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The role of spatial variation of the nonlocal parameter on the free vibration of functionally graded sandwich nanoplates

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Abstract

The role of the spatial variation of the nonlocal parameter on the free vibration of functionally graded sandwich nanoplates is investigated in this study. The key achievement of this work is that the classical nonlocal elasticity theory is modified to take into account the dependence of nonlocal parameters on the varying of materials through the thickness of the functionally graded sandwich nanoplates. Hamilton’s principle is adopted to establish the governing equations of motion using a new inverse hyperbolic shear deformation theory. Numerical results are carried out via Navier’s solution for the fully simply supported rectangular functionally graded sandwich nanoplates, and they are compared with the available results to confirm the accuracy and efficiency of the proposed algorithm. Besides, the effects of some parameters such as the spatial variation of the nonlocal parameters, the aspect ratio, the side-to-thickness ratio as well as the power-law index on the free vibration of the nanoplates are also investigated cautiously. The results show that the variation of the nonlocal parameters plays a significant role in the free vibration of the functionally graded sandwich nanoplates, which is never investigated in the literature. The present methodology could be applied to the design and application of the micro/nanostructures.

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References

  1. Yang F, Chong ACM, Lam DCC, Tong P (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39(10):2731–2743. https://doi.org/10.1016/S0020-7683(02)00152-X

    Article  MATH  Google Scholar 

  2. Arefi M, Firouzeh S, Bidgoli EMR, Civalek O (2022) Analysis of porous micro-plates reinforced with FG-GNPs based on Reddy plate theory. Compos Struct 247:112391. https://doi.org/10.1016/j.compstruct.2020.112391

    Article  Google Scholar 

  3. Gurtin ME, Markenscoff X, Thurston RN (1976) Effect of surface stress on the natural frequency of thin crystals. Appl Phys Lett 29(9):529–530. https://doi.org/10.1063/1.89173

    Article  Google Scholar 

  4. Guo J-G, Zhao Y-P (2007) The size-dependent bending elastic properties of nanobeams with surface effects. Nanotechnology 18(29):295701. https://doi.org/10.1088/0957-4484/18/29/295701

    Article  Google Scholar 

  5. Abo-Bakr RM, Eltaher MA, Attia MA (2020) Pull-in and freestanding instability of actuated functionally graded nanobeams including surface and stiffening effects. Eng Comput. https://doi.org/10.1007/s00366-020-01146-0

    Article  Google Scholar 

  6. Abdelrahman AA, Eltaher MA (2020) On bending and buckling responses of perforated nanobeams including surface energy for different beams theories. Eng Comput. https://doi.org/10.1007/s00366-020-01211-8

    Article  Google Scholar 

  7. Farshi B, Assadi A, Alinia-ziazi A (2010) Frequency analysis of nanotubes with consideration of surface effects. Appl Phys Lett 96(9):93105. https://doi.org/10.1063/1.3332579

    Article  Google Scholar 

  8. Lu P, He LH, Lee HP, Lu C (2006) Thin plate theory including surface effects. Int J Solids Struct 43(16):4631–4647. https://doi.org/10.1016/j.ijsolstr.2005.07.036

    Article  MATH  Google Scholar 

  9. Fleck NA, Muller GM, Ashby MF, Hutchinson JW (1994) Strain gradient plasticity: Theory and experiment. Acta Metall Mater 42(2):475–487. https://doi.org/10.1016/0956-7151(94)90502-9

    Article  Google Scholar 

  10. Aifantis EC (1999) Strain gradient interpretation of size effects. Int J Fract 95(1):299. https://doi.org/10.1023/A:1018625006804

    Article  Google Scholar 

  11. Phung-Van P, Chien CH (2021) A novel size-dependent nonlocal strain gradient isogeometric model for functionally graded carbon nanotube-reinforced composite nanoplates. Eng Comput. https://doi.org/10.1007/s00366-021-01353-3

    Article  Google Scholar 

  12. Esen I, Abdelrhmaan AA, Eltaher MA (2021) 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

    Article  Google Scholar 

  13. Eringen AC (1967) Theory of micropolar plates. Zeitschrift für Angew Math und Phys ZAMP 18(1):12–30. https://doi.org/10.1007/BF01593891

