Steel and Composite Structures

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

An efficient higher order shear deformation theory for free vibration analysis of functionally graded shells

  • Belabed, Zakaria (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Selim, Mahmoud M. (Department of Mathematics, Al-Aflaj College of Science and Humanities, Prince Sattam bin Abdulaziz University) ;
  • Slimani, Omar (FIMAS Laboratory, Department of Civil Engineering, Faculty of Technology, Tahri Mohamed University) ;
  • Taibi, Noureddine (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) ;
  • Hussain, Muzamal (Department of Mathematics, Govt. College University Faisalabad)
  • Received : 2021.03.09
  • Accepted : 2021.07.05
  • Published : 2021.07.25
⟨ Previous Next ⟩

Abstract

In this study, a simple and efficient higher order shear deformation theory is formulated for free vibration analysis of functionally graded (FG) shells. By introducing the undetermined integral terms in displacement field, the number of generated unknowns and their related governing equations is reduced in contrast to previously published theories, and therefore the differentiability of governing motion equations is decreased , this motivation turns the present theory simpler and easily exploited for functionally graded shell mechanical simulation. Both strains and stress rise through the thickness coordinate as function of hyperbolical distribution. The Hamilton's principle is deployed to derive the governing and motion equations. Closed form solutions are obtained for free vibration problems using Navier's method. Furthermore, detailed comparisons with other shear deformation theories are presented to illustrate the efficiency and accuracy of the developed theory. From this perspective, various perceptions on the impact of some important parameters such as material distribution, geometrical configuration, thickness and curvature ratios are studied and discussed. The non-trivial aspects in predicting the free vibration responses of FG shells are also pointed out.

Keywords

References

  1. Abo-Bakr, H.M., Abo-Bakr, R.M., Mohamed, S.A. and Eltaher, M. A. (2020), "Weight optimization of axially functionally graded microbeams under buckling and vibration behaviors", Mech. Based. Des. Struct., 1-22. https://doi.org/10.1080/15397734.2020.1838298.
  2. Abo-bakr, R.M., Abo-bakr, H.M., Mohamed, S.A. and Eltaher, M.A. (2021), "Optimal weight for buckling of FG beam under variable axial load using Pareto optimality", Compos. Struct., 258, 113193. https://doi.org/10.1016/j.compstruct.2020.113193.
  3. Abdul Kareem Abed, Z. and Ibraheem Majeed, W. (2020), "Effect of boundary conditions on harmonic response of laminated plates", Compos. Mater. Eng., 2(2), 125-140. https://doi.org/10.12989/cme.2020.2.2.125.
  4. Abdulrazzaq, M.A., Fenjan, R.M., Ahmed, R.A. and Faleh, N.M. (2020), "Thermal buckling of nonlocal clamped exponentially graded plate according to a secant function based refined theory", Steel Compos. Struct., 35(1), 147-157. https://doi.org/10.12989/scs.2020.35.1.147.
  5. Akbas, S.D. (2015), "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.
  6. Akbas, S.D., Bashiri, A.H., Assie, A.E. and Eltaher, M.A. (2021), "Dynamic analysis of thick beams with functionally graded porous layers and viscoelastic support", J. Vib. Control., 27(13-14), 1644-1655. https://doi:10.1177/1077546320947302.
  7. Akgoz, B. and Civalek, O. (2015), "A microstructure-dependent sinusoidal plate model based on the strain gradient elasticity theory", Acta Mech., 226(7), 2277-2294. https://doi.org/10.1007/s00707-015-1308-4.
  8. Akgoz, B. and Civalek, O. (2017), "Effects of thermal and shear deformation on vibration response of functionally graded thick composite microbeams", Compos. Part B-Eng., 129, 77-87. https://doi.org/10.1016/j.compositesb.2017.07.024.
  9. Alazwari, M.A., Abdelrahman, A.A., Wagih, A., Eltaher, M.A., and Abd-El-Mottaleb, H.E. (2021). "Static analysis of cutout microstructures incorporating the microstructure and surface effects.", Steel Compos. Struct., 38(5), 583-597. https://doi.org/10.12989/SCS.2021.38.5.583.
