janghorban M, Li L (2017) Dynamic characteristics of viscoelastic nanoplates under moving load embedded within visco-Pasternak substrate and hygrothermal environment. Mater Res Express 4(8):085013Google Scholar
  50. 50.
    Ebrahimi F, Barati MR, Zenkour AM (2018) A new nonlocal elasticity theory with graded nonlocality for thermo-mechanical vibration of FG nanobeams via a nonlocal third-order shear deformation theory. Mech Adv Mater Struct 25(6):512–522Google Scholar
  51. 51.
    Zenkour AM (2018) A novel mixed nonlocal elasticity theory for thermoelastic vibration of nanoplates. Compos Struct 185:821–833Google Scholar
  52. 52.
    Rong D, Fan J, Lim C, Xu X, Zhou Z (2018) A new analytical approach for free vibration, buckling and forced vibration of rectangular nanoplates based on nonlocal elasticity theory. Int J Struct Stab Dyn 18(04):1850055MathSciNetGoogle Scholar
  53. 53.
    Zhang D, Lei Y, Adhikari S (2018) Flexoelectric effect on vibration responses of piezoelectric nanobeams embedded in viscoelastic medium based on nonlocal elasticity theory. Acta Mech 229(6):2379–2392MathSciNetzbMATHGoogle Scholar
  54. 54.
    Mehar K, Mahapatra TR, Panda SK, Katariya PV, Tompe UK (2018) Finite-element solution to nonlocal elasticity and scale effect on frequency behavior of shear deformable nanoplate structure. J Eng Mech 144(9):04018094Google Scholar
  55. 55.
    Karami B, Shahsavari D, Nazemosadat SMR, Li L, Ebrahimi A (2018) Thermal buckling of smart porous functionally graded nanobeam rested on Kerr foundation. Steel Compos Struct 29(3):349–362Google Scholar
  56. 56.
    Askes H, Aifantis EC (2009) Gradient elasticity and flexural wave dispersion in carbon nanotubes. Phys Rev B 80(19):195412Google Scholar
  57. 57.
    Zhu X, Li L (2017) Closed form solution for a nonlocal strain gradient rod in tension. Int J Eng Sci 119:16–28MathSciNetzbMATHGoogle Scholar
  58. 58.
    Li L, Li X, Hu Y (2016) Free vibration analysis of nonlocal strain gradient beams made of functionally graded material. Int J Eng Sci 102:77–92zbMATHGoogle Scholar
  59. 59.
    Shahsavari D, Karami B, Fahham HR, Li L (2018) On the shear buckling of porous nanoplates using a new size-dependent quasi-3D shear deformation theory. Acta Mech 229(11):4549–4573MathSciNetGoogle Scholar
  60. 60.
    El-Borgi S, Rajendran P, Friswell M, Trabelssi M, Reddy J (2018) Torsional vibration of size-dependent viscoelastic rods using nonlocal strain and velocity gradient theory. Compos Struct 186:274–292Google Scholar
  61. 61.
    Zeighampour H, Beni YT, Dehkordi MB (2018) Wave propagation in viscoelastic thin cylindrical nanoshell resting on a visco-Pasternak foundation based on nonlocal strain gradient theory. Thin Walled Struct 122:378–386Google Scholar
  62. 62.
    Karami B, Shahsavari D, Li L (2018) Temperature-dependent flexural wave propagation in nanoplate-type porous heterogenous material subjected to in-plane magnetic field. J Therm Stresses 41(4):483–499Google Scholar
  63. 63.
    Sahmani S, Aghdam MM, Rabczuk T (2018) Nonlinear bending of functionally graded porous micro/nano-beams reinforced with graphene platelets based upon nonlocal strain gradient theory. Compos Struct 186:68–78Google Scholar
  64. 64.
    Mohammadi K, Rajabpour A, Ghadiri M (2018) Calibration of nonlocal strain gradient shell model for vibration analysis of a CNT conveying viscous fluid using molecular dynamics simulation. Comput Mater Sci 148:104–115Google Scholar
  65. 65.
    Ebrahimi F, Barati MR (2018) Vibration analysis of piezoelectrically actuated curved nanosize FG beams via a nonlocal strain-electric field gradient theory. Mech Adv Mater Struct 25(4):350–359Google Scholar
  66. 66.
    Li X, Li L, Hu Y, Deng W, Ding Z (2017) A Refined Nonlocal Strain Gradient Theory for Assessing Scaling-Dependent Vibration Behavior of Microbeams. World Acad Sci Eng Technol Int J Mech Aerosp Ind Mech Manuf Eng 11(3):551–561Google Scholar
  67. 67.
    She G-L, Yuan F-G, Ren Y-R, Liu H-B, Xiao W-S (2018) Nonlinear bending and vibration analysis of functionally graded porous tubes via a nonlocal strain gradient theory. Compos Struct 203:614–623Google Scholar
  68. 68.
