Advances in materials Research

  • 제9권1호
  • /
  • Pages.63-98
  • /
  • 2020
  • /
  • 2234-0912(pISSN)
  • /
  • 2234-179X(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Bending and free vibration analysis of functionally graded beams on elastic foundations with analytical validation

  • Hadji, Lazreg (Laboratory of Geomatics and Sustainable Development, University of Tiaret) ;
  • Bernard, Fabrice (University of Rennes, INSA Rennes, Laboratory of Civil Engineering and Mechanical Engineering)
  • 투고 : 2019.11.03
  • 심사 : 2020.03.13
  • 발행 : 2020.03.25
⟨ 이전 논문 다음 논문 ⟩

초록

The novelty of this paper is the use of a simple higher order shear and normal deformation theory for bending and free vibration analysis of functionally graded material (FGM) beams on two-parameter elastic foundation. To this aim, a new shear strain shape function is considered. Moreover, the proposed theory considers a novel displacement field which includes undetermined integral terms and contains fewer unknowns with taking into account the effects of both transverse shear and thickness stretching. 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 for which properties vary continuously across the thickness according to a simple power law. Hamilton's principle is used to derive the governing equations of motion. Navier type analytical solutions are obtained for the bending and vibration problems. Numerical results are obtained to investigate the effects of power-law index, length-to-thickness ratio, foundation parameter, the volume fraction of porosity and micromechanical models on the displacements, stresses, and frequencies.

키워드

참고문헌

  1. Abualnour, M., Chikh, A., Hebali, H., Kaci, A., Tounsi, A., Bousahla, A.A. and Tounsi, A. (2019), "Thermomechanical analysis of antisymmetric laminated reinforced composite plates using a new four variable trigonometric refined plate theory", Comput. Concrete, Int. J., 24(6), 489-498. https://doi.org/10.12989/cac.2019.24.6.489
  2. Adda Bedia, W., Houari, M.S.A., Bessaim, A., Bousahla, A.A., Tounsi, A., Saeed, T. and Alhodaly, M.S. (2019), "A new hyperbolic two-unknown beam model for bending and buckling analysis of a nonlocal strain gradient nanobeams", J. Nano Res., 57, 175-191. https://doi.org/10.4028/www.scientific.net/JNanoR.57.175
  3. Addou, F.Y., Meradjah, M., Bousahla, A.A., Benachour, A., Bourada, F., Tounsi, A. and Mahmoud, S.R. (2019), "Influences of porosity on dynamic response of FG plates resting on Winkler/Pasternak/Kerr foundation using quasi 3D HSDT", Comput. Concrete, Int. J., 24(4), 347-367. https://doi.org/10.12989/cac.2019.24.4.347
  4. Ait Atmane, 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(1), 71-84. https://doi.org/10.1007/s10999-015-9318-x
  5. 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
  6. 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
  7. Akbas, S.D. (2017), "Thermal effects on the vibration of functionally graded deep beams with porosity", Int. J. Appl. Mech.nics, 9(5), 1750076. https://doi.org/10.1142/S1758825117500764
  8. Alimirzaei, S., Mohammadimehr, M. and Tounsi, A. (2019), "Nonlinear analysis of viscoelastic microcomposite beam with geometrical imperfection using FEM: MSGT electro-magneto-elastic bending, buckling and vibration solutions", Struct. Eng. Mech., Int. J., 71(5), 485-502. https://doi.org/10.12989/sem.2019.71.5.485
  9. Barretta, R., Feo, L., Luciano, R., Marotti de Sciarra, F. and Penna, R. (2016), "Functionally graded Timoshenko nanobeams: A novel nonlocal gradient formulation", Compos. Part B, 100(1), 208-219. https://doi.org/10.1016/j.compositesb.2016.05.052
