DOI QR코드

DOI QR Code

An analytical solution for bending and free vibration responses of functionally graded beams with porosities: Effect of the micromechanical models

  • Hadji, Lazreg (Laboratory of Geomatics and Sustainable Development, Ibn Khaldoun University of Tiaret) ;
  • Zouatnia, Nafissa (Department of Civil Engineering, Laboratory of Structures, Geotechnics and Risks (LSGR), Hassiba Benbouali University of Chlef) ;
  • Bernard, Fabrice (University of Rennes, INSA Rennes, Laboratory of Civil Engineering and Mechanical Engineering)
  • Received : 2018.10.16
  • Accepted : 2018.12.15
  • Published : 2019.01.25

Abstract

In this paper, a new higher order shear deformation model is developed for static and free vibration analysis of functionally graded beams with considering porosities that may possibly occur inside the functionally graded materials (FGMs) during their fabrication. Different patterns of porosity distributions (including even and uneven distribution patterns, and the logarithmic-uneven pattern) are considered. In addition, the effect of different micromechanical models on the bending and free vibration response of these beams is studied. Various micromechanical models are used to evaluate the mechanical characteristics of the FG beams whose properties vary continuously across the thickness according to a simple power law. Based on the present higher-order shear deformation model, the equations of motion are derived from Hamilton's principle. Navier type solution method was used to obtain displacement, stresses and frequencies, and the numerical results are compared with those available in the literature. A comprehensive parametric study is carried out to assess the effects of volume fraction index, porosity fraction index, micromechanical models, mode numbers, and geometry on the bending and natural frequencies of imperfect FG beams.

