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http://dx.doi.org/10.12989/scs.2022.42.5.599

Computational mathematical modeling of the nonlinear vibration characteristics of AFG truncated conical nano pipe based on the nonlocal strain gradient theory  

Zhang, Ruihua (College of Mechanical Engineering, Nantong Vocational University)
Cao, Yiqing (School of Mechanical and Electrical Engineering, Putian University)
Publication Information
Steel and Composite Structures / v.42, no.5, 2022 , pp. 599-615 More about this Journal
Abstract
In the present paper, the numerical dynamic analysis of a functionally graded nano-scale nonuniform tube was investigated according to the high-order beam theory coupled with the nonlocal gradient strain theory. The supposed cross-section is changed along the pipe length, and the material distribution, which combines both metal and ceramics, is smoothly changed in the pipe length direction, which is called axially functionally graded (AFG) pipe. Moreover, the porosity voids are dispersed in the cross-section and the radial pattern that the existence of both material distribution along the tube length and porosity voids make a two-dimensional functionally graded (2D-FG) truncated conical pipe. On the basis of the Hamilton principle, the governing equations and the associated boundary conditions equations are derived, and then a numerical approach is applied to solve the obtained equations.
Keywords
2D-FG; AFG; dynamic analysis; nonlocal strain gradient theory; nonuniform pipe; porous material;
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87 Lee, H.-L. and Chang, W.-J. (2008), "Free transverse vibration of the fluid-conveying single-walled carbon nanotube using nonlocal elastic theory", J. Appl. Phys., 103(2), 024302. https://doi.org/10.1063/1.2822099.   DOI
88 Li, Y., Li, S., Guo, K., Fang, X. and Habibi, M. (2020b), "On the modeling of bending responses of graphene-reinforced higher order annular plate via two-dimensional continuum mechanics approach", Eng. Comput., 2020, https://doi.org/10.1007/s00366-020-01166-w.   DOI
89 Zhao, J. and Yu, Z. (2021), "On the modeling and simulation of the nonlinear dynamic response of NEMS via a couple of nonlocal strain gradient theory and classical beam theory", Adv. Nano Res., 11(5), 547-563. https://doi.org/10.12989/ANR.2021.11.5.547.   DOI
90 Zhang, X., Tang, Y., Zhang, F. and Lee, C.-S. (2016), "A Novel Aluminum-Graphite Dual-Ion Battery", Adv. Energy Mater., 6(11), 1502588. https://doi.org/10.1002/aenm.201502588.   DOI
91 Zhao, J., Choe, K., Shuai, C., Wang, A. and Wang, Q. (2019), "Free vibration analysis of functionally graded carbon nanotube reinforced composite truncated conical panels with general boundary conditions", Compos. B Eng., 160, 225-240. https://doi.org/10.1016/j.compositesb.2018.09.105.   DOI
92 Moradi-Dastjerdi, R. and Payganeh, G. (2017), "Thermoelastic dynamic analysis of wavy carbon nanotube reinforced cylinders under thermal loads", Steel Compos. Struct., 25(3), 315-326. https://doi.org/10.12989/scs.2017.25.3.315.   DOI
93 Mueller, E., Drasar, C., Schilz, J. and Kaysser, W. (2003), "Functionally graded materials for sensor and energy applications", Mater. Sci. Eng. A., 362(1-2), 17-39. https://doi.org/10.1016/S0921-5093(03)00581-1.   DOI
94 Murmu, T. and Pradhan, S.C. (2009), "Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory", Comput. Mater. Sci., 46(4), 854-859. https://doi.org/10.1016/j.commatsci.2009.04.019.   DOI
