Abstract
The objective of this paper was the investigation of vibration characteristics of both nonlinear symmetric power and sigmoid functionally graded nonlocal nanobeams. The volume fractions of metal and ceramic are assumed to be distributed through a beam thickness by sigmoid law distribution and symmetric power function. Structures with symmetric distribution with mid-plane such as ceramic–metal–ceramic and metal–ceramic–metal are proposed. Nonlocal differential Eringen’s elasticity is exploited to incorporate size dependency of nanobeam. The kinematic relations of Euler–Bernoulli beam are proposed, with the assumption of a small strain. A nonlocal equation of motion of nanobeam is derived by using principle of virtual work and then discretized by finite element method to obtain numerical solution. Numerical results show the effects of the function distribution, gradient index and nonlocal parameter on natural frequencies of macro- and nanobeam. This model is helpful in the mechanical design of nanoelectromechanical systems manufactured from FGM.
Similar content being viewed by others
References
M.A. Agwa, M.A. Eltaher, Vibration of a carbyne nanomechanical mass sensor with surface effect. Appl. Phys. A 122(4), 1–8 (2016)
M. Ahouel, M.S.A. Houari, E.A. Bedia, A. Tounsi, Size-dependent mechanical behavior of functionally graded trigonometric shear deformable nanobeams including neutral surface position concept. Steel Compos. Struct. 20(5), 963–981 (2016)
A.E. Alshorbagy, M.A. Eltaher, F.F. Mahmoud, Free vibration characteristics of a functionally graded beam by finite element method. Appl. Math. Model. 35(1), 412–425 (2011)
H.A. Atmane, A. Tounsi, S.A. Meftah, H.A. Belhadj, Free vibration behavior of exponential functionally graded beams with varying cross-section. J. Vib. Control 17(2), 311–318 (2010)
Z. Belabed, M.S.A. Houari, A. Tounsi, S.R. Mahmoud, O.A. Bég, An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates. Compos. Part B Eng. 60, 274–283 (2014)
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. J. Braz. Soc. Mech. Sci. Eng. 38(1), 265–275 (2016)
S. Benguediab, A. Tounsi, M. Zidour, A. Semmah, Chirality and scale effects on mechanical buckling properties of zigzag double-walled carbon nanotubes. Compos. Part B Eng. 57, 21–24 (2014)
M. Bennoun, M.S.A. Houari, A. Tounsi, A novel five-variable refined plate theory for vibration analysis of functionally graded sandwich plates. Mech. Adv. Mater. Struct. 23(4), 423–431 (2016)
S. Ben-Oumrane, T. Abedlouahed, M. Ismail, B.B. Mohamed, M. Mustapha, A.B. El Abbas, A theoretical analysis of flexional bending of Al/Al 2 O 3 S-FGM thick beams. Comput. Mater. Sci. 44(4), 1344–1350 (2009)
A. Besseghier, H. Heireche, A.A. Bousahla, A. Tounsi, A. Benzair, Nonlinear vibration properties of a zigzag single-walled carbon nanotube embedded in a polymer matrix. Adv. Nano Res. 3(1), 29–37 (2015)
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 Compos. Struct. 20(2), 227–249 (2016)
B. Bouderba, M.S.A. Houari, A. Tounsi, Thermomechanical bending response of FGM thick plates resting on Winkler–Pasternak elastic foundations. Steel Compos. Struct. 14(1), 85–104 (2013)
M. Bourada, A. Kaci, M.S.A. Houari, A. Tounsi, A new simple shear and normal deformations theory for functionally graded beams. Steel Compos. Struct. 18(2), 409–423 (2015)
F.L. Chaht, A. Kaci, M.S.A. Houari, A. Tounsi, O.A. Beg, S.R. Mahmoud, Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect. Steel Compos. Struct. 18(2), 425–442 (2015)
S.H. Chi, Y.L. Chung, Cracking in sigmoid functionally graded coating. J. Mech. 18, 41–53 (2002)
F. Delale, F. Erdogan, The crack problem for a nonhomogeneous plane. J. Appl. Mech. 50(3), 609–614 (1983)
