Steel and Composite Structures

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Dynamic response of FG porous nanobeams subjected thermal and magnetic fields under moving load

  • Esen, Ismail (Department of Mechanical Engineering, Karabuk University) ;
  • Alazwari, Mashhour A. (Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University) ;
  • Eltaher, Mohamed A (Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University) ;
  • Abdelrahman, Alaa A. (Mechanical Design and Production Department, Faculty of Engineering, Zagazig University)
  • Received : 2021.12.31
  • Accepted : 2022.03.29
  • Published : 2022.03.25
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Abstract

The free and live load-forced vibration behaviour of porous functionally graded (PFG) higher order nanobeams in the thermal and magnetic fields is investigated comprehensively through this work in the framework of nonlocal strain gradient theory (NLSGT). The porosity effects on the dynamic behaviour of FG nanobeams is investigated using four different porosity distribution models. These models are exploited; uniform, symmetrical, condensed upward, and condensed downward distributions. The material characteristics gradation in the thickness direction is estimated using the power-law. The magnetic field effect is incorporated using Maxwell's equations. The third order shear deformation beam theory is adopted to incorporate the shear deformation effect. The Hamilton principle is adopted to derive the coupled thermomagnetic dynamic equations of motion of the whole system and the associated boundary conditions. Navier method is used to derive the analytical solution of the governing equations. The developed methodology is verified and compared with the available results in the literature and good agreement is observed. Parametric studies are conducted to show effects of porosity parameter; porosity distribution, temperature rise, magnetic field intensity, material gradation index, non-classical parameters, and the applied moving load velocity on the vibration behavior of nanobeams. It has been showed that all the analyzed conditions have significant effects on the dynamic behavior of the nanobeams. Additionally, it has been observed that the negative effects of moving load, porosity and thermal load on the nanobeam dynamics can be reduced by the effect of the force induced from the directed magnetic field or can be kept within certain desired design limits by controlling the intensity of the magnetic field.

Keywords

Acknowledgement

The Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, Saudi Arabia has funded this project, under grant No. (FP-130-43).

References

  1. Abdelrahman, A.A., Esen, I. and Eltaher, M.A. (2021a), "Vibration response of Timoshenko perforated microbeams under accelerating load and thermal environment", Appl. Math. Comput., 407, 126307. https://doi.org/10.1016/j.amc.2021.126307.
  2. Abdelrahman, A.A., Esen, I., Ozarpa, C., Shaltout, R., Eltaher, M. A. and Assie, A.E. (2021b), "Dynamics of perforated higher order nanobeams subject to moving load using the nonlocal strain gradient theory", Smart Struct. Syst., 28(4), 515-533. https://doi.org/10.12989/sss.2021.28.4.515.
  3. Abdelrahman, A.A., Esen, I., Daikh, A.A. and Eltaher, M.A. (2021c), "Dynamic analysis of FG nanobeam reinforced by carbon nanotubes and resting on elastic foundation under moving load", Mech. Based Design Struct. Machines, 1-24. https://doi.org/10.1080/15397734.2021.1999263.
  4. Akbas, S.D., Dastjerdi, S., Akgoz, B. and Civalek, O. (2021), "Dynamic Analysis of Functionally Graded Porous Microbeams under Moving Load", Transport Porous Media, 0123456789. https://doi.org/10.1007/s11242-021-01686-z.
  5. Al-shujairi, M. and Mollamahmutoglu, C. (2018), "Buckling and free vibration analysis of functionally graded sandwich micro-beams resting on elastic foundation by using nonlocal strain gradient theory in conjunction with higher order shear theories under thermal effect", Compos. Part B Eng., 154, 292-312. https://doi.org/10.1016/j.compositesb.2018.08.103.
  6. Alazwari, M.A., Daikh, A.A., Houari, M.S.A., Tounsi, A. and Eltaher, M.A. (2021), "On static buckling of multilayered carbon nanotubes reinforced composite nanobeams supported on non-linear elastic foundations", Steel Compos. Struct., 40(3), 389-404. https://doi.org/10.12989/scs.2021.40.3.389.
