Buckling analysis of 2D functionally graded porous beams using novel higher order theory

Document Type : Research Paper

Authors

Mechanical Engineering, School of Technology, GITAM, Hyderabad, 502329, India

Abstract

Functionally graded material is an in-homogeneous composite, constructed from various phases of material elements, often ceramic and metal and is employed in high-temperature applications. Aim of this work is to examine the behaviour of buckling in porous Functionally Graded Material Beams (FGBs) in 2 directions (2D) with help of fifth order shear deformation theory. With help of potential energy principle and Reddy’s beam theory, equilibrium equations for linear buckling were derived. Boundary conditions such as simply supported – Simply supported (SS), Clamped – clamped (CC) and Clamped-Free (CF) were employed. An unique shear shape function was derived and 5th order theory was adapted to take into account the effect of transverse shear deformation to get the zero shear stress conditions at top and bottom surfaces of the beam. Based on power law, FGB material properties were changed in length and thickness directions. The displacement functions in axial directions were articulated in algebraic polynomials, including admissible functions which were used to fulfil different boundary conditions. Convergence and verification were performed on computed results with results of previous studies. It was found that the results obtained using 5th order theory were in agreement and allows for better buckling analysis for porous material.

Keywords

[1]          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.
[2]          M. Mohammadi, M. Safarabadi, A. Rastgoo, A. Farajpour, Hygro-mechanical vibration analysis of a rotating viscoelastic nanobeam embedded in a visco-Pasternak elastic medium and in a nonlinear thermal environment, Acta Mechanica, Vol. 227, No. 8, pp. 2207-2232, 2016.
[3]          M. Mohammadi, A. Farajpour, M. Goodarzi, R. Heydarshenas, Levy Type Solution for Nonlocal Thermo-Mechanical Vibration of Orthotropic Mono-Layer Graphene Sheet Embedded in an Elastic Medium, Journal of Solid Mechanics, Vol. 5, No. 2, pp. 116-132, 2013.
[4]          A. Farajpour, A. Rastgoo, M. Mohammadi, Surface effects on the mechanical characteristics of microtubule networks in living cells, Mechanics Research Communications, Vol. 57, pp. 18-26, 2014/04/01/, 2014.
[5]          M. Mohammadi, A. Farajpour, A. Moradi, M. Hosseini, Vibration analysis of the rotating multilayer piezoelectric Timoshenko nanobeam, Engineering Analysis with Boundary Elements, Vol. 145, pp. 117-131, 2022/12/01/, 2022.
[6]          M. Mohammadi, M. Ghayour, A. Farajpour, Free transverse vibration analysis of circular and annular graphene sheets with various boundary conditions using the nonlocal continuum plate model, Composites Part B: Engineering, Vol. 45, No. 1, pp. 32-42, 2013.
[7]          M. Mohammadi, M. Hosseini, M. Shishesaz, A. Hadi, A. Rastgoo, Primary and secondary resonance analysis of porous functionally graded nanobeam resting on a nonlinear foundation subjected to mechanical and electrical loads, European Journal of Mechanics - A/Solids, Vol. 77, pp. 103793, 2019/09/01/, 2019.
[8]          H. Asemi, S. Asemi, A. Farajpour, M. Mohammadi, Nanoscale mass detection based on vibrating piezoelectric ultrathin films under thermo-electro-mechanical loads, Physica E: Low-dimensional Systems and Nanostructures, Vol. 68, pp. 112-122, 2015.
[9]          S. R. Asemi, A. Farajpour, H. R. Asemi, M. Mohammadi, Influence of initial stress on the vibration of double-piezoelectric-nanoplate systems with various boundary conditions using DQM, Physica E: Low-dimensional Systems and Nanostructures, Vol. 63, pp. 169-179, 2014.
[10]        S. R. Asemi, M. Mohammadi, A. Farajpour, A study on the nonlinear stability of orthotropic single-layered graphene sheet based on nonlocal elasticity theory, Latin American Journal of Solids and Structures, Vol. 11, No. 9, pp. 1515-1540, 2014.
[11]        M. Aydogdu, V. Taskin, Free vibration analysis of functionally graded beams with simply supported edges, Materials & design, Vol. 28, No. 5, pp. 1651-1656, 2007.
[12]        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. 08, No. 04, pp. 1650048, 2016.
[13]        A. Barati, M. M. Adeli, A. Hadi, Static torsion of bi-directional functionally graded microtube based on the couple stress theory under magnetic field, International Journal of Applied Mechanics, Vol. 12, No. 02, pp. 2050021, 2020.
[14]        A. Barati, A. Hadi, M. Z. Nejad, R. Noroozi, On vibration of bi-directional functionally graded nanobeams under magnetic field, Mechanics Based Design of Structures and Machines, Vol. 50, No. 2, pp. 468-485, 2022.
[15]        S. R. Bathini, Free vibration behaviour of bi-directional functionally graded plates with porosities using a refined first order shear deformation theory, Journal of Computational Applied Mechanics, Vol. 51, No. 2, pp. 374-388, 2020.
[16]        H. Bellifa, K. H. Benrahou, L. Hadji, M. S. A. Houari, A. Tounsi, Bending and free vibration analysis of functionally graded plates using a simple shear deformation theory and the concept the neutral surface position, Journal of the Brazilian Society of Mechanical Sciences and Engineering, Vol. 38, No. 1, pp. 265-275, 2016.
