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http://dx.doi.org/10.12989/sss.2018.21.1.065

Dynamic analysis of nanoscale beams including surface stress effects  

Youcef, Djamel Ould (Laboratoire de Modelisation et Simulation Multi-echelle, Departement de Physique, Faculte des Sciences Exactes, Departement de Physique, Universite de Sidi Bel Abbes)
Kaci, Abdelhakim (Universite Dr Tahar Moulay, Faculte de Technologie, Departement de Genie Civil et Hydraulique)
Benzair, Abdelnour (Laboratoire de Modelisation et Simulation Multi-echelle, Departement de Physique, Faculte des Sciences Exactes, Departement de Physique, Universite de Sidi Bel Abbes)
Bousahla, Abdelmoumen Anis (Laboratoire de Modelisation et Simulation Multi-echelle, Departement de Physique, Faculte des Sciences Exactes, Departement de Physique, Universite de Sidi Bel Abbes)
Tounsi, Abdelouahed (Laboratoire de Modelisation et Simulation Multi-echelle, Departement de Physique, Faculte des Sciences Exactes, Departement de Physique, Universite de Sidi Bel Abbes)
Publication Information
Smart Structures and Systems / v.21, no.1, 2018 , pp. 65-74 More about this Journal
Abstract
In this article, an analytic non-classical model for the free vibrations of nanobeams accounting for surface stress effects is developed. The classical continuum mechanics fails to capture the surface energy effects and hence is not directly applicable at nanoscale. A general beam model based on Gurtin-Murdoch continuum surface elasticity theory is developed for the analysis of thin and thick beams. Thus, surface energy has a significant effect on the response of nanoscale structures, and is associated with their size-dependent behavior. To check the validity of the present analytic solution, the numerical results are compared with those obtained in the scientific literature. The influences of beam thickness, surface density, surface residual stress and surface elastic constants on the natural frequencies of nanobeams are also investigated. It is indicated that the effect of surface stress on the vibrational response of a nanobeam is dependent on its aspect ratio and thickness.
Keywords
surface stress effects; nanoscales; free vibration; Gurtin-Murdoch;
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Times Cited By KSCI : 38  (Citation Analysis)
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1 Abualnour, M., Houari, M.S.A., Tounsi, A., Adda Bedia, E.A. and Mahmoud, S.R. (2018), "A novel quasi-3D trigonometric plate theory for free vibration analysis of advanced composite plates", Compos. Struct., 184, 688-697.   DOI
2 Ahouel, M., Houari, M.S.A., AddaBedia, 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
3 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
4 AitYahia, S., AitAtmane, 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
5 Akavci, S.S. (2016), "Mechanical behavior of functionally graded sandwich plates on elastic foundation", Compos.: Part B, 96, 136-152.   DOI
6 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
7 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
8 Bessaim, A., Houari, M.S.A., Tounsi, A., Mahmoud, S.R. and Adda Bedia, E.A. (2013), "A new higher order shear and normal deformation theory for the static and free vibration analysis of sandwich plates with functionally graded isotropic face sheets", J. Sandw. Struct. Mater., 15, 671-703.   DOI
9 Besseghier, A., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2017), "Free vibration analysis of embedded nanosize FG plates using a new nonlocal trigonometric shear deformation theory", Smart Struct. Syst., 19(6), 601-614.   DOI
10 Bouafia, K., Kaci, A., Houari, M.S.A., Benzair, A. and Tounsi, A. (2017), "A nonlocal quasi-3D theory for bending and free flexural vibration behaviors of functionally graded nanobeams", Smart Struct. Syst., 19(2), 115-126.   DOI
11 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
12 Boukhari, A., AitAtmane, H., Tounsi, A., AddaBedia, 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
13 Baseri, V., Jafari, G.S. and Kolahchi, R. (2016), "Analytical solution for buckling of embedded laminated plates based on higher order shear deformation plate theory", Steel Compos. Struct., 21(4), 883-919.   DOI
14 Ansari, R., Hosseini, K., Darvizeh, A. and Daneshian, B. (2013), "A sixth-order compact finite difference method for nonclassical vibration analysis of nanobeams including surface stress effects", Appl. Math. Comput., 219, 4977-4991.