  10. Alibeigloo, A. and Liew, K.M. (2014), "Free vibration analysis of sandwich cylindrical panel with functionally graded core using three-dimensional theory of elasticity", Compos. Struct., 113, 23-30. https://doi.org/10.1016/j.compstruct.2014.03.004.
  11. Alijani, F., Amabili, M., Karagiozis, K. and Bakhtiari-Nejad, F. (2011), "Nonlinear vibrations of functionally graded doubly curved shallow shells", J. Sound Vib., 330(7), 1432-1454. https://doi.org/10.1016/j.jsv.2010.10.003.
  12. Aminipour, H., Janghorban, M. and Li, L. (2018), "A new model for wave propagation in functionally graded anisotropic doublycurved shells", Compos. Struct., 190, 91-111. https://doi.org/10.1016/j.compstruct.2018.02.003.
  13. Asiri, S.A., Akbas, S.D. and Eltaher, M.A. (2020a), "Damped dynamic responses of a layered functionally graded thick beam under a pulse load", Struct. Eng. Mech., 75(6),713-722. https://doi.org/10.12989/SEM.2020.75.6.713.
  14. Asiri, S.A., Akbas, S.D. and Eltaher, M.A. (2020b), "Dynamic Analysis of Layered Functionally Graded Viscoelastic Deep Beams with Different Boundary Conditions Due to a Pulse Load", Int. J. Appl. Mech., 12(5), 2050055. https://doi.org/10.1142/s1758825120500556.
  15. Attia, M.A. (2017), "On the mechanics of functionally graded nanobeams with the account of surface elasticity", Int. J. Eng. Sci., 115, 73101. https://doi.org/10.1016/j.ijengsci.2017.03.011.
  16. Avcar, M. (2019), "Free vibration of imperfect sigmoid and power law functionally graded beams", Steel Compos. Struct., 30(6), 603-615. https://doi.org/10.12989/scs.2019.30.6.603.
  17. Brischetto, S. (2017), "A general exact elastic shell solution for bending analysis of functionally graded structures", Compos. Struct., 175, 70-85. https://doi.org/10.1016/j.compstruct.2017.04.002.
  18. Brischetto, S., Tornabene, F., Fantuzzi, N. and Viola, E. (2016), "3D exact and 2D generalized differential quadrature models for free vibration analysis of functionally graded plates and cylinders", Meccanica, 51(9), 2059-2098. https://doi.org/10.1007/s11012-016-0361-y.
  19. Carrera, E., Brischetto, S., Cinefra, M. and Soave, M. (2011), "Effects of thickness stretching in functionally graded plates and shells", Compos. Part B-Eng., 42(2), 123-133. https://doi.org/10.1016/j.compositesb.2010.10.005.
  20. Civalek, O. and Acar, M.H. (2007), "Discrete singular convolution method for the analysis of Mindlin plates on elastic foundations", Int. J. Pres. Ves. Pip., 84(9), 527-535. https://doi.org/10.1016/j.ijpvp.2007.07.001.
  21. Civalek, O. (2009), "Fundamental frequency of isotropic and orthotropic rectangular plates with linearly varying thickness by discrete singular convolution method", Appl. Math. Model., 33(10), 3825-3835. https://doi.org/10.1016/j.apm.2008.12.019.
  22. Dash, S., Mehar, K., Sharma, N., Mahapatra, T.R. and Panda, S.K. (2018). "Modal analysis of FG sandwich doubly curved shell structure", Struct. Eng. Mech., 68(6), 721-733. https://doi.org/10.12989/sem.2018.68.6.721.
  23. Demir, C. and Civalek, O. (2017a), "A new nonlocal FEM via Hermitian cubic shape functions for thermal vibration of nanobeams surrounded by an elastic matrix", Compos. Struct., 168, 872-884. https://doi.org/10.1016/j.compstruct.2017.02.091.
  24. Demir, C. and Civalek, O. (2017b), "On the analysis of microbeams", Int. J. Eng. Sci., 121, 14-33. https://doi.org/10.1016/j.ijengsci.2017.08.016.
