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http://dx.doi.org/10.12989/sss.2017.19.6.601

Free vibration analysis of embedded nanosize FG plates using a new nonlocal trigonometric shear deformation theory  

Besseghier, Abderrahmane (Faculty of Technology, Department of Civil Engineering, Material and Hydrology Laboratory, University of Sidi Bel Abbes)
Houari, Mohammed Sid Ahmed (Faculty of Technology, Department of Civil Engineering, Material and Hydrology Laboratory, University of Sidi Bel Abbes)
Tounsi, Abdelouahed (Faculty of Technology, Department of Civil Engineering, Material and Hydrology Laboratory, University of Sidi Bel Abbes)
Mahmoud, S.R. (Department of Mathematics, Faculty of Science, King Abdulaziz University)
Publication Information
Smart Structures and Systems / v.19, no.6, 2017 , pp. 601-614 More about this Journal
Abstract
In this work, free vibration analysis of size-dependent functionally graded (FG) nanoplates resting on two-parameter elastic foundation is investigated based on a novel nonlocal refined trigonometric shear deformation theory for the first time. This theory includes undetermined integral variables and contains only four unknowns, with is even less than the conventional first shear deformation theory (FSDT). Mori-Tanaka model is employed to describe gradually distribution of material properties along the plate thickness. Size-dependency of nanosize FG plate is captured via the nonlocal elasticity theory of Eringen. By implementing Hamilton's principle the equations of motion are obtained for a refined four-variable shear deformation plate theory and then solved analytically. To show the accuracy of the present theory, our research results in specific cases are compared with available results in the literature and a good agreement will be demonstrated. Finally, the influence of various parameters such as nonlocal parameter, power law indexes, elastic foundation parameters, aspect ratio, and the thickness ratio on the non-dimensional frequency of rectangular FG nanoscale plates are presented and discussed in detail.
Keywords
nonlocal elasticity theory; FG nanoplate; free vibration; refined theory; elastic foundation;
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Times Cited By KSCI : 35  (Citation Analysis)
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1 Houari, M.S.A., Tounsi, A., Bessaim, A. and Mahmoud, S.R. (2016), "A new simple three-unknown sinusoidal shear deformation theory for functionally graded plates", Steel Compos. Struct., 22(2), 257-276.   DOI
2 Hosseini-Hashemi, S., Bedroud, M. and Nazemnezhad, R. (2013), "An exact analytical solution for free vibration of functionally graded circular/annular Mindlin nanoplates via nonlocal elasticity", Compos Struct, 103, 108-118.   DOI
3 Jung, W.Y., Han, S.C. and Park, W.T. (2014), "A modified couple stress theory for buckling analysis of S-FGM nanoplates embedded in Pasternak elastic medium", Compos. B Eng., 60, 746-756.   DOI
4 Koiter, W.T. (1969), "Couple-stresses in the theory of elasticity, I & II", Philos. T. R. Soc., 67, 17-44.
5 Lam, D.C.C., Yang, F., Chong, A.C.M. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solids, 51, 1477-1508.   DOI
6 Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solids, 51(8), 1477-1508.   DOI
7 Larbi Chaht, F., Kaci, A., Houari, M.S.A., Tounsi, A., Anwar Beg, O. and Mahmoud, S.R. (2015), "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.   DOI
8 Lee, Z., Ophus, C., Fischer, L., Nelson-Fitzpatrick, N., Westra, K.L., Evoy, S., Radmilovic, V., Lu, C., Wu, D. and Chen, W. (2011), "Non-linear responses of nano-scale FGM films including the effects of surface energies", IEEE T. Nanotechnol., 10(6), 1321-1327.   DOI