95 Najaafi, N., Jamali, M., Habibi, M., Sadeghi, S., Jung, D.w. and Nabipour, N. (2021), "Dynamic instability responses of the substructure living biological cells in the cytoplasm environment using stress-strain size-dependent theory", J. Biomolecular Struct. Dynam., 39(7), 2543-2554. https://doi.org/10.1080/07391102.2020.1751297.   DOI
96 Ma, L., Liu, X. and Moradi, Z. (2021), "On the chaotic behavior of graphene-reinforced annular systems under harmonic excitation", Eng. Comput., 2021, https://doi.org/10.1007/s00366-020-01210-9.   DOI
97 Liu, H., Zhao, Y., Pishbin, M., Habibi, M., Bashir, M.O. and Issakhov, A. (2021b), "A comprehensive mathematical simulation of the composite size-dependent rotary 3D microsystem via two-dimensional generalized differential quadrature method", Eng. Comput., 2021, https://doi.org/10.1007/s00366-021-01419-2.   DOI
98 Liu, Y. (2020), "Marine Oil Spill Control Based on Discrete Mathematical Model", J. Coastal Res., 103(SI), 387-391. https://doi.org/10.2112/SI103-079.1.   DOI
99 Liu, Z., Su, S., Xi, D. and Habibi, M. (2020a), "Vibrational responses of a MHC viscoelastic thick annular plate in thermal environment using GDQ method", Mech. Based Design Struct. Machines, 2020, 1-26. https://doi.org/10.1080/15397734.2020.1784201.   DOI
100 Ghadiri, M., Shafiei, N. and Alavi, H. (2017c), "Vibration analysis of a rotating nanoplate using nonlocal elasticity theory", J. Solid Mech., 9(2), 319-337. https://doi.org/20.1001.1.20083505.2017.9.2.8.5.
101 Zhao, Y., Zhu, Y. and Song, J. (2021b), "Analytical modeling of the linear and nonlinear dynamic characteristics of the nonuniform axially functionally graded cylindrical beam based on the strain gradient theory", Waves Random Complex Media, 1-32. https://doi.org/10.1080/17455030.2021.1965672.   DOI
102 Zhen, Y., Gong, Y. and Tang, Y. (2021), "Nonlinear vibration analysis of a supercritical fluid-conveying pipe made of functionally graded material with initial curvature", Compos. Struct., 268, 113980. https://doi.org/10.1016/j.compstruct.2021.113980.   DOI
103 Zhao, Y., Moradi, Z., Davoudi, M. and Zhuang, J. (2021a), "Bending and stress responses of the hybrid axisymmetric system via state-space method and 3D-elasticity theory", Eng. Comput., 2021, https://doi.org/10.1007/s00366-020-01242-1.   DOI
104 Oyarhossein, M.A., Alizadeh, A.a., Habibi, M., Makkiabadi, M., Daman, M., Safarpour, H. and Jung, D.W. (2020), "Dynamic response of the nonlocal strain-stress gradient in laminated polymer composites microtubes", Scientific Reports, 10(1), 5616. https://doi.org/10.1038/s41598-020-61855-w.   DOI
105 Peng, D., Chen, S., Darabi, R., Ghabussi, A. and Habibi, M. (2021), "Prediction of the bending and out-of-plane loading effects on formability response of the steel sheets", Archives Civil Mech. Eng., 21(2), 74. https://doi.org/10.1007/s43452-021-00227-1.   DOI
106 Moradi-Dastjerdi, R. and Payganeh, G. (2018), "Thermoelastic vibration analysis of functionally graded wavy carbon nanotube-reinforced cylinders", Polymer Compos., 39(S2), E826-E834. https://doi.org/10.1002/pc.24278.   DOI
107 Shafiei, N., Kazemi, M. and Ghadiri, M. (2016c), "Nonlinear vibration of axially functionally graded tapered microbeams", J. Eng. Sci., 102, 12-26. https://doi.org/10.1016/j.ijengsci.2016.02.007.   DOI
108 Ebrahimi, F. and Shafiei, N. (2017), "Influence of initial shear stress on the vibration behavior of single-layered graphene sheets embedded in an elastic medium based on Reddy's higher-order shear deformation plate theory", Mech. Adv. Mater. Struct., 24(9), 761-772. https://doi.org/10.1080/15376494.2016.1196781.   DOI
109 Zhong, R., Wang, Q., Tang, J., Shuai, C. and Qin, B. (2018), "Vibration analysis of functionally graded carbon nanotube reinforced composites (FG-CNTRC) circular, annular and sector plates", Compos. Struct., 194, 49-67. https://doi.org/10.1016/j.compstruct.2018.03.104.   DOI
