Browse > Article
http://dx.doi.org/10.12989/scs.2021.40.3.389

On static buckling of multilayered carbon nanotubes reinforced composite nanobeams supported on non-linear elastic foundations  

Alazwari, Mashhour A. (Mechanical Engineering Dept., Faculty of Engineering, King Abdulaziz University)
Daikh, Ahmed Amine (Laboratoire d'Etude des Structures et de Mecanique des Materiaux, Department of Civil Engineering)
Houari, Mohammed Sid Ahmed (Laboratoire d'Etude des Structures et de Mecanique des Materiaux, Department of Civil Engineering)
Tounsi, Abdelouahed (Department of Civil and Environmental Engineering, King Fahd University of Petroleum and Minerals)
Eltaher, Mohamed A. (Mechanical Engineering Dept., Faculty of Engineering, King Abdulaziz University)
Publication Information
Steel and Composite Structures / v.40, no.3, 2021 , pp. 389-404 More about this Journal
Abstract
This paper introduces a comprehensive buckling response of cross-ply orientation of carbon nanotube reinforced composite (CNTRC) multilayered nanobeams with different boundary conditions. The nonlocal strain gradient (NLSG) stress-strain governing relations are utilized to include the size-dependence and microstructure effects. Novel hyperbolic higher shear deformation beam theory including thickness stretching effect is used to fulfill both parabolic shear distribution through the thickness and the zero-shear at free boundaries. Parametric studies are performed to inspect the influences of arrangement of reinforcement material distributions functions, different functionally graded (FG) functions, and uniform distribution (UD). The balance equilibrium equations are derived, and Fourier functions are utilized to obtain the critical buckling loads of nanobeam under mechanical loadings. Mechanical properties are assumed to be temperature-dependent by using Touloukian principal. An exact solution is performed satisfying the edge boundary conditions. A detailed numerical analysis is illustrated to examine the impact of CNTs patterns, lamination, side-to-thickness, aspect ratios, microstructure and size scale parameters on critical buckling loads of CNTRC laminated nanobeams.
Keywords
buckling; CNTRC laminated nanobeams; cross-ply; nonlocal strain gradient theory; quasi-2D higher-order shear; various boundary conditions;
Citations & Related Records
Times Cited By KSCI : 5  (Citation Analysis)
연도 인용수 순위
1 Wu, H., Kitipornchai, S. and Yang, J. (2016), Thermal Buckling and Postbuckling Analysis of Functionally Graded Carbon Nanotube-Reinforced Composite Beams", Appl. Mech. Mater., 846, 182-187. http://dx.doi.org/10.4028/www.scientific.net/AMM.846.182.   DOI
2 Merzouki, T., Ahmed, H.M.S., Bessaim, A., Haboussi, M., Dimitri, R. and Tornabene, F. (2021), "Bending analysis of functionally graded porous nanocomposite beams based on a non-local strain gradient theory", Math. Mech. Solid., 10812865211011759.https://doi/abs/10.1177/10812865211011759   DOI
3 Tam, M., Yang, Z., Zhao, S. and Yang, J. (2019), "Vibration and buckling characteristics of functionally graded graphene nanoplatelets reinforced composite beams with open edge cracks", Materials, 12, 1412. https://doi.org/10.3390/ma12091412.   DOI
4 Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54, 4703-4710. https://doi.org/10.1063/1.332803.   DOI
