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http://dx.doi.org/10.12989/sss.2020.25.2.197

A simple nth-order shear deformation theory for thermomechanical bending analysis of different configurations of FG sandwich plates  

Boussoula, Abderrafik (Laboratory of Mechanics of Structures and Solids (LMSS), Faculty of Technology, Department of Mechanical Engineering University)
Boucham, Belhadj (Laboratory of Mechanics of Structures and Solids (LMSS), Faculty of Technology, Department of Mechanical Engineering University)
Bourada, Mohamed (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department)
Bourada, Fouad (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department)
Tounsi, Abdeldjebbar (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department)
Bousahla, Abdelmoumen Anis (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department)
Tounsi, Abdelouahed (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department)
Publication Information
Smart Structures and Systems / v.25, no.2, 2020 , pp. 197-218 More about this Journal
Abstract
In this work, thermomechanical flexural analysis of functionally graded material sandwich plates with P-FGM face sheets and E-FGM and symmetric S-FGM core is performed by employing a nth-order shear deformation theory. A novel type of S-FGM sandwich plates, namely, both P-FGM face sheets and a symmetric S-FGM hard core are considered. By employing only four unknown variables, the governing equations are obtained based on the principle of virtual work and then Navier method is used to solve these equations. Analytical solutions are deduced to compute the stresses and deflections of simply supported S-FGM sandwich plates. The effects of volume fraction variation, geometrical parameters and thermal load on thermomechanical flexural behavior of the symmetric FGM sandwich plates are investigated.
Keywords
sandwich plate; thermomechanical; analytical modeling; functionally graded material;
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Times Cited By KSCI : 142  (Citation Analysis)
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83 Karami, B., Janghorban, M. and Tounsi, A. (2018a), "Galerkin's approach for buckling analysis of functionally graded anisotropic nanoplates/different boundary conditions", Eng. Comput., 35(4), 1297-1316. https://doi.org/10.1007/s00366-018-0664-9.
84 Rezaiee-Pajand, M., Masoodi, A.R. and Mokhtari, M. (2018), "Static analysis of functionally graded non-prismatic sandwich beams", Adv. Computat. Des., Int. J., 3(2), 165-190. https://doi.org/10.12989/acd.2018.3.2.165
85 Kadari, B., Bessaim, A., Tounsi, A., Heireche, H., Bousahla, A.A. and Houari, M.S.A. (2018), "Buckling analysis of orthotropic nanoscale plates resting on elastic foundations", J. Nano Res., 55, 42-56. https://doi.org/10.4028/www.scientific.net/JNanoR.55.42   DOI
86 Kar, V.R., Mahapatra, T.R. and Panda, S.K. (2017), "Effect of different temperature load on thermal postbuckling behaviour of functionally graded shallow curved shell panels", Compos. Struct., 160, 1236-1247. https://doi.org/10.1016/j.compstruct.2016.10.125   DOI
87 Karami, B. and Shahsavari, D. (2019), "Nonlocal strain gradient model for thermal stability of FG nanoplates integrated with piezoelectric layers", Smart Struct. Syst., Int. J., 23(3), 215-225. https://doi.org/10.12989/sss.2019.23.3.215
88 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. https://doi.org/10.12989/scs.2017.25.3.361
89 Karami, B., Janghorban, M., Shahsavari, D. and Tounsi, A. (2018b), "A size-dependent quasi-3D model for wave dispersion analysis of FG nanoplates", Steel Compos. Struct., Int. J., 28(1), 99-110. https://doi.org/10.12989/scs.2018.28.1.099
