Steel and Composite Structures

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Finite element analysis for functionally graded porous nano-plates resting on elastic foundation

  • Pham, Quoc-Hoa (Faculty of Mechanical Engineering, Industrial University of Ho Chi Minh City) ;
  • Nguyen, Phu-Cuong (Advanced Structural Engineering Laboratory, Faculty of Civil Engineering, Ho Chi Minh City Open University) ;
  • Tran, Van-Ke (Department of Mechanics, Le Quy Don Technical University) ;
  • Nguyen-Thoi, Trung (Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University)
  • Received : 2020.11.29
  • Accepted : 2021.08.20
  • Published : 2021.10.25
⟨ Previous Next ⟩

Abstract

This paper proposes an improved triangular element based on the strain approach and the Reissner-Mindlin theory to investigate the static, free vibration, and buckling response of functionally graded porous (FGP) nano-plates resting on the Parternak's two-parameter elastic medium foundation. The internal pores of nano-plates are described by two distribution laws, including uneven porosity distribution and logarithmic-uneven porosity distribution. Using Hamilton's principle, equilibrium equations of FGP nano-plates lying on a two-parameter foundation are obtained. The most remarkable feature of the improved triangular element is the degrees of freedom of elements approximated by Lagrange functions for the membrane strain and by the high-degree polynomial functions for the bending strain. The numerical results of the present work are compared with the available results in the literature to evaluate the performance of the proposed approach. Effects of geometrical and material properties such as the power-law index n, the porosity coefficient ξ, the nonlocal coefficient μ, and the parameters of the elastic foundation on the static, free vibration, and buckling behavior of the FGP nano-plates are examined in detail.

Keywords

Acknowledgement

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 107.02-2019.330.