  7. 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.
  8. Alazwari, M.A., Daikh and Eltaher, M.A. (2022), "Novel Quasi 3D Theory for Mechanical Responses of FG-CNTs Reinforced Composite Nanoplates", Adv. Nano Res., 12 (2), 117-137. https://doi.org/https://doi.org/10.12989/anr.2022.12.2.117.
  9. Almitani, K.H., Abdelrahman, A.A. and Eltaher, M.A. (2020), "Stability of perforated nanobeams incorporating surface energy effects", Steel Compos. Struct., 35(4), 555-566. https://doi.org/10.12989/scs.2020.35.4.555.
  10. Alshorbagy, A.E., Eltaher, M.A. and Mahmoud, F.F. (2013), "Static analysis of nanobeams using nonlocal FEM", J. Mech. Sci. Technol., 27(7), 2035-2041. https://doi.org/10.1007/s12206-013-0212-x.
  11. Ansari, R., Shojaei, M.F. and Rouhi, H. (2015), "Small-scale Timoshenko beam element", European J. Mech. A/Solids, 53, 19-33. https://doi.org/10.1016/j.euromechsol.2015.02.005.
  12. Arani, A.G. and Jalaei, M.H. (2017), "Investigation of the longitudinal magnetic field effect on dynamic response of viscoelastic graphene sheet based on sinusoidal shear deformation theory", Physica B Condensed Matter, 506, 94-104. https://doi.org/10.1016/j.physb.2016.11.004.
  13. Asghari, M., Rahaeifard, M., Kahrobaiyan, M.H. and Ahmadian, M.T. (2011), "The modified couple stress functionally graded Timoshenko beam formulation", Mater. Design, 32(3), 1435-1443. https://doi.org/10.1016/J.MATDES.2010.08.046.
  14. Barretta, R., Faghidian, S.A., 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 Eng., 154, 20-32. https://doi.org/10.1016/j.compositesb.2018.07.036.
  15. Barretta, R., Faghidian, S.A., Marotti de Sciarra, F. and Vaccaro, M.S. (2020), "Nonlocal strain gradient torsion of elastic beams: variational formulation and constitutive boundary conditions", Arch. Appl. Mech., 90(4), 691-706. https://doi.org/10.1007/s00419-019-01634-w
  16. Barretta, R. and Marotti de Sciarra, F. (2018), "Constitutive boundary conditions for nonlocal strain gradient elastic nano-beams", J. Eng. Sci., 130, 187-198. https://doi.org/10.1016/j.ijengsci.2018.05.009.
  17. Barretta, R/ and Marotti de Sciarra, F. (2019), "Variational nonlocal gradient elasticity for nano-beams", J. Eng. Sci., 143, 73-91. https://doi.org/https://doi.org/10.1016/j.ijengsci.2019.06.016.
  18. Bensaid, I., Bekhadda, A. and Kerboua, B. (2018), "Dynamic analysis of higher order shear-deformable nanobeams resting on elastic foundation based on nonlocal strain gradient theory", Adv. Nano Res., 6(3), 279. https://doi.org/10.12989/anr.2018.6.3.279.
  19. Bhattacharya, S. and Das, D. (2019), "Free vibration analysis of bidirectional-functionally graded and double-tapered rotating micro-beam in thermal environment using modified couple stress theory", Compos. Struct., 215, 471-492. https://doi.org/10.1016/j.compstruct.2019.01.080.
  20. Chen, D., Yang, J. and Kitipornchai, S. (2016), "Free and forced vibrations of shear deformable functionally graded porous beams", J. Mech. Sci., 108-109, 14-22. https://doi.org/10.1016/j.ijmecsci.2016.01.025.
