Steel and Composite Structures

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Effects of size-dependence on static and free vibration of FGP nanobeams using finite element method based on nonlocal strain gradient theory

  • Pham, Quoc-Hoa (Advanced Structural Engineering Laboratory, Department of Structural Engineering, Faculty of Civil Engineering, Ho Chi Minh City Open University) ;
  • Nguyen, Phu-Cuong (Advanced Structural Engineering Laboratory, Department of Structural Engineering, Faculty of Civil Engineering, Ho Chi Minh City Open University)
  • Received : 2021.10.06
  • Accepted : 2022.11.01
  • Published : 2022.11.10
⟨ Previous Next ⟩

Abstract

The main goal of this article is to develop the finite element formulation based on the nonlocal strain gradient and the refined higher-order deformation theory employing a new function f(z) to investigate the static bending and free vibration of functionally graded porous (FGP) nanobeams. The proposed model considers the simultaneous effects of two parameters: nonlocal and strain gradient coefficients. The nanobeam is made by FGP material that exists in un-even and logarithmic-uneven distribution. The governing equation of the nanobeam is established based on Hamilton's principle. The authors use a 2-node beam element, each node with 8 degrees of freedom (DOFs) approximated by the C1 and C2 continuous Hermit functions to obtain the elemental stiffness matrix and mass matrix. The accuracy of the proposed model is tested by comparison with the results of reputable published works. From here, the influences of the parameters: nonlocal elasticity, strain gradient, porosity, and boundary conditions are studied.

Keywords

References

  1. Aifantis, E.C. (2011), "On the gradient approach-relation to Eringens nonlocal theory", Int. J. Eng. Sci., 49(12), 1367-1377. https://doi.org/10.1016/j.ijengsci.2011.03.016.
  2. Akgoz, B. and Civalek, O. (2012), "Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory", Arch. Appl. Mech., 82(3), 423-443. https://doi.org/10.1007/s00419-011-0565-5.
  3. Akgoz, B. and Civalek, O. (2013), "A size-dependent shear deformation beam model based on the strain gradient elasticity theory", Int. J. Eng. Sci., 70, 1-14. https://doi.org/10.1016/j.ijengsci.2013.04.004 .
  4. Anitescu, C., Atroshchenko, E., Alajlan, N. and Rabczuk, T. (2019), "Artificial neural network methods for the solution of second order boundary value problems", Comput. Mater. Continua., 59(1), 345-359. https://doi.org/10.32604/cmc.2019.06641.
  5. Ansari, R., Gholami, R., Shojaei, M. F., Mohammadi, V. and Sahmani, S. (2013), "Size-dependent bending, buckling and free vibration of functionally graded Timoshenko microbeams based on the most general strain gradient theory", Compos Struct., 100, 385-397. https://doi.org/10.1016/j.compstruct.2012.12.048.
  6. Arefi, M., Pourjamshidian, M. and Arani, A.G. (2017), "Application of nonlocal strain gradient theory and various shear deformation theories to nonlinear vibration analysis of sandwich nano-beam with FG-CNTRCs face-sheets in electrothermal environment", Appl. Phys. A-Mater., 123, 323. https://doi.org/10.1007/s00339-017-0922-5.
  7. Attia, M.A. and Mahmoud, F.F. (2016), "Modeling and analysis of nanobeams based on nonlocal-couple stress elasticity and surface energy theories", Int. J. Mech. Sci., 105, 126-134. https://doi.org/10.1016/j.ijmecsci.2015.11.002.
  8. Attia, M.A. (2017), "On the mechanics of functionally graded nanobeams with the account of surface elasticity", Int. J. Eng. Sci., 115, 73-101. https://doi.org/10.1016/j.ijengsci.2017.03.011.
  9. Aydogdu, M. (2009), "A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration", Physica E., 41, 1651-1655. https://doi.org/10.1016/j.physe.2009.05.014.
