RITZ SOLUTION OF BUCKLING AND VIBRATION PROBLEM OF NANOPLATES EMBEDDED IN AN ELASTIC MEDIUM

In this paper, free vibration and buckling of single-layered isotropic rectangular nanoplate is investigated based on classic plate theory (CPT). Nonlocal elasticity theory accounts for the small-nonlocal effects. Both Winkler-type and Pasternak-type foundation models are employed to simulate the surrounding elastic matrix. Governing differential weak form equations of the plate based on nonlocal elasticity theory are derived. The Ritz method is used to solve the problem of buckling and free vibration nanoplate for various boundary conditions. In order to confirm the accuracy of the results, data are compared with the other results published in literature. The effects of different parameters on the plate behavior, such as nonlocal parameter, aspect ratio, boundary conditions, Winkler and shear modulus are investigated.

Keywords:

Buckling and vibration analysis, ritz method nonlocal effects, elastic medium.,

PDF

___

[1] Gurtin, M.E. and Murdoch, A.I. "Surface stress in solids", International Journal of Solids and Structures, 14, pp.431-440 (1978).
[2] Fleck, N. and Hutchinson, J. "Strain gradient plasticity" Advances in applied mechanics, 33, pp.295-361 (1997).
[3] Tsiatas, G.C. "A new Kirchhoff plate model based on a modified couple stress theory", International Journal of Solids and Structures, 46, pp.2757-2764 (2009).
[4] Eringen, A.C. Edelen, D. " On nonlocal elasticity", International Journal of Engineering Science, 10, pp.233-248 (1972).
[5] Pradhan, S. and Phadikar, J. "Nonlocal elasticity theory for vibration of nanoplates", Journal of Sound and Vibration, 325, pp.206-223 ( 2009).
[6] Ansari, R., Sahmani, S. and Arash, B. "Nonlocal plate model for free vibrations of single-layered graphene sheets", Physics Letters A, 375, pp.53-62 (2010).
[7] Murmu, T. and Adhikari, S. "Nonlocal vibration of bonded double-nanoplate-systems", Composites Part B: Engineering, 42, pp.1901-1911 (2011).
[8] Pradhan, S. and Phadikar, J." Small scale effect on vibration of embedded multilayered graphene sheets based on nonlocal continuum models", Physics Letters A, 373, pp.1062-1069 (2009).
[9] Malekzadeh, P., Setoodeh, A. and Beni, A.A. "Small scale effect on the free vibration of orthotropic arbitrary straight-sided quadrilateral nanoplates", Composite Structures, 93, pp.1631-1639 (2011).
[10] Pouresmaeeli, S., Fazelzadeh, S. and Ghavanloo, E. "Exact solution for nonlocal vibration of double-orthotropic nanoplates embedded in elastic medium", Composites Part B: Engineering, 43 pp.3384-3390 (2012).
[11] Chakraverty, S.and Behera, L. "Free vibration of rectangular nanoplates using Rayleigh–Ritz method", Physica E: Low-dimensional Systems and Nanostructures, 56, pp.357-363 (2014).
[12] Mohammadi, M., Ghayour, M., and Farajpour, A. "Free transverse vibration analysis of circular and annular graphene sheets with various boundary conditions using the nonlocal continuum plate model", Composites Part B: Engineering, 45, pp.32-42 (2013).
[13] Mohammadi, M. Farajpour, A. and Goodarzi, M. "Numerical study of the effect of shear in-plane load on the vibration analysis of graphene sheet embedded in an elastic medium". Computational Materials Science, 82, pp.510-520 (2014).
[14] Aksencer, T. and Aydogdu, M. "Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory", Physica E: Low-dimensional Systems and Nanostructures, 43, pp.954-959 (2011).
[15] Pradhan, S. and Murmu, T. "Small scale effect on the buckling of single-layered graphene sheets under biaxial compression via nonlocal continuum mechanics", Computational Materials Science, 47, pp.268-274 (2009).
[16] Hashemi, S.H. and Samaei, A.T. "Buckling analysis of micro/nanoscale plates via nonlocal elasticity theory", Physica E: Low-dimensional Systems and Nanostructures, 43, pp.1400-1404 (2011).
[17] Murmu, T. and Pradhan, S. "Buckling of biaxially compressed orthotropic plates at small scales", Mechanics Research Communications, 36, pp.933-938 (2009).
[18] 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, 94, pp.1605-1615 (2012).
[19] Babaei, H. and Shahidi, A. "Small-scale effects on the buckling of quadrilateral nanoplates based on nonlocal elasticity theory using the Galerkin method", Archive of Applied Mechanics, 81, pp.1051-1062 (2011).
[20] Farajpour, A., Mohammadi, M., Shahidi, A.R. and Mahzoon, M. "Axisymmetric buckling of the circular graphene sheets with the nonlocal continuum plate model", Physica E: Low-dimensional Systems and Nanostructures, 43, pp.1820-1825 (2011).
[21] Pradhan, S. and Murmu, T. "Small scale effect on the buckling analysis of single-layered graphene sheet embedded in an elastic medium based on nonlocal plate theory", Physica E: Low-dimensional Systems and Nanostructures, 42, pp.1293-1301 (2010).
[22] Murmu, T. and Pradhan, S. "Nonlocal buckling of double-nanoplate-systems under biaxial compression". Composites Part B: Engineering, 44, pp.84-94 (2013).
[23] Behfar, K. and Naghdabadi, R. "Nanoscale vibrational analysis of a multi-layered graphene sheet embedded in an elastic medium". Composites Science and Technology, 65, pp.1159-1164 (2005).
[24] Eringen, A.C. "Nonlocal polar elastic continua". International journal of engineering science, 10, pp.1-16 (1972).