90 Bouderba, B., Houari, M.S.A.,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. https://doi.org/10.12989/sem.2016.58.3.397   DOI
91 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. https://doi.org/10.12989/sem.2018.66.1.061
92 Boukhari, A., Ait Atmane, H., Houari, M.S.A., 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., Int. J., 57(5), 837-859. https://doi.org/10.12989/sem.2016.57.5.837   DOI
93 Boukhlif, Z., Bouremana, M., Bourada, F., Bousahla, A.A., Bourada, M., Tounsi, A. and Al-Osta, M.A. (2019), "A simple quasi-3D HSDT for the dynamics analysis of FG thick plate on elastic foundation", Steel Compos. Struct., Int. J., 31(5), 503-516. https://doi.org/10.12989/scs.2019.31.5.503
94 Sekkal, M., Fahsi, B., Tounsi, A. and Mahmoud, S.R. (2017b), "A new quasi-3D HSDT for buckling and vibration of FG plate", Struct. Eng. Mech., Int. J., 64(6), 737-749. https://doi.org/10.12989/sem.2017.64.6.737
95 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., Int. J., 18(2), 425-442. https://doi.org/10.12989/scs.2015.18.2.425   DOI
96 Li, D., Deng, Z. and Xiao, H. (2016), "Thermomechanical bending analysis of functionally graded sandwich plates using fourvariable refined plate theory", Compos. Part B-Eng., 106, 107-119. https://doi.org/10.1016/j.compositesb.2016.08.041   DOI
97 Li, D., Deng, Z., Chen, G., Xiao, H. and Zhu, L. (2017), "Thermomechanical bending analysis of sandwich plates with both functionally graded face sheets and functionally graded core", Compos. Struct., 169, 29-41. https://doi.org/10.1016/j.compstruct.2017.01.026   DOI
98 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., Int. J., 11(2), 289-307. https://doi.org/10.12989/gae.2016.11.2.289   DOI
99 Sekkal, M., Fahsi, B., Tounsi, A. and Mahmoud, S.R. (2017a), "A novel and simple higher order shear deformation theory for stability and vibration of functionally graded sandwich plate", Steel Compos. Struct., Int. J., 25(4), 389-401. https://doi.org/10.12989/scs.2017.25.4.389
100 Selmi, A. and Bisharat, A. (2018), "Free vibration of functionally graded SWNT reinforced aluminum alloy beam", J. Vibroeng., 20(5), 2151-2164. https://doi.org/10.21595/jve.2018.19445   DOI
101 Shen, H.S. and Li, S.R. (2008), "Postbuckling of sandwich plates with FGM face sheets and temperature-dependent properties", Compos. Part B-Eng., 39(2), 332-344. https://doi.org/10.1016/j.compositesb.2007.01.004   DOI
102 Sobhy, M. (2013), "Buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions", Compos. Struct., 99, 76-87. https://doi.org/10.1016/j.compstruct.2012.11.018   DOI
103 Soliman, A.E., Eltaher, M.A., Attia, M.A. and Alshorbagy, A.E. (2018), "Nonlinear transient analysis of FG pipe subjected to internal pressure and unsteady temperature in a natural gas facility", Struct. Eng. Mech., Int. J., 66(1), 85-96. https://doi.org/10.12989/sem.2018.66.1.085
104 Benmansour, D.L., Kaci, A., Bousahla, A.A., Heireche, H., Tounsi, A., Alwabli, A.S., Alhebshi, A.M., Al-ghmady, K. and Mahmoud, S.R. (2019), "The nano scale bending and dynamic properties of isolated protein microtubules based on modified strain gradient theory", Adv. Nano Res., Int. J., 7(6), 443-457. https://doi.org/10.12989/anr.2019.7.6.443   DOI
105 Benchohra, M., Driz, H., Bakora, A., Tounsi, A., Adda Bedia, E.A. and Mahmoud, S.R. (2018), "A new quasi-3D sinusoidal shear deformation theory for functionally graded plates", Struct. Eng. Mech., Int. J., 65(1), 19-31. https://doi.org/10.12989/sem.2018.65.1.019
