Abstract
The thermal vibration of functionally graded (FG) porous nanocomposite beams reinforced by graphene platelets (GPLs) is studied. The beams are exposed to the thermal gradient with a multilayer structure. The temperature varies linearly across the thickness direction. Three different types of dispersion patterns of GPLs as well as porosity distributions are presented. The material properties vary along the thickness direction. By using the mechanical parameters of closed-cell cellular solid, the variation of Poisson’s ratio and the relation between the porosity coefficient and the mass density under the Gaussian random field (GRF) model are obtained. By using the Halpin-Tsai micromechanics model, the elastic modulus of the nanocomposite is achieved. The equations of motion based on the Timoshenko beam theory are obtained by using Hamilton’s principle. These equations are discretized and solved by using the generalized differential quadrature method (GDQM) to obtain the fundamental frequencies. The effects of the weight fraction, the dispersion model, the geometry, and the size of GPLs, as well as the porosity distribution, the porosity coefficient, the boundary condition, the metal matrix, the slenderness ratio, and the thermal gradient are presented.
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DELERUE, J. F., LOMOV, S. V., PAMAS, R. S., VERPOEST, I., and WEVERS, M. Pore network modeling of permeability for textile reinforcements. Polymer Composites, 24, 344–357 (2003)
BETTS, C. Benefits of metal foams and developments in modelling techniques to assess their materials behaviour: a review. Materials Science and Technology, 28, 129–143 (2012)
CHOHRA, M., ADVANI, S. G., GOKEE, A., and YARLAGADDA, S. Modeling of filtration through multiple layers of dual scale fibrous porous media. Polymer Composites, 27, 570–581 (2006)
MAGNUCKI, K. and STASIEWICZ, P. Elastic buckling of a porous beam. Journal of Theoretical and Applied Mechanics, 42, 859–868 (2004)
GRYGOROWICZ, M., MAGNUCKI, K., and MALINOWSKI, M. Elastic buckling of a sandwich beam with variable mechanical properties of the core. Thin-Walled Structures, 87, 127–132 (2015)
CHEN, D., YANG, J., and KITIPORNCHAI, S. Elastic buckling and static bending of shear deformable functionally graded porous beam. Composite Structures, 133, 54–61 (2015)
CHEN, D., YANG, J., and KITIPORNCHAI, S. Free and forced vibrations of shear deformable functionally graded porous beams. International Journal of Mechanical Science, 108-109, 14–22 (2016)
EBRAHIMI, F., GHASEMI, F., and SALARI, E. Investigating thermal effects on vibration behavior of temperature-dependent compositionally graded Euler beams with porosities. Meccanica, 51, 223–249 (2016)
IIJIMA, S. Helical microtubules of graphitic carbon. nature, 354, 56–58 (1991)
RAFIEE, M. A., RAFIEE, J., WANG, Z., SONG, H., YU, Z. Z., and KORAKAR, N. Enhanced mechanical properties of nanocomposites at low graphene content. ACS Nano, 3, 3884–3890 (2009)
GONG, L., YOUNG, R. J., KINLOCH, I. A., RIAZ, I., JALIL, R., and NOVOSELOV, K. S. Optimizing the reinforcement of polymer-based nanocomposites by graphene. ACS Nano, 6, 2086–2095 (2012)
WU, H. L., YANG, J., and KITIPORNCHAI, S. Nonlinear vibration of functionally graded carbon nanotube-reinforced composite beams with geometric imperfections. Composite Part B: Engineering, 90, 86–96 (2016)
WATTANASAKULPONG, N. and UNGBHAKORN, V. Analytical solutions for bending, buckling and vibration responses of carbon nanotube-reinforced composite beams resting on elastic foundation. Computational Materials Science, 71, 201–208 (2013)
MOHAMMADI, S. and YAS, M. H. Modeling of elastic behavior of carbon nanotube reinforced polymer by accounting the interfacial debonding. Journal of Reinforced Plastics and Composites, 35, 1477–1489 (2016)
YAS, M. H., MOHAMMADI, S., ASTINCHAP, B., and HESHMATI, M. A comprehensive study on the thermo mechanical properties of multi-walled carbon nanotube/epoxy. Journal of Composite Materials, 50, 2025–2034 (2015)
YAS, M. H. and SAMADI, N. Free vibrations and buckling analysis of carbon nanotube-reinforced composite Timoshenko beams on elastic foundation. International Journal of Pressure Vessel and Piping, 98, 119–128 (2012)
LAI, R. and DANGI, C. Thermal vibrations of temperature-dependent functionally graded nonuniform Timoshenko nanobeam using nonlocal elasticity theory. Materials Research Express, 6, 1–15 (2019)
