Abstract
Finite element analysis of functionally graded plates based on a general third-order shear deformation plate theory with a modified couple stress effect and the von Kármán nonlinearity is carried out to bring out the effects of couple stress, geometric nonlinearity and power-law variation of the material composition through the plate thickness on the bending deflections of plates. The theory requires no shear correction factors. The principle of virtual displacements is utilized to develop a nonlinear finite element model. The finite element model requires C 1 continuity of all dependent variables. The microstructural effects are captured using a length scale parameter via the modified couple stress theory. The variation of two-constituent material is assumed through the thickness direction according to a power-law distribution. Numerical results are presented for static bending problems of rectangular plates with various boundary conditions to bring out the parametric effects of the power-law index and length scale parameter on the load–deflection characteristics of plates with various boundary conditions.
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Aliaga W., Reddy J.N.: Nonlinear thermoelastic response of functionally graded plates using the third-order plate theory. Int. J. Comput. Eng. Sci. 5(4), 753–780 (2004)
Asghari M., Kahrobaiyan M.H., Ahmadian M.T.: A nonlinear Timoshenko beam formulation based on the modified couple stress theory. Int. J. Eng. Sci. 48, 1749–1761 (2010)
Chong A.C.M., Lam D.C.C.: Strain gradient plasticity effect in indentation hardness of polymers. J. Mater. Res. 14, 4103–4110 (1999)
Eringen A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)
Fleck N.A., Muller G.M., Ashby M.F., Hutchinson J.W.: Strain gradient plasticity: theory and experiments. Acta Metall. Mater. 42, 475–487 (1994)
Fuchiyama T., Noda N., Tsuji T., Obata Y.: Analysis of thermal stress and stress intensity factor of functionally gradient materials. Ceram. Trans. Funct. Gradient. Mater. 34, 425–432 (1993)
Fukui Y., Yamanaka N.: Elastic analysis for thick-walled tubes of functionally graded material subjected to internal pressure. JSME Int. J. Ser. 1 35, 379–385 (1992)
Fukui Y., Yamanaka N., Wakashima K.: The stress and strains in a thick-walled tube of functionally graded materials under uniform thermal loading. Int. J. Jpn. Soc. Mech. Eng. Ser. A 36, 156–162 (1993)
Hasselaman D.P.H., Youngblood G.E.: Enhanced thermal stress resistance of structural ceramics with thermal conductivity gradient. J. Am. Ceram. Soc. 61(1–2), 49–52 (1978)
Jin Z.H., Noda N.: Minimization of thermal stress intensity factor for a crack in a metal–ceramic mixture. Ceram. Trans. Funct. Gradient. Mater. 34, 47–54 (1993)
Jin Z.H., Noda N.: Transient thermal stress intensity factors for a crack in a semi-infinite plate of a functionally gradient material. Int. J. Solids Struct. 31, 203–218 (1994)
Jomehzadeh E., Noori H.R., Saidi A.R.: The size-dependent vibration analysis of micro-plates based on a modified couple stress theory. Phys. E Low Dimens. Syst. Nanostruct. 43(4), 877–883 (2011)
Kim J., Reddy J.N.: Analytical solutions for bending, vibration, and buckling of {FGM} plates using a couple stress-based third-order theory. Compos. Struct. 103, 86–98 (2013)
Koiter W.T.: Couple stresses in the theory of elasticity. Proc. K. Ned. Acad. Wet. Ser. B Phys. Sci. 67, 17–44 (1964)
Koizumi M.: The concept of FGM. Ceram. Trans. Funct. Graded Mater. 34, 3–10 (1993)
Lu P., Zhang P.Q., Lee H.P., Reddy J.N.: Non-local elastic plate theories. Proc. R. Soc. 463, 3225–3240 (2007)
Ma H.M., Gao X.-L., Reddy J.N.: A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids 56(12), 3379–3391 (2008)