9 Bennoun, M., Houari, M.S.A. and Tounsi, A. (2016), "A novel five variable refined plate theory for vibration analysis of functionally graded sandwich plates", Mech. Adv. Mater. Struct., 23(4), 423-431.   DOI
10 Berrabah, H.M., Tounsi, A., Semmah, A. and Adda Bedia, E.A. (2013), "Comparison of various refined nonlocal beam theories for bending, vibration and buckling analysis of nanobeams", Struct. Eng. Mech., 48(3), 351-365.   DOI
11 Besseghier, A., Heireche, H., Bousahla, A.A., Tounsi, A. and Benzair, A. (2015), "Nonlinear vibration properties of a zigzag single-walled carbon nanotube embedded in a polymer matrix", Adv. Nano Res., 3(1), 29-37.   DOI
12 Bouchafa, A., Bachir Bouiadjra, M., Houari, M.S.A. and Tounsi, A. (2015), "Thermal stresses and deflections of functionally graded sandwich plates using a new refined hyperbolic shear deformation theory", Steel Compos. Struct., 18(6), 1493-1515.   DOI
13 Bouderba, B., Houari, M.S.A. and Tounsi, A. (2013) "Thermomechanical bending response of FGM thick plates resting on Winkler-Pasternak elastic foundations", Steel Compos. Struct., 14(1), 85-104.   DOI
14 Bouderba, B., Houari, M.S.A. and Tounsi, A. and Mahmoud, S.R. (2016), "Thermal stability of functionally graded sandwich plates using a simple shear deformation theory", Struct. Eng. Mech., 58(3), 397-422.   DOI
15 Boukhari, A., Ait Atmane, H., Tounsi, A., Adda Bedia, E.A. and Mahmoud, S.R. (2016), "An efficient shear deformation theory for wave propagation of functionally graded material plates", Struct. Eng. Mech., 57(5), 837-859.   DOI
16 Mindlin, R. and Tiersten, H. (1962), "Effects of couple-stresses in linear elasticity", Arch. Ration. Mech. An., 11, 415-448.   DOI
17 Mahi, A., Adda Bedia, E.A. and Tounsi, A. (2015), "A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates", Appl. Math. Modell., 39, 2489-2508.   DOI
18 Meksi, A., Benyoucef, S., Houari, M.S.A. and Tounsi, A. (2015), "A simple shear deformation theory based on neutral surface position for functionally graded plates resting on Pasternak elastic foundations", Struct. Eng. Mech., 53(6), 1215-1240.   DOI
19 Meradjah, M., Kaci, A., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2015), "A new higher order shear and normal deformation theory for functionally graded beams", Steel Compos. Struct., 18(3), 793-809.   DOI
20 Merdaci, S., Tounsi, A. and Bakora, A. (2016), "A novel four variable refined plate theory for laminated composite plates", Steel Compos. Struct., 22(4), 713-732.   DOI
21 Mohammadi-Alasti, B., Rezazadeh, G., Borgheei, A.M., Minaei, S. and Habibifar, R. (2011), "On the mechanical behavior of a functionally graded micro-beam subjected to a thermal moment and nonlinear electrostatic pressure", Compos. Struct., 93(6), 1516-1525.   DOI
22 Nami, M.R. and Janghorban, M. (2014), "Resonance behavior of FG rectangular micro/nano plate based on nonlocal elasticity theory and strain gradient theory with one gradient constant", Compos. Struct., 111, 349-353.   DOI
23 Natarajan, S., Baiz, P., Ganapathi, M., Kerfriden, P. and Bordas, S. (2011), "Linear free flexural vibration of cracked functionally graded plates in thermal environment", Comput. Struct., 89(15-16), 1535-1546.   DOI
24 Bousahla, A.A., Houari, M.S.A., Tounsi, A. and Adda Bedia, E.A. (2014), "A novel higher order shear and normal deformation theory based on neutral surface position for bending analysis of advanced composite plates", Int. J. Comput. Meth., 11(6), 1350082.   DOI
25 Bounouara, F., Benrahou, K.H., Belkorissat, I. and Tounsi, A. (2016), "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.   DOI
26 Bourada, M., Kaci, A., Houari, M.S.A. and Tounsi, A. (2015), "A new simple shear and normal deformations theory for functionally graded beams", Steel Compos. Struct., 18(2), 409-423.   DOI
