Skip to main content
SpringerLink
Log in

Free vibration analysis of functionally graded porous curved nanobeams on elastic foundation in hygro-thermo-magnetic environment

  • Research Article
  • Published:
Frontiers of Structural and Civil Engineering Aims and scope Submit manuscript

Abstract

Herein, a two-node beam element enriched based on the Lagrange and Hermite interpolation function is proposed to solve the governing equation of a functionally graded porous (FGP) curved nanobeam on an elastic foundation in a hygro-thermo-magnetic environment. The material properties of curved nanobeams change continuously along the thickness via a power-law distribution, and the porosity distributions are described by an uneven porosity distribution. The effects of magnetic fields, temperature, and moisture on the curved nanobeam are assumed to result in axial loads and not affect the mechanical properties of the material. The equilibrium equations of the curved nanobeam are derived using Hamilton’s principle based on various beam theories, including the classical theory, first-order shear deformation theory, and higher-order shear deformation theory, and the nonlocal elasticity theory. The accuracy of the proposed method is verified by comparing the results obtained with those of previous reliable studies. Additionally, the effects of different parameters on the free vibration behavior of the FGP curved nanobeams are investigated comprehensively.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Eringen A C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 1983, 54(9): 4703–4710

    Article  Google Scholar 

  2. Eringen A C. Nonlocal Continuum Field Theories. New York (NY): Springer, 2002

    MATH  Google Scholar 

  3. Yang F, Chong A, Lam D C C, Tong P. Couple stress-based strain gradient theory for elasticity. International Journal of Solids and Structures, 2002, 39(10): 2731–2743

    Article  MATH  Google Scholar 

  4. Lam D C, Yang F, Chong A, Wang J, Tong P. Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids, 2003, 51(8): 1477–1508

    Article  MATH  Google Scholar 

  5. Fleck H A, Hutchinson J W. A phenomenological theory for strain gradient effects in plasticity. Journal of the Mechanics and Physics of Solids, 1993, 41(12): 1825–1857

    Article  MathSciNet  MATH  Google Scholar 

  6. Stölken J S, Evans A G. A microbend test method for measuring the plasticity length scale. Acta Materialia, 1998, 46(14): 5109–5115

    Article  Google Scholar 

  7. Chong A, Yang F, Lam D, Tong P. Torsion and bending of micron-scaled structures. Journal of Materials Research, 2001, 16(4): 1052–1058

    Article  Google Scholar 

  8. Triantafyllidis N, Aifantis E C. A gradient approach to localization of deformation. I. Hyperelastic materials. Journal of Elasticity, 1986, 16(3): 225–237

    Article  MathSciNet  MATH  Google Scholar 

  9. Reddy J N. Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science, 2007, 45(2–8): 288–307

    Article  MATH  Google Scholar 

  10. Reddy J N, Pang S D. Nonlocal continuum theories of beams for the analysis of carbon nanotubes. Journal of Applied Physics, 2008, 2008, 103(2): 023511

    Google Scholar 

  11. Reddy J N. Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. International Journal of Engineering Science, 2010, 48(11): 1507–1518

    Article  MathSciNet  MATH  Google Scholar 

  12. Roque C M C, Ferreira A J M, Reddy J N. Analysis of Timoshenko nanobeams with a nonlocal formulation and meshless method. International Journal of Engineering Science, 2011, 49(9): 976–984

    Article  MATH  Google Scholar 

  13. Lim C W, Zhang G, Reddy J N. A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. Journal of the Mechanics and Physics of Solids, 2015, 78: 298–313

    Article  MathSciNet  MATH  Google Scholar 

  14. Wang C, Zhang Y, He X. Vibration of nonlocal Timoshenko beams. Nanotechnology, 2007, 18(10): 105401

    Article  Google Scholar 

  15. Murmu T, Pradhan S. Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM. Physica E, Low-Dimensional Systems and Nanostructures, 2009, 41(7): 1232–1239

    Article  Google Scholar 

  16. Pradhan S C, Phadikar J K. Nonlocal elasticity theory for vibration of nanoplates. Journal of Sound and Vibration, 2009, 325(1–2): 206–223

    Article  Google Scholar 

  17. Aghababaei R, Reddy J N. Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates. Journal of Sound and Vibration, 2009, 326(1–2): 277–289

    Article  Google Scholar 

  18. Thai H T. A nonlocal beam theory for bending, buckling, and vibration of nanobeams. International Journal of Engineering Science, 2012, 52: 56–64

    Article  MathSciNet  MATH  Google Scholar 

  19. Tran V K, Pham Q H, Nguyen-Thoi T. 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. Engineering with Computers, 2022, 38(2): 1465–1490

