Journal of Computational Applied Mechanics

Buckling analysis of three-dimensional functionally graded Euler-Bernoulli nanobeams based on the nonlocal strain gradient theory

Document Type : Research Paper

Authors

1 Department of Mechanical Engineering, University of Jiroft, Jiroft, Iran

2 Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran

3 Engineering Graphics Center, Sharif University of Technology, Tehran, Iran

Abstract

This paper presents a nonlocal strain gradient theory for capturing size effects in buckling analysis of Euler-Bernoulli nanobeams made of three-dimensional functionally graded materials. The material properties vary according to any function. These models can degenerate to the classical models if the material length-scale parameters is assumed to be zero. The Hamilton's principle applied to drive the governing equation and boundary conditions. Generalized differential quadrature method used to solve the governing equation. The effects of some parameters, such as small-scale parameters and constant material parameters are studied.

Keywords

[1]           A. Hadi, A. Rastgoo, N. Haghighipour, A. Bolhassani, Numerical modelling of a spheroid living cell membrane under hydrostatic pressure, Journal of Statistical Mechanics: Theory and Experiment, Vol. 2018, No. 8, pp. 083501, 2018.
[2]           M. Najafzadeh, M. M. Adeli, E. Zarezadeh, A. Hadi, Torsional vibration of the porous nanotube with an arbitrary cross-section based on couple stress theory under magnetic field, Mechanics Based Design of Structures and Machines, pp. 1-15, 2020.
[3]           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, pp. 1-18, 2020.
[4]           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.
[5]           C. Kafadar, A. C. Eringen, Micropolar media—I the classical theory, International Journal of Engineering Science, Vol. 9, No. 3, pp. 271-305, 1971.
[6]           A. C. Eringen, Nonlocal polar elastic continua, International journal of engineering science, Vol. 10, No. 1, pp. 1-16, 1972.
[7]           A. C. Eringen, 2002, Nonlocal continuum field theories, Springer Science & Business Media,
[8]           A. C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of applied physics, Vol. 54, No. 9, pp. 4703-4710, 1983.
[9]           R. Mindlin, N. Eshel, On first strain-gradient theories in linear elasticity, International Journal of Solids and Structures, Vol. 4, No. 1, pp. 109-124, 1968.
[10]         R. A. Toupin, Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis, Vol. 11, No. 1, pp. 385-414, 1962.
[11]         C. Lim, G. Zhang, J. Reddy, A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation, Journal of the Mechanics and Physics of Solids, Vol. 78, pp. 298-313, 2015.
[12]         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.
[13]         S. Pradhan, J. Phadikar, Bending, buckling and vibration analyses of nonhomogeneous nanotubes using GDQ and nonlocal elasticity theory, Structural Engineering and Mechanics, Vol. 33, No. 2, pp. 193-213, 2009.
[14]         J. Phadikar, S. Pradhan, Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates, Computational materials science, Vol. 49, No. 3, pp. 492-499, 2010.
[15]         J. K. Phadikar, S. C. Pradhan, Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates, Computational Materials Science, Vol. 49, No. 3, pp. 492-499, 9//, 2010.
[16]         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, 2016/08/01/, 2016.
[17]         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, 2016/09/01/, 2016.
[18]         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, 6//, 2016.
[19]         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.
[20]         M. M. Adeli, A. Hadi, M. Hosseini, H. H. Gorgani, Torsional vibration of nano-cone based on nonlocal strain gradient elasticity theory, The European Physical Journal Plus, Vol. 132, No. 9, pp. 393, 2017.
[21]         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.
[22]         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, pp. 1-28, 2017.
[23]         F. Ebrahimi, M. R. Barati, A nonlocal higher-order refined magneto-electro-viscoelastic beam model for dynamic analysis of smart nanostructures, International Journal of Engineering Science, Vol. 107, pp. 183-196, 10//, 2016.
