Browse > Article
http://dx.doi.org/10.12989/anr.2018.6.3.279

Dynamic analysis of higher order shear-deformable nanobeams resting on elastic foundation based on nonlocal strain gradient theory  

Bensaid, Ismail (IS2M Laboratory, Mechanical engineering Department, Faculty of Technology, University of Abou Beckr Belkaid (UABT))
Bekhadda, Ahmed (IS2M Laboratory, Mechanical engineering Department, Faculty of Technology, University of Abou Beckr Belkaid (UABT))
Kerboua, Bachir (Mechanical engineering Department, Faculty of Technology, University of Abou Beckr Belkaid (UABT))
Publication Information
Advances in nano research / v.6, no.3, 2018 , pp. 279-298 More about this Journal
Abstract
Present investigation deals with the free vibration characteristics of nanoscale-beams resting on elastic Pasternak's foundation based on nonlocal strain-gradient theory and a higher order hyperbolic beam model which captures shear deformation effect without using any shear correction factor. The nanobeam is lying on two-parameters elastic foundation consist of lower spring layers as well as a shear layer. Nonlocal strain gradient theory takes into account two scale parameters for modeling the small size effects of nanostructures more accurately. Hamilton's principal is utilized to derive the governing equations of embedded strain gradient nanobeam and, after that, analytical solutions are provided for simply supported conditions to solve the governing equations. The obtained results are compared with those predicted by the previous articles available in literature. Finally, the impacts of nonlocal parameter, length scale parameter, slenderness ratio, elastic medium, on vibration frequencies of nanosize beams are all evaluated.
Keywords
nanobeam; dynamic; nonlocal strain gradient elasticity; pasternak foundation;
Citations & Related Records
Times Cited By KSCI : 31  (Citation Analysis)
연도 인용수 순위
1 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., Int. J., 18(1), 235-253.   DOI
2 Hebali, H., Tounsi, A., Houari, M.S.A., Bessaim, A. and Adda Bedia, E.A. (2014), "A new quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates", ASCE J. Eng. Mech., 140, 374-383.   DOI
3 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, 2791-2799.   DOI
4 Kaci, A., Houari, M.S.A., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2018), "Post-buckling analysis of shear-deformable composite beams using a novel simple two-unknown beam theory", Struct. Eng. Mech., Int. J., 65(5), 621-631.
5 Simsek, M. (2016), "Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach", Int. J. Eng. Sci., 105, 12-27.   DOI
6 Thai, H-T. (2012), "A nonlocal beam theory for bending, buckling, and vibration of nanobeams", Int. J. Eng. Sci., 52, 56-64.   DOI
7 Thai, H-T. and Vo, T.P. (2012), "A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams", Int. J. Eng. Sci., 54, 58-66.   DOI
8 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., Int. J., 20(5), 963-981.   DOI
9 Abdelaziz, H., Ait Amar Meziane, M., Bousahla, A.A., Tounsi, A., Mahmoud, S.R. and Alwabli, 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., Int. J., 25(6), 693-704.
10 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
11 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., Int. J., 55(4), pp.743-762.   DOI
12 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
13 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., Int. J., 53(6), 1143-1165.   DOI
14 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
15 Tounsi, A., Houari, M.S.A. and Benyoucef, S. (2013c), "A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates", Aerosp. Sci. Technol., 24(1), 209-220.   DOI
16 Thai, S., Thai, H.T., Vo, T.P. and Patel, V.I. (2018), "A simple shear deformation theory for nonlocal beams". Comp. Struct., 183, 262-270.   DOI
17 Tounsi, A., Benguediab, S., Adda Bedia, E.A., Semmah, A. and Zidour, M. (2013a), "Nonlocal effects on thermal buckling properties of double-walled carbon nanotubes", Adv. Nano Res., Int. J., 1(1), 1-1.   DOI
18 Tounsi, A., Semmah, A. and Bousahla, A.A. (2013b), "Thermal buckling behavior of nanobeams using an efficient higher-order nonlocal beam theory", ASCE J. Nanomech. Micromech., 3(3), 37-42.   DOI
19 Tounsi, A., Benguediab, S., Houari, M.S.A. and Semmah, A. (2013d), "A new nonlocal beam theory with thickness stretching effect for nanobeams", Int. J. Nanosci., 12(4), 1350025.   DOI
20 Wang, Q. and Liew, K.M. (2007), "Application of nonlocal continuum mechanics to static analysis of micro-and nano structures", Phys. Lett. A, 363(3), 236-242.   DOI
21 Belabed, Z., Bousahla, A.A., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2018), "A new 3-unknown hyperbolic shear deformation theory for vibration of functionally graded sandwich plate", Earthq. Struct., Int. J., 14(2), 103-115.