    Article  Google Scholar 

  20. Tran V K, Tran T T, Phung M V, Pham Q H, Nguyen-Thoi T. A finite element formulation and nonlocal theory for the static and free vibration analysis of the sandwich functionally graded nanoplates resting on elastic foundation. Journal of Nanomaterials, 2020, 2020: 8786373

    Article  Google Scholar 

  21. Tran T T, Tran V K, Pham Q H, Zenkour A M. Extended four-unknown higher-order shear deformation nonlocal theory for bending, buckling and free vibration of functionally graded porous nanoshell resting on elastic foundation. Composite Structures, 2021, 264: 113737

    Article  Google Scholar 

  22. Hosseini S A, Rahmani O, Bayat S. Thermal effect on forced vibration analysis of FG nanobeam subjected to moving load by Laplace transform method. Mechanics Based Design of Structures and Machines, 2021 (in press)

  23. Phung-Van P, Thai C H, Nguyen-Xuan H, Abdel Wahab M. Porosity-dependent nonlinear transient responses of functionally graded nanoplates using isogeometric analysis. Composites. Part B, Engineering, 2019, 164: 215–225

    Article  Google Scholar 

  24. Thanh C L, Nguyen T N, Vu T H, Khatir S, Abdel Wahab M. A geometrically nonlinear size-dependent hypothesis for porous functionally graded micro-plate. Engineering with Computers, 2022, 38(S1): 449–460

    Article  Google Scholar 

  25. Phung-Van P, Ferreira A J M, Nguyen-Xuan H, Abdel Wahab M. An isogeometric approach for size-dependent geometrically nonlinear transient analysis of functionally graded nanoplates. Composites. Part B, Engineering, 2017, 118: 125–134

    Article  Google Scholar 

  26. Cuong-Le T, Nguyen K D, Hoang-Le M, Sang-To T, Phan-Vu P, Wahab M A. Nonlocal strain gradient IGA numerical solution for static bending, free vibration and buckling of sigmoid FG sandwich nanoplate. Physica B, Condensed Matter, 2022, 631: 413726

    Article  Google Scholar 

  27. Mahesh V, Harursampath D. Nonlinear vibration of functionally graded magneto-electro-elastic higher order plates reinforced by CNTs using FEM. Engineering with Computers, 2022, 38(2): 1029–1051

    Article  Google Scholar 

  28. Vinyas M, Harursampath D. Nonlinear vibrations of magneto-electro-elastic doubly curved shells reinforced with carbon nanotubes. Composite Structures, 2020, 253: 112749

    Article  Google Scholar 

  29. Mahesh V. Active control of nonlinear coupled transient vibrations of multifunctional sandwich plates with agglomerated FG-CNTs core/magneto–electro–elastic facesheets. Thin-walled Structures, 2022, 179: 109547

    Article  Google Scholar 

  30. Mahesh V. Nonlinear damping of auxetic sandwich plates with functionally graded magneto-electro-elastic facings under multiphysics loads and electromagnetic circuits. Composite Structures, 2022, 290: 115523

    Article  Google Scholar 

  31. Assadi A, Farshi B. Size dependent vibration of curved nanobeams and rings including surface energies. Physica E, Low-Dimensional Systems and Nanostructures, 2011, 43(4): 975–978

    Article  Google Scholar 

  32. Ansari R, Gholami R, Sahmani S. Size-dependent vibration of functionally graded curved microbeams based on the modified strain gradient elasticity theory. Archive of Applied Mechanics, 2013, 83(10): 1439–1449

    Article  MATH  Google Scholar 

  33. Medina L, Gilat R, Ilic B, Krylov S. Experimental investigation of the snap-through buckling of electrostatically actuated initially curved pre-stressed micro beams. Sensors and Actuators. A, Physical, 2014, 220: 323–332

    Google Scholar 

  34. Ebrahimi F, Barati M. Size-dependent dynamic modeling of inhomogeneous curved nanobeams embedded in elastic medium based on nonlocal strain gradient theory. Proceedings of the institution of mechanical engineers, Part C: Journal of mechanical engineering science, 2017, 231: 4457–4469

    Google Scholar 

  35. Hosseini S A H, Rahmani O. Free vibration of shallow and deep curved FG nanobeam via nonlocal Timoshenko curved beam model. Applied Physics. A, Materials Science & Processing, 2016, 122(3): 169

    Article  Google Scholar 

  36. Zenkour A M, Arefi M, Alshehri N A. Size-dependent analysis of a sandwich curved nanobeam integrated with piezomagnetic face-sheets. Results in Physics, 2017, 7: 2172–2182