  21. Chu, L., Dui, G. and Zheng, Y. (2020), "Thermally induced nonlinear dynamic analysis of temperature-dependent functionally graded flexoelectric nanobeams based on nonlocal simplified strain gradient elasticity theory", European J. Mech. A/Solids, 82, 103999. https://doi.org/https://doi.org/10.1016/j.euromechsol.2020.103999.
  22. Chu, L., Li, Y. and Dui, G. (2020), "Nonlinear analysis of functionally graded flexoelectric nanoscale energy harvesters", J. Mech. Sci., 167, 105282. https://doi.org/https://doi.org/10.1016/j.ijmecsci.2019.105282.
  23. Ding, H.X. and She, G.L. (2021), "A higher-order beam model for the snap-buckling analysis of FG pipes conveying fluid", Struct. Eng. Mech., 80(1), 63-72. https://doi.org/10.12989/sem.2021.80.1.063.
  24. Ebrahimi, F., Dabbagh, A. and Taheri, M. (2021), "Vibration analysis of porous metal foam plates rested on viscoelastic substrate", Eng. Comput., 37(4), 3727-3739. https://doi.org/10.1007/s00366-020-01031-w.
  25. Eltaher, M.A., Alshorbagy, A.E. and Mahmoud, F.F. (2013), "Vibration analysis of Euler-Bernoulli nanobeams by using finite element method", Appl. Math. Modelling, 37(7), 4787-4797. https://doi.org/10.1016/j.apm.2012.10.016.
  26. Eltaher, M. A., Emam, S. A. and Mahmoud, F. F. (2012), "Free vibration analysis of functionally graded size-dependent nanobeams", Appl. Math. Comput., 218(14), 7406-7420. https://doi.org/10.1016/j.amc.2011.12.090.
  27. Eltaher, M.A., Emam, S.A. and Mahmoud, F.F. (2013), "Static and stability analysis of nonlocal functionally graded nanobeams", Compos. Struct., 96, 82-88. https://doi.org/10.1016/j.compstruct.2012.09.030.
  28. Eltaher, M.A. and Abdelrahman, A.A. (2020), "Bending behavior of squared cutout nanobeams incorporating surface stress effects", Steel Compos. Struct., 36(2), 143-161. https://doi.org/10.12989/scs.2020.36.2.143.
  29. Eltaher, M.A., Abdelmoteleb, H.E., Daikh, A.A. and Abdelrahman, A. A. (2021), "Vibrations and stress analysis of rotating perforated beams by using finite elements method", Steel Compos. Struct., 41(4), 505-520. https://doi.org/10.12989/scs.2021.41.4.505.
  30. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54(9), 4703-4710. https://doi.org/10.1063/1.332803.
  31. Eringen, A.C. and Suhubi, E.S. (1964), "Nonlinear theory of simple micro-elastic solids-I", J. Eng. Sci., 2(2), 189-203. https://doi.org/https://doi.org/10.1016/0020-7225(64)90004-7.
  32. Esen, I. (2019), "Dynamic response of a functionally graded Timoshenko beam on two-parameter elastic foundations due to a variable velocity moving mass", J. Mech. Sci., 153-154, 21-35. https://doi.org/10.1016/j.ijmecsci.2019.01.033.
  33. Esen, I. (2020), "Dynamics of size-dependant Timoshenko micro beams subjected to moving loads", J. Mech. Sci., 175, 105501. https://doi.org/https://doi.org/10.1016/j.ijmecsci.2020.105501.
  34. Esen, I., Abdelrhmaan, A. A. and Eltaher, M. A. (2021a), "Free vibration and buckling stability of FG nanobeams exposed to magnetic and thermal fields", Eng. Comput., https://doi.org/10.1007/s00366-021-01389-5.
  35. Esen, I., Ozarpa, C. and Eltaher, M. A. M. A. (2021b), "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.
  36. Esfahani, M., Hoseinzade, M., Shakiba, M., Arbab, F., Yekrangnia, M. and Pachideh, G. (2021), "Experimental investigation of residual flexural capacity of damaged reinforced concrete beams exposed to elevated temperatures", Eng. Struct., 240, 112388. https://doi.org/10.1016/j.engstruct.2021.112388.