[24]         F. Ebrahimi, M. R. Barati, A. Dabbagh, A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates, International Journal of Engineering Science, Vol. 107, pp. 169-182, 10//, 2016.
[25]         F. Ebrahimi, M. R. Barati, A nonlocal strain gradient refined beam model for buckling analysis of size-dependent shear-deformable curved FG nanobeams, Composite Structures, Vol. 159, pp. 174-182, 1/1/, 2017.
[26]         F. Ebrahimi, M. R. Barati, A. M. Zenkour, A new nonlocal elasticity theory with graded nonlocality for thermo-mechanical vibration of FG nanobeams via a nonlocal third-order shear deformation theory, Mechanics of Advanced Materials and Structures, No. just-accepted, 2017.
[27]         F. Ebrahimi, M. R. Barati, Vibration analysis of nonlocal beams made of functionally graded material in thermal environment, The European Physical Journal Plus, Vol. 131, No. 8, pp. 279, 2016.
[28]         F. Ebrahimi, M. R. Barati, Buckling analysis of nonlocal third-order shear deformable functionally graded piezoelectric nanobeams embedded in elastic medium, Journal of the Brazilian Society of Mechanical Sciences and Engineering, pp. 1-16, 2016.
[29]         F. Ebrahimi, M. R. Barati, Magnetic field effects on buckling behavior of smart size-dependent graded nanoscale beams, The European Physical Journal Plus, Vol. 131, No. 7, pp. 1-14, 2016.
[30]         F. Ebrahimi, M. R. Barati, Magneto-electro-elastic buckling analysis of nonlocal curved nanobeams, The European Physical Journal Plus, Vol. 131, No. 9, pp. 346, 2016.
[31]         F. Ebrahimi, M. R. Barati, P. Haghi, Nonlocal thermo-elastic wave propagation in temperature-dependent embedded small-scaled nonhomogeneous beams, The European Physical Journal Plus, Vol. 131, No. 11, pp. 383, 2016.
[32]         F. Ebrahimi, M. Daman, Analytical investigation of the surface effects on nonlocal vibration behavior of nanosize curved beams, ADVANCES IN NANO RESEARCH, Vol. 5, No. 1, pp. 35-47, 2017.
[33]         F. Ebrahimi, A. Dabbagh, Nonlocal strain gradient based wave dispersion behavior of smart rotating magneto-electro-elastic nanoplates, Materials Research Express, Vol. 4, No. 2, pp. 025003, 2017.
[34]         L. Li, Y. Hu, Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory, International Journal of Engineering Science, Vol. 97, pp. 84-94, 12//, 2015.
[35]         A. Farajpour, M. Danesh, M. Mohammadi, Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics, Physica E: Low-dimensional Systems and Nanostructures, Vol. 44, No. 3, pp. 719-727, 2011.
[36]         L. Li, X. Li, Y. Hu, Free vibration analysis of nonlocal strain gradient beams made of functionally graded material, International Journal of Engineering Science, Vol. 102, pp. 77-92, 2016.
[37]         L. Li, Y. Hu, Wave propagation in fluid-conveying viscoelastic carbon nanotubes based on nonlocal strain gradient theory, Computational Materials Science, Vol. 112, Part A, pp. 282-288, 2/1/, 2016.
[38]         L. Li, Y. Hu, X. Li, Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory, International Journal of Mechanical Sciences, Vol. 115–116, pp. 135-144, 9//, 2016.
[39]         L. Li, Y. Hu, Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material, International Journal of Engineering Science, Vol. 107, pp. 77-97, 10//, 2016.
[40]         L. Li, Y. Hu, Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects, International Journal of Mechanical Sciences, Vol. 120, pp. 159-170, 1//, 2017.
[41]         X. Li, L. Li, Y. Hu, Z. Ding, W. Deng, Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory, Composite Structures, Vol. 165, pp. 250-265, 4/1/, 2017.
[42]         G. Romano, R. Barretta, M. Diaco, F. M. de Sciarra, Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams, International Journal of Mechanical Sciences, Vol. 121, pp. 151-156, 2017.
[43]         G. Romano, R. Barretta, Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams, Composites Part B: Engineering, Vol. 114, No. Supplement C, pp. 184-188, 2017/04/01/, 2017.
[44]         G. Romano, R. Barretta, Comment on the paper “Exact solution of Eringen’s nonlocal integral model for bending of Euler–Bernoulli and Timoshenko beams” by Meral Tuna & Mesut Kirca, International Journal of Engineering Science, Vol. 109, No. Supplement C, pp. 240-242, 2016/12/01/, 2016.