    Article  Google Scholar 

  14. Eringen AC (1972) Nonlocal polar elastic continua. Int J Eng Sci 10(1):1–16. https://doi.org/10.1016/0020-7225(72)90070-5

    Article  MathSciNet  MATH  Google Scholar 

  15. Eringen AC, Edelen DGB (1972) On nonlocal elasticity. Int J Eng Sci 10(3):233–248. https://doi.org/10.1016/0020-7225(72)90039-0

    Article  MathSciNet  MATH  Google Scholar 

  16. Eringen AC (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

    Article  Google Scholar 

  17. Reddy JN (2007) Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 45(2):288–307. https://doi.org/10.1016/j.ijengsci.2007.04.004

    Article  MATH  Google Scholar 

  18. Reddy JN, Pang SD (2008) Nonlocal continuum theories of beams for the analysis of carbon nanotubes. J Appl Phys 103(2):23511. https://doi.org/10.1063/1.2833431

    Article  Google Scholar 

  19. Ebrahimi F, Barati MR, Zenkour AM (2017) Vibration analysis of smart embedded shear deformable nonhomogeneous piezoelectric nanoscale beams based on nonlocal elasticity theory. Int J Aeronaut Sp Sci 18(2):255–269. https://doi.org/10.5139/IJASS.2017.18.2.255

    Article  Google Scholar 

  20. Thai H-T, Vo TP (2012) A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams. Int J Eng Sci 54:58–66. https://doi.org/10.1016/j.ijengsci.2012.01.009

    Article  MathSciNet  MATH  Google Scholar 

  21. Eltaher MA, Emam SA, Mahmoud FF (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

    Article  MathSciNet  MATH  Google Scholar 

  22. Nazemnezhad R, Hosseini-Hashemi S (2014) Nonlocal nonlinear free vibration of functionally graded nanobeams. Compos Struct 110:192–199. https://doi.org/10.1016/j.compstruct.2013.12.006

    Article  MATH  Google Scholar 

  23. Ebrahimi F, Barati MR, Civalek O (2020) Application of Chebyshev-Ritz method for static stability and vibration analysis of nonlocal microstructure-dependent nanostructures. Eng Comput 36:953–964. https://doi.org/10.1007/s00366-019-00742-z

    Article  Google Scholar 

  24. Hadji L, Avcar M (2021) Nonlocal free vibration analysis of porous FG nanobeams using hyperbolic shear deformation beam theory. Adv Nano Res 10(3):281–293. https://doi.org/10.12989/anr.2021.10.3.281

    Article  Google Scholar 

  25. Youcef G, Ahmed H, Abdelillah B, Mohamed Z (2020) Porosity-dependent free vibration analysis of FG nanobeam using non-local shear deformation and energy principle. Adv Nano Res 8(1):37–47. https://doi.org/10.12989/anr.2020.8.1.037

    Article  Google Scholar 

  26. Shariati A, Jung DW, Sedighi HM, Zur KK, Habibi M, Safa M (2020) “On the vibrations and stability of moving viscoelastic axially functionally graded nanobeams. Materials 13(7):1707. https://doi.org/10.3390/ma13071707

    Article  Google Scholar 

  27. Zenkour AM (2018) A novel mixed nonlocal elasticity theory for thermoelastic vibration of nanoplates. Compos Struct 185:821–833. https://doi.org/10.1016/j.compstruct.2017.10.085

    Article  Google Scholar 

  28. Aghababaei R, Reddy JN (2009) Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates. J Sound Vib 326(1):277–289. https://doi.org/10.1016/j.jsv.2009.04.044

    Article  Google Scholar 

  29. Aksencer T, Aydogdu M (2011) Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory. Phys E Low-dimens Syst Nanostruct 43(4):954–959. https://doi.org/10.1016/j.physe.2010.11.024

    Article  Google Scholar 

  30. Hosseini-Hashemi S, Bedroud M, Nazemnezhad R (2013) An exact analytical solution for free vibration of functionally graded circular/annular Mindlin nanoplates via nonlocal elasticity. Compos Struct 103:108–118. https://doi.org/10.1016/j.compstruct.2013.02.022

    Article  MATH  Google Scholar 

  31. Zare M, Nazemnezhad R, Hosseini-Hashemi S (2015) Natural frequency analysis of functionally graded rectangular nanoplates with different boundary conditions via an analytical method. Meccanica 50(9):2391–2408. https://doi.org/10.1007/s11012-015-0161-9