15 Attia, A., Tounsi, A., AddaBedia, 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
16 Barati, M.R. and Shahverdi, H. (2016), "A four-variable plate theory for thermal vibration of embedded FG nanoplates under non-uniform temperature distributions with different boundary conditions", Struct. Eng. Mech., 60(4), 707-727.   DOI
17 Becheri, T., Amara, K., Bouazza, M. and Benseddiq, N. (2016), "Buckling of symmetrically laminated plates using nth-order shear deformation theory with curvature effects", Steel Compos. Struct., 21(6), 1347 - 1368.   DOI
18 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", Compos.: Part B, 60, 274-283.   DOI
19 Bourada, M., Kaci, A., Houari, M.S.A. and Tounsi, A. (2015), "Anew simple shear and normal deformations theory for functionally graded beams", Steel Compos. Struct., 18(2), 409-423.   DOI
20 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
21 Chikh, A., Tounsi, A., Hebali, H. and Mahmoud, S.R. (2017), "Thermal buckling analysis of cross-ply laminated plates using a simplified HSDT", Smart Struct. Syst., 19(3), 289-297.   DOI
22 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
23 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
24 Chen, C.Q., Shi, Y., Zhang, Y.S., Zhu, J. and Yan, Y.J. (2006), "Size dependence of Young's modulus in ZnO nanowires", Phys. Rev. Lett., 96(7), 075505.   DOI
25 Gibbs, J.W. (1906), "The scientific papers of J. Willard Gibbs", Dover Publications, Inc, New York.
26 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
27 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
28 El-Haina, F., Bakora, A., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2017), "A simple analytical approach for thermal buckling of thick functionally graded sandwich plates", Struct. Eng. Mech., 63(5), 585-595.   DOI
29 Eltaher, M.A., Khater, M.E., Park, S., Abdel-Rahman, E. and Yavuz, M. (2016), "On the static stability of nonlocalnanobeams using higher-order beam theories", Adv. Nano Res., 4(1), 51-64.   DOI
30 Gao, X.L. and Zhang, G.Y. (2015), "A microstructure- and surface energy-dependent third-order shear deformation beam model", ZAMP, 66(4), 1871-1894.
31 Gurtin, M.E. and Murdoch, A.I. (1975), "A continuum theory of elastic material surfaces", Arch. Ration. Mech. Anal., 57(4), 291-323.   DOI
32 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
33 Belkorissat, I., Houari, M.S.A., Tounsi, A., AddaBedia, 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
34 Bellifa, H., Bakora, A., Tounsi, A., Bousahla, A.A. and Mahmoud, S.R. (2017b), "An efficient and simple four variable refined plate theory for buckling analysis of functionally graded plates", Steel Compos. Struct., 25(3), 257-270.   DOI
35 Bellifa, H., Benrahou, K.H., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2017a), "A nonlocal zeroth-order shear deformation theory for nonlinear postbuckling of nanobeams", Struct. Eng. Mech., 62(6), 695 -702.   DOI
36 Benachour, A, Tahar, H.D., Atmane, H.A., Tounsi, A. and Meftah, S.A. (2011), "A four variable refined plate theory for free vibrations of functionally graded plates with arbitrary gradient", Compos.: Part B; 42, 1386-1394.   DOI
37 Benadouda, M., Ait Atmane, H., Tounsi, A., Bernard, F. and Mahmoud, S.R. (2017), "An efficient shear deformation theory for wave propagation in functionally graded material beams with porosities", Earthq. Struct., 13(3), 255-265.   DOI
38 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
39 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