  37. Fan, F., Cai, X., Sahmani, S. and Safaei, B. (2021), "Isogeometric thermal postbuckling analysis of porous FGM quasi-3D nanoplates having cutouts with different shapes based upon surface stress elasticity", Compos. Struct., 262, 113604. https://doi.org/10.1016/j.compstruct.2021.113604.
  38. Faroughi, S., Rahmani, A. and Friswell, M. I. (2020), "On wave propagation in two-dimensional functionally graded porous rotating nano-beams using a general nonlocal higher-order beam model", Appl. Math. Modelling, 80, 169-190. https://doi.org/10.1016/j.apm.2019.11.040.
  39. Fleck, N. A., Muller, G. M., Ashby, M. F. and Hutchinson, J. W. (1994), "Strain gradient plasticity: Theory and experiment", Acta Metallurgica et Materialia, 42(2), 475-487. https://doi.org/https://doi.org/10.1016/0956-7151(94)90502-9.
  40. Ghandourah, E. E., Ahmed, H. M., Eltaher, M. A., Attia, M. A. and Abdraboh, A. M. (2021), "Free vibration of porous FG nonlocal modified couple nanobeams via a modified porosity model", Adv. Nano Res., 11(4), 405-422. https://doi.org/10.12989/anr.2021.11.4.405.
  41. Ibnorachid, Z., Boutahar, L., El Bikri, K. and Benamar, R. (2019), "Buckling temperature and natural frequencies of thick porous functionally graded beams resting on elastic foundation in a thermal environment", Adv. Acoustics Vib., 22. https://doi.org/10.1155/2019/7986569.
  42. Jalaei, M. H., Arani, A. G. and Nguyen-Xuan, H. (2019), "Investigation of thermal and magnetic field effects on the dynamic instability of FG Timoshenko nanobeam employing nonlocal strain gradient theory", J. Mech. Sci., 161-162. https://doi.org/10.1016/j.ijmecsci.2019.105043.
  43. Jalaei, M. H. and Civalek. (2019). "On dynamic instability of magnetically embedded viscoelastic porous FG nanobeam", J. Eng. Sci., 143, 14-32. https://doi.org/10.1016/j.ijengsci.2019.06.013.
  44. Jalaei, Mohammad Hossein and Thai, H. T. (2019), "Dynamic stability of viscoelastic porous FG nanoplate under longitudinal magnetic field via a nonlocal strain gradient quasi-3D theory", Compos. Part B Eng., 175, 107164. https://doi.org/10.1016/j.compositesb.2019.107164.
  45. Jankowski, P., Kamil Zur, K., Kim, J. and Reddy, J. N. (2020), "On the bifurcation buckling and vibration of porous nanobeams", Compos. Struct., 250. https://doi.org/10.1016/j.compstruct.2020.112632.
  46. Jankowski, P., Zur, K. K., Kim, J., Lim, C. W. and Reddy, J. N. (2021), "On the piezoelectric effect on stability of symmetric FGM porous nanobeams", Compos. Struct., 267. https://doi.org/10.1016/j.compstruct.2021.113880.
  47. Karamanli, A. and Aydogdu, M. (2019), "Size dependent flapwise vibration analysis of rotating two-directional functionally graded sandwich porous microbeams based on a transverse shear and normal deformation theory", J. Mech. Sci., 159, 165-181. https://doi.org/10.1016/j.ijmecsci.2019.05.047.
  48. Khalili, S.M.R., Jafari, A.A. and Eftekhari, S.A. (2010), "A mixed Ritz-DQ method for forced vibration of functionally graded beams carrying moving loads", Compos. Struct., 92(10), 2497-2511. https://doi.org/https://doi.org/10.1016/j.compstruct.2010.02.012
  49. Khadir, A.I., Daikh and Eltaher, M.A. (2021), "Novel four-unknowns quasi 3D theory for bending, buckling and free vibration of functionally graded carbon nanotubes reinforced composite laminated nanoplates", Adv. Nano Res., 11(6), 621-640. https://doi.org/10.12989/anr.2021.11.6.621.
