Thermal buckling of functionally graded triangular microplates

  • Parviz MalekzadehEmail author
  • Amin Ghorbani Shenas
  • Sima Ziaee
Technical Paper


The thermal buckling behavior of thin to moderately thick functionally graded isosceles triangular microplates with temperature-dependent material properties is investigated. The governing equations are derived based on the modified strain gradient theory (MSGT) in conjunction with the first-order shear deformation theory. The adjacent equilibrium criterion and Chebyshev–Ritz method are employed to derive the nonlinear thermal buckling eigenvalue equations, which are solved by a direct iterative method. The fast rate of convergence and accuracy of the method are demonstrated numerically. Then, the effects of length scale parameters, material gradient index, different boundary conditions, apex angle and ratio of width to thickness on the critical temperature rises of the triangular microplates are studied. In addition, comparisons between the results of MSGT and modified couple stress theory and classical theory (CT) are performed. The results show that by increasing the apex angle, the critical temperature rise increases, but increase in the material gradient index and the dimensionless length scale parameter decreases the critical temperature rise. In addition, it is observed that by considering the temperature dependence of material properties, the critical temperature rises decrease significantly. Also, the MSGT and CT yield the highest and the lowest critical temperature rise, respectively.


Thermal buckling Triangular microplates Functionally graded materials Chebyshev–Ritz method Modified strain gradient theory 


