Advertisement

Engineering with Computers

, Volume 35, Issue 1, pp 215–228 | Cite as

Non-classical plate model for FGMs

  • Marzieh Alizadeh
  • A. M. FattahiEmail author
Original Article

Abstract

In this work, we developed a non-classical plate model for the analysis of the behavior of microplates prepared from functionally graded materials (FGMs). It was assumed that the material properties of FGM microplates vary along the direction of thickness and could be estimated using Mori–Tanaka homogenization technique. A variational approach based on Hamilton’s principle was employed to achieve motion equations and corresponding boundary conditions simultaneously. Unlike classical plate theory, the proposed plate model contained three material length scale parameters which allowed size effect to be taken into account effectively. Moreover, the proposed non-classical plate model could be divided into classical and modified couple stress plate models when all material length scale parameters were set at zero. To approve the applicability of the proposed size-dependent plate model, analytical solution for free vibration problem of a simply supported FGM microplate was obtained according to the Navier solution, where generalized displacements were stated as multiplication of undetermined functions with known trigonometric functions to identically satisfy simply supported boundary conditions for all edges. The results obtained by the proposed non-classical plate model were compared with those obtained from reduced ones relevant to different values of material property gradient index and dimensionless length scale parameter. It was found that the differences observed between the sets of predicted natural frequencies obtained from different types of plate models were more significant for lower values of dimensionless length scale parameter.

