Buckling analysis of FG micro-beam resting on elastic foundation via nonlocal strain gradient theory
Keywords:
Functionally graded materials, Buckling analysis, Nonlocal strain gradient, Micro-beamAbstract
In this work, the Euler-Bernoulli beam and the nonlocal strain gradient theories are developed to analyze the buckling behavior of a functionally graded microbeam resting on an elastic foundation. The governing equation of motion for the functionally graded micro-beam is derived by employing the Hamilton’s principle. The analytical expression of critical buckling force for the functionally graded micro-beam with simply supported boundary conditions is obtained in the closed form. The accuracy of the obtained result is verified by comparing the obtained solutions with the published ones. Effects of the power-law index, the material length scale parameter, the nonlocal parameter ,and the coefficients of the elastic foundation on the buckling response of the functionally graded micro-beam are investigated and discussed.
References
M. Koizumi, “Concept of FGM,” Ceramic Trans, Vol. 34, pp. 3-10, (1993)
M. Koizumi, “FGM activities in Japan,” Compos. Part B-Eng., Vol.28(1-2), pp. 1-4, (1997).
R. Ansari, R. Gholami, S. Sahmani, “Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory,” Composite Structures, Vol. 94, pp. 221–228, (2011).
A. Arbind, J.N. Reddy, “Nonlinear analysis of functionally graded microstructure-dependent beams,” Composite Structures, Vol. 98, pp. 272–281, (2013).
H.T. Thai, T.P. Vo, “Bending and free vibration of functionally graded beams using various higherorder shear deformation beam theories,” International Journal of Mechanical Sciences, Vol. 62, pp. 57–66, (2012).
B. Uymaz and M. Aydogdu, “Three dimensional mechanical buckling of FG plates with general boundary conditions,” Composite Structures, Vol. 96, pp. 174-193, (2013).
V.S. Khorasani and M. Bayat, “Bending analysis of FG plates using a general third-order plate theory with modified couple stress effect and MLPG method,” Engineering Analysis with Boundary Elements, Vol. 94, pp.159-171, (2018).
J. Rouzegar and F. Abad, “Free vibration analysis of FG plate with piezoelectric layers using fourvariable refined plate theory,” Thin-Walled Structures, Vol. 89, pp. 76-83, (2015).
J.A. Pelesko, D.H. Bernstein, “Modeling MEMS and NEMS,” Chapman & Hall/CRC, (2003).
A.C, Eringen, D.G.B. Edelen, “On nonlocal elasticity,” Int J Eng Sci., Vol. 10, pp. 233-248, (1972).
A.C. Eringen, “On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves,” J Appl Phys., Vol. 54, pp. 4703-4710, (1983).
R.D. Mindlin, H.F. Tiersten, “Effects of couple-stresses in linear elasticity,” Archive for Rational Mechanics and Analysis, Vol. 11(1), pp. 415–448, (1962).
R.D. Mindlin, “Micro-structure in linear elasticity,” Archive for Rational Mechanics and Analysis, Vol. 16 (1), pp. 51–78, (1964).
R.D. Mindlin, “Second gradient of strain and surface-tension in linear elasticity,” International Journal of Solids and Structures, Vol. 1(1), pp. 417–438, (1965)
R.A.Toupin, “Elastic materials with couple-stresses,” Archive for Rational Mechanics and Analysis, Vol. 11(1), pp. 385–414, (1962).
E.C. Aifantis, “On the role of gradients in the localization of deformation and fracture,” Int. J. Eng. Sci., Vol. 30(10), pp. 1279–1299, (1992).
F. Yang, A.C.M. Chong, D.C.C. Lam, et al., “Couple stress based strain gradient theory for elasticity,” International Journal of Solids and Structures, Vol. 39(10), pp. 2731–2743, (2002).
D.C.C. Lam, F. Yang, A.C.M. Chong, et al., “Experiments and theory in strain gradient elasticity,” Journal of Mechanics and Physics of Solids, Vol. 51(8), pp. 1477–1508, (2003).
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, pp. 298–313, (2015).
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, (2015).
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, (2016).
L. Lu, X. Guo, J. Zhao, “Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory,” International Journal of Engineering Science, Vol. 116, pp. 12–24, (2017).
M. Simsek, “Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach,” International Journal of Engineering Science, Vol. 105, pp. 12–27, (2016).
J.N. Reddy, “Nonlocal theories for bending, buckling and vibration of beams”, International Journal of Engineering Science, Vol. 45, pp. 288–307, (2007).
A. Karamanlia, T.P.Vo, “Bending, vibration, buckling analysis of bi-directional FG porous microbeams with a variable material length scale parameter”, Applied Mathematical Modelling, Vol. 91, pp. 723-748, (2021).
H. Liu, Q. Zhang, “Nonlinear dynamics of two-directional functionally graded microbeam with geometrical imperfection using unified shear deformable beam theory”, Applied Mathematical Modelling, Vol. 98, pp. 783-800, (2021).