Keywords

References

  1. Ait Amar Meziane, M., Abdelaziz, H.H. and Tounsi, A. (2014), "An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions", J. Sandw. Struct. Mater., 16(3), 293-318. https://doi.org/10.1177/1099636214526852
  2. Akbarzadeh, A.H., Abedini, A. and Chen, Z.T. (2015), "Effect of micromechanical models on structural responses of functionally graded plates", Compos. Struct., 119, 598-609. https://doi.org/10.1016/j.compstruct.2014.09.031
  3. Akbas, S.D. (2015a), "Wave propagation of a functionally graded beam in thermal environments", Steel Compos. Struct., 19(6), 1421-1447. https://doi.org/10.12989/scs.2015.19.6.1421
  4. Akbas, S.D. (2015b), "On post-buckling behavior of edge cracked functionally graded beams under axial loads", Int. J. Struct. Stab. Dyn., 15(4), 1450065. https://doi.org/10.1142/S0219455414500655
  5. Akbas, S.D. (2015c), "Post-buckling analysis of axially functionally graded three-dimensional beams", Int. J. Appl. Mech., 7(3), 1550047. https://doi.org/10.1142/S1758825115500477
  6. Akbas, S.D. (2017a), "Nonlinear static analysis of functionally graded porous beams under thermal effect", Coupled Syst. Mech., 6(4), 399-415. https://doi.org/10.12989/CSM.2017.6.4.399
  7. Akbas, S.D. (2017b), "Post-buckling responses of functionally graded beams with porosities", Steel Compos. Struct., 24(5), 579-589. https://doi.org/10.12989/SCS.2017.24.5.579
  8. Akbas, S.D. (2017c), "Thermal effects on the vibration of functionally graded deep beams with porosity", Int. J. Appl. Mech., 9(5), 1750076. https://doi.org/10.1142/S1758825117500764
  9. Akbas, S.D. (2017d), "Vibration and static analysis of functionally graded porous plates", J. Appl. Comput. Mech., 3(3), 199-207.
  10. Akbas, S.D. (2018a), "Forced vibration analysis of functionally graded porous deep beams", Compos. Struct., 186, 293-302. https://doi.org/10.1016/j.compstruct.2017.12.013
  11. Akbas, S.D. (2018b), "Geometrically nonlinear analysis of functionally graded porous beams", Wind Struct., 27(1), 59-70. https://doi.org/10.12989/WAS.2018.27.1.059
  12. Atmane, H.A., Tounsi, A., Bernard, F. and Mahmoud, S.R. (2015), "A computational shear displacement model for vibrational analysis of functionally graded beams with porosities", Steel Compos. Struct., 19(2), 369-384. https://doi.org/10.12989/scs.2015.19.2.369
  13. Belabed, Z., Houari, M.S.A., Tounsi, A., Mahmoud, S.R. and Anwar Beg, O. (2014), "An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates", Compos.: Part B, 60, 274-283. https://doi.org/10.1016/j.compositesb.2013.12.057
  14. Bouiadjra, R.B., Mahmoudi, A., Benyoucef, S., Tounsi, A. and Bernard, F. (2018), "Analytical investigation of bending response of FGM plate using a new quasi 3D shear deformation theory: Effect of the micromechanical models", Struct. Eng. Mech., 66(3), 317-328. https://doi.org/10.12989/SEM.2018.66.3.317
  15. Euler, L. (1744), Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, Lausanne and Geneva: Apud Marcum-Michaelem Bousquet & Socio, 1-322.
  16. Gasik, M. (1995), "Scand. Ch226", Acta Polytech, 72.
  17. Gasik, M.M. (1998), "Micromechanical modeling of functionally graded materials", Comput. Mater. Sci., 13, 42-55. https://doi.org/10.1016/S0927-0256(98)00044-5
  18. Gupta, A. and Talha, M. (2017), "Influence of porosity on the flexural and free vibration responses of functionally graded plates in thermal environment", Int. J. Struct. Stab. Dyn., 18(1), 1850013. https://doi.org/10.1142/S021945541850013X
  19. Hamidi, A., Houari, M.S.A., Mahmoud, S.R. and Tounsi, A. (2015), "A sinusoidal plate theory with 5-unknowns and stretching effect for thermomechanical bending of functionally graded sandwich plates", Steel Compos. Struct., 18(1), 235-253. https://doi.org/10.12989/scs.2015.18.1.235
  20. Hebali, H., Tounsi, A., Houari, M.S.A., Bessaim, A. and Adda Bedia, E.A. (2014), "A new quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates", ASCE J. Eng. Mech., 140(2), 374-383. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000665
  21. Jaesang, Y. and Addis, K. (2014), "Modeling functionally graded materials containing multiple heterogeneities", Acta Mech., 225(7), 1931-1943. https://doi.org/10.1007/s00707-013-1033-9
  22. Jha, D.K., Kant, T. and Singh, R.K. (2013), "Critical review of recent research on functionally graded plates", Compos. Struct., 96, 833-849. https://doi.org/10.1016/j.compstruct.2012.09.001
  23. Ju, J. and Chen, T.M. (1994), "Micromechanics and effective moduli of elastic composites containing randomly dispersed ellipsoidal inhomogeneities", Acta Mech., 103(1-4), 103-121. https://doi.org/10.1007/BF01180221
  24. Kendall, K., Howard, A., Birchall, J., Prat, P., Proctor, A. and Jefferies, S.A. (1983), "The relation between porosity, microstructure and strength, and the approach to advanced cement-based materials", Phil. Trans. Roy. Soc. Lond. A, 310(1511), 139-153. https://doi.org/10.1098/rsta.1983.0073
  25. Kitipornchai, S., Yang, J. and Liew, K.M. (2006), "Random vibration of the functionally graded laminates in thermal environments", Comp. Meth. Appl. Mech. Eng., 195(9-12), 1075-1095. https://doi.org/10.1016/j.cma.2005.01.016
  26. Mahi, A., Adda Bedia, E.A. and Tounsi, A. (2015), "A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates", Appl. Math. Modell., 39(9), 2489-2508. https://doi.org/10.1016/j.apm.2014.10.045
  27. Mishnaevsky, J.L. (2007), Computational Mesomechanics of Composites, John Wiley & Sons, U.K.
  28. Reddy, J.N. (1984), "A simple higher order theory for laminated composite plates", ASME J. Appl. Mech., 51(4), 745-752. https://doi.org/10.1115/1.3167719
  29. Sayyad, A.S. and Ghugal, Y.M. (2018), "Analytical solutions for bending, buckling, and vibration analyses of exponential functionally graded higher order beams", Asian J. Civil Eng., 1-17.
  30. Shen, H.S. and Wang, Z.X. (2012), "Assessment of Voigt and Mori-Tanaka models for vibration analysis of functionally graded plates", Compos. Struct., 94(7), 2197-2208. https://doi.org/10.1016/j.compstruct.2012.02.018
  31. Timoshenko, S.P. (1921), "On the correction for shear of the differential equation for transverse vibrations of prismatic bars", Philosoph. Mag., 41(245), 742-746.
  32. Wang, Y.Q. and Zu, J.W. (2017), "Vibration behaviors of functionally graded rectangular plates with porosities and moving in thermal environment", Aerosp. Sci. Technol., 69, 550-562. https://doi.org/10.1016/j.ast.2017.07.023
  33. Wattanasakulpong, N. and Ungbhakorn, V. (2014), "Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities", Aerosp. Sci. Technol., 32(1), 111-120. https://doi.org/10.1016/j.ast.2013.12.002
  34. Yin, H.M., Paulino, G.H., Buttlar, W.G. and Sun, L.Z. (2007), "Micromechanics-based thermoelastic model for functionally graded particulate materials with particle interactions", J. Mech. Phys. Sol., 55(1), 132-160. https://doi.org/10.1016/j.jmps.2006.05.002
  35. Zhu, J., Lai, Z., Yin, Z., Jeon, J. and Lee, S. (2001), "Fabrication of $ZrO_2$-NiCr functionally graded material by powder metallurgy", Mater. Chem. Phys., 68(1-3), 130-135. https://doi.org/10.1016/S0254-0584(00)00355-2