106 Boulefrakh, L., Hebali, H., Chikh, A., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2019), "The effect of parameters of visco-Pasternak foundation on the bending and vibration properties of a thick FG plate", Geomech. Eng., Int. J., 18(2), 161-178. https://doi.org/10.12989/gae.2019.18.2.161
107 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. https://doi.org/10.12989/scs.2016.20.2.227   DOI
108 Bourada, M., Tounsi, A., Houari, M.S.A. and Adda Bedia, E.A. (2012), "A new four-variable refined plate theory for thermal buckling analysis of functionally graded sandwich plates", J. Sandw. Struct. Mater., 14(1), 5-33. https://doi.org/10.1177/1099636211426386   DOI
109 Bendaho, B., Belabed, Z., Bourada, M., Benatta, M.A., Bourada, F. and Tounsi, A. (2019), "Assessment of new 2D and quasi-3D Nonlocal theories for free vibration analysis of size-dependent functionally graded (FG) nanoplates", Adv. Nano Res., Int. J., 7(4), 277-292. https://doi.org/10.12989/anr.2019.7.4.277
110 Benferhat, R., Hassaine Daouadji, T., Hadji, L. and Said Mansour, M. (2016), "Static analysis of the FGM plate with porosities", Steel Compos. Struct., Int. J., 21(1), 123-136. https://doi.org/10.12989/scs.2016.21.1.123   DOI
111 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. https://doi.org/10.1080/15376494.2014.984088   DOI
112 Bensaid, I., Bekhadda, A. and Kerboua, B. (2018), "Dynamic analysis of higher order shear-deformable nanobeams resting on elastic foundation based on nonlocal strain gradient theory", Adv. Nano Res., Int. J., 6(3), 279-298. https://doi.org/10.12989/anr.2018.6.3.279
113 Bensattalah, T., Zidour, M. and Daouadji, T.S. (2019), "A new nonlocal beam model for free vibration analysis of chiral singlewalled carbon nanotubes", Compos. Mater. Eng., Int. J., 1(1), 21-31. https://doi.org/10.12989/cme.2019.1.1.021
114 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. https://doi.org/10.1177/1099636214554904   DOI
115 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. https://doi.org/10.12989/scs.2017.25.6.717
116 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. https://doi.org/10.12989/scs.2015.18.1.235   DOI
117 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(2), 374-383. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000665   DOI
118 Librescu, L. and Hause, T. (2000), "Recent developments in the modeling and behavior of advanced sandwich constructions: a survey", Compos. Struct., 48(1-3), 1-17. https://doi.org/10.1016/S0263-8223(99)00068-9   DOI
119 Lindstrom, A. and Hallstrom, S. (2010), "Energy absorption of SMC/balsa sandwich panels with geometrical triggering features", Compos. Struct., 92(11), 2676-2684. https://doi.org/10.1016/j.compstruct.2010.03.018   DOI
120 Swaminathan, K., Naveenkumar, D.T., Zenkour, A.M. and Carrera, E. (2015), "Stress, vibration and buckling analyses of FGM plates-A state-of-the-art review", Compos. Struct., 120, 10-31. https://doi.org/10.1016/j.compstruct.2014.09.070   DOI
121 Tounsi, A., Houari, M.S.A. and Benyoucef, S. (2013), "A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates", Aerosp. Sci. Technol., 24(1), 209-220. https://doi.org/10.1016/j.ast.2011.11.009   DOI
122 Tounsi, A., Ait Atmane, H., Khiloun, M., Sekkal, M., Taleb, O. and Bousahla, A.A. (2019), "On buckling behavior of thick advanced composite sandwich plates", Compos. Mater. Eng., Int. J., 1(1), 1-19. https://doi.org/10.12989/cme.2019.1.1.001
123 Vinson, J.R. (2001), "Sandwich structures", Appl. Mech. Rev., 54(3), 201-214. https://doi.org/10.1115/1.3097295   DOI
124 Vinson, J.R. (2005), "Sandwich structures: past, present, and future", In: Sandwich Structures 7: Advancing with Sandwich Structures and Materials, (O.T. Thomsen, E. Bozhevolnaya, A. Lyckegaard Eds.), Springer, Netherlands, pp. 3-12.
