pp 1–21 | Cite as

An improved quadrilateral finite element for nonlinear second-order strain gradient elastic Kirchhoff plates

  • Bishweshwar Babu
  • B. P. PatelEmail author


In this paper, a geometrically nonlinear size-dependent Elastic Kirchhoff nanoplate model is developed based on the second-order negative strain gradient nonlocal theory useful for capturing the size effects. Taking the mid-plane stretching into consideration as the source of the nonlinear behaviour, weak form of the governing partial differential equations of equilibrium and the relevant classical/non-classical boundary conditions are derived using the variational method. For finite element analysis, the weak form requires C1 continuity of the in-plane displacements and C2 continuity of the transverse displacement. In the present work, a new computationally efficient subparametric nonconforming 4-noded finite element of arbitrary quadrilateral shape is presented for the first time for the modelling of nanoplates using the second-order negative strain gradient theory. The performance of the developed finite element is investigated for static bending of rectangular nanoplates with all edges simply supported and all edges clamped boundary conditions. The proposed element is found to be accurate and depicts good convergence characteristics for rectangular and non-rectangular meshes. The strain gradient model with negative nonlocal coefficient predicts results matching with those from the lattice model available in the literature.


Nonlocal Strain gradient Variational Finite Element Analysis Subparametric Nonlinearity 



