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A computational shear displacement model for vibrational analysis of functionally graded beams with porosities

  • Atmane, Hassen Ait (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Tounsi, Abdelouahed (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Bernard, Fabrice (Laboratoire de Genie Civil et Genie Mecanique INSA de Rennes) ;
  • Mahmoud, S.R. (Department of Mathematics, Faculty of Science, King Abdulaziz University)
  • Received : 2014.11.18
  • Accepted : 2015.01.18
  • Published : 2015.08.25

Abstract

This work presents a free vibration analysis of functionally graded metal-ceramic (FG) beams with considering porosities that may possibly occur inside the functionally graded materials (FGMs) during their fabrication. For this purpose, a simple displacement field based on higher order shear deformation theory is implemented. The proposed theory is based on the assumption that the transverse displacements consist of bending and shear components in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments. The most interesting feature of this theory is that it accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the beam without using shear correction factors. In addition, it has strong similarities with Euler-Bernoulli beam theory in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The rule of mixture is modified to describe and approximate material properties of the FG beams with porosity phases. By employing the Hamilton's principle, governing equations of motion for coupled axial-shear-flexural response are determined. The validity of the present theory is investigated by comparing some of the present results with those of the first-order and the other higher-order theories reported in the literature. Illustrative examples are given also to show the effects of varying gradients, porosity volume fraction, aspect ratios, and thickness to length ratios on the free vibration of the FG beams.

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. Ait Yahia, S., Ait Atmane, H., Houari, M.S.A. and Tounsi, A. (2015), "Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories", Struct. Eng. Mech, Int. J., 53(6), 1143-1165. https://doi.org/10.12989/sem.2015.53.6.1143
  3. Amara, K., Tounsi, A., Mechab, I. and Adda Bedia, E.A. (2010), "Nonlocal elasticity effect on column buckling of multiwalled carbon nanotubes under temperature field", Appl. Math. Model., 34(12), 3933-3942. https://doi.org/10.1016/j.apm.2010.03.029
  4. Attia, A, Tounsi, A., Adda Bedia, E.A. and Mahmoud, S.R. (2015), "Free vibration analysis of functionally graded plates with temperature-dependent properties using various four variable refined plate theories", Steel Compos. Struct, Int. J., 18(1), 187-212. https://doi.org/10.12989/scs.2015.18.1.187
  5. Bachir Bouiadjra, R., Adda Bedia, E.A. and Tounsi, A. (2013), "Nonlinear thermal buckling behavior of functionally graded plates using an efficient sinusoidal shear deformation theory", Struct. Eng. Mech, Int. J., 48(4), 547-567. https://doi.org/10.12989/sem.2013.48.4.547
  6. Bedjilili, Y., Tounsi, A., Berrabah, H.M., Mechab, I., Adda Bedia, E.A. and Benaissa, S. (2009), "Natural frequencies of composite beams with a variable fiber volume fraction including rotary inertia and shear deformation", Appl. Math. Mech. - Engl. Ed., 30(6), 1-10. https://doi.org/10.1007/s10483-009-0101-z
  7. 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
  8. Benatta, M.A., Mechab, I., Tounsi, A. and Adda Bedia, E.A. (2008), "Static analysis of functionally graded short beams including warping and shear deformation effects", Comput. Mater. Sci., 44(2), 765-773. https://doi.org/10.1016/j.commatsci.2008.05.020
  9. Benatta, M.A, Tounsi, A., Mechab, I. and Bachir Bouiadjra, M. (2009), "Mathematical solution for bending of short hybrid composite beams with variable fibers spacing", Appl. Math. Comput., 212(2), 337-348. https://doi.org/10.1016/j.amc.2009.02.030
  10. Benguediab, S., Tounsi, A., Zidour, M. and Semmah, A. (2014), "Chirality and scale effects on mechanical buckling properties of zigzag double-walled carbon nanotubes", Compos. Part B, 57, 21-24. https://doi.org/10.1016/j.compositesb.2013.08.020
  11. Benzair, A., Tounsi, A., Besseghier, A., Heireche, H., Moulay, N. and Boumia, L. (2008), "The thermal effect on vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory", J. Phys. D: Appl. Phys., 41(22), 225404. https://doi.org/10.1088/0022-3727/41/22/225404
  12. 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, Int. J., 48(3), 351-365. https://doi.org/10.12989/sem.2013.48.3.351
  13. Bouderba, B., Houari, M.S.A. and Tounsi, A. (2013), "Thermomechanical bending response of FGM thick plates resting on Winkler-Pasternak elastic foundations", Steel Compos. Struct., Int. J., 14(1), 85-104. https://doi.org/10.12989/scs.2013.14.1.085
  14. Bourada, M., Kaci, A., Houari, M.S.A. and Tounsi, A. (2015), "A new simple shear and normal deformations theory for functionally graded beams", Steel Compos. Struct., Int. J., 18(2), 409-423. https://doi.org/10.12989/scs.2015.18.2.409
  15. Bousahla, A.A., Houari, M.S.A., Tounsi, A. and Adda Bedia, E.A. (2014), "A novel higher order shear and normal deformation theory based on neutral surface position for bending analysis of advanced composite plates", Int. J. Comput. Method., 11(6), 1350082. https://doi.org/10.1142/S0219876213500825