  25. Eltaher, M.A. and Akbas, S.D. (2020a) "Transient response of 2D functionally graded beam structure", Struct. Eng. Mech., 75(3), 357-367. https://doi.org/ 10.12989/sem.2020.75.3.357.
  26. Eltaher, M.A., Attia, M.A. and Wagih, A. (2020b), "Predictive model for indentation of elasto-plastic functionally graded composites", Compos. Part B-Eng., 197, 108129. https://doi.org/10.1016/j.compositesb.2020.108129.
  27. Eltaher, M.A. and Mohamed, S.A. (2020c), "Buckling and stability analysis of sandwich beams subjected to varying axial loads", Steel Compos. Struct., 34(2), 241-260. https://doi.org/10.12989/scs.2020.34.2.241.
  28. Esen, I., Ozarpa, C. and Eltaher, M.A. (2021), "Free vibration of a cracked FG microbeam embedded in an elastic matrix and exposed to magnetic field in a thermal environment", Compos. Struct., 261, 113552. https://doi.org/10.1016/j.compstruct.2021.113552.
  29. Fadaee, M., Atashipour, S.R. and Hosseini-Hashemi, S. (2013), "Free vibration analysis of Levy-type functionally graded spherical shell panel using a new exact closed-form solution", Int. J. Mech. Sci., 77, 227-238. https://doi.org/10.1016/j.ijmecsci.2013.10.008.
  30. Fadaee, M., Ilkhani, M.R. and Hosseini-Hashemi, S. (2014), "A new generic exact solution for free vibration of functionally graded moderately thick doubly curved shallow shell panel", J. Vib. Control., 22(15), 3355-3367. https://doi.org/10.1177/1077546314551778.
  31. Fantuzzi, N., Brischetto, S., Tornabene, F. and Viola, E. (2016), "2D and 3D shell models for the free vibration investigation of functionally graded cylindrical and spherical panels", Compos. Struct.. 154, 573-590. https://doi.org/10.1016/j.compstruct.2016.07.076.
  32. Fares, M.E., Elmarghany, M.K., Atta, D. and Salem, M.G. (2018), "Bending and free vibration of multilayered functionally graded doubly curved shells by an improved layerwise theory", Compos. Part B-Eng., 154, 272-284. https://doi.org/10.1016/j.compositesb.2018.07.038.
  33. Fazzolari, F.A. and Carrera, E. (2014), "Refined hierarchical kinematics quasi-3D Ritz models for free vibration analysis of doubly curved FGM shells and sandwich shells with FGM core", J. Sound Vib., 333(5), 1485-1508. https://doi.org/10.1016/j.jsv.2013.10.030.
  34. Fenjan, N.M., Moustafa, N.M. and Faleh, N.M. (2020), "Scaledependent thermal vibration analysis of FG beams having porosities based on DQM", Adv. Nano Res., 8(4), 283-292. https://doi.org/10.12989/anr.2020.8.4.283.
  35. Ghandourah, E., Abdraboh, A.M. (2020), "Dynamic analysis of functionally graded nonlocal nanobeam with different porosity models", Steel Compos. Struct., 36(3), 293-305. http://dx.doi.org/10.12989/scs.2020.36.3.293.
  36. Ghassabi, M., Zarastvand, M.R. and Talebitooti, R. (2020), "Investigation of state vector computational solution on modeling of wave propagation through functionally graded nanocomposite doubly curved thick structures", Eng. Comput., 36(4), 1417-1433. https://doi.org/10.1007/s00366-019-00773-6.
  37. Hadji, L. (2020), "Influence of the distribution shape of porosity on the bending of FGM beam using a new higher order shear deformation model", Smart Struct. Syst., 26(2), 253-262. https://doi.org/10.12989/sss.2020.26.2.253.
  38. Hamed, M.A., Eltaher, M.A., Sadoun, A.M. and Almitani, K.H. (2016), "Free vibration of symmetric and sigmoid functionally graded nanobeams", Appl. Phys. A, 122, 829. https://doi.org/10.1007/s00339-016-0324-0.
