Browse > Article
http://dx.doi.org/10.12989/scs.2022.43.1.091

Three-dimensional vibration analysis of 3D graphene foam curved panels on elastic foundations  

Zhao, Li-Cai (Department of Civil and Construction Engineering, National Taiwan University of Science and Technology)
Chen, Shi-Shuenn (Department of Civil and Construction Engineering, National Taiwan University of Science and Technology)
Khajehzadeh, Mohammad (Department of Civil Engineering, Anar Branch, Islamic Azad University)
Yousif, Mariwan Araz (Department of Architectural Engineering, Cihan University-Erbil)
Tahouneh, Vahid (Young Researchers and Elite Club, Islamshahr Branch, Islamic Azad University)
Publication Information
Steel and Composite Structures / v.43, no.1, 2022 , pp. 91-106 More about this Journal
Abstract
This paper has focused on presenting a three dimensional theory of elasticity for free vibration of 3D-graphene foam reinforced polymer matrix composites (GrF-PMC) cylindrical panels resting on two-parameter elastic foundations. The elastic foundation is considered as a Pasternak model with adding a Shear layer to the Winkler model. The porous graphene foams possessing 3D scaffold structures have been introduced into polymers for enhancing the overall stiffness of the composite structure. Also, 3D graphene foams can distribute uniformly or non-uniformly in the shell thickness direction. The effective Young's modulus, mass density and Poisson's ratio are predicted by the rule of mixture. Three complicated equations of motion for the panel under consideration are semi-analytically solved by using 2-D differential quadrature method. The fast rate of convergence and accuracy of the method are investigated through the different solved examples. Because of using two-dimensional generalized differential quadrature method, the present approach makes possible vibration analysis of cylindrical panels with two opposite axial edges simply supported and arbitrary boundary at the curved edges. It is explicated that 3D-GrF skeleton type and weight fraction can significantly affect the vibrational characteristics of GrF-PMC panel resting on two-parameter elastic foundations.
Keywords
2-D differential quadrature method; 3D-GrF; cylindrical panel; natural frequency; Polymer matrix composite (PMC); three-dimensional theory of elasticity; two-parameter elastic foundation;
Citations & Related Records
Times Cited By KSCI : 17  (Citation Analysis)
연도 인용수 순위
1 Bi, H., Yin, K., Xie, X., Zhou, Y., Wan, N. and Xu, F. (2012), "Low temperature casting of graphene with high compressive strength", Adv. Mater., 24, 5124-5129. https://doi.org/10.1002/adma.201201519.   DOI
2 Brischetto, S., Tornabene, F., Fantuzzi, N. and Bacciocchi, M. (2015), "Refined 2D and exact 3D shell models for the free vibration analysis of single- and double-walled carbon nanotubes", Technologies, 3(4), 259-284. https://doi.org/10.3390/technologies3040259.   DOI
3 Barka, M., Benrahou, K.H., Bakora, A. and Tounsi, A. (2016), "Thermal post-buckling behavior of imperfect temperaturedependent sandwich FGM plates resting on Pasternak elastic foundation", Steel Compos. Struct., 22(1), 91-112. https://doi.org/10.12989/scs.2016.22.1.091.   DOI
4 Tornabene, F., Fantuzzi, N., Bacciocchi, M. and Viola, E. (2016a), "Effect of agglomeration on the natural frequencies of functionally graded carbon nanotube-reinforced laminated composite doubly-curved shells", Compos. Part B, 89(1), 187-218. https://doi.org/10.1016/j.compositesb.2015.11.016.   DOI
5 Tornabene, F., Fantuzzi, N. and Bacciocchi, M. (2016b), "Linear static response of nanocomposite plates and shells reinforced by agglomerated carbon nanotubes", Compos. Part B, 115(1), 449-476. https://doi.org/10.1016/j.compositesb.2016.07.011.   DOI
6 Tsai, S.W. (1965), Strength Characteristics of Composite Materials, Philco Corporation, Newport Beach, CA, USA.
7 Wang, L. and Hu, H. (2014b), "Thermal vibration of a rectangular single-layered graphene sheet with quantum effects", J. Appl. Phys., 115(23), https://doi.org/10.1063/1.4885015.   DOI
8 Wang, L. and Hu, H. (2015), "Thermal vibration of a circular single-layered graphene sheet with simply supported or clamped boundary", J. Sound Vib., 349, 206-215. https://doi.org/10.1016/j.jsv.2015.03.045.   DOI
9 Yang, R., Kameda, H. and Takada, S. (1998), "Shell model FEM analysis of buried pipelines under seismic loading", Bull Disas. Prev Res. Inst., 38, 115-146.