  10. Barretta, R., Ali Faghidian, S., Luciano, R., Medaglia, C.M. and Penna, R. (2018), "Free vibrations of FG elastic Timoshenko nano-beams by strain gradient and stress-driven nonlocal models", Compos. Part B, 154(1), 20-32. 10.1016/j.compositesb.2018.07.036
  11. Batou, B., Nebab, M., Bennai, R., Ait Atmane, H., Tounsi, A., and Bouremana, M., (2019), "Wave dispersion properties in imperfect sigmoid plates using various HSDTs", Steel Compos. Struct., Int. J., 33(5), 699-716. https://doi.org/10.12989/scs.2019.33.5.699
  12. Belbachir, N., Draiche, K., Bousahla, A.A., Bourada, M., Tounsi, A. and Mahmoud, S.R. (2019), "Bending analysis of anti-symmetric cross-ply laminated plates under nonlinear thermal and mechanical loadings", Steel Compos. Struct., Int. J., 33(1), 81-92. https://doi.org/10.12989/scs.2019.33.1.081
  13. Berghouti, H., Adda Bedia, E.A., Benkhedda, A. and Tounsi, A. (2019), "Vibration analysis of nonlocal porous nanobeams made of functionally graded material", Adv. Nano Res., Int. J., 7(5), 351-364. https://doi.org/10.12989/anr.2019.7.5.351
  14. Boukhlif, Z., Bouremana, M., Bourada, F., Bousahla, A.A., Bourada, M., Tounsi, A. and Al-Osta, M.A. (2019), "A simple quasi-3D HSDT for the dynamics analysis of FG thick plate on elastic foundation", Steel Compos. Struct., Int. J., 31(5), 503-516. https://doi.org/10.12989/scs.2019.31.5.503
  15. Boulefrakh, L., Hebali, H., Chikh, A., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2019), "The effect of parameters of visco-Pasternak foundation on the bending and vibration properties of a thick FG plate", Geomech. Eng., Int. J., 18(2), 161-178. https://doi.org/10.12989/gae.2019.18.2.161
  16. Bourada, F., Bousahla, A.A., Bourada, M., Azzaz, A., Zinata, A. and Tounsi, A. (2019), "Dynamic investigation of porous functionally graded beam using a sinusoidal shear deformation theory", Wind Struct., Int. J., 28(1), 19-30. https://doi.org/10.12989/was.2019.28.1.019
  17. Boussoula, A., Boucham, B., Bourada, M., Bourada, F., Tounsi, A., Bousahla, A.A. and Tounsi, A. (2020), "A simple nth-order shear deformation theory for thermomechanical bending analysis of different configurations of FG sandwich plates", Smart Struct. Syst., Int. J., 25(2), 197-218. https://doi.org/10.12989/sss.2020.25.2.197
  18. Boutaleb, S., Benrahou, K.H., Bakora, A., Algarni, A., Bousahla, A.A., Tounsi, A., Tounsi, A. and Mahmoud, S.R. (2019), "Dynamic analysis of nanosize FG rectangular plates based on simple nonlocal quasi 3D HSDT", Adv. Nano Res., Int. J., 7(3), 191-208. https://doi.org/10.12989/anr.2019.7.3.191
  19. Chaabane, L.A., Bourada, F., Sekkal, M., Zerouati, S., Zaoui, F.Z., Tounsi, A., Derras, A., Bousahla, A.A. and Tounsi, A. (2019), "Analytical study of bending and free vibration responses of functionally graded beams resting on elastic foundation", Struct. Eng. Mech., Int. J., 71(2), 185-196. https://doi.org/10.12989/sem.2019.71.2.185
  20. Delale, F. and Erdogan, F. (1983), "The crack problem for a nonhomogeneous plane", J. Appl. Mech., 50(6), 609-614. https://doi.org/10.1115/1.3167098
  21. Ding, J.H., Huang, D.J. and Chen, W.Q. (2007), "Elasticity solutions for plane anisotropic functionally graded beams", Int. J. Solids Struct., 44(1), 176-196. https://doi.org/10.1016/j.ijsolstr.2006.04.026
  22. Draiche, K., Bousahla, A.A., Tounsi, A., Alwabli, A.S., Tounsi, A. and Mahmoud, S.R. (2019), "Static analysis of laminated reinforced composite plates using a simple first-order shear deformation theory", Comput. Concrete, Int. J., 24(4), 369-378. https://doi.org/10.12989/cac.2019.24.4.369
  23. Draoui, A., Zidour, M., Tounsi, A. and Adim, B. (2019), "Static and dynamic behavior of nanotubesreinforced sandwich plates using (FSDT)", J. Nano Res., 57, 117-135. https://doi.org/10.4028/www.scientific.net/JNanoR.57.117
