Abstract
This paper aims to study the influence of temperature variations on the large-amplitude free vibration behaviour of functionally graded (FG) metallic foam arches reinforced with graphene platelets (GPLs) using finite element methodology. The temperature-dependent material properties are considered, and the homogenized effective material properties are estimated using the Halpin–Tsai micromechanics model and Voigt’s rule of mixture. The present formulation is based on higher-order shear deformation theory in conjunction with geometric nonlinearity for the structural analysis. A C0 finite element model is implemented to develop the system of nonlinear governing equations, which are solved numerically using the direct-iterative procedure. The material properties are assumed to be varying along the thickness direction. A convergence and validation study has also been performed to validate the accuracy of the present nonlinear finite element formulation. Various design parameters such as porosity index, the weight fraction of GPLs, porosity distributions, opening angle, temperature dependency and independency etc. are considered for the detailed parametric study. New insights pertaining to the effect of temperature on nonlinear frequencies and mode shapes at higher amplitudes are discussed in detail.
Similar content being viewed by others
Data Availability
The data used to support the findings of this study are included in the article.
References
Betts, C.: Benefits of metal foams and developments in modelling techniques to assess their materials behaviour: A review. Mater. Sci. Technol. 28, 129–143 (2012). https://doi.org/10.1179/026708311X13135950699290
Banhart, J.: Manufacture, characterisation and application of cellular metals and metal foams. Prog. Mater Sci. 46, 559–632 (2001). https://doi.org/10.1016/S0079-6425(00)00002-5
Lefebvre, L.P., Banhart, J., Dunand, D.C.: Porous metals and metallic foams: Current status and recent developments. Adv. Eng. Mater. 10, 775–787 (2008). https://doi.org/10.1002/adem.200800241
Zhao, C.Y.: Review on thermal transport in high porosity cellular metal foams with open cells. Int. J. Heat Mass Transf. 55, 3618–3632 (2012). https://doi.org/10.1016/j.ijheatmasstransfer.2012.03.017
Pollien, A., Conde, Y., Pambaguian, L., Mortensen, A.: Graded open-cell aluminium foam core sandwich beams. Mater. Sci. Eng. A 404, 9–18 (2005). https://doi.org/10.1016/j.msea.2005.05.096
Hangai, Y., Takahashi, K., Utsunomiya, T., Kitahara, S., Kuwazuru, O., Yoshikawa, N.: Fabrication of functionally graded aluminum foam using aluminum alloy die castings by friction stir processing. Mater. Sci. Eng. A 534, 716–719 (2012). https://doi.org/10.1016/j.msea.2011.11.100
Hassani, A., Habibolahzadeh, A., Bafti, H.: Production of graded aluminum foams via powder space holder technique. Mater. Des. 40, 510–515 (2012). https://doi.org/10.1016/j.matdes.2012.04.024
Hangai, Y., Saito, K., Utsunomiya, T., Kitahara, S., Kuwazuru, O., Yoshikawa, N.: Compression properties of Al/Al-Si-Cu alloy functionally graded aluminum foam fabricated by friction stir processing route. Mater. Trans. 54, 405–408 (2013). https://doi.org/10.2320/matertrans.M2012376
Magnucka-Blandzi, E.: Axi-symmetrical deflection and buckling of circular porous-cellular plate. Thin-Walled Struct. 46, 333–337 (2008). https://doi.org/10.1016/j.tws.2007.06.006
Chen, D., Yang, J., Kitipornchai, S.: Elastic buckling and static bending of shear deformable functionally graded porous beam. Compos. Struct. 133, 54–61 (2015). https://doi.org/10.1016/j.compstruct.2015.07.052
Chen, D., Yang, J., Kitipornchai, S.: Free and forced vibrations of shear deformable functionally graded porous beams. Int. J. Mech. Sci. 108–109, 14–22 (2016). https://doi.org/10.1016/j.ijmecsci.2016.01.025