AREFI, M., MOHAMMAD-REZAEI, B. E., DIMITR, R., and TORNABENE, F. Free vibrations of functionally graded polymer composite nanoplates reinforced with graphene nanoplatelets. Aerospace Science Technology, 81, 108–117 (2018)
SHAHRJERDI, A. and YAVAR, I. S. Free vibration analysis of functionally graded graphene-reinforced nanocomposite beams with temperature-dependent properties. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 40, 25 (2018)
SONG, M., KITIPORNCHAI, S., and YANG, J. Free and forced vibrations of functionally graded polymer composite plates reinforced with graphene nanoplatelets. Composite Structures, 159, 579–588 (2017)
ZAMAN, I., KNAN, H. C., DAI, J., KAWASHIMA, N., MICHELMORE, A., SOVI, A., DONG, S., LUONG, L., and MA, J. From carbon nanotubes and silicate layers to graphene platelets for polymer nanocomposites. Nanoscale, 4, 4578–4586 (2012)
LI, Z. L., YOUNG, R. J., WILSON, N. R., KINLOCH, I. A., VALLES, C., and LI, Z. Effect of the orientation of graphene-based nanoplatelets upon the Young’s modulus of nanocomposites. Composite Science and Technology, 123, 125–133 (2016)
ZHENG, H. and JAGANANDHAM, K. Thermal conductivity and interface thermal conductance in composites of titanium with graphene platelets. Journal of Heat Transfer, 136, 061301 (2014)
BAKSHI, S., AKSHI, S., LAHIRI, D., and AGARWAL, A. Carbon nanotube reinforced metal matrix composites — a review. International Materials Reviews, 55, 41–64 (2010)
LAKES, R. Cellular solid structures with unbounded thermal expansion. Journal of Materials Science Letters, 15, 475–477 (1996)
ROBERTS, A. P. and GARBOCZI, E. J. Elastic moduli of model random three-dimensional closed-cell cellular solids. Acta Materials, 49, 189–197 (2001)
ROBERTS, A. P. and GARBOCZI, E. J. Computation of the linear elastic properties of random porous materials with a wide variety of microstructure. Proceeding of Royal Society London, Ser. A, 458, 1033–1054 (2002)
SHOKRIEH, M. M., ESMKHANI, M., SHOKRIEH, Z., and ZHAO, Z. Stiffness prediction of graphene nanoplatelet/epoxy nanocomposites by a combined molecular dynamics-micromechanics method. Computional Materials Science, 92, 444–450 (2014)
AFFDL, J. H. and KARDOS, J. L. The Halpin-Tsai equations: a review. Polymer Engineering and Science, 16, 344–352 (1976)
DE VILLORIA, R. G. and MIRAVETE, A. Mechanical model to evaluate the effect of the dispersion in nanocomposites. Acta Materialia, 55, 3025–3031 (2007)
SHU, C. Differential Quadrature and Its Application in Engineering, Springer-Verlag, London (2000)
SHU, C. and DU, H. Implementation of clamped and simply supported boundary conditions in the GDQ free vibration analysis of beams and plates. International Journal of Solid Structures, 34, 819–835 (1997)
JAGANNADHAM, K. Thermal conductivity of copper-graphene composite films synthesized by electrochemical deposition with exfoliated graphene platelets. Metallurgical and Materials Transactions B, 43, 316–324 (2012)
HACKER, M., BURGHARDT, D., FLETCHER, L., GORDON, A., and PERUZZI, W. Engineering and Technology, the National Academies Press, New York, 50–75 (2015)
TJONG, S. C. Recent progress in the development and properties of novel metal matrix nanocomposites reinforced with carbon nanotubes and graphene nanosheets. Materials Science Engneering, 74, 281–350 (2013)
LUTJERING, G. and WILLIAMS, J. C. Titanium, Springer Science+Business Media, Berlin (2007)
SHAINA, P. R., GEORGE, L., YADAV, V., and JAISWAL, M. Estimating the thermal expansion coefficient of graphene: the role of graphene-substrate interactions. Journal of Physics: Condensed Matter, 28, 085301 (2016)
LIU, F., MING, P., and LI, J. Ab initio calculation of ideal strength and phonon instability of graphene under tension. Physical Review B, 76, 064120 (2007)
KITIPORNCHAI, S., CHEN, D., and YANG, J. Free vibration and elastic buckling of functionally graded porous beams reinforced by graphene platelets. Materials and Design, 116, 656–665 (2017)
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Yas, M.H., Rahimi, S. Thermal vibration of functionally graded porous nanocomposite beams reinforced by graphene platelets. Appl. Math. Mech.-Engl. Ed. 41, 1209–1226 (2020). https://doi.org/10.1007/s10483-020-2634-6
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DOI: https://doi.org/10.1007/s10483-020-2634-6