  16. Bresse, J.A.C. (1859), Cours de Mecanique Applique, Mallet-Bachelier, Paris, France.
  17. Cowper, G.R. (1966), "The shear coefficients in Timoshenko beam theory", ASME, J. Appl. Mech., 33(2), 335-340. https://doi.org/10.1115/1.3625046
  18. Chakraverty, S. and Pradhan, K.K. (2014), "Free vibration of exponential functionally graded rectangular plates in thermal environment with general boundary conditions", Aerosp. Sci. Technol., 36, 132-156. https://doi.org/10.1016/j.ast.2014.04.005
  19. Duc, N.D. and Thang, P.T. (2015), "Nonlinear dynamic response and vibration of shear deformable imperfect eccentrically stiffened S-FGM circular cylindrical shells surrounded on elastic foundations", Aerosp. Sci. Technol., 40, 115-127. https://doi.org/10.1016/j.ast.2014.11.005
  20. Draiche, K., Tounsi, A. and Khalfi, Y. (2014), "A trigonometric four variable plate theory for free vibration of rectangular composite plates with patch mass", Steel Compos. Struct., Int. J., 17(1), 69-81. https://doi.org/10.12989/scs.2014.17.1.069
  21. Fekrar, A., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2014), "A new five-unknown refined theory based on neutral surface position for bending analysis of exponential graded plates", Meccanica, 49(4), 795-810. https://doi.org/10.1007/s11012-013-9827-3
  22. Hadji, L., Daouadji, T.H., Tounsi, A. and Adda Bedia, E.A. (2014), "A higher order shear deformation theory for static and free vibration of FGM beam", Steel Compos. Struct., Int. J., 16(5), 507-519. https://doi.org/10.12989/scs.2014.16.5.507
  23. 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., Int. J., 18(1), 235-253. https://doi.org/10.12989/scs.2015.18.1.235
  24. Hebali, H., Tounsi, A., Houari, M.S.A., Bessaim, A. and Adda Bedia, E.A. (2014), "New quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates", J. Eng. Mech., ASCE, 140(2), 374-383. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000665
  25. Heireche, H., Tounsi, A., Benzair, A., Maachou, M. and Adda Bedia, E.A. (2008), "Sound wave propagation in single-walled carbon nanotubes using nonlocal elasticity", Physica E., 40(8), 2791-2799. https://doi.org/10.1016/j.physe.2007.12.021
  26. Kadoli, R., Akhtar, K. and Ganesan, N. (2008), "Static analysis of functionally graded beams using higher order shear deformation theory", Appl. Math. Model., 32(12), 2509-2525. https://doi.org/10.1016/j.apm.2007.09.015
  27. Karama, M., Afaq, K.S. and Mistou, S. (2003), "Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity", Int. J. Solid. Struct., 40(6), 1525-1546. https://doi.org/10.1016/S0020-7683(02)00647-9
  28. Khalfi, Y., Houari, M.S.A. and Tounsi, A. (2014), "A refined and simple shear deformation theory for thermal buckling of solar functionally graded plates on elastic foundation", Int. J. Comput. Method., 11(5), 135007.