5 Hamed, S. and AliReza, S. (2017), "Nonlinear free vibration and post-buckling of FG-CNTRC beams on nonlinear foundation", Steel Compos. Struct., 24(1), 65-77. https://doi.org/10.12989/scs.2017.24.1.065.   DOI
6 Fahsi, B., Bachir Bouiadjra, R., Mahmoudi, A., Benyoucef, S. and Tounsi, A. (2013), "Assessing the effects of porosity on the bending, buckling, and vibrations of functionally graded beams resting on an elastic foundation by using a new refined quasi-3d theory", Mech. Compos. Mater., 55(2), 219-230. https://doi.org/10.1007/s11029-019-09805-0.   DOI
7 Fleck, N.A., Muller, G.M., Ashby, M.F. and Hutchinson, J.W. (1994), "Strain gradient plasticity: theory and experiment", Acta Metallurgica et materialia., 42(2), 475-487. https://doi.org/10.1016/0956-7151(94)90502-9.   DOI
8 Griebel, M. and Hamaekers, J. (2004), "Molecular dynamics simulations of the elastic moduli of polymer-carbon nanotube composites", Comput. Method. Appl. Mech. Eng., 193, 1773-1788. https://doi.org/10.1016/j.cma.2003.12.025.   DOI
9 Han, Y. and Elliott, J. (2007), "Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites", Comput Mater Sci, 39(2), 315-323. https://doi.org/10.1016/j.commatsci.2006.06.011.   DOI
10 Jiao, P. and Alavi, A.H. (2018), "Buckling analysis of graphene-reinforced mechanical metamaterial beams with periodic webbing patterns", Int. J. Eng. Sci., 131, 1-18. https://doi.org/10.1016/j.ijengsci.2018.06.005.   DOI
11 Karami, B., Janghorban, M. and Tounsi, A. (2018a), "Nonlocal strain gradient 3D elasticity theory for anisotropic spherical nanoparticles", Steel Compos. Struct., 27(2), 201-216. https://doi.org/10.12989/scs.2018.27.2.201.   DOI
12 Ma, L.S. and Lee, D.W. (2012), "Exact solutions for nonlinear static responses of a shear deformable FGM beam under an inplane thermal loading", Eur. J. Mech. A/Solids, 31(1), 13-20. https://doi.org/10.1016/j.euromechsol.2011.06.016.   DOI
13 Karami, B., Shahsavari, D., Janghorban, M. and Li, L. (2019), "On the resonance of functionally graded nanoplates using bi-Helmholtz nonlocal strain gradient theory", Int. J. Eng. Sci., 144, 103143. https://doi.org/10.1016/j.ijengsci.2019.103143.   DOI
14 Touloukian, Y.S. (1967), "Thermophysical Properties of High Temperature Solid Materials", MacMillan, New York.
15 Lam, D.C., Yang, F., Chong, A.C.M., Wang, J. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solids, 51(8), 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X.   DOI
16 Nejati, M. and Eslampanah, A. (2016), "Buckling and vibration analysis of functionally graded carbon nanotube-reinforced beam under axial load", Int. J. Appl. Mech., 8(1), 1650008. http://dx.doi.org/10.1142/S1758825116500083.   DOI
17 Phung-Van, P., Thai, C.H., Nguyen-Xuan, H. and Wahab, M.A. (2019), "Porosity-dependent nonlinear transient responses of functionally graded nanoplates using isogeometric analysis", Compos. Part B: Eng., 164, 215-225. https://doi.org/10.1016/j.compositesb.2018.11.036.   DOI
18 Sayyad, A. and Ghugal, Y. (2018), "An inverse hyperbolic theory for FG beams resting on Winkler-Pasternak elastic foundation", Adv. Aircraft Spacecraft Sci., 5(6), 671-689. https://doi.org/10.12989/aas.2018.5.6.671.   DOI
19 Stolken, J.S. and Evans, A.G. (1998), "A microbend test method for measuring the plasticity length scale", Acta Materialia., 46(14), 5109-5115.   DOI