  50. Kim, J., Zur, K.K. and Reddy, J.N. (2019), "Bending, free vibration and buckling of modified couples stress-based functionally graded porous micro-plates", Compos. Struct., 209, 879-888. https://doi.org/10.1016/j.compstruct.2018.11.023.
  51. Kong, S., Zhou, S., Nie, Z. and Wang, K. (2008), "The size-dependent natural frequency of Bernoulli-Euler micro-beams", J. Eng. Sci., 46(5), 427-437. https://doi.org/10.1016/j.ijengsci.2007.10.002.
  52. Kraus, J. (1992). Electromagnetics, McGraw-Hill, NY, USA.
  53. Le, C.I., Le, N.A.T. and Nguyen, D.K. (2020), "Free vibration and buckling of bidirectional functionally graded sandwich beams using an enriched third-order shear deformation beam element", Compos. Struct., 113309. https://doi.org/10.1016/j.compstruct.2020.113309.
  54. Lei, Y.L., Gao, K., Wang, X. and Yang, J. (2020), "Dynamic behaviors of single- and multi-span functionally graded porous beams with flexible boundary constraints", Appl. Math. Modelling, 83, 754-776. https://doi.org/10.1016/j.apm.2020.03.017.
  55. Li, L., Hu, Y. and Li, X. (2016), "Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory", J. Mech. Sci., 115-116. https://doi.org/10.1016/j.ijmecsci.2016.06.011.
  56. Lim, C.W., Zhang, G. and Reddy, J.N. (2015a), "A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation", J. Mech. Phys. Solids, 78, 298-313. https://doi.org/10.1016/j.jmps.2015.02.001.
  57. Lim, C.W., Zhang, G. and Reddy, J.N. (2015b), "A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation", J. Mech. Phys. Solids, 78, 298-313. https://doi.org/10.1016/j.jmps.2015.02.001.
  58. Lin, F., Tong, L.H., Shen, H.S., Lim, C.W. and Xiang, Y. (2020), "Assessment of first and third order shear deformation beam theories for the buckling and vibration analysis of nanobeams incorporating surface stress effects", J. Mech. Sci., 186, 105873. https://doi.org/10.1016/j.ijmecsci.2020.105873.
  59. Liu, H., Liu, H. and Yang, J. (2018), "Vibration of FG magneto-electro-viscoelastic porous nanobeams on visco-Pasternak foundation", Compos. Part B Eng., 155, 244-256. https://doi.org/10.1016/j.compositesb.2018.08.042.
  60. Lu, L., Guo, X. and Zhao, J. (2017), "Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory", J. Eng. Sci., 116, 12-24. https://doi.org/10.1016/j.ijengsci.2017.03.006.
  61. Lu, L., She, G. L. and Guo, X. (2021), "Size-dependent postbuckling analysis of graphene reinforced composite microtubes with geometrical imperfection", J. Mech. Sci., 199, 106428. https://doi.org/10.1016/j.ijmecsci.2021.106428.
  62. Ma, H.M., Gao, X.L. and Reddy, J.N. (2008), "A microstructure-dependent Timoshenko beam model based on a modified couple stress theory", J. Mech. Phys. Solids, 56(12), 3379-3391. https://doi.org/10.1016/J.JMPS.2008.09.007.
  63. Melaibari, A., Daikh, A.A., Basha, M., Abdalla, A.W., Othman, R., Almitani, K.H., Hamed, M.A., Abdelrahman, A. and Eltaher, M. A. (2022), "Free Vibration of FG-CNTRCs Nano-Plates/Shells with Temperature-Dependent Properties", Mathematics, 10(4), 583. https://doi.org/10.3390/math10040583.