Akbaş, ŞD.: Thermal effects on the vibration of functionally graded deep beams with porosity. Int. J. Appl. Mech. 9, 1–18 (2017). https://doi.org/10.1142/S1758825117500764
Amir, M., Talha, M.: An efficient three nodded finite element formulation for free vibration analysis of sandwich arches with graded metallic cellular core. Int. J. Appl. Mech. (2020). https://doi.org/10.1142/S1758825120500696
Amir, M., Talha, M.: Influence of large amplitude vibration on geometrically imperfect sandwich curved panels embedded with gradient metallic cellular core. Int. J. Appl. Mech. 12, 2050099 (2020). https://doi.org/10.1142/S1758825120500994
Zhao, J., Wang, Q., Deng, X., Choe, K., Xie, F., Shuai, C.: A modified series solution for free vibration analyses of moderately thick functionally graded porous (FGP) deep curved and straight beams. Compos. B Eng. 165, 155–166 (2019). https://doi.org/10.1016/j.compositesb.2018.11.080
Zhao, S., Zhao, Z., Yang, Z., Ke, L.L., Kitipornchai, S., Yang, J.: Functionally graded graphene reinforced composite structures: a review. Eng. Struct. 210, 110339 (2020). https://doi.org/10.1016/j.engstruct.2020.110339
Rafiee, M.A., Rafiee, J., Wang, Z., Song, H., Yu, Z.Z., Koratkar, N.: Enhanced mechanical properties of nanocomposites at low graphene content. ACS Nano 3, 3884–3890 (2009). https://doi.org/10.1021/nn9010472
Chu, K., Jia, C.: Enhanced strength in bulk graphene-copper composites. Phys. Status Solidi (A) Appl. Mater. Sci. 211, 184–190 (2014). https://doi.org/10.1002/pssa.201330051
Kitipornchai, S., Chen, D., Yang, J.: Free vibration and elastic buckling of functionally graded porous beams reinforced by graphene platelets. Mater. Des. 116, 656–665 (2016). https://doi.org/10.1016/j.matdes.2016.12.061
Reza Barati, M., Zenkour, A.M.: Post-buckling analysis of refined shear deformable graphene platelet reinforced beams with porosities and geometrical imperfection. Compos. Struct. 181, 194–202 (2017). https://doi.org/10.1016/j.compstruct.2017.08.082
Chen, D., Yang, J., Kitipornchai, S.: Nonlinear vibration and postbuckling of functionally graded graphene reinforced porous nanocomposite beams. Compos. Sci. Technol. 142, 235–245 (2017). https://doi.org/10.1016/j.compscitech.2017.02.008
Sahmani, S., Aghdam, M.M., Rabczuk, T.: Nonlinear bending of functionally graded porous micro/nano-beams reinforced with graphene platelets based upon nonlocal strain gradient theory. Compos. Struct. 186, 68–78 (2018). https://doi.org/10.1016/j.compstruct.2017.11.082
Polit, O., Anant, C., Anirudh, B., Ganapathi, M.: Functionally graded graphene reinforced porous nanocomposite curved beams: Bending and elastic stability using a higher-order model with thickness stretch effect. Compos. B Eng. 166, 310–327 (2019). https://doi.org/10.1016/j.compositesb.2018.11.074
Liu, Z., Yang, C., Gao, W., Wu, D., Li, G.: Nonlinear behaviour and stability of functionally graded porous arches with graphene platelets reinforcements. Int. J. Eng. Sci. 137, 37–56 (2018). https://doi.org/10.1016/j.ijengsci.2018.12.003
Cong, P.H., Duc, N.D.: New approach to investigate the nonlinear dynamic response and vibration of a functionally graded multilayer graphene nanocomposite plate on a viscoelastic Pasternak medium in a thermal environment. Acta Mech. 229, 3651–3670 (2018). https://doi.org/10.1007/s00707-018-2178-3
Ganapathi, M., Anirudh, B., Anant, C., Polit, O.: Dynamic characteristics of functionally graded graphene reinforced porous nanocomposite curved beams based on trigonometric shear deformation theory with thickness stretch effect. Mech. Adv. Mater. Struct. (2019). https://doi.org/10.1080/15376494.2019.1601310