20 Shen, S.H. (2009), "Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments", Compos. Struct., 91(1), 9-19. https://doi.org/10.1016/j.compstruct.2009.04.026.   DOI
21 Karami, B., Janghorban, M. and Rabczuk, T. (2020c), "Forced Vibration Analysis of Functionally Graded Anisotropic Nanoplates Resting on Winkler/Pasternak-Foundation", Comput. Mater. Continua., 62(2), 607-629. https://doi:10.32604/cmc.2020.08032.   DOI
22 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., 28(1), 99-110. http://dx.doi.org/10.12989/scs.2018.28.1.099.   DOI
23 Hadj Elmerabet, A., Heireche, H., Tounsi A. and Semmah, A. (2017), "Buckling temperature of a single-walled boron nitride nanotubes using a novel nonlocal beam model", Adv. Nano Res., 5(1), 1-12. https://doi.org/10.12989/anr.2017.5.1.001.   DOI
24 Nguyen, H.X., Nguyen, T.N., Abdel-Wahab, M., Bordas, S.P., Nguyen-Xuan, H. and Vo, T.P. (2017), "A refined quasi-3D isogeometric analysis for functionally graded microplates based on the modified couple stress theory", Comput. Method. Appl. M., 313, 904-940. https://doi.org/10.1016/j.cma.2016.10.002.   DOI
25 Yang, S.Y., Ma, C.C.M., Teng, C.C., Huang, Y.W., Liao, S.H., Huang, Y.L., Tien, H.W., Lee, T.M. and Chiou, K.C. (2010), "Effect of functionalized carbon nanotubes on the thermal conductivity of epoxy composites", Carbon, 48(3), 592-603. https://doi.org/10.1016/j.carbon.2009.08.047.   DOI
26 Ebrahimi, F., Nouraei, M., Dabbagh, A. and Civalek, O. (2019), "Buckling analysis of graphene oxide powder-reinforced nanocomposite beams subjected to non-uniform magnetic field", Struct. Eng. Mech., 71(4), 351-361. https://doi.org/10.12989/sem.2019.71.4.351.   DOI
27 Daikh, A.A., Houari, M.S.A. and Tounsi, A. (2019), "Buckling analysis of porous FGM sandwich nanoplates due to heat conduction via nonlocal strain gradient theory", Eng. Res. Express, 1, 015022. https://doi.org/10.1088/2631-8695/ab38f9.   DOI
28 Daikh, A.A., Houari, M.S.A., Belarbi, M.O., Chakraverty, S. and Eltaher, M.A. (2021a), "Analysis of axially temperature-dependent functionally graded carbon nanotube reinforced composite plates", Eng. with Comput., 1-22. https://doi.org/10.1007/s00366-021-01413-8.   DOI
29 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., 28(1), 13-24. https://doi.org/10.12989/scs.2018.28.1.013.   DOI
30 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., 65(5), 621-631. https://doi.org/10.12989/SEM.2018.65.5.621.   DOI
31 Fattahi, A.M. and Safaei, B. (2017), "Buckling analysis of CNT-reinforced beams with arbitrary boundary conditions", Microsyst. Technol., 23(10), 5079-5091. https://doi.org/10.1007/s00542-017-3345-5.   DOI
32 Karami, B. and Shahsavari, D. (2020a), "On the forced resonant vibration analysis of functionally graded polymer composite doubly-curved nanoshells reinforced with graphene-nanoplatelets", Comput. Method. Appl. Mech. Eng., 359, 112767. https://doi.org/10.1016/j.cma.2019.112767.   DOI
33 Lim, C.W., Zhang, G. and Redd, 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. https://doi.org/10.1016/j.jmps.2015.02.001.   DOI
34 Khalaf, B.S., Fenjan, R. and Faleh N. (2019), "Analyzing nonlinear mechanical-thermal buckling of imperfect microscale beam made of graded graphene reinforced composites", Adv. Mater. Res., 8(3), 219-235. https://doi.org/10.12989/amr.2019.8.3.219.   DOI