[17]        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.
[18]        F. Ebrahimi, M. R. Barati, A nonlocal higher-order shear deformation beam theory for vibration analysis of size-dependent functionally graded nanobeams, Arabian Journal for Science and Engineering, Vol. 41, No. 5, pp. 1679-1690, 2016.
[19]        L. Hadji, F. Bernard, Bending and free vibration analysis of functionally graded beams on elastic foundations with analytical validation, Advances in materials Research, Vol. 9, No. 1, pp. 63-98, 2020.
[20]        H. Haghshenas Gorgani, M. Mahdavi Adeli, M. Hosseini, Pull-in behaviour of functionally graded micro/nano-beams for MEMS and NEMS switches, Microsystem Technologies, Vol. 25, No. 8, pp. 3165-3173, 2019.
[21]        N. Hebbar, I. Hebbar, D. Ouinas, M. Bourada, Numerical modeling of bending, buckling, and vibration of functionally graded beams by using a higher-order shear deformation theory, Frattura ed Integrità Strutturale, Vol. 14, No. 52, pp. 230-246, 2020.
[22]        M. Hosseini, M. Shishesaz, A. Hadi, Thermoelastic analysis of rotating functionally graded micro/nanodisks of variable thickness, Thin-Walled Structures, Vol. 134, pp. 508-523, 2019.
[23]        Y. Huang, L.-E. Yang, Q.-Z. Luo, Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section, Composites Part B: Engineering, Vol. 45, No. 1, pp. 1493-1498, 2013.
[24]        D. Jha, T. Kant, R. Singh, Higher order shear and normal deformation theory for natural frequency of functionally graded rectangular plates, Nuclear Engineering and Design, Vol. 250, pp. 8-13, 2012.
[25]        A. Karamanlı, Analytical solutions for buckling behaviour of two directional functionally graded beams using a third order shear deformable beam theory, Academic Platform-Journal of Engineering and Science, Vol. 6, No. 2, pp. 164-178, 2018.
[26]        M. J. Ketabdari, A. Allahverdi, S. Boreyri, M. F. Ardestani, Free vibration analysis of homogeneous and FGM skew plates resting on variable Winkler-Pasternak elastic foundation, Mechanics & Industry, Vol. 17, No. 1, pp. 107, 2016.
[27]        M. M. Khoram, M. Hosseini, A. Hadi, M. Shishehsaz, Bending analysis of bidirectional FGM Timoshenko nanobeam subjected to mechanical and magnetic forces and resting on Winkler–Pasternak foundation, International Journal of Applied Mechanics, Vol. 12, No. 08, pp. 2050093, 2020.
[28]        J. Kim, K. K. Żur, J. Reddy, Bending, free vibration, and buckling of modified couples stress-based functionally graded porous micro-plates, Composite Structures, Vol. 209, pp. 879-888, 2019.
[29]        L. O. Larbi, A. Kaci, M. S. A. Houari, A. Tounsi, An efficient shear deformation beam theory based on neutral surface position for bending and free vibration of functionally graded beams#, Mechanics Based Design of Structures and Machines, Vol. 41, No. 4, pp. 421-433, 2013.
[30]        M. Mohammadi, A. Farajpour, M. Goodarzi, H. Shehni nezhad pour, 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, Vol. 82, pp. 510-520, 2014/02/01/, 2014.
[31]        M. Mohammadi, A. Rastgoo, Nonlinear vibration analysis of the viscoelastic composite nanoplate with three directionally imperfect porous FG core, Structural Engineering and Mechanics, An Int'l Journal, Vol. 69, No. 2, pp. 131-143, 2019.
[32]        M. Mohammadi, A. Rastgoo, Primary and secondary resonance analysis of FG/lipid nanoplate with considering porosity distribution based on a nonlinear elastic medium, Mechanics of Advanced Materials and Structures, Vol. 27, No. 20, pp. 1709-1730, 2020/10/15, 2020.
[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]        T.-K. Nguyen, T. P. Vo, H.-T. Thai, Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory, Composites Part B: Engineering, Vol. 55, pp. 147-157, 2013.
[35]        O. Rahmani, O. Pedram, Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory, International Journal of Engineering Science, Vol. 77, pp. 55-70, 2014.
[36]        M. Safarabadi, M. Mohammadi, A. Farajpour, M. Goodarzi, Effect of surface energy on the vibration analysis of rotating nanobeam, 2015.
[37]        M. Şimşek, Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories, Nuclear Engineering and Design, Vol. 240, No. 4, pp. 697-705, 2010.
[38]        M. Şimşek, Bi-directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions, Composite Structures, Vol. 133, pp. 968-978, 2015.
[39]        M. Slimane, B. Samir, B. Hakima, H. M. Adda, Free vibration analysis of functionally graded plates FG with porosities, International Journal of Engineering Research & Technology (IJERT), Vol. 8, No. 03, 2019.
[40]        M. Talha, B. Singh, Static response and free vibration analysis of FGM plates using higher order shear deformation theory, Applied Mathematical Modelling, Vol. 34, No. 12, pp. 3991-4011, 2010.
[41]        H.-T. Thai, T. P. Vo, Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories, International journal of mechanical sciences, Vol. 62, No. 1, pp. 57-66, 2012.
[42]        T. P. Vo, H.-T. Thai, T.-K. Nguyen, F. Inam, Static and vibration analysis of functionally graded beams using refined shear deformation theory, Meccanica, Vol. 49, No. 1, pp. 155-168, 2014.
Volume 53, Issue 3
September 2022
Pages 393-413
  • Receive Date: 04 July 2022
  • Revise Date: 05 August 2022
  • Accept Date: 05 August 2022