N.D. Duc, P.H. Cong, Nonlinear dynamic response of imperfect symmetric thin sigmoid-functionally graded material plate with metal–ceramic–metal layers on elastic foundation. J. Vib. Control 21, 637–646 (2013)
F. Ebrahimi, E. Salari, Size-dependent free flexural vibrational behavior of functionally graded nanobeams using semi-analytical differential transform method. Compos. Part B Eng. 79, 156–169 (2015)
F. Ebrahimi, M. Boreiry, Investigating various surface effects on nonlocal vibrational behavior of nanobeams. Appl. Phys. A 121(3), 1305–1316 (2015)
F. Ebrahimi, M.R. Barati, Dynamic modeling of a thermo–piezo-electrically actuated nanosize beam subjected to a magnetic field. Appl. Phys. A 122(4), 1–18 (2016)
M.A. Eltaher, S.A. Emam, F.F. Mahmoud, Free vibration analysis of functionally graded size-dependent nanobeams. Appl. Math. Comput. 218(14), 7406–7420 (2012)
M.A. Eltaher, S.A. Emam, F.F. Mahmoud, Static and stability analysis of nonlocal functionally graded nanobeams. Compos. Struct. 96, 82–88 (2013)
M.A. Eltaher, A.E. Alshorbagy, F.F. Mahmoud, Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams. Compos. Struct. 99, 193–201 (2013)
M.A. Eltaher, A.E. Alshorbagy, F.F. Mahmoud, Vibration analysis of Euler-Bernoulli nanobeams by using finite element method. Appl. Math. Model. 37(7), 4787–4797 (2013)
M.A. Eltaher, A. Khairy, A.M. Sadoun, F.A. Omar, Static and buckling analysis of functionally graded Timoshenko nanobeams. Appl. Math. Comput. 229, 283–295 (2014)
M.A. Eltaher, A.A. Abdelrahman, A. Al-Nabawy, M. Khater, A. Mansour, Vibration of nonlinear graduation of nano-Timoshenko beam considering the neutral axis position. Appl. Math. Comput. 235, 512–529 (2014)
M.A. Eltaher, M.E. Khater, S.A. Emam, A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams. Appl. Math. Model. 40(5–6), 4109–4128 (2016)
M.A. Eltaher, S. El-Borgi, J.N. Reddy, Nonlinear analysis of size-dependent and material-dependent nonlocal CNTs. Compos. Struct. 153, 902–913 (2016)
A.C. Eringen, Nonlocal polar elastic continua. Int. J. Eng. Sci. 10(1), 1–16 (1972)
A.C. Eringen, D.G.B. Edelen, On nonlocal elasticity. Int. J. Eng. Sci. 10(3), 233–248 (1972)
A.C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54(9), 4703–4710 (1983)
A. Fereidoon, A. Mohyeddin, Bending analysis of thin functionally graded plates using generalized differential quadrature method. Arch. Appl. Mech. 81(11), 1523–1539 (2011)
S. Filiz, M. Aydogdu, Wave propagation analysis of embedded (coupled) functionally graded nanotubes conveying fluid. Compos. Struct. 132, 1260–1273 (2015)
A. Hamidi, M.S.A. Houari, S.R. Mahmoud, A. Tounsi, A sinusoidal plate theory with 5-unknowns and stretching effect for thermomechanical bending of functionally graded sandwich plates. Steel Compos. Struct. 18(1), 235–253 (2015)
H. Hebali, A. Tounsi, M.S.A. Houari, A. Bessaim, E.A.A. Bedia, New quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates. J. Eng. Mech. 140(2), 374–383 (2014)
S.A.H. Hosseini, O. Rahmani, Free vibration of shallow and deep curved FG nanobeam via nonlocal Timoshenko curved beam model. Appl. Phys. A 122(3), 1–11 (2016)
W.Y. Jung, S.C. Han, Analysis of sigmoid functionally graded material (S-FGM) nanoscale plates using the nonlocal elasticity theory. Math. Problems Eng. 49, 449–458 (2013)
S. Kapuria, M. Bhattacharyya, A.N. Kumar, Bending and free vibration response of layered functionally graded beams: a theoretical model and its experimental validation. Compos. Struct. 82(3), 390–402 (2008)
K. Kiani, Longitudinal and transverse instabilities of moving nanoscale beam-like structures made of functionally graded materials. Compos. Struct. 107, 610–619 (2014)
M. Komijani, S.E. Esfahani, J.N. Reddy, Y.P. Liu, M.R. Eslami, Nonlinear thermal stability and vibration of pre/post-buckled temperature-and microstructure dependent functionally graded beams resting on elastic foundation. Compos. Struct. 112, 292–307 (2014)