  10. Civalek, O. and Demir, C. (2011), "Bending analysis of microtubules using nonlocal Euler-Bernoulli beam theory", Appl. Math Model., 35, 2053-2067. https://doi.org/10.1016/j.apm.2010.11.004
  11. Davood, S., Karami, B., Fahham, H.R. and Li, L. (2018), "On the shear buckling of porous nanoplates using a new size-dependent quasi-3D shear deformation theory", Acta Mech., https://doi.org/10.1007/s00707-018-2247-7.
  12. Doan, T.L., Do, V.T., Tran, T.T., Phung, V.M., Tran, V.K. and Pham, V.V. (2021), "Mechanical analysis of bi-functionally graded sandwich nanobeams", Adv. Nano Res., 11(1), 55-71. https://doi.org/10.12989/anr.2021.11.1.055.
  13. Ebrahimi, F. and Barati, M.R. (2016), "A nonlocal higher-order refined magneto-electro-viscoelastic beam model for dynamic analysis of smart nanostructures", Int. J. Eng. Sci., 107, 183-196. https://doi.org/10.1016/j.ijengsci.2016.08.001,
  14. Ebrahimi, F, and Barati, M.R. (2017a), "A nonlocal strain gradient refined beam model for buckling analysis of size-dependent shear-deformable curved FG nanobeams", Compos Struct., 159, 174-182. https://doi.org/10.1016/j.compstruct.2016.09.058.
  15. Ebrahimi, F. and Barati, M.R. (2017b), "Hygrothermal effects on vibration characteristics of viscoelastic FG nanobeams based on nonlocal strain gradient theory", Compos Struc., 159, 433-444. https://doi.org/10.1016/j.compstruct.2016.09.092.
  16. Ebrahimi, F. and Barati, M.R. (2017c), "Vibration analysis of viscoelastic inhomogeneous nanobeams incorporating surface and thermal effects", Appl. Phys. A-Mater., 123(5). https://doi.org/10.1007/s00707-016-1755-6.
  17. Ebrahimi, F. and Barati, M.R. (2017d), "Vibration analysis of viscoelastic inhomogeneous nanobeams resting on a viscoelastic foundation based on nonlocal strain gradient theory incorporating surface and thermal effects", Acta Mech, 228, 1197-1210. DOI:10.1007/s00707-016-1755-6.
  18. Eltaher, M.A., Khairy, A., Sadoun, A.M. and Omar, F. (2014), "Static and buckling analysis of functionally graded Timoshenko nanobeams", Appl Math Comput., 229, 283-295. https://doi.org/10.1016/j.amc.2013.12.072.
  19. Eltaher, M.A., Khaterb, M.E and Emam, S.A. (2016), "A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams", Appl. Math. Model., 40, 4109-4128. https://doi.org/10.1016/j.apm.2015.11.026.
  20. Emam, S.A. (2013), "A general nonlocal nonlinear model for buckling of nanobeams", Appl Math Model., 37, 6929-6939. https://doi.org/10.1016/j.apm.2013.01.043.
  21. Eringen, A.C (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phy., 54, 4703-4710. https://doi.org/10.1063/1.332803.
  22. Eringen, A.C. (2002), Nonlocal Continuum Field Theories, Springer: New York (NY).
  23. Guo, H., Zhuang, X. and Rabczuk, T. (2019), "A deep collocation method for the bending analysis of Kirchhoff plate", Comput Mater. Continua., 59(2), 433-456. https://doi.org/10.48550/arXiv.2102.02617.
  24. Hughes, T.J.R., Cottrell, J.A. and Bazilevs, Y. (2005), "Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement", Comput Methods Appl Mech Engrg., 194(39-41), 4135-4195. https://doi.org/10.1016/j.cma.2004.10.008.
  25. Khorshidi, M. A., Shariati, M and Emam, S.A. (2016), "Postbuckling of functionally graded nanobeams based on modified couple stress theory under general beam theory", Int. J. Mech. Sci., 110, 160-169. https://doi.org/10.1016/j.ijmecsci.2016.03.006.