  39. Hamed, M.A., Sadoun, A.M. and Eltaher, M.A. (2019), "Effects of porosity models on static behavior of size dependent functionally graded beam", Struct. Eng. Mech., 71(1), 89-98. https://doi.org/10.12989/sem.2019.71.1.089.
  40. Hamed, M.A., Mohamed, S.A. and Eltaher, M.A. (2020), "Buckling analysis of sandwich beam rested on elastic foundation and subjected to varying axial in-plane loads", Steel Compos. Struct., 34(1), 75-89. https://doi.org/10.12989/scs.2020.34.1.075.
  41. Hosseini-Hashemi, S., Ilkhani, M.R. and Fadaee, M. (2012), "Identification of the validity range of Donnell and Sanders shell theories using an exact vibration analysis of functionally graded thick cylindrical shell panel", Acta Mech., 223(5), 1101-1118. https://doi.org/10.1007/s00707-011-0601-0.
  42. Kapuria, S., Patni, M. and Yasin, M.Y. (2015), "A quadrilateral shallow shell element based on the third-order theory for functionally graded plates and shells and the inaccuracy of rule of mixtures", Eur. J. Mech. - A/Solids., 49, 268-282. https://doi.org/10.1016/j.euromechsol.2014.06.010.
  43. Kar, V.R. and Panda, S.K. (2014), "Nonlinear free vibration of functionally graded doubly curved shear deformable panels using finite element method", J. Vib. Control., 22(7), 1935-1949. https://doi.org/10.1177/1077546314545102.
  44. Kar, V.R. and Panda, S.K. (2015), "Thermoelastic analysis of functionally graded doubly curved shell panels using nonlinear finite element method", Compos. Struct., 129, 202-212. https://doi.org/10.1016/j.compstruct.2015.04.006.
  45. Khayat, M., Dehghan, S.M., Najafgholipour, M.A. and Baghlani, A. (2018). "Free vibration analysis of functionally graded cylindrical shells with different shell theories using semianalytical method", Steel Compos. Struct., 28(6), 735-748. https://doi.org/10.12989/scs.2018.28.6.735.
  46. Kiani, Y. (2019), "NURBS-based thermal buckling analysis of graphene platelet reinforced composite laminated skew plates", J. Therm. Stresses., 1-19. https://doi.org/10.1080/01495739.2019.1673687.
  47. Le-Xuan, D., Pham-Quoc, H., Tran-The, V. and Nguyen-Van, N. (2018), "Static and Free Vibration Analysis of Functionally Graded Shells Using a Cell-Based Smoothed Discrete Shear Gap Method and Triangular Elements", 381-394. https://doi.org/10.1007/978-981-10-7149-2_26.
  48. Liang, C. and Wang, Y.Q. (2020), "A quasi-3D trigonometric shear deformation theory for wave propagation analysis of FGM sandwich plates with porosities resting on viscoelastic foundation", Compos. Struct., 247, 112478. https://doi.org/10.1016/j.compstruct.2020.112478.
  49. Madenci, E. (2019), "A refined functional and mixed formulation to static analyses of fgm beams", Struct. Eng. Mech., 69(4), 427-437. https://doi.org/10.12989/sem.2019.69.4.427.
  50. Mantari, J.L. (2015), "Refined and generalized hybrid type quasi3D shear deformation theory for the bending analysis of functionally graded shells", Compos. Part B-Eng., 83, 142-152. https://doi.org/10.1016/j.compositesb.2015.08.048.
  51. Matsunaga, H. (2008), "Free vibration and stability of functionally graded shallow shells according to a 2D higher-order deformation theory", Compos. Struct., 84(2), 132-146. https://doi.org/10.1016/j.compstruct.2007.07.006.
  52. Matsunaga, H. (2009), "Free vibration and stability of functionally graded circular cylindrical shells according to a 2D higher-order deformation theory", Compos. Struct., 88(4), 519-531. https://doi.org/10.1016/j.compstruct.2008.05.019.