    Article  Google Scholar 

  37. She G L, Ren Y R, Yuan F G, Xiao W S. On vibrations of porous nanotubes. International Journal of Engineering Science, 2018, 125: 23–35

    Article  MathSciNet  MATH  Google Scholar 

  38. Ebrahimi F, Barati M R. Hygrothermal effects on vibration characteristics of viscoelastic FG nanobeams based on nonlocal strain gradient theory. Composite Structures, 2017, 159: 433–444

    Article  Google Scholar 

  39. Ganapathi M, Merzouki T, Polit O. Vibration study of curved nanobeams based on nonlocal higher-order shear deformation theory using finite element approach. Composite Structures, 2018, 184: 821–838

    Article  Google Scholar 

  40. Rezaiee-Pajand M, Rajabzadeh-Safaei N, Masoodi A R. An efficient curved beam element for thermo-mechanical nonlinear analysis of functionally graded porous beams. Structures, 2020, 28: 1035–1049

    Article  Google Scholar 

  41. Thanh C L, Tran L V, Vu-Huu T, Abdel-Wahab M. The size-dependent thermal bending and buckling analyses of composite laminate microplate based on new modified couple stress theory. Computer Methods in Applied Mechanics and Engineering, 2019, 350: 337–361

    Article  MathSciNet  MATH  Google Scholar 

  42. Vu-Bac N, Duong T X, Lahmer T, Zhuang X, Sauer R A, Park H S, Rabczuk T. A NURBS-based inverse analysis for reconstruction of nonlinear deformations of thin shell structures. Computer Methods in Applied Mechanics and Engineering, 2018, 331: 427–455

    Article  MathSciNet  MATH  Google Scholar 

  43. Vu-Bac N, Duong T X, Lahmer T, Areias P, Sauer R A, Park H S, Rabczuk T. A NURBS-based inverse analysis of thermal expansion induced morphing of thin shells. Computer Methods in Applied Mechanics and Engineering, 2019, 350: 480–510

    Article  MathSciNet  MATH  Google Scholar 

  44. Vu-Bac N, Rabczuk T, Park H S, Fu X, Zhuang X. A NURBS-based inverse analysis of swelling induced morphing of thin stimuli-responsive polymer gels. Computer Methods in Applied Mechanics and Engineering, 2022, 397: 115049

    Article  MathSciNet  MATH  Google Scholar 

  45. Merzouki T, Ganapathi M, Polit O. A nonlocal higher-order curved beam finite model including thickness stretching effect for bending analysis of curved nanobeams. Mechanics of Advanced Materials and Structures, 2019, 26(7): 614–630

    Article  Google Scholar 

  46. Ganapathi M, Aditya S, Shubhendu S, Polit O, Zineb T B. Nonlinear supersonic flutter study of porous 2D curved panels including graphene platelets reinforcement effect using trigonometric shear deformable fnite element. International Journal of Non-linear Mechanics, 2020, 125: 103543

    Article  Google Scholar 

  47. Belarbi M O, Houari M S A, Hirane H, Daikh A A, Bordas S P A. On the finite element analysis of functionally graded sandwich curved beams via a new refined higher order shear deformation theory. Composite Structures, 2022, 279: 114715

    Article  Google Scholar 

  48. Vinyas M. Interphase effect on the controlled frequency response of three-phase smart magneto-electro-elastic plates embedded with active constrained layer damping: FE study. Materials Research Express, 2020, 6(12): 125707

    Article  Google Scholar 

  49. Mahesh V, Mahesh V, Mukunda S, Harursampath D. Influence of micro-topological textures of BaTiO3-CoFe2O4 composites on the nonlinear pyrocoupled dynamic response of blast loaded magneto-electro-elastic plates in thermal environment. European Physical Journal Plus, 2022, 137(6): 675

    Article  Google Scholar 

  50. Mahesh V, Kattimani S. Subhaschandra Kattimani. Finite element simulation of controlled frequency response of skew multiphase magneto-electro-elastic plates. Journal of Intelligent Material Systems and Structures, 2019, 30(12): 1757–1771

    Article  Google Scholar 

  51. Khoei A R, Vahab M, Hirmand M, Khoei A R, Vahab M, Hirmand M. An enriched-FEM technique for numerical simulation of interacting discontinuities in naturally fractured porous media. Computer Methods in Applied Mechanics and Engineering, 2018, 331: 197–231

    Article  MathSciNet  MATH  Google Scholar 

  52. Tai C Y, Chan Y J. A hierarchic high-order Timoshenko beam finite element. Computers & Structures, 2016, 165: 48–58

    Article  Google Scholar 

  53. Aragón A M, Duarte C A, Geubelle P H. Generalized finite element enrichment functions for discontinuous gradient fields. International Journal for Numerical Methods in Engineering, 2010, 82(2): 242–268