    Article  MathSciNet  MATH  Google Scholar 

  32. Sobhy M (2014) Natural frequency and buckling of orthotropic nanoplates resting on two-parameter elastic foundations with various boundary conditions. J Mech 30(5):443–453. https://doi.org/10.1017/jmech.2014.46

    Article  Google Scholar 

  33. Sobhy M (2015) A comprehensive study on FGM nanoplates embedded in an elastic medium. Compos Struct 134:966–980. https://doi.org/10.1016/j.compstruct.2015.08.102

    Article  Google Scholar 

  34. Sobhy M, Radwan AF (2017) A new quasi 3D nonlocal plate theory for vibration and buckling of FGM nanoplates. Int J Appl Mech 09(01):1750008. https://doi.org/10.1142/S1758825117500089

    Article  Google Scholar 

  35. Hoa LK, Vinh PV, Duc ND, Trung NT, Son LT, Thom DV (2020) “Bending and free vibration analyses of functionally graded material nanoplates via a novel nonlocal single variable shear deformation plate theory. Proc Inst Mech Eng Part C J Mech Eng Sci 15:14. https://doi.org/10.1177/0954406220964522

    Article  Google Scholar 

  36. Akbas SD (2020) Modal analysis of viscoelastic nanorods under an axially harmonic load. Adv Nano Res 8(4):277–282. https://doi.org/10.12989/anr.2020.8.4.277

    Article  Google Scholar 

  37. Ghandourah EE, Abdraboh AM (2020) Dynamic analysis of functionally graded nonlocal nanobeam with different porosity models. Steel Compos Struct 36(3):293–305. https://doi.org/10.12989/scs.2020.36.3.293

    Article  Google Scholar 

  38. Natarajan S, Chakraborty S, Thangavel M, Bordas S, Rabczuk T (2012) Size-dependent free flexural vibration behavior of functionally graded nanoplates. Comput Mater Sci 65:74–80. https://doi.org/10.1016/j.commatsci.2012.06.031

    Article  Google Scholar 

  39. Koizumi M (1997) FGM activities in Japan. Compos Part B Eng 28(1):1–4. https://doi.org/10.1016/S1359-8368(96)00016-9

    Article  Google Scholar 

  40. Swaminathan K, Naveenkumar DT, Zenkour AM, Carrera E (2015) Stress, vibration and buckling analyses of FGM plates—a state-of-the-art review. Compos Struct 120:10–31. https://doi.org/10.1016/j.compstruct.2014.09.070

    Article  Google Scholar 

  41. Sayyad AS, Ghugal YM (2019) Modeling and analysis of functionally graded sandwich beams: a review. Mech Adv Mater Struct 26(21):1776–1795. https://doi.org/10.1080/15376494.2018.1447178

    Article  Google Scholar 

  42. Thom DV, Vinh PV, Nam NH (2020) On the development of refined plate theory for static bending behavior of functionally graded plates. Math Probl Eng. https://doi.org/10.1155/2020/2836763

    Article  MathSciNet  Google Scholar 

  43. Vinh PV, Dung NT, Tho NC, Thom DV, Hoa LK (2021) Modified single variable shear deformation plate theory for free vibration analysis of rectangular FGM plates. Structures 29:1435–1444. https://doi.org/10.1016/j.istruc.2020.12.027

    Article  Google Scholar 

  44. Abouelregal AE, Mohammed WW, Mohammad-Sedighi H (2021) Vibration analysis of functionally graded microbeam under initial stress via a generalized thermoelastic model with dual-phase lags. Arch Appl Mech 91(5):2127–2142. https://doi.org/10.1007/s00419-020-01873-2

    Article  Google Scholar 

  45. Civalek O, Avcar M (2020) Free vibration and buckling analyses of CNT reinforced laminated non-rectangular plates by discrete singular convolution method. Eng Comput. https://doi.org/10.1007/s00366-020-01168-8

    Article  Google Scholar 

  46. Lyashenko I, Borysiuk VN, Popov VL (2020) Dynamical model of the asymmetric actuator of directional motion based on power-law graded materials. Facta Univ Ser Mech Eng 18(2):245–254. https://doi.org/10.22190/FUME200129020L