27 Bourada, F., Amara, K. and Tounsi, A. (2016), "Buckling analysis of isotropic and orthotropic plates using a novel four variable refined plate theory", Steel Compos. Struct., 21(6), 1287-1306.   DOI
28 Bousahla, A.A., Benyoucef, S., Tounsi, A. and Mahmoud, S.R. (2016), "On thermal stability of plates with functionally graded coefficient of thermal expansion", Struct. Eng. Mech., 60(2), 313-335.   DOI
29 Chong, A.C.M., Yang, F., Lam, D.C.C. and Tong, P. (2001), "Torsion and bending of micron-scaled structures", J. Mater. Res., 16(4), 1052-1058.   DOI
30 Dahmen, U. and Mitlin, D. (2006), "Metallic NEMS components fabricated from nanocomposite Al-Mo films", Nanotechnol., 17(12), 3063-3070.   DOI
31 Daneshmehr, A. and Rajabpoor, A. (2014), "Stability of size dependent functionally graded nanoplate based on nonlocal elasticity and higher order plate theories and different boundary conditions", Int. J. Eng. Sci., 82, 84-100.   DOI
32 Draiche, K., Tounsi, A. and Mahmoud, S.R. (2016), "A refined theory with stretching effect for the flexure analysis of laminated composite plates", Geomech. Eng., 11(5), 671-690.   DOI
33 Eltaher, M.A., El-Borgi, S. and Reddy, J.N. (2016b), "Nonlinear analysis of size-dependent and material-dependent nonlocal CNTs", Compos. Struct., 153, 902-913.   DOI
34 Natarajan, S., Chakraborty, S., Thangavel, M., Bordas, S. and Rabczuk, T. (2012), "Size-dependent free flexural vibration behavior of functionally graded nanoplates", Comput. Mater. Sci., 65, 74-80.   DOI
35 Nix, W.D. and Gao, H. (1998), "Indentation size effects in crystalline materials: a law for strain gradient plasticity", J. Mech. Phys. Solids, 46, 411-425.   DOI
36 Ebrahimi, F. and Shaghaghi, G.R. (2016), "Thermal effects on nonlocal vibrational characteristics of nanobeams with nonideal boundary conditions", Smart Struct. Syst., 18(6), 1087-1109.   DOI
37 El-Hassar, S.M., Benyoucef, S., Heireche, H. and Tounsi, A. (2016), "Thermal stability analysis of solar functionally graded plates on elastic foundation using an efficient hyperbolic shear deformation theory", Geomech. Eng., 10(3), 357-386.   DOI
38 Eltaher, M.A., Khater, M.E., Park, S., Abdel-Rahman, E. and Yavuz, M. (2016a), "On the static stability of nonlocal nanobeams using higher-order beam theories", Adv. Nano Res., 4(1), 51-64.   DOI
39 Eltaher, M.A., Abdelrahman, A.A., Al-Nabawy, A., Khater, M. and Mansour, A. (2014a), "Vibration of nonlinear graduation of nano-Timoshenko beam considering the neutral axis position", Appl. Math. Comput., 235, 512-529.
40 Eltaher, M.A., Hamed, M.A., Sadoun, A.M. and Mansour, A. (2014b), "Mechanical analysis of higher order gradient nanobeams", Appl. Math. Comput., 229, 260-272.
41 Eltaher, M.A., Khairy, A., Sadoun, A.M. and Omar, F.A. (2014c), "Static and buckling analysis of functionally graded Timoshenko nanobeams", Appl. Math. Comput., 229, 283-295.
42 Eltaher, M.A., Emam, S.A. and Mahmoud, F.F. (2012), "Free vibration analysis of functionally graded size-dependent nanobeams", Appl. Math. Comput., 218(14), 7406-7420.   DOI
43 Fu, Y., Du, H. and Zhang, S. (2003), "Functionally graded TiN/TiNi shape memory alloy films", Mater Lett., 57(20), 2995-2999.   DOI
44 Eringen, A.C. (1972), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10, 1-16.   DOI
45 Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54(9), 4703-4710.   DOI
46 Rahaeifard, M., Kahrobaiyan, M.H. and Ahmadian, M.T. (2009), "Sensitivity analysis of atomic force microscope cantilever made of functionally graded materials", Proceedings of the 3rd International Conference on Micro-and Nanosystems, San Diego, CA, USA, September.