  64. Mojahedin, A., Jabbari, M. and Rabczuk, T. (2018), "Thermoelastic analysis of functionally graded porous beam", J. Thermal Stresses, 41(8), 937-950. https://doi.org/10.1080/01495739.2018.1446374.
  65. Mollamahmutoglu, C. and Mercan, A. (2019), "A novel functional and mixed finite element analysis of functionally graded micro-beams based on modified couple stress theory", Compos. Struct., 223, 110950. https://doi.org/10.1016/j.compstruct.2019.110950.
  66. Najafi, F., Shojaeefard, M.H. and Googarchin, H.S. (2017), "Nonlinear dynamic response of FGM beams with Winkler-Pasternak foundation subject to noncentral low velocity impact in thermal field", Compos. Struct., 167, 132-143. https://doi.org/https://doi.org/10.1016/j.compstruct.2017.01.063.
  67. Nejad, M.Z. and Hadi, A. (2016), "Non-local analysis of free vibration of bi-directional functionally graded Euler-Bernoulli nano-beams", J. Eng. Sci., 105, 1-11. https://doi.org/10.1016/j.ijengsci.2016.04.011.
  68. Nguyen, D.K., Gan, B.S. and Le, T.H. (2013), "Dynamic Response of Non-Uniform Functionally Graded Beams Subjected to a Variable Speed Moving Load", J. Computational Sci. Technol., 7(1), 12-27. https://doi.org/10.1299/jcst.7.12.
  69. Pachideh, G., Gholhaki, M., Moshtagh, A. and Felaverjani, M.K. (2019), "An Investigation on the Effect of High Temperatures on the Mechanical Properties and Microstructure of Concrete Containing Multiwalled Carbon Nanotubes", Mater. Performance Characterization, 8(3), 503-517. 10.1520/MPC20180061.
  70. Pinnola, F.P., Faghidian, S.A., Barretta, R. and Marotti de Sciarra, F. (2020), "Variationally consistent dynamics of nonlocal gradient elastic beams", J. Eng. Sci., 149, 103220. https://doi.org/10.1016/j.ijengsci.2020.103220.
  71. Rahmani, O. and Pedram, O. (2014), "Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory", J. Eng. Sci., 77, 55-70. https://doi.org/10.1016/j.ijengsci.2013.12.003.
  72. Reddy, J.N. (2007), "Nonlocal theories for bending, buckling and vibration of beams", J. Eng. Sci., 45(2-8), 288-307. https://doi.org/10.1016/j.ijengsci.2007.04.004.
  73. Reddy, JN. and Chin, C. D. (1998a), "Thermomechanical analysis of functionally graded cylinders and plates", J. Thermal Stresses, 21(6), 593-626. https://doi.org/10.1080/01495739808956165.
  74. Salari, E., Sadough Vanini, S.A., Ashoori, A.R. and Akbarzadeh, A.H. (2020), "Nonlinear thermal behavior of shear deformable FG porous nanobeams with geometrical imperfection: Snap-through and postbuckling analysis", J. Mech. Sci., 178, 105615. https://doi.org/10.1016/j.ijmecsci.2020.105615.
  75. Seddighi, F., Pachideh, G. and Salimbahrami, S.B. (2021), "A study of mechanical and microstructures properties of autoclaved aerated concrete containing nano-graphene", J. Building Eng., 43, 103106. https://doi.org/10.1016/j.jobe.2021.103106.
  76. Sedighi, H. M., Koochi, A., Daneshmand, F. and Abadyan, M. (2015), "Non-linear dynamic instability of a double-sided nano-bridge considering centrifugal force and rarefied gas flow", J. Non-Linear Mech., 77, 96-106. https://doi.org/https://doi.org/10.1016/j.ijnonlinmec.2015.08.002.
  77. Shafiei, N. and Kazemi, M. (2017a), "Nonlinear buckling of functionally graded nano-/micro-scaled porous beams", Compos. Struct., 178, 483-492. https://doi.org/10.1016/j.compstruct.2017.07.045.