    Article  Google Scholar 

  47. Daikh AA, Houari MSA, Belarbi MO, Chakraverty S, Eltaher MA (2021) Analysis of axially temperature-dependent functionally graded carbon nanotube reinforced composite plates. Eng Comput. https://doi.org/10.1007/s00366-021-01413-8

    Article  Google Scholar 

  48. Nguyen T-K, Vo TP, Thai H-T (2013) “Vibration and buckling analysis of functionally graded sandwich plates with improved transverse shear stiffness based on the first-order shear deformation theory. Proc Inst Mech Eng Part C J Mech Eng Sci 228(12):2110–2131. https://doi.org/10.1177/0954406213516088

    Article  Google Scholar 

  49. Vinh PV (2021) Formulation of a new mixed four-node quadrilateral element for static bending analysis of variable thickness functionally graded material plates. Math Probl Eng. https://doi.org/10.1155/2021/6653350

    Article  MathSciNet  Google Scholar 

  50. Hassan AH, Kurgan N, Can N (2020) The relations between the various critical temperatures of thin FGM plates. J Appl Comput 6:1404–1419. https://doi.org/10.22055/jacm.2020.34697.2459

    Article  Google Scholar 

  51. AlSaid-Alwan HHS, Avcar M (2020) Analytical solution of free vibration of FG beam utilizing different types of beam theories: a comparative study. Comput Concr 26(3):285–292. https://doi.org/10.12989/cac.2020.26.3.285

    Article  Google Scholar 

  52. Hadji L, Bernard F, Safa A, Tounsi A (2021) Bending and free vibration analysis for FGM plates containing various distribution shape of porosity. Adv Mater Res 10(2):115–135. https://doi.org/10.12989/amr.2021.10.2.115

    Article  Google Scholar 

  53. Hadji L, Avcar M (2021) Free vibration analysis of FG porous sandwich plates under various boundary conditions. J Appl Comput 7(2):505–519. https://doi.org/10.22055/jacm.2020.35328.2628

    Article  Google Scholar 

  54. Zenkour AM (2005) A comprehensive analysis of functionally graded sandwich plates: Part 2-Buckling and free vibration. Int J Solids Struct 42(18):5243–5258. https://doi.org/10.1016/j.ijsolstr.2005.02.016

    Article  MATH  Google Scholar 

  55. Tahir AI, Chikh A, Tounsi A, Al-Osta MA, Al-Dulaijan SU, Al-Zahrani MM (2021) Wave propagation analysis of a ceramic-metal functionally graded sandwich plate with different porosity distributions in a hygro-thermal environment. Compos Struct 269:114030. https://doi.org/10.1016/j.compstruct.2021.114030

    Article  Google Scholar 

  56. Rebai B, Bouhadra A, Bousahla AA, Bourada MM, Tounsi A, Tounsi A, Hussain M (2021) Thermoelastic response of functionally graded sandwich plates using a simple integral HSDT. Arch Appl Mech 91:3403–3420. https://doi.org/10.1007/s00419-021-01973-7

    Article  Google Scholar 

  57. Bennoun M, Houari MSA, Tounsi A (2016) A novel five-variable refined plate theory for vibration analysis of functionally graded sandwich plates. Mech Adv Mater Struct 23(4):423–431. https://doi.org/10.1080/15376494.2014.984088

    Article  Google Scholar 

  58. El Meiche N, Tounsi A, Ziane N, Mechab I, Adda EA (2011) Bedia, “A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate.” Int J Mech Sci 53(4):237–247. https://doi.org/10.1016/j.ijmecsci.2011.01.004

    Article  Google Scholar 

  59. Nguyen V-H, Nguyen T-K, Thai H-T, Vo TP (2014) A new inverse trigonometric shear deformation theory for isotropic and functionally graded sandwich plates. Compos Part B Eng 66:233–246. https://doi.org/10.1016/j.compositesb.2014.05.012

    Article  Google Scholar 

  60. Bessaim A, Houari MSA, Tounsi A, Mahmoud SR, Bedia EAA (2013) A new higher-order shear and normal deformation theory for the static and free vibration analysis of sandwich plates with functionally graded isotropic face sheets. J Sandw Struct Mater 15(6):671–703. https://doi.org/10.1177/1099636213498888