    Karami B, Shahsavari D, Janghorban M (2018) Wave propagation analysis in functionally graded (FG) nanoplates under in-plane magnetic field based on nonlocal strain gradient theory and four variable refined plate theory. Mech Adv Mater Struct 25(12):1047–1057Google Scholar
  69. 69.
    She G-L, Yan K-M, Zhang Y-L, Liu H-B, Ren Y-R (2018) Wave propagation of functionally graded porous nanobeams based on non-local strain gradient theory. Eur Phys J Plus 133(9):368Google Scholar
  70. 70.
    Karami B, Shahsavari D, Li L (2018) Hygrothermal wave propagation in viscoelastic graphene under in-plane magnetic field based on nonlocal strain gradient theory. Phys E 97:317–327Google Scholar
  71. 71.
    Shahsavari D, Karami B, Li L (2018) Damped vibration of a graphene sheet using a higher order nonlocal strain-gradient Kirchhoff plate model. Comptes Rendus Mécanique 346(12):1216–1232Google Scholar
  72. 72.
    Karami B, Janghorban M, Tounsi A (2017) Effects of triaxial magnetic field on the anisotropic nanoplates. Steel Compos Struct 25(3):361–374Google Scholar
  73. 73.
    Li L, Hu Y, Ling L (2015) Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory. Compos Struct 133:1079–1092Google Scholar
  74. 74.
    Nami MR, Janghorban M (2014) Resonance behavior of FG rectangular micro/nano plate based on nonlocal elasticity theory and strain gradient theory with one gradient constant. Compos Struct 111:349–353Google Scholar
  75. 75.
    Şimşek M (2016) Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach. Int J Eng Sci 105:12–27MathSciNetzbMATHGoogle Scholar
  76. 76.
    Shahverdi H, Barati MR (2017) Vibration analysis of porous functionally graded nanoplates. Int J Eng Sci 120:82–99MathSciNetzbMATHGoogle Scholar
  77. 77.
    Li X, Li L, Hu Y, Ding Z, Deng W (2017) Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory. Compos Struct 165:250–265Google Scholar
  78. 78.
    Karami B, Janghorban M, Tounsi A (2018) Variational approach for wave dispersion in anisotropic doubly-curved nanoshells based on a new nonlocal strain gradient higher order shell theory. Thin Walled Struct 129:251–264Google Scholar
  79. 79.
    Karami B, Janghorban M, Tounsi A (2018) Nonlocal strain gradient 3D elasticity theory for anisotropic spherical nanoparticles. Steel Compos Struct 27(2):201–216Google Scholar
  80. 80.
    Sahmani S, Aghdam MM, Rabczuk T (2018) Nonlocal strain gradient plate model for nonlinear large-amplitude vibrations of functionally graded porous micro/nano-plates reinforced with GPLs. Compos Struct 198:51–62Google Scholar
  81. 81.
    She G-L, Ren Y-R, Yuan F-G, Xiao W-S (2018) On vibrations of porous nanotubes. Int J Eng Sci 125:23–35MathSciNetzbMATHGoogle Scholar
  82. 82.
    She G-L, Yuan F-G, Ren Y-R (2018) On wave propagation of porous nanotubes. Int J Eng Sci 130:62–74MathSciNetzbMATHGoogle Scholar
  83. 83.
    Pan E (2003) Exact solution for functionally graded anisotropic elastic composite laminates. J Compos Mater 37(21):1903–1920Google Scholar
  84. 84.
    Shimpi RP (2002) Refined plate theory and its variants. AIAA J 40(1):137–146Google Scholar
  85. 85.
    Ebrahimi F, Jafari A, Barati MR (2017) Vibration analysis of magneto-electro-elastic heterogeneous porous material plates resting on elastic foundations. Thin Walled Struct 119:33–46Google Scholar
  86. 86.
    Barati MR (2018) A general nonlocal stress-strain gradient theory for forced vibration analysis of heterogeneous porous nanoplates. Eur J Mech A/Solids 67:215–230MathSciNetzbMATHGoogle Scholar
  87. 87.
    Natarajan S, Chakraborty S, Thangavel M, Bordas S, Rabczuk T (2012) Size-dependent free flexural vibration behavior of functionally graded nanoplates. Comput Mater Sci 65:74–80Google Scholar
  88. 88.
    Sobhy M (2015) A comprehensive study on FGM nanoplates embedded in an elastic medium. Compos Struct 134:966–980Google Scholar
  89. 89.
    Barati MR, Zenkour AM, Shahverdi H (2016) Thermo-mechanical buckling analysis of embedded nanosize FG plates in thermal environments via an inverse cotangential theory. Compos Struct 141:203–212Google Scholar

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© Springer-Verlag London Ltd., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical Engineering, Marvdasht BranchIslamic Azad UniversityMarvdashtIran
  2. 2.Material and Hydrology Laboratory, Civil Engineering Department, Faculty of TechnologyUniversity of Sidi Bel AbbesSidi Bel AbbèsAlgeria

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