[45]         J. Fernández-Sáez, R. Zaera, J. A. Loya, J. N. Reddy, Bending of Euler–Bernoulli beams using Eringen’s integral formulation: A paradox resolved, International Journal of Engineering Science, Vol. 99, No. Supplement C, pp. 107-116, 2016/02/01/, 2016.
[46]         G. Romano, R. Barretta, Nonlocal elasticity in nanobeams: the stress-driven integral model, International Journal of Engineering Science, Vol. 115, No. Supplement C, pp. 14-27, 2017/06/01/, 2017.
[47]         A. Apuzzo, R. Barretta, R. Luciano, F. Marotti de Sciarra, R. Penna, Free vibrations of Bernoulli-Euler nano-beams by the stress-driven nonlocal integral model, Composites Part B: Engineering, Vol. 123, No. Supplement C, pp. 105-111, 2017/08/15/, 2017.
[48]         C. W. Lim, G. Zhang, J. N. Reddy, A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation, Journal of the Mechanics and Physics of Solids, Vol. 78, No. Supplement C, pp. 298-313, 2015/05/01/, 2015.
[49]         L. Li, Y. Hu, Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory, International Journal of Engineering Science, Vol. 97, No. Supplement C, pp. 84-94, 2015/12/01/, 2015.
[50]         L. Li, Y. Hu, L. Ling, Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory, Composite Structures, Vol. 133, No. Supplement C, pp. 1079-1092, 2015/12/01/, 2015.
[51]         A. Farajpour, M. H. Yazdi, A. Rastgoo, M. Mohammadi, A higher-order nonlocal strain gradient plate model for buckling of orthotropic nanoplates in thermal environment, Acta Mechanica, Vol. 227, No. 7, pp. 1849-1867, 2016.
[52]         F. Ebrahimi, M. R. Barati, A. Dabbagh, A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates, International Journal of Engineering Science, Vol. 107, No. Supplement C, pp. 169-182, 2016/10/01/, 2016.
[53]         M. Tuna, M. Kirca, Exact solution of Eringen's nonlocal integral model for vibration and buckling of Euler–Bernoulli beam, International Journal of Engineering Science, Vol. 107, No. Supplement C, pp. 54-67, 2016/10/01/, 2016.
[54]         M. Tuna, M. Kirca, Exact solution of Eringen's nonlocal integral model for bending of Euler–Bernoulli and Timoshenko beams, International Journal of Engineering Science, Vol. 105, No. Supplement C, pp. 80-92, 2016/08/01/, 2016.
[55]         F. Ebrahimi, M. R. Barati, Dynamic Modeling of Magneto-electrically Actuated Compositionally Graded Nanosize Plates Lying on Elastic Foundation, Arabian Journal for Science and Engineering, Vol. 42, No. 5, pp. 1977-1997, May 01, 2017.
[56]         L. Li, X. Li, Y. Hu, Free vibration analysis of nonlocal strain gradient beams made of functionally graded material, International Journal of Engineering Science, Vol. 102, No. Supplement C, pp. 77-92, 2016/05/01/, 2016.
[57]         L. Li, Y. Hu, Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material, International Journal of Engineering Science, Vol. 107, No. Supplement C, pp. 77-97, 2016/10/01/, 2016.
[58]         J. Romanoff, J. N. Reddy, J. Jelovica, Using non-local Timoshenko beam theories for prediction of micro- and macro-structural responses, Composite Structures, Vol. 156, No. Supplement C, pp. 410-420, 2016/11/15/, 2016.
[59]         F. Ebrahimi, M. R. Barati, Vibration analysis of viscoelastic inhomogeneous nanobeams incorporating surface and thermal effects, Applied Physics A, Vol. 123, No. 1, pp. 5, December 10, 2016.
[60]         F. Ebrahimi, M. R. Barati, A nonlocal strain gradient refined beam model for buckling analysis of size-dependent shear-deformable curved FG nanobeams, Composite Structures, Vol. 159, No. Supplement C, pp. 174-182, 2017/01/01/, 2017.
[61]         X.-J. Xu, X.-C. Wang, M.-L. Zheng, Z. Ma, Bending and buckling of nonlocal strain gradient elastic beams, Composite Structures, Vol. 160, No. Supplement C, pp. 366-377, 2017/01/15/, 2017.
[62]         X. Li, L. Li, Y. Hu, Z. Ding, W. Deng, Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory, Composite Structures, Vol. 165, No. Supplement C, pp. 250-265, 2017/04/01/, 2017.
[63]         L. Li, Y. Hu, Torsional vibration of bi-directional functionally graded nanotubes based on nonlocal elasticity theory, Composite Structures, Vol. 172, No. Supplement C, pp. 242-250, 2017/07/15/, 2017.