10 Abbas, I.A. and Marin, M. (2017), "Analytical solution of thermoelastic interaction in a half-space by pulsed laser heating", Physica E-Low-Dimensional Syst. Nanostruct., 87, 254-260. https://doi.org/10.1016/j.physe.2016.10.048.   DOI
11 Wu, Y., Yi, N., Huang, L., Zhang, T., Fang, S. and Chang, H. (2015), "Three-dimensionally bonded spongy graphene material with super compressive elasticity and near-zero Poisson's ratio", Nat. Commun., 6. https://doi.org/10.1038/ncomms7141.   DOI
12 Hosseini, S.M. and Zhang, C. (2018), "Elastodynamic and wave propagation analysis in a FG Graphene platelets-reinforced nanocomposite cylinder using a modified nonlinear micromechanical model", Steel Compos. Struct., 27(3), 255-271.   DOI
13 Shi, D.L., Huang, Y.Y., Hwang, K.C. and Gao, H., (2004), "The effect of nanotube waviness and agglomeration on the elastic property of carbon nanotube-reinforced composites", J. Eng. Mater. T. ASME, 126, 250-257. https://doi.org/10.1115/1.1751182.   DOI
14 Soldatos, K.P. and Hadjigeorgiou, V.P. (1990), "Three-dimensional solution of the free vibration problem of homogeneous isotropic cylindrical shells and panels", J. Sound Vib., 137(3), 369-384. https://doi.org/10.1016/0022-460X(90)90805-A.   DOI
15 Wang, Y.Q. (2019), "Bending and buckling of three-dimensional graphene foam plates", Results Phys., 13, https://doi.org/10.1016/j.rinp.2019.02.072.   DOI
16 Wang, Y.Q. and Liu, Y.F. (2019), "Free vibration and buckling of polymeric shells reinforced with 3D graphene foams", Results Phys., 14
17 Xu, X., Li, H., Zhang, Q., Hu, H., Zhao, Z. and Li, J. (2015), "Self-sensing, ultralight, and conductive 3D graphene/iron oxide aerogel elastomer Deformable in a Magnetic Field", ACS Nano, 9(4), 3969-3977. https://doi.org/10.1021/nn507426u.   DOI
18 Viola, E. and Tornabene, F. (2009), "Free vibrations of three-parameter functionally graded parabolic panels of revolution", Mech. Res. Commun., 36(5), 587-594. https://doi.org/10.1016/j.mechrescom.2009.02.001.   DOI
19 Qiu, L., Huang, B., He, Z., Wang, Y., Tian, Z. and Liu, J.Z. (2017), "Extremely low density and super compressible graphene cellular materials", Adv. Mater., 29, 1-6.