Polit, O., Pradyumna, B., Ganapathi, M.: Large amplitude free flexural vibrations of functionally graded graphene platelets reinforced porous composite curved beams using finite element based on trigonometric shear deformation theory. Int. J. Non-Linear Mech. 116, 302–317 (2019). https://doi.org/10.1016/j.ijnonlinmec.2019.07.010
Bahranifard, F., Golbahar Haghighi, M.R., Malekzadeh, P.: In-plane responses of multilayer FG-GPLRC curved beams in thermal environment under moving load. Acta Mech. 231, 2679–2696 (2020). https://doi.org/10.1007/s00707-020-02654-2
Shahgholian, D., Safarpour, M., Rahimi, A.R., Alibeigloo, A.: Buckling analyses of functionally graded graphene-reinforced porous cylindrical shell using the Rayleigh-Ritz method. Acta Mech. 231, 1887–1902 (2020). https://doi.org/10.1007/s00707-020-02616-8
Babaei, H., Kiani, Y., Eslami, M.R.: Vibrational behavior of thermally pre-/post-buckled FG-CNTRC beams on a nonlinear elastic foundation: a two-step perturbation technique. Acta Mech. 232, 3897–3915 (2021). https://doi.org/10.1007/s00707-021-03027-z
Babaei, H.: Free vibration and snap-through instability of FG-CNTRC shallow arches supported on nonlinear elastic foundation. Appl. Math. Comput. 413, 126606 (2022). https://doi.org/10.1016/j.amc.2021.126606
Esmaeili, H.R., Kiani, Y., Beni, Y.T.: Vibration characteristics of composite doubly curved shells reinforced with graphene platelets with arbitrary edge supports. Acta Mech. 233, 665–683 (2022). https://doi.org/10.1007/s00707-021-03140-z
Chidamparam, P., Leissa, A.W.: Vibrations of planar curved beams, rings, and arches. Appl. Mech. Rev. 46, 467–483 (1993). https://doi.org/10.1115/1.3120374
Li, S., Peng, G., Ji, M., Cheng, F., Chen, Z., Li, Z.: Impact identification of composite cylinder based on improved deep metric learning model and weighted fusion Tikhonov regularized total least squares. Compos. Struct. 283, 115144 (2022). https://doi.org/10.1016/j.compstruct.2021.115144
Li, Z., Chen, Y., Zheng, J., Sun, Q.: Thermal-elastic buckling of the arch-shaped structures with FGP aluminum reinforced by composite graphene platelets. Thin-Walled Struct. 157, 107142 (2020). https://doi.org/10.1016/j.tws.2020.107142
Li, Z., Zheng, J., Chen, Y., Sun, Q., Zhang, Z.: Effect of temperature variations on the stability mechanism of the confined functionally graded porous arch with nanocomposites reinforcement under mechanical loading. Compos. B Eng. 176, 107330 (2019). https://doi.org/10.1016/j.compositesb.2019.107330
Yas, M.H., Rahimi, S.: Thermal vibration of functionally graded porous nanocomposite beams reinforced by graphene platelets. Appl. Math. Mech. 41, 1209–1226 (2020). https://doi.org/10.1007/s10483-020-2634-6
Yas, M.H., Rahimi, S.: Thermal buckling analysis of porous functionally graded nanocomposite beams reinforced by graphene platelets using Generalized differential quadrature method. Aerosp. Sci. Technol. 107, 106261 (2020). https://doi.org/10.1016/j.ast.2020.106261
Yang, Z., Wu, D., Yang, J., Lai, S.K., Lv, J., Liu, A., Fu, J.: Dynamic buckling of rotationally restrained FG porous arches reinforced with graphene nanoplatelets under a uniform step load. Thin-Walled Struct. 166, 1–11 (2021). https://doi.org/10.1016/j.tws.2021.108103
Wang, Y., Zhang, W.: On the thermal buckling and postbuckling responses of temperature-dependent graphene platelets reinforced porous nanocomposite beams. Compos. Struct. 296, 115880 (2022). https://doi.org/10.1016/j.compstruct.2022.115880
Affdl, J.C.H., Kardos, J.L.: The Halpin-Tsai equations: a review. Polym. Eng. Sci. 16, 344–352 (1976). https://doi.org/10.1002/pen.760160512
Roberts, A.P., Garboczi, E.J.: Elastic moduli of model random three-dimensional closed-cell cellular solids. Acta Mater. 49, 189–197 (2001). https://doi.org/10.1016/S1359-6454(00)00314-1