C.Y. Lee, J.H. Kim, Thermal post-buckling and snap-through instabilities of FGM panels in hypersonic flows. Aerosp. Sci. Technol. 30(1), 175–182 (2013)
S.R. Li, H.D. Su, C.J. Cheng, Free vibration of functionally graded material beams with surface-bonded piezoelectric layers in thermal environment. Appl. Math. Mech. 30, 969–982 (2009)
X.F. Li, Y.A. Kang, J.X. Wu, Exact frequency equations of free vibration of exponentially functionally graded beams. Appl. Acoust. 74(3), 413–420 (2013)
Y. Liu, D.W. Shu, Free vibration analysis of exponential functionally graded beams with a single delamination. Compos. Part B Eng. 59, 166–172 (2014)
A. Mahi, E.A. Bedia, A. Tounsi, I. Mechab, An analytical method for temperature-dependent free vibration analysis of functionally graded beams with general boundary conditions. Compos. Struct. 92(8), 1877–1887 (2010)
A. Mahi, A. Tounsi, A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates. Appl. Math. Model. 39(9), 2489–2508 (2015)
M.A.A. Meziane, H.H. Abdelaziz, A. Tounsi, An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions. J. Sandw. Struct. Mater. 16(3), 293–318 (2014)
T. Mori, K. Tanaka, Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 21(5), 571–574 (1973)
O. Rahmani, A.A. Jandaghian, Buckling analysis of functionally graded nanobeams based on a nonlocal third-order shear deformation theory. Appl. Phys. A 119(3), 1019–1032 (2015)
O. Rahmani, O. Pedram, Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory. Int. J. Eng. Sci. 77, 55–70 (2014)
J.N. Reddy, Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45(2), 288–307 (2007)
J.N. Reddy, S. El-Borgi, Eringen’s nonlocal theories of beams accounting for moderate rotations. Int. J. Eng. Sci. 82, 159–177 (2014)
J.N. Reddy, S. El-Borgi, J. Romanoff, Non-linear analysis of functionally graded microbeams using Eringen's non-local differential model. Int. J. Non-Linear Mech. 67, 308–318 (2014)
H. Salehipour, A.R. Shahidi, H. Nahvi, Modified nonlocal elasticity theory for functionally graded materials. Int. J. Eng. Sci. 90, 44–57 (2015)
M. Şimşek, H.H. Yurtcu, Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory. Compos. Struct. 97, 378–386 (2013)
R. Sourki, S.A.H. Hoseini, Free vibration analysis of size-dependent cracked microbeam based on the modified couple stress theory. Appl. Phys. A 122(4), 1–11 (2016)
T.R. Tauchert, Energy Principles in Structural Mechanics (McGraw-Hill Companies, New York, 1974)
Y. Tomota, K. Kuroki, T. Mori, I. Tamura, Tensile deformation of two-ductile phase alloys: flow curves of α–γ Fe–Cr–Ni alloys. Mater. Sci. Eng. 24(1), 85–94 (1976)
A. Tounsi, S. Benguediab, B. Adda, A. Semmah, M. Zidour, Nonlocal effects on thermal buckling properties of double-walled carbon nanotubes. Adv. Nano Res. 1(1), 1–11 (2013)
B. Uymaz, Forced vibration analysis of functionally graded beams using nonlocal elasticity. Compos. Struct. 105, 227–239 (2013)
S.A. Yahia, H.A. Atmane, M.S.A. Houari, A. Tounsi, Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories. Struct. Eng. Mech. 53(6), 1143–1165 (2015)
M. Zidi, A. Tounsi, M.S.A. Houari, O.A. Bég, Bending analysis of FGM plates under hygro-thermo-mechanical loading using a four variable refined plate theory. Aerosp. Sci. Technol. 34, 24–34 (2014)
Acknowledgments
This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. (135-790-D1435). The authors, therefore, acknowledge DSR technical and financial support.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hamed, M.A., Eltaher, M.A., Sadoun, A.M. et al. Free vibration of symmetric and sigmoid functionally graded nanobeams. Appl. Phys. A 122, 829 (2016). https://doi.org/10.1007/s00339-016-0324-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00339-016-0324-0