Ma H.M., Gao X.-L., Reddy J.N.: A non-classical mindlin plate model based on a modified couple stress. Acta Mech. 220(1–4), 217–235 (2011)
Ma Q., Clarke D.R.: Size dependent hardness of silver single crystals. J. Mater. Res. 10, 853–863 (1995)
Mindlin R.D., Tiersten H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11, 415–448 (1962)
Noda N.: Thermal stresses in materials with temperature-dependent properties. Appl. Mech. Rev. 44(9), 383–397 (1991)
Noda N., Jin Z.H.: Thermal stress intensity factors for a crack in a strip of a functionally gradient material. Int. J. Solids Struct. 30, 1039–1056 (1993)
Noda N., Tsuji T.: Steady thermal stresses in a plate of functionally gradient material with temperature-dependent properties. Trans. Jpn. Soc. Mech. Eng. Ser. A 57(9), 625–631 (1991)
Obata Y., Noda N., Tsuji T.: Steady thermal stresses in a functionally gradient material plate. Trans. Jpn. Soc. Mech. Eng. 58(553), 1689–1695 (1992)
Park S.K., Gao X.-L.: Bernoulli–Euler beam model based on a modified couple stress theory. J. Micromec. Microeng. 16(11), 2355–2359 (2006)
Praveen G.N., Reddy J.N.: Nonlinear transient thermoelastic analysis of functionally graded ceramic–metal plates. Int. J. Solids Struct. 35(33), 4457–4476 (1998)
Reddy J.N.: A refined nonlinear theory of plates with transverse shear deformation. Int. J. Solids Struct. 20, 881–896 (1984)
Reddy J.N.: Analysis of functionally graded plates. Int. J. Numer. Methods Eng. 47(1–3), 663–684 (2000)
Reddy J.N.: An Introduction to Nonlinear Finite Element Analysis, 2nd edn. Oxford University Press, Oxford (2015)
Reddy J.N.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45(11), 288–307 (2007)
Reddy J.N.: An Introduction to Continuum Mechanics with Applications, 2nd edn. Cambridge University Press, New York (2013)
Reddy J.N.: Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. Int. J. Eng. Sci. 48, 1507–1518 (2010)
Reddy J.N., Berry J.: Nonlinear theories of axisymmetric bending of functionally graded circular plates with modified couple stress. Compos. Struct. 94(12), 3664–3668 (2012)
Reddy J.N., Kim J.: A nonlinear modified couple stress-based third-order theory of functionally graded plates. Compos. Struct. 94(3), 1128–1143 (2012)
Reddy, J.N., Romanoff, J., Loya, J.A.: Nonlinear finite element analysis of functionally graded circular plates with modified couple stress theory. Eur. J. Mech. A Solids (in review)
Saidi A.R., Bodaghi M., Atashipour S.R.: Levy-type solution for bending-stretching of thick functionally graded rectangular plates based on third-order shear deformation theory. Mech. Adv. Mater. Struct. 19(8), 577–589 (2012)
Shen H.-S.: Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments. Int. J. Mech. Sci. 44(3), 561–584 (2002)
Stölken J.S., Evans A.G.: A microbend test method for measuring the plasticity length scale. Acta Metall. Mater. 46, 5109–5115 (1998)
Toupin R.A.: Elastic materials with couple stresses. Arch. Ration. Mech. Anal. 11, 385–414 (1962)
Tsiatas G.C.: A new Kirchhoff plate model based on a modified couple stress theory. Int. J. Solids Struct. 46, 2757–2764 (2009)
Yang F., Chong A.C.M., Lam D.C.C., Tong P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39(10), 2731–2743 (2002)
Yang J., Shen H.S.: Non-linear bending analysis of shear deformable functionally graded plates subjected to thermo-mechanical loads under various boundary conditions. Compos. Part B 34, 103–115 (2003)
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Kim, J., Reddy, J.N. A general third-order theory of functionally graded plates with modified couple stress effect and the von Kármán nonlinearity: theory and finite element analysis. Acta Mech 226, 2973–2998 (2015). https://doi.org/10.1007/s00707-015-1370-y
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DOI: https://doi.org/10.1007/s00707-015-1370-y