  53. Mehala, T., Belabed, Z., Tounsi, A. and Beg, O.A. (2018), "Investigation of influence of homogenization models on stability and dynamic of FGM plates on elastic foundations", Geomech. Eng., 16(3), 257-271. http://dx.doi.org/10.12989/gae.2018.16.3.257.
  54. Melaibari, A., Abo-bakr, R.M., Mohamed, S.A. and Eltaher, M.A. (2020a), "Static stability of higher order functionally graded beam under variable axial load", Alex. Eng. J., 59(3), 1661-1675. https://doi.org/10.1016/j.aej.2020.04.012.
  55. Melaibari, A., Khoshaim, A.B., Mohamed, S.A., Eltaher, M.A., (2020b), "Static stability and of symmetric and sigmoid functionally graded beam under variable axial load", Steel Compos. Struct., 35(5), 671-685. http://dx.doi.org/10.12989/scs.2020.35.5.671.
  56. Nath, J.K. and Das, T. (2017), "Static and free vibration analysis of multilayered functionally graded shells and plates using an efficient zigzag theory", Mech. Adv. Mat. Struct., 26(9), 770-788. https://doi.org/10.1080/15376494.2017.1410915.
  57. Neves, A.M.A., Ferreira, A.J.M., Carrera, E., Cinefra, M., Roque, C.M.C., Jorge, R.M.N. and Soares, C.M.M. (2013), "Free vibration analysis of functionally graded shells by a higherorder shear deformation theory and radial basis functions collocation, accounting for through-the-thickness deformations", Eur. J. Mech. - A/Solids., 37, 24-34. https://doi.org/10.1016/j.euromechsol.2012.05.005.
  58. Oktem, A.S., Mantari, J.L. and Soares, C.G. (2012), "Static response of functionally graded plates and doubly-curved shells based on a higher order shear deformation theory", Eur. J. Mech. - A/Solids., 36, 163-172. https://doi.org/10.1016/j.euromechsol.2012.03.002.
  59. Oner, E., Yaylaci, M. and Birinci, A. (2015), "Analytical solution of a contact problem and comparison with the results from FEM", Struct. Eng. Mech., 54(4), 607-622. https://doi.org/10.12989/sem.2015.54.4.607.
  60. Pradyumna, S. and Bandyopadhyay, J.N. (2008), "Free vibration analysis of functionally graded curved panels using a higherorder finite element formulation", J. Sound Vib., 318(1-2), 176-192. https://doi.org/10.1016/j.jsv.2008.03.056.
  61. Punera, D. and Kant, T. (2017), "Free vibration of functionally graded open cylindrical shells based on several refined higher order displacement models", Thin. Wall. Struct., 119, 707-726. https://doi.org/10.1016/j.tws.2017.07.016.
  62. Punera, D. and Kant, T. (2019), "A critical review of stress and vibration analyses of functionally graded shell structures", Compos. Struct., 210, 787-809. https://doi.org/10.1016/j.compstruct.2018.11.084.
  63. Reddy, J.N. (2004), Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, (2nd Ed.), CRC Press.
  64. Selmi, A. (2019), "Effectiveness of SWNT in reducing the crack effect on the dynamic behavior of aluminium alloy", Adv. Nano Res., 7(5), 365-377. https://doi.org/10.12989/anr.2019.7.5.365.
  65. Selmi, A. (2020), "Exact solution for nonlinear vibration of clamped-clamped functionally graded buckled beam", Smart Struct. Syst., 26(3), 361-371. https://doi.org/10.12989/sss.2020.26.3.361.
  66. Shahmohammadi, M.A., Azhari, M. and Saadatpour, M.M. (2020), "Free vibration analysis of sandwich FGM shells using isogeometric B-spline finite strip method", Steel Compos. Struct., 34(3), 361-376. https://doi.org/10.12989/scs.2020.34.3.361.
  67. Shah, P.H. and Batra, R.C. (2016), "Stretching and bending deformations due to normal and shear tractions of doubly curved shells using third-order shear and normal deformable theory", Mech. Adv. Mat. Struct., 25(15-16), 1276-1296. https://doi.org/10.1080/15376494.2016.1194505.