  24. Gasik, M. (1995), "Scand. Ch226", Acta Polytech, 72.
  25. Guerroudj, H.Z, Yeghnem, R., Kaci, A., Zaoui, F.Z., Benyoucef, S. and Tounsi, A. (2018), "Eigenfrequencies of advanced composite plates using an efficient hybrid quasi-3D shear deformation theory", Smart Struct. Syst., Int. J., 22(1), 121-132. https://doi.org/10.12989/sss.2018.22.1.121
  26. Hassaine Daouadji, T., Henni, A.H., Tounsi, A. and Bedia, E.A.A. (2013), "Elasticity solution of a cantilever functionally graded beam", Appl. Compos. Mater., 20(1), 1-15. https://doi.org/10.1007/s10443-011-9243-6
  27. Hassaine Daouadji, T., Adim, B. and Benferhat, R. (2016), "Bending analysis of an imperfect FGM plates under hygro-thermo-mechanical loading with analytical validation", Adv. Mater. Res., Int. J., 5(1), 35-53. https://doi.org/10.12989/amr.2016.5.1.035
  28. Hellal, H., Bourada, M., Hebali, H., Bourda, F., Tounsi, A., Bousahla, A.A. and Mahmour, S.R. (2019), "Dynamic and stability analysis of functionally graded material sandwich plates in hygro-thermal environment using a simple higher shear deformation theory", J. Sandw. Struct. Mater. https://doi.org/10.1177/1099636219845841
  29. Hussain, M., Naeem, M.N., Tounsi, A. and Taj, M. (2019), "Nonlocal effect on the vibration of armchair and zigzag SWCNTs with bending rigidity", Adv. Nano Res., Int. J., 7(6), 431-442. https://doi.org/10.12989/anr.2019.7.6.431
  30. 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
  31. 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
  32. 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
  33. Karama, M., Afaq, K.S. and Mistou, S. (2003), "Mechanical behavior of laminated composite beam by new multi-layered laminated Compos Struct model with transverse shear stress continuity", Int. J. Solids Struct., 40(6), 1525-1546. https://doi.org/10.1016/S0020-7683(02)00647-9
  34. Karami, B., Janghorban, M. and Li, L. (2017), "On guided wave propagation in fully clamped porous functionally graded nanoplates", Acta Astronautica, 143, 380-390. https://doi.org/10.1016/j.actaastro.2017.12.011
  35. Karami, B., Janghorban, M., Shahsavari, D. and Tounsi, A. (2018a), "A size-dependent quasi-3D model for wave dispersion analysis of FG nanoplates", Steel Compos. Struct., Int. J., 28 (1), 99-110. https://doi.org/10.12989/scs.2018.28.1.099
  36. Karami, B., Janghorban, M. and Janghorban, M. (2018b), "Wave propagation analysis in functionally graded (FG) nanoplates under in-plane magnetic field based on nonlocal strain gradient theory and four variable refined plate theory", Mech. Adv. Mater. Struct., 25(12), 1047-1057. https://doi.org/10.1080/15376494.2017.1323143
  37. Karami, B., Janghorban, M. and Tounsi, A. (2019a), "Galerkin's approach for buckling analysis of functionally graded anisotropic nanoplates/different boundary conditions", Eng. Comput., 35, 1297-1316. https://doi.org/10.1007/s00366-018-0664-9
  38. Karami, B., Shahsavari, D., Janghorban, M. and Tounsi, A. (2019b), "Resonance behavior of functionally graded polymer composite nanoplates reinforced with grapheme nanoplatelets", Int. J. Mech. Sci., 156, 94-105. https://doi.org/10.1016/j.ijmecsci.2019.03.036
  39. Karami, B., Janghorban, M. and Tounsi, A. (2019c), "On exact wave propagation analysis of triclinic material using three dimensional bi-Helmholtz gradient plate model", Struct. Eng. Mech., Int. J., 69(5), 487-497. https://doi.org/10.12989/sem.2019.69.5.487
  40. Karami, B., Janghorban, M. and Tounsi, A. (2019d), "On pre stressed functionally graded anisotropic nanoshell in magnetic field", J. Brazil. Soc. Mech. Sci. Eng., 41(11), 495. https://doi.org/10.1007/s40430-019-1996-0