  26. Kong, S., Zhou, S., Nie, Z. and Wang, K. (2009), "Static and dynamic analysis of micro beams based on strain gradient elasticity theory", Int J Eng Sci., 47(4), 87(4) 98. https://doi.org/10.1016/j.ijengsci.2008.08.008.
  27. Lam, D., Yang, F., Chong, A., Wang, J. and Tong, P. (2003). "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solids., 51, 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X.
  28. Lei, J., He, Y., Zhang, B., Gan, Z. and Zeng, P. (2013), "Bending and vibration of functionally graded sinusoidal microbeams based on the strain gradient elasticity theory", Int. J. Eng. Sci., 72, 36-52. https://doi.org/10.1016/j.ijengsci.2013.06.012.
  29. Li, L. and Hu, Y. (2015), "Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory", INT J. Eng. Sci., 97, 84-94. https://doi.org/10.1016/j.ijengsci.2015.08.013.
  30. Li, L. and Hu, Y. (2016a), "Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material", Int. J. Eng. Sci., 107, 77-97. https://doi.org/10.1016/j.ijengsci.2016.07.011.
  31. Li, L. and Hu, Y. (2017a), "Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects", Int. J. Mech. Sci., 120, 159-170. https://doi.org/10.1016/j.ijmecsci.2016.11.025.
  32. Li, L., Hu, Y. and Li, X. (2016b), "Longitudinal vibration of sizedependent rods via nonlocal strain gradient theory", Int J Mech Sc., 115-116, 135-144. https://doi.org/10.1016/j.ijmecsci.2016.06.011.
  33. Li, L., Li, X. and Hu, Y. (2016c), "Free vibration analysis of nonlocal strain gradient beams made of functionally graded material", INT J. Eng. Sci., 102, 77-92. https://doi.org/10.1016/j.ijengsci.2016.02.010.
  34. Li, X., Li, L., Hu, Y., Ding, Z. and Deng, W. (2017b), "Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory", Compos Struct., 165, 250-265. https://doi.org/10.1016/j.compstruct.2017.01.032.
  35. Li, Y.S., Feng, W.J. and Cai, Z.Y. (2014), "Bending and free vibration of functionally graded piezoelectric beam based on modified strain gradient theory", Compos Struc., 115, 41-50. https://doi.org/10.1016/j.compstruct.2014.04.005.
  36. Li, Y.S., Ma, P. and Wang, W. (2016d), "Bending, buckling, and free vibration of magnetoelectroelastic nanobeam based on nonlocal theory", J, Intel. Mat. Syst. Str., 27, 1139-1149. https://doi.org/10.1177/1045389X15585899.
  37. Lim, C.W., Zhang, G. and Reddy, J.N. (2015), "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.
  38. Lim, C.W., Zhang, G. and Reddy, J.N. (2015), "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.
  39. Lu, L., Guo, X and Zhao, J. (2017a), "A unified nonlocal strain gradient model for nanobeams and the importance of higher order terms", Int J Eng Sci., 119, 265-277. https://doi.org/10.1016/j.ijengsci.2017.06.024.
  40. Lu, L., Guo, X. and Zhao, J. (2017b), "Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory", Int J Eng Sci., 116, 12-24. https://doi.org/10.1016/j.ijengsci.2017.03.006.
  41. Ma, H.M., Gao, X.L and Reddy, J.N. (2008), "A microstructuredependent Timoshenko beam model based on a modified couple stress theory", J Mech Phys Solids., 56, 3379-3391. https://doi.org/10.1016/j.jmps.2008.09.007.
  42. Mindlin, R.D. (1964), "Micro-structure in linear elasticity", Arch Ration Mech. An., 16, 51-78. https://doi.org/10.1007/BF00248490.
  43. Mindlin, R.D. (1965), "Second gradient of strain and surfacetension in linear elasticity", Int J Solids Struct., 1, 417-438. https://doi.org/10.1016/0020-7683(65)90006-5.