  1. 1.
    Gammel P, Fischer G, Bouchaud J (2005) RF MEMS and NEMS technology, devices, and applications. Bell Labs Tech J 10:29–59CrossRefGoogle Scholar
  2. 2.
    Lajimi AM, Rahmany EA, Heppler GR (2009) On natural frequencies and mode shapes of microbeams. Proc Int Multi Conf Eng Comput Sci 2:18–20Google Scholar
  3. 3.
    Pei J, Tian F, Thundat T (2004) Glucose biosensor based on the microcantilever. Anal Chem 76:292–297CrossRefGoogle Scholar
  4. 4.
    Arlett JL, Myers EB, Roukes ML (2011) Comparative advantages of mechanical biosensor. Nat Nanotech 6:203–215CrossRefGoogle Scholar
  5. 5.
    Filippini D, Andersson TPM, Svensson SPS, Lundstrom I (2003) Microplate based biosensing with a computer screen aided technique. Biosens Bioelectron 19:35–41CrossRefGoogle Scholar
  6. 6.
    Singh PM (2009) Application of biolog FF micro plate for substrate utilization and metabolite profiling of closely related fungi. J Microbiol Methods 77:102–108CrossRefGoogle Scholar
  7. 7.
    Bruin GJ, Waldmeier F, Boernsen KO, Pfaar U, Gross G, Zollinger M (2006) A microplate solid scintillation counter as a radioactivity detector for high performance liquid chromatography in drug metabolism: validation and applications. J Chromatogr A 1133:184–194CrossRefGoogle Scholar
  8. 8.
    Fu YQ, Du HJ, Huang WM, Zhang S, Hu M (2004) TiNi-based thin films in MEMS applications: a review. Sens Act A Phys 112:395–408CrossRefGoogle Scholar
  9. 9.
    Witvrouw A, Mehta A (2005) The use of functionally graded poly-SiGe layers for MEMS applications. Mater Sci Forum 492–493:255–260CrossRefGoogle Scholar
  10. 10.
    Hasanyan DJ, Batra RC, Harutyunyan S (2008) Pull-in instabilities in functionally graded microthermoelectromechanical systems. J Therm Stress 31:1006–1021CrossRefGoogle Scholar
  11. 11.
    Lü CF, Lim CW, Chen WQ (2009) Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory. Int J Solids Struct 46:1176–1185CrossRefGoogle Scholar
  12. 12.
    Rahaeifard M, Kahrobaiyan M, Ahmadian MT (2009) Sensitivity analysis of atomic force microscope cantilever made of functionally graded materials. In: ASME proceedings 3rd international conference on micro- and nanosystems, vol 6, pp 539–544Google Scholar
  13. 13.
    Moser Y, Gijs MAM (2007) Miniaturized flexible temperature sensor. J Microelectromech Syst 16:1349–1354CrossRefGoogle Scholar
  14. 14.
    Qu H, Xie H (2007) Process development for CMOS-MEMS sensors with robust electrically isolated bulk silicon microstructures. J Microelectromech Syst 16:1152–1161CrossRefGoogle Scholar
  15. 15.
    Chong ACM, Yang F, Lam DCC, Tong P (2001) Torsion and bending of micron-scaled structures. J Mater Res 16:1052–1058CrossRefGoogle Scholar
  16. 16.
    Lam DCC, Yang F, Chong ACM, Wang J, Tong P (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51:1477–1508CrossRefGoogle Scholar
  17. 17.
    Mcfarland AW, Colton JS (2005) Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J Micromech Microeng 15:1060–1067CrossRefGoogle Scholar
  18. 18.
    Toupin RA (1960) Elastic materials with couple-stress. Arch Ration Mech Anal 11:385–414MathSciNetCrossRefGoogle Scholar
  19. 19.
    Mindlin R, Tiersten H (1962) Effects of couple-stresses in linear elasticity. Arch Ration Mech Anal 11:415–448MathSciNetCrossRefGoogle Scholar
  20. 20.
    Koiter WT (1964) Couple stresses in the theory of elasticity. Proc Konink Nederl Akad Wetensch B 67:17–44MathSciNetzbMATHGoogle Scholar
  21. 21.
    Mindlin RD (1964) Micro-structure in linear elasticity. Arch Ration Mech Anal 16(1):51–78MathSciNetCrossRefGoogle Scholar
  22. 22.
    Mindlin RD (1965) Second gradient of strain and surface tension in linear elasticity. Int J Solids Struct 1:417–438CrossRefGoogle Scholar
  23. 23.
    Eringen AC, Edelen DGB (1972) On nonlocal elasticity. Int J Eng Sci 10:233–248MathSciNetCrossRefGoogle Scholar
  24. 24.
    Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54:4703–4710CrossRefGoogle Scholar
  25. 25.
    Eringen AC (2002) Nonlocal continuum field theories. Springer, New YorkzbMATHGoogle Scholar
  26. 26.
    Shu J, Fleck N (1998) The prediction of a size effect in microindentation. Int J Solids Struct 35:1363–1383CrossRefGoogle Scholar
  27. 27.
    Akgoz B, Civalek O (2014) A new trigonometric beam model for buckling of strain gradient microbeams. Int J Mech Sci 81:88–94CrossRefGoogle Scholar
  28. 28.
    Setoodeh AR, Afrahim S (2014) Nonlinear dynamic analysis of FG micro-pipes conveying fluid based on strain gradient theory. Compos Struct 116:128–135CrossRefGoogle Scholar
  29. 29.
    Zhang B, He Y, Liu D, Shen L, Lei J (2015) An efficient size-dependent plate theory for bending, buckling and free vibration analyses of functionally graded microplates resting on elastic foundation. Appl Math Model 39:3814–3845MathSciNetCrossRefGoogle Scholar
  30. 30.
    Zhang B, He Y, Liu D, Lei J, Lei S, Wang L (2015) A size-dependent third-order shear deformable plate model incorporating strain gradient effects for mechanical analysis of functionally graded circular/annular microplates. Compos Part B 79:553–580CrossRefGoogle Scholar