  41. Karami, B., Shahsavari, D., Janghorban, J. and Li, L. (2019e), "Influence of homogenization schemes on vibration of functionally graded curved microbeams", Compos. Struct., 216(15), 67-79. https://doi.org/10.1016/j.compstruct.2019.02.089
  42. Karami, B., Janghorban, M. and Rabczuk, T. (2019f), "Static analysis of functionally graded anisotropic nanoplates using nonlocal strain gradient theory", Compos. Struct., 227, 111249. https://doi.org/10.1016/j.compstruct.2019.111249
  43. Karami, B., Janghorban, M. and Tounsi, A. (2019g), "Wave propagation of functionally graded anisotropic nanoplates resting on Winkler-Pasternak foundation", Struct. Eng. Mech., Int. J., 7(1), 55-66. https://doi.org/10.12989/sem.2019.70.1.055
  44. Karami, B., Janghorban, M. and Rabczuk, T. (2020a), "Dynamics of two-dimensional functionally graded tapered Timoshenko nanobeam in thermal environment using nonlocal strain gradient theory", Compos. Part B: Eng., 182(1), 107622. https://doi.org/10.1016/j.compositesb.2019.107622
  45. Karami, B., Janghorban, M. and Tounsi, A. (2020b), "Novel study on functionally graded anisotropic doubly curved nanoshells", Eur. Phys. J. Plus, 135(1), 103. https://doi.org/10.1140/epjp/s13360-019-00079-y
  46. Karami, B., Shahsavari, D., Janghorban, M. and Li, L. (2020c), "Free vibration analysis of FG nanoplate with poriferous imperfection in hygrothermal environment", Struct. Eng. Mech., Int. J., 73(2), 191-207. https://doi.org/10.12989/sem.2020.73.2.191
  47. 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
  48. Khelifa, Z., Hadji, L., Hassaine Daouadji, T. and Bourada, M. (2018), "Buckling response with stretching effect of carbon nanotube-reinforced composite beams resting on elastic foundation", Struct. Eng. Mech., Int. J., 67(2), 125-130. https://doi.org/10.12989/sem.2018.67.2.125
  49. Khiloun, M., Bousahla, A.A., Kaci, A., Bessaim, A., Tounsi, A. and Mahmoud, S.R. (2019), "Analytical modeling of bending and vibration of thick advanced composite plates using a four-variable quasi 3D HSDT", Eng. Comput., 1-15. https://doi.org/10.1007/s00366-019-00732-1
  50. Kitipornchai, S., Yang, J. and Liew, K.M. (2006), "Random vibration of the functionally graded laminates in thermal environments", Comput. Meth. Appl. Mech. Eng., 195, 1075-1095. https://doi.org/10.1016/j.cma.2005.01.016
  51. Mahmoud, S.R. and Tounsi, A. (2019), "On the stability of isotropic and composite thick plates", Steel Compos. Struct., Int. J., 33(4), 551-568. https://doi.org/10.12989/scs.2019.33.4.551
  52. Mahmoudi, A., Benyoucef, S., Tounsi, A., Benacour, A., Adda Bedia, E.A. and Mahmoud, S.R. (2019), "A refined quasi-3D shear deformation theory for thermo-mechanical behavior of functionally graded sandwich plates on elastic foundations", J. Sandw. Struct. Mater., 21(6), 1906-1926. https://doi.org/10.1177/1099636217727577
  53. Mantari, J.L., Oktem, A.S. and Soares, C.G. (2012), "A new higher order shear deformation theory for sandwich and composite laminated plates" Compos. Part B Eng., 43(3), 1489-1499. https://doi.org/10.1016/j.compositesb.2011.07.017
  54. Mantari, J.L., Bonilla, E.M. and Guedes, S.C. (2014), "A new tangential-exponential higher order shear deformation theory for advanced composite plates", Compos. Part B, 60, 319-328. https://doi.org/10.1016/j.compositesb.2013.12.001
  55. Mantari, J.L., Ramos, I.A., Carrera, E. and Petrolo, M. (2016), "Static analysis of functionally graded plates using new nonpolynomial displacement fields via Carrera Unified Formulation", Compos. Part B, 89, 127-142. https://doi.org/10.1016/j.compositesb.2015.11.025