  44. Mirfatah, S.M., Shahmohammadi, M.A., Salehipour, H. and Civalek, O. (2022), "Size-dependent dynamic stability of nanocomposite enriched micro-shell panels in thermal environment using the modified couple stress theory", Eng. Anal. Bound. Elements, 143, 483-500. https://doi.org/10.1016/j.enganabound.2022.07.004.
  45. Mirfatah, S.M., Tayebikhorami, S., Shahmohammadi, M.A., Salehipour, H. and Civalek, O. (2022), "Thermo-elastic damped nonlinear dynamic response of the initially stressed hybrid GPL/CNT/fiber/polymer composite toroidal shells surrounded by elastic foundation", Compos Struc., 283, 115047. https://doi.org/10.1016/j.compstruct.2021.115047.
  46. Mohammad-Abadi, M. and Daneshmehr, A.R. (2014), "Size dependent buckling analysis of microbeams based on modified couple stress theory with high order theories and general boundary conditions", Int J Eng Sci., 74, 1-14. https://doi.org/10.1016/j.ijengsci.2013.08.010.
  47. Nateghi, A., Salamat-talab, M., Rezapour, J. and Daneshian, B. (2012), "Size dependent buckling analysis of functionally graded micro beams based on modified couple stress theory", Appl. Math Model., 36(4), 971-974. https://doi.org/10.1016/j.apm.2011.12.035.
  48. Nejad, M.Z. and Hadi, A. (2016a), "Non-local analysis of free vibration of bi-directional functionally graded Euler-Bernoulli nano-beams", Int. J. Eng. Sci., 105, 1-11. https://doi.org/10.1016/ j.ijengsci.2016.04.011.
  49. Nejad, M.Z. and Hadi, A. (2016b), "Eringen's non-local elasticity theory for bending analysis of bi-directional functionally graded Euler-Bernoulli nano-beams", Int J Eng Sci., 106, 1-9. https://doi.org/10.1016/j.ijengsci.2016.05.005.
  50. Nejad, M.Z., Hadi, A. and Rastgoo, A. (2016c), "Buckling analysis of arbitrary two-directional functionally graded Euler-Bernoulli nano-beams based on nonlocal elasticity theory", Int. J. Eng. Sci., 103, 1-10. https://doi.org/10.1016/j.ijengsci.2016.03.001.
  51. Nguyen-Thanh, N., Zhou, K., Zhuang, X., Areias, P., NguyenXuan, H., Bazilevs, Y. and Rabczuk, T. (2017), "Isogeometric analysis of large-deformation thin shells using RHT-splines for multiple-patch coupling", Comput. Meth. Appl. Mech. Engrg., 316, 1157-1178. https://doi.org/10.1016/j.cma.2016.12.002.
  52. Nguyen, T.N., Lee, S., Nguyen, P.C., Nguyen-Xuan, H. and Lee, J. (2020), "Geometrically nonlinear postbuckling behavior of imperfect FG-CNTRC shells under axial compression using isogeometric analysis", Europ. J. Mech. - A/Solids, 84, 1040-1066. https://doi.org/10.1016/j.euromechsol.2020.104066 .
  53. Nguyen, T.D., Tran, V.K., Truong, T.T.H. and Phung, V.M. (2022), "Nonlinear static bending analysis of microplates resting on imperfect two-parameter elastic foundations using modified couple stress theory", Compt. Rendus. Mecanique., 350(G1), 121-141. https://doi.org/10.5802/crmeca.105.
  54. Park, S.K. and Gao, X.L. (2006), "Bernoulli-Euler beam model based on a modified couple stress theory", J. Micromech Microeng., 16, 2355-2359. https://doi.org/10.1088/0960-1317/16/11/015.
  55. Pham, Q.H. and Nguyen, P.C. (2022c), "Dynamic stability analysis of porous functionally graded microplates using a refined isogeometric approach", Compos. Struct., 284, 115086. https://doi.org/10.1016/j.compstruct.2021.115086.