  68. She, G.L. (2020), "Wave propagation of FG polymer composite nanoplates reinforced with GNPs", Steel Compos. Struct., 37(1), 27-35. https://doi.org/10.12989/scs.2020.37.1.027 27.
  69. Shmatko, T. and Bhaskar, A. (2017), "R-functions theory applied to investigation of nonlinear free vibrations of functionally graded shallow shells", Nonlinear Dynam., 93(1), 189-204. https://doi.org/10.1007/s11071-017-3922-2.
  70. Soliman, A.E., Eltaher, M.A., Attia, M.A. and Alshorbagy, A.E. (2018a), "Analysis of crack occurs under unsteady pressure and temperature in a natural gas facility by applying FGM", Struct. Eng. Mech., 66(1), 97-111. https://doi.org/10.12989/sem.2018.66.1.097.
  71. Soliman, A.E., Eltaher, M.A., Attia, M.A. and Alshorbagy, A.E. (2018b), "Nonlinear transient analysis of FG pipe subjected to internal pressure and unsteady temperature in a natural gas facility", Struct. Eng. Mech., 66(1), 85-96. https://doi.org/10.12989/sem.2018.66.1.085.
  72. Su, Z., Jin, G. and Ye, T. (2014), "Free vibration analysis of moderately thick functionally graded open shells with general boundary conditions", Compos. Struct., 117, 169-186. https://doi.org/10.1016/j.compstruct.2014.06.026.
  73. Talebitooti, R., Gohari, H.D. and Zarastvand, M.R. (2017), "Multi objective optimization of sound transmission across laminated composite cylindrical shell lined with porous core investigating Non-dominated Sorting Genetic Algorithm", Aerosp. Sci. Tech., 69, 269-280. https://doi.org/10.1016/j.ast.2017.06.008.
  74. Teng, M.W. and Wang, Y.Q. (2021), "Nonlinear forced vibration of simply supported functionally graded porous nanocomposite thin plates reinforced with graphene platelets", Thin Wall. Struct., 164, 107799. https://doi.org/10.1016/j.tws.2021.107799.
  75. Tornabene, F., Fantuzzi, N. and Bacciocchi, M. (2014), "Free vibrations of free-form doubly-curved shells made of functionally graded materials using higher-order equivalent single layer theories", Compos. Part B-Eng., 67, 490-509. https://doi.org/10.1016/j.compositesb.2014.08.012.
  76. Trinh, M.C. and Kim, S.E. (2019), "A three variable refined shear deformation theory for porous functionally graded doublycurved shell analysis", Aerosp. Sci. Tech., 94 105356. https://doi.org/10.1016/j.ast.2019.105356.
  77. Vinyas, M. (2020), "On frequency response of porous functionally graded magneto-electro-elastic circular and annular plates with different electro-magnetic conditions using HSDT", Compos. Struct., 240, 112044. https://doi.org/10.1016/j.compstruct.2020.112044.
  78. Vinyas, M., Harursampath, D. and Kattimani, S.C. (2021), "On vibration analysis of functionally graded carbon nanotube reinforced magneto-electro-elastic plates with different electromagnetic conditions using higher order finite element methods", Def. Tech., 17(1), 287-303. https://doi.org/10.1016/j.dt.2020.03.012.
  79. Yaghoobi, H. and Yaghoobi, P. (2013), "Buckling analysis of sandwich plates with FGM face sheets resting on elastic foundation with various boundary conditions: an analytical approach", Meccanica, 48, 2019-2035. https://doi.org/10.1007/s11012-013-9720-0.
  80. YaminiPoorna, N. and Sai Ram, K.S. (2019), "Free vibration of functionally graded cylindrical and spherical shell panels integrated with piezoelectric layers", ISSS J. Micro Smart Sys., 8(2), 113-125.https://doi.org/10.1007/s41683-019-00041-1.
  81. Yaylaci, M. and Birinci, A. (2013), "The receding contact problem of two elastic layers supported by two elastic quarter planes", Struct. Eng. Mech., 48(2), 241-255. https://doi.org/10.12989/sem.2013.48.2.241.