  56. Mechab, I., El Meiche, N. and Bernard, F. (2017), "Analytical study for the development of a new warping function for high order beam theory", Compos. Part B, 119, 18-31. https://doi.org/10.1016/j.compositesb.2017.03.006
  57. Medani, M., Benahmed, A., Zidour, M., Heireche, H., Tounsi, A., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2019), "Static and dynamic behavior of (FG-CNT) reinforced porous sandwich plate", Steel Compos. Struct., Int. J., 32(5), 595-610. https://doi.org/10.12989/scs.2019.32.5.595
  58. Meksi, R., Benyoucef, S., Mahmoudi, A., Tounsi, A., Adda Bedia, E.A. and Mahmoud, S.R. (2019), "An analytical solution for bending, buckling and vibration responses of FGM sandwich plates", J. Sandw. Struct. Mater., 21(2), 727-757. https://doi.org/10.1177/1099636217698443
  59. Mishnaevsky, J.L. (2007), Computational Mesomechanics of Composites: Numerical analysis of the effect of microstructures of composites on their strength and damage resistance, John Wiley & Sons, UK. https://doi.org/10.1002/9780470513170
  60. Neves, A.M.A., Ferreira, A.J.M., Carrera, E., Roque, C.M.C., Cinefra, M., Jorge, R.M.N. and Soares, C.M.M. (2012), "A quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates", Compos. Struct., 94(5), 1814-1825. https://doi.org/10.1016/j.compstruct.2011.12.005
  61. 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. Bas. Des. Struct. Mach., 41, 421-433. https://doi.org/10.1080/15397734.2013.763713
  62. 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
  63. Sahla, M., Saidi, H., Draiche, K., Bousahla, A.A., Bourada, F. and Tounsi, A. (2019), "Free vibration analysis of angle-ply laminated composite and soft core sandwich plates", Steel Compos. Struct., Int. J., 33(5), 663-679. https://doi.org/10.12989/scs.2019.33.5.663
  64. Salah, F., Boucham, B., Bourada, F., Benzair, A., Bousahla, A.A. and Tounsi, A. (2019), "Investigation of thermal buckling properties of ceramic-metal FGM sandwich plates using 2D integral plate model", Steel Compos. Struct., Int. J., 33(6), 805-822. https://doi.org/10.12989/scs.2019.33.6.805
  65. Sallai, B.O., Tounsi, A., Mechab, I., Bachir, B.M., Meradjah, M. and Adda Bedia, E.A. (2009), "A theoretical analysis of flexional bending of Al/Al2O3 S-FGM thick beams", Computat. Mater. Sci., 44(4), 1344-1350. https://doi.org/10.1016/j.commatsci.2008.09.001
  66. 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
  67. She, G.L., Yuan, F.G., Karami, B., Ren, Y.R. and Xiao, W.S. (2019), "On nonlinear bending behavior of FG porous curved nanotubes", Int. J. Eng. Sci., 135, 58-74. https://doi.org/10.1016/j.ijengsci.2018.11.005
  68. Sayyad, A.S. and Ghugal, Y.M. (2017), "A unified shear deformation theory for the bending of isotropic, functionally graded, laminated and sandwich beams and plates", Int. J. Appl. Mech., 9(1), 1-36. https://doi.org/10.1142/S1758825117500077
  69. Sayyad, A.S. and Ghugal, Y.M. (2018), "An inverse hyperbolic theory for FG beams resting on Winkler-Pasternak elastic foundation", Adv. Aircr. Spacecr. Sci., Int. J., 5(6), 671-689. https://doi.org/10.12989/aas.2018.5.6.671
  70. Sayyad, A.S., Ghugal, Y.M. and Naik, N.S. (2015), "Bending analysis of laminated composite and sandwich beams according to refined trigonometric beam theory", Curved Layer. Struct., 2(1), 279-289. https://doi.org/10.1515/cls-2015-0015
  71. Semmah, A., Heireche, H., Bousahla, A.A. and Tounsi, A. (2019), "Thermal buckling analysis of SWBNNT on Winkler foundation by non local FSDT", Adv. Nano Res., Int. J., 7(2), 89-98. https://doi.org/10.12989/anr.2019.7.2.089