    Article  Google Scholar 

  61. Pham VV, Le QH (2021) Finite element analysis of functionally graded sandwich plates with porosity via a new hyperbolic shear deformation theory. Def Technol. https://doi.org/10.1016/j.dt.2021.03.006

    Article  Google Scholar 

  62. Neves AMA et al (2013) Static, free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3D higher-order shear deformation theory and a meshless technique. Compos Part B Eng 44(1):657–674. https://doi.org/10.1016/j.compositesb.2012.01.089

    Article  Google Scholar 

  63. Natarajan S, Manickam G (2012) Bending and vibration of functionally graded material sandwich plates using an accurate theory. Finite Elem Anal Des 57:32–42. https://doi.org/10.1016/j.finel.2012.03.006

    Article  Google Scholar 

  64. Vinh PV (2021) Deflections, stresses and free vibration analysis of bi-functionally graded sandwich plates resting on Pasternak’s elastic foundations via a hybrid quasi-3D theory. Mech Based Des Struct Mach. https://doi.org/10.1080/15397734.2021.1894948

    Article  Google Scholar 

  65. Li Q, Iu VP, Kou KP (2008) Three-dimensional vibration analysis of functionally graded material sandwich plates. J Sound Vib 311(1):498–515. https://doi.org/10.1016/j.jsv.2007.09.018

    Article  Google Scholar 

  66. Iurlaro L, Gherlone M, Di Sciuva M (2014) Bending and free vibration analysis of functionally graded sandwich plates using the Refined Zigzag Theory. J Sandw Struct Mater 16(6):669–699. https://doi.org/10.1177/1099636214548618

    Article  Google Scholar 

  67. Arefi M, Zenkour AM (2017) Size-dependent free vibration and dynamic analyses of piezo-electro-magnetic sandwich nanoplates resting on viscoelastic foundation. Phys B Condens Matter 521:188–197. https://doi.org/10.1016/j.physb.2017.06.066

    Article  Google Scholar 

  68. Arefi M, Kiani M, Zamani MH (2018) Nonlocal strain gradient theory for the magneto-electro-elastic vibration response of a porous FG-core sandwich nanoplate with piezomagnetic face sheets resting on an elastic foundation. J Sandw Struct Mater 22(7):2157–2185. https://doi.org/10.1177/1099636218795378

    Article  Google Scholar 

  69. Zeng S, Wang BL, Wang KF (2019) Nonlinear vibration of piezoelectric sandwich nanoplates with functionally graded porous core with consideration of flexoelectric effect. Compos Struct 207:340–351. https://doi.org/10.1016/j.compstruct.2018.09.040

    Article  Google Scholar 

  70. Daikh AA, Drai A, Bensaid I, Houari MSA, Tounsi A (2020) On vibration of functionally graded sandwich nanoplates in the thermal environment. J Sandw Struct Mater. https://doi.org/10.1177/1099636220909790

    Article  Google Scholar 

  71. Salehipour H, Shahidi AR, Nahvi H (2015) 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

    Article  MathSciNet  MATH  Google Scholar 

  72. Batra RC (2021) Misuse of Eringen’s nonlocal elasticity theory for functionally graded materials. Int J Eng Sci 159:103425. https://doi.org/10.1016/j.ijengsci.2020.103425

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Solid Mechanics, Le Quy Don Technical University, 236 Hoang Quoc Viet, Hanoi, Vietnam

    Pham Van Vinh

  2. YFL (Yonsei Frontier Lab), Yonsei University, Seoul, South Korea

    Abdelouahed Tounsi

  3. Department of Civil and Environmental Engineering, King Fahd University of Petroleum & Minerals, Dhahran, 31261, Eastern Province, Saudi Arabia

    Abdelouahed Tounsi

  4. Material and Hydrology Laboratory, Civil Engineering Department, Faculty of Technology, University of Sidi Bel Abbes, Sidi Bel Abbès, Algeria

    Abdelouahed Tounsi

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Van Vinh, P., Tounsi, A. The role of spatial variation of the nonlocal parameter on the free vibration of functionally graded sandwich nanoplates. Engineering with Computers 38 (Suppl 5), 4301–4319 (2022). https://doi.org/10.1007/s00366-021-01475-8

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