[64]         M. Z. Nejad, P. Fatehi, Exact elasto-plastic analysis of rotating thick-walled cylindrical pressure vessels made of functionally graded materials, International Journal of Engineering Science, Vol. 86, pp. 26-43, 2015.
[65]         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.
[66]         Z. Mazarei, M. Nejad, A. Hadi, Thermo-elasto-plastic analysis of thick-walled spherical pressure vessels made of functionally graded materials, International Journal of Applied Mechanics, 2016.
[67]         M. 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, Vol. 6, No. 4, pp. 366-377, 2014.
[68]         M. Ghannad, G. H. Rahimi, M. Z. Nejad, Elastic analysis of pressurized thick cylindrical shells with variable thickness made of functionally graded materials, Composites Part B: Engineering, Vol. 45, No. 1, pp. 388-396, 2013.
[69]         M. Z. Nejad, M. Jabbari, M. Ghannad, Elastic analysis of FGM rotating thick truncated conical shells with axially-varying properties under non-uniform pressure loading, Composite Structures, Vol. 122, pp. 561-569, 2015.
[70]         M. Jabbari, M. Z. Nejad, M. Ghannad, Thermo-elastic analysis of axially functionally graded rotating thick cylindrical pressure vessels with variable thickness under mechanical loading, International journal of engineering science, Vol. 96, pp. 1-18, 2015.
[71]         M. Jabbari, M. Z. Nejad, M. Ghannad, Thermo-elastic analysis of axially functionally graded rotating thick truncated conical shells with varying thickness, Composites Part B: Engineering, Vol. 96, pp. 20-34, 2016.
[72]         M. Z. Nejad, G. Rahimi, M. Ghannad, Set of field equations for thick shell of revolution made of functionally graded materials in curvilinear coordinate system, Mechanics, Vol. 77, No. 3, pp. 18-26, 2016.
[73]         M. Ghannad, M. Z. Nejad, Elastic analysis of pressurized thick hollow cylindrical shells with clamped-clamped ends, Mechanics, Vol. 85, No. 5, pp. 11-18, 2016.
[74]         M. Z. Nejad, G. Rahimi, Deformations and stresses in rotating FGM pressurized thick hollow cylinder under thermal load, Scientific Research and Essays, Vol. 4, No. 3, pp. 131-140, 2009.
[75]         M. Z. Nejad, G. H. Rahimi, Elastic analysis of FGM rotating cylindrical pressure vessels, Journal of the Chinese institute of engineers, Vol. 33, No. 4, pp. 525-530, 2010.
[76]         P. Fatehi, M. Z. Nejad, Effects of material gradients on onset of yield in FGM rotating thick cylindrical shells, International Journal of Applied Mechanics, Vol. 6, No. 04, pp. 1450038, 2014.
[77]         C. Lü, C. W. Lim, W. Chen, Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory, International Journal of Solids and Structures, Vol. 46, No. 5, pp. 1176-1185, 2009.
[78]         M. Lezgy-Nazargah, Fully coupled thermo-mechanical analysis of bi-directional FGM beams using NURBS isogeometric finite element approach, Aerospace Science and Technology, 2015.
[79]         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.
[80]         S. Sahmani, M. M. Aghdam, Size dependency in axial postbuckling behavior of hybrid FGM exponential shear deformable nanoshells based on the nonlocal elasticity theory, Composite Structures, Vol. 166, pp. 104-113, 4/15/, 2017.
[81]         N. Tutuncu, Stresses in thick-walled FGM cylinders with exponentially-varying properties, Engineering Structures, Vol. 29, No. 9, pp. 2032-2035, 2007.
[82]         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.
[83]         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.
[84]         M. A. Steinberg, Materials for aerospace, Sci. Am.;(United States), Vol. 255, No. 4, 1986.
[85]         A. Hadi, M. Z. Nejad, M. Hosseini, Vibrations of three-dimensionally graded nanobeams, International Journal of Engineering Science, Vol. 128, pp. 12-23, 2018/07/01/, 2018.
[86]         M. R. Farajpour, A. Rastgoo, A. Farajpour, M. Mohammadi, Vibration of piezoelectric nanofilm‐based electromechanical sensors via higher‐order non‐local strain gradient theory, Micro & Nano Letters, Vol. 11, No. 6, pp. 302-307, 2016.
[87]         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, An International Journal, Vol. 26, No. 6, pp. 663-672, 2018.
Journal of Computational Applied Mechanics
Volume 53, Issue 1
March 2022
Pages 24-40
  • Receive Date: 03 November 2021
  • Revise Date: 31 January 2022
  • Accept Date: 02 February 2022