  29. Koizumi, M. (1997), "FGM activities in Japan", Compos. Part B, 28(1-2), 1-4. https://doi.org/10.1016/S1359-8368(96)00016-9
  30. Koochaki, G.R. (2011), "Free vibration analysis of functionally graded beams", World Acad. Sci., Eng. Technol., 74, 366-369.
  31. Larbi Chaht, F., Kaci, A., Houari, M.S.A., Tounsi, A., Anwar Beg, O. 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., Int. J., 18(2), 425-442. https://doi.org/10.12989/scs.2015.18.2.425
  32. Li, X.F. (2008), "A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler-Bernoulli beams", J. Sound Vib., 318(4-5), 1210-1229. https://doi.org/10.1016/j.jsv.2008.04.056
  33. 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. Model., 39(9), 2489-2508. https://doi.org/10.1016/j.apm.2014.10.045
  34. Mahmoud, S.R., Abd-Alla, A.M., Tounsi, A. and Marin, M. (2014), "The problem of wave propagation in magneto-rotating orthotropic non-homogeneous medium", J. Vib. Control, 1077546314521443. https://doi.org/10.1177/1077546314521443
  35. Najafov, A.M., Sofivev, A.H., Hui, D., Karaca, Z., Kalpakci, V. and Ozcelik, M. (2014), "Stability of EG cylindrical shells with shear stresses on a Pasternak foundation", Steel Compos. Struct., Int. J., 17(4), 453-470. https://doi.org/10.12989/scs.2014.17.4.453
  36. Nedri, K., El Meiche, N. and Tounsi, A. (2014), "Free vibration analysis of laminated composite plates resting on elastic foundations by using a refined hyperbolic shear deformation theory", Mech. Compos. Mater., 49(6), 641-650. https://doi.org/10.1007/s11029-013-9380-0
  37. Ould Larbi, L., Kaci, A., Houari, M.S.A. and Tounsi, A. (2013), "An efficient shear deformation beam theory based on neutral surface position for bending and free vibration of functionally graded beams", Mech. Based Des. Struct. Mach., 41(4), 421-433. https://doi.org/10.1080/15397734.2013.763713
  38. Rayleigh, J.W.S. (1880), The Theory of Sound, Macmillan Publishers, London, UK.
  39. 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
  40. Reddy, J.N. (1999), Theory and Anaslysis of Elastic Plates, Taylor & Francis Publication, PA, USA.
  41. Saidi, H., Houari, M.S.A., Tounsi, A. and Adda Bedia, E.A. (2013), "Thermo-mechanical bending response with stretching effect of functionally graded sandwich plates using a novel shear deformation theory", Steel Compos. Struct., Int. J., 15(2), 221-245. https://doi.org/10.12989/scs.2013.15.2.221
  42. Sallai, B.O., Tounsi, A., Mechab, I., Bachir, B.M., Meradjah, M. and Adda, B.E.A. (2009), "A theoretical analysis of flexional bending of $Al/Al_2O_3$ S-FGM thick beams", Comput. Mater. Sci., 44(4), 1344-1350. https://doi.org/10.1016/j.commatsci.2008.09.001
  43. Sankar, B.V. (2001), "An elasticity solution for functionally graded beams", Compos. Sci. Technol., 61(5), 689-696. https://doi.org/10.1016/S0266-3538(01)00007-0
  44. Semmah, A., Tounsi, A., Zidour, M., Heireche, H. and Naceri, M. (2014), "Effect of chirality on critical buckling temperature of a zigzag single-walled carbon nanotubes using nonlocal continuum theory", Fuller. Nanotub. Car. N., 23(6), 518-522. https://doi.org/10.1080/1536383X.2012.749457
  45. Shimpi, R.P. and Patel, H.G. (2006a), "A two variable refined plate theory for orthotropic plate analysis", Int. J. Solid. Struct., 43(22), 6783-6799. https://doi.org/10.1016/j.ijsolstr.2006.02.007
  46. Shimpi, R.P. and Patel, H.G. (2006b), "Free vibrations of plate using two variable refined plate theory", J. Sound Vib., 296(4-5), 979-999. https://doi.org/10.1016/j.jsv.2006.03.030
  47. Sina, S.A., Navazi, H.M. and Haddadpour H. (2009), "An analytical method for free vibration analysis of functionally graded beams", Mater. Des., 30(3), 741-747. https://doi.org/10.1016/j.matdes.2008.05.015
  48. Swaminathan, K. and Naveenkumar, D.T. (2014), "Higher order refined computational models for the stability analysis of FGM plates - Analytical solutions", Eur. J. Mech. A/Solids, 47, 349-361. https://doi.org/10.1016/j.euromechsol.2014.06.003
  49. Timoshenko, S.P. (1921), On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philosophical Magazine, Series 6, pp. 742-746.