  1. 1.
    Bunch JS, Van Der Zande AM, Verbridge SS, Frank IW, Tanenbaum DM, Parpia JM, Craighead HG, McEuen PL (2007) Electromechanical resonators from graphene sheets. Science 315(5811):490–493ADSCrossRefGoogle Scholar
  2. 2.
    Sakhaee-Pour A, Ahmadian MT, Vafai A (2008) Application of single-layered graphene sheets as mass sensors and atomistic dust detectors. Solid State Commun 145(4):168–172ADSCrossRefGoogle Scholar
  3. 3.
    Arash B, Wang Q, Duan WH (2011) Detection of gas atoms via vibration of graphenes. Phys Lett A 375(24):2411–2415ADSCrossRefGoogle Scholar
  4. 4.
    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
  5. 5.
    Stölken JS, Evans AG (1998) A microbend test method for measuring the plasticity length scale. Acta Mater 46(14):5109–5115CrossRefGoogle Scholar
  6. 6.
    Lam DC, Yang F, Chong ACM, Wang J, Tong P (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51(8):1477–1508ADSzbMATHCrossRefGoogle Scholar
  7. 7.
    McFarland AW, Colton JS (2005) Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J Micromech Microeng 15(5):1060–1067CrossRefGoogle Scholar
  8. 8.
    Allinger NL (1977) Conformational analysis. 130. MM2. A hydrocarbon force field utilizing V1 and V2 torsional terms. J Am Chem Soc 99(25):8127–8134CrossRefGoogle Scholar
  9. 9.
    Lii JH, Allinger NL (1989) Molecular mechanics. The MM3 force field for hydrocarbons. 3. The van der Waals’ potentials and crystal data for aliphatic and aromatic hydrocarbons. J Am Chem Soc 111(23):8576–8582CrossRefGoogle Scholar
  10. 10.
    Tersoff J (1988) New empirical approach for the structure and energy of covalent systems. Phys Rev B 37(12):6991–7000ADSCrossRefGoogle Scholar
  11. 11.
    Brenner DW (1990) Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films. Phys Rev B 42(15):9458–9471ADSCrossRefGoogle Scholar
  12. 12.
    Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54(9):4703–4710ADSCrossRefGoogle Scholar
  13. 13.
    Ghorbanpour-Arani AH, Rastgoo A, Sharafi MM, Kolahchi R, Arani AG (2016) Nonlocal viscoelasticity based vibration of double viscoelastic piezoelectric nanobeam systems. Meccanica 51(1):25–40MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Zhong J, Fu Y, Tao C (2016) Linear free vibration in pre/post-buckled states and nonlinear dynamic stability of lipid tubules based on nonlocal beam model. Meccanica 51(6):1481–1489MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Mohammadsalehi M, Zargar O, Baghani M (2017) Study of non-uniform viscoelastic nanoplates vibration based on nonlocal first-order shear deformation theory. Meccanica 52(4–5):1063–1077MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Aifantis EC (1992) On the role of gradients in the localization of deformation and fracture. Int J Eng Sci 30(10):1279–1299zbMATHCrossRefGoogle Scholar
  17. 17.
    Ru CQ, Aifantis EC (1993) A simple approach to solve boundary-value problems in gradient elasticity. Acta Mech 101(1–4):59–68MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Chang CS, Gao J (1995) Second-gradient constitutive theory for granular material with random packing structure. Int J Solids Struct 32(16):2279–2293zbMATHCrossRefGoogle Scholar
  19. 19.
    Mühlhaus HB, Oka F (1996) Dispersion and wave propagation in discrete and continuous models for granular materials. Int J Solids Struct 33(19):2841–2858zbMATHCrossRefGoogle Scholar
  20. 20.
    Gutkin MY, Aifantis EC (1999) Dislocations in the theory of gradient elasticity. Scr Mater 5(40):559–566CrossRefGoogle Scholar
  21. 21.
    Aifantis EC (2003) Update on a class of gradient theories. Mech Mater 35(3–6):259–280CrossRefGoogle Scholar
  22. 22.
    Yang FACM, Chong ACM, Lam DCC, Tong P (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39(10):2731–2743zbMATHCrossRefGoogle Scholar
  23. 23.
    Farokhi H, Ghayesh MH, Kosasih B, Akaber P (2016) On the nonlinear resonant dynamics of Timoshenko microbeams: effects of axial load and geometric imperfection. Meccanica 51(1):155–169MathSciNetCrossRefGoogle Scholar
  24. 24.
    Farokhi H, Ghayesh MH, Hussain S (2016) Dynamic stability in parametric resonance of axially excited Timoshenko microbeams. Meccanica 51(10):2459–2472MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ghasabi SA, Shahgholi M, Arbabtafti M (2018) Dynamic bifurcations analysis of a micro rotating shaft considering non-classical theory and internal damping. Meccanica 53(15):3795–3814MathSciNetCrossRefGoogle Scholar
  26. 26.
    Gholami R, Darvizeh A, Ansari R, Hosseinzadeh M (2014) Size-dependent axial buckling analysis of functionally graded circular cylindrical microshells based on the modified strain gradient elasticity theory. Meccanica 49(7):1679–1695MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Krishnan NA, Ghosh D (2017) Buckling analysis of cylindrical thin-shells using strain gradient elasticity theory. Meccanica 52(6):1369–1379MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Di Paola M, Failla G, Zingales M (2009) Physically-based approach to the mechanics of strong non-local linear elasticity theory. J Elast 97(2):103–130MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Di Paola M, Failla G, Pirrotta A, Sofi A, Zingales M (2013) The mechanically based non-local elasticity: an overview of main results and future challenges. Philos Trans R Soc A 371(1993):20120433MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Peddieson J, Buchanan GR, McNitt RP (2003) Application of nonlocal continuum models to nanotechnology. Int J Eng Sci 41(3–5):305–312CrossRefGoogle Scholar
  31. 31.
    Reddy JN, Pang SD (2008) Nonlocal continuum theories of beams for the analysis of carbon nanotubes. J Appl Phys 103(2):023511ADSCrossRefGoogle Scholar
  32. 32.
    Babu B, Patel BP (2019) On the finite element formulation for second-order strain gradient nonlocal beam theories. Mech Adv Mater Struct 26(15):1316–1332CrossRefGoogle Scholar
  33. 33.
    Gitman IM, Askes H, Aifantis EC (2005) The representative volume size in static and dynamic micro-macro transitions. Int J Fract 135(1–4):L3–L9zbMATHCrossRefGoogle Scholar
  34. 34.
    Tsepoura KG, Papargyri-Beskou S, Polyzos D, Beskos DE (2002) Static and dynamic analysis of a gradient-elastic bar in tension. Arch Appl Mech 72(6):483–497zbMATHCrossRefGoogle Scholar
  35. 35.
    Papargyri-Beskou S, Tsepoura KG, Polyzos D, Beskos DE (2003) Bending and stability analysis of gradient elastic beams. Int J Solids Struct 40(2):385–400zbMATHCrossRefGoogle Scholar
  36. 36.
    Papargyri-Beskou S, Beskos DE (2008) Static, stability and dynamic analysis of gradient elastic flexural Kirchhoff plates. Arch Appl Mech 78(8):625–635zbMATHCrossRefGoogle Scholar
  37. 37.
    Papargyri-Beskou S, Giannakopoulos AE, Beskos DE (2010) Variational analysis of gradient elastic flexural plates under static loading. Int J Solids Struct 47(20):2755–2766zbMATHCrossRefGoogle Scholar
  38. 38.
    Babu B, Patel BP (2019) Analytical solution for strain gradient elastic Kirchhoff rectangular plates under transverse static loading. Eur J Mech A Solids 73:101–111MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Pegios IP, Papargyri-Beskou S, Beskos DE (2015) Finite element static and stability analysis of gradient elastic beam structures. Acta Mech 226(3):745–768MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Babu B, Patel BP (2019) A new computationally efficient finite element formulation for nanoplates using second-order strain gradient Kirchhoff’s plate theory. Compos Part B Eng 168:302–311CrossRefGoogle Scholar
  41. 41.
    Niiranen J, Kiendl J, Niemi AH, Reali A (2017) Isogeometric analysis for sixth-order boundary value problems of gradient-elastic Kirchhoff plates. Comput Methods Appl Mech Eng 316:328–348ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    Timoshenko S, Woinowsky-Krieger S (1959) Theory of plates and shells. McGraw-Hill, New YorkzbMATHGoogle Scholar
  43. 43.
    Reddy JN (2014) An introduction to nonlinear finite element analysis: with applications to heat transfer, fluid mechanics, and solid mechanics. OUP Oxford, OxfordzbMATHCrossRefGoogle Scholar
  44. 44.
    Lachemi M, Lahoud AE (1991) A refined quadrilateral element for the finite-element analysis of plates. Commun Appl Numer Methods 7(7):527–537zbMATHCrossRefGoogle Scholar
  45. 45.
    Scarpa F, Adhikari S, Gil AJ, Remillat C (2010) The bending of single layer graphene sheets: the lattice versus continuum approach. Nanotechnology 21(12):125702ADSCrossRefGoogle Scholar
  46. 46.
    Reddy JN (1997) Mechanics of laminated composite plates: theory and analysis. CRC Press, Boca RatonzbMATHGoogle Scholar
  47. 47.
    Ming PG, Fa LS (1987) A new element used in the non-orthogonal boundary plate bending theory—an arbitrarily quadrilateral element. Int J Numer Methods Eng 24(6):1031–1042zbMATHCrossRefGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Applied MechanicsIndian Institute of Technology DelhiNew DelhiIndia

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