References

  1. Abderahim, B., Sadok B., Mohamed, N.H and Lamine, B. (2019), "Static, free vibration, and buckling analysis of plates using strain-based Reissner-Mindlin elements", Int. J. Adv. Struct. Eng., 11(2),211-230. https://doi.org/10.1007/s40091-019-0226-4
  2. Abderahim, B., Sadok, B. and Lamine, B. (2018), "Strain based triangular finite element for plate bending analysis", Mech. Adv. Mater. Struct., 27(8), 1-13. https://doi.org/10.1080/15376494.2018. 1488310
  3. Ali, R.S., Amin, A., Sayyed, H.S. and Mehdi, K. (2013), "Fundamental size dependent natural frequencies of nonuniform orthotropic nano scaled plates using nonlocal variational principle and finite element method", Appl. Math. Model., 37, 7047-7061. https://doi.org/10.1016/j.apm.2013.02.015.
  4. Barati, M.R. (2017), "Nonlocal-strain gradient forced vibration analysis of metal foam nano-plates with uniform and graded porosities", Adv. Nano Res., 5(4), 393. https://doi.org/10.12989/anr.2017.5.4.393.
  5. Chen, D., Yang, J. and Kitipornchai, S. (2015), "Elastic buckling and static bending of shear deformable functionally graded porous beam", Compos. Struct., 133, 54-61. https://doi.org/10.1016/j. compstruct.2015.07.052.
  6. Doan, T.L., Le, P. B., Tran, T.T., Trai, V.K. and Pham, Q.H. (2021). "Free vibration analysis of functionally graded porous nano-plates with different shapes resting on elastic foundation", J. Appl. Comput. Mech., 7(3), 1593-1605. https://doi.org/10.22055/JACM.2021.36181.2807
  7. Ebrahimi, F., Ghasemi, F. and Salari, E. (2016), "Investigating thermal effects on vibration behavior of temperature-dependent compositionally graded Euler beams with porosities", Meccanica., 51(1), 223-249. https://doi.org/10.1007/ s11012-015-0208-y.
  8. Ebraihimi, F. and Habibi, S. (2016), "Deflection and vibration analysis of higher-order shear deformable compositionally graded porous plate", Steel Compos. Struct., 20(1), 205-225. http://doi.org/10.12989/scs.2016.20.1.205.
  9. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54, 4703-4710. https://doi.org/10.1063/1.332803.
  10. Eringen, A.C. (2002), Nonlocal Continuum Field Theories, Springer, New York, NY, USA.
  11. Galeban, M.R., Mojahedin, A., Taghavi, Y. and Jabbari, M. (2016), "Free vibration of functionally graded thin beams made of saturated porous materials", Steel Compos. Struct., 21(5), 999-1016. http://doi.org/10.12989/scs.2016.21.5.999.
  12. Gao, Y., Xiao, W.S. and Zu, H. (2019), "Nonlinear bending of functionally graded porous nanobeam subjected to multiple physical load based on nonlocal strain gradient theory", Steel Compos. Struct., 32(5), 469-488. http://doi.org/10.12989/scs.2019.31.5.469.
  13. Guo, H., Zhuang, X. and Rabczuk, T. (2019), "A deep collocation method for the bending analysis of Kirchhoff plate", CMC, 59(2), 433-456. http://doi.org/10.32604/cmc.2019.06660.
  14. Hadjiconstantinou, N.G. and Patera, A.T. (1997), "Heterogeneous atomistic-continuum representations for dense fluid systems", Int. J. Mod Phys. C., 8(4), 967-976. https://doi.org/10.1142/S0129183197000837.
  15. Hashemi, S.H., Bedroud, M and Nazemnezhad, R (2013), "An exact analytical solution for free vibration of functionally graded circular/annular Mindlin nano-plates via nonlocal elasticity", Compos. Struct., 103, 108-118. http://doi.org/10.1016/j.compstruct. 2013.02.022.
  16. Jung, W.Y. and Han, S.C. (2013), "Analysis of sigmoid functionally graded material (S-FGM) nanoscale plates using the nonlocal elasticity theory", Math. Probl. Eng., 49, 449-458. https://doi.org/10.1. 155/2013/476131. https://doi.org/10.1.155/2013/476131
  17. Li, K., Wu, D., Chen, X., Cheng, J., Liu, Z., Gao, W. and Liu, M. (2018), "Isogeometric analysis of functionally graded porous plates reinforced by graphene platelets", Compos. Struct., 204, 114-130. https://doi.org/10.1016/j.compstruct.2018.07.059.
  18. Mechab, B., Mechab, I., Benaissa, S., Ameri, M. and Serier, B. (2016), "Probabilistic analysis of effect of the porosities in functionally graded material nano-plate resting on Winkler-Pasternak elastic foundations", Appl. Math. Model., 40(2), 738-749. http://doi.org/10.1016/j.apm.2015.09.093