    Article  MathSciNet  MATH  Google Scholar 

  54. Arndt M, Machado R D, Scremin A. The generalized finite element method applied to free vibration of framed structures. IntechOpen, 2011, 187–212

  55. Le C I, Ngoc A T, Nguyen D K. Free vibration and buckling of bidirectional functionally graded sandwich beams using an enriched third-order shear deformation beam element. Composite Structures, 2021, 261: 113309

    Article  Google Scholar 

  56. Nguyen D K, Vu A N T, Pham V N, Truong T T. Vibration of a three-phase bidirectional functionally graded sandwich beam carrying a moving mass using an enriched beam element. Engineering with Computers, 2022, 38(5): 4629–4650

    Article  Google Scholar 

  57. Shahsavari D, Karami B, Fahham H R, Li L. On the shear buckling of porous nanoplates using a new size-dependent quasi-3D shear deformation theory. Acta Mechanica, 2018, 229(11): 4549–4573

    Article  MathSciNet  Google Scholar 

  58. Ebrahimi F, Barati M R. A nonlocal strain gradient refined beam model for buckling analysis of size-dependent shear-deformable curved FG nanobeams. Composite Structures, 2017, 159: 174–182

    Article  Google Scholar 

  59. Nguyen H N, Hong T T, Vinh P V, Quang N D, Thom D V. A refined simple first-order shear deformation theory for static bending and free vibration analysis of advanced composite plates. Materials (Basel), 2019, 12(15): 2385

    Article  Google Scholar 

  60. Ebrahimi F, Barati M R. Wave propagation analysis of quasi-3D FG nanobeams in thermal environment based on nonlocal strain gradient theory. Applied Physics. A, Materials Science & Processing, 2016, 122(9): 843

    Article  Google Scholar 

  61. Anjomshoa A, Shahidi A R, Hassani B, Jomehzadeh E. Finite element buckling analysis of multi-layered graphene sheets on elastic substrate based on nonlocal elasticity theory. Applied Mathematical Modelling, 2014, 38(24): 5934–5955

    Article  MathSciNet  MATH  Google Scholar 

  62. Ganapathi M, Polit O. Dynamic characteristics of curved nanobeams using nonlocal higher-order curved beam theory. Physica E, Low-Dimensional Systems and Nanostructures, 2017, 91: 190–202

    Article  Google Scholar 

  63. Reddy J N. Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. 2nd ed. Boca Raton: CRC Press, 2003

    Book  Google Scholar 

  64. Zienkiewicz O C, Taylor R L. The Finite Element Method. 4th ed. London: Mc Graw-Hill Book Company, 1997

    MATH  Google Scholar 

  65. Solin P. Partial Differential Equations and the Finite Element Method. Hoboken: John Wiley & Sons Inc., 2006

    MATH  Google Scholar 

Download references

Acknowledgements

This study was supported by Bualuang ASEAN Chair Professor Fund.

Author information

Authors and Affiliations

  1. Faculty of Engineering and Technology, Nguyen Tat Thanh University, Ho Chi Minh City, 700000, Vietnam

    Quoc-Hoa Pham

  2. Department of Mechanical Engineering, Persian Gulf University, Bushehr, 7516913817, Iran

    Parviz Malekzadeh

  3. Faculty of Mechanical Engineering, Le Quy Don Technical University, Hanoi, 100000, Vietnam

    Van Ke Tran

  4. Laboratory for Applied and Industrial Mathematics, Institute for Computational Science and Artificial Intelligence, Van Lang University, Ho Chi Minh City, 700000, Vietnam

    Trung Nguyen-Thoi

  5. Faculty of Mechanical-Electrical and Computer Engineering, School of Technology, Van Lang University, Ho Chi Minh City, 700000, Vietnam

    Trung Nguyen-Thoi

  6. Thammasat School of Engineering, Thammasat University, Pathumtani, 12120, Thailand

    Trung Nguyen-Thoi

Authors
  1. Quoc-Hoa Pham
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Parviz Malekzadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Van Ke Tran
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Trung Nguyen-Thoi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Trung Nguyen-Thoi.

Appendices for

11709_2023_916_MOESM1_ESM.pdf

Free vibration analysis of functionally graded porous curved nanobeams on elastic foundation in hygro-thermal-magnetic environment

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pham, QH., Malekzadeh, P., Tran, V.K. et al. Free vibration analysis of functionally graded porous curved nanobeams on elastic foundation in hygro-thermo-magnetic environment. Front. Struct. Civ. Eng. 17, 584–605 (2023). https://doi.org/10.1007/s11709-023-0916-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11709-023-0916-7

Keywords

Search

Navigation