22 Attia, A., Bousahla, A.A., Tounsi, A., Mahmoud, S.R. and Alwabli, A.S. (2018), "A refined four variable plate theory for thermoelastic analysis of FGM plates resting on variable elastic foundations", Struct. Eng. Mech., Int. J., 65(4), 453-464.
23 Behera, L. and Chakraverty, S. (2013), "Free vibration of Euler and Tomoshenko nanobeams using boundary characteristic orthogonal polynomials", Appl. Nanosci., 4(3), 347-358.   DOI
24 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
25 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., Int. J., 18(4), 755-786.   DOI
26 Karami, B., Janghorban, M., Shahsavari, D. and Tounsi, A. (2018c), "A size-dependent quasi-3D model for wave dispersion analysis of FG nanoplates", Steel Compos. Struct., Int. J., 28(1), 99-110.
27 Houari, M.S.A., Bessaim, A., Bernard, F., Tounsi, A. and Mahmoud, S.R. (2018), "Buckling analysis of new quasi-3D FG nanobeams based on nonlocal strain gradient elasticity theory and variable length scale parameter", Steel Compos. Struct., Int. J., 28(1), 13-24.
28 Karami, B., Janghorban, M. and Tounsi, A. (2017), "Effects of triaxial magnetic field on the anisotropic nanoplates", Steel Compos. Struct., Int. J., 25(3), 361-374.
29 Karami, B., Janghorban, M. and Tounsi, A. (2018a), "Nonlocal strain gradient 3D elasticity theory for anisotropic spherical nanoparticles", Steel Compos. Struct., Int. J., 27(2), 201-216.
30 Karami, B., Janghorban, M. and Tounsi, A. (2018b), "Variational approach for wave dispersion in anisotropic doubly-curved nanoshells based on a new nonlocal strain gradient higher order shell theory", Thin-Wall. Struct., 129, 251-264.   DOI
31 Karami, B., Shahsarvi, D., Janghorban, M. and Li, L. (2018d), "Wave dispersion of mounted graphene with initial stress", Thin-Wall. Struct., 122, 102-111.   DOI
32 Karami, B., Shahsarvi, D. and Li, L. (2018e), "Hygrothermal wave propagation in viscoelastic graphene under in-plane magnetic field based on nonlocal strain gradient theory", Phys. E: Low Dimens. Syst. Nanostruct., 97, 317-327.   DOI
33 Khalfi, Y., Houari, M.S.A. and Tounsi, A. (2014), "A refined and simple shear deformation theory for thermal buckling of solar functionally graded plates on elastic foundation", Int. J. Comput. Meth., 11(5), 135007.
34 Kheroubi, B., Benzair, A., Tounsi, A. and Semmah, A. (2016), "A new refined nonlocal beam theory accounting for effect of thickness stretching in nanoscale beams", Adv. Nano Res., Int. J., 4(4), 35-47.   DOI
35 Pradhan, S.C., Bazant, Z.P. and Murmu, T. (2009), "Small-scale effect on vibration analysis of singlewalled carbon nanotubes embedded in an elastic medium using nonlocal elasticity theory", J. Appl. Phys., 105, 124306.   DOI
36 Mokhtar, Y., Heireche, H., Tounsi, A., Bousahla, A.A. and Houari, M.S.A. (2018), "A novel shear deformation theory for buckling analysis of single layer graphene sheet based on nonlocal elasticity theory", Smart Struct. Syst., Int. J., 21(4), 397-405.
37 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., Int. J., 20(3), 369-383.
