Free Vibration Analysis of Nanoplates Made of Functionally Graded Materials Based On Nonlocal Elasticity Theory Using Finite Element Method

Document Type: Research Paper

Authors

School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran

Abstract

In this paper, an analysis of free vibration in functionally graded nanoplate is presented. Third-order shear deformation plate theory is used to reach more accuracy in results. Small-scale effects are investigated using Eringen`s nonlocal theory. The governing equations of motion are obtained by Hamilton`s principle. It is assumed that the properties of nanoplates vary through their thicknesses according to a volume fraction power law distribution. The finite element method (FEM) is presented to model the functionally graded nanoplate and solve mathematical equations accurately. The finite element formulation for HSDT nanoplate is also presented. Natural frequencies of FG nanoplate with various boundary conditions are compared with available results in the literature. At the end some numerical results are presented to evaluate the influence of different parameters, such as power law index, nonlocal parameter, aspect ratio and aspect of length to thickness of nanoplate. In addition, all combinations of simply supported and clamped boundary conditions are considered.

Keywords

Main Subjects


[1]    T. Aksencer, M. Aydogdu, Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory, Physica E: Low-dimensional Systems and Nanostructures, Vol. 43, No. 4, pp. 954-959, 2011.

[2]    R. Ansari, R. Rajabiehfard, B. Arash, Nonlocal finite element model for vibrations of embedded multi-layered graphene sheets, Computational Materials Science, Vol. 49, No. 4, pp. 831-838, 2010.

[3]    S. H. Hashemi, A. T. Samaei, Buckling analysis of micro/nanoscale plates via nonlocal elasticity theory, Physica E: Low-dimensional Systems and Nanostructures, Vol. 43, No. 7, pp. 1400-1404, 2011.

[4]    S. Narendar, Buckling analysis of micro-/nano-scale plates based on two-variable refined plate theory incorporating nonlocal scale effects, Composite Structures, Vol. 93, No. 12, pp. 3093-3103, 2011.

[5]    A. Daneshmehr, A. Rajabpoor, A. Hadi, Size dependent free vibration analysis of nanoplates made of functionally graded materials based on nonlocal elasticity theory with high order theories, International Journal of Engineering Science, Vol. 95, pp. 23-35, 2015.

[6]    A. BAKHSHESHY, K. KHORSHIDI, Free Vibration of Functionally Graded Rectangular Nanoplates in Thermal Environment Based on the Modified Couple Stress Theory, 2015.

[7]    S. Hosseini-Hashemi, H. R. D. Taher, H. Akhavan, M. Omidi, Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory, Applied Mathematical Modelling, Vol. 34, No. 5, pp. 1276-1291, 2010.

[8]    M. Zare, R. Nazemnezhad, S. Hosseini-Hashemi, Natural frequency analysis of functionally graded rectangular nanoplates with different boundary conditions via an analytical method, Meccanica, Vol. 50, No. 9, pp. 2391-2408, 2015.

[9]    F. Bounouara, K. H. Benrahou, I. Belkorissat, A. Tounsi, A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundation, Steel and Composite Structures, Vol. 20, No. 2, pp. 227-249, 2016.

[10]  H. Salehipour, H. Nahvi, A. Shahidi, Exact closed-form free vibration analysis for functionally graded micro/nano plates based on modified couple stress and three-dimensional elasticity theories, Composite Structures, Vol. 124, pp. 283-291, 2015.

[11]  I. Belkorissat, M. S. A. Houari, A. Tounsi, E. Bedia, S. Mahmoud, On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable model, Steel and Composite Structures, Vol. 18, No. 4, pp. 1063-1081, 2015.

[12]  R. Ansari, M. Ashrafi, T. Pourashraf, S. Sahmani, Vibration and buckling characteristics of functionally graded nanoplates subjected to thermal loading based on surface elasticity theory, Acta Astronautica, Vol. 109, pp. 42-51, 2015.