47 Rahmani, O. and Pedram, O. (2014), "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.   DOI
48 Sallai, B., Hadji, L., Hassaine Daouadji, T. and Adda Bedia, E.A. (2015), "Analytical solution for bending analysis of functionally graded beam", Steel Compos. Struct., 19(4), 829-841.   DOI
49 Fekrar, A., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2014), "A new five-unknown refined theory based on neutral surface position for bending analysis of exponential graded plates", Meccanica, 49(4), 795-810.   DOI
50 Fleck, N.A., Muller, G.M., Ashby, M.F. and Hutchinson, J.W. (1994), "Strain gradient plasticity: theory and experiment", Acta Metall. Mater., 42(2), 475-487.   DOI
51 Hamed, M.A., Eltaher, M.A., Sadoun, A.M. and Almitani, K.H. (2016), "Free vibration of symmetric and sigmoid functionally graded nanobeams", Appl. Phys. A, 122(9), 829.   DOI
52 Hamidi, A., Houari, M.S.A., Mahmoud, S.R. and Tounsi, A. (2015), "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.   DOI
53 Hasanyan, D.J., Batra, R.C. and Harutyunyan, S. (2008), "Pull-in instabilities in functionally graded microthermoelectromechanical systems", J. Therm. Stress., 31(10), 1006-1021.   DOI
54 Hebali, H., Tounsi, A., Houari, M.S.A., Bessaim, A. and Adda Bedia, E.A. (2014), "New quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates", J. Eng. Mech.-ASCE, 140, 374-383.   DOI
55 Hebali, H., Bakora, A., Tounsi, A. and Kaci, A. (2016), "A novel four variable refined plate theory for bending, buckling, and vibration of functionally graded plates", Steel Compos. Struct., 22(3), 473-495.   DOI
56 Tounsi, A., Houari, M.S.A. and Bessaim, A. (2016), "A new 3-unknowns non-polynomial plate theory for buckling and vibration of functionally graded sandwich plate", Struct. Eng. Mech., 60(4), 547-565.   DOI
57 Stolken, J.S. and Evans, A.G. (1998), "A microbend test method for measuring the plasticity length scale", Acta Mater., 46(14), 5109-5115.   DOI
58 Tagrara, S.H., Benachour, A., Bachir Bouiadjra, M. and Tounsi, A. (2015), "On bending, buckling and vibration responses of functionally graded carbon nanotube-reinforced composite beams", Steel Compos. Struct., 19(5), 1259-1277.   DOI
59 Tebboune, W., Benrahou, K.H., Houari, M.S.A. and Tounsi, A. (2015), "Thermal buckling analysis of FG plates resting on elastic foundation based on an efficient and simple trigonometric shear deformation theory", Steel Compos. Struct., 18(2), 443-465.   DOI
60 Tounsi, A., Houari, M.S.A., Benyoucef, S. and Adda Bedia, E.A. (2013), "A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates", Aerosp. Sci. Technol., 24(1), 209-220.   DOI
61 Toupin, R.A. (1962), "Elastic materials with couple-stresses", Arch. Ration. Mech. An., 11, 385-414.   DOI
62 Ahouel, M., Houari, M.S.A., Adda Bedia, E.A. and Tounsi, A. (2016), "Size-dependent mechanical behavior of functionally graded trigonometric shear deformable nanobeams including neutral surface position concept", Steel Compos. Struct., 20(5), 963-981.   DOI
63 Heireche, H., Tounsi, A., Benzair, A., Maachou, M. and Adda Bedia, E.A. (2008), "Sound wave propagation in single-walled carbon nanotubes using nonlocal elasticity", Physica E., 40(8), 2791-2799.   DOI
64 Valizadeh, N., Natarajan, S., Gonzalez-Estrada, O.A., Rabczuk, T., Bui, T.Q. and Bordas, S.P.A. (2013), "NURBS-based finite element analysis of functionally graded plates: static bending, vibration, buckling and flutter", Compos. Struct., 99, 309-326.   DOI
65 Adda Bedia, W., Benzair, A., Semmah, A., Tounsi, A. and Mahmoud, S.R. (2015), "On the thermal buckling characteristics of armchair single-walled carbon nanotube embedded in an elastic medium based on nonlocal continuum elasticity", Brazil. J. Phys., 45(2), 225-233.   DOI
66 Witvrouw, A. and Mehta, A. (2005), "The use of functionally graded poly-SiGe layers for MEMS applications", Mater. Sci. Forum., 492, 255-260.