110 Zhou, C., Zhao, Y., Zhang, J., Fang, Y. and Habibi, M. (2020), "Vibrational characteristics of multi-phase nanocomposite reinforced circular/annular system", Adv. Nano Res., 9(4), 295-307. https://doi.org/10.12989/ANR.2020.9.4.295.   DOI
111 Habibi, M., Hashemi, R., Fallah Tafti, M. and Assempour, A. (2018), "Experimental investigation of mechanical properties, formability and forming limit diagrams for tailor-welded blanks produced by friction stir welding", J. Manufacturing Processes, 31, 310-323. https://doi.org/10.1016/j.jmapro.2017.11.009.   DOI
112 Ghadiri, M., Shafiei, N. and Safarpour, H. (2017e), "Influence of surface effects on vibration behavior of a rotary functionally graded nanobeam based on Eringen's nonlocal elasticity", Microsyst. Technol., 23(4), 1045-1065. https://doi.org/10.1007/s00542-016-2822-6.   DOI
113 Ghadiri, M., Shafiei, N., Salekdeh, S.H., Mottaghi, P. and Mirzaie, T. (2016d), "Investigation of the dental implant geometry effect on stress distribution at dental implant-bone interface", J. Brazilian Soc. Mech. Sci. Eng., 38(2), 335-343. https://doi.org/10.1007/s40430-015-0472-8.   DOI
114 Ghazanfari, A., Soleimani, S.S., Keshavarzzadeh, M., Habibi, M., Assempuor, A. and Hashemi, R. (2020), "Prediction of FLD for sheet metal by considering through-thickness shear stresses", Mech. Based Design Struct. Machines, 48(6), 755-772. https://doi.org/10.1080/15397734.2019.1662310.   DOI
115 Habibi, M., Hashemi, R., Ghazanfari, A., Naghdabadi, R. and Assempour, A. (2016), "Forming limit diagrams by including the M-K model in finite element simulation considering the effect of bending", Proc. Institution of Mech. Engineers, Part L: J. Mater.: Design Appl., 232(8), 625-636. https://doi.org/10.1177/1464420716642258.   DOI
116 Gheshlaghi, B. and Hasheminejad, S.M. (2011), "Surface effects on nonlinear free vibration of nanobeams", Compos. B Eng., 42(4), 934-937. https://doi.org/10.1016/j.compositesb.2010.12.026.   DOI
117 Zine, A., Bousahla, A.A., Bourada, F., Benrahou, K.H., Tounsi, A., Adda Bedia, E.A., Mahmoud, S.R. and Tounsi, A. (2020), "Bending analysis of functionally graded porous plates via a refined shear deformation theory", Comput. Concrete, 26(1), 63-74. https://doi.org/10.12989/cac.2020.26.1.063.   DOI
118 Pompe, W., Worch, H., Epple, M., Friess, W., Gelinsky, M., Greil, P., Hempel, U., Scharnweber, D. and Schulte, K. (2003), "Functionally graded materials for biomedical applications", Mater. Sci. Eng. A., 362(1-2), 40-60. https://doi.org/10.1016/S0921-5093(03)00580-X.   DOI
119 Shafiei, N., Mousavi, A. and Ghadiri, M. (2016g), "Vibration behavior of a rotating non-uniform FG microbeam based on the modified couple stress theory and GDQEM", Compos. Struct., 149, 157-169. https://doi.org/10.1016/j.compstruct.2016.04.024.   DOI
120 Shen, Z.-B., Li, X.-F., Sheng, L.-P. and Tang, G.-J. (2012), "Transverse vibration of nanotube-based micro-mass sensor via nonlocal Timoshenko beam theory", Comput. Mater. Sci., 53(1), 340-346. https://doi.org/10.1016/j.commatsci.2011.09.023.   DOI
121 Simsek, M. (2011), "Forced vibration of an embedded single-walled carbon nanotube traversed by a moving load using nonlocal Timoshenko beam theory", Steel Compos. Struct., 11(1), 59-76. https://doi.org/10.12989/scs.2011.11.1.059.   DOI
122 Sola, A., Bellucci, D. and Cannillo, V. (2016), "Functionally graded materials for orthopedic applications-an update on design and manufacturing", Biotech. Adv., 34(5), 504-531. https://doi.org/10.1016/j.biotechadv.2015.12.013.   DOI