[13]  M. Hosseini, A. Jamalpoor, Analytical solution for thermomechanical vibration of double-viscoelastic nanoplate-systems made of functionally graded materials, Journal of Thermal Stresses, Vol. 38, No. 12, pp. 1428-1456, 2015.

[14]  R. Ansari, M. F. Shojaei, A. Shahabodini, M. Bazdid-Vahdati, Three-dimensional bending and vibration analysis of functionally graded nanoplates by a novel differential quadrature-based approach, Composite Structures, Vol. 131, pp. 753-764, 2015.

[15]  R. Aghababaei, J. Reddy, Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates, Journal of Sound and Vibration, Vol. 326, No. 1, pp. 277-289, 2009.

[16]  R. B. Bouiadjra, E. Bedia, A. Tounsi, Nonlinear thermal buckling behavior of functionally graded plates using an efficient sinusoidal shear deformation theory, Structural Engineering and Mechanics, Vol. 48, No. 4, pp. 547-567, 2013.

[17]  N.-T. Nguyen, D. Hui, J. Lee, H. Nguyen-Xuan, An efficient computational approach for size-dependent analysis of functionally graded nanoplates, Computer Methods in Applied Mechanics and Engineering, Vol. 297, pp. 191-218, 2015.

[18]  A. Daneshmehr, A. Rajabpoor, Stability of size dependent functionally graded nanoplate based on nonlocal elasticity and higher order plate theories and different boundary conditions, International Journal of Engineering Science, Vol. 82, pp. 84-100, 2014.

[19]  A. Ghorbanpour Arani, H. Baba Akbar Zarei, E. Haghparast, Application of Halpin-Tsai Method in Modelling and Size-dependent Vibration Analysis of CNTs/fiber/polymer Composite Microplates, Journal of Computational Applied Mechanics, Vol. 47, No. 1, pp. 45-52, 2016.

[20]  M. Goodarzi, M. N. Bahrami, V. Tavaf, Refined plate theory for free vibration analysis of FG nanoplates using the nonlocal continuum plate model, Journal of Computational Applied Mechanics, Vol. 48, No. 1, pp. 123-136, 2017.

[21]  H. Raissi, M. Shishehsaz, S. Moradi, Applications of higher order shear deformation theories on stress distribution in a five layer sandwich plate.

[22]  M. H. Ghayesh, H. Farokhi, A. Gholipour, M. Tavallaeinejad, Nonlinear oscillations of functionally graded microplates, International Journal of Engineering Science, Vol. 122, pp. 56-72, 2018.

[23]  M. Baghani, M. Mohammadi, A. Farajpour, Dynamic and stability analysis of the rotating nanobeam in a nonuniform magnetic field considering the surface energy, International Journal of Applied Mechanics, Vol. 8, No. 04, pp. 1650048, 2016.

[24]  M. H. Ghayesh, H. Farokhi, A. Gholipour, Oscillations of functionally graded microbeams, International Journal of Engineering Science, Vol. 110, pp. 35-53, 2017.

[25]  N. Kordani, A. Fereidoon, M. Divsalar, A. Farajpour, Influence of surface piezoelectricity on the forced vibration of piezoelectric nanowires based on nonlocal elasticity theory, Journal of Computational Applied Mechanics, Vol. 47, No. 2, pp. 137-150, 2016.

[26]  A. Farajpour, A. Rastgoo, Influence of carbon nanotubes on the buckling of microtubule bundles in viscoelastic cytoplasm using nonlocal strain gradient theory, Results in physics, Vol. 7, pp. 1367-1375, 2017.

[27]  M. Hosseini, H. H. Gorgani, M. Shishesaz, A. Hadi, Size-Dependent Stress Analysis of Single-Wall Carbon Nanotube Based on Strain Gradient Theory, International Journal of Applied Mechanics, Vol. 9, No. 06, pp. 1750087, 2017.