20 Lemaitre, J. and Chaboche, J.L. (1990), Mechanics of Solid Materials, Cambridge University Press: New York, NY, USA.
21 Hill, R. (1964a), "Theory ofmechanical properties of fibre-strengthened materials Elastic behavior", J. Mech. Phys. Solids, 12, 199-212. https://doi.org/10.1016/0022-5096(64)90019-5.   DOI
22 Affdl Halpin, J.C. and Kardos, J.L. (1976), "The Halpin-Tsai equations: A review", Polym. Eng. Sci., 16(5), 344-352. https://doi.org/10.1002/pen.760160512.   DOI
23 Yang, J. and Shen, S.H. (2003), "Free vibration and parametric resonance of shear deformable functionally graded cylindrical panels", J. Sound Vib., 261(5), 871-893. https://doi.org/10.1016/S0022-460X(02)01015-5.   DOI
24 Zenkour, A.M. (2005a), "A comprehensive analysis of functionally graded sandwich plates. Part 1-deflection and stresses", Int. J. Solid Struct., 42(1), 5224-5242. https://doi.org/10.1016/j.ijsolstr.2005.02.015.   DOI
25 Zenkour, A.M. (2005b), "A comprehensive analysis of functionally graded sandwich plates. Part 1-buckling and free vibration deflection and stresses", Int. J. Solid Struct., 42(18), 5243-5258. https://doi.org/10.1016/j.ijsolstr.2005.02.016.   DOI
26 Zhao, X., Liew, K.M. and Ng, T.Y. (2003), "Vibration analysis of laminated composite cylindrical panels via a meshfree approach", Int. J. Solids Struct., 40(1), 161-180. https://doi.org/10.1016/S0020-7683(02)00475-4.   DOI
27 Zhu, C., Han, T.Y.J., Duoss, E.B., Golobic, A.M., Kuntz, J.D.a nd Spadaccini, C.M. (2015), "Highly compressible 3D periodic graphene aerogel microlattices", Nat. Commun., 6, 1-8. https://doi.org/10.1038/ncomms7962.   DOI
28 Civalek, O. (2005), "Geometrically nonlinear dynamic analysis of doubly curved isotropic shells resting on elastic foundation by a combination of HDQ-FD methods", Int. J. Press Vessel Pip., 82(6), 470-479. https://doi.org/10.1016/j.ijpvp.2004.12.003.   DOI
29 Fantuzzi, N., Tornabene, F., Bacciocchi, M. and Dimitri, R. (2016), "Free vibration analysis of arbitrarily shaped functionally carbon nanotube-reinforced plates", Compos. Part B, 115(1), 384-408. https://doi.org/10.1016/j.compositesb.2016.09.021.   DOI
30 Gunawan, H., Mikami, T., Kanie, S. and Sato, M. (2006), "Free vibration characteristics of cylindrical shells partially buried in elastic foundations", J. Sound Vib., 290(3-5), 785-793. https://doi.org/10.1016/j.jsv.2005.04.014.   DOI
31 Hu, H., Zhao, Z., Wan, W., Gogotsi, Y. and Qiu, J. (2013), "Ultralight and highly compressible graphene aerogels", Adv. Mater., 25(15), 2219-2223. https://doi.org/10.1002/adma.201204530.   DOI
32 Marin, M., Ellahi, R. and Chirila, A. (2017), "On solutions of saint-venant's problem for elastic dipolar bodies with voids", Carpathian J. Mathem., 33(2), 219-232. https://doi.org/10.37193/CJM.2017.02.09.   DOI
33 Marin, M., Craciun, E.M. and Pop, N. (2016), "Considerations on mixed initial-boundary value problems for micropolar porous bodies", Dyn. Syst. Appl., 25 (1-2), 175-196.
34 Embrey, L., Nautiyal, P., Loganathan, A., Idowu, A., Boesl, B. and Agarwal, A. (2017), "Three-dimensional graphene foam induces multifunctionality in epoxy nanocomposites by simultaneous improvement in mechanical, thermal, and electrical properties", ACS Appl. Mater. Interfaces, 9(45), 39717-39727. https://doi.org/10.1021/acsami.7b14078.   DOI
35 Loy, C.T., Lam, K.Y. and Reddy, J.N. (1999), "Vibration of functionally graded cylindrical shells", Int. J. Mech. Sci., 41(3), 309-324.   DOI