125 Wang, Z.X. and Shen, H.S. (2011), "Nonlinear analysis of sandwich plates with FGM face sheets resting on elastic foundations", Compos. Struct., 93(10), 2521-2532. https://doi.org/10.1016/j.compstruct.2011.04.014   DOI
126 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. https://doi.org/10.12989/sem.2015.53.6.1143   DOI
127 Houari, M.S.A., Tounsi, A. and Anwar Beg, O. (2013), "Thermoelastic bending analysis of functionally graded sandwich plates using a new higher order shear and normal deformation theory", Int. J. Mech. Sci., 76, 102-111. https://doi.org/10.1016/j.ijmecsci.2013.09.004   DOI
128 Chaabane, L.A., Bourada, F., Sekkal, M., Zerouati, S., Zaoui, F.Z., Tounsi, A., Derras, A., Bousahla, A.A. and Tounsi, A. (2019), "Analytical study of bending and free vibration responses of functionally graded beams resting on elastic foundation", Struct. Eng. Mech., Int. J., 71(2), 185-196. https://doi.org/10.12989/sem.2019.71.2.185
129 Hellal, H., Bourada, M., Hebali, H., Bourada, F., Tounsi, A., Bousahla, A.A. and Mahmoud, S.R. (2019), "Dynamic and stability analysis of functionally graded material sandwich plates in hygro-thermal environment using a simple higher shear deformation theory", J. Sandw. Struct. Mater. https://doi.org/10.1177/1099636219845841
130 Houari, M.S.A., Benyoucef, S., Mechab, I., Tounsi, A. and Bedia, E.A.A. (2011), "Two-variable refined plate theory for thermoelastic bending analysis of functionally graded sandwich plates", J. Therm. Stress., 34(4), 315-334. https://doi.org/10.1080/01495739.2010.550806   DOI
131 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., Int. J., 22(2), 257-276. https://doi.org/10.12989/scs.2016.22.2.257   DOI
132 Mantari, J.L. and Granados, E.V. (2015b), "Thermoelastic analysis of advanced sandwich plates based on a new quasi-3D hybrid type HSDT with 5 unknowns", Compos. Part B-Eng., 69, 317-334. https://doi.org/10.1016/j.compositesb.2014.10.009   DOI
133 Liu, B., Ferreira, A.J.M., Xing, Y.F. and Neves, A.M.A. (2016), "Analysis of functionally graded sandwich and laminated shells using a layerwise theory and a differential quadrature finite element method", Compos Struct., 136, 546-553. https://doi.org/10.1016/j.compstruct.2015.10.044   DOI
134 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(9), 2489-2508. https://doi.org/10.1016/j.apm.2014.10.045   DOI
135 Mantari, J.L. and Granados, E.V. (2015a), "Thermoelastic behavior of advanced composite sandwich plates by using a new 6 unknown quasi-3D hybrid type HSDT", Compos. Struct., 126, 132-144. https://doi.org/10.1016/j.compstruct.2015.01.055   DOI
136 Mantari, J.L. and Monge, J.C. (2016), "Buckling, free vibration and bending analysis of functionally graded sandwich plates based on an optimized hyperbolic unified formulation", Int. J. Mech. Sci., 119, 170-186. https://doi.org/10.1016/j.ijmecsci.2016.10.015   DOI
137 Medani, M., Benahmed, A., Zidour, M., Heireche, H., Tounsi, A., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2019), "Static and dynamic behavior of (FG-CNT) reinforced porous sandwich plate", Steel Compos. Struct., Int. J., 32(5), 595-610. https://doi.org/10.12989/scs.2019.32.5.595
138 Daouadji, T.H. (2017), "Analytical and numerical modeling of interfacial stresses in beams bonded with a thin plate", Adv. Computat. Des., Int. J., 2(1), 57-69. https://doi.org/10.12989/acd.2017.2.1.057