Cited by

  1. A simple quasi-3D HSDT for the dynamics analysis of FG thick plate on elastic foundation vol.31, pp.5, 2019, https://doi.org/10.12989/scs.2019.31.5.503
  2. Influence of boundary conditions on the bending and free vibration behavior of FGM sandwich plates using a four-unknown refined integral plate theory vol.25, pp.3, 2020, https://doi.org/10.12989/cac.2020.25.3.225
  3. Effects of hygro-thermo-mechanical conditions on the buckling of FG sandwich plates resting on elastic foundations vol.25, pp.4, 2019, https://doi.org/10.12989/cac.2020.25.4.311
  4. Porosity-dependent mechanical behaviors of FG plate using refined trigonometric shear deformation theory vol.26, pp.5, 2020, https://doi.org/10.12989/cac.2020.26.5.439
  5. Influences of porosity distributions and boundary conditions on mechanical bending response of functionally graded plates resting on Pasternak foundation vol.38, pp.1, 2019, https://doi.org/10.12989/scs.2021.38.1.001
  6. The nano scale buckling properties of isolated protein microtubules based on modified strain gradient theory and a new single variable trigonometric beam theory vol.10, pp.1, 2019, https://doi.org/10.12989/anr.2021.10.1.015
  7. Exact third-order static and free vibration analyses of functionally graded porous curved beam vol.39, pp.1, 2021, https://doi.org/10.12989/scs.2021.39.1.001
  8. Bending analysis of functionally graded plates using a new refined quasi-3D shear deformation theory and the concept of the neutral surface position vol.39, pp.1, 2021, https://doi.org/10.12989/scs.2021.39.1.051
  9. On the free vibration response of laminated composite plates via FEM vol.39, pp.2, 2019, https://doi.org/10.12989/scs.2021.39.2.149
  10. Mechanics of anisotropic cardiac muscles embedded in viscoelastic medium vol.12, pp.1, 2019, https://doi.org/10.12989/acc.2021.12.1.057