40 Hadji, L., Atmane, H.A., Tounsi, A., Mechab, I. and Adda Bedia, E.A. (2011), "Free vibration of functionally graded sandwich plates using four variable refined plate theory", Appl. Math. Mech., 32(7), 925-942.   DOI
41 Hanifi Hachemi Amar, L., Kaci, A. and Tounsi, A. (2017), "On the size-dependent behavior of functionally graded micro-beams with porosities", Struct. Eng. Mech., 64(5), 527-541.   DOI
42 He, J. and Lilley, C.M. (2008), "Surface effect on the elastic behavior of static bending nanowires", Nano Lett., 8, 1798-1802.   DOI
43 He, J., and Lilley, C. M., (2008), "Surface Stress Effect on Bending Resonance of Nanowires With Different Boundary Conditions," Appl. Phys. Lett., 93,263108.   DOI
44 Hebali, H., Tounsi, A., Houari, M.S.A., Bessaim, A. and AddaBedia, E.A. (2014), "A new quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates", J. Eng. Mech. - ASCE, 140(2), 374-383.   DOI
45 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
46 Kar, V.R. and Panda, S.K. (2015), "Large deformation bending analysis of functionally graded spherical shell using FEM", Struct. Eng. Mech., 53(4), 661-679.   DOI
47 Karami, B., Janghorban, M. and Tounsi, A. (2017), "Effects of triaxial magnetic field on the anisotropic nanoplates", Steel Compos. Struct., 25(3), 361-374.   DOI
48 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
49 Gurtin, M.E. and Murdoch, A.I. (1978), "Surface stress in solids", Int. J. Solids Struct., 14(6), 431-440.   DOI
50 Khetir, H., Bachir Bouiadjra, M., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2017), "A new nonlocal trigonometric shear deformation theory for thermal buckling analysis of embedded nanosize FG plates", Struct. Eng. Mech., 64(4), 391-402.   DOI
51 Liu, C., Rajapakse, R.K.N.D. and Phani, A.S. (2011), "Finite element modelling of beams with surface energy effects", ASME J. Appl. Mech., 78, 031014, see also Erratum, (2012), ASME J. Appl. Mech., 79, 017001.
52 Liu, L. and Zhang, Y. (2004), "Multi-wall carbon nanotube as a new infrared detected material", Sensors Actuat., 116, 394-397.   DOI
53 Maede, R.D. and Vanderbilt, D. (1989), "Origins of stress on elemental and chemisorbed semiconductor surfaces", Phys. Rev. Lett., 63(13), 1404-1407.   DOI
54 Mahi, A., AddaBedia, 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. Model., 39(9), 2489-2508.   DOI
55 Menasria, A., Bouhadra, A., Tounsi, A., Bousahla, A.A. and Mahmoud, S.R. (2017), "A new and simple HSDT for thermal stability analysis of FG sandwich plates", Steel Compos. Struct., 25(2), 157-175.   DOI
56 Miller, R.E. and Shenoy, V.B. (2000), "Size-dependent elastic properties of nanosized structural elements", Nanotechnology, 11(3), 139-147.   DOI
57 Qin, C., Shen, J., Hu, Y. and Ye, M. (2009), "Facile attachment of magnetic nanoparticles to carbon nanotubes via robust linkages and its fabrication of magneticnanocomposites", Compos. Sci. Technol., 69, 427-431.   DOI
58 Mouffoki, A., Adda Bedia, E.A., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2017), "Vibration analysis of nonlocal advanced nanobeams in hygro-thermal environment using a new two-unknown trigonometric shear deformation beam theory", Smart Struct. Syst., 20(3), 369-383.   DOI
59 Needs, R.J. (1987), "Calculations of the surface stress tensor at aluminum (111) and (110) surfaces", Phys. Rev. Lett., 58(1),. 53-56.   DOI
60 Ould Youcef, D., Kaci, A., Houari, M.S.A., Tounsi, A., Benzair, A. and Heireche, H. (2015), "On the bending and stability of nanowire using various HSDTs", Adv. Nano Res., 3(3), 177-191.   DOI