36 Tahouneh, V., Naei, M.H. and Mosavi Mashhadi, M. (2019), "Using IGA and trimming approaches for vibrational analysis of L-shape graphene sheets via nonlocal elasticity theory", Steel Compos. Struct., 33(5), 717-727. https://doi.org/10.12989/scs.2019.33.5.717.   DOI
37 Jam, J.E., Noorabadi, M. and Namdaran, N. (2017), "Nonlinear free vibration analysis of micro-beams resting on viscoelastic foundation based on the modified couple stress theory", Archive Mech. Eng., https://doi.org/10.1515/meceng-2017-0015.   DOI
38 Lv, L., Zhang, P., Cheng, H., Zhao, Y., Zhang, Z. and Shi, G. (2016), "Solution-Processed Ultraelastic and Strong Air-Bubbled Graphene Foams", Small, 12(24), 3229-3234. https://doi.org/10.1002/smll.201600509.   DOI
39 Chen, C.S., Liu, F.H. and Chen, W.R. (2017), "Vibration and stability of initially stressed sandwich plates with FGM face sheets in thermal environments", Steel Compos. Struct., 23(3), 251-261. https://doi.org/10.12989/scs.2017.23.3.251.   DOI
40 Itu, C., Ochsner, A. and Vlase, S., Marin, M. (2019), "Improved rigidity of composite circular plates through radial ribs", Proceedings of the Institution of Mechanical Engineers Part L-Journal of Materials-Design and Applications, 233(8), 1585-1593. https://doi.org/10.1177/1464420718768049.   DOI
41 Anderson, T.A. (2003), "3D elasticity solution for a sandwich composite with functionally graded core subjected to transverse loading by a rigid sphere", Compos. Struct., 60(3), 265-274. https://doi.org/10.1016/S0263-8223(03)00013-8.   DOI
42 Bacciocchi, M. and Tarantino, A.M., (2019), "Time-dependent behavior of viscoelastic three-phase composite plates reinforced by carbon nanotubes", Compos. Struct., 216, 20-31. https://doi.org/10.1016/j.compstruct.2019.02.083.   DOI
43 Baltacioglu, A.K. and Civalek, O. (2018), "Vibration analysis of circular cylindrical panels with CNT reinforced and FGM composites", 202, 374-388. https://doi.org/10.1016/j.compstruct.2018.02.024.   DOI
44 Bennai, R., Ait Atmane, H. and Tounsi, A. (2015), "A new higher-order shear and normal deformation theory for functionally graded sandwich beams", Steel Compos. Struct., 19(3), 521-546. DOI: https://doi.org/10.12989/scs.2015.19.3.521.   DOI
45 Cai, J.B., Chen W.Q., Ye, G.R. and Ding, H.J. (2000), "On natural frequencies of a transversely isotropic cylindrical panel on a kerr foundation", J. Sound Vib., 232(5), 997-1004. https://doi.org/10.1006/jsvi.1999.2703.   DOI
46 Hosseini-Hashemi, S., Abaei, A.R. and Ilkhani, M.R. (2015), "Free vibrations of functionally graded viscoelastic cylindrical panel under various boundary conditions", Compos. Struct., 126, 1-15. https://doi.org/10.1016/j.compstruct.2015.02.031.   DOI
47 Hosseini-Hashemi, S., Ilkhani, M.R. and Fadaee, M. (2012), "Identification of the validity range of Donnell and Sanders shell theories using an exact vibration analysis of functionally graded thick cylindrical shell panel", Acta Mech., 223(5), 1101-1118.   DOI
48 Bi, H., Xie, X., Yin, K., Zhou, Y., Wan, S., He, L. and Ruoff, R.S. (2012), "Spongy graphene as a highly efficient and recyclable sorbent for oils and organic solvents", Adv. Funct. Mater., 22, 4421-4425. https://doi.org/10.1002/adfm.201200888.   DOI
49 Afrookhteh, S.S., Shakeri, M., Baniassadi, M. and Alizadeh Sahraei, A. (2018), "Microstructure Reconstruction and Characterization of the Porous GDLs for PEMFC Based on Fibers Orientation Distribution", Fuel Cells, 18(2), https://doi.org/10.1002/fuce.201700239.   DOI