  50. Tounsi, A., Houari, M.S.A., Benyoucef, S. and Adda Bedia, E.A. (2013a), "A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates", Aerosp. Sci. Technol., 24(1), 209-220. https://doi.org/10.1016/j.ast.2011.11.009
  51. Tounsi, A., Semmah, A. and Bousahla, A.A. (2013b), "Thermal buckling behavior of nanobeam using an efficient higher-order nonlocal beam theory", J. Nanomech. Micromech., ASCE, 3(3), 37-42. https://doi.org/10.1061/(ASCE)NM.2153-5477.0000057
  52. Tounsi, A., Benguediab, S., Adda Bedia, E.A., Semmah, A. and Zidour, M. (2013c), "Nonlocal effects on thermal buckling properties of double-walled carbon nanotubes", Adv. Nano Res., Int. J., 1(1), 1-11. https://doi.org/10.12989/anr.2013.1.1.001
  53. Tounsi, A., Benguediab, S., Houari, M.S.A. and Semmah, A. (2013d), "A new nonlocal beam theory with thickness stretching effect for nanobeams", Int. J. Nanosci., 12(4), 1350025. https://doi.org/10.1142/S0219581X13500257
  54. Al-Basyouni, K.S., Tounsi, A. and Mahmoud, S.R. (2015), "Size dependent bending and vibration analysis of functionally graded micro beams based on modified couple stress theory and neutral surface position", Compos. Struct., 125, 621-630. https://doi.org/10.1016/j.compstruct.2014.12.070
  55. Touratier, M. (1991), "An efficient standard plate theory", Int. J. Eng. Sci., 29(8), 901-916. https://doi.org/10.1016/0020-7225(91)90165-Y
  56. 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
  57. Wattanasakulpong, N., Prusty, B.G., Kelly, D.W. and Hoffman, M. (2012), "Free vibration analysis of layered functionally graded beams with experimental validation", Mater. Des., 36, 182-190. https://doi.org/10.1016/j.matdes.2011.10.049
  58. Yaghoobi, H. and Torabi, M. (2013a), "Post-buckling and nonlinear free vibration analysis of geometrically imperfect functionally graded beams resting on nonlinear elastic foundation", Appl. Math. Model., 37(18-19), 8324-8340. https://doi.org/10.1016/j.apm.2013.03.037
  59. Yaghoobi, H. and Fereidoon, A. (2014), "Mechanical and thermal buckling analysis of functionally graded plates resting on elastic foundations: An assessment of a simple refined nth-order shear deformation theory", Compos. part B, 62, 54-64 https://doi.org/10.1016/j.compositesb.2014.02.014
  60. Yaghoobi, H. and Torabi, M. (2013b), "An analytical approach to large amplitude vibration and post-buckling of functionally graded beams rest on non-linear elastic foundation", J. Theor. Appl. Mech., 51(1), 39-52.
  61. Yaghoobi, H., Valipour, M.S., Fereidoon, A. and Khoshnevisrad, P. (2014), "Analytical study on post-buckling and nonlinear free vibration analysis of FG beams resting on nonlinear elastic foundation under thermo-mechanical loading using VIM", Steel Compos. Struct., Int. J., 17(5), 753-776. https://doi.org/10.12989/scs.2014.17.5.753
  62. Zhong, Z. and Yu, T. (2007), "Analytical solution of a cantilever functionally graded beam", Compos. Sci. Technol., 67(3-4), 481-488. https://doi.org/10.1016/j.compscitech.2006.08.023
  63. 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
  64. Zidi, M., Tounsi, A., Houari, M.S.A., Adda Bedia, E.A. and Anwar Beg, O. (2014), "Bending analysis of FGM plates under hygro-thermo-mechanical loading using a four variable refined plate theory", Aerosp. Sci. Technol., 34, 24-34. https://doi.org/10.1016/j.ast.2014.02.001

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