35 AitAtmane, H., Tounsi, A. and Bernard, F. (2017), "Effect of thickness stretching and porosity on mechanical response of a functionally graded beams resting on elastic foundations", Int J Mech. Mater. Des., 13, 71-84. http://dx.doi.org/10.1007/s10999-015-9318-x.   DOI
36 Bellifa, H., Benrahou, K.H., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2017), "A nonlocal zeroth-order shear deformation theory for nonlinear post-buckling of nanobeams", Struct. Eng. Mech., 62(6), 695-702. https://doi.org/10.12989/SEM.2017.62.6.695   DOI
37 Benahmed, A., Fahsi, B., Benzair, A., Zidour, M., Bourada, F., and Tounsi, A. (2019), "Critical buckling of functionally graded nanoscale beam with porosities using nonlocal higher-order shear deformation", Struct. Eng. Mech.,69(4), 457-466. https://doi.org/10.12989/sem.2019.69.4.457.   DOI
38 Bensaid, I. and Guenanou, A. (2017), "Bending and stability analysis of size-dependent compositionally graded Timoshenko nanobeams with porosities", Adv. Mater. Res., 6(1), 45-63. https://doi.org/10.12989/amr.2017.6.1.045.   DOI
39 Ahouel, M., Houari, M.S.A., Bedia, E.A.A. and Tounsi, A. (2016), "Size-dependent mechanical behavior of functionally graded trigonometric shear deformable nanobeams including neutral surface position concept", Steel Compos. Struct., 20(5), 963-981. https://doi.org/10.12989/SCS.2016.20.5.963.   DOI
40 Thanh, C.L., Tran, L.V., Vu-Huu, T. and Abdel-Wahab, M. (2019), "The size-dependent thermal bending and buckling analyses of composite laminate microplate based on new modified couple stress theory and isogeometric analysis", Comput. Method. Appl. M., 350, 337-361. https://doi.org/10.1016/j.cma.2019.02.028.   DOI
41 Daikh, A.A. (2019), "Temperature dependent vibration analysis of functionally graded sandwich plates resting on Winkler/Pasternak/Kerr foundation", Mater. Res. Express, 6, 065702. https://doi.org/10.1088/2053-1591/ab097b.   DOI
42 Vodenitcharova, T. and Zhang, L.C. (2003), "Effective wall thickness of a single-walled carbon nanotube", Phys. Rev. B, 68, 165401. https://doi.org/10.1103/PhysRevB.68.165401.   DOI
43 Wattanasakulpong, N. and Ungbhakorn, V. (2013), "Analytical solutions for bending, buckling and vibration responses of carbon nanotube-reinforced composite beams resting on elastic foundation", Comput. Mater. Sci., 71, 201-208. http://dx.doi.org/10.1016/j.commatsci.2013.01.028.   DOI
44 Wu, H. and Kitipornchai, S. (2015), "Free vibration and buckling analysis of sandwich beams with functionally graded carbon nanotube-reinforced composite face sheets", Int. J. Struct. Stab. Dy., 15(7), 1540011. http://dx.doi.org/10.1142/S0219455415400118.   DOI
45 Karami, B., Janghorban, M. and Rabczuk, T. (2020b), "Dynamics of two-dimensional functionally graded tapered Timoshenko nanobeam in thermal environment using nonlocal strain gradient theory". Compos. Part B: Eng., 182, 107622. https://doi.org/10.1016/j.compositesb.2019.107622.   DOI
46 kumar, P.P., Subbarao, V.V., Sarath, C.S.,and Malathi, B. (2017), "Deflection and Buckling behaviour of simply supported nanocomposite beams under FSDT approach", IOP Conf. Ser.: Mater. Sci. Eng., 225, 012284. http://dx.doi.org/10.1088/1757-899X/225/1/012284   DOI
47 Li, C., Thostenson, E.T. and Chou, T.W. (2008), "Sensors and actuators based on carbon nanotubes and their composites: a review", Compos. Sci. Technol.., 68(6), 1227-1249. https://doi.org/10.1016/j.compscitech.2008.01.006.   DOI