Piggott, M.R., Taplin, D.M.R.: Load Bearing Fiber Composites. Springer, Berlin (1980)
Heyliger, P.R., Reddy, J.N.: A higher order beam finite element for bending and vibration problems. J. Sound Vib. 126, 309–326 (1988). https://doi.org/10.1016/0022-460X(88)90244-1
Cook, R.D.: Application of finite element analysis, (1987)
Talha, M., Singh, B.N.: Static response and free vibration analysis of FGM plates using higher order shear deformation theory. Appl. Math. Model. 34, 3991–4011 (2010). https://doi.org/10.1016/j.apm.2010.03.034
Qatu, M.S.: Theories and analyses of thin and moderately thick laminated composite curved beams. Int. J. Solids Struct. 30, 2743–2756 (1993). https://doi.org/10.1016/0020-7683(93)90152-W
Kundu, C.K., Sinha, P.K.: Nonlinear transient analysis of laminated composite shells. J. Reinf. Plast. Compos. 25, 1129–1147 (2006). https://doi.org/10.1177/0731684406065196
Amir, M., Talha, M.: Thermoelastic vibration of shear deformable functionally graded curved beams with microstructural defects. Int. J Struct. Stab. Dyn. (2018). https://doi.org/10.1142/S0219455418501353
Nguyen, T.K., Nguyen, B.D., Vo, T.P., Thai, H.T.: Hygro-thermal effects on vibration and thermal buckling behaviours of functionally graded beams. Compos. Struct. 176, 1050–1060 (2017). https://doi.org/10.1016/j.compstruct.2017.06.036
Wattanasakulpong, N., Gangadhara Prusty, B., Kelly, D.W.: Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams. Int. J. Mech. Sci. 53, 734–743 (2011). https://doi.org/10.1016/j.ijmecsci.2011.06.005
Hahn, T.A.: Thermal expansion of copper from 20 to 800 k—standard reference material 736. J. Appl. Phys. 41, 5096–5101 (1970). https://doi.org/10.1063/1.1658614
Shaina, P.R., George, L., Yadav, V., Jaiswal, M.: Estimating the thermal expansion coefficient of graphene: the role of graphene-substrate interactions. J. Phys. Condens. Matter. (2016). https://doi.org/10.1088/0953-8984/28/8/085301
Li, W., Kou, H., Zhang, X., Ma, J., Li, Y., Geng, P., Wu, X., Chen, L., Fang, D.: Temperature-dependent elastic modulus model for metallic bulk materials. Mech. Mater. 139, 103194 (2019). https://doi.org/10.1016/j.mechmat.2019.103194
Shakir, M., Talha, M.: Influence of material uncertainty on higher-order FG-GPLs reinforced porous spherical panels under blast loading. Thin-Walled Struct. 176, 109319 (2022). https://doi.org/10.1016/j.tws.2022.109319
Mukhopadhyay, M., Sheikh, A.H.: Large amplitude vibration of horizontally curved beams: a finite element approach. J. Sound Vib. 180, 239–251 (1995). https://doi.org/10.1006/jsvi.1995.0077
Malekzadeh, P.: Two-dimensional in-plane free vibrations of functionally graded circular arches with temperature-dependent properties. Compos. Struct. 91, 38–47 (2009). https://doi.org/10.1016/j.compstruct.2009.04.034
Li, S.R., Teng, Z.C., Zhou, Y.H.: Free vibration of heated Euler-Bernoulli beams with thermal postbuckling deformations. J. Therm. Stresses 27, 843–856 (2004). https://doi.org/10.1080/01495730490486352
Ebrahimi, F., Ghasemi, F., Salari, E.: Investigating thermal effects on vibration behavior of temperature-dependent compositionally graded Euler beams with porosities. Meccanica 51, 223–249 (2016). https://doi.org/10.1007/s11012-015-0208-y
Funding
The authors have not disclosed any funding
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Mohd, F., Talha, M. The influence of temperature variations on large-amplitude vibration of functionally graded metallic foam arches reinforced with graphene platelets. Acta Mech 234, 425–450 (2023). https://doi.org/10.1007/s00707-022-03398-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-022-03398-x