  82. Yaylaci, M. (2016), "The investigation crack problem through numerical analysis", Struct. Eng. Mech., 57(6), 1143-1156. https://doi.org/10.12989/sem.2016.57.6.1143.
  83. Yaylaci, M., Terzi, C. and Avcar, M. (2019). "Numerical analysis of the receding contact problem of two bonded layers resting on an elastic half plane", Struct. Eng. Mech., 72(6), 775-783. https://doi.org/10.12989/sem.2019.72.6.775.
  84. Yaylaci, E.U., Yaylaci, M., Olmez, H. and Birinci, A. (2020a), "Artificial neural network calculations for a receding contact problem", Comput. Concrete, 25(6), 551-563. https://doi.org/10.12989/cac.2020.25.6.551.
  85. Yaylaci, M. and Avcar, M. (2020b), "Finite element modeling of contact between an elastic layer and two elastic quarter planes", Comput. Concrete, 26(2), 107-114. https://doi.org/10.12989/cac.2020.26.2.107.
  86. Yaylaci, M., Eyuboglu, A., Adiyaman, G., Yaylaci, E.U., Oner, E. and Birinci, A. (2021), "Assessment of different solution methods for receding contact problems in functionally graded layered mediums", Mech. Mater., 154, 103730. https://doi.org/10.1016/j.mechmat.2020.103730.
  87. Wang, Y.Q. and Zu, J.W. (2017), "Vibration behaviors of functionally graded rectangular plates with porosities and moving in thermal environment", Aeros. Sci. Tech., 69, 550-562. https://doi.org/https://doi.org/10.1016/j.ast.2017.07.023.
  88. Wang, Y.Q. (2018a), "Electro-mechanical vibration analysis of functionally graded piezoelectric porous plates in the translation state", Acta Astronaut., 143, 263-271. https://doi.org/10.1016/j.actaastro.2017.12.004.
  89. Wang, Y., Ye, C. and Zu, J.W. (2018b), "Identifying the temperature effect on the vibrations of functionally graded cylindrical shells with porosities", Appl. Math. Mech., 39(11), 1587-1604. https://doi.org/10.1007/s10483-018-2388-6.
  90. Wang, Y.Q., Wan, Y.H. and Zu, J.W. (2019a), "Nonlinear dynamic characteristics of functionally graded sandwich thin nanoshells conveying fluid incorporating surface stress influence", Thin Wall. Struct., 135, 537-547. https://doi.org/10.1016/j.tws.2018.11.023.
  91. Wang, Y.Q., Ye, C. and Zu, J.W. (2019b), "Nonlinear vibration of metal foam cylindrical shells reinforced with graphene platelets", Aerosp. Sci. Tech., 85, 359-370. https://doi.org/10.1016/j.ast.2018.12.022.
  92. Ye, C. and Wang, Y.Q. (2021), "Nonlinear forced vibration of functionally graded graphene platelet-reinforced metal foam cylindrical shells: internal resonances", Nonlinear Dynam., 104(3), 2051-2069. https://doi.org/10.1007/s11071-021-06401-7.
  93. Zarastvand, M.R., Ghassabi, M. and Talebitooti, R. (2019), "Acoustic Insulation Characteristics of Shell Structures: A Review", Arch. Comput. Meth. Eng., https://doi.org/10.1007/s11831-019-09387-z.
  94. Zhao, J., Xie, F., Wang, A., Shuai, C., Tang, J. and Wang, Q. (2019), "Vibration behavior of the functionally graded porous (FGP) doubly-curved panels and shells of revolution by using a semi-analytical method", Compos. Part B-Eng., 157, 219-238. https://doi.org/10.1016/j.compositesb.2018.08.087.
  95. Zhao, X., Lee, Y.Y. and Liew, K.M. (2009), "Thermoelastic and vibration analysis of functionally graded cylindrical shells", Int. J. Mech. Sci., 51(9-10), 694-707. https://doi.org/10.1016/j.ijmecsci.2009.08.001.