48 Mayandi, K. and Jeyaraj, P. (2015), "Bending, buckling and free vibration characteristics of FG-CNT-reinforced polymer composite beam under non-uniform thermal load", J. Mater.: Des. Appl., 229(1) 13-28. https://doi.org/10.1177/1464420713493720.   DOI
49 Daikh, A.A., Houari, M.S.A., Karami, B., Eltaher, M.A., Dimitri, R. and Tornabene, F. (2021b), "Buckling Analysis of CNTRC Curved Sandwich Nanobeams in Thermal Environment", Appl. Sci., 11(7), 3250. https://doi.org/10.3390/app11073250.   DOI
50 Yang, J., Wu, H. and Kitipornchai, S. (2016), "Buckling and postbuckling of functionally graded multilayer graphene plateletreinforced composite beams", Compos. Struct., 161(1), 111-118. http://dx.doi.org/10.1016/j.compstruct.2016.11.048.   DOI
51 Mohammadimehr, M., Mohammadi-Dehabadi, A.A., Akhavan Alavi, S.M., Alambeigi, K., Bamdad, M., Yazdani, R. and Hanifehlou, S. (2018), "Bending, buckling, and free vibration analyses of carbon nanotube reinforced composite beams and experimental tensile test to obtain the mechanical properties of nanocomposite", Steel Compos. Struct., 29(3), 405-422. https://doi.org/10.12989/scs.2018.29.3.405.   DOI
52 Bennai, R., Atmane, H.A. 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. https://doi.org/10.12989/scs.2015.19.3.521.   DOI
53 Bensattalah, T., Zidour, M., Ait Amar Meziane, M., Hassaine Daouadji, T. and Tounsi, A. (2018), "Critical buckling load of carbon nanotube with non-local timoshenko beam using the differential transform method", Int. J. Civil Environ. Eng., 12(6). https://doi.org/10.5281/zenodo.1317114   DOI
54 Fattahi, A.M and Safaei, B. (2017), "Buckling analysis of CNT-reinforced beams with arbitrary boundary conditions", Microsyst. Technol., 23, 5079-5091. https://doi.org/10.1007/s00542-017-3345-5   DOI
55 Bekhadda, A., Cheikh, A., Bensaid, I., Hadjoui, A. and Daikh, A.A. (2019), "A novel first order refined shear-deformation beam theory for vibration and buckling analysis of continuously graded beams", Adv. Aircraft Spacecraft Sci., 6(3), 189-206. https://doi.org/10.12989/aas.2019.6.3.189.   DOI
56 Daikh, A.A., Drai, A., Bensaid, I., Houari, M.S.A. and Tounsi, A. (2020), "On vibration of functionally graded sandwich nanoplates in the thermal environment", J. Sandw. Struct. Mater., https://doi.org/10.1177/1099636220909790.   DOI
57 Daikh, A.A., Bensaid, I. and Zenkour, A.M., (2020), "Temperature dependent thermomechanical bending response of functionally graded sandwich plates", Eng. Res. Express, 2, 015006. https://doi.org0.1088/2631-8695/ab638c.   DOI
58 Aissani, K., Bouiadjra, Mohamed B., Ahouel, M. and Tounsi, A. (2015), "A new nonlocal hyperbolic shear deformation theory for nanobeams embedded in an elastic medium", Struct. Eng. Mech., 55(4), 743-763. https://doi.org/10.12989/SEM.2015.55.4.743.   DOI
59 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., 21(1), 15-25. https://doi.org/10.12989/sss.2018.21.1.015.   DOI
60 Houari. M.S.A., Bousahla. A.A., Bessaim., A., Adda Bedia. E.A. and Tounsi. A. (2014), "Buckling of functionally graded nanobeams based on the nonlocal new first-order shear deformation beam theory", Proceedings of the MATEC Web of Conferences 11, 01024, http://dx.doi.org/10.1051/matecconf/20141101024.   DOI
61 Yas, M.H. and Samadi N. (2012), "Free vibrations and buckling analysis of carbon nanotube-reinforced composite Timoshenko beams on elastic foundation", Int. J. Pressure Vess. Piping, 98, 119-128. https://doi.org/10.1016/j.ijpvp.2012.07.012.   DOI