  56. Pham, Q.H., Nguyen, P.C. and Tran, T.T (2022d), "Dynamic response of porous functionally graded sandwich nanoplates using nonlocal higher-order isogeometric analysis", Compos. Struct., 290, 115565. https://doi.org/10.1016/j.compstruct.2022.115565.
  57. Pham, Q.H., Nguyen, P.C. and Tran, T.T. (2022f), "Free vibration response of auxetic honeycomb sandwich plates using an improved higher-order ES-MITC3 element and artificial neural network", Thin-Wall. Struct., 175, 109203. https://doi.org/10.1016/j.tws.2022.109203.
  58. Pham, Q.H., Nguyen, P.C., Tran, T.T. and Nguyen-Thoi, T. (2021d), "Free vibration analysis of nanoplates with auxetic honeycomb core using a new third-order finite element method and nonlocal elasticity theory", Eng. Comput, 1-19. https://doi.org/10.1007/s00366-021-01531-3.
  59. Pham, Q.H., Nguyen, P.-C., Tran, V.K and Nguyen-Thoi, T. (2021a), "Finite element analysis for functionally graded porous nano-plates resting on elastic foundation", Steel Compos. Struct., 41(2), 149-166. https://doi.org/10.12989/scs.2021.41.2.149.
  60. Pham, Q.H., Nguyen, P.-C., Tran, V.K., Lieu, X.Q. and Tran, T.T (2022b), "Modified nonlocal couple stress isogeometric approach for bending and free vibration analysis of functionally graded nanoplates", Eng. Comput., 1-26. https://doi.org/10.1007/s00366-022-01726-2.
  61. Pham, Q.H., Nguyen, P.C., Tran, V.K., Nguyen-Thoi, T. (2021), "Isogeometric analysis for free vibration of bidirectional functionally graded plates in the fluid medium", Defence Technol., 18(8), 1311-1329. https://doi.org/10.1016/j.dt.2021.09.006.
  62. Pham, Q.H., Tran, T.T., Tran, V.K., Nguyen, P.C. and NguyenThoi, T. (2022e), "Free vibration of functionally graded porous non-uniform thickness annular-nanoplates resting on elastic foundation using ES-MITC3 element", Alexandria Eng. J., 61(3), 1788-1802. https://doi.org/10.1016/j.aej.2021.06.082.
  63. Pham, Q.H., Tran, T.T., Tran, V.K., Nguyen, P.C., Nguyen-Thoi, T. and Zenkour, A.M. (2021c), "Bending and hygro thermo mechanical vibration analysis of a functionally graded porous sandwich nanoshell resting on elastic foundation", Mech. Adv. Mater. Struct., 28, https://doi.org/10.1080/15376494.2021.1968549.
  64. Pham, Q.H., Tran, V.K., Tran, T.T., Nguyen-Thoi, T., Nguyen, P.C. and Pham, V.D. (2021b), "A nonlocal quasi-3D theory for thermal free vibration analysis of functionally graded material nanoplates resting on elastic foundation", Case Studies Therm. Eng., 26, 101170. https://doi.org/10.1016/j.csite.2021.101170.
  65. Pham, Q.H., Tran, V.K., Tran, T.T., Nguyen, P.C. and Malekzadeh, P. (2022a), "Dynamic instability of magnetically embedded functionally graded porous nanobeams using the strain gradient theory", Alexandria Eng. J., 61(12), 10025-10044. https://doi.org/10.1016/j.aej.2022.03.007.
  66. Salehipour, H., Emadi, S., Tayebikhorami, S. and Shahmohammadi, M.A. (2022), "A semi-analytical solution for dynamic stability analysis of nanocomposite/fibre-reinforced doubly-curved panels resting on the elastic foundation in thermal environment", Europ. Phys. J. Plus., 137(1), 1-36. https://doi.org/10.1140/epjp/s13360-021-02190-5.
  67. Samaniego, E., Anitescu, C., Goswami, S., Nguyen-Thanh, V.M., Guo, H., Hamdia, K. and Rabczuk, T. (2020), "An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications", Comput. Meth. Appl. Mech. Engrg., 362, 112790. https://doi.org/10.1016/j.cma.2019.112790.