123 Shariati, A., Jung, D.w., Mohammad-Sedighi, H., Zur, K.K., Habibi, M. and Safa, M. (2020a), "On the Vibrations and Stability of Moving Viscoelastic Axially Functionally Graded Nanobeams", Materials, 13(7), 1707. https://doi.org/10.3390/ma13071707.   DOI
124 Guo, J., Baharvand, A., Tazeddinova, D., Habibi, M., Safarpour, H., Roco-Videla, A. and Selmi, A. (2021a), "An intelligent computer method for vibration responses of the spinning multilayer symmetric nanosystem using multi-physics modeling", Eng. Comput., https://doi.org/10.1007/s00366-021-01433-4.   DOI
125 Ehyaei, J., Akbarshahi, A. and Shafiei, N.J.A.i.n.r. (2017), "Influence of porosity and axial preload on vibration behavior of rotating FG nanobeam", 5(2), 141. https://doi.org/10.12989/anr.2017.5.2.141.   DOI
126 Cheshmeh, E., Karbon, M., Eyvazian, A., Jung, D.w., Habibi, M. and Safarpour, M. (2020), "Buckling and vibration analysis of FG-CNTRC plate subjected to thermo-mechanical load based on higher order shear deformation theory", Mech. Based Design Struct. Machines, 1-24. https://doi.org/10.1080/15397734.2020.1744005.   DOI
127 Shafiei, N., Mirjavadi, S.S., Afshari, B.M., Rabby, S. and Hamouda, A.M.S. (2017c), "Nonlinear thermal buckling of axially functionally graded micro and nanobeams", Compos. Struct., 168, 428-439. https://doi.org/10.1016/j.compstruct.2017.02.048.   DOI
128 Ansari, R., Mohammadi, V., Faghih Shojaei, M., Gholami, R. and Rouhi, H. (2014), "Nonlinear vibration analysis of Timoshenko nanobeams based on surface stress elasticity theory", European J. Mech. A/Solids, 45, 143-152. https://doi.org/10.1016/j.euromechsol.2013.11.002.   DOI
129 Azimi, M., Mirjavadi, S.S., Shafiei, N., Hamouda, A.M.S. and Davari, E. (2018), "Vibration of rotating functionally graded Timoshenko nano-beams with nonlinear thermal distribution", Mech. Adv. Mater. Struct., 25(6), 467-480. https://doi.org/10.1080/15376494.2017.1285455.   DOI
130 Balubaid, M., Tounsi, A., Dakhel, B. and Mahmoud, S.R. (2019), "Free vibration investigation of FG nanoscale plate using nonlocal two variables integral refined plate theory", Comput. Concrete, 24(6), 579-586. https://doi.org/10.12989/cac.2019.24.6.579.   DOI
131 Civalek, O. (2020), "Vibration of functionally graded carbon nanotube reinforced quadrilateral plates using geometric transformation discrete singular convolution method", J. Numerical Methods Eng., 121(5), 990-1019. https://doi.org/10.1002/nme.6254.   DOI
132 Al-Furjan, M.S.H., hatami, A., Habibi, M., Shan, L. and Tounsi, A. (2021), "On the vibrations of the imperfect sandwich higher-order disk with a lactic core using generalize differential quadrature method", Compos. Struct., 257, 113150. https://doi.org/10.1016/j.compstruct.2020.113150.   DOI
133 He, X., Ding, J., Habibi, M., Safarpour, H. and Safarpour, M. (2021), "Non-polynomial framework for bending responses of the multi-scale hybrid laminated nanocomposite reinforced circular/annular plate", Thin-Walled Struct., 166, 108019. https://doi.org/10.1016/j.tws.2021.108019.   DOI
134 Tang, Y. and Yang, T. (2018a), "Bi-Directional Functionally Graded Nanotubes: Fluid Conveying Dynamics", J. Appl. Mech., 10(04), 1850041. https://doi.org/10.1142/S1758825118500412.   DOI
135 Dai, Z., Zhang, L., Bolandi, S.Y. and Habibi, M. (2021b), "On the vibrations of the non-polynomial viscoelastic composite open-type shell under residual stresses", Compos. Struct., 263, 113599. https://doi.org/10.1016/j.compstruct.2021.113599.   DOI
136 Shafiei, N., Hamisi, M. and Ghadiri, M. (2020), "Vibration analysis of rotary tapered axially functionally graded Timoshenko nanobeam in thermal environment", J. Solid Mech., 12(1), 16-32.