50 Bouguenina, O., Belakhdar, K., Tounsi, A. and Bedia, E.A.A. (2015), "Numerical analysis of FGM plates with variable thickness subjected to thermal buckling", Steel Compos. Struct., 19(3), 679-695. https://doi.org/10.12989/scs.2015.19.3.679.   DOI
51 Chen, S., Bao, P., Huang, X., Sun, B. and Wang, G. (2014), "Hierarchical 3D mesoporous silicon@graphene nanoarchitectures for lithium ion batteries with superior performance", Nano Res., 7, 85-94.   DOI
52 Chen, W.Q., Bian, Z.G. and Ding, H.U. (2004), "Three-dimensional vibration analysis of fluid-filled orthotropic FGM cylindrical shells", Int. J. Mech. Sci., 46(1), 159-171. https://doi.org/10.1016/j.ijmecsci.2003.12.005.   DOI
53 Hong, M. and Lee, U. (2015), "Dynamics of a functionally graded material axial bar, Spectral element modeling and analysis", Compos.: Part B, 69, 427-434. https://doi.org/10.1016/j.compositesb.2014.10.022.   DOI
54 Moradi-Dastjerdi, R. and Momeni-Khabisi, H. (2016), "Dynamic analysis of functionally graded nanocomposite plates reinforced by wavy carbon nanotube", Steel Compos. Struct., 22(2), 277-299. https://doi.org/10.12989/scs.2016.22.2.277.   DOI
55 Odegard, G.M., Gates, T.S., Wise, K.E., Park, C. and Siochi, E.J. (2003), "Constitutive modeling of nanotube-reinforced polymer composites", Compos. Sci. Technol., 63, 1671-1687. https://doi.org/10.1016/S0266-3538(03)00063-0.   DOI
56 Paliwal, D.N., Pandey, R.K. and Nath, T. (1996), "Free vibration of circular cylindrical shell on Winkler and Pasternak foundation", Int. J. Press. Vessel Pip., 69(1), 79-89. https://doi.org/10.1016/0308-0161(95)00010-0.   DOI
57 Idowu, A., Boesl, B. and Agarwal, A. (2018), "3D graphene foam-reinforced polymer composites - A review", Carbon, 135, 52-71. https://doi.org/10.1016/j.carbon.2018.04.024.   DOI
58 Kasim, M.N. and Mahmoud, M.I. (2019), "Sediment Yield Problems in Khassa Chai Watershed Using Hydrologic Models", Cihan University-Erbil Sci. J., 3(1), 34-41. https://doi.org/10.24086/cuesj.v3n1y2019.pp34-41.   DOI
59 Eshelby, J.D. (1957), "The determination of the elastic field of an ellipsoidal inclusion, and related problems", P. Roy. Soc. Lond. A Mat., 241, 376-396. https://www.jstor.org/stable/100095.   DOI
60 Chen, Z., Ren, W., Gao, L., Liu, B., Pei, S. and Cheng, H.M. (2011), "Three-dimensional flexible and conductive interconnected graphene networks grown by chemical vapour deposition", Nat. Mater., 10, 424-428. https://doi.org/10.1038/nmat3001.   DOI
61 Park, W.T., Han, S.C., Jung, W.Y. and Lee, W.H. (2016), "Dynamic instability analysis for S-FGM plates embedded in Pasternak elastic medium using the modified couple stress theory", Steel Compos. Struct., 22(6), 1239-1259. https://doi.org/10.12989/scs.2016.22.6.1239.   DOI
62 Pelletier Jacob, L. and Vel Senthil, S. (2006), "An exact solution for the steady state thermo elastic response of functionally graded orthotropic cylindrical shells", Int. J. Solid Struct., 43(5), 1131-1158. https://doi.org/10.1016/j.ijsolstr.2005.03.079.   DOI
63 Pradhan, S.C., Loy, C.T., Lam, K.Y. and Reddy, J.N. (2000), "Vibration characteristic of functionally graded cylindrical shells under various boundary conditions", Appl. Acoust., 61(1), 119-129. https://doi.org/10.1016/S0003-682X(99)00063-8.   DOI
64 Rumeng, L. and Wang, Lifeng (2016), "Thermal vibration of a double-layered graphene sheet with initial stress at low temperature", Chinese Sci. Bull., 62(4), 245-253. https://doi.org/10.1360/N972016-00927.   DOI