  68. Shahmohammadi, M.A., Azhari, M., Saadatpour, M.M., Salehipour, H. and Civalek, O. (2021), "Dynamic instability analysis of general shells reinforced with polymeric matrix and carbon fibers using a coupled IG-SFSM formulation", Compos. Struct., 263, 113720. https://doi.org/10.1016/j.compstruct.2021.113720.
  69. Shahmohammadi, M.A., Azhari, M., Salehipour, H., Fantuzzi, N., Amabili, M. and Civalek, O. (2022), "Nonlinear analysis of fiber-reinforced folded shells enriched by nano-additives using a coupled FEM-IGA formulation", Compos. Struct, 116221. https://doi.org/10.1016/j.compstruct.2022.116221.
  70. Shahmohammadi, M.A., Mirfatah, S.M., Salehipour, H., Azhari, F. and Civalek, O. (2022), "Dynamic stability of hybrid fiber/nanocomposite-reinforced toroidal shells subjected to the periodic axial and pressure loadings", Mech. Adv. Mater. Struct., 1-17. https://doi.org/10.1080/15376494.2022.2037172.
  71. Thai, H.T. (2012), "A nonlocal beam theory for bending, buckling, and vibration of nanobeams", Int J Eng Sci., 52, 56-64. https://doi.org/10.1016/j.ijengsci.2011.11.011.
  72. Tran, T.T., Tran, V.K., Pham, Q.H. and Zenkour, A.M. (2021a), "Extended four-unknown higher-order shear deformation nonlocal theory for bending, buckling and free vibration of functionally graded porous nanoshell resting on elastic foundation", Compos. Struct., 264(15), 113737. https://doi.org/10.1016/j.compstruct.2021.113737.
  73. Tran, T.T., Nguyen, P.C. and Pham, Q.H. (2021b), "Vibration analysis of FGM plates in thermal environment resting on elastic foundation using ES-MITC3 element and prediction of ANN", Case Studies Thermal Eng., 24, 100852. https://doi.org/10.1016/j.csite.2021.100852.
  74. Tran, V.K., Pham, Q.H. and Nguyen-Thoi, T. (2020b), "A finite element formulation using four-unknown incorporating nonlocal theory for bending and free vibration analysis of functionally graded nanoplates resting on elastic medium foundations", Eng. with Comput., 1-26. http://doi.org/10.1007/s00366-020-01107-7.
  75. Tran, V.K., Tran, T.T., Phung, M.V., Pham, Q.H. and Nguyen-Thoi, T. (2020a), "A finite element formulation and nonlocal theory for the static and free vibration analysis of the sandwich functionally graded nanoplates resting on elastic foundation", J. Nanomater., 2020. https://doi.org/10.1155/2020/8786373.
  76. Triantafyllidis, N. and Bardenhagen, S. (1993), "On higher order gradient continuum theories in 1-D nonlinear elasticity. Derivation from and comparison to the corresponding discrete models", J. Elasticity., 33, 259-293. https://doi.org/10.1007/BF00043251.
  77. Vu-Bac, N., Duong, T.X., Lahmer, T., Zhuang, X., Sauer, R.A., Park, H.S. and Rabczuk, T. (2018), "A NURBS-based inverse analysis for reconstruction of nonlinear deformations of thin shell structures", Comput Methods Appl Mech Engrg., 331, 427-455. https://doi.org/10.1016/j.cma.2017.09.034.
  78. Yang, F., Chong, A.C.M., Lam, D.C.C. and Tong, P. (2002), "Couple stress based strain gradient theory for elasticity", Int J. Solids Struct., 39, 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X.
  79. Zhuang, X., Guo, H., Alajlan, N., Zhu, H. and Rabczuk, T. (2021), "Deep autoencoder based energy method for the bending, vibration, and buckling analysis of Kirchhoff plates with transfer learning", Europ. J. Mech. A/Solids, 87, 104225. https://doi.org/10.1016/j.euromechsol.2021.104225.