62 Chaht, F.L., Kaci, A., Houari, M.S.A., Tounsi, A., Beg, O.A. and Mahmoud, S.R. (2015), Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect", Steel Compos. Struct., 18(2), 425-442. https://doi.org/10.12989/scs.2015.18.2.425.   DOI
63 Bensaid, I. and Kerboua, B. (2019),"Improvement of thermal buckling response of FG-CNT reinforced composite beams with temperature-dependent material properties resting on elastic foundations", Adv. Aircraft Spacecraft Sci., 6(3), 207-223. https://doi.org/10.12989/aas.2019.6.3.207.   DOI
64 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., 48(3), 351-365. https://doi.org/10.12989/sem.2013.48.3.351.   DOI
65 Bessaim, A., Houari1, M.S.A., Bousahla, A.A., Kaci, A., Tounsi, A. and Adda Bedia, E.A. (2018), "Buckling analysis of embedded nanosizefg beams based on a refined hyperbolic shear deformation theory", J. Appl. Comput. Mech., 4(3), 140-146. https://doi.org/10.22055/jacm.2017.22996.1146.   DOI
66 Daikh A.A., Bensaid, I. and Zenkour A.M. (2020), "Temperature dependent thermomechanical bending response of functionally graded sandwich plates", Eng. Res.Express, 2(1), 015006. https://doi.org/10.1088/2631-8695/ab638c.   DOI
67 Ebrahimi, F. and Fardshad, R.E. (2018), "Dynamic modeling of nonlocal compositionally graded temperature-dependent beams", Adv. Aircraft Spacecraft Sci., 5(1), 141-164. https://doi.org/10.12989/aas.2018.5.1.141.   DOI
68 Zemri, A., Houari, MSA., Bousahla, A.A. and Tounsi, A. (2015), "A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory", Struct. Eng. Mech., 54(4), 693-710. https://doi.org/10.12989/sem.2015.54.4.693.   DOI
69 Zian, N., Mefta, S.A., Ruta, G. and Tounsi, A. (2017), "Thermal effects on the instabilities of porous FGM box beams", Eng. Struct., 134(1), 150-158. http://dx.doi.org/10.1016/j.engstruct.2016.12.039.   DOI
70 Daikh, A.A., Guerroudj, M., Elajrami, M. and Megueni, A. (2019a), "Thermal Buckling of Functionally Graded Sandwich Beams", Adv. Mater. Res., 1156, 43-59. https://doi.org/10.4028/www.scientific.net/AMR.1156.43   DOI
71 Karami, B., Janghorban, M. and Tounsi, A. (2017), "Effects of triaxial magnetic field on the anisotropic nanoplates", Steel Compos. Struct., 25(3), 361-374. http://dx.doi.org/10.12989/scs.2017.25.3.361.   DOI
72 Tagrara, S.H., 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
73 Mohseni, A. and Shakouri, M. (2019), "Vibration and stability analysis of functionally graded CNT-reinforced composite beams with variable thickness on elastic foundation", J Mater. Des. Appl., 233(12), 2478-2489. https://doi.org/10.1177/1464420719866222.   DOI
74 Nan, C.W., Liu, G., Lin, Y. and Li, M. (2004), "Interface effect on thermal conductivity of carbon nanotube composites", Appl. Phys. Lett., 85,3549. https://doi.org/10.1063/1.1808874.   DOI
75 Shahsavari, D., Shahsavari, M., Li, L. and Karami, B. (2018), "A novel quasi-3D hyperbolic theory for free vibration of FG plates with porosities resting on Winkler/Pasternak/Kerr foundation", Aerosp. Sci. Technol., 72, 134-149. https://doi.org/10.1016/j.ast.2017.11.004.   DOI
76 Wan, Y., Feng, C., Santiuste, C., Zhao, Z. and Yang, J. (2019), "Buckling and postbuckling of dielectric composite beam reinforced with Graphene Platelets (GPLs)", Aerosp. Sci. Technol., 91, 208-218. https://doi.org/10.1016/j.ast.2019.05.008.   DOI