[28]  M. Hosseini, M. Shishesaz, K. N. Tahan, A. Hadi, Stress analysis of rotating nano-disks of variable thickness made of functionally graded materials, International Journal of Engineering Science, Vol. 109, pp. 29-53, 2016.

[29]  M. Z. Nejad, A. Rastgoo, A. Hadi, Effect of exponentially-varying properties on displacements and stresses in pressurized functionally graded thick spherical shells with using iterative technique, Journal of Solid Mechanics, Vol. 6, No. 4, pp. 366-377, 2014.

[30]  M. Z. Nejad, A. Hadi, Eringen's non-local elasticity theory for bending analysis of bi-directional functionally graded Euler–Bernoulli nano-beams, International Journal of Engineering Science, Vol. 106, pp. 1-9, 2016.

[31]  M. Z. Nejad, A. Hadi, Non-local analysis of free vibration of bi-directional functionally graded Euler–Bernoulli nano-beams, International Journal of Engineering Science, Vol. 105, pp. 1-11, 2016.

[32]  M. Z. Nejad, A. Hadi, A. Farajpour, Consistent couple-stress theory for free vibration analysis of Euler-Bernoulli nano-beams made of arbitrary bi-directional functionally graded materials, Structural Engineering and Mechanics, Vol. 63, No. 2, pp. 161-169, 2017.

[33]  M. Z. Nejad, A. Hadi, A. Rastgoo, Buckling analysis of arbitrary two-directional functionally graded Euler–Bernoulli nano-beams based on nonlocal elasticity theory, International Journal of Engineering Science, Vol. 103, pp. 1-10, 2016.

[34]  M. Z. Nejad, A. Rastgoo, A. Hadi, Exact elasto-plastic analysis of rotating disks made of functionally graded materials, International Journal of Engineering Science, Vol. 85, pp. 47-57, 2014.

[35]  M. Shishesaz, M. Hosseini, K. N. Tahan, A. Hadi, Analysis of functionally graded nanodisks under thermoelastic loading based on the strain gradient theory, Acta Mechanica, Vol. 228, No. 12, pp. 4141-4168, 2017.

[36]  A. Hadi, M. Z. Nejad, A. Rastgoo, M. Hosseini, Buckling analysis of FGM Euler-Bernoulli nano-beams with 3D-varying properties based on consistent couple-stress theory, Steel and Composite Structures, Vol. 26, No. 6, pp. 663-672, 2018.

[37]  M. Zamani Nejad, M. Jabbari, A. Hadi, A review of functionally graded thick cylindrical and conical shells, Journal of Computational Applied Mechanics, Vol. 48, No. 2, pp. 357-370, 2017.

[38]  A. Hadi, M. Z. Nejad, M. Hosseini, Vibrations of three-dimensionally graded nanobeams, International Journal of Engineering Science, Vol. 128, pp. 12-23, 2018.

[39]  J. N. Reddy, A simple higher-order theory for laminated composite plates, Journal of applied mechanics, Vol. 51, No. 4, pp. 745-752, 1984.

[40]  H.-S. Shen, Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments, International Journal of Mechanical Sciences, Vol. 44, No. 3, pp. 561-584, 2002.

[41]  A. C. Eringen, 2002, Nonlocal continuum field theories, Springer Science & Business Media,

[42]  S. Tajalli, M. M. Zand, M. Ahmadian, Effect of geometric nonlinearity on dynamic pull-in behavior of coupled-domain microstructures based on classical and shear deformation plate theories, European Journal of Mechanics-A/Solids, Vol. 28, No. 5, pp. 916-925, 2009.

[43]  S. Natarajan, S. Chakraborty, M. Thangavel, S. Bordas, T. Rabczuk, Size-dependent free flexural vibration behavior of functionally graded nanoplates, Computational Materials Science, Vol. 65, pp. 74-80, 2012.