65 Halpin, J.C. and Tsai, S.W. (1969), "Effects of environmental factors on composite materials", AFML-TR-67-423.
66 Bouchafa, A., Bouiadjra, M.B., 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. https://doi.org/10.12989/scs.2015.18.6.1493.   DOI
67 Kashtalyan, M. and Menshykova, M. (2009), "Three-dimensional elasticity solution for sandwich panels with a functionally graded core", Compos. Struct., 87(1), 36-43. https://doi.org/10.1016/j.compstruct.2007.12.003.   DOI
68 Li, Q., Iu, V.P. and Kou, K.P. (2008), "Three-dimensional vibration analysis of functionally graded material sandwich plates", J. Sound Vib., 311(1-2), 498-515. https://doi.org/10.1016/j.jsv.2007.09.018.   DOI
69 Ghavamian, A., Rahmandoust, M. and Ochsner, A. (2012), "A numerical evaluation of the influence of defects on the elastic modulus of single and multi-walled carbon nanotubes", Comput. Mater. Sci., 62, 110-116. https://doi.org/10.1016/j.commatsci.2012.05.003.   DOI
70 Guan, L.Z., Zhao, L., Wan, Y.J. and Tang, L.C. (2018), "Three-dimensional graphene-based polymer nanocomposites: Preparation, properties and applications", Nanoscale, 10(31), 14788-14811. https://doi.org/10.1039/C8NR03044H.   DOI
71 Hill, R. (1964b), "Theory of mechanical properties of fibre-strengthened materials: II. Inelastic behavior", J. Mech. Phys. Solids, 12, 213-218. https://doi.org/10.1016/0022-5096(64)90020-1.   DOI
72 Gang, S.W., Lam, K.Y. and Reddy, J.N. (1999), "The elastic response of functionally graded cylindrical shells to low-velocity", Int. J. Impact Eng., 22(4), 397-417. https://doi.org/10.1016/S0734-743X(98)00058-X.   DOI
73 Sha, J., Li, Y., Villegas Salvatierra R., Wang, T., Dong, P. and Ji, Y. (2017), "Three-dimensional printed graphene foams", ACS Nano, 11(7), 6860-6867. https://doi.org/10.1021/acsnano.7b01987.   DOI
74 Afrookhteh, S.S., Fathi, A., Naghdipour, M. and Alizadeh Sahraei, A. (2016), "An experimental investigation of the effects of weight fractions of reinforcement and timing of hardener addition on the strain sensitivity of carbon nanotube/polymer composites", U.P.B. Sci. Bull., Series B, 78(4), 121-130.
75 Arefi, M. (2015), "Elastic solution of a curved beam made of functionally graded materials with different cross sections", Steel Compos. Struct., 18(3), 659-672. https://doi.org/10.12989/scs.2015.18.3.659.   DOI
76 Tahouneh, V., Naei, M.H. and Mosavi Mashhadi, M. (2020), "Influence of vacancy defects on vibration analysis of graphene sheets applying isogeometric method: Molecular and continuum approaches", Steel Compos. Struct., 34(2), 261-277. https://doi.org/10.12989/scs.2020.34.2.261.   DOI
77 Tornabene, F. and Ceruti, A. (2013), "Mixed Static and Dynamic Optimization of Four-Parameter Functionally Graded Completely Doubly Curved and Degenerate Shells and Panels Using GDQ Method", Math. Probl. Eng., 1-33. https://doi.org/10.1155/2013/867079.   DOI
78 Tornabene, F. (2009), "Free vibration analysis of functionally graded conical cylindrical shell and annular plate structures with a four-parameter power-law distribution", Comput. Meth. Appl. Mech. Eng., 198(37), 2911-2935. https://doi.org/10.1016/j.cma.2009.04.011.   DOI
79 Patel, B.P., Gupta, S.S., Loknath, M.S.B. and Kadu, C.P. (2005), "Free vibration analysis of functionally graded elliptical cylindrical shells using higher-order theory", Compos. Struct., 69(3), 259-270. https://doi.org/10.1016/j.compstruct.2004.07.002.   DOI
80 Pradyumna, S. and Bandyopadhyay, J.N. (2008), "Free vibration analysis of functionally graded panels using higher-order finite-element formulation", J. Sound Vib., 318(1-2), 176-192. https://doi.org/10.1016/j.jsv.2008.03.056.   DOI
81 Strek, W., Tomala, R., Lukaszewicz, M., Cichy, B., Gerasymchuk, Y. and Gluchowski, P. (2017), "Laser induced white lighting of graphene foam", Sci. Rep., 7(41281). https://doi.org/10.1038/srep41281.   DOI
82 Tornabene, F., Bacciocchi, M., Fantuzzi, N. and Reddy, J.N. (2019), "Multiscale approach for three-phase cnt/polymer/fiber laminated nanocomposite structures", Polym. Compos., 2019, 40, 102-126. https://doi.org/10.1002/pc.24520.   DOI