139 Cherif, R.H., Meradjah, M., Zidour, M., Tounsi, A., Belmahi, H. and Bensattalah, T. (2018), "Vibration analysis of nano beam using differential transform method including thermal effect", J. Nano Res., 54, 1-14. https://doi.org/10.4028/www.scientific.net/JNanoR.54.1   DOI
140 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), 289-297. https://doi.org/10.12989/sss.2017.19.3.289   DOI
141 Dean, J., Fallah, A.S., Brown, P.M., Louca, L.A. and Clyne, T.W. (2011), "Energy absorption during projectile perforation of lightweight sandwich panels with metallic fibre cores", Compos. Struct., 93(3), 1089-1095. https://doi.org/10.1016/j.compstruct.2010.09.019   DOI
142 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. https://doi.org/10.12989/gae.2016.11.5.671   DOI
143 Draiche, K., Bousahla, A.A., Tounsi, A., Alwabli, A.S., Tounsi, A. and Mahmoud, S.R. (2019), "Static analysis of laminated reinforced composite plates using a simple first-order shear deformation theory", Comput. Concrete, Int. J., 24(4), 369-378. https://doi.org/10.12989/cac.2019.24.4.369
144 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., 14(6), 519-532. https://doi.org/10.12989/gae.2018.14.6.519
145 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. https://doi.org/10.12989/sss.2018.21.1.015
146 Yeghnem, R., Guerroudj, H.Z., Amar, L.H.H., Meftah, S.A., Benyoucef, S., Tounsi, A. and Adda Bedia, E.A. (2017), "Numerical modeling of the aging effects of RC shear walls strengthened by CFRP plates: A comparison of results from different "code type" models", Comput. Concrete, Int. J., 19(5), 579-588. https://doi.org/10.12989/cac.2017.19.5.579   DOI
147 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. https://doi.org/10.12989/sss.2018.21.1.065
148 Zaoui, F.Z., Ouinas, D. and Tounsi, A. (2019), "New 2D and quasi-3D shear deformation theories for free vibration of functionally graded plates on elastic foundations", Compos. Part B, 159, 231-247. https://doi.org/10.1016/j.compositesb.2018.09.051   DOI
149 Zarga, D., Tounsi, A., Bousahla, A.A., Bourada, F. and Mahmoud, S.R. (2019), "Thermomechanical bending study for functionally graded sandwich plates using a simple quasi-3D shear deformation theory", Steel Compos. Struct., Int. J., 32(3), 389-410. https://doi.org/10.12989/scs.2019.32.3.389
150 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. https://doi.org/10.12989/sem.2015.54.4.693   DOI
151 Addou, F.Y., Meradjah, M., M.A.A, Bousahla, Benachour, A., Bourada, F., Tounsi, A. and Mahmoud, S.R. (2019), "Influences of porosity on dynamic response of FG plates resting on Winkler/Pasternak/Kerr foundation using quasi 3D HSDT", Comput. Concrete, Int. J., 24(4), 347-367. https://doi.org/10.12989/cac.2019.24.4.347
152 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., Int. J., 25(6), 693-704. https://doi.org/10.12989/scs.2017.25.6.693
153 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. https://doi.org/10.1016/j.compstruct.2017.10.047   DOI
154 Adda Bedia, W., Houari, M.S.A., Bessaim, A., Bousahla, A.A., Tounsi, A., Saeed, T. and Alhodaly, M.S. (2019), "A new hyperbolic two-unknown beam model for bending and buckling analysis of a nonlocal strain gradient nanobeams", J. Nano Res., 57, 175-191. https://doi.org/10.4028/www.scientific.net/JNanoR.57.175   DOI
155 Ahmed, A. (2014), "Post buckling analysis of sandwich beams with functionally graded faces using a consistent higher order theory", Int. J. Civil Struct. Environ., 4(2), 59-64.
156 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., Int. J., 18(1), 187-212. https://doi.org/10.12989/scs.2015.18.1.187   DOI