  72. Shahsavari, D., Karami, B. and Li, L. (2018), "A high-order gradient model for wave propagation analysis of porous FG nanoplates", Steel Compos. Struct., Int. J., 29(1), 53-66. https://doi.org/10.12989/scs.2018.29.1.053
  73. 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
  74. Simsek, M. (2010), "Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories", Nucl. Eng. Des., 240(4), 697-705. https://doi.org/10.1016/j.nucengdes.2009.12.013
  75. Soldatos, K.P. (1992), "A transverse shear deformation theory for homogeneous monoclinic plates", Acta Mech., 94(3-4), 195-220. https://doi.org/10.1007/BF01176650
  76. Thai, H.T. and Choi, D.H. (2012), "A refined shear deformation theory for free vibration of functionally graded plates on elastic foundation", Compos. Part B, 43, 2335-2347. https://doi.org/10.1016/j.compositesb.2011.11.062
  77. Timoshenko, S.P. (1921), "On the correction for shear of the differential equation for transverse vibrations of prismatic bars", Philos. Mag., 41(245), 742-746. https://doi.org/10.1080/14786442108636264
  78. Tlidji, Y., Zidour, M., Draiche, K., Safa, A., Bourada, M., Tounsi, A., Bousahla, A.A. and Mahmoud, S.R. (2019), "Vibration analysis of different material distributions of functionally graded microbeam", Struct. Eng. Mech., Int. J., 69(6), 637-649. https://doi.org/10.12989/sem.2019.69.6.637
  79. 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
  80. 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
  81. 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
  82. Ying, J., Lu, C.F. and Chen, W.Q. (2008), "Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations", Compos. Struct., 84(3), 209-219. https://doi.org/10.1016/j.compstruct.2007.07.004
  83. Zaoui, F.Z., Ouinas, D. and Tounsi, A. (2019), "New 2D and quasi-3D shear deformation theories for free vibration of functionally graded plates on elastic foundations", Compos. Part B, 159, 231-247. https://doi.org/10.1016/j.compositesb.2018.09.051
  84. Zarga, D., Tounsi, A., Bousahla, A.A., Bourada, F. and Mahmoud, S.R. (2019), "Thermomechanical bending study for functionally graded sandwich plates using a simple quasi-3D shear deformation theory", Steel Compos. Struct., Int. J., 32(3), 389-410. https://doi.org/10.12989/scs.2019.32.3.389
  85. 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
  86. Zimmerman, R.W. (1994), "Behavior of the Poisson ratio of a two-phase composite material in the highconcentration limit", Appl. Mech. Rev., 47(1), 38-44. https://doi.org/10.1115/1.3122819
  87. Zouatnia, N. and Hadji, L. (2019), "Effect of the micromechanical models on the bending of FGM beam using a new hyperbolic shear deformation theory", Earthq. Struct., Int. J., 16(2), 177-183. https://doi.org/10.12989/eas.2019.16.2.177
  88. Zouatnia, N., Hadji, L. and Kassoul, A. (2017), "An analytical solution for bending and vibration responses of functionally graded beams with porosities", Wind Struct., Int. J., 25(4), 329-3420. https://doi.org/10.12989/was.2017.25.4.329

피인용 문헌

  1. Influences of porosity distributions and boundary conditions on mechanical bending response of functionally graded plates resting on Pasternak foundation vol.38, pp.1, 2020, https://doi.org/10.12989/scs.2021.38.1.001
  2. Investigation on the dynamic response of porous FGM beams resting on variable foundation using a new higher order shear deformation theory vol.39, pp.1, 2020, https://doi.org/10.12989/scs.2021.39.1.095