  19. Mirjavadi, S.S., Afshari, B.M., Shafiei, N., Hamouda, A.M.S. and Kazemi, M. (2017)," Thermal vibration of two-dimensional functionally graded (2D-FG) porous Timoshenko nanobeams", Steel Compos. Struct., 25(4), 415-426. http://doi.org/10.12989/scs.2017.25.4.415
  20. Monchai, P., Boonme, C. and Somchai, C. (2016), "Free vibration analysis of FG nano-plates embedded in elastic medium based on second-order shear deformation plate theory and nonlocal elasticity", Compos Struct., 41(2), 666-686. http://doi.org/10.1016 /j.compstruct.2016.06.045 https://doi.org/10.1016/j.compstruct.2016.06.045
  21. Nami, M.R., Janghorban, M. and Damadam, M. (2015), "Thermal buckling analysis of functional graded rectangular nano-plates based on nonlocal third-order shear deformation theory", Aero. Sci. Tech., 41, 7-15. https://doi.org/10.1016/j.ast.2014.12.001.
  22. Natarajan, S., Chakraborty, S., Thangavel, M., Bordas, S. and Rabczuk, T. (2012), "Sizedependent free flexural vibration behavior of functionally graded nano-plates", Compos. Mater. Sci., 65, 74-80. https://doi.org/10.1016/j.commatsci.2012.06.031.
  23. Nguyen, N. V., Nguyen-Xuan, H., Lee, D. and Lee, J. (2020), "A novel computational approach to functionally graded porous plates with graphene platelets reinforcement", Thin Wall. Struct., 150, 106684. https://doi.org/10.1016/j.tws.2020.106684.
  24. Nguyen, N.V., Nguyen, H.X., Lee, S. and Nguyen-Xuan, H. (2018), "Geometrically nonlinear polygonal finite element analysis of functionally graded porous plates", Adv. Eng. Softw., 126, 110-126. https://doi.org/10.1016/j.advengsoft.2018.11.005.
  25. 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", Eur. J. Mech.-A/Solids, 84, 104066. https://doi.org/10.1016/j.euromechsol.2020.104066.
  26. 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 Technology, (Articles in Press). https://doi.org/10.1016/j.dt.2021.09.006.
  27. Pham, Q.H., Tran, T.T., Tran, V.K., Nguyen, P.C., Nguyen-Thoi, T. and Zenkour, A.M. (2021), "Bending and hygro thermo mechanical vibration analysis of a functionally graded porous sandwich nanoshell resting on elastic foundation", Mech. Adv. Mater. Struct., 28, (Articles in Press). https://doi.org/10.1080/15376494.2021.1968549.
  28. Pham, Q.H., Tran, T.T., Tran, V.K., Nguyen, P.C. and Nguyen-Thoi, T. (2021), "Free vibration of functionally graded porous non-uniform thickness annular-nanoplates resting on elastic foundation using ES-MITC3 element", Alexandria Eng. J., (Articles in Press). https://doi.org/10.1016/j.aej.2021.06.082.
  29. Pham, Q.H., Tran, V.K., Tran, T.T., Nguyen-Thoi, T., Nguyen, P.C. and Pham, V.D. (2021), "A nonlocal quasi-3D theory for thermal free vibration analysis of functionally graded material nanoplates resting on elastic foundation", Case Studies in Therm. Eng., 26, 101170. https://doi.org/10.1016/j.csite.2021.101170.
  30. Pradhan, S.C. and Murmu. T. (2009), "Small scale effect on the buckling of single-layered graphene sheets under biaxial compression via nonlocal continuum mechanics", Comput. Mater. Sci., 47, 268-274. https://doi.org/10.1016/j.commatsci.2009.08.001.
  31. Prandhan, S.C. and Phadikar, J.K. (2011), "Nonlocal theory for buckling of nano-plates", Int. J. Struct. Stab. Dynam., 11(3), 411-429. https://doi.org/10.1142/S021945541100418X.
  32. Reddy, J.N. (2004), Mechanics of laminated composite plate and shell, Theory and Analysis, CRC Press, New York, NY, USA.
  33. Salehipour, H., Nahvi, H. and Shahidi, A.S. (2015a), "Exact analytical solution for free vibration of functionally graded micro/nano-plates via three-dimensional nonlocal elasticity", Phys. E., 66, 350-358. https://doi.org/10.1016/j.physe.2014.10.001
  34. Salehipour, H., Shahidi, A.S. and Nahvi, H. (2015b), "Modified nonlocal elasticity theory for functionally graded materials", Int. J. Eng. Sci., 90, 44-57. https://doi.org/10.1016/j .ijengsci.2015.01.005.