83 Tsai, S.W. (1964), Structural Behavior of Composite Materials; Philco Corporation, Newport Beach, CA, USA.
84 Tornabene, F., Fantuzzi, N. and Bacciocchi, M. (2014), "Free vibrations of free-form doubly curved shells made of functionally graded materials using higher-order equivalent single layer theories", Compos. Part B, 67(1), 490-509. https://doi.org/10.1016/j.compositesb.2014.08.012.   DOI
85 Wang, Lifeng. and Hu, H. (2014a), "Thermal vibration single-walled carbon nanotubes with quantum effects", Proc. Math. Phys. Eng. Sci., 470(2168). https://doi.org/10.1098/rspa.2014.0087.   DOI
86 Wu, C.P. and Liu, Y.C. (2016), "A state space meshless method for the 3D analysis of FGM axisymmetric circular plates", Steel Compos. Struct., 22(1), 161-182. https://doi.org/10.12989/scs.2016.22.1.161.   DOI
87 Matsunaga, H. (2008), "Free vibration and stability of functionally graded shallow shells according to a 2-D higher-order deformation theory", Compos. Struct., 84(2), 132-146. https://doi.org/10.1016/j.compstruct.2007.07.006.   DOI
88 Mori, T. and Tanaka, K. (1973), "Average stress in matrix and average elastic energy of materials with misfitting inclusions", Acta Metall., 21, 571-574. https://doi.org/10.1016/0001-6160(73)90064-3.   DOI
89 Ni, Y., Chen, L., Teng, K., Shi, J., Qian, X. and Xu, Z. (2015), "Superior mechanical properties of epoxy composites reinforced by 3D interconnected graphene skeleton", ACS Appl. Mater. Interfaces, 7(21), 11583-11591. https://doi.org/10.1021/acsami.5b02552.   DOI
90 Zhang, Q., Xu, X., Li, H., Xiong, G., Hu, H. and Fisher, T.S. (2015), "Mechanically robust honeycomb graphene aerogel multifunctional polymer composites", Carbon, 93, 659-670. https://doi.org/10.1016/j.carbon.2015.05.102.   DOI
91 Wang, C., Zhang, C. and Chen, S. (2016), "The microscopic deformation mechanism of 3D graphene foam materials under uniaxial compression", Carbon, 109, 666-672. https://doi.org/10.1016/j.carbon.2016.08.084.   DOI
92 Paliwal, D.N., Kanagasabapathy, H. and Gupta, K.M. (1995), "The large deflection of an orthotropic cylindrical shell on a Pasternak foundation", Compos. Struct., 31, 31-37. https://doi.org/10.1016/0263-8223(94)00068-9.   DOI
93 Yavari, F., Chen, Z., Thomas, A.V., Ren, W., Cheng, H.M. and Koratkar, N. (2011), "High sensitivity gas detection using a macroscopic three-dimensional graphene foam network", Sci. Rep., 1, 1-5. https://doi.org/10.1038/srep00166.   DOI
94 Xu, Y., Sheng, K., Li, C. and Shi, G. (2010), "Self-assembled graphene hydrogel via a one-step hydrothermal process", ACS Nano, 4(7), 4324-4330. https://doi.org/10.1021/nn101187z.   DOI
95 Marin, M. (1994), "The Lagrange identity method in thermoelasticity of bodies with microstructure", Int. J. Eng. Sci., 32(8), 1229-1240. https://doi.org/10.1016/0020-7225(94)90034-5.   DOI
96 Marin, M., Agarwal, R.P. and Mahmoud, S.R. (2013), "Nonsimple material problems addressed by the Lagrange's identity", Bound. Value Prob., 135, 1-14. https://doi.org/10.1186/1687-2770-2013-135.   DOI