137 Shafiei, N., Kazemi, M. and Fatahi, L. (2017b), "Transverse vibration of rotary tapered microbeam based on modified couple stress theory and generalized differential quadrature element method", Mech. Adv. Mater. Struct., 24(3), 240-252. https://doi.org/10.1080/15376494.2015.1128025.   DOI
138 Shafiei, N., Kazemi, M. and Ghadiri, M. (2016b), "Nonlinear vibration behavior of a rotating nanobeam under thermal stress using Eringen's nonlocal elasticity and DQM", Appl. Phys. A, 122(8), 728. https://doi.org/10.1007/s00339-016-0245-y.   DOI
139 Ghabussi, A., Habibi, M., NoormohammadiArani, O., Shavalipour, A., Moayedi, H. and Safarpour, H. (2020), "Frequency characteristics of a viscoelastic graphene nanoplatelet-reinforced composite circular microplate", J. Vib. Control, 27(1-2), 101-118. https://doi.org/10.1177/1077546320923930.   DOI
140 Hussain, M., Naeem, M.N., Tounsi, A. and Taj, M. (2019), "Nonlocal effect on the vibration of armchair and zigzag SWCNTs with bending rigidity", Adv. Nano Res., 7(6), 431-442. https://doi.org/10.12989/anr.2019.7.6.431.   DOI
141 Guellil, M., Saidi, H., Bourada, F., Bousahla, A.A., Tounsi, A., AlZahrani, M.M., Hussain, M. and Mahmoud, S. (2021), "Influences of porosity distributions and boundary conditions on mechanical bending response of functionally graded plates resting on Pasternak foundation", Steel Compos. Struct., 38(1), 1-15. https://doi.org/10.12989/scs.2021.38.1.001.   DOI
142 Liu, Z., Wu, X., Yu, M. and Habibi, M. (2020b), "Large-amplitude dynamical behavior of multilayer graphene platelets reinforced nanocomposite annular plate under thermo-mechanical loadings", Mech. Based Design Struct. Machines, 2020, 1-25. https://doi.org/10.1080/15397734.2020.1815544.   DOI
143 Moayedi, H., Darabi, R., Ghabussi, A., Habibi, M. and Foong, L.K. (2020b), "Weld orientation effects on the formability of tailor welded thin steel sheets", Thin-Walled Struct., 149, 106669. https://doi.org/10.1016/j.tws.2020.106669.   DOI
144 Simsek, M. (2019), "Some closed-form solutions for static, buckling, free and forced vibration of functionally graded (FG) nanobeams using nonlocal strain gradient theory", Compos. Struct., 224, 111041. https://doi.org/10.1016/j.compstruct.2019.111041.   DOI
145 Addou, F.Y., Meradjah, M., Bousahla, A.A., 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, 24(4), 347-367. https://doi.org/10.12989/cac.2019.24.4.347.   DOI
146 Shafiei, N., Kazemi, M. and Ghadiri, M. (2016d), "On sizedependent vibration of rotary axially functionally graded microbeam", J. Eng. Sci., 101, 29-44. https://doi.org/10.1016/j.ijengsci.2015.12.008.   DOI
147 Shafiei, N., Kazemi, M., Safi, M. and Ghadiri, M. (2016e), "Nonlinear vibration of axially functionally graded non-uniform nanobeams", J. Eng. Sci., 106, 77-94. https://doi.org/10.1016/j.ijengsci.2016.05.009.   DOI
