Bending and Vibration Analysis of a Mindlin Rectangular Nanoplate using Modified Couple Stress Theory and Navier's Solution

محورهای موضوعی : micro and nano mechanics

Majid Eskandari Shahraki ¹ ( Department of Aerospace Engineering, Ferdowsi University of Mashhad, Mashhad, Iran )
Mahmoud Shariati ² ( Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran )
naser asiaban ³ ( Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran )

تاریخ دریافت : 1399/11/22 تاریخ پذیرش : 1400/04/19 تاریخ انتشار : 1400/12/10

کلید واژه: Modified couple stress theory, Bending, Mindlin nanoplate, Vibration, Navier's Solution,

چکیده مقاله :

In this paper, a Mindlin rectangular nanoplate model is developed for the bending and vibration analysis of a graphene nanoplate based on a modified couple stress theory. In order to consider the small scale effects, the modified couple stress theory, with one length scale parameter, is used. In modified couple stress theory, strain energy density is a function of strain tensor, curvature tensor, stress tensor and symmetric part of couple stress tensor. After obtaining the strain and kinetic energy, external work and substituting them in the Hamilton’s principle, the main and auxiliary equations of the nanoplate are obtained. Then, by manipulating the boundary conditions the governing equations are solved using Navier approach for bending and vibration of the nanoplate. The bending rates and dimensionless bending values under uniform surface traction and sinusoidal load and different mode frequencies are all obtained for various plate's dimensional ratios and material length scale to thickness ratios. The effect of material length scale, length, width and thickness of the nanoplate on the bending and vibration ratios are investigated and the results are presented and discussed in details.

منابع و مأخذ:

• 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, No. 11, 1962, pp. 415–448.


• Koiter, W. T., Couple Stresses in the Theory of Elasticity, I and II Proc K. Ned. Akad, Wet. (B), No. 67, 1964, pp. 17–44.


• Mindlin, R. D., Micro-Structure in Linear Elasticity, Arch. Rational Mech. Anal, 16, 1964, pp. 51–78.


• 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.


• T., HoChoi. 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.


• 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.


• 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: 10.1016/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: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.