38 Murmu, T. and Adhikari, S. (2010), "Nonlocal transverse vibration of double nanobeamsystems", J. Appl. Phys., 108, 083514.   DOI
39 Nguyen, T.N., Thai, C.H. and Nguyen-Xuan, H. (2016), "On the general framework of high order shear deformation theories for laminated composite plate structures: A novel unified approach", Int. J. Mech. Sci., 110, 242-255.   DOI
40 Peddieson, J., Buchanan, G.R. and McNitt, R.P. (2003), "Application of nonlocal continuum models to nanotechnology", Int. J. Eng. Sci., 41(3-5), 305-312.   DOI
41 Reddy, J.N. (2007), "Nonlocal theories for bending, buckling and vibration of beams", Int. J. Eng. Sci., 5(2-8), 288-307.   DOI
42 Reddy, J.N. and Pang, S.D. (2008), "Nonlocal continuum theories of beams for the analysis of carbon nanotubes", J. Appl. Phys., 103, 023511-1-023511-16.   DOI
43 Shahsavari, D., Karami, B. and Tounsi, A. (2018), "Damped vibration of a graphene sheet using a higher order nonlocal strain-gradient Kirchhoff plate model", Compos. Rend. Mech. [In Press] DOI: 10.1016/j.crme.2018.08.011
44 Ebrahimi, F. and Barati, M. (2017c), "Size-dependent dynamic modeling of inhomogeneous curved nanobeams embedded in elastic medium based on nonlocal strain gradient theory", Proc. I. Mech. E Part C: J. Mech. Eng. Sci., 1-13. DOI: https://doi.org/10.1177/0954406216668912
45 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., Int. J., 11(5), 671-690.   DOI
46 Ebrahimi, F. and Barati, M.R. (2017a), "A Nonlocal Strain Gradient Rened Beam Model for Buckling Analysis of Sizedependent Shear-deformable Curved FG Nanobeams", Compos. Struct., 159, 174-182.   DOI
47 Ebrahimi, F. and Barati, M. (2017b), "Vibration analysis of embedded size dependent FG nanobeams based on third-order shear deformation beam theory", Struct. Eng. Mech., Int. J., 61(6), 721-736.   DOI
48 Ebrahimi, F. and Daman, M. (2017), "Analytical investigation of the surface effects on nonlocal vibration behavior of nanosize curved beams", Adv. Nano Res., Int. J., 4(1), 35-47.
49 Ebrahimi, F., Barati, M.R. and Dabbagh, A. (2016), "A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates", Int. J. Eng. Sci., 107, 169-182.   DOI
50 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., Int. J., 63(5), 585-595.
51 El Meiche, N., Tounsi, A., Ziane, N., Mechab, I. and Adda Bedia, E.A. (2011), "A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate", Int. J. Mech. Sci., 53(4), 237-247.   DOI
52 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., Int. J., 54(4), 693-710.   DOI
53 Eltaher, M.A., Khater, M.E., Park, S., Abdel-Rahman, E. and Yavuz, M. (2016), "On the static stability of nonlocal nanobeams using higher-order beam theories", Adv. Nano Res., Int. J., 4(1), 51-64.
54 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., Int. J., 21(1), 15-25.
55 Youcef, D.O., Kaci, A., Benzair, A., Bousahla, A.A. and Tounsi, A. (2018), "Dynamic analysis of nanoscale beams including surface stress effects", Smart Struct. Syst., Int. J., 21(1), 65-74.
56 Younsi, A., Tounsi, A, Zaoui, F.Z., Bousahla, A.A. and Mahmoud, S.R. (2018), "Novel quasi-3D and 2D shear deformation theories for bending and free vibration analysis of FGM plates", Geomech. Eng., Int. J., 4(6), 519-532.
57 Zeighampour, H., Beni, Y.T. and Karimipour, I. (2017) "Wave propagation in double-walled carbon nanotube conveying fluid considering slip boundary condition and shell model based on nonlocal strain gradient theory", Microfl Nanofl., 21(5), 85.   DOI
58 Zenkour, A.M. and Sobhy, M. (2015), "A simplified shear and normal deformations nonlocal theory for bending of nanobeams in thermal environment", Phys. E Low Dimens. Syst. Nanostruct., 70, 121-128.   DOI
59 Zhu, X. and Li, L. (2017a), "Closed form solution for a nonlocal strain gradient rod in tension", Int. J. Eng. Sci., 119, 16-28.   DOI
60 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., Int. J., 18(4), 1063-1081.   DOI
61 Li, L., Li, X. and Hu, Y. (2016a), "Free vibration analysis of nonlocal strain gradient beams made of functionally graded material", Int. J. Eng. Sci., 102, 77-92.   DOI
62 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., Int. J., 64(4), 391-402.
63 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. Solid, 51(8), 1477-1508.   DOI
64 Li, L. and Hu, Y. (2017), "Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects", Int. J. Mech. Sci., 120, 159-170.   DOI
65 Li, L., Hu, Y. and Li, X. (2016b), "Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory", Int. J. Mech. Sci., 115, 135-144.
66 Li, L., Tang, H. and Hu, Y. (2018), "The effect of thickness on the mechanics of nanobeams", Int. J. Eng. Sci., 123, 81-91.   DOI
67 Lim, C.W., Zhang, G. and Reddy, J.N. (2015), "A Higher-order Nonlocal Elasticity and Strain Gradient Theory and its Applications in Wave Propagation", J. Mech. Phys. Solids, 78, 298-313.   DOI
68 Zidi, M., Tounsi, A., Houari, M.S.A., Adda Bedia, E.A. and Anwar Beg, O. (2014), "Bending analysis of FGM plates under hygro-thermo-mechanical loading using a four variable refined plate theory", Aerosp. Sci. Tech., 34, 24-34.   DOI
69 Zhu, X. and Li, L. (2017b), "Twisting statics of functionally graded nanotubes using eringens nonlocal integral model", Compos. Struct., 178, 87-96.   DOI
70 Zhu, X. and Li, L. (2017c), "On longitudinal dynamics of nanorods", Int. J. Eng. Sci., 120, 129-145.   DOI
71 Zine, A. Tounsi, A. Draiche, K., Sekkal, M. and Mahmoud, S.R. (2018), "A novel higher-order shear deformation theory for bending and free vibration analysis of isotropic and multilayered plates and shells", Steel Compos. Struct., Int. J., 26(2), 125-137.