148 Tagrara, S., Benachour, A., Bouiadjra, M.B. 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. https://doi.org/10.12989/scs.2015.19.5.1259.   DOI
149 Tang, Y. and Yang, T. (2018b), "Post-buckling behavior and nonlinear vibration analysis of a fluid-conveying pipe composed of functionally graded material", Compos. Struct., 185, 393-400. https://doi.org/10.1016/j.compstruct.2017.11.032.   DOI
150 Srinivas, P., Babu, P.R. and Balakrishna, B. (2019), "Effect of silicon carbide, magnesium oxide as reinforcing elements and zinc sterate as binding agent in the characterization of Al functionally graded materials for automotive applications", Mater. Today: Proc., 27, 460-466. https://doi.org/10.1016/j.matpr.2019.11.275.   DOI
151 Shafiei, N., Mirjavadi, S.S., MohaselAfshari, B., Rabby, S. and Kazemi, M. (2017d), "Vibration of two-dimensional imperfect functionally graded (2D-FG) porous nano-/micro-beams", Comput. Methods Appl. Mech. Eng., 322, 615-632. https://doi.org/10.1016/j.cma.2017.05.007.   DOI
152 Shao, Y., Zhao, Y., Gao, J. and Habibi, M. (2021), "Energy absorption of the strengthened viscoelastic multi-curved composite panel under friction force", Archives Civil Mech. Eng., 21(4), 141. https://doi.org/10.1007/s43452-021-00279-3.   DOI
153 Hou, F., Wu, S., Moradi, Z. and Shafiei, N. (2021), "The computational modeling for the static analysis of axially functionally graded micro-cylindrical imperfect beam applying the computer simulation", Eng. Comput., https://doi.org/10.1007/s00366-021-01456-x.   DOI
154 Liu, K., Ke, F., Huang, X., Yu, R., Lin, F., Wu, Y. and Ng, D.W.K. (2021c), "DeepBAN: A Temporal Convolution-Based Communication Framework for Dynamic WBANs", IEEE Transactions on Communications, 69(10), 6675-6690. https://doi.org/10.1109/TCOMM.2021.3094581.   DOI
155 Huang, X., Zhang, Y., Moradi, Z. and Shafiei, N. (2021b), "Computer simulation via a couple of homotopy perturbation methods and the generalized differential quadrature method for nonlinear vibration of functionally graded non-uniform micro-tube", Eng. Comput., https://doi.org/10.1007/s00366-021-01395-7.   DOI
156 Berghouti, H., Adda Bedia, E.A., Benkhedda, A. and Tounsi, A. (2019), "Vibration analysis of nonlocal porous nanobeams made of functionally graded material", Adv. Nano Res., 7(5), 351-364. https://doi.org/10.12989/anr.2019.7.5.351.   DOI
157 Hosseini, S.M.R., HABIBI, M. and ASSEMPOUR, A. (2018), "Experimental and numerical determination of forming limit diagram of steel-copper two-layer sheet considering the interface between the layers", Modares Mech. Eng., 18(6), 174-181. https://www.sid.ir/en/journal/ViewPaper.aspx?id=734559.