61 Reddy, J.N. (2002), "Energy principles and variational methods in applied mechanics", Wiley, NewYork.
62 Reddy, J.N. (1984), "A simple higher-order theory for laminated composite plates", J Appl. Mech., 51(4), 745-752.   DOI
63 Saidi, H., Tounsi, A. and Bousahla, A.A. (2016), "A simple hyperbolic shear deformation theory for vibration analysis of thick functionally graded rectangular plates resting on elastic foundations", Geomech. Eng., 11(2), 289-307.   DOI
64 Sekkal, M., Fahsi, B., Tounsi, A. and Mahmoud, S.R. (2017), "A new quasi-3D HSDT for buckling and vibration of FG plate", Struct. Eng. Mech., 64(6), 737-749.   DOI
65 Shenoy, V.B. (2005), "Atomistic calculations of elastic properties of metallic Fcc crystal surfaces", Phys. Rev. B, 71(9), 094104.   DOI
66 Wang, G.F. and Feng, X.Q. (2009a), "Surface effects on buckling of nanowiresunder uniaxial compression", Appl. Phys. Lett., 94, 141913.   DOI
67 Song, F., Huang, G.L., Park, H.S. and Liu, X.N. (2011), "A continuum model for the mechanical behavior of nanowires including surface and surface induced initial stresses", Int. J. Solids Struct., 48, 2154-2163.   DOI
68 Taibi, F.Z., Benyoucef, S., Tounsi, A., Bachir Bouiadjra, R., Adda Bedia, E.A. and Mahmoud, S.R. (2015), "A simple shear deformation theory for thermo-mechanical behaviour of functionally graded sandwich plates on elastic foundations", J. Sandw. Struct. Mater., 17(2), 99-129.   DOI
69 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
70 Wang, G.F. and Feng, X.Q. (2009b), "Timoshenko beam model for bucklingand vibration of nanowires with surface effects," J. Phys. D: Appl. Phys., 42,155411.   DOI
71 Wang, G.F. and Feng, X.Q. (2007), "Effects of Surface elasticity and residualsurface tension on the natural frequency of microbeams," Appl. Phys. Lett., 90, 231904.   DOI
72 Wong, E.W., Sheehan, P.E. and Lieber, C.M. (1997), "Nanobeam mechanics: elasticity,strength, and toughness of nanorods and nanotubes", Science, 277, 1971-1975.   DOI
73 Xu, F., Qin, Q., Mishra, A., Gu, Y. and Zhu, Y. (2010), "Mechanical properties of ZnO nanowires under different loading modes", Nano Res., 3, 271-280.   DOI
74 Yan, X.B., Chen, X.J., Tay, B.K. and Khor, K.A. (2007), "Transparent and flexible glucose via layer-by-layerassembly of multi-wall carbon nanotubes and glucoseoxidase", Electrochem. Commun., 9, 1269-1275.   DOI
75 Zhao, C., Song, Y., Ren, J. and Qu, X. (2009), "A DNA nanomachine induced by single-walled carbon nanotubes on gold surface", Biomaterials, 30, 1739-1745.   DOI
76 Yazid, M., Heireche, H., Tounsi, A., Bousahla, A.A. and Houari, M.S.A. (2018), "A novel nonlocal refined plate theory for stability response of orthotropic single-layer graphene sheet resting on elastic medium", Smart Struct. Syst., (In press).
77 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
78 Zhang, W.D., Wen, Y., Liu, S.M., Tjiu, W.C., Xu, G.Q. and Gan. L.M. (2002), "Synthesis of vertically alignedcarbon nanotubes on metal deposited quartz plates", Carbon, 40, 1981-1989.   DOI
79 Zidi, M., Houari, M.S.A., Tounsi, A., Bessaim, A. and Mahmoud, S.R. (2017), "A novel simple two-unknown hyperbolic shear deformation theory for functionally graded beams", Struct. Eng. Mech., 64(2), 145-153.
80 Zidi, M., Tounsi, A., Houari M.S.A., AddaBedia, 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
81 Abdelaziz, H.H., Ait Amar Meziane, M., Bousahla, A.A., Tounsi, A., Mahmoud, S.R. and Alwabli, A.S. (2017), "An efficient hyperbolic shear deformation theory for bending, buckling and free vibration of FGM sandwich plates with various boundary conditions", Steel Compos. Struct., 25(6), 693-704.   DOI