  31. 31.
    Hosseini M, Bahaadini R (2016) Size dependent stability analysis of cantilever micro-pipes conveying fluid based on modified strain gradient theory. Int J Eng Sci 101:1–13CrossRefGoogle Scholar
  32. 32.
    Ghorbani Shenas A, Malekzadeh P (2017) Thermal environmental effects on free vibration of functionally graded isosceles triangular microplates. Mech Adv Mater Struct 24:885–907CrossRefGoogle Scholar
  33. 33.
    Farahmand H, Ahmadi AR, Arabnejad S (2011) Thermal buckling analysis of rectangular microplates using higher continuity p-version finite element method. Thin Wall Struct 49:1584–1591CrossRefGoogle Scholar
  34. 34.
    Lie J, He Y, Guo S, Li Z, Liu D (2016) Size-dependent vibration of nickel cantilever microbeams: experiment and gradient elasticity. AIP Adv 6(10):105202CrossRefGoogle Scholar
  35. 35.
    Lie J, He Y, Guo S, Li Z, Liu D (2016) Thermal buckling and vibration of functionally graded sinusoidal microbeams incorporating nonlinear temperature distribution using DQM. J Therm Stress 40(6):1–25Google Scholar
  36. 36.
    Ke LL, Wang YS, Wang ZD (2011) Thermal effect on free vibration and buckling of size-dependent microbeams. Phys E 43:1387–1393CrossRefGoogle Scholar
  37. 37.
    Mohammadi H, Mahzon M (2013) Thermal effects on postbuckling of nonlinear microbeams based on the modified strain gradient theory. Compos Struct 106:764–776CrossRefGoogle Scholar
  38. 38.
    Akgöz B, Civalek Ö (2014) Thermo-mechanical buckling behavior of functionally graded microbeams embedded in elastic medium. Int J Eng Sci 85:90–104CrossRefGoogle Scholar
  39. 39.
    Tao C, Fu Y (2017) Thermal buckling and postbuckling analysis of size-dependent composite laminated microbeams based on a new modified couple stress theory. Acta Mech 228:1711–1724MathSciNetCrossRefGoogle Scholar
  40. 40.
    Ghorbani Shenas A, Malekzadeh P, Ziaee S (2017) Thermal buckling of rotating pre-twisted functionally graded microbeams with temperature-dependent material properties. Acta Mech 228:1115–1133MathSciNetCrossRefGoogle Scholar
  41. 41.
    Cheung YK, Zhou D (2002) Three-dimensional vibration analysis of cantilevered and completely free isosceles triangular plates. Int J Solids Struct 39:673–687CrossRefGoogle Scholar
  42. 42.
    Dong CY (2008) Three-dimensional vibration analysis of functionally graded annular plates using the Chebyshev–Ritz method. Mater Des 29:1518–1525CrossRefGoogle Scholar
  43. 43.
    Malekzadeh P, Bahranifard F, Ziaee S (2013) Three-dimensional free vibration analysis of functionally graded cylindrical panels with cut-out using Chebyshev–Ritz method. Compos Struct 105:1–13CrossRefGoogle 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 93:1246–1254CrossRefGoogle Scholar
  45. 45.
    Shen HS, Wang ZX (2012) Assessment of Voigt and Mori–Tanaka models for vibration analysis of functionally graded plates. Compos Struct 94:2197–2208CrossRefGoogle Scholar
  46. 46.
    Kim YW (2005) Temperature dependent vibration analysis of functionally graded rectangular plates. J Sound Vib 284:531–549CrossRefGoogle Scholar
  47. 47.
    Tornabene F, Viol E (2009) Free vibration analysis of functionally graded panels and shells of revolution. Meccanica 44:255–281CrossRefGoogle Scholar
  48. 48.
    Tornabene F (2009) Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution. Comput Methods Appl Mech Eng 198:2911–2935CrossRefGoogle Scholar
  49. 49.
    Tornabene F, Viola E, Inman DJ (2009) 2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annular plate structures. J Sound Vib 328:259–290CrossRefGoogle Scholar
  50. 50.
    Tornabene F, Fantuzzi N, Bacciocchi M (2014) Free vibrations of free-form doubly-curved shells made of functionally graded materials using higher-order equivalent single layer theories. Compos Part B 67:490–509CrossRefGoogle Scholar
  51. 51.
    Tornabene F, Fantuzzi N, Bacciocchi M (2017) A new doubly-curved shell element for the free vibrations of arbitrarily shaped laminated structures based on weak formulation isogeometric analysis. Compos Struct 171:429–461CrossRefGoogle Scholar
  52. 52.
    Fantuzzi N, Tornabene F (2016) Strong formulation isogeometric analysis (SFIGA) for laminated composite arbitrarily shaped plates. Compos Part B 96:173–203CrossRefGoogle Scholar
  53. 53.
    Tornabene F, Fantuzzi N, Bacciocchi M (2016) The GDQ method for the free vibration analysis of arbitrarily shaped laminated composite shells using a NURBS-based isogeometric approach. Compos Struct 154:190–218CrossRefGoogle Scholar
  54. 54.
    Xiang Y, Wang CM, Kitipornchai S, Liew KM (1994) Buckling of triangular Mindlin plates under isotropic inplane compression. Acta Mech 102:123–135CrossRefGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  1. 1.Department of Mechanical Engineering, School of EngineeringPersian Gulf UniversityBushehrIran
  2. 2.Department of Mechanical EngineeringYasouj UniversityYasoujIran

Personalised recommendations