Keywords

Microplates Free vibration Functionally graded materials Size effect 

References

  1. 1.
    Ansari R, Gholami R, Sahmani S (2011) Free vibration of size-dependent functionally graded microbeams based on a strain gradient theory. Compos Struct.  https://doi.org/10.1016/j.compstruct.2011.06.024 zbMATHGoogle Scholar
  2. 2.
    Asghari M, Rahaeifard M, Kahrobaiyan MH, Ahmadian MT (2011) The modified couple stress functionally graded Timoshenko beam formulation. Mater Des 32:1435–1443CrossRefzbMATHGoogle Scholar
  3. 3.
    Batista M (2010) An elementary derivation of basic equations of the Reissner and Mindlin plate theories. Eng Struct 32:906–909CrossRefGoogle Scholar
  4. 4.
    Chasiotis I, Knauss WG (2003) The mechanical strength of polysilicon films: part 2. Size effects associated with elliptical and circular perforations. J Mech Phys Solids 51:1551–1572CrossRefGoogle Scholar
  5. 5.
    Eringen AC, Suhubi ES (1964) Nonlinear theory of simple microelastic solid-I. Int J Eng Sci 2:89–203Google Scholar
  6. 6.
    Eringen AC, Suhubi ES (1964) Nonlinear theory of simple microelastic solid-II. Int J Eng Sci 2:389–404CrossRefGoogle Scholar
  7. 7.
    Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54:4703–4710CrossRefGoogle Scholar
  8. 8.
    Fares ME, Elmarghany MK, Atta D (2009) An efficient and simple refined theory for bending and vibration of functionally graded plates. Compos Struct 91:296–305CrossRefGoogle Scholar
  9. 9.
    Fleck NA, Hutchinson JW (1993) Phenomenological theory for strain gradient effects in plasticity. J Mech Phys Solids 41:1825–1857MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fleck NA, Muller GM, Ashby MF, Hutchinson JW (1994) Strain gradient plasticity—theory and experiment. Acta Metall Mater 42:475–484CrossRefGoogle Scholar
  11. 11.
    Fu YQ, Du HJ, Huang WM, Zhang S, Hu M (2004) TiNi-based thin films in MEMS applications: a review. J Sensors Actuators A 112:395–408CrossRefGoogle Scholar
  12. 12.
    Ganapathi M (2007) Dynamic stability characteristics of functionally graded materials shallow spherical shells. Compos Struct 79:338–343CrossRefGoogle Scholar
  13. 13.
    Hasanyan DJ, Batra RC, Harutyunyan RC (2008) Pull-in instabilities in functionally graded microthermoelectromechanical systems. J Therm Stresses 31:1006–1021CrossRefGoogle Scholar
  14. 14.
    Ke LL, Wang YS (2011) Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory. Compos Struct 93:342–350CrossRefGoogle Scholar
  15. 15.
    Ke LL, Wang YS, Wang ZD (2011) Thermal effect on free vibration and buckling of size-dependent microbeams. Phys E 43(7):1387–1393CrossRefGoogle Scholar
  16. 16.
    Ke LL, Wang YS, Yang J, Kitipornchai S (2011) Nonlinear free vibration of size-dependent functionally graded microbeams. Int J Eng Sci 50(1):256–267MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Koiter WT (1964) couple stresses in the theory of elasticity I and II. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 67:17–44MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kong S, Zhou S, Nie Z, Wang K (2009) Static and dynamic analysis of micro beams based on strain gradient elasticity theory. Int J Eng Sci 47:487–498MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lam DCC, Chong ACM (1999) Indentation model and strain gradient plasticity law for glassy polymers. J Mater Res 14:3784–3788CrossRefGoogle Scholar
  20. 20.
    Lam DCC, Yang F, Chong ACM, Wang J, Tong P (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51:1477–1508CrossRefzbMATHGoogle Scholar
  21. 21.
    Lee Z, Ophus C, Fischer LM, Nelson-Fitzpatrick N, Westra KL, Evoy S (2006) Metallic NEMS components fabricated from nanocomposite Al-Mo films. Nanotechnology 17:3063–3070CrossRefGoogle Scholar
  22. 22.
    Liu Y, Soh C-K (2007) Shear correction for Mindlin type plate and shell elements. Int J Num Methods Eng 69:2789–2806MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ma HM, Gao XL, Reddy JN (2011) A non-classical Mindlin plate model based on a modified couple stress theory. Acta Mech 220:217–235CrossRefzbMATHGoogle Scholar
  24. 24.
    McFarland AW, Colton JS (2005) Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J Micromech Microeng 15:1060–1067CrossRefGoogle Scholar
  25. 25.
    Mindlin RD (1951) American Society of Mechanical Engineers. J Appl Mech 73Google Scholar
  26. 26.
    Mindlin RD, Tiersten HF (1962) Effects of couple-stresses in linear elasticity. Arch Ration Mech Anal 11:415–448MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mindlin RD (1964) Micro-structure in linear elasticity. Arch Ration Mech Anal 16:51–78MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Mindlin RD (1965) Second gradient of strain and surface tension in linear elasticity. Int J Solids Struct 1:417–438CrossRefGoogle Scholar
  29. 29.
    Mindlin RD, Eshel NN (1968) On first strain-gradient theories in linear elasticity. Int J Solids Struct 4:109–124CrossRefzbMATHGoogle Scholar
  30. 30.
    Stephen NG (1997) Mindlin plate theory: best shear coefficient and higher spectra validity. J Sound Vib 202:539–553CrossRefGoogle Scholar
  31. 31.
    Timoshenko SP, Goodier JN (1970) Theory of elasticity, 3rd edn. McGraw-Hill, New YorkzbMATHGoogle Scholar
  32. 32.
    Toupin RA (1964) Theory of elasticity with couple stresses. Arch Ration Mech Anal 17:85–112CrossRefzbMATHGoogle Scholar
  33. 33.
    Tsiatas GC (2009) A new Krichhoff plate model based on a modified couple stress theory. Int J Solids Struct 46:2757–2764CrossRefzbMATHGoogle Scholar
  34. 34.
    Vardoulakis I, Exadaktylos G, Kourkoulis SK (1998) Bending of marble with intrinsic length scales: a gradient theory with surface energy and size effects. J Phys IV 8:399–406Google Scholar
  35. 35.
    Wang CM, Lim GT, Reddy JN, Lee KH (2001) Relationships between bending solutions of Reissner and Mindlin plate theories. Eng Struct 23:838–849CrossRefGoogle Scholar
  36. 36.
    Wang B, Zhao J, Zhou S (2010) A micro scale Timoshenko beam model based on strain gradient elasticity theory. Eur J Mech A/Solids 29:591–599CrossRefGoogle Scholar
  37. 37.
    Witvrouw A, Mehta A (2005) The use of functionally graded Poly-SiGe layers for MEMS applications. Mater Sci Forum 4992–493:255–260CrossRefGoogle Scholar
  38. 38.
    Xia W, Wang L, Yin L (2010) Nonlinear non-classical microscale beams: static bending, postbuckling and free vibration. Int J Eng Sci 48:2044–2053MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Yang F, Chong ACM, Lam DCC et al (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39:2731–2743CrossRefzbMATHGoogle Scholar
  40. 40.
    Fattahi AM, Sahmani S (2017) Size dependency in the axial postbuckling behavior of nanopanels made of functionally graded material considering surface elasticity. Arab J Sci Eng 1–17Google Scholar
  41. 41.
    Sahmani S, Fattahi AM (2017) An anisotropic calibrated nonlocal plate model for biaxial instability analysis of 3D metallic carbon nanosheets using molecular dynamics simulations. Mater Res Express 4(6):1–14CrossRefGoogle Scholar
  42. 42.
    Fattahi AM, Safaei B (2017) Buckling analysis of CNT-reinforced beams with arbitrary boundary conditions. Microsyst Technol 23(10):5079–5091CrossRefGoogle Scholar
  43. 43.
    Sahmani S, Fattahi AM (2017) Thermo-electro-mechanical size-dependent postbuckling response of axially loaded piezoelectric shear deformable nanoshells via nonlocal elasticity theory. Microsyst Technol 23(10):5105–5119CrossRefGoogle Scholar
  44. 44.
    Fattahi AM, Sahmani S (2017) Nonlocal temperature-dependent postbuckling behavior of FG-CNT reinforced nanoshells under hydrostatic pressure combined with heat conduction. Microsyst Technol 23(10):5121–5137CrossRefGoogle Scholar
  45. 45.
    Sahmani S, Fattahi AM (2016) Size-dependent nonlinear instability of shear deformable cylindrical nanopanels subjected to axial compression in thermal environments. Microsyst Technol 23(10):4717–4731CrossRefGoogle Scholar
  46. 46.
    Fattahi AM, Najipour A (2017) Experimental study on mechanical properties of PE/CNT composites. J Theor Appl Mech 55(2):719–726Google Scholar
  47. 47.
    Azizi S, Safaei B, Fattahi AM, Tekere M (2015) Nonlinear vibrational analysis of nanobeams embedded in an elastic medium including surface stress effects. Adv Mater Sci Eng 1–7Google Scholar
  48. 48.
    Azizi S, Fattahi AM, Kahnamouei JT (2015) Evaluating mechanical properties of nanoplatelet reinforced composites undermechanical and thermal loads. Compu Theor Nanosci 12:4179–4185CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringTabriz Branch, Islamic Azad UniversityTabrizIran

Personalised recommendations