158 Tang, Y., Yang, T. and Fang, B. (2018a), "Fractional Dynamics of Fluid-Conveying Pipes Made of Polymer-Like Materials", Acta Mechanica Solida Sinica. 31(2), 243-258. https://doi.org/10.1007/s10338-018-0007-9.   DOI
159 Shafiei, N., Ghadiri, M. and Mahinzare, M. (2019), "Flapwise bending vibration analysis of rotary tapered functionally graded nanobeam in thermal environment", Mech. Adv. Mater. Struct., 26(2), 139-155. https://doi.org/10.1080/15376494.2017.1365982.   DOI
160 Huang, X., Hao, H., Oslub, K., Habibi, M. and Tounsi, A. (2021a), "Dynamic stability/instability simulation of the rotary size-dependent functionally graded microsystem", Eng. Comput., https://doi.org/10.1007/s00366-021-01399-3.   DOI
161 Huo, J., Zhang, G., Ghabussi, A. and Habibi, M. (2021), "Bending analysis of FG-GPLRC axisymmetric circular/annular sector plates by considering elastic foundation and horizontal friction force using 3D-poroelasticity theory", Compos. Struct., 276, 114438. https://doi.org/10.1016/j.compstruct.2021.114438.   DOI
162 Hussain, M. and Naeem, M.N. (2019), "Effects of ring supports on vibration of armchair and zigzag FGM rotating carbon nanotubes using Galerkin's method", Compos. B Eng., 163, 548-561. https://doi.org/10.1016/j.compositesb.2018.12.144.   DOI
163 Jiao, J., Ghoreishi, S.-m., Moradi, Z. and Oslub, K. (2021), "Coupled particle swarm optimization method with genetic algorithm for the static-dynamic performance of the magneto-electro-elastic nanosystem", Eng. Comput., 2021, https://doi.org/10.1007/s00366-021-01391-x.   DOI
164 Shafiei, N. and Kazemi, M. (2017b), "Nonlinear buckling of functionally graded nano-/micro-scaled porous beams", Compos. Struct., 178, 483-492. https://doi.org/10.1016/j.compstruct.2017.07.045.   DOI
165 Rezaiee-Pajand, M., Masoodi, A.R. and Rajabzadeh-Safaei, N. (2019), "Nonlinear vibration analysis of carbon nanotube reinforced composite plane structures", Steel Compos. Struct., 30(6), 493-516. https://doi.org/10.12989/scs.2019.30.6.493.   DOI
166 Rouabhia, A., Chikh, A., Bousahla, A.A., Bourada, F., Heireche, H., Tounsi, A., Kouider Halim, B., Tounsi, A. and Al-Zahrani, M.M. (2020), "Physical stability response of a SLGS resting on viscoelastic medium using nonlocal integral first-order theory", Steel Compos. Struct., 37(6), 695-709. https://doi.org/10.12989/scs.2020.37.6.695.   DOI
167 Shafiei, N. and Kazemi, M. (2017a), "Buckling analysis on the bidimensional functionally graded porous tapered nano-/micro-scale beams", Aerosp. Sci. Technol., 66, 1-11. https://doi.org/10.1016/j.ast.2017.02.019.   DOI
168 Shafiei, N. and She, G.L. (2018), "On vibration of functionally graded nano-tubes in the thermal environment", J. Eng. Sci., 133, 84-98. https://doi.org/10.1016/j.ijengsci.2018.08.004.   DOI
169 Karami, B., Janghorban, M. and Tounsi, A. (2019a), "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.   DOI
170 Mahmoud, D. and Elbestawi, M.A. (2017), "Lattice structures and functionally graded materials applications in additive manufacturing of orthopedic implants: A review", J. Manufact. Mater. Process., 1(2), 13. https://doi.org/10.3390/jmmp1020013.   DOI
171 Boutaleb, S., Benrahou, K.H., Bakora, A., Algarni, A., Bousahla, A.A., Tounsi, A., Tounsi, A. and Mahmoud, S.R. (2019), "Dynamic analysis of nanosize FG rectangular plates based on simple nonlocal quasi 3D HSDT", Adv. Nano Res., 7(3), 191. https://doi.org/10.12989/anr.2019.7.3.191.   DOI
172 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 using energy principle", Steel Compos. Struct., 32(5), 595-610. https://doi.org/10.12989/scs.2019.32.5.595.   DOI
173 Matouk, H., Bousahla, A.A., Heireche, H., Bourada, F., Bedia, E.A., Tounsi, A., Mahmoud, S.R., Tounsi, A. and Benrahou, K.H. (2020), "Investigation on hygro-thermal vibration of P-FG and symmetric S-FG nanobeam using integral Timoshenko beam theory", Adv. Nano Res., 8(4), 293-305. https://doi.org/10.12989/anr.2020.8.4.293.   DOI