67 Zare, M., Nazemnezhad, R. and Hosseini-Hashemi, S. (2015), "Natural frequency analysis of functionally graded rectangular nanoplates with different boundary conditions via an analytical method", Meccanica, 50(9), 2391-2408.   DOI
68 Zhang, J. and Fu, Y. (2012), "Pull-in analysis of electrically actuated viscoelastic microbeams based on a modified couple stress theory", Meccanica, 47(7), 1649-1658.   DOI
69 Zemri, A., Houari, M.S.A., Bousahla, A.A. and Tounsi, A. (2015), "A mechanical response of functionally graded nanoscale beam: An assessment of a refined nonlocal shear deformation theory beam theory", Struct. Eng. Mech., 54(4), 693-710.   DOI
70 Zidi, M., Tounsi, A., Houari, M.S.A., Adda Bedia, E.A. and Anwar Beg, O. (2014), "Bending analysis of FGM plates under hygrothermo-mechanical loading using a four variable refined plate theory", Aerosp. Sci. Technol., 34, 24-34.   DOI
71 Akbas, S.D. (2016), "Forced vibration analysis of viscoelastic nanobeams embedded in an elastic medium", Smart Struct. Syst., 18(6), 1125-1143.   DOI
72 Aifantis, E. (1999), "Strain gradient interpretation of size effects", Int. J. Fracture, 95, 299-314.   DOI
73 Aissani, K., Bachir Bouiadjra, M., Ahouel, M. and Tounsi, A. (2015), "A new nonlocal hyperbolic shear deformation theory for nanobeams embedded in an elastic medium", Struct. Eng. Mech., 55(4), 743-762.   DOI
74 Al-Basyouni, K.S., Tounsi, A. and Mahmoud, S.R. (2015), "Size dependent bending and vibration analysis of functionally graded micro beams based on modified couple stress theory and neutral surface position", Compos. Struct., 125, 621-630.   DOI
75 Ait Amar Meziane, M., Abdelaziz, H.H. and Tounsi, A. (2014), "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.   DOI
76 Ait Atmane, H., Tounsi, A., Bernard, F. and Mahmoud, S.R. (2015), "A computational shear displacement model for vibrational analysis of functionally graded beams with porosities", Steel Compos. Struct., 19(2), 369-384.   DOI
77 Ait Yahia, S., Ait Atmane, H., Houari, M.S.A. and Tounsi, A. (2015), "Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories", Struct. Eng. Mech., 53(6), 1143-1165.   DOI
78 Alshorbagy, A.E., Eltaher, M.A. and Mahmoud, F.F. (2011), "Free vibration characteristics of a functionally graded beam by finite element method", Appl. Math. Modell., 35, 412-425.   DOI
79 Bedroud, M., Nazemnezhad, R. and Hosseini-Hashemi, S. (2015), "Axisymmetric/asymmetric buckling of functionally graded circular/annular Mindlin nanoplates via nonlocal elasticity", Meccanica, 50(7), 1791-1806.   DOI
80 Attia, A., Tounsi, A., Adda Bedia, E.A. and Mahmoud, S.R. (2015), "Free vibration analysis of functionally graded plates with temperature-dependent properties using various four variable refined plate theories", Steel Compos. Struct., 18(1), 187-212.   DOI
81 Bellifa, H., Benrahou, K.H., Hadji, L., Houari, M.S.A. and Tounsi, A. (2016), "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, 265-275.   DOI
82 Belabed, Z., Houari, M.S.A., Tounsi, A., Mahmoud, S.R. and Anwar Beg, O. (2014), "An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates", Composites: Part B, 60, 274-283.   DOI
83 Beldjelili, Y., Tounsi, A. and Mahmoud, S.R. (2016), "Hygrothermo-mechanical bending of S-FGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory", Smart Struct. Syst., 18(4), 755-786.   DOI
84 Belkorissat, I., Houari, M.S.A., Tounsi, A., Adda Bedia, E.A. and Mahmoud, S.R. (2015), "On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable model", Steel Compos. Struct., 18(4), 1063-1081.   DOI
85 Benachour, A., Daouadji, H.T., Ait Atmane, H., Tounsi, A. and Meftah, S.A. (2011), "A four variable refined plate theory for free vibrations of functionally graded plates with arbitrary gradient", Composites Part B, 42(6), 1386-1394.   DOI
86 Benguediab, S., Tounsi, A., Zidour, M. and Semmah, A. (2014), "Chirality and scale rffects on mechanical buckling properties of zigzag double-walled carbon nanotubes", Composites Part B, 57, 21-24.   DOI
87 Bennai, R., Ait Atmane, H. and Tounsi, A. (2015), "A new higherorder shear and normal deformation theory for functionally graded sandwich beams", Steel Compos. Struct., 19(3), 521-546.   DOI