Bending and Buckling Analysis of a Nth-Order Shear Deformation Nanoplate using Modified Couple Stress Theory

Document Type : Original Article

Authors

1 Department of Aerospace Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

2 Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

10.30495/admt.2021.1915086.1229

Abstract

In this paper a Nth order nanoplate model is developed for the bending and buckling analysis of a graphene nanoplate based on a modified couple stress theory. The strain energy, external work and buckling equations are solved. Also using Hamilton’ principle, main and auxiliary equations of nano plate are obtained. The bending rates and dimensionless bending values under uniform surface traction and sinusoidal load, the dimensionless critical force under a uniaxial surface force in x direction are all obtained for various plate's dimensional ratios and material length scale to thickness ratios. The governing equations are numerically solved. The effect of material length scale, length, width and thickness of the nanoplate on the bending and buckling ratios are investigated and the results are presented and discussed in details.

Keywords


  • Daghigh, H., Daghigh, V., Milani, A., Tannant, D., Lacy, T. E., and Reddy, J. N, Nonlocal Bending and Buckling of Agglomerated CNT-Reinforced Composite Nanoplates, Composites Part B: Engineering, 183, 2020, 107716. DOI:10.1016/j.compositesb.2019.107716.
  • Daikh, A. A., Houari, M. S. A., and Eltaher, M. A, A Novel Nonlocal Strain Gradient Quasi-3D Bending Analysis of Sigmoid Functionally Graded Sandwich Nanoplates. Composite Structures, In Press, 2020, 113347. DOI:10.1016/j.compstruct.2020.113347.
  • Ruocco, E., & Reddy, J. N, Buckling analysis of elastic–plastic nanoplates resting on a Winkler–Pasternak foundation based on nonlocal third-order plate theory. International Journal of Non-Linear Mechanics, 121, 2020, 103453, DOI:10.1016/j.ijnonlinmec.2020.103453.
  • Banh Thien, T., Dang Trung, H., Le Anh, L., Ho Huu, V., and Nguyen Thoi, T., Buckling Analysis of Non-Uniform Thickness Nanoplates in an Elastic Medium Using the Isogeometric Analysis. Composite Structures, 162, 2017, pp. 182-193. DOI:10.1016/j.compstruct.2016.11.092.
  • Yang, F., Chong, A. C. M., Lam, D. C. C., and Tong, P., Couple Stress Based Strain Gradient Theory for Elasticity, Int. J. Solids Struct, 39, 2002, pp. 2731–2743, DOI: 10.1016/S0020-7683(02)00152-X.
  • Toupin, R. A., Elastic Materials with Couple Stresses, Arch. Rational Mech. Anal, 11, 1962, pp. 385–414.
  • Mindlin, R. D., Tiersten, H. F., Effects of Couple-Stresses in Linear Elasticity, Arch. Rational Mech. Anal, 11,1962, pp. 415–448.
  • Koiter, W. T., Couple Stresses in The Theory of Elasticity, I and II. Proc. K. Ned. Akad. Wet. (B), 67, 1964, pp. 17–44.
  • Mindlin, R. D., Micro-Structure in Linear Elasticity, Arch. Rational Mech. Anal, 16, 1964, pp. 51–78.
  • Tsiatas, G. C., A New Kirchhoff Model Based On a Modified Couple Stress Theory, International Journal of solids and structures, No. 46, 2009, pp. 2757-2764. DOI: 10.1016/j.ijsolstr.2009.03.004.
  • Wang, B., Zhou, S., Zhao, J., and Chen, X., Asize-Dependent Kirchhoff Micro-Plate Model Based On Strain Gradient Elasticity Theory, European Journal of Mechanics A/Solids, No. 30, 2011, pp. 517-524. DOI: 10.1016/j.euromechsol.2011.04.001.
  • Farajpour, A., Shahidi, A. R., Mohammadi, M., and Mahzoon, M., Buckling of Orthotropic Micro/Nanoscale Plates Under Linearly Varying In-Plane Load Via Nonlocal Continuum Mechanics, Composite Structures, No. 94, 2012, pp. 1605-1615. DOI:10.1016/j.compstruct.2011.12.032.
  • Tai, T., Ho Choi, D., Size-Dependent Functionally Graded Kirchhoff and Mindlin Plate Theory Based On a Modified Couple Stress Theory, Composite Structures, No. 95, 2013, pp.142-153. DOI: 10.1016/j.compstruct.2012.08.023.
  • Akgoz, B., Civalek, O., Free Vibration Analysis for Single –Layered Graphene Sheets in an Elastic Matrix Via Modified Couple Stress Theory, Materials and Design, No. 42, 2012, pp. 164-171. DOI: 10.1016/j.matdes.2012.06.002
  • Roque, C. M. C., Ferreira, A. J. M., and Reddy, J. N., Analysis of Mindlin Micro Plates with A Modified Couple Stress Theory and Meshlessmethod, Applied Mathematical Modeling, No. 37, 2013, pp. 4626-4633. DOI: 1016/j.apm.2012.09.063.
  • Thai, H. T., Kim, S. E., A Size-Dependent Functionally Graded Reddy Plate Model Based On a Modified Couple Stress Theory, Composites Part B: Engineering, Vol. 45, 2013, pp. 1636-1645. DOI:10.1016/j.compositesb.2012.09.065.
  • Lou, J., He, L., and Du, J., A Unified Higher Order Plate Theory for Functionally Graded Microplates Based On the Modified Couple Stress Theory, Composite Structures, No. 133, 2015, 1036-1047. DOI: 10.1016/j.compstruct.2015.08.009.
  • Xiang, S., Kang, G. w., A Nth-Order Shear Deformation Theory for The Bending Analysis On the Functionally Graded Plates, European Journal of Mechanics - A/Solids, No. 37, 2013, 336-343. DOI: 10.1016/j.euromechsol.2012.08.005.