72 Benguediab, S., Tounsi, A., Zidour, M. and Semmah, A. (2014b), "Chirality and scale rffects on mechanical buckling properties of zigzag double-walled carbon nanotubes", Compos. Part B, 57, 21-24.   DOI
73 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(1), 265-275.   DOI
74 Bellifa, H., Benrahou, K.H., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2017), "A nonlocal zerothorder shear deformation theory for nonlinear postbuckling of nanobeams", Struct. Eng. Mech., Int. J., 62(6), 695-702.
75 Benguediab, S., Semmah, A., Larbi chaht, F., Mouaz, S. and Tounsi, A. (2014a), "An investigation on the characteristics of bending, buckling and vibration of nanobeams via nonlocal beam theory", Int. J. Compos. Meth., 11(6), pp.1350085-1-14.   DOI
76 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
77 Bensaid, I. (2017), "A refined nonlocal hyperbolic shear deformation beam model for bending and dynamic analysis of nanoscale beams", Adv. Nano Res., Int. J., 5(2), 113-126.   DOI
78 Bensaid, I. and Bekhadda, A. (2018), "Thermal stability analysis of temperature dependent inhomogeneous size-dependent nano-scale beams", Adv. Mater. Res., Int. J., 7(1), 363-378.
79 Bensaid, I., Cheikh, A., Mangouchi, A. and Kerboua, B. (2017), "Static deflection and dynamic behavior of higher-order hyperbolic shear deformable compositionally graded beams", Adv. Mater. Res., Int. J., 6(1), 13-26.
80 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. Model., 39, 2489-2508.   DOI
81 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., Int. J., 25(2), 157-175.
82 Besseghier, A., Houari, M.S.A., Tounsi, A. and Mahmoud, S. (2017), "Free vibration analysis of embedded nanosize FG plates using a new nonlocal trigonometric shear deformation theory", Smart Struct. Syst., Int. J., 19(6), 601-614.
83 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., Int. J., 48(3), 351-365.   DOI
84 Besseghier, A., Tounsi, A., Houari, M.S.A., Benzair, A., Boumia, L. and Heireche, H. (2011), "Thermal effect on wave propagation in double-walled carbon nanotubes embedded in a polymer matrix using nonlocal elasticity". Physica E, 43, 1379-1386.   DOI
85 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., Int. J., 3(1), 29-37.   DOI
86 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., Int. J., 19(2), 115-126.   DOI
87 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., Int. J., 14(1), 85-104.   DOI
88 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., Int. J., 58(3), 397-422.   DOI
89 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
90 Bouhadra, A., Tounsi, A., Bousahla, A.A., Benyoucef, S. and Mahmoud, S.R. (2018), "Improved HSDT accounting for effect of thickness stretching in advanced composite plates", Struct. Eng. Mech., Int. J., 66(1), 61-73.
91 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., Int. J., 20(2), 227-249.   DOI
92 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., Int. J., 18(2), 409-423.   DOI
93 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
94 Fourn, H., Ait Atmane, H., Bourada, M., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2018), "A novel four variable refined plate theory for wave propagation in functionally graded material plates", Steel Compos. Struct., Int. J., 27(1), 109-122.
95 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., Int. J., 60(2), 313-335.   DOI
96 Chakraverty, S. and Behera, L (2015), "Vibration and buckling analyses of nanobeams embedded in an elastic medium", Chinese Phys. B, 24(9), 097305.   DOI
97 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., Int. J., 19(3), 289-276.   DOI
98 Eringen, A.C. (1972), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10, 1-16.   DOI
99 Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54, 4703-4710.   DOI
100 Hachemi, H., Kaci, A., Houari, M.S.A., Bourada, A., Tounsi, A. and Mahmoud, S.R. (2017), "A new simple three unknown shear deformation theory for bending analysis of FG plates resting on elastic foundations", Steel Compos. Struct., Int. J., 25(6), 717-726.
101 Hadj Elmerabet, A., Heireche, A., Tounsi, A. and Semmah, A. (2017), "Buckling temperature of a singlewalled boron nitride nanotubes using a novel nonlocal beam model", Adv. Nano Res., Int. J., 5(1), 1-12.