  35. Samaniego, E., Anitescu, C., Goswami, S., Nguyen-Thanh, V.M., Guo, H., Hamdia, K., Zhuang, X. 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. Method. Appl. M., 362, 112790. https://doi.org/10.1016/j.cma.2019.112790
  36. Seref, D. (2018), "Forced vibration analysis of functionally graded porous deep beams", Compos. Struct., 186, 293-302. https://doi.org/10.1016/j.compstruct.2017.12.013.
  37. Shahsavari, D., Karami, B., Fahham, H.R. and Li, L. (2018), "On the shear buckling of porous nano-plates using a new size-dependent quasi-3D shear deformation theory", Acta Mech., 229, 4549-4573. https://doi.org/10.1007/s00707-018-2247-7.
  38. Si, H., Shen, D., Xia, J. and Tahouned, V. (2020), "Vibration behavior of functionally graded sandwich beam with porous core and nanocomposite layers", Steel Compos. Struct., 36(1), 1-16. http://doi.org/10.12989/scs.2020.36.1.001.
  39. Sobhy, M. (2015), "A comprehensive study on FGM nano-plates embedded in an elastic medium", Compos. Struct., 134, 966-980. http://doi.org/10.1016/j. compstruct.2015.08.102.
  40. Sobhy, M. (2017), "A new quasi 3D nonlocal plate theory for vibration and buckling of FGM nano-plates", IJAM., 220, 289-303. https://doi.org/10.1142/S1758825117500089.
  41. Ta, H.D. and Nguyen, P.C. (2020), "Perturbation based stochastic isogeometric analysis for bending of functionally graded plates with the randomness of elastic modulus", Latin Am. J. Solids Struct., 17(7), e306. https://doi.org/10.1590/1679-78256066.
  42. Tran, T.V., Tran, T.D., Pham, Q.H., Nguyen-Thoi, T. and Tran, V.K. (2020), "An ES-MITC3 finite element method based on higherorder shear deformation theory for static and free vibration analyses of FG porous plates reinforced by GPLs", Math. Probl. Eng., 2020. https://doi.org/10.1155/2020/ 7520209.
  43. Tran, T.T., Nguyen, P.C. and Pham, Q.H. (2021), "Vibration analysis of FGM plates in thermal environment resting on elastic foundation using ES-MITC3 element and prediction of ANN", Case Studies in Therm. Eng., 24, 100852. https://doi.org/10.1016/j.csite.2021.100852.
  44. Tran, T.T., Pham, Q.H. and Nguyen-Thoi, T. (2020), "An edgebased smoothed finite element for free vibration analysis of functionally graded porous (FGP) plates on elastic foundation taking into mass (EFTIM)", Math. Probl. Eng., 2020. https://doi.org/10.1155/2020/8278743
  45. Tran, T.T., Pham, Q.H. and Nguyen-Thoi, T. (2020), "Dynamic analysis of functionally graded porous plates resting on elastic foundation taking into mass subjected to moving loads using an edge-based smoothed finite element method", Shock Vib., 2020. https://doi.org/10.1155/2020/8853920.
  46. Tran, T.T., Pham, Q.H. and Nguyen-Thoi, T. (2020), "Static and free vibration analyses of functionally graded porous variable-thickness plates using an edge-based smoothed finite element method", Defence Technol., http://doi.org/10.1016/j.dt.2020.06.001.
  47. Tran, T.T., Tran, V.K., Pham, Q.H. and Zenkour, A.M. (2021), "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, 113737. http://doi.org/10.1016/ j.compstruct.2021.113737.
  48. Tran, V.K., Pham, Q.H. and Nguyen-Thoi, T. (2020), "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.
  49. Tran, V.K., Tran, T.T., Phung, M.V., Pham, Q.H. and Nguyen-Thoi, T. (2020), "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., http://doi.org/10.1155/2020 /8786373.
  50. Wang, Q. and Varadan, V.K. (2006), "Wave characteristics of carbon nanotubes", Int. J. Solids Struct., 43(2), 254-265. https://doi.org/10.1016/j.ijsolstr.2005.02.047.
  51. Zenkour, A.M. and Sobhy, M. (2013), "Nonlocal elasticity theory for thermal buckling of nano-plates lying on Winkler-Pasternak elastic substrate medium", Phys. E., 53, 251-259. https://doi.org/10.1016/j.physe.2013.04.022.
  52. Zhou, C., Zhang, Z., Zang, J., Fang, Y. and Tahouned, V. (2020), "Vibration analysis of FG porous rectangular plates reinforced by graphene platelets", Steel Compos/ Struct., 34(2), 215-226. http://doi.org/10.12989/scs.2020.34.2.215.