  78. She, G.L. (2021), "Guided wave propagation of porous functionally graded plates: The effect of thermal loadings", J. Thermal Stresses, 44(10), 1289-1305. https://doi.org/10.1080/01495739.2021.1974323.
  79. She, G.L., Liu, H.B. and Karami, B. (2021), "Resonance analysis of composite curved microbeams reinforced with graphene nanoplatelets", Thin-Walled Struct., 160, 107407. https://doi.org/10.1016/j.tws.2020.107407.
  80. Shen, J.P., Wang, P.Y., Li, C. and Wang, Y.Y. (2019), "New observations on transverse dynamics of microtubules based on nonlocal strain gradient theory", Compos. Struct., 225. https://doi.org/10.1016/j.compstruct.2019.111036.
  81. Simsek, M. and Kocaturk, T. (2009), "Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load", Compos. Struct., 90(4), 465-473. https://doi.org/10.1016/j.compstruct.2009.04.024.
  82. Sina, S.A., Navazi, H.M. and Haddadpour, H. (2009), "An analytical method for free vibration analysis of functionally graded beams", Mater. Design, 30(3), 741-747. https://doi.org/10.1016/j.matdes.2008.05.015.
  83. Sobhy, M. and Zenkour, A.M. (2018), "Magnetic field effect on thermomechanical buckling and vibration of viscoelastic sandwich nanobeams with CNT reinforced face sheets on a viscoelastic substrate", Compos. Part B Eng., 154, 492-506. https://doi.org/10.1016/j.compositesb.2018.09.011.
  84. Touloukian, Y. S. (1967). Thermophysical Properties of High Temperature Solid Materials, Macmillan, NY, USA.
  85. Wang, Y., Zhou, A., Fu, T. and Zhang, W. (2020), "Transient response of a sandwich beam with functionally graded porous core traversed by a non-uniformly distributed moving mass", J. Mech. Mater. Design, 16(3), 519-540. https://doi.org/10.1007/s10999-019-09483-9.
  86. 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.
  87. Wu, D., Liu, A., Huang, Y., Huang, Y., Pi, Y. and Gao, W. (2018), "Dynamic analysis of functionally graded porous structures through finite element analysis", Eng. Struct., 165, 287-301. https://doi.org/10.1016/j.engstruct.2018.03.023.
  88. Xie, K., Wang, Y. and Fu, T. (2020), "Nonlinear vibration analysis of third-order shear deformable functionally graded beams by a new method based on direct numerical integration technique", J. Mech. Mater. Design, 16(4), 839-855. https://doi.org/10.1007/s10999-020-09493-y.
  89. Yas, M.H. and Heshmati, M. (2012), "Dynamic analysis of functionally graded nanocomposite beams reinforced by randomly oriented carbon nanotube under the action of moving load", Appl. Math. Modelling, 36(4), 1371-1394. https://doi.org/https://doi.org/10.1016/j.apm.2011.08.037.
  90. Yurtcu, H.H. (2013), "Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory", Compos. Struct., 97, 378-386. https://doi.org/10.1016/j.compstruct.2012.10.038.
  91. Zhang, Q. and Liu, H. (2020), "On the dynamic response of porous functionally graded microbeam under moving load", J. Eng. Sci., 153, 103317. https://doi.org/10.1016/j.ceicie.2012.10.038.
  92. Zhang, Y.Y., Wang, Y. X., Zhang, X., Shen, H.M. and She, G.L. (2021), "On snap-buckling of FG-CNTR curved nanobeams considering surface effects", Steel Compos. Struct., 38(3), 293-304. https://doi.org/10.12989/scs.2021.38.3.293.
  93. Zhou, Y., Yang, X., Pan, D. and Wang, B. (2018), "Improved incorporation of strain gradient elasticity in the flexoelectricity based energy harvesting from nanobeams", Physica E: Low-Dimensional Syst. Nanostruct., 98, 148-158. https://doi.org/10.1016/j.physe.2017.12.037.